[{"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/ProdLp.lean", "full_name": "WithLp.prod_nnnorm_eq_add", "start": [610, 1], "end": [613, 59], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "p : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp\u271d : Fact (1 \u2264 p)\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : SeminormedAddCommGroup \u03b2\nhp : p \u2260 \u22a4\nf : WithLp p (\u03b1 \u00d7 \u03b2)\n\u22a2 \u2016f\u2016\u208a = (\u2016f.1\u2016\u208a ^ p.toReal + \u2016f.2\u2016\u208a ^ p.toReal) ^ (1 / p.toReal)", "state_after": "case a\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp\u271d : Fact (1 \u2264 p)\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : SeminormedAddCommGroup \u03b2\nhp : p \u2260 \u22a4\nf : WithLp p (\u03b1 \u00d7 \u03b2)\n\u22a2 \u2191\u2016f\u2016\u208a = \u2191((\u2016f.1\u2016\u208a ^ p.toReal + \u2016f.2\u2016\u208a ^ p.toReal) ^ (1 / p.toReal))"}, {"tactic": "simp [prod_norm_eq_add (p.toReal_pos_iff_ne_top.mpr hp)]", "annotated_tactic": ["simp [<a>prod_norm_eq_add</a> (p.toReal_pos_iff_ne_top.mpr hp)]", [{"full_name": "WithLp.prod_norm_eq_add", "def_path": "Mathlib/Analysis/NormedSpace/ProdLp.lean", "def_pos": [272, 9], "def_end_pos": [272, 25]}]], "state_before": "case a\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nhp\u271d : Fact (1 \u2264 p)\ninst\u271d\u00b9 : SeminormedAddCommGroup \u03b1\ninst\u271d : SeminormedAddCommGroup \u03b2\nhp : p \u2260 \u22a4\nf : WithLp p (\u03b1 \u00d7 \u03b2)\n\u22a2 \u2191\u2016f\u2016\u208a = \u2191((\u2016f.1\u2016\u208a ^ p.toReal + \u2016f.2\u2016\u208a ^ p.toReal) ^ (1 / p.toReal))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "MeasureTheory.measurableSet_of_null", "start": [442, 1], "end": [443, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/ShortComplex/ShortExact.lean", "full_name": "CategoryTheory.ShortComplex.isIso\u2082_of_shortExact_of_isIso\u2081\u2083", "start": [142, 1], "end": [150, 29], "traced_tactics": [{"tactic": "have := h\u2081.mono_f", "annotated_tactic": ["have := h\u2081.mono_f", []], "state_before": "C : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{u_3, u_1} C\ninst\u271d\u2074 : Category.{?u.12449, u_2} D\ninst\u271d\u00b3 : Preadditive C\ninst\u271d\u00b2 : Balanced C\nS\u2081 S\u2082 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.ShortExact\nh\u2082 : S\u2082.ShortExact\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2081\ninst\u271d : IsIso \u03c6.\u03c4\u2083\n\u22a2 IsIso \u03c6.\u03c4\u2082", "state_after": "C : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{u_3, u_1} C\ninst\u271d\u2074 : Category.{?u.12449, u_2} D\ninst\u271d\u00b3 : Preadditive C\ninst\u271d\u00b2 : Balanced C\nS\u2081 S\u2082 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.ShortExact\nh\u2082 : S\u2082.ShortExact\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2081\ninst\u271d : IsIso \u03c6.\u03c4\u2083\nthis : Mono S\u2081.f\n\u22a2 IsIso \u03c6.\u03c4\u2082"}, {"tactic": "have := h\u2082.mono_f", "annotated_tactic": ["have := h\u2082.mono_f", []], "state_before": "C : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{u_3, u_1} C\ninst\u271d\u2074 : Category.{?u.12449, u_2} D\ninst\u271d\u00b3 : Preadditive C\ninst\u271d\u00b2 : Balanced C\nS\u2081 S\u2082 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.ShortExact\nh\u2082 : S\u2082.ShortExact\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2081\ninst\u271d : IsIso \u03c6.\u03c4\u2083\nthis : Mono S\u2081.f\n\u22a2 IsIso \u03c6.\u03c4\u2082", "state_after": "C : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{u_3, u_1} C\ninst\u271d\u2074 : Category.{?u.12449, u_2} D\ninst\u271d\u00b3 : Preadditive C\ninst\u271d\u00b2 : Balanced C\nS\u2081 S\u2082 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.ShortExact\nh\u2082 : S\u2082.ShortExact\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2081\ninst\u271d : IsIso \u03c6.\u03c4\u2083\nthis\u271d : Mono S\u2081.f\nthis : Mono S\u2082.f\n\u22a2 IsIso \u03c6.\u03c4\u2082"}, {"tactic": "have := h\u2081.epi_g", "annotated_tactic": ["have := h\u2081.epi_g", []], "state_before": "C : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{u_3, u_1} C\ninst\u271d\u2074 : Category.{?u.12449, u_2} D\ninst\u271d\u00b3 : Preadditive C\ninst\u271d\u00b2 : Balanced C\nS\u2081 S\u2082 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.ShortExact\nh\u2082 : S\u2082.ShortExact\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2081\ninst\u271d : IsIso \u03c6.\u03c4\u2083\nthis\u271d : Mono S\u2081.f\nthis : Mono S\u2082.f\n\u22a2 IsIso \u03c6.\u03c4\u2082", "state_after": "C : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{u_3, u_1} C\ninst\u271d\u2074 : Category.{?u.12449, u_2} D\ninst\u271d\u00b3 : Preadditive C\ninst\u271d\u00b2 : Balanced C\nS\u2081 S\u2082 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.ShortExact\nh\u2082 : S\u2082.ShortExact\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2081\ninst\u271d : IsIso \u03c6.\u03c4\u2083\nthis\u271d\u00b9 : Mono S\u2081.f\nthis\u271d : Mono S\u2082.f\nthis : Epi S\u2081.g\n\u22a2 IsIso \u03c6.\u03c4\u2082"}, {"tactic": "have := h\u2082.epi_g", "annotated_tactic": ["have := h\u2082.epi_g", []], "state_before": "C : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{u_3, u_1} C\ninst\u271d\u2074 : Category.{?u.12449, u_2} D\ninst\u271d\u00b3 : Preadditive C\ninst\u271d\u00b2 : Balanced C\nS\u2081 S\u2082 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.ShortExact\nh\u2082 : S\u2082.ShortExact\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2081\ninst\u271d : IsIso \u03c6.\u03c4\u2083\nthis\u271d\u00b9 : Mono S\u2081.f\nthis\u271d : Mono S\u2082.f\nthis : Epi S\u2081.g\n\u22a2 IsIso \u03c6.\u03c4\u2082", "state_after": "C : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{u_3, u_1} C\ninst\u271d\u2074 : Category.{?u.12449, u_2} D\ninst\u271d\u00b3 : Preadditive C\ninst\u271d\u00b2 : Balanced C\nS\u2081 S\u2082 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.ShortExact\nh\u2082 : S\u2082.ShortExact\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2081\ninst\u271d : IsIso \u03c6.\u03c4\u2083\nthis\u271d\u00b2 : Mono S\u2081.f\nthis\u271d\u00b9 : Mono S\u2082.f\nthis\u271d : Epi S\u2081.g\nthis : Epi S\u2082.g\n\u22a2 IsIso \u03c6.\u03c4\u2082"}, {"tactic": "have := mono_\u03c4\u2082_of_exact_of_mono \u03c6 h\u2081.exact", "annotated_tactic": ["have := <a>mono_\u03c4\u2082_of_exact_of_mono</a> \u03c6 h\u2081.exact", [{"full_name": "CategoryTheory.ShortComplex.mono_\u03c4\u2082_of_exact_of_mono", "def_path": "Mathlib/Algebra/Homology/ShortComplex/Exact.lean", "def_pos": [796, 7], "def_end_pos": [796, 31]}]], "state_before": "C : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{u_3, u_1} C\ninst\u271d\u2074 : Category.{?u.12449, u_2} D\ninst\u271d\u00b3 : Preadditive C\ninst\u271d\u00b2 : Balanced C\nS\u2081 S\u2082 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.ShortExact\nh\u2082 : S\u2082.ShortExact\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2081\ninst\u271d : IsIso \u03c6.\u03c4\u2083\nthis\u271d\u00b2 : Mono S\u2081.f\nthis\u271d\u00b9 : Mono S\u2082.f\nthis\u271d : Epi S\u2081.g\nthis : Epi S\u2082.g\n\u22a2 IsIso \u03c6.\u03c4\u2082", "state_after": "C : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{u_3, u_1} C\ninst\u271d\u2074 : Category.{?u.12449, u_2} D\ninst\u271d\u00b3 : Preadditive C\ninst\u271d\u00b2 : Balanced C\nS\u2081 S\u2082 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.ShortExact\nh\u2082 : S\u2082.ShortExact\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2081\ninst\u271d : IsIso \u03c6.\u03c4\u2083\nthis\u271d\u00b3 : Mono S\u2081.f\nthis\u271d\u00b2 : Mono S\u2082.f\nthis\u271d\u00b9 : Epi S\u2081.g\nthis\u271d : Epi S\u2082.g\nthis : Mono \u03c6.\u03c4\u2082\n\u22a2 IsIso \u03c6.\u03c4\u2082"}, {"tactic": "have := epi_\u03c4\u2082_of_exact_of_epi \u03c6 h\u2082.exact", "annotated_tactic": ["have := <a>epi_\u03c4\u2082_of_exact_of_epi</a> \u03c6 h\u2082.exact", [{"full_name": "CategoryTheory.ShortComplex.epi_\u03c4\u2082_of_exact_of_epi", "def_path": "Mathlib/Algebra/Homology/ShortComplex/Exact.lean", "def_pos": [808, 7], "def_end_pos": [808, 29]}]], "state_before": "C : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{u_3, u_1} C\ninst\u271d\u2074 : Category.{?u.12449, u_2} D\ninst\u271d\u00b3 : Preadditive C\ninst\u271d\u00b2 : Balanced C\nS\u2081 S\u2082 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.ShortExact\nh\u2082 : S\u2082.ShortExact\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2081\ninst\u271d : IsIso \u03c6.\u03c4\u2083\nthis\u271d\u00b3 : Mono S\u2081.f\nthis\u271d\u00b2 : Mono S\u2082.f\nthis\u271d\u00b9 : Epi S\u2081.g\nthis\u271d : Epi S\u2082.g\nthis : Mono \u03c6.\u03c4\u2082\n\u22a2 IsIso \u03c6.\u03c4\u2082", "state_after": "C : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{u_3, u_1} C\ninst\u271d\u2074 : Category.{?u.12449, u_2} D\ninst\u271d\u00b3 : Preadditive C\ninst\u271d\u00b2 : Balanced C\nS\u2081 S\u2082 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.ShortExact\nh\u2082 : S\u2082.ShortExact\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2081\ninst\u271d : IsIso \u03c6.\u03c4\u2083\nthis\u271d\u2074 : Mono S\u2081.f\nthis\u271d\u00b3 : Mono S\u2082.f\nthis\u271d\u00b2 : Epi S\u2081.g\nthis\u271d\u00b9 : Epi S\u2082.g\nthis\u271d : Mono \u03c6.\u03c4\u2082\nthis : Epi \u03c6.\u03c4\u2082\n\u22a2 IsIso \u03c6.\u03c4\u2082"}, {"tactic": "apply isIso_of_mono_of_epi", "annotated_tactic": ["apply <a>isIso_of_mono_of_epi</a>", [{"full_name": "CategoryTheory.isIso_of_mono_of_epi", "def_path": "Mathlib/CategoryTheory/Balanced.lean", "def_pos": [39, 9], "def_end_pos": [39, 29]}]], "state_before": "C : Type u_1\nD : Type u_2\ninst\u271d\u2075 : Category.{u_3, u_1} C\ninst\u271d\u2074 : Category.{?u.12449, u_2} D\ninst\u271d\u00b3 : Preadditive C\ninst\u271d\u00b2 : Balanced C\nS\u2081 S\u2082 : ShortComplex C\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.ShortExact\nh\u2082 : S\u2082.ShortExact\ninst\u271d\u00b9 : IsIso \u03c6.\u03c4\u2081\ninst\u271d : IsIso \u03c6.\u03c4\u2083\nthis\u271d\u2074 : Mono S\u2081.f\nthis\u271d\u00b3 : Mono S\u2082.f\nthis\u271d\u00b2 : Epi S\u2081.g\nthis\u271d\u00b9 : Epi S\u2082.g\nthis\u271d : Mono \u03c6.\u03c4\u2082\nthis : Epi \u03c6.\u03c4\u2082\n\u22a2 IsIso \u03c6.\u03c4\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "nhdsWithin_pi_univ_eq", "start": [326, 1], "end": [328, 60], "traced_tactics": [{"tactic": "simpa [nhdsWithin] using nhdsWithin_pi_eq finite_univ s x", "annotated_tactic": ["simpa [<a>nhdsWithin</a>] using <a>nhdsWithin_pi_eq</a> <a>finite_univ</a> s x", [{"full_name": "nhdsWithin", "def_path": "Mathlib/Topology/Defs/Filter.lean", "def_pos": [123, 5], "def_end_pos": [123, 15]}, {"full_name": "nhdsWithin_pi_eq", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [316, 9], "def_end_pos": [316, 25]}, {"full_name": "Set.finite_univ", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [727, 9], "def_end_pos": [727, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d : Finite \u03b9\ns : (i : \u03b9) \u2192 Set (\u03c0 i)\nx : (i : \u03b9) \u2192 \u03c0 i\n\u22a2 \ud835\udcdd[univ.pi s] x = \u2a05 i, comap (fun x => x i) (\ud835\udcdd[s i] x i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/ProjIcc.lean", "full_name": "Monotone.IicExtend", "start": [321, 11], "end": [322, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean", "full_name": "mem_adjoin_of_smul_prime_pow_smul_of_minpoly_isEisensteinAt", "start": [378, 1], "end": [386, 98], "traced_tactics": [{"tactic": "induction' n with n hn", "annotated_tactic": ["induction' n with n hn", []], "state_before": "R : Type u\nK : Type v\nL : Type z\np : R\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra R L\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsScalarTower R K L\ninst\u271d\u00b3 : IsSeparable K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsIntegrallyClosed R\nB : PowerBasis K L\nhp : _root_.Prime p\nhBint : IsIntegral R B.gen\nn : \u2115\nz : L\nhzint : IsIntegral R z\nhz : p ^ n \u2022 z \u2208 adjoin R {B.gen}\nhei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})\n\u22a2 z \u2208 adjoin R {B.gen}", "state_after": "case zero\nR : Type u\nK : Type v\nL : Type z\np : R\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra R L\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsScalarTower R K L\ninst\u271d\u00b3 : IsSeparable K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsIntegrallyClosed R\nB : PowerBasis K L\nhp : _root_.Prime p\nhBint : IsIntegral R B.gen\nz : L\nhzint : IsIntegral R z\nhei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})\nhz : p ^ 0 \u2022 z \u2208 adjoin R {B.gen}\n\u22a2 z \u2208 adjoin R {B.gen}\n\ncase succ\nR : Type u\nK : Type v\nL : Type z\np : R\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra R L\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsScalarTower R K L\ninst\u271d\u00b3 : IsSeparable K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsIntegrallyClosed R\nB : PowerBasis K L\nhp : _root_.Prime p\nhBint : IsIntegral R B.gen\nz : L\nhzint : IsIntegral R z\nhei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})\nn : \u2115\nhn : p ^ n \u2022 z \u2208 adjoin R {B.gen} \u2192 z \u2208 adjoin R {B.gen}\nhz : p ^ (n + 1) \u2022 z \u2208 adjoin R {B.gen}\n\u22a2 z \u2208 adjoin R {B.gen}"}, {"tactic": "simpa using hz", "annotated_tactic": ["simpa using hz", []], "state_before": "case zero\nR : Type u\nK : Type v\nL : Type z\np : R\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra R L\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsScalarTower R K L\ninst\u271d\u00b3 : IsSeparable K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsIntegrallyClosed R\nB : PowerBasis K L\nhp : _root_.Prime p\nhBint : IsIntegral R B.gen\nz : L\nhzint : IsIntegral R z\nhei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})\nhz : p ^ 0 \u2022 z \u2208 adjoin R {B.gen}\n\u22a2 z \u2208 adjoin R {B.gen}", "state_after": "no goals"}, {"tactic": "rw [_root_.pow_succ', mul_smul] at hz", "annotated_tactic": ["rw [<a>_root_.pow_succ'</a>, <a>mul_smul</a>] at hz", [{"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [667, 34], "def_end_pos": [667, 43]}, {"full_name": "MulAction.mul_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [114, 3], "def_end_pos": [114, 11]}]], "state_before": "case succ\nR : Type u\nK : Type v\nL : Type z\np : R\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra R L\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsScalarTower R K L\ninst\u271d\u00b3 : IsSeparable K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsIntegrallyClosed R\nB : PowerBasis K L\nhp : _root_.Prime p\nhBint : IsIntegral R B.gen\nz : L\nhzint : IsIntegral R z\nhei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})\nn : \u2115\nhn : p ^ n \u2022 z \u2208 adjoin R {B.gen} \u2192 z \u2208 adjoin R {B.gen}\nhz : p ^ (n + 1) \u2022 z \u2208 adjoin R {B.gen}\n\u22a2 z \u2208 adjoin R {B.gen}", "state_after": "case succ\nR : Type u\nK : Type v\nL : Type z\np : R\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra R L\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsScalarTower R K L\ninst\u271d\u00b3 : IsSeparable K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsIntegrallyClosed R\nB : PowerBasis K L\nhp : _root_.Prime p\nhBint : IsIntegral R B.gen\nz : L\nhzint : IsIntegral R z\nhei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})\nn : \u2115\nhn : p ^ n \u2022 z \u2208 adjoin R {B.gen} \u2192 z \u2208 adjoin R {B.gen}\nhz : p \u2022 p ^ n \u2022 z \u2208 adjoin R {B.gen}\n\u22a2 z \u2208 adjoin R {B.gen}"}, {"tactic": "exact\n  hn (mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt hp hBint (hzint.smul _) hz hei)", "annotated_tactic": ["exact\n      hn (<a>mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt</a> hp hBint (hzint.smul _) hz hei)", [{"full_name": "mem_adjoin_of_smul_prime_smul_of_minpoly_isEisensteinAt", "def_path": "Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean", "def_pos": [237, 9], "def_end_pos": [237, 64]}]], "state_before": "case succ\nR : Type u\nK : Type v\nL : Type z\np : R\ninst\u271d\u00b9\u2070 : CommRing R\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : Field L\ninst\u271d\u2077 : Algebra K L\ninst\u271d\u2076 : Algebra R L\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsScalarTower R K L\ninst\u271d\u00b3 : IsSeparable K L\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsFractionRing R K\ninst\u271d : IsIntegrallyClosed R\nB : PowerBasis K L\nhp : _root_.Prime p\nhBint : IsIntegral R B.gen\nz : L\nhzint : IsIntegral R z\nhei : (minpoly R B.gen).IsEisensteinAt (Submodule.span R {p})\nn : \u2115\nhn : p ^ n \u2022 z \u2208 adjoin R {B.gen} \u2192 z \u2208 adjoin R {B.gen}\nhz : p \u2022 p ^ n \u2022 z \u2208 adjoin R {B.gen}\n\u22a2 z \u2208 adjoin R {B.gen}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/ShortComplex/Basic.lean", "full_name": "CategoryTheory.ShortComplex.zero_\u03c4\u2083", "start": [122, 1], "end": [122, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Sups.lean", "full_name": "Finset.image_subset_sups_left", "start": [104, 1], "end": [104, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/MvPolynomial/Degrees.lean", "full_name": "MvPolynomial.degrees_map_of_injective", "start": [214, 1], "end": [216, 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/Algebra/MvPolynomial/Degrees.lean", "def_pos": [79, 5], "def_end_pos": [79, 12]}, {"full_name": "MvPolynomial.support_map_of_injective", "def_path": "Mathlib/Algebra/MvPolynomial/Basic.lean", "def_pos": [1450, 9], "def_end_pos": [1450, 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 \u21d1f\n\u22a2 ((map f) p).degrees = p.degrees", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Interval/Basic.lean", "full_name": "NonemptyInterval.toProd_mul", "start": [162, 1], "end": [163, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean", "full_name": "CategoryTheory.ShortComplex.SnakeInput.w\u2081\u2083_\u03c4\u2081", "start": [124, 1], "end": [125, 33], "traced_tactics": [{"tactic": "rw [\u2190 comp_\u03c4\u2081, S.w\u2081\u2083, zero_\u03c4\u2081]", "annotated_tactic": ["rw [\u2190 <a>comp_\u03c4\u2081</a>, S.w\u2081\u2083, <a>zero_\u03c4\u2081</a>]", [{"full_name": "CategoryTheory.ShortComplex.comp_\u03c4\u2081", "def_path": "Mathlib/Algebra/Homology/ShortComplex/Basic.lean", "def_pos": [107, 18], "def_end_pos": [107, 25]}, {"full_name": "CategoryTheory.ShortComplex.zero_\u03c4\u2081", "def_path": "Mathlib/Algebra/Homology/ShortComplex/Basic.lean", "def_pos": [120, 15], "def_end_pos": [120, 22]}]], "state_before": "C : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Abelian C\nS : SnakeInput C\n\u22a2 S.v\u2081\u2082.\u03c4\u2081 \u226b S.v\u2082\u2083.\u03c4\u2081 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean", "full_name": "AffineEquiv.injective_pointReflection_left_of_injective_bit0", "start": [587, 1], "end": [589, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "full_name": "MeasureTheory.lintegral_nnnorm_condexpL2_le", "start": [161, 1], "end": [182, 71], "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": [246, 9], "def_end_pos": [246, 37]}, {"full_name": "MeasureTheory.condexpL2", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean", "def_pos": [72, 19], "def_end_pos": [72, 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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((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((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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((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((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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((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((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": [72, 19], "def_end_pos": [72, 28]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1530, 9], "def_end_pos": [1530, 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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((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((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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\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": [72, 19], "def_end_pos": [72, 28]}, {"full_name": "MeasureTheory.ae_restrict_of_ae", "def_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "def_pos": [657, 9], "def_end_pos": [657, 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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\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": [72, 19], "def_end_pos": [72, 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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1da0[ae (\u03bc.restrict s)] fun x => \u2191\u2016\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\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": [328, 9], "def_end_pos": [328, 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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1da0[ae (\u03bc.restrict s)] fun x => \u2191\u2016\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 \u222b\u207b (x : \u03b1) in s, \u2191\u2016\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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1da0[ae (\u03bc.restrict s)] fun x => \u2191\u2016\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 lintegral_nnnorm_le_of_forall_fin_meas_integral_eq\n  hm (Lp.stronglyMeasurable f) ?_ ?_ ?_ ?_ hs h\u03bcs", "annotated_tactic": ["refine <a>lintegral_nnnorm_le_of_forall_fin_meas_integral_eq</a>\n    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": [218, 9], "def_end_pos": [218, 59]}, {"full_name": "MeasureTheory.Lp.stronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [213, 19], "def_end_pos": [213, 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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1da0[ae (\u03bc.restrict s)] fun x => \u2191\u2016\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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1da0[ae (\u03bc.restrict s)] fun x => \u2191\u2016\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 IntegrableOn (\u2191\u2191f) s \u03bc\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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1da0[ae (\u03bc.restrict s)] fun x => \u2191\u2016\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 StronglyMeasurable g\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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1da0[ae (\u03bc.restrict s)] fun x => \u2191\u2016\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 IntegrableOn g s \u03bc\n\ncase refine_4\n\u03b1 : Type u_1\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b3 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1da0[ae (\u03bc.restrict s)] fun x => \u2191\u2016\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 \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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\n\u22a2 (fun x => \u2191\u2016g x\u2016\u208a) =\u1da0[ae (\u03bc.restrict s)] fun x => \u2191\u2016\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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nx : \u03b1\nhx : g x = \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((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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nx : \u03b1\nhx : g x = \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((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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nx : \u03b1\nhx : g x = \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f) x\n\u22a2 \u2191\u2016g x\u2016\u208a = \u2191\u2016\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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nx : \u03b1\nhx : g x = \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f) x\n\u22a2 \u2191\u2016g x\u2016\u208a = \u2191\u2016\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": [373, 9], "def_end_pos": [373, 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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1da0[ae (\u03bc.restrict s)] fun x => \u2191\u2016\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 IntegrableOn (\u2191\u2191f) s \u03bc", "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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1da0[ae (\u03bc.restrict s)] fun x => \u2191\u2016\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 StronglyMeasurable g", "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": [485, 9], "def_end_pos": [485, 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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1da0[ae (\u03bc.restrict s)] fun x => \u2191\u2016\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 IntegrableOn g s \u03bc", "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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1da0[ae (\u03bc.restrict s)] fun x => \u2191\u2016\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 Integrable (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) (\u03bc.restrict s)"}, {"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": [85, 9], "def_end_pos": [85, 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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1da0[ae (\u03bc.restrict s)] fun x => \u2191\u2016\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 Integrable (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) (\u03bc.restrict s)", "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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1da0[ae (\u03bc.restrict s)] fun x => \u2191\u2016\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f) x\u2016\u208a\n\u22a2 \u2200 (t : Set \u03b1), MeasurableSet t \u2192 \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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1da0[ae (\u03bc.restrict s)] fun x => \u2191\u2016\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f) x\u2016\u208a\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \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": [152, 9], "def_end_pos": [152, 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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1da0[ae (\u03bc.restrict s)] fun x => \u2191\u2016\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f) x\u2016\u208a\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1da0[ae (\u03bc.restrict s)] fun x => \u2191\u2016\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f) x\u2016\u208a\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u03bc t < \u22a4\n\u22a2 \u222b (x : \u03b1) in t, g x \u2202\u03bc = \u222b (x : \u03b1) in t, \u2191\u2191\u2191((condexpL2 \u211d \u211d ?m.61340) 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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1da0[ae (\u03bc.restrict s)] fun x => \u2191\u2016\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f) x\u2016\u208a\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u03bc t < \u22a4\n\u22a2 m \u2264 m0"}, {"tactic": "exact setIntegral_congr_ae (hm t ht) (hg_eq.mono fun x hx _ => hx)", "annotated_tactic": ["exact <a>setIntegral_congr_ae</a> (hm t ht) (hg_eq.mono fun x hx _ => hx)", [{"full_name": "MeasureTheory.setIntegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [79, 9], "def_end_pos": [79, 29]}]], "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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1da0[ae (\u03bc.restrict s)] fun x => \u2191\u2016\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f) x\u2016\u208a\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u03bc t < \u22a4\n\u22a2 \u222b (x : \u03b1) in t, g x \u2202\u03bc = \u222b (x : \u03b1) in t, \u2191\u2191\u2191((condexpL2 \u211d \u211d ?m.61340) 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 : RCLike \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 : \u03bc s \u2260 \u22a4\nf : \u21a5(Lp \u211d 2 \u03bc)\nh_meas : AEStronglyMeasurable' m (\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)) \u03bc := lpMeas.aeStronglyMeasurable' ((condexpL2 \u211d \u211d hm) f)\ng : \u03b1 \u2192 \u211d := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\nhg_eq : g =\u1da0[ae \u03bc] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_eq_restrict : g =\u1da0[ae (\u03bc.restrict s)] \u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f)\nhg_nnnorm_eq : (fun x => \u2191\u2016g x\u2016\u208a) =\u1da0[ae (\u03bc.restrict s)] fun x => \u2191\u2016\u2191\u2191\u2191((condexpL2 \u211d \u211d hm) f) x\u2016\u208a\nt : Set \u03b1\nht : MeasurableSet t\nh\u03bct : \u03bc t < \u22a4\n\u22a2 m \u2264 m0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "full_name": "Zsqrtd.sub_re", "start": [175, 1], "end": [176, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "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": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Star.lean", "full_name": "CliffordAlgebra.star_smul", "start": [57, 1], "end": [58, 46], "traced_tactics": [{"tactic": "rw [star_def, star_def, map_smul, map_smul]", "annotated_tactic": ["rw [<a>star_def</a>, <a>star_def</a>, <a>map_smul</a>, <a>map_smul</a>]", [{"full_name": "CliffordAlgebra.star_def", "def_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Star.lean", "def_pos": [41, 9], "def_end_pos": [41, 17]}, {"full_name": "CliffordAlgebra.star_def", "def_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Star.lean", "def_pos": [41, 9], "def_end_pos": [41, 17]}, {"full_name": "map_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Hom.lean", "def_pos": [111, 9], "def_end_pos": [111, 17]}, {"full_name": "map_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Hom.lean", "def_pos": [111, 9], "def_end_pos": [111, 17]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : CommRing R\nM : Type u_2\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nr : R\nx : CliffordAlgebra Q\n\u22a2 star (r \u2022 x) = r \u2022 star x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Bitwise.lean", "full_name": "Nat.lor_comm", "start": [327, 1], "end": [328, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Submodule.lean", "full_name": "LieSubmodule.coe_toSubmodule_mk", "start": [132, 1], "end": [133, 92], "traced_tactics": [{"tactic": "cases p", "annotated_tactic": ["cases p", []], "state_before": "R : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nN N' : LieSubmodule R L M\np : Submodule R M\nh : \u2200 {x : L} {m : M}, m \u2208 p.carrier \u2192 \u2045x, m\u2046 \u2208 p.carrier\n\u22a2 \u2191{ toSubmodule := p, lie_mem := h } = p", "state_after": "case mk\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nN N' : LieSubmodule R L M\ntoAddSubmonoid\u271d : AddSubmonoid M\nsmul_mem'\u271d : \u2200 (c : R) {x : M}, x \u2208 toAddSubmonoid\u271d.carrier \u2192 c \u2022 x \u2208 toAddSubmonoid\u271d.carrier\nh :\n  \u2200 {x : L} {m : M},\n    m \u2208 { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }.carrier \u2192\n      \u2045x, m\u2046 \u2208 { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }.carrier\n\u22a2 \u2191{ toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d, lie_mem := h } =\n    { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case mk\nR : Type u\nL : Type v\nM : Type w\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra R L\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\nN N' : LieSubmodule R L M\ntoAddSubmonoid\u271d : AddSubmonoid M\nsmul_mem'\u271d : \u2200 (c : R) {x : M}, x \u2208 toAddSubmonoid\u271d.carrier \u2192 c \u2022 x \u2208 toAddSubmonoid\u271d.carrier\nh :\n  \u2200 {x : L} {m : M},\n    m \u2208 { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }.carrier \u2192\n      \u2045x, m\u2046 \u2208 { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }.carrier\n\u22a2 \u2191{ toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d, lie_mem := h } =\n    { toAddSubmonoid := toAddSubmonoid\u271d, smul_mem' := smul_mem'\u271d }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Complex/Abs.lean", "full_name": "Complex.abs_im_div_abs_le_one", "start": [237, 1], "end": [240, 75], "traced_tactics": [{"tactic": "simp [hz, zero_le_one]", "annotated_tactic": ["simp [hz, <a>zero_le_one</a>]", [{"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}]], "state_before": "z : \u2102\nhz : z = 0\n\u22a2 |z.im / abs z| \u2264 1", "state_after": "no goals"}, {"tactic": "simp_rw [_root_.abs_div, abs_abs,\ndiv_le_iff (AbsoluteValue.pos Complex.abs hz), one_mul, abs_im_le_abs]", "annotated_tactic": ["simp_rw [<a>_root_.abs_div</a>, <a>abs_abs</a>,\n    <a>div_le_iff</a> (<a>AbsoluteValue.pos</a> <a>Complex.abs</a> hz), <a>one_mul</a>, <a>abs_im_le_abs</a>]", [{"full_name": "abs_div", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [1000, 9], "def_end_pos": [1000, 16]}, {"full_name": "Complex.abs_abs", "def_path": "Mathlib/Data/Complex/Abs.lean", "def_pos": [199, 9], "def_end_pos": [199, 16]}, {"full_name": "div_le_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [61, 9], "def_end_pos": [61, 19]}, {"full_name": "AbsoluteValue.pos", "def_path": "Mathlib/Algebra/Order/AbsoluteValue.lean", "def_pos": [121, 19], "def_end_pos": [121, 22]}, {"full_name": "Complex.abs", "def_path": "Mathlib/Data/Complex/Abs.lean", "def_pos": [56, 19], "def_end_pos": [56, 37]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "Complex.abs_im_le_abs", "def_path": "Mathlib/Data/Complex/Abs.lean", "def_pos": [165, 9], "def_end_pos": [165, 22]}]], "state_before": "z : \u2102\nhz : \u00acz = 0\n\u22a2 |z.im / abs z| \u2264 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/NumberField/Embeddings.lean", "full_name": "NumberField.InfinitePlace.not_isReal_iff_isComplex", "start": [401, 1], "end": [402, 33], "traced_tactics": [{"tactic": "rw [isComplex_iff, isReal_iff]", "annotated_tactic": ["rw [<a>isComplex_iff</a>, <a>isReal_iff</a>]", [{"full_name": "NumberField.InfinitePlace.isComplex_iff", "def_path": "Mathlib/NumberTheory/NumberField/Embeddings.lean", "def_pos": [385, 9], "def_end_pos": [385, 22]}, {"full_name": "NumberField.InfinitePlace.isReal_iff", "def_path": "Mathlib/NumberTheory/NumberField/Embeddings.lean", "def_pos": [378, 9], "def_end_pos": [378, 19]}]], "state_before": "k : Type u_1\ninst\u271d\u00b2 : Field k\nK : Type u_2\ninst\u271d\u00b9 : Field K\nF : Type u_3\ninst\u271d : Field F\nw : InfinitePlace K\n\u22a2 \u00acw.IsReal \u2194 w.IsComplex", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/CharP/ExpChar.lean", "full_name": "LinearMap.iterateFrobenius_def", "start": [389, 1], "end": [390, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Tactic/Ring/Basic.lean", "full_name": "Mathlib.Tactic.Ring.add_pos_right", "start": [616, 1], "end": [616, 99], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Finprod.lean", "full_name": "smul_finsum", "start": [340, 1], "end": [344, 83], "traced_tactics": [{"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": [212, 9], "def_end_pos": [212, 17]}]], "state_before": "G : Type u_1\nM\u271d : Type u_2\nN : Type u_3\n\u03b1 : Sort u_4\n\u03b2 : Sort u_5\n\u03b9 : Sort u_6\ninst\u271d\u2075 : CommMonoid M\u271d\ninst\u271d\u2074 : CommMonoid N\nR : Type u_7\nM : Type u_8\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nc : R\nf : \u03b9 \u2192 M\n\u22a2 c \u2022 \u2211\u1da0 (i : \u03b9), f i = \u2211\u1da0 (i : \u03b9), c \u2022 f i", "state_after": "case inl\nG : Type u_1\nM\u271d : Type u_2\nN : Type u_3\n\u03b1 : Sort u_4\n\u03b2 : Sort u_5\n\u03b9 : Sort u_6\ninst\u271d\u2075 : CommMonoid M\u271d\ninst\u271d\u2074 : CommMonoid N\nR : Type u_7\nM : Type u_8\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nf : \u03b9 \u2192 M\n\u22a2 0 \u2022 \u2211\u1da0 (i : \u03b9), f i = \u2211\u1da0 (i : \u03b9), 0 \u2022 f i\n\ncase inr\nG : Type u_1\nM\u271d : Type u_2\nN : Type u_3\n\u03b1 : Sort u_4\n\u03b2 : Sort u_5\n\u03b9 : Sort u_6\ninst\u271d\u2075 : CommMonoid M\u271d\ninst\u271d\u2074 : CommMonoid N\nR : Type u_7\nM : Type u_8\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nc : R\nf : \u03b9 \u2192 M\nhc : c \u2260 0\n\u22a2 c \u2022 \u2211\u1da0 (i : \u03b9), f i = \u2211\u1da0 (i : \u03b9), c \u2022 f i"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case inl\nG : Type u_1\nM\u271d : Type u_2\nN : Type u_3\n\u03b1 : Sort u_4\n\u03b2 : Sort u_5\n\u03b9 : Sort u_6\ninst\u271d\u2075 : CommMonoid M\u271d\ninst\u271d\u2074 : CommMonoid N\nR : Type u_7\nM : Type u_8\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nf : \u03b9 \u2192 M\n\u22a2 0 \u2022 \u2211\u1da0 (i : \u03b9), f i = \u2211\u1da0 (i : \u03b9), 0 \u2022 f i", "state_after": "no goals"}, {"tactic": "exact (smulAddHom R M c).map_finsum_of_injective (smul_right_injective M hc) _", "annotated_tactic": ["exact (<a>smulAddHom</a> R M c).<a>map_finsum_of_injective</a> (<a>smul_right_injective</a> M hc) _", [{"full_name": "smulAddHom", "def_path": "Mathlib/Algebra/Module/Defs.lean", "def_pos": [187, 5], "def_end_pos": [187, 15]}, {"full_name": "AddMonoidHom.map_finsum_of_injective", "def_path": "Mathlib/Algebra/BigOperators/Finprod.lean", "def_pos": [316, 3], "def_end_pos": [316, 14]}, {"full_name": "smul_right_injective", "def_path": "Mathlib/Algebra/Module/Defs.lean", "def_pos": [556, 9], "def_end_pos": [556, 29]}]], "state_before": "case inr\nG : Type u_1\nM\u271d : Type u_2\nN : Type u_3\n\u03b1 : Sort u_4\n\u03b2 : Sort u_5\n\u03b9 : Sort u_6\ninst\u271d\u2075 : CommMonoid M\u271d\ninst\u271d\u2074 : CommMonoid N\nR : Type u_7\nM : Type u_8\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : NoZeroSMulDivisors R M\nc : R\nf : \u03b9 \u2192 M\nhc : c \u2260 0\n\u22a2 c \u2022 \u2211\u1da0 (i : \u03b9), f i = \u2211\u1da0 (i : \u03b9), c \u2022 f i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.insert_ne_self", "start": [1163, 1], "end": [1164, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Tilted.lean", "full_name": "MeasureTheory.tilted_apply_eq_ofReal_integral", "start": [114, 1], "end": [120, 58], "traced_tactics": [{"tactic": "by_cases hf : Integrable (fun x \u21a6 exp (f x)) \u03bc", "annotated_tactic": ["by_cases hf : <a>Integrable</a> (fun x \u21a6 <a>exp</a> (f x)) \u03bc", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [438, 5], "def_end_pos": [438, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\n\u22a2 (\u03bc.tilted f) s = ENNReal.ofReal (\u222b (a : \u03b1) in s, rexp (f a) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc \u2202\u03bc)", "state_after": "case pos\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Integrable (fun x => rexp (f x)) \u03bc\n\u22a2 (\u03bc.tilted f) s = ENNReal.ofReal (\u222b (a : \u03b1) in s, rexp (f a) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc \u2202\u03bc)\n\ncase neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : \u00acIntegrable (fun x => rexp (f x)) \u03bc\n\u22a2 (\u03bc.tilted f) s = ENNReal.ofReal (\u222b (a : \u03b1) in s, rexp (f a) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc \u2202\u03bc)"}, {"tactic": "rw [tilted_apply _ _, \u2190 ofReal_integral_eq_lintegral_ofReal]", "annotated_tactic": ["rw [<a>tilted_apply</a> _ _, \u2190 <a>ofReal_integral_eq_lintegral_ofReal</a>]", [{"full_name": "MeasureTheory.tilted_apply", "def_path": "Mathlib/MeasureTheory/Measure/Tilted.lean", "def_pos": [101, 7], "def_end_pos": [101, 19]}, {"full_name": "MeasureTheory.ofReal_integral_eq_lintegral_ofReal", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1204, 9], "def_end_pos": [1204, 44]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Integrable (fun x => rexp (f x)) \u03bc\n\u22a2 (\u03bc.tilted f) s = ENNReal.ofReal (\u222b (a : \u03b1) in s, rexp (f a) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc \u2202\u03bc)", "state_after": "case pos.hfi\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Integrable (fun x => rexp (f x)) \u03bc\n\u22a2 Integrable (fun a => rexp (f a) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) (\u03bc.restrict s)\n\ncase pos.f_nn\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Integrable (fun x => rexp (f x)) \u03bc\n\u22a2 0 \u2264\u1da0[ae (\u03bc.restrict s)] fun a => rexp (f a) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc"}, {"tactic": "exact hf.integrableOn.div_const _", "annotated_tactic": ["exact hf.integrableOn.div_const _", []], "state_before": "case pos.hfi\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Integrable (fun x => rexp (f x)) \u03bc\n\u22a2 Integrable (fun a => rexp (f a) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) (\u03bc.restrict s)", "state_after": "no goals"}, {"tactic": "exact ae_of_all _ (fun _ \u21a6 by positivity)", "annotated_tactic": ["exact <a>ae_of_all</a> _ (fun _ \u21a6 by positivity)", [{"full_name": "MeasureTheory.ae_of_all", "def_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean", "def_pos": [94, 9], "def_end_pos": [94, 18]}]], "state_before": "case pos.f_nn\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Integrable (fun x => rexp (f x)) \u03bc\n\u22a2 0 \u2264\u1da0[ae (\u03bc.restrict s)] fun a => rexp (f a) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : Integrable (fun x => rexp (f x)) \u03bc\nx\u271d : \u03b1\n\u22a2 0 x\u271d \u2264 (fun a => rexp (f a) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) x\u271d", "state_after": "no goals"}, {"tactic": "simp [tilted_of_not_integrable hf, integral_undef hf]", "annotated_tactic": ["simp [<a>tilted_of_not_integrable</a> hf, <a>integral_undef</a> hf]", [{"full_name": "MeasureTheory.tilted_of_not_integrable", "def_path": "Mathlib/MeasureTheory/Measure/Tilted.lean", "def_pos": [42, 7], "def_end_pos": [42, 31]}, {"full_name": "MeasureTheory.integral_undef", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [833, 9], "def_end_pos": [833, 23]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d : \u03b1 \u2192 \u211d\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u2192 \u211d\ns : Set \u03b1\nhf : \u00acIntegrable (fun x => rexp (f x)) \u03bc\n\u22a2 (\u03bc.tilted f) s = ENNReal.ofReal (\u222b (a : \u03b1) in s, rexp (f a) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc \u2202\u03bc)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Nilpotent.lean", "full_name": "nilpotencyClass_prod", "start": [729, 1], "end": [734, 46], "traced_tactics": [{"tactic": "refine eq_of_forall_ge_iff fun k => ?_", "annotated_tactic": ["refine <a>eq_of_forall_ge_iff</a> fun k => ?_", [{"full_name": "eq_of_forall_ge_iff", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [543, 9], "def_end_pos": [543, 28]}]], "state_before": "G : Type u_1\ninst\u271d\u2075 : Group G\nH : Subgroup G\ninst\u271d\u2074 : H.Normal\nG\u2081 : Type u_2\nG\u2082 : Type u_3\ninst\u271d\u00b3 : Group G\u2081\ninst\u271d\u00b2 : Group G\u2082\ninst\u271d\u00b9 : Group.IsNilpotent G\u2081\ninst\u271d : Group.IsNilpotent G\u2082\n\u22a2 Group.nilpotencyClass (G\u2081 \u00d7 G\u2082) = max (Group.nilpotencyClass G\u2081) (Group.nilpotencyClass G\u2082)", "state_after": "G : Type u_1\ninst\u271d\u2075 : Group G\nH : Subgroup G\ninst\u271d\u2074 : H.Normal\nG\u2081 : Type u_2\nG\u2082 : Type u_3\ninst\u271d\u00b3 : Group G\u2081\ninst\u271d\u00b2 : Group G\u2082\ninst\u271d\u00b9 : Group.IsNilpotent G\u2081\ninst\u271d : Group.IsNilpotent G\u2082\nk : \u2115\n\u22a2 Group.nilpotencyClass (G\u2081 \u00d7 G\u2082) \u2264 k \u2194 max (Group.nilpotencyClass G\u2081) (Group.nilpotencyClass G\u2082) \u2264 k"}, {"tactic": "simp only [max_le_iff, \u2190 lowerCentralSeries_eq_bot_iff_nilpotencyClass_le,\n  lowerCentralSeries_prod, prod_eq_bot_iff]", "annotated_tactic": ["simp only [<a>max_le_iff</a>, \u2190 <a>lowerCentralSeries_eq_bot_iff_nilpotencyClass_le</a>,\n    <a>lowerCentralSeries_prod</a>, <a>prod_eq_bot_iff</a>]", [{"full_name": "max_le_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [48, 9], "def_end_pos": [48, 19]}, {"full_name": "lowerCentralSeries_eq_bot_iff_nilpotencyClass_le", "def_path": "Mathlib/GroupTheory/Nilpotent.lean", "def_pos": [433, 9], "def_end_pos": [433, 57]}, {"full_name": "lowerCentralSeries_prod", "def_path": "Mathlib/GroupTheory/Nilpotent.lean", "def_pos": [704, 9], "def_end_pos": [704, 32]}, {"full_name": "Subgroup.prod_eq_bot_iff", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [1783, 9], "def_end_pos": [1783, 24]}]], "state_before": "G : Type u_1\ninst\u271d\u2075 : Group G\nH : Subgroup G\ninst\u271d\u2074 : H.Normal\nG\u2081 : Type u_2\nG\u2082 : Type u_3\ninst\u271d\u00b3 : Group G\u2081\ninst\u271d\u00b2 : Group G\u2082\ninst\u271d\u00b9 : Group.IsNilpotent G\u2081\ninst\u271d : Group.IsNilpotent G\u2082\nk : \u2115\n\u22a2 Group.nilpotencyClass (G\u2081 \u00d7 G\u2082) \u2264 k \u2194 max (Group.nilpotencyClass G\u2081) (Group.nilpotencyClass G\u2082) \u2264 k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Finiteness.lean", "full_name": "Ideal.fg_ker_comp", "start": [499, 1], "end": [508, 99], "traced_tactics": [{"tactic": "letI : Algebra R S := RingHom.toAlgebra f", "annotated_tactic": ["letI : <a>Algebra</a> R S := <a>RingHom.toAlgebra</a> f", [{"full_name": "Algebra", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [101, 7], "def_end_pos": [101, 14]}, {"full_name": "RingHom.toAlgebra", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [248, 5], "def_end_pos": [248, 22]}]], "state_before": "R\u271d : Type u_1\nM : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R\u271d M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing A\nf : R \u2192+* S\ng : S \u2192+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective \u21d1f\n\u22a2 (RingHom.ker (g.comp f)).FG", "state_after": "R\u271d : Type u_1\nM : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R\u271d M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing A\nf : R \u2192+* S\ng : S \u2192+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective \u21d1f\nthis : Algebra R S := f.toAlgebra\n\u22a2 (RingHom.ker (g.comp f)).FG"}, {"tactic": "letI : Algebra R A := RingHom.toAlgebra (g.comp f)", "annotated_tactic": ["letI : <a>Algebra</a> R A := <a>RingHom.toAlgebra</a> (g.comp f)", [{"full_name": "Algebra", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [101, 7], "def_end_pos": [101, 14]}, {"full_name": "RingHom.toAlgebra", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [248, 5], "def_end_pos": [248, 22]}]], "state_before": "R\u271d : Type u_1\nM : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R\u271d M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing A\nf : R \u2192+* S\ng : S \u2192+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective \u21d1f\nthis : Algebra R S := f.toAlgebra\n\u22a2 (RingHom.ker (g.comp f)).FG", "state_after": "R\u271d : Type u_1\nM : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R\u271d M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing A\nf : R \u2192+* S\ng : S \u2192+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective \u21d1f\nthis\u271d : Algebra R S := f.toAlgebra\nthis : Algebra R A := (g.comp f).toAlgebra\n\u22a2 (RingHom.ker (g.comp f)).FG"}, {"tactic": "letI : Algebra S A := RingHom.toAlgebra g", "annotated_tactic": ["letI : <a>Algebra</a> S A := <a>RingHom.toAlgebra</a> g", [{"full_name": "Algebra", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [101, 7], "def_end_pos": [101, 14]}, {"full_name": "RingHom.toAlgebra", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [248, 5], "def_end_pos": [248, 22]}]], "state_before": "R\u271d : Type u_1\nM : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R\u271d M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing A\nf : R \u2192+* S\ng : S \u2192+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective \u21d1f\nthis\u271d : Algebra R S := f.toAlgebra\nthis : Algebra R A := (g.comp f).toAlgebra\n\u22a2 (RingHom.ker (g.comp f)).FG", "state_after": "R\u271d : Type u_1\nM : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R\u271d M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing A\nf : R \u2192+* S\ng : S \u2192+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective \u21d1f\nthis\u271d\u00b9 : Algebra R S := f.toAlgebra\nthis\u271d : Algebra R A := (g.comp f).toAlgebra\nthis : Algebra S A := g.toAlgebra\n\u22a2 (RingHom.ker (g.comp f)).FG"}, {"tactic": "letI : IsScalarTower R S A := IsScalarTower.of_algebraMap_eq fun _ => rfl", "annotated_tactic": ["letI : <a>IsScalarTower</a> R S A := <a>IsScalarTower.of_algebraMap_eq</a> fun _ => <a>rfl</a>", [{"full_name": "IsScalarTower", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [249, 7], "def_end_pos": [249, 20]}, {"full_name": "IsScalarTower.of_algebraMap_eq", "def_path": "Mathlib/Algebra/Algebra/Tower.lean", "def_pos": [110, 9], "def_end_pos": [110, 25]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "R\u271d : Type u_1\nM : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R\u271d M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing A\nf : R \u2192+* S\ng : S \u2192+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective \u21d1f\nthis\u271d\u00b9 : Algebra R S := f.toAlgebra\nthis\u271d : Algebra R A := (g.comp f).toAlgebra\nthis : Algebra S A := g.toAlgebra\n\u22a2 (RingHom.ker (g.comp f)).FG", "state_after": "R\u271d : Type u_1\nM : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R\u271d M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing A\nf : R \u2192+* S\ng : S \u2192+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective \u21d1f\nthis\u271d\u00b2 : Algebra R S := f.toAlgebra\nthis\u271d\u00b9 : Algebra R A := (g.comp f).toAlgebra\nthis\u271d : Algebra S A := g.toAlgebra\nthis : IsScalarTower R S A := IsScalarTower.of_algebraMap_eq fun x => rfl\n\u22a2 (RingHom.ker (g.comp f)).FG"}, {"tactic": "let f\u2081 := Algebra.linearMap R S", "annotated_tactic": ["let f\u2081 := <a>Algebra.linearMap</a> R S", [{"full_name": "Algebra.linearMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [412, 15], "def_end_pos": [412, 24]}]], "state_before": "R\u271d : Type u_1\nM : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R\u271d M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing A\nf : R \u2192+* S\ng : S \u2192+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective \u21d1f\nthis\u271d\u00b2 : Algebra R S := f.toAlgebra\nthis\u271d\u00b9 : Algebra R A := (g.comp f).toAlgebra\nthis\u271d : Algebra S A := g.toAlgebra\nthis : IsScalarTower R S A := IsScalarTower.of_algebraMap_eq fun x => rfl\n\u22a2 (RingHom.ker (g.comp f)).FG", "state_after": "R\u271d : Type u_1\nM : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R\u271d M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing A\nf : R \u2192+* S\ng : S \u2192+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective \u21d1f\nthis\u271d\u00b2 : Algebra R S := f.toAlgebra\nthis\u271d\u00b9 : Algebra R A := (g.comp f).toAlgebra\nthis\u271d : Algebra S A := g.toAlgebra\nthis : IsScalarTower R S A := IsScalarTower.of_algebraMap_eq fun x => rfl\nf\u2081 : R \u2192\u2097[R] S := Algebra.linearMap R S\n\u22a2 (RingHom.ker (g.comp f)).FG"}, {"tactic": "let g\u2081 := (IsScalarTower.toAlgHom R S A).toLinearMap", "annotated_tactic": ["let g\u2081 := (<a>IsScalarTower.toAlgHom</a> R S A).<a>toLinearMap</a>", [{"full_name": "IsScalarTower.toAlgHom", "def_path": "Mathlib/Algebra/Algebra/Tower.lean", "def_pos": [143, 5], "def_end_pos": [143, 13]}, {"full_name": "AlgHom.toLinearMap", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [349, 5], "def_end_pos": [349, 16]}]], "state_before": "R\u271d : Type u_1\nM : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R\u271d M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing A\nf : R \u2192+* S\ng : S \u2192+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective \u21d1f\nthis\u271d\u00b2 : Algebra R S := f.toAlgebra\nthis\u271d\u00b9 : Algebra R A := (g.comp f).toAlgebra\nthis\u271d : Algebra S A := g.toAlgebra\nthis : IsScalarTower R S A := IsScalarTower.of_algebraMap_eq fun x => rfl\nf\u2081 : R \u2192\u2097[R] S := Algebra.linearMap R S\n\u22a2 (RingHom.ker (g.comp f)).FG", "state_after": "R\u271d : Type u_1\nM : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R\u271d M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing A\nf : R \u2192+* S\ng : S \u2192+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective \u21d1f\nthis\u271d\u00b2 : Algebra R S := f.toAlgebra\nthis\u271d\u00b9 : Algebra R A := (g.comp f).toAlgebra\nthis\u271d : Algebra S A := g.toAlgebra\nthis : IsScalarTower R S A := IsScalarTower.of_algebraMap_eq fun x => rfl\nf\u2081 : R \u2192\u2097[R] S := Algebra.linearMap R S\ng\u2081 : S \u2192\u2097[R] A := (IsScalarTower.toAlgHom R S A).toLinearMap\n\u22a2 (RingHom.ker (g.comp f)).FG"}, {"tactic": "exact Submodule.fg_ker_comp f\u2081 g\u2081 hf (Submodule.fg_restrictScalars (RingHom.ker g) hg hsur) hsur", "annotated_tactic": ["exact <a>Submodule.fg_ker_comp</a> f\u2081 g\u2081 hf (<a>Submodule.fg_restrictScalars</a> (<a>RingHom.ker</a> g) hg hsur) hsur", [{"full_name": "Submodule.fg_ker_comp", "def_path": "Mathlib/RingTheory/Finiteness.lean", "def_pos": [349, 9], "def_end_pos": [349, 20]}, {"full_name": "Submodule.fg_restrictScalars", "def_path": "Mathlib/RingTheory/Finiteness.lean", "def_pos": [360, 9], "def_end_pos": [360, 27]}, {"full_name": "RingHom.ker", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [600, 5], "def_end_pos": [600, 8]}]], "state_before": "R\u271d : Type u_1\nM : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R\u271d M\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing A\nf : R \u2192+* S\ng : S \u2192+* A\nhf : (RingHom.ker f).FG\nhg : (RingHom.ker g).FG\nhsur : Surjective \u21d1f\nthis\u271d\u00b2 : Algebra R S := f.toAlgebra\nthis\u271d\u00b9 : Algebra R A := (g.comp f).toAlgebra\nthis\u271d : Algebra S A := g.toAlgebra\nthis : IsScalarTower R S A := IsScalarTower.of_algebraMap_eq fun x => rfl\nf\u2081 : R \u2192\u2097[R] S := Algebra.linearMap R S\ng\u2081 : S \u2192\u2097[R] A := (IsScalarTower.toAlgHom R S A).toLinearMap\n\u22a2 (RingHom.ker (g.comp f)).FG", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean", "full_name": "GromovHausdorff.one_le_maxVar", "start": [76, 9], "end": [79, 90], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "X : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\n\u22a2 1 = 2 * 0 + 1 + 2 * 0", "state_after": "no goals"}, {"tactic": "gcongr <;> positivity", "annotated_tactic": ["gcongr <;> positivity", []], "state_before": "X : Type u\nY : Type v\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\n\u22a2 2 * 0 + 1 + 2 * 0 \u2264 2 * diam univ + 1 + 2 * diam univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Lattice.lean", "full_name": "sup_right_comm", "start": [249, 1], "end": [250, 40], "traced_tactics": [{"tactic": "rw [sup_assoc, sup_assoc, sup_comm b]", "annotated_tactic": ["rw [<a>sup_assoc</a>, <a>sup_assoc</a>, <a>sup_comm</a> b]", [{"full_name": "sup_assoc", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [229, 9], "def_end_pos": [229, 18]}, {"full_name": "sup_assoc", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [229, 9], "def_end_pos": [229, 18]}, {"full_name": "sup_comm", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [224, 9], "def_end_pos": [224, 17]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : SemilatticeSup \u03b1\na\u271d b\u271d c\u271d d a b c : \u03b1\n\u22a2 a \u2294 b \u2294 c = a \u2294 c \u2294 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/LocalizedModule.lean", "full_name": "IsLocalizedModule.iso_apply_mk", "start": [850, 1], "end": [852, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/EraseLead.lean", "full_name": "Polynomial.lt_natDegree_of_mem_eraseLead_support", "start": [95, 1], "end": [98, 54], "traced_tactics": [{"tactic": "rw [eraseLead_support, mem_erase] at h", "annotated_tactic": ["rw [<a>eraseLead_support</a>, <a>mem_erase</a>] at h", [{"full_name": "Polynomial.eraseLead_support", "def_path": "Mathlib/Algebra/Polynomial/EraseLead.lean", "def_pos": [42, 9], "def_end_pos": [42, 26]}, {"full_name": "Finset.mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1895, 9], "def_end_pos": [1895, 18]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\nf : R[X]\na : \u2115\nh : a \u2208 f.eraseLead.support\n\u22a2 a < f.natDegree", "state_after": "R : Type u_1\ninst\u271d : Semiring R\nf : R[X]\na : \u2115\nh : a \u2260 f.natDegree \u2227 a \u2208 f.support\n\u22a2 a < f.natDegree"}, {"tactic": "exact (le_natDegree_of_mem_supp a h.2).lt_of_ne h.1", "annotated_tactic": ["exact (<a>le_natDegree_of_mem_supp</a> a h.2).<a>lt_of_ne</a> h.1", [{"full_name": "Polynomial.le_natDegree_of_mem_supp", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [187, 9], "def_end_pos": [187, 33]}, {"full_name": "LE.le.lt_of_ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [127, 7], "def_end_pos": [127, 21]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\nf : R[X]\na : \u2115\nh : a \u2260 f.natDegree \u2227 a \u2208 f.support\n\u22a2 a < f.natDegree", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/ShortComplex/ModuleCat.lean", "full_name": "CategoryTheory.ShortComplex.Exact.moduleCat_of_range_eq_ker", "start": [95, 1], "end": [98, 58], "traced_tactics": [{"tactic": "rw [hfg]", "annotated_tactic": ["rw [hfg]", []], "state_before": "R : Type u\ninst\u271d : Ring R\nS : ShortComplex (ModuleCat R)\nX\u2081 X\u2082 X\u2083 : ModuleCat R\nf : X\u2081 \u27f6 X\u2082\ng : X\u2082 \u27f6 X\u2083\nhfg : LinearMap.range f = LinearMap.ker g\n\u22a2 LinearMap.range f \u2264 LinearMap.ker g", "state_after": "no goals"}, {"tactic": "simpa only [moduleCat_exact_iff_range_eq_ker] using hfg", "annotated_tactic": ["simpa only [<a>moduleCat_exact_iff_range_eq_ker</a>] using hfg", [{"full_name": "CategoryTheory.ShortComplex.moduleCat_exact_iff_range_eq_ker", "def_path": "Mathlib/Algebra/Homology/ShortComplex/ModuleCat.lean", "def_pos": [57, 7], "def_end_pos": [57, 39]}]], "state_before": "R : Type u\ninst\u271d : Ring R\nS : ShortComplex (ModuleCat R)\nX\u2081 X\u2082 X\u2083 : ModuleCat R\nf : X\u2081 \u27f6 X\u2082\ng : X\u2082 \u27f6 X\u2083\nhfg : LinearMap.range f = LinearMap.ker g\n\u22a2 (moduleCatMkOfKerLERange f g \u22ef).Exact", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.not_mem_range_self", "start": [2963, 1], "end": [2964, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean", "full_name": "eval\u2082_minpolyDiv_of_eval\u2082_eq_zero", "start": [68, 1], "end": [76, 44], "traced_tactics": [{"tactic": "split_ifs with h", "annotated_tactic": ["split_ifs with h", []], "state_before": "R : Type u_3\nK : Type ?u.34864\nL : Type ?u.34867\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra K L\nx\u271d : S\nhx : IsIntegral R x\u271d\nT : Type u_1\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : IsDomain T\ninst\u271d : DecidableEq T\nx : S\ny : T\n\u03c3 : S \u2192+* T\nhy : eval\u2082 (\u03c3.comp (algebraMap R S)) y (minpoly R x) = 0\n\u22a2 eval\u2082 \u03c3 y (minpolyDiv R x) = if \u03c3 x = y then \u03c3 ((aeval x) (derivative (minpoly R x))) else 0", "state_after": "case pos\nR : Type u_3\nK : Type ?u.34864\nL : Type ?u.34867\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra K L\nx\u271d : S\nhx : IsIntegral R x\u271d\nT : Type u_1\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : IsDomain T\ninst\u271d : DecidableEq T\nx : S\ny : T\n\u03c3 : S \u2192+* T\nhy : eval\u2082 (\u03c3.comp (algebraMap R S)) y (minpoly R x) = 0\nh : \u03c3 x = y\n\u22a2 eval\u2082 \u03c3 y (minpolyDiv R x) = \u03c3 ((aeval x) (derivative (minpoly R x)))\n\ncase neg\nR : Type u_3\nK : Type ?u.34864\nL : Type ?u.34867\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra K L\nx\u271d : S\nhx : IsIntegral R x\u271d\nT : Type u_1\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : IsDomain T\ninst\u271d : DecidableEq T\nx : S\ny : T\n\u03c3 : S \u2192+* T\nhy : eval\u2082 (\u03c3.comp (algebraMap R S)) y (minpoly R x) = 0\nh : \u00ac\u03c3 x = y\n\u22a2 eval\u2082 \u03c3 y (minpolyDiv R x) = 0"}, {"tactic": "rw [\u2190 h, eval\u2082_hom, eval_minpolyDiv_self]", "annotated_tactic": ["rw [\u2190 h, <a>eval\u2082_hom</a>, <a>eval_minpolyDiv_self</a>]", [{"full_name": "Polynomial.eval\u2082_hom", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [1052, 9], "def_end_pos": [1052, 18]}, {"full_name": "eval_minpolyDiv_self", "def_path": "Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean", "def_pos": [59, 7], "def_end_pos": [59, 27]}]], "state_before": "case pos\nR : Type u_3\nK : Type ?u.34864\nL : Type ?u.34867\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra K L\nx\u271d : S\nhx : IsIntegral R x\u271d\nT : Type u_1\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : IsDomain T\ninst\u271d : DecidableEq T\nx : S\ny : T\n\u03c3 : S \u2192+* T\nhy : eval\u2082 (\u03c3.comp (algebraMap R S)) y (minpoly R x) = 0\nh : \u03c3 x = y\n\u22a2 eval\u2082 \u03c3 y (minpolyDiv R x) = \u03c3 ((aeval x) (derivative (minpoly R x)))", "state_after": "no goals"}, {"tactic": "rw [\u2190 eval\u2082_map, \u2190 minpolyDiv_spec] at hy", "annotated_tactic": ["rw [\u2190 <a>eval\u2082_map</a>, \u2190 <a>minpolyDiv_spec</a>] at hy", [{"full_name": "Polynomial.eval\u2082_map", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [959, 9], "def_end_pos": [959, 18]}, {"full_name": "minpolyDiv_spec", "def_path": "Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean", "def_pos": [31, 7], "def_end_pos": [31, 22]}]], "state_before": "case neg\nR : Type u_3\nK : Type ?u.34864\nL : Type ?u.34867\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra K L\nx\u271d : S\nhx : IsIntegral R x\u271d\nT : Type u_1\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : IsDomain T\ninst\u271d : DecidableEq T\nx : S\ny : T\n\u03c3 : S \u2192+* T\nhy : eval\u2082 (\u03c3.comp (algebraMap R S)) y (minpoly R x) = 0\nh : \u00ac\u03c3 x = y\n\u22a2 eval\u2082 \u03c3 y (minpolyDiv R x) = 0", "state_after": "case neg\nR : Type u_3\nK : Type ?u.34864\nL : Type ?u.34867\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra K L\nx\u271d : S\nhx : IsIntegral R x\u271d\nT : Type u_1\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : IsDomain T\ninst\u271d : DecidableEq T\nx : S\ny : T\n\u03c3 : S \u2192+* T\nhy : eval\u2082 \u03c3 y (minpolyDiv R x * (X - C x)) = 0\nh : \u00ac\u03c3 x = y\n\u22a2 eval\u2082 \u03c3 y (minpolyDiv R x) = 0"}, {"tactic": "simpa [sub_eq_zero, Ne.symm h] using hy", "annotated_tactic": ["simpa [<a>sub_eq_zero</a>, <a>Ne.symm</a> h] using hy", [{"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1070, 3], "def_end_pos": [1070, 14]}, {"full_name": "Ne.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [704, 9], "def_end_pos": [704, 16]}]], "state_before": "case neg\nR : Type u_3\nK : Type ?u.34864\nL : Type ?u.34867\nS : Type u_2\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : Field L\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\ninst\u271d\u00b3 : Algebra K L\nx\u271d : S\nhx : IsIntegral R x\u271d\nT : Type u_1\ninst\u271d\u00b2 : CommRing T\ninst\u271d\u00b9 : IsDomain T\ninst\u271d : DecidableEq T\nx : S\ny : T\n\u03c3 : S \u2192+* T\nhy : eval\u2082 \u03c3 y (minpolyDiv R x * (X - C x)) = 0\nh : \u00ac\u03c3 x = y\n\u22a2 eval\u2082 \u03c3 y (minpolyDiv R x) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "full_name": "MeasureTheory.SignedMeasure.toJordanDecomposition_eq", "start": [491, 1], "end": [493, 48], "traced_tactics": [{"tactic": "rw [h, toJordanDecomposition_toSignedMeasure]", "annotated_tactic": ["rw [h, <a>toJordanDecomposition_toSignedMeasure</a>]", [{"full_name": "MeasureTheory.JordanDecomposition.toJordanDecomposition_toSignedMeasure", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [429, 9], "def_end_pos": [429, 46]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns : SignedMeasure \u03b1\nj : JordanDecomposition \u03b1\nh : s = j.toSignedMeasure\n\u22a2 s.toJordanDecomposition = j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "full_name": "Subalgebra.mem_toSubring", "start": [229, 1], "end": [231, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Invertible/Defs.lean", "full_name": "mul_invOf_mul_self_cancel", "start": [136, 1], "end": [137, 19], "traced_tactics": [{"tactic": "simp [mul_assoc]", "annotated_tactic": ["simp [<a>mul_assoc</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "\u03b1 : Type u\ninst\u271d\u00b9 : Monoid \u03b1\na b : \u03b1\ninst\u271d : Invertible b\n\u22a2 a * \u215fb * b = a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Regular/RegularSequence.lean", "full_name": "RingTheory.Sequence.isWeaklyRegular_append_iff'", "start": [383, 1], "end": [389, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Finprod.lean", "full_name": "finprod_nonneg", "start": [278, 1], "end": [280, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean", "full_name": "MvPolynomial.isWeightedHomogeneous_one", "start": [234, 1], "end": [235, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Spectrum.lean", "full_name": "spectrum.singleton_add_eq", "start": [309, 1], "end": [311, 94], "traced_tactics": [{"tactic": "rw [singleton_add, image_add_left, mem_preimage, add_comm, add_mem_iff, map_neg, neg_neg]", "annotated_tactic": ["rw [<a>singleton_add</a>, <a>image_add_left</a>, <a>mem_preimage</a>, <a>add_comm</a>, <a>add_mem_iff</a>, <a>map_neg</a>, <a>neg_neg</a>]", [{"full_name": "Set.singleton_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [405, 3], "def_end_pos": [405, 14]}, {"full_name": "Set.image_add_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1203, 3], "def_end_pos": [1203, 14]}, {"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [120, 9], "def_end_pos": [120, 21]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "spectrum.add_mem_iff", "def_path": "Mathlib/Algebra/Algebra/Spectrum.lean", "def_pos": [222, 9], "def_end_pos": [222, 20]}, {"full_name": "map_neg", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [437, 3], "def_end_pos": [437, 14]}, {"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [869, 3], "def_end_pos": [869, 14]}]], "state_before": "R : Type u\nA : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Ring A\ninst\u271d : Algebra R A\na : A\nr x : R\n\u22a2 x \u2208 {r} + \u03c3 a \u2194 x \u2208 \u03c3 (\u2191\u2090 r + a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Quaternion.lean", "full_name": "Quaternion.sub_im", "start": [1003, 16], "end": [1003, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "IntervalIntegrable.neg", "start": [187, 1], "end": [188, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/BilinearForm/Basic.lean", "full_name": "LinearMap.BilinForm.add_right", "start": [76, 1], "end": [76, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/CoprodI.lean", "full_name": "Monoid.CoprodI.NeWord.inv_head", "start": [856, 1], "end": [857, 32], "traced_tactics": [{"tactic": "induction w <;> simp [inv, *]", "annotated_tactic": ["induction w <;> simp [<a>inv</a>, *]", [{"full_name": "Monoid.CoprodI.NeWord.inv", "def_path": "Mathlib/GroupTheory/CoprodI.lean", "def_pos": [845, 5], "def_end_pos": [845, 8]}]], "state_before": "\u03b9 : Type u_1\nM : \u03b9 \u2192 Type u_2\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Monoid (M i)\nN : Type u_3\ninst\u271d\u00b9 : Monoid N\nG : \u03b9 \u2192 Type u_4\ninst\u271d : (i : \u03b9) \u2192 Group (G i)\ni j : \u03b9\nw : NeWord G i j\n\u22a2 w.inv.head = w.last\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.sdiff_inter_self_left", "start": [2157, 1], "end": [2158, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "IsOpen.mul_closure_one_eq", "start": [1578, 1], "end": [1581, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Order/Lattice.lean", "full_name": "AEMeasurable.inf_const", "start": [194, 1], "end": [196, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM_zero_left'", "start": [895, 1], "end": [898, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "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": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/ToIntervalMod.lean", "full_name": "toIcoMod_toIocMod", "start": [739, 1], "end": [740, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/Basic.lean", "full_name": "Set.Iio_def", "start": [91, 1], "end": [92, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Subsemiring/Basic.lean", "full_name": "Subsemiring.mem_closure_iff", "start": [841, 1], "end": [843, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SumIntegralComparisons.lean", "full_name": "MonotoneOn.integral_le_sum_Ico", "start": [168, 1], "end": [171, 39], "traced_tactics": [{"tactic": "rw [\u2190 neg_le_neg_iff, \u2190 Finset.sum_neg_distrib, \u2190 intervalIntegral.integral_neg]", "annotated_tactic": ["rw [\u2190 <a>neg_le_neg_iff</a>, \u2190 <a>Finset.sum_neg_distrib</a>, \u2190 <a>intervalIntegral.integral_neg</a>]", [{"full_name": "neg_le_neg_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [342, 3], "def_end_pos": [342, 14]}, {"full_name": "Finset.sum_neg_distrib", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [2116, 3], "def_end_pos": [2116, 14]}, {"full_name": "intervalIntegral.integral_neg", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [613, 16], "def_end_pos": [613, 28]}]], "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : MonotoneOn f (Icc \u2191a \u2191b)\n\u22a2 \u222b (x : \u211d) in \u2191a..\u2191b, f x \u2264 \u2211 i \u2208 Finset.Ico a b, f \u2191(i + 1)", "state_after": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : MonotoneOn f (Icc \u2191a \u2191b)\n\u22a2 \u2211 x \u2208 Finset.Ico a b, -f \u2191(x + 1) \u2264 \u222b (x : \u211d) in \u2191a..\u2191b, -f x"}, {"tactic": "exact hf.neg.sum_le_integral_Ico hab", "annotated_tactic": ["exact hf.neg.sum_le_integral_Ico hab", []], "state_before": "x\u2080 : \u211d\na b : \u2115\nf : \u211d \u2192 \u211d\nhab : a \u2264 b\nhf : MonotoneOn f (Icc \u2191a \u2191b)\n\u22a2 \u2211 x \u2208 Finset.Ico a b, -f \u2191(x + 1) \u2264 \u222b (x : \u211d) in \u2191a..\u2191b, -f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Hom/Basic.lean", "full_name": "MonoidHom.mul_comp", "start": [217, 1], "end": [219, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.vector_ofFn'", "start": [1419, 1], "end": [1420, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UrysohnsLemma.lean", "full_name": "Urysohns.CU.continuous_lim", "start": [278, 1], "end": [312, 37], "traced_tactics": [{"tactic": "obtain \u27e8h0, h1234, h1\u27e9 : 0 < (2\u207b\u00b9 : \u211d) \u2227 (2\u207b\u00b9 : \u211d) < 3 / 4 \u2227 (3 / 4 : \u211d) < 1 := by norm_num", "annotated_tactic": ["obtain \u27e8h0, h1234, h1\u27e9 : 0 < (2\u207b\u00b9 : \u211d) \u2227 (2\u207b\u00b9 : \u211d) < 3 / 4 \u2227 (3 / 4 : \u211d) < 1 := by norm_num", []], "state_before": "X : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nc : CU P\n\u22a2 Continuous c.lim", "state_after": "case intro.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nc : CU P\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\n\u22a2 Continuous c.lim"}, {"tactic": "refine\n  continuous_iff_continuousAt.2 fun x =>\n    (Metric.nhds_basis_closedBall_pow (h0.trans h1234) h1).tendsto_right_iff.2 fun n _ => ?_", "annotated_tactic": ["refine\n    <a>continuous_iff_continuousAt</a>.2 fun x =>\n      (<a>Metric.nhds_basis_closedBall_pow</a> (h0.trans h1234) h1).<a>tendsto_right_iff</a>.2 fun n _ => ?_", [{"full_name": "continuous_iff_continuousAt", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1664, 9], "def_end_pos": [1664, 36]}, {"full_name": "Metric.nhds_basis_closedBall_pow", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [1003, 9], "def_end_pos": [1003, 34]}, {"full_name": "Filter.HasBasis.tendsto_right_iff", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [881, 9], "def_end_pos": [881, 35]}]], "state_before": "case intro.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nc : CU P\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\n\u22a2 Continuous c.lim", "state_after": "case intro.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nc : CU P\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nn : \u2115\nx\u271d : True\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, c.lim x_1 \u2208 Metric.closedBall (c.lim x) ((3 / 4) ^ n)"}, {"tactic": "simp only [Metric.mem_closedBall]", "annotated_tactic": ["simp only [<a>Metric.mem_closedBall</a>]", [{"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [468, 17], "def_end_pos": [468, 31]}]], "state_before": "case intro.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nc : CU P\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nn : \u2115\nx\u271d : True\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, c.lim x_1 \u2208 Metric.closedBall (c.lim x) ((3 / 4) ^ n)", "state_after": "case intro.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nc : CU P\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nn : \u2115\nx\u271d : True\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n"}, {"tactic": "induction' n with n ihn generalizing c", "annotated_tactic": ["induction' n with n ihn generalizing c", []], "state_before": "case intro.intro\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nc : CU P\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nn : \u2115\nx\u271d : True\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n", "state_after": "case intro.intro.zero\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nc : CU P\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ 0\n\ncase intro.intro.succ\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ (n + 1)"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "X : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nc : CU P\n\u22a2 0 < 2\u207b\u00b9 \u2227 2\u207b\u00b9 < 3 / 4 \u2227 3 / 4 < 1", "state_after": "no goals"}, {"tactic": "filter_upwards with y", "annotated_tactic": ["filter_upwards with y", []], "state_before": "case intro.intro.zero\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nc : CU P\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ 0", "state_after": "case intro.intro.zero.h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nc : CU P\ny : X\n\u22a2 dist (c.lim y) (c.lim x) \u2264 (3 / 4) ^ 0"}, {"tactic": "rw [pow_zero]", "annotated_tactic": ["rw [<a>pow_zero</a>]", [{"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [651, 9], "def_end_pos": [651, 17]}]], "state_before": "case intro.intro.zero.h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nc : CU P\ny : X\n\u22a2 dist (c.lim y) (c.lim x) \u2264 (3 / 4) ^ 0", "state_after": "case intro.intro.zero.h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nc : CU P\ny : X\n\u22a2 dist (c.lim y) (c.lim x) \u2264 1"}, {"tactic": "exact Real.dist_le_of_mem_Icc_01 (c.lim_mem_Icc _) (c.lim_mem_Icc _)", "annotated_tactic": ["exact <a>Real.dist_le_of_mem_Icc_01</a> (c.lim_mem_Icc _) (c.lim_mem_Icc _)", [{"full_name": "Real.dist_le_of_mem_Icc_01", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean", "def_pos": [38, 7], "def_end_pos": [38, 33]}]], "state_before": "case intro.intro.zero.h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nc : CU P\ny : X\n\u22a2 dist (c.lim y) (c.lim x) \u2264 1", "state_after": "no goals"}, {"tactic": "by_cases hxl : x \u2208 c.left.U", "annotated_tactic": ["by_cases hxl : x \u2208 c.left.U", []], "state_before": "case intro.intro.succ\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ (n + 1)", "state_after": "case pos\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\nhxl : x \u2208 c.left.U\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ (n + 1)\n\ncase neg\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\nhxl : x \u2209 c.left.U\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ (n + 1)"}, {"tactic": "filter_upwards [IsOpen.mem_nhds c.left.open_U hxl, ihn c.left] with _ hyl hyd", "annotated_tactic": ["filter_upwards [<a>IsOpen.mem_nhds</a> c.left.open_U hxl, ihn c.left] with _ hyl hyd", [{"full_name": "IsOpen.mem_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [870, 9], "def_end_pos": [870, 24]}]], "state_before": "case pos\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\nhxl : x \u2208 c.left.U\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ (n + 1)", "state_after": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\nhxl : x \u2208 c.left.U\na\u271d : X\nhyl : a\u271d \u2208 c.left.U\nhyd : dist (c.left.lim a\u271d) (c.left.lim x) \u2264 (3 / 4) ^ n\n\u22a2 dist (c.lim a\u271d) (c.lim x) \u2264 (3 / 4) ^ (n + 1)"}, {"tactic": "rw [pow_succ', c.lim_eq_midpoint, c.lim_eq_midpoint,\n  c.right.lim_of_mem_C _ (c.left_U_subset_right_C hyl),\n  c.right.lim_of_mem_C _ (c.left_U_subset_right_C hxl)]", "annotated_tactic": ["rw [<a>pow_succ'</a>, c.lim_eq_midpoint, c.lim_eq_midpoint,\n        c.right.lim_of_mem_C _ (c.left_U_subset_right_C hyl),\n        c.right.lim_of_mem_C _ (c.left_U_subset_right_C hxl)]", [{"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [667, 34], "def_end_pos": [667, 43]}]], "state_before": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\nhxl : x \u2208 c.left.U\na\u271d : X\nhyl : a\u271d \u2208 c.left.U\nhyd : dist (c.left.lim a\u271d) (c.left.lim x) \u2264 (3 / 4) ^ n\n\u22a2 dist (c.lim a\u271d) (c.lim x) \u2264 (3 / 4) ^ (n + 1)", "state_after": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\nhxl : x \u2208 c.left.U\na\u271d : X\nhyl : a\u271d \u2208 c.left.U\nhyd : dist (c.left.lim a\u271d) (c.left.lim x) \u2264 (3 / 4) ^ n\n\u22a2 dist (midpoint \u211d (c.left.lim a\u271d) 0) (midpoint \u211d (c.left.lim x) 0) \u2264 3 / 4 * (3 / 4) ^ n"}, {"tactic": "refine (dist_midpoint_midpoint_le _ _ _ _).trans ?_", "annotated_tactic": ["refine (<a>dist_midpoint_midpoint_le</a> _ _ _ _).<a>trans</a> ?_", [{"full_name": "dist_midpoint_midpoint_le", "def_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean", "def_pos": [273, 9], "def_end_pos": [273, 34]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}]], "state_before": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\nhxl : x \u2208 c.left.U\na\u271d : X\nhyl : a\u271d \u2208 c.left.U\nhyd : dist (c.left.lim a\u271d) (c.left.lim x) \u2264 (3 / 4) ^ n\n\u22a2 dist (midpoint \u211d (c.left.lim a\u271d) 0) (midpoint \u211d (c.left.lim x) 0) \u2264 3 / 4 * (3 / 4) ^ n", "state_after": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\nhxl : x \u2208 c.left.U\na\u271d : X\nhyl : a\u271d \u2208 c.left.U\nhyd : dist (c.left.lim a\u271d) (c.left.lim x) \u2264 (3 / 4) ^ n\n\u22a2 (dist (c.left.lim a\u271d) (c.left.lim x) + dist 0 0) / 2 \u2264 3 / 4 * (3 / 4) ^ n"}, {"tactic": "rw [dist_self, add_zero, div_eq_inv_mul]", "annotated_tactic": ["rw [<a>dist_self</a>, <a>add_zero</a>, <a>div_eq_inv_mul</a>]", [{"full_name": "dist_self", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [173, 9], "def_end_pos": [173, 18]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [482, 3], "def_end_pos": [482, 14]}, {"full_name": "div_eq_inv_mul", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 23]}]], "state_before": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\nhxl : x \u2208 c.left.U\na\u271d : X\nhyl : a\u271d \u2208 c.left.U\nhyd : dist (c.left.lim a\u271d) (c.left.lim x) \u2264 (3 / 4) ^ n\n\u22a2 (dist (c.left.lim a\u271d) (c.left.lim x) + dist 0 0) / 2 \u2264 3 / 4 * (3 / 4) ^ n", "state_after": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\nhxl : x \u2208 c.left.U\na\u271d : X\nhyl : a\u271d \u2208 c.left.U\nhyd : dist (c.left.lim a\u271d) (c.left.lim x) \u2264 (3 / 4) ^ n\n\u22a2 2\u207b\u00b9 * dist (c.left.lim a\u271d) (c.left.lim x) \u2264 3 / 4 * (3 / 4) ^ n"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\nhxl : x \u2208 c.left.U\na\u271d : X\nhyl : a\u271d \u2208 c.left.U\nhyd : dist (c.left.lim a\u271d) (c.left.lim x) \u2264 (3 / 4) ^ n\n\u22a2 2\u207b\u00b9 * dist (c.left.lim a\u271d) (c.left.lim x) \u2264 3 / 4 * (3 / 4) ^ n", "state_after": "no goals"}, {"tactic": "replace hxl : x \u2208 c.left.right.C\u1d9c :=\n  compl_subset_compl.2 c.left.right.subset hxl", "annotated_tactic": ["replace hxl : x \u2208 c.left.right.C\u1d9c :=\n        <a>compl_subset_compl</a>.2 c.left.right.subset hxl", [{"full_name": "Set.compl_subset_compl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1707, 9], "def_end_pos": [1707, 27]}]], "state_before": "case neg\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\nhxl : x \u2209 c.left.U\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ (n + 1)", "state_after": "case neg\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\nhxl : x \u2208 c.left.right.C\u1d9c\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ (n + 1)"}, {"tactic": "filter_upwards [IsOpen.mem_nhds (isOpen_compl_iff.2 c.left.right.closed_C) hxl,\n  ihn c.left.right, ihn c.right] with y hyl hydl hydr", "annotated_tactic": ["filter_upwards [<a>IsOpen.mem_nhds</a> (<a>isOpen_compl_iff</a>.2 c.left.right.closed_C) hxl,\n        ihn c.left.right, ihn c.right] with y hyl hydl hydr", [{"full_name": "IsOpen.mem_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [870, 9], "def_end_pos": [870, 24]}, {"full_name": "isOpen_compl_iff", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [160, 17], "def_end_pos": [160, 33]}]], "state_before": "case neg\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\nhxl : x \u2208 c.left.right.C\u1d9c\n\u22a2 \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ (n + 1)", "state_after": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\nhxl : x \u2208 c.left.right.C\u1d9c\ny : X\nhyl : y \u2208 c.left.right.C\u1d9c\nhydl : dist (c.left.right.lim y) (c.left.right.lim x) \u2264 (3 / 4) ^ n\nhydr : dist (c.right.lim y) (c.right.lim x) \u2264 (3 / 4) ^ n\n\u22a2 dist (c.lim y) (c.lim x) \u2264 (3 / 4) ^ (n + 1)"}, {"tactic": "replace hxl : x \u2209 c.left.left.U :=\n  compl_subset_compl.2 c.left.left_U_subset_right_C hxl", "annotated_tactic": ["replace hxl : x \u2209 c.left.left.U :=\n        <a>compl_subset_compl</a>.2 c.left.left_U_subset_right_C hxl", [{"full_name": "Set.compl_subset_compl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1707, 9], "def_end_pos": [1707, 27]}]], "state_before": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\nhxl : x \u2208 c.left.right.C\u1d9c\ny : X\nhyl : y \u2208 c.left.right.C\u1d9c\nhydl : dist (c.left.right.lim y) (c.left.right.lim x) \u2264 (3 / 4) ^ n\nhydr : dist (c.right.lim y) (c.right.lim x) \u2264 (3 / 4) ^ n\n\u22a2 dist (c.lim y) (c.lim x) \u2264 (3 / 4) ^ (n + 1)", "state_after": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\ny : X\nhyl : y \u2208 c.left.right.C\u1d9c\nhydl : dist (c.left.right.lim y) (c.left.right.lim x) \u2264 (3 / 4) ^ n\nhydr : dist (c.right.lim y) (c.right.lim x) \u2264 (3 / 4) ^ n\nhxl : x \u2209 c.left.left.U\n\u22a2 dist (c.lim y) (c.lim x) \u2264 (3 / 4) ^ (n + 1)"}, {"tactic": "replace hyl : y \u2209 c.left.left.U :=\n  compl_subset_compl.2 c.left.left_U_subset_right_C hyl", "annotated_tactic": ["replace hyl : y \u2209 c.left.left.U :=\n        <a>compl_subset_compl</a>.2 c.left.left_U_subset_right_C hyl", [{"full_name": "Set.compl_subset_compl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1707, 9], "def_end_pos": [1707, 27]}]], "state_before": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\ny : X\nhyl : y \u2208 c.left.right.C\u1d9c\nhydl : dist (c.left.right.lim y) (c.left.right.lim x) \u2264 (3 / 4) ^ n\nhydr : dist (c.right.lim y) (c.right.lim x) \u2264 (3 / 4) ^ n\nhxl : x \u2209 c.left.left.U\n\u22a2 dist (c.lim y) (c.lim x) \u2264 (3 / 4) ^ (n + 1)", "state_after": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\ny : X\nhydl : dist (c.left.right.lim y) (c.left.right.lim x) \u2264 (3 / 4) ^ n\nhydr : dist (c.right.lim y) (c.right.lim x) \u2264 (3 / 4) ^ n\nhxl : x \u2209 c.left.left.U\nhyl : y \u2209 c.left.left.U\n\u22a2 dist (c.lim y) (c.lim x) \u2264 (3 / 4) ^ (n + 1)"}, {"tactic": "simp only [pow_succ, c.lim_eq_midpoint, c.left.lim_eq_midpoint,\n  c.left.left.lim_of_nmem_U _ hxl, c.left.left.lim_of_nmem_U _ hyl]", "annotated_tactic": ["simp only [<a>pow_succ</a>, c.lim_eq_midpoint, c.left.lim_eq_midpoint,\n        c.left.left.lim_of_nmem_U _ hxl, c.left.left.lim_of_nmem_U _ hyl]", [{"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [657, 9], "def_end_pos": [657, 17]}]], "state_before": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\ny : X\nhydl : dist (c.left.right.lim y) (c.left.right.lim x) \u2264 (3 / 4) ^ n\nhydr : dist (c.right.lim y) (c.right.lim x) \u2264 (3 / 4) ^ n\nhxl : x \u2209 c.left.left.U\nhyl : y \u2209 c.left.left.U\n\u22a2 dist (c.lim y) (c.lim x) \u2264 (3 / 4) ^ (n + 1)", "state_after": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\ny : X\nhydl : dist (c.left.right.lim y) (c.left.right.lim x) \u2264 (3 / 4) ^ n\nhydr : dist (c.right.lim y) (c.right.lim x) \u2264 (3 / 4) ^ n\nhxl : x \u2209 c.left.left.U\nhyl : y \u2209 c.left.left.U\n\u22a2 dist (midpoint \u211d (midpoint \u211d 1 (c.left.right.lim y)) (c.right.lim y))\n      (midpoint \u211d (midpoint \u211d 1 (c.left.right.lim x)) (c.right.lim x)) \u2264\n    (3 / 4) ^ n * (3 / 4)"}, {"tactic": "refine (dist_midpoint_midpoint_le _ _ _ _).trans ?_", "annotated_tactic": ["refine (<a>dist_midpoint_midpoint_le</a> _ _ _ _).<a>trans</a> ?_", [{"full_name": "dist_midpoint_midpoint_le", "def_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean", "def_pos": [273, 9], "def_end_pos": [273, 34]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}]], "state_before": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\ny : X\nhydl : dist (c.left.right.lim y) (c.left.right.lim x) \u2264 (3 / 4) ^ n\nhydr : dist (c.right.lim y) (c.right.lim x) \u2264 (3 / 4) ^ n\nhxl : x \u2209 c.left.left.U\nhyl : y \u2209 c.left.left.U\n\u22a2 dist (midpoint \u211d (midpoint \u211d 1 (c.left.right.lim y)) (c.right.lim y))\n      (midpoint \u211d (midpoint \u211d 1 (c.left.right.lim x)) (c.right.lim x)) \u2264\n    (3 / 4) ^ n * (3 / 4)", "state_after": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\ny : X\nhydl : dist (c.left.right.lim y) (c.left.right.lim x) \u2264 (3 / 4) ^ n\nhydr : dist (c.right.lim y) (c.right.lim x) \u2264 (3 / 4) ^ n\nhxl : x \u2209 c.left.left.U\nhyl : y \u2209 c.left.left.U\n\u22a2 (dist (midpoint \u211d 1 (c.left.right.lim y)) (midpoint \u211d 1 (c.left.right.lim x)) +\n        dist (c.right.lim y) (c.right.lim x)) /\n      2 \u2264\n    (3 / 4) ^ n * (3 / 4)"}, {"tactic": "refine (div_le_div_of_nonneg_right (add_le_add_right (dist_midpoint_midpoint_le _ _ _ _) _)\n  zero_le_two).trans ?_", "annotated_tactic": ["refine (<a>div_le_div_of_nonneg_right</a> (<a>add_le_add_right</a> (<a>dist_midpoint_midpoint_le</a> _ _ _ _) _)\n        <a>zero_le_two</a>).<a>trans</a> ?_", [{"full_name": "div_le_div_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [273, 7], "def_end_pos": [273, 33]}, {"full_name": "add_le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [66, 32], "def_end_pos": [66, 48]}, {"full_name": "dist_midpoint_midpoint_le", "def_path": "Mathlib/Analysis/NormedSpace/AddTorsor.lean", "def_pos": [273, 9], "def_end_pos": [273, 34]}, {"full_name": "zero_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [32, 7], "def_end_pos": [32, 18]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}]], "state_before": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\ny : X\nhydl : dist (c.left.right.lim y) (c.left.right.lim x) \u2264 (3 / 4) ^ n\nhydr : dist (c.right.lim y) (c.right.lim x) \u2264 (3 / 4) ^ n\nhxl : x \u2209 c.left.left.U\nhyl : y \u2209 c.left.left.U\n\u22a2 (dist (midpoint \u211d 1 (c.left.right.lim y)) (midpoint \u211d 1 (c.left.right.lim x)) +\n        dist (c.right.lim y) (c.right.lim x)) /\n      2 \u2264\n    (3 / 4) ^ n * (3 / 4)", "state_after": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\ny : X\nhydl : dist (c.left.right.lim y) (c.left.right.lim x) \u2264 (3 / 4) ^ n\nhydr : dist (c.right.lim y) (c.right.lim x) \u2264 (3 / 4) ^ n\nhxl : x \u2209 c.left.left.U\nhyl : y \u2209 c.left.left.U\n\u22a2 ((dist 1 1 + dist (c.left.right.lim y) (c.left.right.lim x)) / 2 + dist (c.right.lim y) (c.right.lim x)) / 2 \u2264\n    (3 / 4) ^ n * (3 / 4)"}, {"tactic": "rw [dist_self, zero_add]", "annotated_tactic": ["rw [<a>dist_self</a>, <a>zero_add</a>]", [{"full_name": "dist_self", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [173, 9], "def_end_pos": [173, 18]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [476, 3], "def_end_pos": [476, 14]}]], "state_before": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\ny : X\nhydl : dist (c.left.right.lim y) (c.left.right.lim x) \u2264 (3 / 4) ^ n\nhydr : dist (c.right.lim y) (c.right.lim x) \u2264 (3 / 4) ^ n\nhxl : x \u2209 c.left.left.U\nhyl : y \u2209 c.left.left.U\n\u22a2 ((dist 1 1 + dist (c.left.right.lim y) (c.left.right.lim x)) / 2 + dist (c.right.lim y) (c.right.lim x)) / 2 \u2264\n    (3 / 4) ^ n * (3 / 4)", "state_after": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\ny : X\nhydl : dist (c.left.right.lim y) (c.left.right.lim x) \u2264 (3 / 4) ^ n\nhydr : dist (c.right.lim y) (c.right.lim x) \u2264 (3 / 4) ^ n\nhxl : x \u2209 c.left.left.U\nhyl : y \u2209 c.left.left.U\n\u22a2 (dist (c.left.right.lim y) (c.left.right.lim x) / 2 + dist (c.right.lim y) (c.right.lim x)) / 2 \u2264\n    (3 / 4) ^ n * (3 / 4)"}, {"tactic": "set r := (3 / 4 : \u211d) ^ n", "annotated_tactic": ["set r := (3 / 4 : \u211d) ^ n", []], "state_before": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 (3 / 4) ^ n\nc : CU P\ny : X\nhydl : dist (c.left.right.lim y) (c.left.right.lim x) \u2264 (3 / 4) ^ n\nhydr : dist (c.right.lim y) (c.right.lim x) \u2264 (3 / 4) ^ n\nhxl : x \u2209 c.left.left.U\nhyl : y \u2209 c.left.left.U\n\u22a2 (dist (c.left.right.lim y) (c.left.right.lim x) / 2 + dist (c.right.lim y) (c.right.lim x)) / 2 \u2264\n    (3 / 4) ^ n * (3 / 4)", "state_after": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nc : CU P\ny : X\nhxl : x \u2209 c.left.left.U\nhyl : y \u2209 c.left.left.U\nr : \u211d := (3 / 4) ^ n\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 r\nhydl : dist (c.left.right.lim y) (c.left.right.lim x) \u2264 r\nhydr : dist (c.right.lim y) (c.right.lim x) \u2264 r\n\u22a2 (dist (c.left.right.lim y) (c.left.right.lim x) / 2 + dist (c.right.lim y) (c.right.lim x)) / 2 \u2264 r * (3 / 4)"}, {"tactic": "calc _ \u2264 (r / 2 + r) / 2 := by gcongr\n  _ = _ := by field_simp; ring", "annotated_tactic": ["calc _ \u2264 (r / 2 + r) / 2 := by gcongr\n        _ = _ := by field_simp; ring", []], "state_before": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nc : CU P\ny : X\nhxl : x \u2209 c.left.left.U\nhyl : y \u2209 c.left.left.U\nr : \u211d := (3 / 4) ^ n\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 r\nhydl : dist (c.left.right.lim y) (c.left.right.lim x) \u2264 r\nhydr : dist (c.right.lim y) (c.right.lim x) \u2264 r\n\u22a2 (dist (c.left.right.lim y) (c.left.right.lim x) / 2 + dist (c.right.lim y) (c.right.lim x)) / 2 \u2264 r * (3 / 4)", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "X : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nc : CU P\ny : X\nhxl : x \u2209 c.left.left.U\nhyl : y \u2209 c.left.left.U\nr : \u211d := (3 / 4) ^ n\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 r\nhydl : dist (c.left.right.lim y) (c.left.right.lim x) \u2264 r\nhydr : dist (c.right.lim y) (c.right.lim x) \u2264 r\n\u22a2 (dist (c.left.right.lim y) (c.left.right.lim x) / 2 + dist (c.right.lim y) (c.right.lim x)) / 2 \u2264 (r / 2 + r) / 2", "state_after": "no goals"}, {"tactic": "field_simp", "annotated_tactic": ["field_simp", []], "state_before": "X : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nc : CU P\ny : X\nhxl : x \u2209 c.left.left.U\nhyl : y \u2209 c.left.left.U\nr : \u211d := (3 / 4) ^ n\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 r\nhydl : dist (c.left.right.lim y) (c.left.right.lim x) \u2264 r\nhydr : dist (c.right.lim y) (c.right.lim x) \u2264 r\n\u22a2 (r / 2 + r) / 2 = r * (3 / 4)", "state_after": "X : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nc : CU P\ny : X\nhxl : x \u2209 c.left.left.U\nhyl : y \u2209 c.left.left.U\nr : \u211d := (3 / 4) ^ n\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 r\nhydl : dist (c.left.right.lim y) (c.left.right.lim x) \u2264 r\nhydr : dist (c.right.lim y) (c.right.lim x) \u2264 r\n\u22a2 (r + r * 2) * 4 = r * 3 * (2 * 2)"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "X : Type u_1\ninst\u271d : TopologicalSpace X\nP : Set X \u2192 Prop\nh0 : 0 < 2\u207b\u00b9\nh1234 : 2\u207b\u00b9 < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx\u271d : True\nn : \u2115\nc : CU P\ny : X\nhxl : x \u2209 c.left.left.U\nhyl : y \u2209 c.left.left.U\nr : \u211d := (3 / 4) ^ n\nihn : \u2200 (c : CU P), \u2200\u1da0 (x_1 : X) in \ud835\udcdd x, dist (c.lim x_1) (c.lim x) \u2264 r\nhydl : dist (c.left.right.lim y) (c.left.right.lim x) \u2264 r\nhydr : dist (c.right.lim y) (c.right.lim x) \u2264 r\n\u22a2 (r + r * 2) * 4 = r * 3 * (2 * 2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Module/Multilinear/Bounded.lean", "full_name": "Bornology.IsVonNBounded.image_multilinear", "start": [90, 1], "end": [96, 43], "traced_tactics": [{"tactic": "cases isEmpty_or_nonempty \u03b9 with\n| inl h =>\n  exact (isBounded_iff_isVonNBounded _).1 <|\n    @Set.Finite.isBounded _ (vonNBornology \ud835\udd5c F) _ (s.toFinite.image _)\n| inr h => exact hs.image_multilinear' f", "annotated_tactic": ["cases <a>isEmpty_or_nonempty</a> \u03b9 with\n  | <a>inl</a> h =>\n    exact (<a>isBounded_iff_isVonNBounded</a> _).1 <|\n      @<a>Set.Finite.isBounded</a> _ (<a>vonNBornology</a> \ud835\udd5c F) _ (s.toFinite.image _)\n  | <a>inr</a> h => exact hs.image_multilinear' f", [{"full_name": "isEmpty_or_nonempty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [221, 9], "def_end_pos": [221, 28]}, {"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "Bornology.isBounded_iff_isVonNBounded", "def_path": "Mathlib/Analysis/LocallyConvex/Bounded.lean", "def_pos": [351, 9], "def_end_pos": [351, 36]}, {"full_name": "Set.Finite.isBounded", "def_path": "Mathlib/Topology/Bornology/Basic.lean", "def_pos": [316, 9], "def_end_pos": [316, 29]}, {"full_name": "Bornology.vonNBornology", "def_path": "Mathlib/Analysis/LocallyConvex/Bounded.lean", "def_pos": [343, 8], "def_end_pos": [343, 21]}, {"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nF : Type u_3\nE : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommGroup (E i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : ContinuousSMul \ud835\udd5c F\ns : Set ((i : \u03b9) \u2192 E i)\nhs : IsVonNBounded \ud835\udd5c s\nf : ContinuousMultilinearMap \ud835\udd5c E F\n\u22a2 IsVonNBounded \ud835\udd5c (\u21d1f '' s)", "state_after": "no goals"}, {"tactic": "exact (isBounded_iff_isVonNBounded _).1 <|\n  @Set.Finite.isBounded _ (vonNBornology \ud835\udd5c F) _ (s.toFinite.image _)", "annotated_tactic": ["exact (<a>isBounded_iff_isVonNBounded</a> _).1 <|\n      @<a>Set.Finite.isBounded</a> _ (<a>vonNBornology</a> \ud835\udd5c F) _ (s.toFinite.image _)", [{"full_name": "Bornology.isBounded_iff_isVonNBounded", "def_path": "Mathlib/Analysis/LocallyConvex/Bounded.lean", "def_pos": [351, 9], "def_end_pos": [351, 36]}, {"full_name": "Set.Finite.isBounded", "def_path": "Mathlib/Topology/Bornology/Basic.lean", "def_pos": [316, 9], "def_end_pos": [316, 29]}, {"full_name": "Bornology.vonNBornology", "def_path": "Mathlib/Analysis/LocallyConvex/Bounded.lean", "def_pos": [343, 8], "def_end_pos": [343, 21]}]], "state_before": "case inl\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nF : Type u_3\nE : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommGroup (E i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : ContinuousSMul \ud835\udd5c F\ns : Set ((i : \u03b9) \u2192 E i)\nhs : IsVonNBounded \ud835\udd5c s\nf : ContinuousMultilinearMap \ud835\udd5c E F\nh : IsEmpty \u03b9\n\u22a2 IsVonNBounded \ud835\udd5c (\u21d1f '' s)", "state_after": "no goals"}, {"tactic": "exact hs.image_multilinear' f", "annotated_tactic": ["exact hs.image_multilinear' f", []], "state_before": "case inr\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nF : Type u_3\nE : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommGroup (E i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : TopologicalSpace F\ninst\u271d : ContinuousSMul \ud835\udd5c F\ns : Set ((i : \u03b9) \u2192 E i)\nhs : IsVonNBounded \ud835\udd5c s\nf : ContinuousMultilinearMap \ud835\udd5c E F\nh : Nonempty \u03b9\n\u22a2 IsVonNBounded \ud835\udd5c (\u21d1f '' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Type.lean", "full_name": "exists_prime_addOrderOf_dvd_card", "start": [509, 1], "end": [511, 78], "traced_tactics": [{"tactic": "convert hdvd", "annotated_tactic": ["convert hdvd", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : Fintype \u03b1\nG\u271d : Type u_2\ninst\u271d\u00b2 : Group G\u271d\nn : \u2115\nG : Type u_3\ninst\u271d\u00b9 : AddGroup G\ninst\u271d : Fintype G\np : \u2115\nhp : Fact (Nat.Prime p)\nhdvd : p \u2223 Fintype.card G\n\u22a2 p \u2223 Fintype.card (Multiplicative G)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Int/Basic.lean", "full_name": "Int.sq_of_coprime", "start": [72, 1], "end": [74, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Deprecated/Subgroup.lean", "full_name": "IsSubgroup.iInter", "start": [113, 1], "end": [117, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean", "full_name": "MeasureTheory.ext_of_forall_lintegral_eq_of_IsFiniteMeasure", "start": [225, 1], "end": [234, 75], "traced_tactics": [{"tactic": "have key := @measure_isClosed_eq_of_forall_lintegral_eq_of_isFiniteMeasure \u03a9 _ _ _ _ \u03bc \u03bd _ h", "annotated_tactic": ["have key := @<a>measure_isClosed_eq_of_forall_lintegral_eq_of_isFiniteMeasure</a> \u03a9 _ _ _ _ \u03bc \u03bd _ h", [{"full_name": "MeasureTheory.measure_isClosed_eq_of_forall_lintegral_eq_of_isFiniteMeasure", "def_path": "Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean", "def_pos": [207, 9], "def_end_pos": [207, 70]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : HasOuterApproxClosed \u03a9\ninst\u271d\u00b9 : BorelSpace \u03a9\n\u03bc \u03bd : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u03bc = \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u03bd\n\u22a2 \u03bc = \u03bd", "state_after": "\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : HasOuterApproxClosed \u03a9\ninst\u271d\u00b9 : BorelSpace \u03a9\n\u03bc \u03bd : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u03bc = \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u03bd\nkey : \u2200 {F : Set \u03a9}, IsClosed F \u2192 \u03bc F = \u03bd F\n\u22a2 \u03bc = \u03bd"}, {"tactic": "apply ext_of_generate_finite _ ?_ isPiSystem_isClosed", "annotated_tactic": ["apply <a>ext_of_generate_finite</a> _ ?_ <a>isPiSystem_isClosed</a>", [{"full_name": "MeasureTheory.ext_of_generate_finite", "def_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "def_pos": [1372, 9], "def_end_pos": [1372, 31]}, {"full_name": "isPiSystem_isClosed", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [99, 7], "def_end_pos": [99, 26]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : HasOuterApproxClosed \u03a9\ninst\u271d\u00b9 : BorelSpace \u03a9\n\u03bc \u03bd : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u03bc = \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u03bd\nkey : \u2200 {F : Set \u03a9}, IsClosed F \u2192 \u03bc F = \u03bd F\n\u22a2 \u03bc = \u03bd", "state_after": "case h\u03bc\u03bd\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : HasOuterApproxClosed \u03a9\ninst\u271d\u00b9 : BorelSpace \u03a9\n\u03bc \u03bd : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u03bc = \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u03bd\nkey : \u2200 {F : Set \u03a9}, IsClosed F \u2192 \u03bc F = \u03bd F\n\u22a2 \u2200 s \u2208 {s | IsClosed s}, \u03bc s = \u03bd s\n\ncase h_univ\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : HasOuterApproxClosed \u03a9\ninst\u271d\u00b9 : BorelSpace \u03a9\n\u03bc \u03bd : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u03bc = \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u03bd\nkey : \u2200 {F : Set \u03a9}, IsClosed F \u2192 \u03bc F = \u03bd F\n\u22a2 \u03bc univ = \u03bd univ\n\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : HasOuterApproxClosed \u03a9\ninst\u271d\u00b9 : BorelSpace \u03a9\n\u03bc \u03bd : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u03bc = \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u03bd\nkey : \u2200 {F : Set \u03a9}, IsClosed F \u2192 \u03bc F = \u03bd F\n\u22a2 inst\u271d\u2074 = MeasurableSpace.generateFrom {s | IsClosed s}"}, {"tactic": "exact fun F F_closed \u21a6 key F_closed", "annotated_tactic": ["exact fun F F_closed \u21a6 key F_closed", []], "state_before": "case h\u03bc\u03bd\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : HasOuterApproxClosed \u03a9\ninst\u271d\u00b9 : BorelSpace \u03a9\n\u03bc \u03bd : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u03bc = \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u03bd\nkey : \u2200 {F : Set \u03a9}, IsClosed F \u2192 \u03bc F = \u03bd F\n\u22a2 \u2200 s \u2208 {s | IsClosed s}, \u03bc s = \u03bd s", "state_after": "no goals"}, {"tactic": "exact key isClosed_univ", "annotated_tactic": ["exact key <a>isClosed_univ</a>", [{"full_name": "isClosed_univ", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [176, 17], "def_end_pos": [176, 30]}]], "state_before": "case h_univ\n\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : HasOuterApproxClosed \u03a9\ninst\u271d\u00b9 : BorelSpace \u03a9\n\u03bc \u03bd : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u03bc = \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u03bd\nkey : \u2200 {F : Set \u03a9}, IsClosed F \u2192 \u03bc F = \u03bd F\n\u22a2 \u03bc univ = \u03bd univ", "state_after": "no goals"}, {"tactic": "rw [BorelSpace.measurable_eq (\u03b1 := \u03a9), borel_eq_generateFrom_isClosed]", "annotated_tactic": ["rw [<a>BorelSpace.measurable_eq</a> (\u03b1 := \u03a9), <a>borel_eq_generateFrom_isClosed</a>]", [{"full_name": "BorelSpace.measurable_eq", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [135, 3], "def_end_pos": [135, 16]}, {"full_name": "borel_eq_generateFrom_isClosed", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [102, 9], "def_end_pos": [102, 39]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2074 : MeasurableSpace \u03a9\ninst\u271d\u00b3 : TopologicalSpace \u03a9\ninst\u271d\u00b2 : HasOuterApproxClosed \u03a9\ninst\u271d\u00b9 : BorelSpace \u03a9\n\u03bc \u03bd : Measure \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\nh : \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u03bc = \u222b\u207b (x : \u03a9), \u2191(f x) \u2202\u03bd\nkey : \u2200 {F : Set \u03a9}, IsClosed F \u2192 \u03bc F = \u03bd F\n\u22a2 inst\u271d\u2074 = MeasurableSpace.generateFrom {s | IsClosed s}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/PartialHomeomorph.lean", "full_name": "PartialHomeomorph.transHomeomorph_eq_trans", "start": [1443, 1], "end": [1445, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "full_name": "Embedding.isSeparable_preimage", "start": [1493, 11], "end": [1495, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/JacobsonIdeal.lean", "full_name": "Ideal.eq_jacobson_iff_not_mem", "start": [165, 1], "end": [176, 17], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\n\u22a2 I.jacobson = I \u2194 \u2200 x \u2209 I, \u2203 M, (I \u2264 M \u2227 M.IsMaximal) \u2227 x \u2209 M", "state_after": "case mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\n\u22a2 I.jacobson = I \u2192 \u2200 x \u2209 I, \u2203 M, (I \u2264 M \u2227 M.IsMaximal) \u2227 x \u2209 M\n\ncase mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\n\u22a2 (\u2200 x \u2209 I, \u2203 M, (I \u2264 M \u2227 M.IsMaximal) \u2227 x \u2209 M) \u2192 I.jacobson = I"}, {"tactic": "intro h x hx", "annotated_tactic": ["intro h x hx", []], "state_before": "case mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\n\u22a2 I.jacobson = I \u2192 \u2200 x \u2209 I, \u2203 M, (I \u2264 M \u2227 M.IsMaximal) \u2227 x \u2209 M", "state_after": "case mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nh : I.jacobson = I\nx : R\nhx : x \u2209 I\n\u22a2 \u2203 M, (I \u2264 M \u2227 M.IsMaximal) \u2227 x \u2209 M"}, {"tactic": "erw [\u2190 h, mem_sInf] at hx", "annotated_tactic": ["erw [\u2190 h, <a>mem_sInf</a>] at hx", [{"full_name": "Ideal.mem_sInf", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [429, 9], "def_end_pos": [429, 17]}]], "state_before": "case mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nh : I.jacobson = I\nx : R\nhx : x \u2209 I\n\u22a2 \u2203 M, (I \u2264 M \u2227 M.IsMaximal) \u2227 x \u2209 M", "state_after": "case mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nh : I.jacobson = I\nx : R\nhx : \u00ac\u2200 \u2983I_1 : Ideal R\u2984, I_1 \u2208 {J | I \u2264 J \u2227 J.IsMaximal} \u2192 x \u2208 I_1\n\u22a2 \u2203 M, (I \u2264 M \u2227 M.IsMaximal) \u2227 x \u2209 M"}, {"tactic": "push_neg at hx", "annotated_tactic": ["push_neg at hx", []], "state_before": "case mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nh : I.jacobson = I\nx : R\nhx : \u00ac\u2200 \u2983I_1 : Ideal R\u2984, I_1 \u2208 {J | I \u2264 J \u2227 J.IsMaximal} \u2192 x \u2208 I_1\n\u22a2 \u2203 M, (I \u2264 M \u2227 M.IsMaximal) \u2227 x \u2209 M", "state_after": "case mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nh : I.jacobson = I\nx : R\nhx : \u2203 I_1 \u2208 {J | I \u2264 J \u2227 J.IsMaximal}, x \u2209 I_1\n\u22a2 \u2203 M, (I \u2264 M \u2227 M.IsMaximal) \u2227 x \u2209 M"}, {"tactic": "exact hx", "annotated_tactic": ["exact hx", []], "state_before": "case mp\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nh : I.jacobson = I\nx : R\nhx : \u2203 I_1 \u2208 {J | I \u2264 J \u2227 J.IsMaximal}, x \u2209 I_1\n\u22a2 \u2203 M, (I \u2264 M \u2227 M.IsMaximal) \u2227 x \u2209 M", "state_after": "no goals"}, {"tactic": "refine fun h => le_antisymm (fun x hx => ?_) le_jacobson", "annotated_tactic": ["refine fun h => <a>le_antisymm</a> (fun x hx => ?_) <a>le_jacobson</a>", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "Ideal.le_jacobson", "def_path": "Mathlib/RingTheory/JacobsonIdeal.lean", "def_pos": [65, 9], "def_end_pos": [65, 20]}]], "state_before": "case mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\n\u22a2 (\u2200 x \u2209 I, \u2203 M, (I \u2264 M \u2227 M.IsMaximal) \u2227 x \u2209 M) \u2192 I.jacobson = I", "state_after": "case mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nh : \u2200 x \u2209 I, \u2203 M, (I \u2264 M \u2227 M.IsMaximal) \u2227 x \u2209 M\nx : R\nhx : x \u2208 I.jacobson\n\u22a2 x \u2208 I"}, {"tactic": "contrapose hx", "annotated_tactic": ["contrapose hx", []], "state_before": "case mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nh : \u2200 x \u2209 I, \u2203 M, (I \u2264 M \u2227 M.IsMaximal) \u2227 x \u2209 M\nx : R\nhx : x \u2208 I.jacobson\n\u22a2 x \u2208 I", "state_after": "case mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nh : \u2200 x \u2209 I, \u2203 M, (I \u2264 M \u2227 M.IsMaximal) \u2227 x \u2209 M\nx : R\nhx : x \u2209 I\n\u22a2 x \u2209 I.jacobson"}, {"tactic": "erw [mem_sInf]", "annotated_tactic": ["erw [<a>mem_sInf</a>]", [{"full_name": "Ideal.mem_sInf", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [429, 9], "def_end_pos": [429, 17]}]], "state_before": "case mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nh : \u2200 x \u2209 I, \u2203 M, (I \u2264 M \u2227 M.IsMaximal) \u2227 x \u2209 M\nx : R\nhx : x \u2209 I\n\u22a2 x \u2209 I.jacobson", "state_after": "case mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nh : \u2200 x \u2209 I, \u2203 M, (I \u2264 M \u2227 M.IsMaximal) \u2227 x \u2209 M\nx : R\nhx : x \u2209 I\n\u22a2 \u00ac\u2200 \u2983I_1 : Ideal R\u2984, I_1 \u2208 {J | I \u2264 J \u2227 J.IsMaximal} \u2192 x \u2208 I_1"}, {"tactic": "push_neg", "annotated_tactic": ["push_neg", []], "state_before": "case mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nh : \u2200 x \u2209 I, \u2203 M, (I \u2264 M \u2227 M.IsMaximal) \u2227 x \u2209 M\nx : R\nhx : x \u2209 I\n\u22a2 \u00ac\u2200 \u2983I_1 : Ideal R\u2984, I_1 \u2208 {J | I \u2264 J \u2227 J.IsMaximal} \u2192 x \u2208 I_1", "state_after": "case mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nh : \u2200 x \u2209 I, \u2203 M, (I \u2264 M \u2227 M.IsMaximal) \u2227 x \u2209 M\nx : R\nhx : x \u2209 I\n\u22a2 \u2203 I_1 \u2208 {J | I \u2264 J \u2227 J.IsMaximal}, x \u2209 I_1"}, {"tactic": "exact h x hx", "annotated_tactic": ["exact h x hx", []], "state_before": "case mpr\nR : Type u\nS : Type v\ninst\u271d\u00b9 : Ring R\ninst\u271d : Ring S\nI : Ideal R\nh : \u2200 x \u2209 I, \u2203 M, (I \u2264 M \u2227 M.IsMaximal) \u2227 x \u2209 M\nx : R\nhx : x \u2209 I\n\u22a2 \u2203 I_1 \u2208 {J | I \u2264 J \u2227 J.IsMaximal}, x \u2209 I_1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Ideal/Basic.lean", "full_name": "Ideal.isCompactElement_top", "start": [168, 1], "end": [169, 86], "traced_tactics": [{"tactic": "simpa only [\u2190 span_singleton_one] using Submodule.singleton_span_isCompactElement 1", "annotated_tactic": ["simpa only [\u2190 <a>span_singleton_one</a>] using <a>Submodule.singleton_span_isCompactElement</a> 1", [{"full_name": "Ideal.span_singleton_one", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [164, 9], "def_end_pos": [164, 27]}, {"full_name": "Submodule.singleton_span_isCompactElement", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [732, 9], "def_end_pos": [732, 40]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d : Semiring \u03b1\nI : Ideal \u03b1\na b : \u03b1\n\u22a2 CompleteLattice.IsCompactElement \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/AlexandrovDiscrete.lean", "full_name": "gc_exterior_interior", "start": [182, 1], "end": [183, 63], "traced_tactics": [{"tactic": "simp [exterior_subset_iff, subset_interior_iff]", "annotated_tactic": ["simp [<a>exterior_subset_iff</a>, <a>subset_interior_iff</a>]", [{"full_name": "exterior_subset_iff", "def_path": "Mathlib/Topology/AlexandrovDiscrete.lean", "def_pos": [169, 7], "def_end_pos": [169, 26]}, {"full_name": "subset_interior_iff", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [281, 9], "def_end_pos": [281, 28]}]], "state_before": "\u03b9 : Sort u_1\n\u03ba : \u03b9 \u2192 Sort u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ns\u271d t\u271d : Set \u03b1\na x y : \u03b1\ninst\u271d\u00b9 : AlexandrovDiscrete \u03b1\ninst\u271d : AlexandrovDiscrete \u03b2\ns t : Set \u03b1\n\u22a2 exterior s \u2264 t \u2194 s \u2264 interior t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "Continuous.congr", "start": [1555, 1], "end": [1556, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Fourier/FourierTransformDeriv.lean", "full_name": "Real.hasDerivAt_fourierChar", "start": [93, 1], "end": [98, 93], "traced_tactics": [{"tactic": "have h1 (y : \u211d) : \ud835\udc1e y = fourier 1 (y : UnitAddCircle) := by\n  rw [fourierChar_apply, fourier_coe_apply]\n  push_cast\n  ring_nf", "annotated_tactic": ["have h1 (y : \u211d) : \ud835\udc1e y = <a>fourier</a> 1 (y : <a>UnitAddCircle</a>) := by\n    rw [<a>fourierChar_apply</a>, <a>fourier_coe_apply</a>]\n    push_cast\n    ring_nf", [{"full_name": "fourier", "def_path": "Mathlib/Analysis/Fourier/AddCircle.lean", "def_pos": [106, 5], "def_end_pos": [106, 12]}, {"full_name": "UnitAddCircle", "def_path": "Mathlib/Topology/Instances/AddCircle.lean", "def_pos": [562, 8], "def_end_pos": [562, 21]}, {"full_name": "Real.fourierChar_apply", "def_path": "Mathlib/Analysis/Fourier/FourierTransform.lean", "def_pos": [337, 9], "def_end_pos": [337, 26]}, {"full_name": "fourier_coe_apply", "def_path": "Mathlib/Analysis/Fourier/AddCircle.lean", "def_pos": [117, 9], "def_end_pos": [117, 26]}]], "state_before": "x : \u211d\n\u22a2 HasDerivAt (fun x => \u2191(\ud835\udc1e x)) (2 * \u2191\u03c0 * I * \u2191(\ud835\udc1e x)) x", "state_after": "x : \u211d\nh1 : \u2200 (y : \u211d), \u2191(\ud835\udc1e y) = (fourier 1) \u2191y\n\u22a2 HasDerivAt (fun x => \u2191(\ud835\udc1e x)) (2 * \u2191\u03c0 * I * \u2191(\ud835\udc1e x)) x"}, {"tactic": "simpa only [h1, Int.cast_one, ofReal_one, div_one, mul_one] using hasDerivAt_fourier 1 1 x", "annotated_tactic": ["simpa only [h1, <a>Int.cast_one</a>, <a>ofReal_one</a>, <a>div_one</a>, <a>mul_one</a>] using <a>hasDerivAt_fourier</a> 1 1 x", [{"full_name": "Int.cast_one", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [79, 9], "def_end_pos": [79, 17]}, {"full_name": "Complex.ofReal_one", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [180, 9], "def_end_pos": [180, 19]}, {"full_name": "div_one", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [490, 9], "def_end_pos": [490, 16]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "hasDerivAt_fourier", "def_path": "Mathlib/Analysis/Fourier/AddCircle.lean", "def_pos": [487, 9], "def_end_pos": [487, 27]}]], "state_before": "x : \u211d\nh1 : \u2200 (y : \u211d), \u2191(\ud835\udc1e y) = (fourier 1) \u2191y\n\u22a2 HasDerivAt (fun x => \u2191(\ud835\udc1e x)) (2 * \u2191\u03c0 * I * \u2191(\ud835\udc1e x)) x", "state_after": "no goals"}, {"tactic": "rw [fourierChar_apply, fourier_coe_apply]", "annotated_tactic": ["rw [<a>fourierChar_apply</a>, <a>fourier_coe_apply</a>]", [{"full_name": "Real.fourierChar_apply", "def_path": "Mathlib/Analysis/Fourier/FourierTransform.lean", "def_pos": [337, 9], "def_end_pos": [337, 26]}, {"full_name": "fourier_coe_apply", "def_path": "Mathlib/Analysis/Fourier/AddCircle.lean", "def_pos": [117, 9], "def_end_pos": [117, 26]}]], "state_before": "x y : \u211d\n\u22a2 \u2191(\ud835\udc1e y) = (fourier 1) \u2191y", "state_after": "x y : \u211d\n\u22a2 cexp (\u2191(2 * \u03c0 * y) * I) = cexp (2 * \u2191\u03c0 * I * \u21911 * \u2191y / \u21911)"}, {"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "x y : \u211d\n\u22a2 cexp (\u2191(2 * \u03c0 * y) * I) = cexp (2 * \u2191\u03c0 * I * \u21911 * \u2191y / \u21911)", "state_after": "x y : \u211d\n\u22a2 cexp (2 * \u2191\u03c0 * \u2191y * I) = cexp (2 * \u2191\u03c0 * I * 1 * \u2191y / 1)"}, {"tactic": "ring_nf", "annotated_tactic": ["ring_nf", []], "state_before": "x y : \u211d\n\u22a2 cexp (2 * \u2191\u03c0 * \u2191y * I) = cexp (2 * \u2191\u03c0 * I * 1 * \u2191y / 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Disjoint.lean", "full_name": "Disjoint.left_le_of_le_sup_left", "start": [215, 1], "end": [216, 50], "traced_tactics": [{"tactic": "rwa [sup_comm]", "annotated_tactic": ["rwa [<a>sup_comm</a>]", [{"full_name": "sup_comm", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [224, 9], "def_end_pos": [224, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DistribLattice \u03b1\ninst\u271d : OrderBot \u03b1\na b c : \u03b1\nh : a \u2264 c \u2294 b\nhd : Disjoint a c\n\u22a2 a \u2264 b \u2294 c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/MonoidAlgebra/Basic.lean", "full_name": "MonoidAlgebra.single_commute", "start": [471, 1], "end": [475, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/ContDiff/FiniteDimension.lean", "full_name": "contDiff_clm_apply_iff", "start": [46, 1], "end": [48, 52], "traced_tactics": [{"tactic": "simp_rw [\u2190 contDiffOn_univ, contDiffOn_clm_apply]", "annotated_tactic": ["simp_rw [\u2190 <a>contDiffOn_univ</a>, <a>contDiffOn_clm_apply</a>]", [{"full_name": "contDiffOn_univ", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "def_pos": [1455, 9], "def_end_pos": [1455, 24]}, {"full_name": "contDiffOn_clm_apply", "def_path": "Mathlib/Analysis/Calculus/ContDiff/FiniteDimension.lean", "def_pos": [35, 9], "def_end_pos": [35, 29]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\ninst\u271d\u00b9 : CompleteSpace \ud835\udd5c\nn : \u2115\u221e\nf : E \u2192 F \u2192L[\ud835\udd5c] G\ninst\u271d : FiniteDimensional \ud835\udd5c F\n\u22a2 ContDiff \ud835\udd5c n f \u2194 \u2200 (y : F), ContDiff \ud835\udd5c n fun x => (f x) y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/FiberBundle/Basic.lean", "full_name": "FiberBundle.mem_trivializationAt_proj_source", "start": [281, 1], "end": [283, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finmap.lean", "full_name": "Finmap.mem_list_toFinmap", "start": [524, 1], "end": [534, 36], "traced_tactics": [{"tactic": "induction' xs with x xs", "annotated_tactic": ["induction' xs with x xs", []], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nxs : List (Sigma \u03b2)\n\u22a2 a \u2208 xs.toFinmap \u2194 \u2203 b, \u27e8a, b\u27e9 \u2208 xs", "state_after": "case nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\n\u22a2 a \u2208 [].toFinmap \u2194 \u2203 b, \u27e8a, b\u27e9 \u2208 []\n\ncase cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nx : Sigma \u03b2\nxs : List (Sigma \u03b2)\ntail_ih\u271d : a \u2208 xs.toFinmap \u2194 \u2203 b, \u27e8a, b\u27e9 \u2208 xs\n\u22a2 a \u2208 (x :: xs).toFinmap \u2194 \u2203 b, \u27e8a, b\u27e9 \u2208 x :: xs"}, {"tactic": "cases' x with fst_i snd_i", "annotated_tactic": ["cases' x with fst_i snd_i", []], "state_before": "case cons\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nx : Sigma \u03b2\nxs : List (Sigma \u03b2)\ntail_ih\u271d : a \u2208 xs.toFinmap \u2194 \u2203 b, \u27e8a, b\u27e9 \u2208 xs\n\u22a2 a \u2208 (x :: xs).toFinmap \u2194 \u2203 b, \u27e8a, b\u27e9 \u2208 x :: xs", "state_after": "case cons.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nxs : List (Sigma \u03b2)\ntail_ih\u271d : a \u2208 xs.toFinmap \u2194 \u2203 b, \u27e8a, b\u27e9 \u2208 xs\nfst_i : \u03b1\nsnd_i : \u03b2 fst_i\n\u22a2 a \u2208 (\u27e8fst_i, snd_i\u27e9 :: xs).toFinmap \u2194 \u2203 b, \u27e8a, b\u27e9 \u2208 \u27e8fst_i, snd_i\u27e9 :: xs"}, {"tactic": "simp only [toFinmap_cons, *, exists_or, mem_cons, mem_insert, exists_and_left, Sigma.mk.inj_iff]", "annotated_tactic": ["simp only [<a>toFinmap_cons</a>, *, <a>exists_or</a>, <a>mem_cons</a>, <a>mem_insert</a>, <a>exists_and_left</a>, <a>Sigma.mk.inj_iff</a>]", [{"full_name": "Finmap.toFinmap_cons", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [519, 9], "def_end_pos": [519, 22]}, {"full_name": "exists_or", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [259, 9], "def_end_pos": [259, 18]}, {"full_name": "List.mem_cons", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [285, 17], "def_end_pos": [285, 25]}, {"full_name": "Finmap.mem_insert", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [492, 9], "def_end_pos": [492, 19]}, {"full_name": "exists_and_left", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [288, 17], "def_end_pos": [288, 32]}, {"full_name": 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?_).symm", "annotated_tactic": ["refine (<a>or_congr_left</a> <| <a>and_iff_left_of_imp</a> ?_).<a>symm</a>", [{"full_name": "or_congr_left", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [78, 9], "def_end_pos": [78, 22]}, {"full_name": "and_iff_left_of_imp", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [161, 9], "def_end_pos": [161, 28]}, {"full_name": "Iff.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [813, 9], "def_end_pos": [813, 17]}]], "state_before": "case cons.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nxs : List (Sigma \u03b2)\ntail_ih\u271d : a \u2208 xs.toFinmap \u2194 \u2203 b, \u27e8a, b\u27e9 \u2208 xs\nfst_i : \u03b1\nsnd_i : \u03b2 fst_i\n\u22a2 (a = fst_i \u2228 \u2203 b, \u27e8a, b\u27e9 \u2208 xs) \u2194 (a = fst_i \u2227 \u2203 x, HEq x snd_i) \u2228 \u2203 x, \u27e8a, x\u27e9 \u2208 xs", "state_after": "case cons.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nxs : List (Sigma \u03b2)\ntail_ih\u271d : a \u2208 xs.toFinmap \u2194 \u2203 b, \u27e8a, b\u27e9 \u2208 xs\nfst_i : \u03b1\nsnd_i : \u03b2 fst_i\n\u22a2 a = fst_i \u2192 \u2203 x, HEq x snd_i"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "case cons.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nxs : List (Sigma \u03b2)\ntail_ih\u271d : a \u2208 xs.toFinmap \u2194 \u2203 b, \u27e8a, b\u27e9 \u2208 xs\nfst_i : \u03b1\nsnd_i : \u03b2 fst_i\n\u22a2 a = fst_i \u2192 \u2203 x, HEq x snd_i", "state_after": "case cons.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nxs : List (Sigma \u03b2)\ntail_ih\u271d : a \u2208 xs.toFinmap \u2194 \u2203 b, \u27e8a, b\u27e9 \u2208 xs\nsnd_i : \u03b2 a\n\u22a2 \u2203 x, HEq x snd_i"}, {"tactic": "simp only [exists_eq, heq_iff_eq]", "annotated_tactic": ["simp only [<a>exists_eq</a>, <a>heq_iff_eq</a>]", [{"full_name": "exists_eq", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [278, 17], "def_end_pos": [278, 26]}, {"full_name": "heq_iff_eq", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [37, 9], "def_end_pos": [37, 19]}]], "state_before": "case cons.mk\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nxs : List (Sigma \u03b2)\ntail_ih\u271d : a \u2208 xs.toFinmap \u2194 \u2203 b, \u27e8a, b\u27e9 \u2208 xs\nsnd_i : \u03b2 a\n\u22a2 \u2203 x, HEq x snd_i", "state_after": "no goals"}, {"tactic": "simp only [toFinmap_nil, not_mem_empty, find?, not_mem_nil, exists_false]", "annotated_tactic": ["simp only [<a>toFinmap_nil</a>, <a>not_mem_empty</a>, <a>find?</a>, <a>not_mem_nil</a>, <a>exists_false</a>]", [{"full_name": "Finmap.toFinmap_nil", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [221, 9], "def_end_pos": [221, 21]}, {"full_name": "Finmap.not_mem_empty", "def_path": "Mathlib/Data/Finmap.lean", "def_pos": [225, 9], "def_end_pos": [225, 22]}, {"full_name": "List.find?", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [980, 5], "def_end_pos": [980, 10]}, {"full_name": "List.not_mem_nil", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [283, 17], "def_end_pos": [283, 28]}, {"full_name": "exists_false", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [263, 17], "def_end_pos": [263, 29]}]], "state_before": "case nil\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\na : \u03b1\n\u22a2 a \u2208 [].toFinmap \u2194 \u2203 b, \u27e8a, b\u27e9 \u2208 []", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Equiv/PartialEquiv.lean", "full_name": "PartialEquiv.transEquiv_transEquiv", "start": [1151, 1], "end": [1153, 75], "traced_tactics": [{"tactic": "simp only [transEquiv_eq_trans, trans_assoc, Equiv.trans_toPartialEquiv]", "annotated_tactic": ["simp only [<a>transEquiv_eq_trans</a>, <a>trans_assoc</a>, <a>Equiv.trans_toPartialEquiv</a>]", [{"full_name": "PartialEquiv.transEquiv_eq_trans", "def_path": "Mathlib/Logic/Equiv/PartialEquiv.lean", "def_pos": [1145, 9], "def_end_pos": [1145, 28]}, {"full_name": "PartialEquiv.trans_assoc", "def_path": "Mathlib/Logic/Equiv/PartialEquiv.lean", "def_pos": [749, 9], "def_end_pos": [749, 20]}, {"full_name": "Equiv.trans_toPartialEquiv", "def_path": "Mathlib/Logic/Equiv/PartialEquiv.lean", "def_pos": [1096, 9], "def_end_pos": [1096, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ne : PartialEquiv \u03b1 \u03b2\nf' : \u03b2 \u2243 \u03b3\nf'' : \u03b3 \u2243 \u03b4\n\u22a2 (e.transEquiv f').transEquiv f'' = e.transEquiv (f'.trans f'')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean", "full_name": "groupCohomology.H1LequivOfIsTrivial_comp_H1_\u03c0", "start": [675, 1], "end": [677, 11], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "k G : Type u\ninst\u271d\u00b2 : CommRing k\ninst\u271d\u00b9 : Group G\nA : Rep k G\ninst\u271d : A.IsTrivial\n\u22a2 \u2191(H1LequivOfIsTrivial A) \u2218\u2097 H1_\u03c0 A = \u2191(oneCocyclesLequivOfIsTrivial A)", "state_after": "case h.h\nk G : Type u\ninst\u271d\u00b2 : CommRing k\ninst\u271d\u00b9 : Group G\nA : Rep k G\ninst\u271d : A.IsTrivial\nx\u271d\u00b9 : \u21a5(oneCocycles A)\nx\u271d : Additive G\n\u22a2 ((\u2191(H1LequivOfIsTrivial A) \u2218\u2097 H1_\u03c0 A) x\u271d\u00b9) x\u271d = (\u2191(oneCocyclesLequivOfIsTrivial A) x\u271d\u00b9) x\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.h\nk G : Type u\ninst\u271d\u00b2 : CommRing k\ninst\u271d\u00b9 : Group G\nA : Rep k G\ninst\u271d : A.IsTrivial\nx\u271d\u00b9 : \u21a5(oneCocycles A)\nx\u271d : Additive G\n\u22a2 ((\u2191(H1LequivOfIsTrivial A) \u2218\u2097 H1_\u03c0 A) x\u271d\u00b9) x\u271d = (\u2191(oneCocyclesLequivOfIsTrivial A) x\u271d\u00b9) x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/FiberBundle/Basic.lean", "full_name": "FiberBundleCore.mem_localTriv_target", "start": [691, 1], "end": [693, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean", "full_name": "NonUnitalAlgebra.mem_iInf", "start": [763, 1], "end": [764, 89], "traced_tactics": [{"tactic": "simp only [iInf, mem_sInf, Set.forall_mem_range]", "annotated_tactic": ["simp only [<a>iInf</a>, <a>mem_sInf</a>, <a>Set.forall_mem_range</a>]", [{"full_name": "iInf", "def_path": "Mathlib/Order/SetNotation.lean", "def_pos": [65, 5], "def_end_pos": [65, 9]}, {"full_name": "NonUnitalAlgebra.mem_sInf", "def_path": "Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean", "def_pos": [746, 9], "def_end_pos": [746, 17]}, {"full_name": "Set.forall_mem_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [662, 9], "def_end_pos": [662, 25]}]], "state_before": "F : Type u_1\nR : Type u\nA : Type v\nB : Type w\ninst\u271d\u00b9\u2070 : CommSemiring R\ninst\u271d\u2079 : NonUnitalNonAssocSemiring A\ninst\u271d\u2078 : Module R A\ninst\u271d\u2077 : IsScalarTower R A A\ninst\u271d\u2076 : SMulCommClass R A A\ninst\u271d\u2075 : NonUnitalNonAssocSemiring B\ninst\u271d\u2074 : Module R B\ninst\u271d\u00b3 : IsScalarTower R B B\ninst\u271d\u00b2 : SMulCommClass R B B\ninst\u271d\u00b9 : FunLike F A B\ninst\u271d : NonUnitalAlgHomClass F R A B\n\u03b9 : Sort u_2\nS : \u03b9 \u2192 NonUnitalSubalgebra R A\nx : A\n\u22a2 x \u2208 \u2a05 i, S i \u2194 \u2200 (i : \u03b9), x \u2208 S i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/OrderClosed.lean", "full_name": "continuous_min", "start": [856, 1], "end": [857, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Sets/Compacts.lean", "full_name": "TopologicalSpace.PositiveCompacts.map_id", "start": [405, 1], "end": [406, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Finset/Fin.lean", "full_name": "Fin.map_valEmbedding_Iic", "start": [205, 1], "end": [206, 86], "traced_tactics": [{"tactic": "simp [Iic_eq_finset_subtype, Finset.fin, Finset.map_map, Iic_filter_lt_of_lt_right]", "annotated_tactic": ["simp [<a>Iic_eq_finset_subtype</a>, <a>Finset.fin</a>, <a>Finset.map_map</a>, <a>Iic_filter_lt_of_lt_right</a>]", [{"full_name": "Fin.Iic_eq_finset_subtype", "def_path": "Mathlib/Order/Interval/Finset/Fin.lean", "def_pos": [166, 9], "def_end_pos": [166, 30]}, {"full_name": "Finset.fin", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [797, 15], "def_end_pos": [797, 18]}, {"full_name": "Finset.map_map", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [147, 9], "def_end_pos": [147, 16]}, {"full_name": "Finset.Iic_filter_lt_of_lt_right", "def_path": "Mathlib/Order/Interval/Finset/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 34]}]], "state_before": "n : \u2115\na b : Fin n\n\u22a2 map valEmbedding (Iic b) = Iic \u2191b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/WithBot.lean", "full_name": "WithTop.ofDual_map", "start": [835, 1], "end": [837, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Sets/Opens.lean", "full_name": "TopologicalSpace.Opens.mk_coe", "start": [121, 9], "end": [121, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.map_le_iff_le_comap", "start": [2342, 1], "end": [2343, 92], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/DedekindDomain/Different.lean", "full_name": "Submodule.le_traceDual_mul_iff", "start": [69, 1], "end": [71, 51], "traced_tactics": [{"tactic": "simp_rw [le_traceDual_iff_map_le_one, mul_assoc]", "annotated_tactic": ["simp_rw [<a>le_traceDual_iff_map_le_one</a>, <a>mul_assoc</a>]", [{"full_name": "Submodule.le_traceDual_iff_map_le_one", "def_path": "Mathlib/RingTheory/DedekindDomain/Different.lean", "def_pos": [63, 7], "def_end_pos": [63, 34]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst\u271d\u00b9\u2077 : CommRing A\ninst\u271d\u00b9\u2076 : Field K\ninst\u271d\u00b9\u2075 : CommRing B\ninst\u271d\u00b9\u2074 : Field L\ninst\u271d\u00b9\u00b3 : Algebra A K\ninst\u271d\u00b9\u00b2 : Algebra B L\ninst\u271d\u00b9\u00b9 : Algebra A B\ninst\u271d\u00b9\u2070 : Algebra K L\ninst\u271d\u2079 : Algebra A L\ninst\u271d\u2078 : IsScalarTower A K L\ninst\u271d\u2077 : IsScalarTower A B L\ninst\u271d\u2076 : IsDomain A\ninst\u271d\u2075 : IsDomain B\ninst\u271d\u2074 : IsFractionRing A K\ninst\u271d\u00b3 : IsIntegralClosure B A L\ninst\u271d\u00b2 : IsFractionRing B L\ninst\u271d\u00b9 : FiniteDimensional K L\ninst\u271d : IsSeparable K L\nI J J' : Submodule B L\n\u22a2 I \u2264 (J * J')\u1d5b \u2194 I * J \u2264 J'\u1d5b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Multiplicity.lean", "full_name": "multiplicity.multiplicity_eq_zero", "start": [202, 1], "end": [204, 83], "traced_tactics": [{"tactic": "rw [\u2190 Nat.cast_zero, eq_coe_iff]", "annotated_tactic": ["rw [\u2190 <a>Nat.cast_zero</a>, <a>eq_coe_iff</a>]", [{"full_name": "Nat.cast_zero", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}, {"full_name": "multiplicity.eq_coe_iff", "def_path": "Mathlib/RingTheory/Multiplicity.lean", "def_pos": [156, 9], "def_end_pos": [156, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Monoid \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\na b : \u03b1\n\u22a2 multiplicity a b = 0 \u2194 \u00aca \u2223 b", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Monoid \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\na b : \u03b1\n\u22a2 a ^ 0 \u2223 b \u2227 \u00aca ^ (0 + 1) \u2223 b \u2194 \u00aca \u2223 b"}, {"tactic": "simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and]", "annotated_tactic": ["simp only [<a>_root_.pow_zero</a>, <a>isUnit_one</a>, <a>IsUnit.dvd</a>, <a>zero_add</a>, <a>pow_one</a>, <a>true_and</a>]", [{"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [651, 9], "def_end_pos": [651, 17]}, {"full_name": "isUnit_one", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [695, 9], "def_end_pos": [695, 19]}, {"full_name": "IsUnit.dvd", "def_path": "Mathlib/Algebra/Divisibility/Units.lean", "def_pos": [81, 9], "def_end_pos": [81, 12]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [476, 3], "def_end_pos": [476, 14]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [663, 7], "def_end_pos": [663, 14]}, {"full_name": "true_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [105, 17], "def_end_pos": [105, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Monoid \u03b1\ninst\u271d\u00b2 : Monoid \u03b2\ninst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d : DecidableRel fun x x_1 => x \u2223 x_1\na b : \u03b1\n\u22a2 a ^ 0 \u2223 b \u2227 \u00aca ^ (0 + 1) \u2223 b \u2194 \u00aca \u2223 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finsupp/Basic.lean", "full_name": "Finsupp.snd_sumFinsuppAddEquivProdFinsupp", "start": [1373, 1], "end": [1375, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/ZPow.lean", "full_name": "Matrix.zpow_bit0'", "start": [312, 1], "end": [313, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/LSeries/Linearity.lean", "full_name": "LSeriesSummable.neg_iff", "start": [66, 1], "end": [69, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/IsAdjoinRoot.lean", "full_name": "IsAdjoinRootMonic.coeff_apply_coe", "start": [526, 1], "end": [527, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Periodic.lean", "full_name": "Function.Periodic.image_uIcc", "start": [315, 1], "end": [321, 77], "traced_tactics": [{"tactic": "cases hc.lt_or_lt with\n| inl hc =>\n  rw [uIcc_of_ge (add_le_of_nonpos_right hc.le), \u2190 h.neg.image_Icc (neg_pos.2 hc) (a + c),\n    add_neg_cancel_right]\n| inr hc => rw [uIcc_of_le (le_add_of_nonneg_right hc.le), h.image_Icc hc]", "annotated_tactic": ["cases hc.lt_or_lt with\n  | <a>inl</a> hc =>\n    rw [<a>uIcc_of_ge</a> (<a>add_le_of_nonpos_right</a> hc.le), \u2190 h.neg.image_Icc (<a>neg_pos</a>.2 hc) (a + c),\n      <a>add_neg_cancel_right</a>]\n  | <a>inr</a> hc => rw [<a>uIcc_of_le</a> (<a>le_add_of_nonneg_right</a> hc.le), h.image_Icc hc]", [{"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "Set.uIcc_of_ge", "def_path": "Mathlib/Order/Interval/Set/UnorderedInterval.lean", "def_pos": [75, 7], "def_end_pos": [75, 17]}, {"full_name": "add_le_of_nonpos_right", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [400, 15], "def_end_pos": [400, 37]}, {"full_name": "neg_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [677, 24], "def_end_pos": [677, 31]}, {"full_name": "add_neg_cancel_right", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1261, 3], "def_end_pos": [1261, 14]}, {"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}, {"full_name": "Set.uIcc_of_le", "def_path": "Mathlib/Order/Interval/Set/UnorderedInterval.lean", "def_pos": [71, 7], "def_end_pos": [71, 17]}, {"full_name": "le_add_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [391, 15], "def_end_pos": [391, 37]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf g : \u03b1 \u2192 \u03b2\nc c\u2081 c\u2082 x : \u03b1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup \u03b1\ninst\u271d : Archimedean \u03b1\nh : Periodic f c\nhc : c \u2260 0\na : \u03b1\n\u22a2 f '' uIcc a (a + c) = range f", "state_after": "no goals"}, {"tactic": "rw [uIcc_of_ge (add_le_of_nonpos_right hc.le), \u2190 h.neg.image_Icc (neg_pos.2 hc) (a + c),\n  add_neg_cancel_right]", "annotated_tactic": ["rw [<a>uIcc_of_ge</a> (<a>add_le_of_nonpos_right</a> hc.le), \u2190 h.neg.image_Icc (<a>neg_pos</a>.2 hc) (a + c),\n      <a>add_neg_cancel_right</a>]", [{"full_name": "Set.uIcc_of_ge", "def_path": "Mathlib/Order/Interval/Set/UnorderedInterval.lean", "def_pos": [75, 7], "def_end_pos": [75, 17]}, {"full_name": "add_le_of_nonpos_right", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [400, 15], "def_end_pos": [400, 37]}, {"full_name": "neg_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [677, 24], "def_end_pos": [677, 31]}, {"full_name": "add_neg_cancel_right", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1261, 3], "def_end_pos": [1261, 14]}]], "state_before": "case inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf g : \u03b1 \u2192 \u03b2\nc c\u2081 c\u2082 x : \u03b1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup \u03b1\ninst\u271d : Archimedean \u03b1\nh : Periodic f c\nhc\u271d : c \u2260 0\na : \u03b1\nhc : c < 0\n\u22a2 f '' uIcc a (a + c) = range f", "state_after": "no goals"}, {"tactic": "rw [uIcc_of_le (le_add_of_nonneg_right hc.le), h.image_Icc hc]", "annotated_tactic": ["rw [<a>uIcc_of_le</a> (<a>le_add_of_nonneg_right</a> hc.le), h.image_Icc hc]", [{"full_name": "Set.uIcc_of_le", "def_path": "Mathlib/Order/Interval/Set/UnorderedInterval.lean", "def_pos": [71, 7], "def_end_pos": [71, 17]}, {"full_name": "le_add_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [391, 15], "def_end_pos": [391, 37]}]], "state_before": "case inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf g : \u03b1 \u2192 \u03b2\nc c\u2081 c\u2082 x : \u03b1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup \u03b1\ninst\u271d : Archimedean \u03b1\nh : Periodic f c\nhc\u271d : c \u2260 0\na : \u03b1\nhc : 0 < c\n\u22a2 f '' uIcc a (a + c) = range f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicTopology/SimplexCategory.lean", "full_name": "SimplexCategory.Hom.mk_toOrderHom", "start": [120, 1], "end": [121, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "full_name": "Matrix.toLinearMapRight'_mul_apply", "start": [165, 1], "end": [169, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Connected/PathConnected.lean", "full_name": "Filter.Tendsto.path_extend", "start": [255, 1], "end": [259, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matroid/Restrict.lean", "full_name": "Matroid.Restriction.finite", "start": [335, 1], "end": [337, 53], "traced_tactics": [{"tactic": "obtain \u27e8R, hR, rfl\u27e9 := h", "annotated_tactic": ["obtain \u27e8R, hR, rfl\u27e9 := h", []], "state_before": "\u03b1 : Type u_1\nM\u271d : Matroid \u03b1\nR I J X Y : Set \u03b1\nN M : Matroid \u03b1\ninst\u271d : M.Finite\nh : N \u2264r M\n\u22a2 N.Finite", "state_after": "case intro.intro\n\u03b1 : Type u_1\nM\u271d : Matroid \u03b1\nR\u271d I J X Y : Set \u03b1\nM : Matroid \u03b1\ninst\u271d : M.Finite\nR : Set \u03b1\nhR : R \u2286 M.E\n\u22a2 (M \u21be R).Finite"}, {"tactic": "exact restrict_finite <| M.ground_finite.subset hR", "annotated_tactic": ["exact <a>restrict_finite</a> <| M.ground_finite.subset hR", [{"full_name": "Matroid.restrict_finite", "def_path": "Mathlib/Data/Matroid/Restrict.lean", "def_pos": [133, 9], "def_end_pos": [133, 24]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\nM\u271d : Matroid \u03b1\nR\u271d I J X Y : Set \u03b1\nM : Matroid \u03b1\ninst\u271d : M.Finite\nR : Set \u03b1\nhR : R \u2286 M.E\n\u22a2 (M \u21be R).Finite", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Bicategory/Coherence.lean", "full_name": "CategoryTheory.FreeBicategory.normalize_naturality", "start": [161, 1], "end": [183, 14], "traced_tactics": [{"tactic": "rcases \u03b7 with \u27e8\u03b7'\u27e9", "annotated_tactic": ["rcases \u03b7 with \u27e8\u03b7'\u27e9", []], "state_before": "B : Type u\ninst\u271d : Quiver B\na b c : B\np : Path a b\nf g : Hom b c\n\u03b7 : f \u27f6 g\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 \u03b7 \u226b (normalizeIso p g).hom =\n    (normalizeIso p f).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)", "state_after": "case mk\nB : Type u\ninst\u271d : Quiver B\na b c : B\np : Path a b\nf g : Hom b c\n\u03b7 : f \u27f6 g\n\u03b7' : Hom\u2082 f g\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g).hom =\n    (normalizeIso p f).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)"}, {"tactic": "clear \u03b7", "annotated_tactic": ["clear \u03b7", []], "state_before": "case mk\nB : Type u\ninst\u271d : Quiver B\na b c : B\np : Path a b\nf g : Hom b c\n\u03b7 : f \u27f6 g\n\u03b7' : Hom\u2082 f g\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g).hom =\n    (normalizeIso p f).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)", "state_after": "case mk\nB : Type u\ninst\u271d : Quiver B\na b c : B\np : Path a b\nf g : Hom b c\n\u03b7' : Hom\u2082 f g\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g).hom =\n    (normalizeIso p f).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)"}, {"tactic": "induction \u03b7' with\n| id => simp\n| vcomp \u03b7 \u03b8 ihf ihg =>\n  simp only [mk_vcomp, Bicategory.whiskerLeft_comp]\n  slice_lhs 2 3 => rw [ihg]\n  slice_lhs 1 2 => rw [ihf]\n  simp\n| whisker_left _ _ ih =>\n  dsimp\n  rw [associator_inv_naturality_right_assoc, whisker_exchange_assoc, ih]\n  simp\n| whisker_right h \u03b7' ih =>\n  dsimp\n  rw [associator_inv_naturality_middle_assoc, \u2190 comp_whiskerRight_assoc, ih, comp_whiskerRight]\n  have := dcongr_arg (fun x => (normalizeIso x h).hom) (normalizeAux_congr p (Quot.mk _ \u03b7'))\n  dsimp at this; simp [this]\n| _ => simp", "annotated_tactic": ["induction \u03b7' with\n  | <a>id</a> => simp\n  | <a>vcomp</a> \u03b7 \u03b8 ihf ihg =>\n    simp only [<a>mk_vcomp</a>, <a>Bicategory.whiskerLeft_comp</a>]\n    slice_lhs 2 3 => rw [ihg]\n    slice_lhs 1 2 => rw [ihf]\n    simp\n  -- p \u2260 nil required! See the docstring of `normalizeAux`.\n  | <a>whisker_left</a> _ _ ih =>\n    dsimp\n    rw [<a>associator_inv_naturality_right_assoc</a>, <a>whisker_exchange_assoc</a>, ih]\n    simp\n  | <a>whisker_right</a> h \u03b7' ih =>\n    dsimp\n    rw [<a>associator_inv_naturality_middle_assoc</a>, \u2190 <a>comp_whiskerRight_assoc</a>, ih, <a>comp_whiskerRight</a>]\n    have := <a>dcongr_arg</a> (fun x => (<a>normalizeIso</a> x h).<a>hom</a>) (<a>normalizeAux_congr</a> p (<a>Quot.mk</a> _ \u03b7'))\n    dsimp at this; simp [this]\n  | _ => simp", [{"full_name": "CategoryTheory.FreeBicategory.Hom\u2082.id", "def_path": "Mathlib/CategoryTheory/Bicategory/Free.lean", "def_pos": [70, 5], "def_end_pos": [70, 7]}, {"full_name": "CategoryTheory.FreeBicategory.Hom\u2082.vcomp", "def_path": "Mathlib/CategoryTheory/Bicategory/Free.lean", "def_pos": [71, 5], "def_end_pos": [71, 10]}, {"full_name": "CategoryTheory.FreeBicategory.mk_vcomp", "def_path": "Mathlib/CategoryTheory/Bicategory/Free.lean", "def_pos": [225, 9], "def_end_pos": [225, 17]}, {"full_name": "CategoryTheory.Bicategory.whiskerLeft_comp", "def_path": "Mathlib/CategoryTheory/Bicategory/Basic.lean", "def_pos": [75, 3], "def_end_pos": [75, 19]}, {"full_name": "CategoryTheory.FreeBicategory.Hom\u2082.whisker_left", "def_path": "Mathlib/CategoryTheory/Bicategory/Free.lean", "def_pos": [72, 5], "def_end_pos": [72, 17]}, {"full_name": "CategoryTheory.Bicategory.associator_inv_naturality_right_assoc", "def_path": "Mathlib/CategoryTheory/Bicategory/Basic.lean", "def_pos": [373, 3], "def_end_pos": [373, 10]}, {"full_name": "CategoryTheory.Bicategory.whisker_exchange_assoc", "def_path": "Mathlib/CategoryTheory/Bicategory/Basic.lean", "def_pos": [177, 12], "def_end_pos": [177, 19]}, {"full_name": "CategoryTheory.FreeBicategory.Hom\u2082.whisker_right", "def_path": "Mathlib/CategoryTheory/Bicategory/Free.lean", "def_pos": [74, 5], "def_end_pos": [74, 18]}, {"full_name": "CategoryTheory.Bicategory.associator_inv_naturality_middle_assoc", "def_path": "Mathlib/CategoryTheory/Bicategory/Basic.lean", "def_pos": [358, 3], "def_end_pos": [358, 10]}, {"full_name": "CategoryTheory.Bicategory.comp_whiskerRight_assoc", "def_path": "Mathlib/CategoryTheory/Bicategory/Basic.lean", "def_pos": [177, 12], "def_end_pos": [177, 19]}, {"full_name": "CategoryTheory.Bicategory.comp_whiskerRight", "def_path": "Mathlib/CategoryTheory/Bicategory/Basic.lean", "def_pos": [91, 3], "def_end_pos": [91, 20]}, {"full_name": "CategoryTheory.dcongr_arg", "def_path": "Mathlib/CategoryTheory/EqToHom.lean", "def_pos": [334, 9], "def_end_pos": [334, 19]}, {"full_name": "CategoryTheory.FreeBicategory.normalizeIso", "def_path": "Mathlib/CategoryTheory/Bicategory/Coherence.lean", "def_pos": [136, 5], "def_end_pos": [136, 17]}, {"full_name": "CategoryTheory.Iso.hom", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [53, 3], "def_end_pos": [53, 6]}, {"full_name": "CategoryTheory.FreeBicategory.normalizeAux_congr", "def_path": "Mathlib/CategoryTheory/Bicategory/Coherence.lean", "def_pos": [148, 9], "def_end_pos": [148, 27]}, {"full_name": "Quot.mk", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [419, 14], "def_end_pos": [419, 21]}]], "state_before": "case mk\nB : Type u\ninst\u271d : Quiver B\na b c : B\np : Path a b\nf g : Hom b c\n\u03b7' : Hom\u2082 f g\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g).hom =\n    (normalizeIso p f).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case mk.id\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.id f\u271d) \u226b (normalizeIso p f\u271d).hom =\n    (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)", "state_after": "no goals"}, {"tactic": "simp only [mk_vcomp, Bicategory.whiskerLeft_comp]", "annotated_tactic": ["simp only [<a>mk_vcomp</a>, <a>Bicategory.whiskerLeft_comp</a>]", [{"full_name": "CategoryTheory.FreeBicategory.mk_vcomp", "def_path": "Mathlib/CategoryTheory/Bicategory/Free.lean", "def_pos": [225, 9], "def_end_pos": [225, 17]}, {"full_name": "CategoryTheory.Bicategory.whiskerLeft_comp", "def_path": "Mathlib/CategoryTheory/Bicategory/Basic.lean", "def_pos": [75, 3], "def_end_pos": [75, 19]}]], "state_before": "case mk.vcomp\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\nihg :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (\u03b7.vcomp \u03b8) \u226b (normalizeIso p h\u271d).hom =\n    (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)", "state_after": "case mk.vcomp\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\nihg :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\np : Path a a\u271d\n\u22a2 ((\u2191(preinclusion B)).map { as := p } \u25c1 \u03b7.mk \u226b (\u2191(preinclusion B)).map { as := p } \u25c1 \u03b8.mk) \u226b (normalizeIso p h\u271d).hom =\n    (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)"}, {"tactic": "slice_lhs 2 3 => rw [ihg]", "annotated_tactic": ["slice_lhs 2 3 => rw [ihg]", []], "state_before": "case mk.vcomp\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\nihg :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\np : Path a a\u271d\n\u22a2 ((\u2191(preinclusion B)).map { as := p } \u25c1 \u03b7.mk \u226b (\u2191(preinclusion B)).map { as := p } \u25c1 \u03b8.mk) \u226b (normalizeIso p h\u271d).hom =\n    (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)", "state_after": "case mk.vcomp\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\nihg :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 \u03b7.mk \u226b (normalizeIso p g\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef) =\n    (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)"}, {"tactic": "slice_lhs 1 2 => rw [ihf]", "annotated_tactic": ["slice_lhs 1 2 => rw [ihf]", []], "state_before": "case mk.vcomp\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\nihg :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 \u03b7.mk \u226b (normalizeIso p g\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef) =\n    (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)", "state_after": "case mk.vcomp\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\nihg :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\np : Path a a\u271d\n\u22a2 ((normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)) \u226b (preinclusion B).map\u2082 (eqToHom \u22ef) =\n    (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case mk.vcomp\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d g\u271d h\u271d : a\u271d \u27f6 b\u271d\n\u03b7 : Hom\u2082 f\u271d g\u271d\n\u03b8 : Hom\u2082 g\u271d h\u271d\nihf :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7 \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\nihg :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b8 \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\np : Path a a\u271d\n\u22a2 ((normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)) \u226b (preinclusion B).map\u2082 (eqToHom \u22ef) =\n    (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)", "state_after": "no goals"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case mk.whisker_left\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d h\u271d : b\u271d \u27f6 c\u271d\n\u03b7\u271d : Hom\u2082 g\u271d h\u271d\nih :\n  \u2200 (p : Path a b\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7\u271d \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.whisker_left f\u271d \u03b7\u271d) \u226b (normalizeIso p (f\u271d \u226b h\u271d)).hom =\n    (normalizeIso p (f\u271d \u226b g\u271d)).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)", "state_after": "case mk.whisker_left\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d h\u271d : b\u271d \u27f6 c\u271d\n\u03b7\u271d : Hom\u2082 g\u271d h\u271d\nih :\n  \u2200 (p : Path a b\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7\u271d \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 f\u271d \u25c1 \u03b7\u271d.mk \u226b\n      (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h\u271d).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 h\u271d \u226b (normalizeIso (normalizeAux p f\u271d) h\u271d).hom =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d g\u271d).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 g\u271d \u226b (normalizeIso (normalizeAux p f\u271d) g\u271d).hom) \u226b\n      (preinclusion B).map\u2082 (eqToHom \u22ef)"}, {"tactic": "rw [associator_inv_naturality_right_assoc, whisker_exchange_assoc, ih]", "annotated_tactic": ["rw [<a>associator_inv_naturality_right_assoc</a>, <a>whisker_exchange_assoc</a>, ih]", [{"full_name": "CategoryTheory.Bicategory.associator_inv_naturality_right_assoc", "def_path": "Mathlib/CategoryTheory/Bicategory/Basic.lean", "def_pos": [373, 3], "def_end_pos": [373, 10]}, {"full_name": "CategoryTheory.Bicategory.whisker_exchange_assoc", "def_path": "Mathlib/CategoryTheory/Bicategory/Basic.lean", "def_pos": [177, 12], "def_end_pos": [177, 19]}]], "state_before": "case mk.whisker_left\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d h\u271d : b\u271d \u27f6 c\u271d\n\u03b7\u271d : Hom\u2082 g\u271d h\u271d\nih :\n  \u2200 (p : Path a b\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7\u271d \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 f\u271d \u25c1 \u03b7\u271d.mk \u226b\n      (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h\u271d).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 h\u271d \u226b (normalizeIso (normalizeAux p f\u271d) h\u271d).hom =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d g\u271d).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 g\u271d \u226b (normalizeIso (normalizeAux p f\u271d) g\u271d).hom) \u226b\n      (preinclusion B).map\u2082 (eqToHom \u22ef)", "state_after": "case mk.whisker_left\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d h\u271d : b\u271d \u27f6 c\u271d\n\u03b7\u271d : Hom\u2082 g\u271d h\u271d\nih :\n  \u2200 (p : Path a b\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7\u271d \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\np : Path a a\u271d\n\u22a2 (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d g\u271d).inv \u226b\n      (normalizeIso p f\u271d).hom \u25b7 g\u271d \u226b (normalizeIso (normalizeAux p f\u271d) g\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef) =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d g\u271d).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 g\u271d \u226b (normalizeIso (normalizeAux p f\u271d) g\u271d).hom) \u226b\n      (preinclusion B).map\u2082 (eqToHom \u22ef)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case mk.whisker_left\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\ng\u271d h\u271d : b\u271d \u27f6 c\u271d\n\u03b7\u271d : Hom\u2082 g\u271d h\u271d\nih :\n  \u2200 (p : Path a b\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7\u271d \u226b (normalizeIso p h\u271d).hom =\n      (normalizeIso p g\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\np : Path a a\u271d\n\u22a2 (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d g\u271d).inv \u226b\n      (normalizeIso p f\u271d).hom \u25b7 g\u271d \u226b (normalizeIso (normalizeAux p f\u271d) g\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef) =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d g\u271d).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 g\u271d \u226b (normalizeIso (normalizeAux p f\u271d) g\u271d).hom) \u226b\n      (preinclusion B).map\u2082 (eqToHom \u22ef)", "state_after": "no goals"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.whisker_right h \u03b7') \u226b (normalizeIso p (Hom.comp g\u271d h)).hom =\n    (normalizeIso p (Hom.comp f\u271d h)).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)", "state_after": "case mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 \u03b7'.mk \u25b7 h \u226b\n      (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) g\u271d h).inv \u226b\n        (normalizeIso p g\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p g\u271d) h).hom =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p f\u271d) h).hom) \u226b\n      (preinclusion B).map\u2082 (eqToHom \u22ef)"}, {"tactic": "rw [associator_inv_naturality_middle_assoc, \u2190 comp_whiskerRight_assoc, ih, comp_whiskerRight]", "annotated_tactic": ["rw [<a>associator_inv_naturality_middle_assoc</a>, \u2190 <a>comp_whiskerRight_assoc</a>, ih, <a>comp_whiskerRight</a>]", [{"full_name": "CategoryTheory.Bicategory.associator_inv_naturality_middle_assoc", "def_path": "Mathlib/CategoryTheory/Bicategory/Basic.lean", "def_pos": [358, 3], "def_end_pos": [358, 10]}, {"full_name": "CategoryTheory.Bicategory.comp_whiskerRight_assoc", "def_path": "Mathlib/CategoryTheory/Bicategory/Basic.lean", "def_pos": [177, 12], "def_end_pos": [177, 19]}, {"full_name": "CategoryTheory.Bicategory.comp_whiskerRight", "def_path": "Mathlib/CategoryTheory/Bicategory/Basic.lean", "def_pos": [91, 3], "def_end_pos": [91, 20]}]], "state_before": "case mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 \u03b7'.mk \u25b7 h \u226b\n      (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) g\u271d h).inv \u226b\n        (normalizeIso p g\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p g\u271d) h).hom =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p f\u271d) h).hom) \u226b\n      (preinclusion B).map\u2082 (eqToHom \u22ef)", "state_after": "case mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\np : Path a a\u271d\n\u22a2 (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n      ((normalizeIso p f\u271d).hom \u25b7 h \u226b (preinclusion B).map\u2082 (eqToHom \u22ef) \u25b7 h) \u226b (normalizeIso (normalizeAux p g\u271d) h).hom =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p f\u271d) h).hom) \u226b\n      (preinclusion B).map\u2082 (eqToHom \u22ef)"}, {"tactic": "have := dcongr_arg (fun x => (normalizeIso x h).hom) (normalizeAux_congr p (Quot.mk _ \u03b7'))", "annotated_tactic": ["have := <a>dcongr_arg</a> (fun x => (<a>normalizeIso</a> x h).<a>hom</a>) (<a>normalizeAux_congr</a> p (<a>Quot.mk</a> _ \u03b7'))", [{"full_name": "CategoryTheory.dcongr_arg", "def_path": "Mathlib/CategoryTheory/EqToHom.lean", "def_pos": [334, 9], "def_end_pos": [334, 19]}, {"full_name": "CategoryTheory.FreeBicategory.normalizeIso", "def_path": "Mathlib/CategoryTheory/Bicategory/Coherence.lean", "def_pos": [136, 5], "def_end_pos": [136, 17]}, {"full_name": "CategoryTheory.Iso.hom", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [53, 3], "def_end_pos": [53, 6]}, {"full_name": "CategoryTheory.FreeBicategory.normalizeAux_congr", "def_path": "Mathlib/CategoryTheory/Bicategory/Coherence.lean", "def_pos": [148, 9], "def_end_pos": [148, 27]}, {"full_name": "Quot.mk", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [419, 14], "def_end_pos": [419, 21]}]], "state_before": "case mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\np : Path a a\u271d\n\u22a2 (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n      ((normalizeIso p f\u271d).hom \u25b7 h \u226b (preinclusion B).map\u2082 (eqToHom \u22ef) \u25b7 h) \u226b (normalizeIso (normalizeAux p g\u271d) h).hom =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p f\u271d) h).hom) \u226b\n      (preinclusion B).map\u2082 (eqToHom \u22ef)", "state_after": "case mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\np : Path a a\u271d\nthis : (normalizeIso (normalizeAux p f\u271d) h).hom = eqToHom \u22ef \u226b (normalizeIso (normalizeAux p g\u271d) h).hom \u226b eqToHom \u22ef\n\u22a2 (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n      ((normalizeIso p f\u271d).hom \u25b7 h \u226b (preinclusion B).map\u2082 (eqToHom \u22ef) \u25b7 h) \u226b (normalizeIso (normalizeAux p g\u271d) h).hom =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p f\u271d) h).hom) \u226b\n      (preinclusion B).map\u2082 (eqToHom \u22ef)"}, {"tactic": "simp [this]", "annotated_tactic": ["simp [this]", []], "state_before": "case mk.whisker_right\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d c\u271d : FreeBicategory B\nf\u271d g\u271d : a\u271d \u27f6 b\u271d\nh : b\u271d \u27f6 c\u271d\n\u03b7' : Hom\u2082 f\u271d g\u271d\nih :\n  \u2200 (p : Path a a\u271d),\n    (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel \u03b7' \u226b (normalizeIso p g\u271d).hom =\n      (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)\np : Path a a\u271d\nthis : (normalizeIso (normalizeAux p f\u271d) h).hom = eqToHom \u22ef \u226b (normalizeIso (normalizeAux p g\u271d) h).hom \u226b eqToHom \u22ef\n\u22a2 (\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n      ((normalizeIso p f\u271d).hom \u25b7 h \u226b (preinclusion B).map\u2082 (eqToHom \u22ef) \u25b7 h) \u226b (normalizeIso (normalizeAux p g\u271d) h).hom =\n    ((\u03b1_ ((\u2191(preinclusion B)).map { as := p }) f\u271d h).inv \u226b\n        (normalizeIso p f\u271d).hom \u25b7 h \u226b (normalizeIso (normalizeAux p f\u271d) h).hom) \u226b\n      (preinclusion B).map\u2082 (eqToHom \u22ef)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case mk.left_unitor_inv\nB : Type u\ninst\u271d : Quiver B\na b c : B\nf g : Hom b c\na\u271d b\u271d : FreeBicategory B\nf\u271d : a\u271d \u27f6 b\u271d\np : Path a a\u271d\n\u22a2 (\u2191(preinclusion B)).map { as := p } \u25c1 Quot.mk Rel (Hom\u2082.left_unitor_inv f\u271d) \u226b (normalizeIso p (\ud835\udfd9 a\u271d \u226b f\u271d)).hom =\n    (normalizeIso p f\u271d).hom \u226b (preinclusion B).map\u2082 (eqToHom \u22ef)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "full_name": "LinearMap.neg_comp", "start": [941, 1], "end": [942, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "full_name": "PMF.toMeasure_apply_eq_zero_iff", "start": [269, 1], "end": [271, 88], "traced_tactics": [{"tactic": "rw [toMeasure_apply_eq_toOuterMeasure_apply p s hs, toOuterMeasure_apply_eq_zero_iff]", "annotated_tactic": ["rw [<a>toMeasure_apply_eq_toOuterMeasure_apply</a> p s hs, <a>toOuterMeasure_apply_eq_zero_iff</a>]", [{"full_name": "PMF.toMeasure_apply_eq_toOuterMeasure_apply", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "def_pos": [255, 9], "def_end_pos": [255, 48]}, {"full_name": "PMF.toOuterMeasure_apply_eq_zero_iff", "def_path": "Mathlib/Probability/ProbabilityMassFunction/Basic.lean", "def_pos": [196, 9], "def_end_pos": [196, 41]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : MeasurableSpace \u03b1\np : PMF \u03b1\ns t : Set \u03b1\nhs : MeasurableSet s\n\u22a2 p.toMeasure s = 0 \u2194 Disjoint p.support s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "full_name": "PiNat.cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff", "start": [569, 1], "end": [590, 58], "traced_tactics": [{"tactic": "rw [l_eq, \u2190 mem_cylinder_iff_eq]", "annotated_tactic": ["rw [l_eq, \u2190 <a>mem_cylinder_iff_eq</a>]", [{"full_name": "PiNat.mem_cylinder_iff_eq", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [134, 9], "def_end_pos": [134, 28]}]], "state_before": "E : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\nl_eq : longestPrefix y s = longestPrefix x s\n\u22a2 cylinder x (longestPrefix x s) = cylinder y (longestPrefix y s)", "state_after": "E : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\nl_eq : longestPrefix y s = longestPrefix x s\n\u22a2 x \u2208 cylinder y (longestPrefix x s)"}, {"tactic": "exact cylinder_anti y H.le (mem_cylinder_firstDiff x y)", "annotated_tactic": ["exact <a>cylinder_anti</a> y H.le (<a>mem_cylinder_firstDiff</a> x y)", [{"full_name": "PiNat.cylinder_anti", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [122, 9], "def_end_pos": [122, 22]}, {"full_name": "PiNat.mem_cylinder_firstDiff", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [164, 9], "def_end_pos": [164, 31]}]], "state_before": "E : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\nl_eq : longestPrefix y s = longestPrefix x s\n\u22a2 x \u2208 cylinder y (longestPrefix x s)", "state_after": "no goals"}, {"tactic": "rcases lt_trichotomy (longestPrefix y s) (longestPrefix x s) with (L | L | L)", "annotated_tactic": ["rcases <a>lt_trichotomy</a> (<a>longestPrefix</a> y s) (<a>longestPrefix</a> x s) with (L | L | L)", [{"full_name": "lt_trichotomy", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [313, 9], "def_end_pos": [313, 22]}, {"full_name": "PiNat.longestPrefix", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [531, 5], "def_end_pos": [531, 18]}, {"full_name": "PiNat.longestPrefix", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [531, 5], "def_end_pos": [531, 18]}]], "state_before": "E : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\n\u22a2 longestPrefix y s = longestPrefix x s", "state_after": "case inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix y s < longestPrefix x s\n\u22a2 longestPrefix y s = longestPrefix x s\n\ncase inr.inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix y s = longestPrefix x s\n\u22a2 longestPrefix y s = longestPrefix x s\n\ncase inr.inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix x s < longestPrefix y s\n\u22a2 longestPrefix y s = longestPrefix x s"}, {"tactic": "have Ax : (s \u2229 cylinder x (longestPrefix x s)).Nonempty :=\n  inter_cylinder_longestPrefix_nonempty hs hne x", "annotated_tactic": ["have Ax : (s \u2229 <a>cylinder</a> x (<a>longestPrefix</a> x s)).<a>Nonempty</a> :=\n        <a>inter_cylinder_longestPrefix_nonempty</a> hs hne x", [{"full_name": "PiNat.cylinder", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [108, 5], "def_end_pos": [108, 13]}, {"full_name": "PiNat.longestPrefix", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [531, 5], "def_end_pos": [531, 18]}, {"full_name": "Set.Nonempty", "def_path": "Mathlib/Init/Set.lean", "def_pos": [218, 15], "def_end_pos": [218, 23]}, {"full_name": "PiNat.inter_cylinder_longestPrefix_nonempty", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [542, 9], "def_end_pos": [542, 46]}]], "state_before": "case inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix y s < longestPrefix x s\n\u22a2 longestPrefix y s = longestPrefix x s", "state_after": "case inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix y s < longestPrefix x s\nAx : (s \u2229 cylinder x (longestPrefix x s)).Nonempty\n\u22a2 longestPrefix y s = longestPrefix x s"}, {"tactic": "have Z := disjoint_cylinder_of_longestPrefix_lt hs ys L", "annotated_tactic": ["have Z := <a>disjoint_cylinder_of_longestPrefix_lt</a> hs ys L", [{"full_name": "PiNat.disjoint_cylinder_of_longestPrefix_lt", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [557, 9], "def_end_pos": [557, 46]}]], "state_before": "case inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix y s < longestPrefix x s\nAx : (s \u2229 cylinder x (longestPrefix x s)).Nonempty\n\u22a2 longestPrefix y s = longestPrefix x s", "state_after": "case inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix y s < longestPrefix x s\nAx : (s \u2229 cylinder x (longestPrefix x s)).Nonempty\nZ : Disjoint s (cylinder y (longestPrefix x s))\n\u22a2 longestPrefix y s = longestPrefix x s"}, {"tactic": "rw [firstDiff_comm] at H", "annotated_tactic": ["rw [<a>firstDiff_comm</a>] at H", [{"full_name": "PiNat.firstDiff_comm", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [88, 9], "def_end_pos": [88, 23]}]], "state_before": "case inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix y s < longestPrefix x s\nAx : (s \u2229 cylinder x (longestPrefix x s)).Nonempty\nZ : Disjoint s (cylinder y (longestPrefix x s))\n\u22a2 longestPrefix y s = longestPrefix x s", "state_after": "case inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff y x\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix y s < longestPrefix x s\nAx : (s \u2229 cylinder x (longestPrefix x s)).Nonempty\nZ : Disjoint s (cylinder y (longestPrefix x s))\n\u22a2 longestPrefix y s = longestPrefix x s"}, {"tactic": "rw [cylinder_eq_cylinder_of_le_firstDiff _ _ H.le] at Z", "annotated_tactic": ["rw [<a>cylinder_eq_cylinder_of_le_firstDiff</a> _ _ H.le] at Z", [{"full_name": "PiNat.cylinder_eq_cylinder_of_le_firstDiff", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [168, 9], "def_end_pos": [168, 45]}]], "state_before": "case inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff y x\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix y s < longestPrefix x s\nAx : (s \u2229 cylinder x (longestPrefix x s)).Nonempty\nZ : Disjoint s (cylinder y (longestPrefix x s))\n\u22a2 longestPrefix y s = longestPrefix x s", "state_after": "case inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff y x\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix y s < longestPrefix x s\nAx : (s \u2229 cylinder x (longestPrefix x s)).Nonempty\nZ : Disjoint s (cylinder x (longestPrefix x s))\n\u22a2 longestPrefix y s = longestPrefix x s"}, {"tactic": "exact (Ax.not_disjoint Z).elim", "annotated_tactic": ["exact (Ax.not_disjoint Z).<a>elim</a>", [{"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "case inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff y x\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix y s < longestPrefix x s\nAx : (s \u2229 cylinder x (longestPrefix x s)).Nonempty\nZ : Disjoint s (cylinder x (longestPrefix x s))\n\u22a2 longestPrefix y s = longestPrefix x s", "state_after": "no goals"}, {"tactic": "exact L", "annotated_tactic": ["exact L", []], "state_before": "case inr.inl\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix y s = longestPrefix x s\n\u22a2 longestPrefix y s = longestPrefix x s", "state_after": "no goals"}, {"tactic": "have Ay : (s \u2229 cylinder y (longestPrefix y s)).Nonempty :=\n  inter_cylinder_longestPrefix_nonempty hs hne y", "annotated_tactic": ["have Ay : (s \u2229 <a>cylinder</a> y (<a>longestPrefix</a> y s)).<a>Nonempty</a> :=\n        <a>inter_cylinder_longestPrefix_nonempty</a> hs hne y", [{"full_name": "PiNat.cylinder", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [108, 5], "def_end_pos": [108, 13]}, {"full_name": "PiNat.longestPrefix", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [531, 5], "def_end_pos": [531, 18]}, {"full_name": "Set.Nonempty", "def_path": "Mathlib/Init/Set.lean", "def_pos": [218, 15], "def_end_pos": [218, 23]}, {"full_name": "PiNat.inter_cylinder_longestPrefix_nonempty", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [542, 9], "def_end_pos": [542, 46]}]], "state_before": "case inr.inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix x s < longestPrefix y s\n\u22a2 longestPrefix y s = longestPrefix x s", "state_after": "case inr.inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix x s < longestPrefix y s\nAy : (s \u2229 cylinder y (longestPrefix y s)).Nonempty\n\u22a2 longestPrefix y s = longestPrefix x s"}, {"tactic": "have A'y : (s \u2229 cylinder y (longestPrefix x s).succ).Nonempty :=\n  Ay.mono (inter_subset_inter_right s (cylinder_anti _ L))", "annotated_tactic": ["have A'y : (s \u2229 <a>cylinder</a> y (<a>longestPrefix</a> x s).<a>succ</a>).<a>Nonempty</a> :=\n        Ay.mono (<a>inter_subset_inter_right</a> s (<a>cylinder_anti</a> _ L))", [{"full_name": "PiNat.cylinder", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [108, 5], "def_end_pos": [108, 13]}, {"full_name": "PiNat.longestPrefix", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [531, 5], "def_end_pos": [531, 18]}, {"full_name": "Nat.succ", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1083, 5], "def_end_pos": [1083, 9]}, {"full_name": "Set.Nonempty", "def_path": "Mathlib/Init/Set.lean", "def_pos": [218, 15], "def_end_pos": [218, 23]}, {"full_name": "Set.inter_subset_inter_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [999, 9], "def_end_pos": [999, 33]}, {"full_name": "PiNat.cylinder_anti", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [122, 9], "def_end_pos": [122, 22]}]], "state_before": "case inr.inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix x s < longestPrefix y s\nAy : (s \u2229 cylinder y (longestPrefix y s)).Nonempty\n\u22a2 longestPrefix y s = longestPrefix x s", "state_after": "case inr.inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix x s < longestPrefix y s\nAy : (s \u2229 cylinder y (longestPrefix y s)).Nonempty\nA'y : (s \u2229 cylinder y (longestPrefix x s).succ).Nonempty\n\u22a2 longestPrefix y s = longestPrefix x s"}, {"tactic": "have Z := disjoint_cylinder_of_longestPrefix_lt hs xs (Nat.lt_succ_self _)", "annotated_tactic": ["have Z := <a>disjoint_cylinder_of_longestPrefix_lt</a> hs xs (<a>Nat.lt_succ_self</a> _)", [{"full_name": "PiNat.disjoint_cylinder_of_longestPrefix_lt", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [557, 9], "def_end_pos": [557, 46]}, {"full_name": "Nat.lt_succ_self", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [358, 17], "def_end_pos": [358, 29]}]], "state_before": "case inr.inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix x s < longestPrefix y s\nAy : (s \u2229 cylinder y (longestPrefix y s)).Nonempty\nA'y : (s \u2229 cylinder y (longestPrefix x s).succ).Nonempty\n\u22a2 longestPrefix y s = longestPrefix x s", "state_after": "case inr.inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix x s < longestPrefix y s\nAy : (s \u2229 cylinder y (longestPrefix y s)).Nonempty\nA'y : (s \u2229 cylinder y (longestPrefix x s).succ).Nonempty\nZ : Disjoint s (cylinder x (longestPrefix x s).succ)\n\u22a2 longestPrefix y s = longestPrefix x s"}, {"tactic": "rw [cylinder_eq_cylinder_of_le_firstDiff _ _ H] at Z", "annotated_tactic": ["rw [<a>cylinder_eq_cylinder_of_le_firstDiff</a> _ _ H] at Z", [{"full_name": "PiNat.cylinder_eq_cylinder_of_le_firstDiff", "def_path": "Mathlib/Topology/MetricSpace/PiNat.lean", "def_pos": [168, 9], "def_end_pos": [168, 45]}]], "state_before": "case inr.inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix x s < longestPrefix y s\nAy : (s \u2229 cylinder y (longestPrefix y s)).Nonempty\nA'y : (s \u2229 cylinder y (longestPrefix x s).succ).Nonempty\nZ : Disjoint s (cylinder x (longestPrefix x s).succ)\n\u22a2 longestPrefix y s = longestPrefix x s", "state_after": "case inr.inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix x s < longestPrefix y s\nAy : (s \u2229 cylinder y (longestPrefix y s)).Nonempty\nA'y : (s \u2229 cylinder y (longestPrefix x s).succ).Nonempty\nZ : Disjoint s (cylinder y (longestPrefix x s).succ)\n\u22a2 longestPrefix y s = longestPrefix x s"}, {"tactic": "exact (A'y.not_disjoint Z).elim", "annotated_tactic": ["exact (A'y.not_disjoint Z).<a>elim</a>", [{"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "case inr.inr\nE : \u2115 \u2192 Type u_1\ninst\u271d\u00b9 : (n : \u2115) \u2192 TopologicalSpace (E n)\ninst\u271d : \u2200 (n : \u2115), DiscreteTopology (E n)\nx y : (n : \u2115) \u2192 E n\ns : Set ((n : \u2115) \u2192 E n)\nhs : IsClosed s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x \u2209 s\nys : y \u2209 s\nL : longestPrefix x s < longestPrefix y s\nAy : (s \u2229 cylinder y (longestPrefix y s)).Nonempty\nA'y : (s \u2229 cylinder y (longestPrefix x s).succ).Nonempty\nZ : Disjoint s (cylinder y (longestPrefix x s).succ)\n\u22a2 longestPrefix y s = longestPrefix x s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/RCLike/Basic.lean", "full_name": "RCLike.inv_re", "start": [542, 1], "end": [543, 81], "traced_tactics": [{"tactic": "rw [inv_def, normSq_eq_def', mul_comm, re_ofReal_mul, conj_re, div_eq_inv_mul]", "annotated_tactic": ["rw [<a>inv_def</a>, <a>normSq_eq_def'</a>, <a>mul_comm</a>, <a>re_ofReal_mul</a>, <a>conj_re</a>, <a>div_eq_inv_mul</a>]", [{"full_name": "RCLike.inv_def", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [533, 9], "def_end_pos": [533, 16]}, {"full_name": "RCLike.normSq_eq_def'", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [452, 9], "def_end_pos": [452, 23]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "RCLike.re_ofReal_mul", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [253, 9], "def_end_pos": [253, 22]}, {"full_name": "RCLike.conj_re", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [322, 9], "def_end_pos": [322, 16]}, {"full_name": "div_eq_inv_mul", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 23]}]], "state_before": "K : Type u_1\nE : Type u_2\ninst\u271d : RCLike K\nz\u271d z : K\n\u22a2 re z\u207b\u00b9 = re z / normSq z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Ordmap/Ordset.lean", "full_name": "Ordnode.size_balance'", "start": [443, 1], "end": [449, 8], "traced_tactics": [{"tactic": "unfold balance'", "annotated_tactic": ["unfold <a>balance'</a>", [{"full_name": "Ordnode.balance'", "def_path": "Mathlib/Data/Ordmap/Ordset.lean", "def_pos": [305, 5], "def_end_pos": [305, 13]}]], "state_before": "\u03b1 : Type u_1\nl : Ordnode \u03b1\nx : \u03b1\nr : Ordnode \u03b1\nhl : l.Sized\nhr : r.Sized\n\u22a2 (l.balance' x r).size = l.size + r.size + 1", "state_after": "\u03b1 : Type u_1\nl : Ordnode \u03b1\nx : \u03b1\nr : Ordnode \u03b1\nhl : l.Sized\nhr : r.Sized\n\u22a2 (if l.size + r.size \u2264 1 then l.node' x r\n      else\n        if r.size > delta * l.size then l.rotateL x r\n        else if l.size > delta * r.size then l.rotateR x r else l.node' x r).size =\n    l.size + r.size + 1"}, {"tactic": "split_ifs", "annotated_tactic": ["split_ifs", []], "state_before": "\u03b1 : Type u_1\nl : Ordnode \u03b1\nx : \u03b1\nr : Ordnode \u03b1\nhl : l.Sized\nhr : r.Sized\n\u22a2 (if l.size + r.size \u2264 1 then l.node' x r\n      else\n        if r.size > delta * l.size then l.rotateL x r\n        else if l.size > delta * r.size then l.rotateR x r else l.node' x r).size =\n    l.size + r.size + 1", "state_after": "case pos\n\u03b1 : Type u_1\nl : Ordnode \u03b1\nx : \u03b1\nr : Ordnode \u03b1\nhl : l.Sized\nhr : r.Sized\nh\u271d : l.size + r.size \u2264 1\n\u22a2 (l.node' x r).size = l.size + r.size + 1\n\ncase pos\n\u03b1 : Type u_1\nl : Ordnode \u03b1\nx : \u03b1\nr : Ordnode \u03b1\nhl : l.Sized\nhr : r.Sized\nh\u271d\u00b9 : \u00acl.size + r.size \u2264 1\nh\u271d : r.size > delta * l.size\n\u22a2 (l.rotateL x r).size = l.size + r.size + 1\n\ncase pos\n\u03b1 : Type u_1\nl : Ordnode \u03b1\nx : \u03b1\nr : Ordnode \u03b1\nhl : l.Sized\nhr : r.Sized\nh\u271d\u00b2 : \u00acl.size + r.size \u2264 1\nh\u271d\u00b9 : \u00acr.size > delta * l.size\nh\u271d : l.size > delta * r.size\n\u22a2 (l.rotateR x r).size = l.size + r.size + 1\n\ncase neg\n\u03b1 : Type u_1\nl : Ordnode \u03b1\nx : \u03b1\nr : Ordnode \u03b1\nhl : l.Sized\nhr : r.Sized\nh\u271d\u00b2 : \u00acl.size + r.size \u2264 1\nh\u271d\u00b9 : \u00acr.size > delta * l.size\nh\u271d : \u00acl.size > delta * r.size\n\u22a2 (l.node' x r).size = l.size + r.size + 1"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case pos\n\u03b1 : Type u_1\nl : Ordnode \u03b1\nx : \u03b1\nr : Ordnode \u03b1\nhl : l.Sized\nhr : r.Sized\nh\u271d : l.size + r.size \u2264 1\n\u22a2 (l.node' x r).size = l.size + r.size + 1", "state_after": "no goals"}, {"tactic": "exact hr.rotateL_size", "annotated_tactic": ["exact hr.rotateL_size", []], "state_before": "case pos\n\u03b1 : Type u_1\nl : Ordnode \u03b1\nx : \u03b1\nr : Ordnode \u03b1\nhl : l.Sized\nhr : r.Sized\nh\u271d\u00b9 : \u00acl.size + r.size \u2264 1\nh\u271d : r.size > delta * l.size\n\u22a2 (l.rotateL x r).size = l.size + r.size + 1", "state_after": "no goals"}, {"tactic": "exact hl.rotateR_size", "annotated_tactic": ["exact hl.rotateR_size", []], "state_before": "case pos\n\u03b1 : Type u_1\nl : Ordnode \u03b1\nx : \u03b1\nr : Ordnode \u03b1\nhl : l.Sized\nhr : r.Sized\nh\u271d\u00b2 : \u00acl.size + r.size \u2264 1\nh\u271d\u00b9 : \u00acr.size > delta * l.size\nh\u271d : l.size > delta * r.size\n\u22a2 (l.rotateR x r).size = l.size + r.size + 1", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg\n\u03b1 : Type u_1\nl : Ordnode \u03b1\nx : \u03b1\nr : Ordnode \u03b1\nhl : l.Sized\nhr : r.Sized\nh\u271d\u00b2 : \u00acl.size + r.size \u2264 1\nh\u271d\u00b9 : \u00acr.size > delta * l.size\nh\u271d : \u00acl.size > delta * r.size\n\u22a2 (l.node' x r).size = l.size + r.size + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Pointwise.lean", "full_name": "Filter.coe_pureOneHom", "start": [157, 1], "end": [158, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/FunctorCategory.lean", "full_name": "CategoryTheory.Limits.colimit_obj_ext", "start": [310, 1], "end": [315, 18], "traced_tactics": [{"tactic": "apply (cancel_epi (colimitObjIsoColimitCompEvaluation H k).inv).1", "annotated_tactic": ["apply (<a>cancel_epi</a> (<a>colimitObjIsoColimitCompEvaluation</a> H k).<a>inv</a>).1", [{"full_name": "CategoryTheory.cancel_epi", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 19]}, {"full_name": "CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation", "def_path": "Mathlib/CategoryTheory/Limits/FunctorCategory.lean", "def_pos": [264, 5], "def_end_pos": [264, 39]}, {"full_name": "CategoryTheory.Iso.inv", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [55, 3], "def_end_pos": [55, 6]}]], "state_before": "C : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\nH : J \u2964 K \u2964 C\ninst\u271d : HasColimitsOfShape J C\nk : K\nW : C\nf g : (colimit H).obj k \u27f6 W\nw : \u2200 (j : J), (colimit.\u03b9 H j).app k \u226b f = (colimit.\u03b9 H j).app k \u226b g\n\u22a2 f = g", "state_after": "C : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\nH : J \u2964 K \u2964 C\ninst\u271d : HasColimitsOfShape J C\nk : K\nW : C\nf g : (colimit H).obj k \u27f6 W\nw : \u2200 (j : J), (colimit.\u03b9 H j).app k \u226b f = (colimit.\u03b9 H j).app k \u226b g\n\u22a2 (colimitObjIsoColimitCompEvaluation H k).inv \u226b f = (colimitObjIsoColimitCompEvaluation H k).inv \u226b g"}, {"tactic": "ext j", "annotated_tactic": ["ext j", []], "state_before": "C : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\nH : J \u2964 K \u2964 C\ninst\u271d : HasColimitsOfShape J C\nk : K\nW : C\nf g : (colimit H).obj k \u27f6 W\nw : \u2200 (j : J), (colimit.\u03b9 H j).app k \u226b f = (colimit.\u03b9 H j).app k \u226b g\n\u22a2 (colimitObjIsoColimitCompEvaluation H k).inv \u226b f = (colimitObjIsoColimitCompEvaluation H k).inv \u226b g", "state_after": "case w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\nH : J \u2964 K \u2964 C\ninst\u271d : HasColimitsOfShape J C\nk : K\nW : C\nf g : (colimit H).obj k \u27f6 W\nw : \u2200 (j : J), (colimit.\u03b9 H j).app k \u226b f = (colimit.\u03b9 H j).app k \u226b g\nj : J\n\u22a2 colimit.\u03b9 (H \u22d9 (evaluation K C).obj k) j \u226b (colimitObjIsoColimitCompEvaluation H k).inv \u226b f =\n    colimit.\u03b9 (H \u22d9 (evaluation K C).obj k) j \u226b (colimitObjIsoColimitCompEvaluation H k).inv \u226b g"}, {"tactic": "simpa using w j", "annotated_tactic": ["simpa using w j", []], "state_before": "case w\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\nD : Type u'\ninst\u271d\u00b3 : Category.{v', u'} D\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\nH : J \u2964 K \u2964 C\ninst\u271d : HasColimitsOfShape J C\nk : K\nW : C\nf g : (colimit H).obj k \u27f6 W\nw : \u2200 (j : J), (colimit.\u03b9 H j).app k \u226b f = (colimit.\u03b9 H j).app k \u226b g\nj : J\n\u22a2 colimit.\u03b9 (H \u22d9 (evaluation K C).obj k) j \u226b (colimitObjIsoColimitCompEvaluation H k).inv \u226b f =\n    colimit.\u03b9 (H \u22d9 (evaluation K C).obj k) j \u226b (colimitObjIsoColimitCompEvaluation H k).inv \u226b g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.Measure.toENNRealVectorMeasure_ennrealToMeasure", "start": [523, 1], "end": [526, 77], "traced_tactics": [{"tactic": "rw [toENNRealVectorMeasure_apply_measurable hs, ennrealToMeasure_apply hs]", "annotated_tactic": ["rw [<a>toENNRealVectorMeasure_apply_measurable</a> hs, <a>ennrealToMeasure_apply</a> hs]", [{"full_name": "MeasureTheory.Measure.toENNRealVectorMeasure_apply_measurable", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [478, 9], "def_end_pos": [478, 48]}, {"full_name": "MeasureTheory.VectorMeasure.ennrealToMeasure_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [517, 9], "def_end_pos": [517, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc : VectorMeasure \u03b1 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u2191\u03bc.ennrealToMeasure.toENNRealVectorMeasure s = \u2191\u03bc s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.atTop_basis_Ioi", "start": [274, 1], "end": [277, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "full_name": "zero_le_four", "start": [44, 1], "end": [47, 45], "traced_tactics": [{"tactic": "rw [\u2190 three_add_one_eq_four]", "annotated_tactic": ["rw [\u2190 <a>three_add_one_eq_four</a>]", [{"full_name": "three_add_one_eq_four", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [242, 9], "def_end_pos": [242, 30]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : AddMonoidWithOne \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : ZeroLEOneClass \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 0 \u2264 4", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b3 : AddMonoidWithOne \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : ZeroLEOneClass \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 0 \u2264 3 + 1"}, {"tactic": "exact add_nonneg zero_le_three zero_le_one", "annotated_tactic": ["exact <a>add_nonneg</a> <a>zero_le_three</a> <a>zero_le_one</a>", [{"full_name": "add_nonneg", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [1092, 24], "def_end_pos": [1092, 34]}, {"full_name": "zero_le_three", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [38, 7], "def_end_pos": [38, 20]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : AddMonoidWithOne \u03b1\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : ZeroLEOneClass \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\n\u22a2 0 \u2264 3 + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/InfiniteSum/Ring.lean", "full_name": "tsum_mul_tsum_eq_tsum_sum_range", "start": [249, 1], "end": [253, 57], "traced_tactics": [{"tactic": "simp_rw [\u2190 Nat.sum_antidiagonal_eq_sum_range_succ fun k l \u21a6 f k * g l]", "annotated_tactic": ["simp_rw [\u2190 <a>Nat.sum_antidiagonal_eq_sum_range_succ</a> fun k l \u21a6 f k * g l]", [{"full_name": "Finset.Nat.sum_antidiagonal_eq_sum_range_succ", "def_path": "Mathlib/Algebra/BigOperators/NatAntidiagonal.lean", "def_pos": [69, 3], "def_end_pos": [69, 14]}]], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nhf : Summable f\nhg : Summable g\nhfg : Summable fun x => f x.1 * g x.2\n\u22a2 (\u2211' (n : \u2115), f n) * \u2211' (n : \u2115), g n = \u2211' (n : \u2115), \u2211 k \u2208 range (n + 1), f k * g (n - k)", "state_after": "\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nhf : Summable f\nhg : Summable g\nhfg : Summable fun x => f x.1 * g x.2\n\u22a2 (\u2211' (n : \u2115), f n) * \u2211' (n : \u2115), g n = \u2211' (n : \u2115), \u2211 ij \u2208 antidiagonal n, f ij.1 * g ij.2"}, {"tactic": "exact tsum_mul_tsum_eq_tsum_sum_antidiagonal hf hg hfg", "annotated_tactic": ["exact <a>tsum_mul_tsum_eq_tsum_sum_antidiagonal</a> hf hg hfg", [{"full_name": "tsum_mul_tsum_eq_tsum_sum_antidiagonal", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Ring.lean", "def_pos": [221, 9], "def_end_pos": [221, 47]}]], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\nR : Type u_3\n\u03b1 : Type u_4\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : NonUnitalNonAssocSemiring \u03b1\nf g : \u2115 \u2192 \u03b1\ninst\u271d\u00b9 : T3Space \u03b1\ninst\u271d : TopologicalSemiring \u03b1\nhf : Summable f\nhg : Summable g\nhfg : Summable fun x => f x.1 * g x.2\n\u22a2 (\u2211' (n : \u2115), f n) * \u2211' (n : \u2115), g n = \u2211' (n : \u2115), \u2211 ij \u2208 antidiagonal n, f ij.1 * g ij.2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/Irrational.lean", "full_name": "Irrational.sub_rat", "start": [313, 1], "end": [314, 61], "traced_tactics": [{"tactic": "simpa only [sub_eq_add_neg, cast_neg] using h.add_rat (-q)", "annotated_tactic": ["simpa only [<a>sub_eq_add_neg</a>, <a>cast_neg</a>] using h.add_rat (-q)", [{"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1057, 3], "def_end_pos": [1057, 14]}, {"full_name": "Rat.cast_neg", "def_path": "Mathlib/Data/Rat/Cast/Defs.lean", "def_pos": [190, 26], "def_end_pos": [190, 34]}]], "state_before": "q : \u211a\nx y : \u211d\nh : Irrational x\n\u22a2 Irrational (x - \u2191q)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecificLimits/Normed.lean", "full_name": "NormedField.tendsto_zero_smul_of_tendsto_zero_of_bounded", "start": [72, 1], "end": [77, 85], "traced_tactics": [{"tactic": "rw [\u2190 isLittleO_one_iff \ud835\udd5c] at h\u03b5 \u22a2", "annotated_tactic": ["rw [\u2190 <a>isLittleO_one_iff</a> \ud835\udd5c] at h\u03b5 \u22a2", [{"full_name": "Asymptotics.isLittleO_one_iff", "def_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "def_pos": [1362, 9], "def_end_pos": [1362, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b9 : Type u_4\n\ud835\udd5c : Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup \ud835\udd38\ninst\u271d\u00b9 : Module \ud835\udd5c \ud835\udd38\ninst\u271d : BoundedSMul \ud835\udd5c \ud835\udd38\nl : Filter \u03b9\n\u03b5 : \u03b9 \u2192 \ud835\udd5c\nf : \u03b9 \u2192 \ud835\udd38\nh\u03b5 : Tendsto \u03b5 l (\ud835\udcdd 0)\nhf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l (norm \u2218 f)\n\u22a2 Tendsto (\u03b5 \u2022 f) l (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b9 : Type u_4\n\ud835\udd5c : Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup \ud835\udd38\ninst\u271d\u00b9 : Module \ud835\udd5c \ud835\udd38\ninst\u271d : BoundedSMul \ud835\udd5c \ud835\udd38\nl : Filter \u03b9\n\u03b5 : \u03b9 \u2192 \ud835\udd5c\nf : \u03b9 \u2192 \ud835\udd38\nh\u03b5 : \u03b5 =o[l] fun _x => 1\nhf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l (norm \u2218 f)\n\u22a2 (\u03b5 \u2022 f) =o[l] fun _x => 1"}, {"tactic": "simpa using IsLittleO.smul_isBigO h\u03b5 (hf.isBigO_const (one_ne_zero : (1 : \ud835\udd5c) \u2260 0))", "annotated_tactic": ["simpa using <a>IsLittleO.smul_isBigO</a> h\u03b5 (hf.isBigO_const (<a>one_ne_zero</a> : (1 : \ud835\udd5c) \u2260 0))", [{"full_name": "Asymptotics.IsLittleO.smul_isBigO", "def_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "def_pos": [1822, 9], "def_end_pos": [1822, 30]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [58, 15], "def_end_pos": [58, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9\u271d : Type u_3\n\u03b9 : Type u_4\n\ud835\udd5c : Type u_5\n\ud835\udd38 : Type u_6\ninst\u271d\u00b3 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b2 : NormedAddCommGroup \ud835\udd38\ninst\u271d\u00b9 : Module \ud835\udd5c \ud835\udd38\ninst\u271d : BoundedSMul \ud835\udd5c \ud835\udd38\nl : Filter \u03b9\n\u03b5 : \u03b9 \u2192 \ud835\udd5c\nf : \u03b9 \u2192 \ud835\udd38\nh\u03b5 : \u03b5 =o[l] fun _x => 1\nhf : IsBoundedUnder (fun x x_1 => x \u2264 x_1) l (norm \u2218 f)\n\u22a2 (\u03b5 \u2022 f) =o[l] fun _x => 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "full_name": "hasSum_geometric_two'", "start": [329, 1], "end": [335, 13], "traced_tactics": [{"tactic": "convert HasSum.mul_left (a / 2)\n    (hasSum_geometric_of_lt_one (le_of_lt one_half_pos) one_half_lt_one) using 1", "annotated_tactic": ["convert <a>HasSum.mul_left</a> (a / 2)\n      (<a>hasSum_geometric_of_lt_one</a> (<a>le_of_lt</a> <a>one_half_pos</a>) <a>one_half_lt_one</a>) using 1", [{"full_name": "HasSum.mul_left", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Ring.lean", "def_pos": [34, 9], "def_end_pos": [34, 24]}, {"full_name": "hasSum_geometric_of_lt_one", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [263, 9], "def_end_pos": [263, 35]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "one_half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [449, 9], "def_end_pos": [449, 21]}, {"full_name": "one_half_lt_one", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [472, 9], "def_end_pos": [472, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\na : \u211d\n\u22a2 HasSum (fun n => a / 2 / 2 ^ n) a", "state_after": "case h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\na : \u211d\n\u22a2 (fun n => a / 2 / 2 ^ n) = fun i => a / 2 * (1 / 2) ^ i\n\ncase h.e'_6\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\na : \u211d\n\u22a2 a = a / 2 * (1 - 1 / 2)\u207b\u00b9"}, {"tactic": "funext n", "annotated_tactic": ["funext n", []], "state_before": "case h.e'_5\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\na : \u211d\n\u22a2 (fun n => a / 2 / 2 ^ n) = fun i => a / 2 * (1 / 2) ^ i", "state_after": "case h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\na : \u211d\nn : \u2115\n\u22a2 a / 2 / 2 ^ n = a / 2 * (1 / 2) ^ n"}, {"tactic": "simp only [one_div, inv_pow]", "annotated_tactic": ["simp only [<a>one_div</a>, <a>inv_pow</a>]", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 16]}, {"full_name": "inv_pow", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [578, 7], "def_end_pos": [578, 14]}]], "state_before": "case h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\na : \u211d\nn : \u2115\n\u22a2 a / 2 / 2 ^ n = a / 2 * (1 / 2) ^ n", "state_after": "case h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\na : \u211d\nn : \u2115\n\u22a2 a / 2 / 2 ^ n = a / 2 * (2 ^ n)\u207b\u00b9"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.e'_5.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\na : \u211d\nn : \u2115\n\u22a2 a / 2 / 2 ^ n = a / 2 * (2 ^ n)\u207b\u00b9", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "case h.e'_6\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\na : \u211d\n\u22a2 a = a / 2 * (1 - 1 / 2)\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean", "full_name": "MeasureTheory.SimpleFunc.measure_support_lt_top_of_mem\u2112p", "start": [396, 1], "end": [398, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Opposites.lean", "full_name": "CategoryTheory.NatTrans.unop_id", "start": [330, 1], "end": [331, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Seminorm.lean", "full_name": "Seminorm.closedBall_add_closedBall_subset", "start": [789, 1], "end": [793, 58], "traced_tactics": [{"tactic": "rintro x \u27e8y\u2081, hy\u2081, y\u2082, hy\u2082, rfl\u27e9", "annotated_tactic": ["rintro x \u27e8y\u2081, hy\u2081, y\u2082, hy\u2082, rfl\u27e9", []], "state_before": "R : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u00b2 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : SMul \ud835\udd5c E\np\u271d : Seminorm \ud835\udd5c E\nx y : E\nr : \u211d\np : Seminorm \ud835\udd5c E\nr\u2081 r\u2082 : \u211d\nx\u2081 x\u2082 : E\n\u22a2 p.closedBall x\u2081 r\u2081 + p.closedBall x\u2082 r\u2082 \u2286 p.closedBall (x\u2081 + x\u2082) (r\u2081 + r\u2082)", "state_after": "case intro.intro.intro.intro\nR : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u00b2 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : SMul \ud835\udd5c E\np\u271d : Seminorm \ud835\udd5c E\nx y : E\nr : \u211d\np : Seminorm \ud835\udd5c E\nr\u2081 r\u2082 : \u211d\nx\u2081 x\u2082 y\u2081 : E\nhy\u2081 : y\u2081 \u2208 p.closedBall x\u2081 r\u2081\ny\u2082 : E\nhy\u2082 : y\u2082 \u2208 p.closedBall x\u2082 r\u2082\n\u22a2 (fun x x_1 => x + x_1) y\u2081 y\u2082 \u2208 p.closedBall (x\u2081 + x\u2082) (r\u2081 + r\u2082)"}, {"tactic": "rw [mem_closedBall, add_sub_add_comm]", "annotated_tactic": ["rw [<a>mem_closedBall</a>, <a>add_sub_add_comm</a>]", [{"full_name": "Seminorm.mem_closedBall", "def_path": "Mathlib/Analysis/Seminorm.lean", "def_pos": [679, 9], "def_end_pos": [679, 23]}, {"full_name": "add_sub_add_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [833, 3], "def_end_pos": [833, 14]}]], "state_before": "case intro.intro.intro.intro\nR : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u00b2 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : SMul \ud835\udd5c E\np\u271d : Seminorm \ud835\udd5c E\nx y : E\nr : \u211d\np : Seminorm \ud835\udd5c E\nr\u2081 r\u2082 : \u211d\nx\u2081 x\u2082 y\u2081 : E\nhy\u2081 : y\u2081 \u2208 p.closedBall x\u2081 r\u2081\ny\u2082 : E\nhy\u2082 : y\u2082 \u2208 p.closedBall x\u2082 r\u2082\n\u22a2 (fun x x_1 => x + x_1) y\u2081 y\u2082 \u2208 p.closedBall (x\u2081 + x\u2082) (r\u2081 + r\u2082)", "state_after": "case intro.intro.intro.intro\nR : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u00b2 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : SMul \ud835\udd5c E\np\u271d : Seminorm \ud835\udd5c E\nx y : E\nr : \u211d\np : Seminorm \ud835\udd5c E\nr\u2081 r\u2082 : \u211d\nx\u2081 x\u2082 y\u2081 : E\nhy\u2081 : y\u2081 \u2208 p.closedBall x\u2081 r\u2081\ny\u2082 : E\nhy\u2082 : y\u2082 \u2208 p.closedBall x\u2082 r\u2082\n\u22a2 p (y\u2081 - x\u2081 + (y\u2082 - x\u2082)) \u2264 r\u2081 + r\u2082"}, {"tactic": "exact (map_add_le_add p _ _).trans (add_le_add hy\u2081 hy\u2082)", "annotated_tactic": ["exact (<a>map_add_le_add</a> p _ _).<a>trans</a> (<a>add_le_add</a> hy\u2081 hy\u2082)", [{"full_name": "SubadditiveHomClass.map_add_le_add", "def_path": "Mathlib/Algebra/Order/Hom/Basic.lean", "def_pos": [86, 3], "def_end_pos": [86, 17]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [205, 32], "def_end_pos": [205, 42]}]], "state_before": "case intro.intro.intro.intro\nR : Type u_1\nR' : Type u_2\n\ud835\udd5c : Type u_3\n\ud835\udd5c\u2082 : Type u_4\n\ud835\udd5c\u2083 : Type u_5\n\ud835\udd5d : Type u_6\nE : Type u_7\nE\u2082 : Type u_8\nE\u2083 : Type u_9\nF : Type u_10\nG : Type u_11\n\u03b9 : Type u_12\ninst\u271d\u00b2 : SeminormedRing \ud835\udd5c\ninst\u271d\u00b9 : AddCommGroup E\ninst\u271d : SMul \ud835\udd5c E\np\u271d : Seminorm \ud835\udd5c E\nx y : E\nr : \u211d\np : Seminorm \ud835\udd5c E\nr\u2081 r\u2082 : \u211d\nx\u2081 x\u2082 y\u2081 : E\nhy\u2081 : y\u2081 \u2208 p.closedBall x\u2081 r\u2081\ny\u2082 : E\nhy\u2082 : y\u2082 \u2208 p.closedBall x\u2082 r\u2082\n\u22a2 p (y\u2081 - x\u2081 + (y\u2082 - x\u2082)) \u2264 r\u2081 + r\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "full_name": "hasFDerivWithinAt_iff_hasDerivWithinAt", "start": [171, 1], "end": [173, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Classes/Order.lean", "full_name": "Ordering.then_eq_lt", "start": [20, 1], "end": [21, 35], "traced_tactics": [{"tactic": "cases o\u2081 <;> cases o\u2082 <;> decide", "annotated_tactic": ["cases o\u2081 <;> cases o\u2082 <;> decide", []], "state_before": "o\u2081 o\u2082 : Ordering\n\u22a2 o\u2081.then o\u2082 = lt \u2194 o\u2081 = lt \u2228 o\u2081 = eq \u2227 o\u2082 = lt", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Lattice.lean", "full_name": "inf_le_right", "start": [366, 1], "end": [367, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.condexpL1CLM_lpMeas", "start": [491, 1], "end": [512, 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": [481, 19], "def_end_pos": [481, 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\n\u22a2 (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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\n\u22a2 (condexpL1CLM F' hm \u03bc) \u2191f = \u2191f"}, {"tactic": "have hfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g := by\n  simp only [g, 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 [g, <a>LinearIsometryEquiv.symm_apply_apply</a>]", [{"full_name": "MeasureTheory.lpMeasToLpTrimLie", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [481, 19], "def_end_pos": [481, 36]}, {"full_name": "LinearIsometryEquiv.symm", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [749, 5], "def_end_pos": [749, 9]}, {"full_name": "LinearIsometryEquiv.symm_apply_apply", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [760, 9], "def_end_pos": [760, 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\n\u22a2 (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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\n\u22a2 (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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\n\u22a2 (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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\n\u22a2 (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g) = \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm 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": [911, 9], "def_end_pos": [911, 21]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [325, 17], "def_end_pos": [325, 27]}, {"full_name": "MeasureTheory.condexpL1CLM", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [381, 5], "def_end_pos": [381, 17]}, {"full_name": "MeasureTheory.lpMeasToLpTrimLie", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [481, 19], "def_end_pos": [481, 36]}, {"full_name": "LinearIsometryEquiv.symm", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [749, 5], "def_end_pos": [749, 9]}, {"full_name": "MeasureTheory.lpMeasToLpTrimLie", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [481, 19], "def_end_pos": [481, 36]}, {"full_name": "LinearIsometryEquiv.symm", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [749, 5], "def_end_pos": [749, 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\n\u22a2 (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g) = \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\n\u22a2 \u2200 (c : F') {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : (\u03bc.trim hm) s < \u22a4),\n    (fun g =>\n        (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g) = \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g))\n      \u2191(simpleFunc.indicatorConst 1 hs \u22ef 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192 F'\u2984 (hf : Mem\u2112p f 1 (\u03bc.trim hm)) (hg : Mem\u2112p g 1 (\u03bc.trim hm)),\n    Disjoint (Function.support f) (Function.support g) \u2192\n      (fun g =>\n            (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g) =\n              \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g))\n          (Mem\u2112p.toLp f hf) \u2192\n        (fun g =>\n              (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g) =\n                \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g))\n            (Mem\u2112p.toLp g hg) \u2192\n          (fun g =>\n              (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g) =\n                \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\n\u22a2 IsClosed\n    {f |\n      (fun g =>\n          (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g) = \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g))\n        f}"}, {"tactic": "simp only [g, LinearIsometryEquiv.symm_apply_apply]", "annotated_tactic": ["simp only [g, <a>LinearIsometryEquiv.symm_apply_apply</a>]", [{"full_name": "LinearIsometryEquiv.symm_apply_apply", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [760, 9], "def_end_pos": [760, 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\n\u22a2 f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\n\u22a2 \u2200 (c : F') {s : Set \u03b1} (hs : MeasurableSet s) (h\u03bcs : (\u03bc.trim hm) s < \u22a4),\n    (fun g =>\n        (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g) = \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g))\n      \u2191(simpleFunc.indicatorConst 1 hs \u22ef 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns\u271d : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\nc : F'\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : (\u03bc.trim hm) s < \u22a4\n\u22a2 (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm \u2191(simpleFunc.indicatorConst 1 hs \u22ef c)) =\n    \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm \u2191(simpleFunc.indicatorConst 1 hs \u22ef 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": [688, 9], "def_end_pos": [688, 27]}, {"full_name": "MeasureTheory.lpMeasToLpTrimLie_symm_indicator", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [539, 9], "def_end_pos": [539, 41]}, {"full_name": "MeasureTheory.condexpL1CLM_indicatorConstLp", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [394, 9], "def_end_pos": [394, 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns\u271d : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\nc : F'\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : (\u03bc.trim hm) s < \u22a4\n\u22a2 (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm \u2191(simpleFunc.indicatorConst 1 hs \u22ef c)) =\n    \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm \u2191(simpleFunc.indicatorConst 1 hs \u22ef 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns\u271d : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\nc : F'\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : (\u03bc.trim hm) s < \u22a4\n\u22a2 (condexpInd F' hm \u03bc s) c = indicatorConstLp 1 \u22ef \u22ef 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": [347, 9], "def_end_pos": [347, 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": [119, 7], "def_end_pos": [119, 21]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [147, 7], "def_end_pos": [147, 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns\u271d : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\nc : F'\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : (\u03bc.trim hm) s < \u22a4\n\u22a2 (condexpInd F' hm \u03bc s) c = indicatorConstLp 1 \u22ef \u22ef 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\n\u22a2 \u2200 \u2983f g : \u03b1 \u2192 F'\u2984 (hf : Mem\u2112p f 1 (\u03bc.trim hm)) (hg : Mem\u2112p g 1 (\u03bc.trim hm)),\n    Disjoint (Function.support f) (Function.support g) \u2192\n      (fun g =>\n            (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g) =\n              \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g))\n          (Mem\u2112p.toLp f hf) \u2192\n        (fun g =>\n              (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g) =\n                \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g))\n            (Mem\u2112p.toLp g hg) \u2192\n          (fun g =>\n              (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g) =\n                \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d\u00b9 g\u271d\u00b9 : \u03b1 \u2192 F'\ns : Set \u03b1\nf\u271d : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng\u271d : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\u271d\nhfg : f\u271d = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\u271d\nf g : \u03b1 \u2192 F'\nhf : Mem\u2112p f 1 (\u03bc.trim hm)\nhg : Mem\u2112p g 1 (\u03bc.trim hm)\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf_eq :\n  (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf)) =\n    \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf))\nhg_eq :\n  (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp g hg)) =\n    \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp g hg))\n\u22a2 (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)) =\n    \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (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": [968, 9], "def_end_pos": [968, 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d\u00b9 g\u271d\u00b9 : \u03b1 \u2192 F'\ns : Set \u03b1\nf\u271d : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng\u271d : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\u271d\nhfg : f\u271d = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\u271d\nf g : \u03b1 \u2192 F'\nhf : Mem\u2112p f 1 (\u03bc.trim hm)\nhg : Mem\u2112p g 1 (\u03bc.trim hm)\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf_eq :\n  (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf)) =\n    \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf))\nhg_eq :\n  (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp g hg)) =\n    \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp g hg))\n\u22a2 (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf + Mem\u2112p.toLp g hg)) =\n    \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (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 : RCLike \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 (\u03bc.trim hm)\nf\u271d\u00b9 g\u271d\u00b9 : \u03b1 \u2192 F'\ns : Set \u03b1\nf\u271d : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng\u271d : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\u271d\nhfg : f\u271d = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\u271d\nf g : \u03b1 \u2192 F'\nhf : Mem\u2112p f 1 (\u03bc.trim hm)\nhg : Mem\u2112p g 1 (\u03bc.trim hm)\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf_eq :\n  (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf)) =\n    \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf))\nhg_eq :\n  (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp g hg)) =\n    \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp g hg))\n\u22a2 (condexpL1CLM F' hm \u03bc)\n      \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf) +\n          (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp g hg)) =\n    \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf) + (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (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 : RCLike \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 (\u03bc.trim hm)\nf\u271d\u00b9 g\u271d\u00b9 : \u03b1 \u2192 F'\ns : Set \u03b1\nf\u271d : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng\u271d : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\u271d\nhfg : f\u271d = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\u271d\nf g : \u03b1 \u2192 F'\nhf : Mem\u2112p f 1 (\u03bc.trim hm)\nhg : Mem\u2112p g 1 (\u03bc.trim hm)\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf_eq :\n  (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf)) =\n    \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf))\nhg_eq :\n  (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp g hg)) =\n    \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp g hg))\n\u22a2 (condexpL1CLM F' hm \u03bc)\n      \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf) +\n          (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp g hg)) =\n    \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf) + (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (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 : RCLike \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 (\u03bc.trim hm)\nf\u271d\u00b9 g\u271d\u00b9 : \u03b1 \u2192 F'\ns : Set \u03b1\nf\u271d : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng\u271d : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\u271d\nhfg : f\u271d = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\u271d\nf g : \u03b1 \u2192 F'\nhf : Mem\u2112p f 1 (\u03bc.trim hm)\nhg : Mem\u2112p g 1 (\u03bc.trim hm)\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf_eq :\n  (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf)) =\n    \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf))\nhg_eq :\n  (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp g hg)) =\n    \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp g hg))\n\u22a2 (condexpL1CLM F' hm \u03bc)\n      (\u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf)) +\n        \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp g hg))) =\n    \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf)) +\n      \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (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/Group/Hom/Defs.lean", "def_pos": [308, 3], "def_end_pos": [308, 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d\u00b9 g\u271d\u00b9 : \u03b1 \u2192 F'\ns : Set \u03b1\nf\u271d : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng\u271d : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\u271d\nhfg : f\u271d = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\u271d\nf g : \u03b1 \u2192 F'\nhf : Mem\u2112p f 1 (\u03bc.trim hm)\nhg : Mem\u2112p g 1 (\u03bc.trim hm)\na\u271d : Disjoint (Function.support f) (Function.support g)\nhf_eq :\n  (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf)) =\n    \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf))\nhg_eq :\n  (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp g hg)) =\n    \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp g hg))\n\u22a2 (condexpL1CLM F' hm \u03bc)\n      (\u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf)) +\n        \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp g hg))) =\n    \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (Mem\u2112p.toLp f hf)) +\n      \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm (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": [1822, 9], "def_end_pos": [1822, 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\n\u22a2 IsClosed\n    {f |\n      (fun g =>\n          (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g) = \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\n\u22a2 Continuous fun f => (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\n\u22a2 Continuous fun f => \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm 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": [381, 5], "def_end_pos": [381, 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\n\u22a2 Continuous fun f => (condexpL1CLM F' hm \u03bc) \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\n\u22a2 Continuous \u21d1(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm"}, {"tactic": "exact LinearIsometryEquiv.continuous _", "annotated_tactic": ["exact <a>LinearIsometryEquiv.continuous</a> _", [{"full_name": "LinearIsometryEquiv.continuous", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [690, 19], "def_end_pos": [690, 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\n\u22a2 Continuous \u21d1(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm", "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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\n\u22a2 Continuous fun f => \u2191((lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\n\u22a2 Continuous \u21d1(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm"}, {"tactic": "exact LinearIsometryEquiv.continuous _", "annotated_tactic": ["exact <a>LinearIsometryEquiv.continuous</a> _", [{"full_name": "LinearIsometryEquiv.continuous", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [690, 19], "def_end_pos": [690, 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 : RCLike \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 (\u03bc.trim hm)\nf\u271d g\u271d : \u03b1 \u2192 F'\ns : Set \u03b1\nf : \u21a5(lpMeas F' \u211d m 1 \u03bc)\ng : \u21a5(Lp F' 1 (\u03bc.trim hm)) := (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm) f\nhfg : f = (lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm g\n\u22a2 Continuous \u21d1(lpMeasToLpTrimLie F' \u211d 1 \u03bc hm).symm", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "full_name": "MeasureTheory.Integrable.withDensity\u1d65_trim_absolutelyContinuous", "start": [217, 1], "end": [223, 70], "traced_tactics": [{"tactic": "refine VectorMeasure.AbsolutelyContinuous.mk fun j hj\u2081 hj\u2082 => ?_", "annotated_tactic": ["refine <a>VectorMeasure.AbsolutelyContinuous.mk</a> fun j hj\u2081 hj\u2082 => ?_", [{"full_name": "MeasureTheory.VectorMeasure.AbsolutelyContinuous.mk", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [1053, 9], "def_end_pos": [1053, 11]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : 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 g : \u03b1 \u2192 E\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhfi : Integrable f \u03bc\n\u22a2 (\u03bc.withDensity\u1d65 f).trim hm \u226a\u1d65 (\u03bc.trim hm).toENNRealVectorMeasure", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : 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 g : \u03b1 \u2192 E\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhfi : Integrable f \u03bc\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : \u2191(\u03bc.trim hm).toENNRealVectorMeasure j = 0\n\u22a2 \u2191((\u03bc.withDensity\u1d65 f).trim hm) j = 0"}, {"tactic": "rw [Measure.toENNRealVectorMeasure_apply_measurable hj\u2081, trim_measurableSet_eq hm hj\u2081] at hj\u2082", "annotated_tactic": ["rw [<a>Measure.toENNRealVectorMeasure_apply_measurable</a> hj\u2081, <a>trim_measurableSet_eq</a> hm hj\u2081] at hj\u2082", [{"full_name": "MeasureTheory.Measure.toENNRealVectorMeasure_apply_measurable", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [478, 9], "def_end_pos": [478, 48]}, {"full_name": "MeasureTheory.trim_measurableSet_eq", "def_path": "Mathlib/MeasureTheory/Measure/Trim.lean", "def_pos": [53, 9], "def_end_pos": [53, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : 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 g : \u03b1 \u2192 E\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhfi : Integrable f \u03bc\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : \u2191(\u03bc.trim hm).toENNRealVectorMeasure j = 0\n\u22a2 \u2191((\u03bc.withDensity\u1d65 f).trim hm) j = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : 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 g : \u03b1 \u2192 E\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhfi : Integrable f \u03bc\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : \u03bc j = 0\n\u22a2 \u2191((\u03bc.withDensity\u1d65 f).trim hm) j = 0"}, {"tactic": "rw [VectorMeasure.trim_measurableSet_eq hm hj\u2081, withDensity\u1d65_apply hfi (hm _ hj\u2081)]", "annotated_tactic": ["rw [<a>VectorMeasure.trim_measurableSet_eq</a> hm hj\u2081, <a>withDensity\u1d65_apply</a> hfi (hm _ hj\u2081)]", [{"full_name": "MeasureTheory.VectorMeasure.trim_measurableSet_eq", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [1269, 9], "def_end_pos": [1269, 30]}, {"full_name": "MeasureTheory.withDensity\u1d65_apply", "def_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "def_pos": [59, 9], "def_end_pos": [59, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : 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 g : \u03b1 \u2192 E\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhfi : Integrable f \u03bc\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : \u03bc j = 0\n\u22a2 \u2191((\u03bc.withDensity\u1d65 f).trim hm) j = 0", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : 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 g : \u03b1 \u2192 E\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhfi : Integrable f \u03bc\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : \u03bc j = 0\n\u22a2 \u222b (x : \u03b1) in j, f x \u2202\u03bc = 0"}, {"tactic": "simp only [Measure.restrict_eq_zero.mpr hj\u2082, integral_zero_measure]", "annotated_tactic": ["simp only [Measure.restrict_eq_zero.mpr hj\u2082, <a>integral_zero_measure</a>]", [{"full_name": "MeasureTheory.integral_zero_measure", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1603, 9], "def_end_pos": [1603, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm\u271d : 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 g : \u03b1 \u2192 E\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nhm : m \u2264 m0\nhfi : Integrable f \u03bc\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : \u03bc j = 0\n\u22a2 \u222b (x : \u03b1) in j, f x \u2202\u03bc = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Normed/Group/Hom.lean", "full_name": "NormedAddGroupHom.opNorm_eq_of_bounds", "start": [282, 1], "end": [286, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Coprime/Basic.lean", "full_name": "IsCoprime.mul_left", "start": [114, 1], "end": [121, 8], "traced_tactics": [{"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "R : Type u\ninst\u271d : CommSemiring R\nx y z : R\nH1 : IsCoprime x z\nH2 : IsCoprime y z\na b : R\nh1 : a * x + b * z = 1\nc d : R\nh2 : c * y + d * z = 1\n\u22a2 a * c * (x * y) + (a * x * d + b * c * y + b * d * z) * z = (a * x + b * z) * (c * y + d * z)", "state_after": "no goals"}, {"tactic": "rw [h1, h2, mul_one]", "annotated_tactic": ["rw [h1, h2, <a>mul_one</a>]", [{"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "R : Type u\ninst\u271d : CommSemiring R\nx y z : R\nH1 : IsCoprime x z\nH2 : IsCoprime y z\na b : R\nh1 : a * x + b * z = 1\nc d : R\nh2 : c * y + d * z = 1\n\u22a2 (a * x + b * z) * (c * y + d * z) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Hom/Defs.lean", "full_name": "NonUnitalRingHom.coe_mulHom_mk", "start": [134, 1], "end": [136, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean", "full_name": "MeasureTheory.snorm'_smul_le_mul_snorm'", "start": [295, 1], "end": [299, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.castAdd_injective", "start": [735, 1], "end": [735, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Order/Field.lean", "full_name": "Filter.Tendsto.atBot_mul_neg", "start": [102, 1], "end": [105, 58], "traced_tactics": [{"tactic": "have := (tendsto_neg_atBot_atTop.comp hf).atTop_mul_neg hC hg", "annotated_tactic": ["have := (tendsto_neg_atBot_atTop.comp hf).<a>atTop_mul_neg</a> hC hg", [{"full_name": "Filter.Tendsto.atTop_mul_neg", "def_path": "Mathlib/Topology/Algebra/Order/Field.lean", "def_pos": [79, 9], "def_end_pos": [79, 37]}]], "state_before": "\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\nl : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nC : \ud835\udd5c\nhC : C < 0\nhf : Tendsto f l atBot\nhg : Tendsto g l (\ud835\udcdd C)\n\u22a2 Tendsto (fun x => f x * g x) l atTop", "state_after": "\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\nl : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nC : \ud835\udd5c\nhC : C < 0\nhf : Tendsto f l atBot\nhg : Tendsto g l (\ud835\udcdd C)\nthis : Tendsto (fun x => (Neg.neg \u2218 f) x * g x) l atBot\n\u22a2 Tendsto (fun x => f x * g x) l atTop"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "full_name": "Asymptotics.IsBigO.prod_left_snd", "start": [1050, 1], "end": [1051, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/BoxIntegral/Partition/Filter.lean", "full_name": "BoxIntegral.IntegrationParams.toFilteriUnion_mono", "start": [459, 1], "end": [461, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/TensorProduct/Tower.lean", "full_name": "TensorProduct.AlgebraTensorModule.congr_symm", "start": [273, 1], "end": [273, 99], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Tactic/ReduceModChar.lean", "full_name": "Tactic.ReduceModChar.CharP.intCast_eq_mod", "start": [47, 1], "end": [51, 51], "traced_tactics": [{"tactic": "calc\n  (k : R) = \u2191(k % p + p * (k / p)) := by rw [Int.emod_add_ediv]\n  _ = \u2191(k % p) := by simp [CharP.cast_eq_zero R]", "annotated_tactic": ["calc\n    (k : R) = \u2191(k % p + p * (k / p)) := by rw [<a>Int.emod_add_ediv</a>]\n    _ = \u2191(k % p) := by simp [<a>CharP.cast_eq_zero</a> R]", [{"full_name": "Int.emod_add_ediv", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean", "def_pos": [201, 9], "def_end_pos": [201, 22]}, {"full_name": "CharP.cast_eq_zero", "def_path": "Mathlib/Algebra/CharP/Defs.lean", "def_pos": [60, 15], "def_end_pos": [60, 27]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9 : Ring R\np : \u2115\ninst\u271d : CharP R p\nk : \u2124\n\u22a2 \u2191k = \u2191(k % \u2191p)", "state_after": "no goals"}, {"tactic": "rw [Int.emod_add_ediv]", "annotated_tactic": ["rw [<a>Int.emod_add_ediv</a>]", [{"full_name": "Int.emod_add_ediv", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean", "def_pos": [201, 9], "def_end_pos": [201, 22]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9 : Ring R\np : \u2115\ninst\u271d : CharP R p\nk : \u2124\n\u22a2 \u2191k = \u2191(k % \u2191p + \u2191p * (k / \u2191p))", "state_after": "no goals"}, {"tactic": "simp [CharP.cast_eq_zero R]", "annotated_tactic": ["simp [<a>CharP.cast_eq_zero</a> R]", [{"full_name": "CharP.cast_eq_zero", "def_path": "Mathlib/Algebra/CharP/Defs.lean", "def_pos": [60, 15], "def_end_pos": [60, 27]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9 : Ring R\np : \u2115\ninst\u271d : CharP R p\nk : \u2124\n\u22a2 \u2191(k % \u2191p + \u2191p * (k / \u2191p)) = \u2191(k % \u2191p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/FLT/Four.lean", "full_name": "Fermat42.coprime_of_minimal", "start": [89, 1], "end": [105, 65], "traced_tactics": [{"tactic": "apply Int.gcd_eq_one_iff_coprime.mp", "annotated_tactic": ["apply Int.gcd_eq_one_iff_coprime.mp", []], "state_before": "a b c : \u2124\nh : Minimal a b c\n\u22a2 IsCoprime a b", "state_after": "a b c : \u2124\nh : Minimal a b c\n\u22a2 a.gcd b = 1"}, {"tactic": "by_contra hab", "annotated_tactic": ["by_contra hab", []], "state_before": "a b c : \u2124\nh : Minimal a b c\n\u22a2 a.gcd b = 1", "state_after": "a b c : \u2124\nh : Minimal a b c\nhab : \u00aca.gcd b = 1\n\u22a2 False"}, {"tactic": "obtain \u27e8p, hp, hpa, hpb\u27e9 := Nat.Prime.not_coprime_iff_dvd.mp hab", "annotated_tactic": ["obtain \u27e8p, hp, hpa, hpb\u27e9 := Nat.Prime.not_coprime_iff_dvd.mp hab", []], "state_before": "a b c : \u2124\nh : Minimal a b c\nhab : \u00aca.gcd b = 1\n\u22a2 False", "state_after": "case intro.intro.intro\na b c : \u2124\nh : Minimal a b c\nhab : \u00aca.gcd b = 1\np : \u2115\nhp : Nat.Prime p\nhpa : p \u2223 a.natAbs\nhpb : p \u2223 b.natAbs\n\u22a2 False"}, {"tactic": "obtain \u27e8a1, rfl\u27e9 := Int.natCast_dvd.mpr hpa", "annotated_tactic": ["obtain \u27e8a1, rfl\u27e9 := Int.natCast_dvd.mpr hpa", []], "state_before": "case intro.intro.intro\na b c : \u2124\nh : Minimal a b c\nhab : \u00aca.gcd b = 1\np : \u2115\nhp : Nat.Prime p\nhpa : p \u2223 a.natAbs\nhpb : p \u2223 b.natAbs\n\u22a2 False", "state_after": "case intro.intro.intro.intro\nb c : \u2124\np : \u2115\nhp : Nat.Prime p\nhpb : p \u2223 b.natAbs\na1 : \u2124\nh : Minimal (\u2191p * a1) b c\nhab : \u00ac(\u2191p * a1).gcd b = 1\nhpa : p \u2223 (\u2191p * a1).natAbs\n\u22a2 False"}, {"tactic": "obtain \u27e8b1, rfl\u27e9 := Int.natCast_dvd.mpr hpb", "annotated_tactic": ["obtain \u27e8b1, rfl\u27e9 := Int.natCast_dvd.mpr hpb", []], "state_before": "case intro.intro.intro.intro\nb c : \u2124\np : \u2115\nhp : Nat.Prime p\nhpb : p \u2223 b.natAbs\na1 : \u2124\nh : Minimal (\u2191p * a1) b c\nhab : \u00ac(\u2191p * a1).gcd b = 1\nhpa : p \u2223 (\u2191p * a1).natAbs\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro\nc : \u2124\np : \u2115\nhp : Nat.Prime p\na1 : \u2124\nhpa : p \u2223 (\u2191p * a1).natAbs\nb1 : \u2124\nhpb : p \u2223 (\u2191p * b1).natAbs\nh : Minimal (\u2191p * a1) (\u2191p * b1) c\nhab : \u00ac(\u2191p * a1).gcd (\u2191p * b1) = 1\n\u22a2 False"}, {"tactic": "have hpc : (p : \u2124) ^ 2 \u2223 c := by\n  rw [\u2190 Int.pow_dvd_pow_iff two_ne_zero, \u2190 h.1.2.2]\n  apply Dvd.intro (a1 ^ 4 + b1 ^ 4)\n  ring", "annotated_tactic": ["have hpc : (p : \u2124) ^ 2 \u2223 c := by\n    rw [\u2190 <a>Int.pow_dvd_pow_iff</a> <a>two_ne_zero</a>, \u2190 h.1.2.2]\n    apply <a>Dvd.intro</a> (a1 ^ 4 + b1 ^ 4)\n    ring", [{"full_name": "Int.pow_dvd_pow_iff", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [344, 9], "def_end_pos": [344, 24]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [65, 7], "def_end_pos": [65, 18]}, {"full_name": "Dvd.intro", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [49, 9], "def_end_pos": [49, 18]}]], "state_before": "case intro.intro.intro.intro.intro\nc : \u2124\np : \u2115\nhp : Nat.Prime p\na1 : \u2124\nhpa : p \u2223 (\u2191p * a1).natAbs\nb1 : \u2124\nhpb : p \u2223 (\u2191p * b1).natAbs\nh : Minimal (\u2191p * a1) (\u2191p * b1) c\nhab : \u00ac(\u2191p * a1).gcd (\u2191p * b1) = 1\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro\nc : \u2124\np : \u2115\nhp : Nat.Prime p\na1 : \u2124\nhpa : p \u2223 (\u2191p * a1).natAbs\nb1 : \u2124\nhpb : p \u2223 (\u2191p * b1).natAbs\nh : Minimal (\u2191p * a1) (\u2191p * b1) c\nhab : \u00ac(\u2191p * a1).gcd (\u2191p * b1) = 1\nhpc : \u2191p ^ 2 \u2223 c\n\u22a2 False"}, {"tactic": "obtain \u27e8c1, rfl\u27e9 := hpc", "annotated_tactic": ["obtain \u27e8c1, rfl\u27e9 := hpc", []], "state_before": "case intro.intro.intro.intro.intro\nc : \u2124\np : \u2115\nhp : Nat.Prime p\na1 : \u2124\nhpa : p \u2223 (\u2191p * a1).natAbs\nb1 : \u2124\nhpb : p \u2223 (\u2191p * b1).natAbs\nh : Minimal (\u2191p * a1) (\u2191p * b1) c\nhab : \u00ac(\u2191p * a1).gcd (\u2191p * b1) = 1\nhpc : \u2191p ^ 2 \u2223 c\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro.intro\np : \u2115\nhp : Nat.Prime p\na1 : \u2124\nhpa : p \u2223 (\u2191p * a1).natAbs\nb1 : \u2124\nhpb : p \u2223 (\u2191p * b1).natAbs\nhab : \u00ac(\u2191p * a1).gcd (\u2191p * b1) = 1\nc1 : \u2124\nh : Minimal (\u2191p * a1) (\u2191p * b1) (\u2191p ^ 2 * c1)\n\u22a2 False"}, {"tactic": "have hf : Fermat42 a1 b1 c1 :=\n  (Fermat42.mul (Int.natCast_ne_zero.mpr (Nat.Prime.ne_zero hp))).mpr h.1", "annotated_tactic": ["have hf : <a>Fermat42</a> a1 b1 c1 :=\n    (<a>Fermat42.mul</a> (Int.natCast_ne_zero.mpr (<a>Nat.Prime.ne_zero</a> hp))).<a>mpr</a> h.1", [{"full_name": "Fermat42", "def_path": "Mathlib/NumberTheory/FLT/Four.lean", "def_pos": [26, 5], "def_end_pos": [26, 13]}, {"full_name": "Fermat42.mul", "def_path": "Mathlib/NumberTheory/FLT/Four.lean", "def_pos": [38, 9], "def_end_pos": [38, 12]}, {"full_name": "Nat.Prime.ne_zero", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [61, 9], "def_end_pos": [61, 22]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}]], "state_before": "case intro.intro.intro.intro.intro.intro\np : \u2115\nhp : Nat.Prime p\na1 : \u2124\nhpa : p \u2223 (\u2191p * a1).natAbs\nb1 : \u2124\nhpb : p \u2223 (\u2191p * b1).natAbs\nhab : \u00ac(\u2191p * a1).gcd (\u2191p * b1) = 1\nc1 : \u2124\nh : Minimal (\u2191p * a1) (\u2191p * b1) (\u2191p ^ 2 * c1)\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro.intro\np : \u2115\nhp : Nat.Prime p\na1 : \u2124\nhpa : p \u2223 (\u2191p * a1).natAbs\nb1 : \u2124\nhpb : p \u2223 (\u2191p * b1).natAbs\nhab : \u00ac(\u2191p * a1).gcd (\u2191p * b1) = 1\nc1 : \u2124\nh : Minimal (\u2191p * a1) (\u2191p * b1) (\u2191p ^ 2 * c1)\nhf : Fermat42 a1 b1 c1\n\u22a2 False"}, {"tactic": "apply Nat.le_lt_asymm (h.2 _ _ _ hf)", "annotated_tactic": ["apply <a>Nat.le_lt_asymm</a> (h.2 _ _ _ hf)", [{"full_name": "Nat.le_lt_asymm", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [444, 19], "def_end_pos": [444, 30]}]], "state_before": "case intro.intro.intro.intro.intro.intro\np : \u2115\nhp : Nat.Prime p\na1 : \u2124\nhpa : p \u2223 (\u2191p * a1).natAbs\nb1 : \u2124\nhpb : p \u2223 (\u2191p * b1).natAbs\nhab : \u00ac(\u2191p * a1).gcd (\u2191p * b1) = 1\nc1 : \u2124\nh : Minimal (\u2191p * a1) (\u2191p * b1) (\u2191p ^ 2 * c1)\nhf : Fermat42 a1 b1 c1\n\u22a2 False", "state_after": "case intro.intro.intro.intro.intro.intro\np : \u2115\nhp : Nat.Prime p\na1 : \u2124\nhpa : p \u2223 (\u2191p * a1).natAbs\nb1 : \u2124\nhpb : p \u2223 (\u2191p * b1).natAbs\nhab : \u00ac(\u2191p * a1).gcd (\u2191p * b1) = 1\nc1 : \u2124\nh : Minimal (\u2191p * a1) (\u2191p * b1) (\u2191p ^ 2 * c1)\nhf : Fermat42 a1 b1 c1\n\u22a2 c1.natAbs < (\u2191p ^ 2 * c1).natAbs"}, {"tactic": "rw [Int.natAbs_mul, lt_mul_iff_one_lt_left, Int.natAbs_pow, Int.natAbs_ofNat]", "annotated_tactic": ["rw [<a>Int.natAbs_mul</a>, <a>lt_mul_iff_one_lt_left</a>, <a>Int.natAbs_pow</a>, <a>Int.natAbs_ofNat</a>]", [{"full_name": "Int.natAbs_mul", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Order.lean", "def_pos": [438, 9], "def_end_pos": [438, 19]}, {"full_name": "lt_mul_iff_one_lt_left", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [702, 9], "def_end_pos": [702, 31]}, {"full_name": "Int.natAbs_pow", "def_path": "Mathlib/Data/Int/Defs.lean", "def_pos": [432, 7], "def_end_pos": [432, 17]}, {"full_name": "Int.natAbs_ofNat", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Order.lean", "def_pos": [413, 17], "def_end_pos": [413, 29]}]], "state_before": "case intro.intro.intro.intro.intro.intro\np : \u2115\nhp : Nat.Prime p\na1 : \u2124\nhpa : p \u2223 (\u2191p * a1).natAbs\nb1 : \u2124\nhpb : p \u2223 (\u2191p * b1).natAbs\nhab : \u00ac(\u2191p * a1).gcd (\u2191p * b1) = 1\nc1 : \u2124\nh : Minimal (\u2191p * a1) (\u2191p * b1) (\u2191p ^ 2 * c1)\nhf : Fermat42 a1 b1 c1\n\u22a2 c1.natAbs < (\u2191p ^ 2 * c1).natAbs", "state_after": "case intro.intro.intro.intro.intro.intro\np : \u2115\nhp : Nat.Prime p\na1 : \u2124\nhpa : p \u2223 (\u2191p * a1).natAbs\nb1 : \u2124\nhpb : p \u2223 (\u2191p * b1).natAbs\nhab : \u00ac(\u2191p * a1).gcd (\u2191p * b1) = 1\nc1 : \u2124\nh : Minimal (\u2191p * a1) (\u2191p * b1) (\u2191p ^ 2 * c1)\nhf : Fermat42 a1 b1 c1\n\u22a2 1 < p ^ 2\n\ncase intro.intro.intro.intro.intro.intro\np : \u2115\nhp : Nat.Prime p\na1 : \u2124\nhpa : p \u2223 (\u2191p * a1).natAbs\nb1 : \u2124\nhpb : p \u2223 (\u2191p * b1).natAbs\nhab : \u00ac(\u2191p * a1).gcd (\u2191p * b1) = 1\nc1 : \u2124\nh : Minimal (\u2191p * a1) (\u2191p * b1) (\u2191p ^ 2 * c1)\nhf : Fermat42 a1 b1 c1\n\u22a2 0 < c1.natAbs"}, {"tactic": "rw [\u2190 Int.pow_dvd_pow_iff two_ne_zero, \u2190 h.1.2.2]", "annotated_tactic": ["rw [\u2190 <a>Int.pow_dvd_pow_iff</a> <a>two_ne_zero</a>, \u2190 h.1.2.2]", [{"full_name": "Int.pow_dvd_pow_iff", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [344, 9], "def_end_pos": [344, 24]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [65, 7], "def_end_pos": [65, 18]}]], "state_before": "c : \u2124\np : \u2115\nhp : Nat.Prime p\na1 : \u2124\nhpa : p \u2223 (\u2191p * a1).natAbs\nb1 : \u2124\nhpb : p \u2223 (\u2191p * b1).natAbs\nh : Minimal (\u2191p * a1) (\u2191p * b1) c\nhab : \u00ac(\u2191p * a1).gcd (\u2191p * b1) = 1\n\u22a2 \u2191p ^ 2 \u2223 c", "state_after": "c : \u2124\np : \u2115\nhp : Nat.Prime p\na1 : \u2124\nhpa : p \u2223 (\u2191p * a1).natAbs\nb1 : \u2124\nhpb : p \u2223 (\u2191p * b1).natAbs\nh : Minimal (\u2191p * a1) (\u2191p * b1) c\nhab : \u00ac(\u2191p * a1).gcd (\u2191p * b1) = 1\n\u22a2 (\u2191p ^ 2) ^ 2 \u2223 (\u2191p * a1) ^ 4 + (\u2191p * b1) ^ 4"}, {"tactic": "apply Dvd.intro (a1 ^ 4 + b1 ^ 4)", "annotated_tactic": ["apply <a>Dvd.intro</a> (a1 ^ 4 + b1 ^ 4)", [{"full_name": "Dvd.intro", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [49, 9], "def_end_pos": [49, 18]}]], "state_before": "c : \u2124\np : \u2115\nhp : Nat.Prime p\na1 : \u2124\nhpa : p \u2223 (\u2191p * a1).natAbs\nb1 : \u2124\nhpb : p \u2223 (\u2191p * b1).natAbs\nh : Minimal (\u2191p * a1) (\u2191p * b1) c\nhab : \u00ac(\u2191p * a1).gcd (\u2191p * b1) = 1\n\u22a2 (\u2191p ^ 2) ^ 2 \u2223 (\u2191p * a1) ^ 4 + (\u2191p * b1) ^ 4", "state_after": "c : \u2124\np : \u2115\nhp : Nat.Prime p\na1 : \u2124\nhpa : p \u2223 (\u2191p * a1).natAbs\nb1 : \u2124\nhpb : p \u2223 (\u2191p * b1).natAbs\nh : Minimal (\u2191p * a1) (\u2191p * b1) c\nhab : \u00ac(\u2191p * a1).gcd (\u2191p * b1) = 1\n\u22a2 (\u2191p ^ 2) ^ 2 * (a1 ^ 4 + b1 ^ 4) = (\u2191p * a1) ^ 4 + (\u2191p * b1) ^ 4"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "c : \u2124\np : \u2115\nhp : Nat.Prime p\na1 : \u2124\nhpa : p \u2223 (\u2191p * a1).natAbs\nb1 : \u2124\nhpb : p \u2223 (\u2191p * b1).natAbs\nh : Minimal (\u2191p * a1) (\u2191p * b1) c\nhab : \u00ac(\u2191p * a1).gcd (\u2191p * b1) = 1\n\u22a2 (\u2191p ^ 2) ^ 2 * (a1 ^ 4 + b1 ^ 4) = (\u2191p * a1) ^ 4 + (\u2191p * b1) ^ 4", "state_after": "no goals"}, {"tactic": "exact Nat.one_lt_pow two_ne_zero (Nat.Prime.one_lt hp)", "annotated_tactic": ["exact <a>Nat.one_lt_pow</a> <a>two_ne_zero</a> (<a>Nat.Prime.one_lt</a> hp)", [{"full_name": "Nat.one_lt_pow", "def_path": "Mathlib/Data/Nat/Defs.lean", "def_pos": [793, 7], "def_end_pos": [793, 17]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [65, 7], "def_end_pos": [65, 18]}, {"full_name": "Nat.Prime.one_lt", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [75, 9], "def_end_pos": [75, 21]}]], "state_before": "case intro.intro.intro.intro.intro.intro\np : \u2115\nhp : Nat.Prime p\na1 : \u2124\nhpa : p \u2223 (\u2191p * a1).natAbs\nb1 : \u2124\nhpb : p \u2223 (\u2191p * b1).natAbs\nhab : \u00ac(\u2191p * a1).gcd (\u2191p * b1) = 1\nc1 : \u2124\nh : Minimal (\u2191p * a1) (\u2191p * b1) (\u2191p ^ 2 * c1)\nhf : Fermat42 a1 b1 c1\n\u22a2 1 < p ^ 2", "state_after": "no goals"}, {"tactic": "exact Nat.pos_of_ne_zero (Int.natAbs_ne_zero.2 (ne_zero hf))", "annotated_tactic": ["exact <a>Nat.pos_of_ne_zero</a> (<a>Int.natAbs_ne_zero</a>.2 (<a>ne_zero</a> hf))", [{"full_name": "Nat.pos_of_ne_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [354, 19], "def_end_pos": [354, 33]}, {"full_name": "Int.natAbs_ne_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Order.lean", "def_pos": [1000, 9], "def_end_pos": [1000, 23]}, {"full_name": "Fermat42.ne_zero", "def_path": "Mathlib/NumberTheory/FLT/Four.lean", "def_pos": [58, 9], "def_end_pos": [58, 16]}]], "state_before": "case intro.intro.intro.intro.intro.intro\np : \u2115\nhp : Nat.Prime p\na1 : \u2124\nhpa : p \u2223 (\u2191p * a1).natAbs\nb1 : \u2124\nhpb : p \u2223 (\u2191p * b1).natAbs\nhab : \u00ac(\u2191p * a1).gcd (\u2191p * b1) = 1\nc1 : \u2124\nh : Minimal (\u2191p * a1) (\u2191p * b1) (\u2191p ^ 2 * c1)\nhf : Fermat42 a1 b1 c1\n\u22a2 0 < c1.natAbs", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Comma/Presheaf.lean", "full_name": "CategoryTheory.OverPresheafAux.counitForward_naturality\u2082", "start": [507, 1], "end": [516, 12], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nA : C\u1d52\u1d56 \u2964 Type v\nF : (CostructuredArrow yoneda A)\u1d52\u1d56 \u2964 Type v\nX : C\ns t : (CostructuredArrow yoneda A)\u1d52\u1d56\nf : t \u27f6 s\nx : F.obj t\n\u22a2 yoneda.map f.unop.left \u226b t.unop.hom = s.unop.hom", "state_after": "no goals"}, {"tactic": "refine OverArrows.ext <| YonedaCollection.ext (by simp) ?_", "annotated_tactic": ["refine <a>OverArrows.ext</a> <| <a>YonedaCollection.ext</a> (by simp) ?_", [{"full_name": "CategoryTheory.OverPresheafAux.OverArrows.ext", "def_path": "Mathlib/CategoryTheory/Comma/Presheaf.lean", "def_pos": [131, 7], "def_end_pos": [131, 10]}, {"full_name": "CategoryTheory.OverPresheafAux.YonedaCollection.ext", "def_path": "Mathlib/CategoryTheory/Comma/Presheaf.lean", "def_pos": [290, 7], "def_end_pos": [290, 10]}]], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nA : C\u1d52\u1d56 \u2964 Type v\nF : (CostructuredArrow yoneda A)\u1d52\u1d56 \u2964 Type v\nX : C\ns t : (CostructuredArrow yoneda A)\u1d52\u1d56\nf : t \u27f6 s\nx : F.obj t\n\u22a2 counitForward F s.unop (F.map f x) = (counitForward F t.unop x).map\u2082 f.unop.left \u22ef", "state_after": "C : Type u\ninst\u271d : Category.{v, u} C\nA : C\u1d52\u1d56 \u2964 Type v\nF : (CostructuredArrow yoneda A)\u1d52\u1d56 \u2964 Type v\nX : C\ns t : (CostructuredArrow yoneda A)\u1d52\u1d56\nf : t \u27f6 s\nx : F.obj t\n\u22a2 F.map (eqToHom \u22ef) (YonedaCollection.snd ((counitForward F t.unop x).map\u2082 f.unop.left \u22ef).val) =\n    YonedaCollection.snd (counitForward F s.unop (F.map f x)).val"}, {"tactic": "have : (CostructuredArrow.mkPrecomp t.unop.hom f.unop.left).op =\n    f \u226b eqToHom (by simp [\u2190 CostructuredArrow.eq_mk]) := by\n  apply Quiver.Hom.unop_inj\n  aesop_cat", "annotated_tactic": ["have : (<a>CostructuredArrow.mkPrecomp</a> t.unop.hom f.unop.left).<a>op</a> =\n      f \u226b <a>eqToHom</a> (by simp [\u2190 <a>CostructuredArrow.eq_mk</a>]) := by\n    apply <a>Quiver.Hom.unop_inj</a>\n    aesop_cat", [{"full_name": "CategoryTheory.CostructuredArrow.mkPrecomp", "def_path": "Mathlib/CategoryTheory/Comma/StructuredArrow.lean", "def_pos": [531, 5], "def_end_pos": [531, 14]}, {"full_name": "Quiver.Hom.op", "def_path": "Mathlib/Combinatorics/Quiver/Basic.lean", "def_pos": [152, 5], "def_end_pos": [152, 11]}, {"full_name": "CategoryTheory.eqToHom", "def_path": "Mathlib/CategoryTheory/EqToHom.lean", "def_pos": [43, 5], "def_end_pos": [43, 12]}, {"full_name": "CategoryTheory.CostructuredArrow.eq_mk", "def_path": "Mathlib/CategoryTheory/Comma/StructuredArrow.lean", "def_pos": [586, 9], "def_end_pos": [586, 14]}, {"full_name": "Quiver.Hom.unop_inj", "def_path": "Mathlib/CategoryTheory/Opposites.lean", "def_pos": [42, 9], "def_end_pos": [42, 28]}]], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nA : C\u1d52\u1d56 \u2964 Type v\nF : (CostructuredArrow yoneda A)\u1d52\u1d56 \u2964 Type v\nX : C\ns t : (CostructuredArrow yoneda A)\u1d52\u1d56\nf : t \u27f6 s\nx : F.obj t\n\u22a2 F.map (eqToHom \u22ef) (YonedaCollection.snd ((counitForward F t.unop x).map\u2082 f.unop.left \u22ef).val) =\n    YonedaCollection.snd (counitForward F s.unop (F.map f x)).val", "state_after": "C : Type u\ninst\u271d : Category.{v, u} C\nA : C\u1d52\u1d56 \u2964 Type v\nF : (CostructuredArrow yoneda A)\u1d52\u1d56 \u2964 Type v\nX : C\ns t : (CostructuredArrow yoneda A)\u1d52\u1d56\nf : t \u27f6 s\nx : F.obj t\nthis : (CostructuredArrow.mkPrecomp t.unop.hom f.unop.left).op = f \u226b eqToHom \u22ef\n\u22a2 F.map (eqToHom \u22ef) (YonedaCollection.snd ((counitForward F t.unop x).map\u2082 f.unop.left \u22ef).val) =\n    YonedaCollection.snd (counitForward F s.unop (F.map f x)).val"}, {"tactic": "aesop_cat", "annotated_tactic": ["aesop_cat", []], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nA : C\u1d52\u1d56 \u2964 Type v\nF : (CostructuredArrow yoneda A)\u1d52\u1d56 \u2964 Type v\nX : C\ns t : (CostructuredArrow yoneda A)\u1d52\u1d56\nf : t \u27f6 s\nx : F.obj t\nthis : (CostructuredArrow.mkPrecomp t.unop.hom f.unop.left).op = f \u226b eqToHom \u22ef\n\u22a2 F.map (eqToHom \u22ef) (YonedaCollection.snd ((counitForward F t.unop x).map\u2082 f.unop.left \u22ef).val) =\n    YonedaCollection.snd (counitForward F s.unop (F.map f x)).val", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nA : C\u1d52\u1d56 \u2964 Type v\nF : (CostructuredArrow yoneda A)\u1d52\u1d56 \u2964 Type v\nX : C\ns t : (CostructuredArrow yoneda A)\u1d52\u1d56\nf : t \u27f6 s\nx : F.obj t\n\u22a2 YonedaCollection.fst (counitForward F s.unop (F.map f x)).val =\n    YonedaCollection.fst ((counitForward F t.unop x).map\u2082 f.unop.left \u22ef).val", "state_after": "no goals"}, {"tactic": "simp [\u2190 CostructuredArrow.eq_mk]", "annotated_tactic": ["simp [\u2190 <a>CostructuredArrow.eq_mk</a>]", [{"full_name": "CategoryTheory.CostructuredArrow.eq_mk", "def_path": "Mathlib/CategoryTheory/Comma/StructuredArrow.lean", "def_pos": [586, 9], "def_end_pos": [586, 14]}]], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nA : C\u1d52\u1d56 \u2964 Type v\nF : (CostructuredArrow yoneda A)\u1d52\u1d56 \u2964 Type v\nX : C\ns t : (CostructuredArrow yoneda A)\u1d52\u1d56\nf : t \u27f6 s\nx : F.obj t\n\u22a2 s = { unop := CostructuredArrow.mk (yoneda.map f.unop.left \u226b t.unop.hom) }", "state_after": "no goals"}, {"tactic": "apply Quiver.Hom.unop_inj", "annotated_tactic": ["apply <a>Quiver.Hom.unop_inj</a>", [{"full_name": "Quiver.Hom.unop_inj", "def_path": "Mathlib/CategoryTheory/Opposites.lean", "def_pos": [42, 9], "def_end_pos": [42, 28]}]], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nA : C\u1d52\u1d56 \u2964 Type v\nF : (CostructuredArrow yoneda A)\u1d52\u1d56 \u2964 Type v\nX : C\ns t : (CostructuredArrow yoneda A)\u1d52\u1d56\nf : t \u27f6 s\nx : F.obj t\n\u22a2 (CostructuredArrow.mkPrecomp t.unop.hom f.unop.left).op = f \u226b eqToHom \u22ef", "state_after": "case a\nC : Type u\ninst\u271d : Category.{v, u} C\nA : C\u1d52\u1d56 \u2964 Type v\nF : (CostructuredArrow yoneda A)\u1d52\u1d56 \u2964 Type v\nX : C\ns t : (CostructuredArrow yoneda A)\u1d52\u1d56\nf : t \u27f6 s\nx : F.obj t\n\u22a2 (CostructuredArrow.mkPrecomp t.unop.hom f.unop.left).op.unop = (f \u226b eqToHom \u22ef).unop"}, {"tactic": "aesop_cat", "annotated_tactic": ["aesop_cat", []], "state_before": "case a\nC : Type u\ninst\u271d : Category.{v, u} C\nA : C\u1d52\u1d56 \u2964 Type v\nF : (CostructuredArrow yoneda A)\u1d52\u1d56 \u2964 Type v\nX : C\ns t : (CostructuredArrow yoneda A)\u1d52\u1d56\nf : t \u27f6 s\nx : F.obj t\n\u22a2 (CostructuredArrow.mkPrecomp t.unop.hom f.unop.left).op.unop = (f \u226b eqToHom \u22ef).unop", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Prime.lean", "full_name": "Nat.minFac_has_prop", "start": [324, 1], "end": [335, 30], "traced_tactics": [{"tactic": "by_cases n0 : n = 0", "annotated_tactic": ["by_cases n0 : n = 0", []], "state_before": "n\u271d n : \u2115\nn1 : n \u2260 1\n\u22a2 Nat.minFacProp n n.minFac", "state_after": "case pos\nn\u271d n : \u2115\nn1 : n \u2260 1\nn0 : n = 0\n\u22a2 Nat.minFacProp n n.minFac\n\ncase neg\nn\u271d n : \u2115\nn1 : n \u2260 1\nn0 : \u00acn = 0\n\u22a2 Nat.minFacProp n n.minFac"}, {"tactic": "have n2 : 2 \u2264 n := by\n  revert n0 n1\n  rcases n with (_ | _ | _) <;> simp [succ_le_succ]", "annotated_tactic": ["have n2 : 2 \u2264 n := by\n    revert n0 n1\n    rcases n with (_ | _ | _) <;> simp [<a>succ_le_succ</a>]", [{"full_name": "Nat.succ_le_succ", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1667, 9], "def_end_pos": [1667, 25]}]], "state_before": "case neg\nn\u271d n : \u2115\nn1 : n \u2260 1\nn0 : \u00acn = 0\n\u22a2 Nat.minFacProp n n.minFac", "state_after": "case neg\nn\u271d n : \u2115\nn1 : n \u2260 1\nn0 : \u00acn = 0\nn2 : 2 \u2264 n\n\u22a2 Nat.minFacProp n n.minFac"}, {"tactic": "simp only [minFac_eq, Nat.isUnit_iff]", "annotated_tactic": ["simp only [<a>minFac_eq</a>, <a>Nat.isUnit_iff</a>]", [{"full_name": "Nat.minFac_eq", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [287, 9], "def_end_pos": [287, 18]}, {"full_name": "Nat.isUnit_iff", "def_path": "Mathlib/Algebra/Group/Nat.lean", "def_pos": [181, 25], "def_end_pos": [181, 35]}]], "state_before": "case neg\nn\u271d n : \u2115\nn1 : n \u2260 1\nn0 : \u00acn = 0\nn2 : 2 \u2264 n\n\u22a2 Nat.minFacProp n n.minFac", "state_after": "case neg\nn\u271d n : \u2115\nn1 : n \u2260 1\nn0 : \u00acn = 0\nn2 : 2 \u2264 n\n\u22a2 Nat.minFacProp n (if 2 \u2223 n then 2 else n.minFacAux 3)"}, {"tactic": "by_cases d2 : 2 \u2223 n <;> simp [d2]", "annotated_tactic": ["by_cases d2 : 2 \u2223 n <;> simp [d2]", []], "state_before": "case neg\nn\u271d n : \u2115\nn1 : n \u2260 1\nn0 : \u00acn = 0\nn2 : 2 \u2264 n\n\u22a2 Nat.minFacProp n (if 2 \u2223 n then 2 else n.minFacAux 3)", "state_after": "case pos\nn\u271d n : \u2115\nn1 : n \u2260 1\nn0 : \u00acn = 0\nn2 : 2 \u2264 n\nd2 : 2 \u2223 n\n\u22a2 Nat.minFacProp n 2\n\ncase neg\nn\u271d n : \u2115\nn1 : n \u2260 1\nn0 : \u00acn = 0\nn2 : 2 \u2264 n\nd2 : \u00ac2 \u2223 n\n\u22a2 Nat.minFacProp n (n.minFacAux 3)"}, {"tactic": "simp [n0, minFacProp, GE.ge]", "annotated_tactic": ["simp [n0, <a>minFacProp</a>, <a>GE.ge</a>]", [{"full_name": "_private.Mathlib.Data.Nat.Prime.0.Nat.minFacProp", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [290, 13], "def_end_pos": [290, 23]}, {"full_name": "GE.ge", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1122, 18], "def_end_pos": [1122, 23]}]], "state_before": "case pos\nn\u271d n : \u2115\nn1 : n \u2260 1\nn0 : n = 0\n\u22a2 Nat.minFacProp n n.minFac", "state_after": "no goals"}, {"tactic": "revert n0 n1", "annotated_tactic": ["revert n0 n1", []], "state_before": "n\u271d n : \u2115\nn1 : n \u2260 1\nn0 : \u00acn = 0\n\u22a2 2 \u2264 n", "state_after": "n\u271d n : \u2115\n\u22a2 n \u2260 1 \u2192 \u00acn = 0 \u2192 2 \u2264 n"}, {"tactic": "rcases n with (_ | _ | _) <;> simp [succ_le_succ]", "annotated_tactic": ["rcases n with (_ | _ | _) <;> simp [<a>succ_le_succ</a>]", [{"full_name": "Nat.succ_le_succ", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1667, 9], "def_end_pos": [1667, 25]}]], "state_before": "n\u271d n : \u2115\n\u22a2 n \u2260 1 \u2192 \u00acn = 0 \u2192 2 \u2264 n", "state_after": "no goals"}, {"tactic": "exact \u27e8le_rfl, d2, fun k k2 _ => k2\u27e9", "annotated_tactic": ["exact \u27e8<a>le_rfl</a>, d2, fun k k2 _ => k2\u27e9", [{"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}]], "state_before": "case pos\nn\u271d n : \u2115\nn1 : n \u2260 1\nn0 : \u00acn = 0\nn2 : 2 \u2264 n\nd2 : 2 \u2223 n\n\u22a2 Nat.minFacProp n 2", "state_after": "no goals"}, {"tactic": "refine\n  minFacAux_has_prop n2 3 0 rfl fun m m2 d => (Nat.eq_or_lt_of_le m2).resolve_left (mt ?_ d2)", "annotated_tactic": ["refine\n      <a>minFacAux_has_prop</a> n2 3 0 <a>rfl</a> fun m m2 d => (<a>Nat.eq_or_lt_of_le</a> m2).<a>resolve_left</a> (<a>mt</a> ?_ d2)", [{"full_name": "Nat.minFacAux_has_prop", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [293, 9], "def_end_pos": [293, 27]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "Nat.eq_or_lt_of_le", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1719, 19], "def_end_pos": [1719, 37]}, {"full_name": "Or.resolve_left", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [555, 9], "def_end_pos": [555, 24]}, {"full_name": "mt", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [647, 9], "def_end_pos": [647, 11]}]], "state_before": "case neg\nn\u271d n : \u2115\nn1 : n \u2260 1\nn0 : \u00acn = 0\nn2 : 2 \u2264 n\nd2 : \u00ac2 \u2223 n\n\u22a2 Nat.minFacProp n (n.minFacAux 3)", "state_after": "case neg\nn\u271d n : \u2115\nn1 : n \u2260 1\nn0 : \u00acn = 0\nn2 : 2 \u2264 n\nd2 : \u00ac2 \u2223 n\nm : \u2115\nm2 : 2 \u2264 m\nd : m \u2223 n\n\u22a2 2 = m \u2192 2 \u2223 n"}, {"tactic": "exact fun e => e.symm \u25b8 d", "annotated_tactic": ["exact fun e => e.symm \u25b8 d", []], "state_before": "case neg\nn\u271d n : \u2115\nn1 : n \u2260 1\nn0 : \u00acn = 0\nn2 : 2 \u2264 n\nd2 : \u00ac2 \u2223 n\nm : \u2115\nm2 : 2 \u2264 m\nd : m \u2223 n\n\u22a2 2 = m \u2192 2 \u2223 n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "full_name": "MeasureTheory.snorm'_mono_nnnorm_ae", "start": [375, 1], "end": [380, 9], "traced_tactics": [{"tactic": "simp only [snorm']", "annotated_tactic": ["simp only [<a>snorm'</a>]", [{"full_name": "MeasureTheory.snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [71, 5], "def_end_pos": [71, 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\nf : \u03b1 \u2192 F\ng : \u03b1 \u2192 G\nhq : 0 \u2264 q\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 \u2016g x\u2016\u208a\n\u22a2 snorm' f q \u03bc \u2264 snorm' g q \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\ng : \u03b1 \u2192 G\nhq : 0 \u2264 q\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 \u2016g x\u2016\u208a\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) \u2264 (\u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q)"}, {"tactic": "gcongr ?_ ^ (1/q)", "annotated_tactic": ["gcongr ?_ ^ (1/q)", []], "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\ng : \u03b1 \u2192 G\nhq : 0 \u2264 q\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 \u2016g x\u2016\u208a\n\u22a2 (\u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q) \u2264 (\u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ q \u2202\u03bc) ^ (1 / q)", "state_after": "case h\u2081\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \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\nhq : 0 \u2264 q\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 \u2016g x\u2016\u208a\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ q \u2202\u03bc"}, {"tactic": "refine lintegral_mono_ae (h.mono fun x hx => ?_)", "annotated_tactic": ["refine <a>lintegral_mono_ae</a> (h.mono fun x hx => ?_)", [{"full_name": "MeasureTheory.lintegral_mono_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [274, 9], "def_end_pos": [274, 26]}]], "state_before": "case h\u2081\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \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\nhq : 0 \u2264 q\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 \u2016g x\u2016\u208a\n\u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ q \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), \u2191\u2016g a\u2016\u208a ^ q \u2202\u03bc", "state_after": "case h\u2081\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \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\nhq : 0 \u2264 q\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 \u2016g x\u2016\u208a\nx : \u03b1\nhx : \u2016f x\u2016\u208a \u2264 \u2016g x\u2016\u208a\n\u22a2 \u2191\u2016f x\u2016\u208a ^ q \u2264 \u2191\u2016g x\u2016\u208a ^ q"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "case h\u2081\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \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\nhq : 0 \u2264 q\nh : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2016f x\u2016\u208a \u2264 \u2016g x\u2016\u208a\nx : \u03b1\nhx : \u2016f x\u2016\u208a \u2264 \u2016g x\u2016\u208a\n\u22a2 \u2191\u2016f x\u2016\u208a ^ q \u2264 \u2191\u2016g x\u2016\u208a ^ q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/Deriv/Comp.lean", "full_name": "HasDerivAt.iterate", "start": [317, 18], "end": [319, 74], "traced_tactics": [{"tactic": "rwa [hx] at this", "annotated_tactic": ["rwa [hx] at this", []], "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nF : Type v\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\nE : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\nf\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F\nf'\u271d f\u2080' f\u2081' g' : F\nx : \ud835\udd5c\ns t : Set \ud835\udd5c\nL L\u2081 L\u2082 : Filter \ud835\udd5c\n\ud835\udd5c' : Type u_1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c'\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c' F\ninst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F\ns' t' : Set \ud835\udd5c'\nh : \ud835\udd5c \u2192 \ud835\udd5c'\nh\u2081 : \ud835\udd5c \u2192 \ud835\udd5c\nh\u2082 : \ud835\udd5c' \u2192 \ud835\udd5c'\nh' h\u2082' : \ud835\udd5c'\nh\u2081' : \ud835\udd5c\ng\u2081 : \ud835\udd5c' \u2192 F\ng\u2081' : F\nL' : Filter \ud835\udd5c'\ny : \ud835\udd5c'\nf : \ud835\udd5c \u2192 \ud835\udd5c\nf' : \ud835\udd5c\nhf : HasDerivAt f f' x\nhx : f x = x\nn : \u2115\nthis : Tendsto f (\ud835\udcdd x) (\ud835\udcdd (f x))\n\u22a2 Tendsto f (\ud835\udcdd x) (\ud835\udcdd x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order.lean", "full_name": "continuous_iSup_dom", "start": [755, 1], "end": [757, 53], "traced_tactics": [{"tactic": "simp only [continuous_iff_le_induced, iSup_le_iff]", "annotated_tactic": ["simp only [<a>continuous_iff_le_induced</a>, <a>iSup_le_iff</a>]", [{"full_name": "continuous_iff_le_induced", "def_path": "Mathlib/Topology/Order.lean", "def_pos": [683, 9], "def_end_pos": [683, 34]}, {"full_name": "iSup_le_iff", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [849, 9], "def_end_pos": [849, 20]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type u_1\nf : \u03b1 \u2192 \u03b2\n\u03b9 : Sort u_2\nt\u2081 : \u03b9 \u2192 TopologicalSpace \u03b1\nt\u2082 : TopologicalSpace \u03b2\n\u22a2 Continuous f \u2194 \u2200 (i : \u03b9), Continuous f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean", "full_name": "DifferentiableOn.csin", "start": [396, 1], "end": [397, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "closure_diff_frontier", "start": [686, 1], "end": [687, 93], "traced_tactics": [{"tactic": "rw [frontier, diff_diff_right_self, inter_eq_self_of_subset_right interior_subset_closure]", "annotated_tactic": ["rw [<a>frontier</a>, <a>diff_diff_right_self</a>, <a>inter_eq_self_of_subset_right</a> <a>interior_subset_closure</a>]", [{"full_name": "frontier", "def_path": "Mathlib/Topology/Defs/Basic.lean", "def_pos": [121, 5], "def_end_pos": [121, 13]}, {"full_name": "Set.diff_diff_right_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [2035, 9], "def_end_pos": [2035, 29]}, {"full_name": "Set.inter_eq_self_of_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [960, 9], "def_end_pos": [960, 38]}, {"full_name": "interior_subset_closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [541, 9], "def_end_pos": [541, 32]}]], "state_before": "X : Type u\nY : Type v\n\u03b9 : Sort w\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nx : X\ns\u271d s\u2081 s\u2082 t : Set X\np p\u2081 p\u2082 : X \u2192 Prop\ninst\u271d : TopologicalSpace X\ns : Set X\n\u22a2 closure s \\ frontier s = interior s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/BigOperators/Group/Finset.lean", "full_name": "Finset.one_le_prod''", "start": [143, 1], "end": [144, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/UnionFind/Lemmas.lean", "full_name": "Batteries.UnionFind.parent_empty", "start": [11, 9], "end": [11, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Kleene.lean", "full_name": "Prod.kstar_def", "start": [307, 1], "end": [308, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Vandermonde.lean", "full_name": "Matrix.eq_zero_of_forall_index_sum_pow_mul_eq_zero", "start": [170, 1], "end": [173, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicTopology/FundamentalGroupoid/SimplyConnected.lean", "full_name": "simply_connected_iff_paths_homotopic", "start": [85, 1], "end": [90, 96], "traced_tactics": [{"tactic": "intro", "annotated_tactic": ["intro", []], "state_before": "Y : Type u_1\ninst\u271d : TopologicalSpace Y\n\u22a2 SimplyConnectedSpace Y \u2192 PathConnectedSpace Y \u2227 \u2200 (x y : Y), Subsingleton (Path.Homotopic.Quotient x y)", "state_after": "Y : Type u_1\ninst\u271d : TopologicalSpace Y\na\u271d : SimplyConnectedSpace Y\n\u22a2 PathConnectedSpace Y \u2227 \u2200 (x y : Y), Subsingleton (Path.Homotopic.Quotient x y)"}, {"tactic": "constructor <;> infer_instance", "annotated_tactic": ["constructor <;> infer_instance", []], "state_before": "Y : Type u_1\ninst\u271d : TopologicalSpace Y\na\u271d : SimplyConnectedSpace Y\n\u22a2 PathConnectedSpace Y \u2227 \u2200 (x y : Y), Subsingleton (Path.Homotopic.Quotient x y)", "state_after": "no goals"}, {"tactic": "cases h", "annotated_tactic": ["cases h", []], "state_before": "Y : Type u_1\ninst\u271d : TopologicalSpace Y\nh : PathConnectedSpace Y \u2227 \u2200 (x y : Y), Subsingleton (Path.Homotopic.Quotient x y)\n\u22a2 SimplyConnectedSpace Y", "state_after": "case intro\nY : Type u_1\ninst\u271d : TopologicalSpace Y\nleft\u271d : PathConnectedSpace Y\nright\u271d : \u2200 (x y : Y), Subsingleton (Path.Homotopic.Quotient x y)\n\u22a2 SimplyConnectedSpace Y"}, {"tactic": "rw [simply_connected_iff_unique_homotopic]", "annotated_tactic": ["rw [<a>simply_connected_iff_unique_homotopic</a>]", [{"full_name": "simply_connected_iff_unique_homotopic", "def_path": "Mathlib/AlgebraicTopology/FundamentalGroupoid/SimplyConnected.lean", "def_pos": [42, 9], "def_end_pos": [42, 46]}]], "state_before": "case intro\nY : Type u_1\ninst\u271d : TopologicalSpace Y\nleft\u271d : PathConnectedSpace Y\nright\u271d : \u2200 (x y : Y), Subsingleton (Path.Homotopic.Quotient x y)\n\u22a2 SimplyConnectedSpace Y", "state_after": "case intro\nY : Type u_1\ninst\u271d : TopologicalSpace Y\nleft\u271d : PathConnectedSpace Y\nright\u271d : \u2200 (x y : Y), Subsingleton (Path.Homotopic.Quotient x y)\n\u22a2 Nonempty Y \u2227 \u2200 (x y : Y), Nonempty (Unique (Path.Homotopic.Quotient x y))"}, {"tactic": "exact \u27e8inferInstance, fun x y => \u27e8uniqueOfSubsingleton \u27e6PathConnectedSpace.somePath x y\u27e7\u27e9\u27e9", "annotated_tactic": ["exact \u27e8<a>inferInstance</a>, fun x y => \u27e8<a>uniqueOfSubsingleton</a> \u27e6<a>PathConnectedSpace.somePath</a> x y\u27e7\u27e9\u27e9", [{"full_name": "inferInstance", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [99, 8], "def_end_pos": [99, 21]}, {"full_name": "uniqueOfSubsingleton", "def_path": "Mathlib/Logic/Unique.lean", "def_pos": [82, 8], "def_end_pos": [82, 28]}, {"full_name": "PathConnectedSpace.somePath", "def_path": "Mathlib/Topology/Connected/PathConnected.lean", "def_pos": [1151, 5], "def_end_pos": [1151, 13]}]], "state_before": "case intro\nY : Type u_1\ninst\u271d : TopologicalSpace Y\nleft\u271d : PathConnectedSpace Y\nright\u271d : \u2200 (x y : Y), Subsingleton (Path.Homotopic.Quotient x y)\n\u22a2 Nonempty Y \u2227 \u2200 (x y : Y), Nonempty (Unique (Path.Homotopic.Quotient x y))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Sigma.lean", "full_name": "List.lookupAll_length_le_one", "start": [311, 1], "end": [315, 39], "traced_tactics": [{"tactic": "have := Nodup.sublist ((lookupAll_sublist a l).map _) h", "annotated_tactic": ["have := <a>Nodup.sublist</a> ((<a>lookupAll_sublist</a> a l).<a>map</a> _) h", [{"full_name": "List.Nodup.sublist", "def_path": "Mathlib/Data/List/Nodup.lean", "def_pos": [75, 19], "def_end_pos": [75, 32]}, {"full_name": "List.lookupAll_sublist", "def_path": "Mathlib/Data/List/Sigma.lean", "def_pos": [300, 9], "def_end_pos": [300, 26]}, {"full_name": "List.Sublist.map", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [2824, 9], "def_end_pos": [2824, 20]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : l.NodupKeys\n\u22a2 (lookupAll a l).length \u2264 1", "state_after": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : l.NodupKeys\nthis : (map Sigma.fst (map (Sigma.mk a) (lookupAll a l))).Nodup\n\u22a2 (lookupAll a l).length \u2264 1"}, {"tactic": "rw [map_map] at this", "annotated_tactic": ["rw [<a>map_map</a>] at this", [{"full_name": "List.map_map", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [842, 17], "def_end_pos": [842, 24]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : l.NodupKeys\nthis : (map Sigma.fst (map (Sigma.mk a) (lookupAll a l))).Nodup\n\u22a2 (lookupAll a l).length \u2264 1", "state_after": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : l.NodupKeys\nthis : (map (Sigma.fst \u2218 Sigma.mk a) (lookupAll a l)).Nodup\n\u22a2 (lookupAll a l).length \u2264 1"}, {"tactic": "rwa [\u2190 nodup_replicate, \u2190 map_const]", "annotated_tactic": ["rwa [\u2190 <a>nodup_replicate</a>, \u2190 <a>map_const</a>]", [{"full_name": "List.nodup_replicate", "def_path": "Mathlib/Data/List/Nodup.lean", "def_pos": [197, 9], "def_end_pos": [197, 24]}, {"full_name": "List.map_const", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [1356, 17], "def_end_pos": [1356, 26]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\nl\u271d l\u2081 l\u2082 : List (Sigma \u03b2)\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl : List (Sigma \u03b2)\nh : l.NodupKeys\nthis : (map (Sigma.fst \u2218 Sigma.mk a) (lookupAll a l)).Nodup\n\u22a2 (lookupAll a l).length \u2264 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Modular.lean", "full_name": "ModularGroup.abs_c_le_one", "start": [478, 1], "end": [509, 42], "traced_tactics": [{"tactic": "let c' : \u2124 := (\u2191\u2098g) 1 0", "annotated_tactic": ["let c' : \u2124 := (\u2191\u2098g) 1 0", []], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\n\u22a2 |\u2191g 1 0| \u2264 1", "state_after": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\n\u22a2 |\u2191g 1 0| \u2264 1"}, {"tactic": "let c : \u211d := (c' : \u211d)", "annotated_tactic": ["let c : \u211d := (c' : \u211d)", []], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\n\u22a2 |\u2191g 1 0| \u2264 1", "state_after": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\n\u22a2 |\u2191g 1 0| \u2264 1"}, {"tactic": "suffices 3 * c ^ 2 < 4 by\n  rw [\u2190 Int.cast_pow, \u2190 Int.cast_three, \u2190 Int.cast_four, \u2190 Int.cast_mul, Int.cast_lt] at this\n  replace this : c' ^ 2 \u2264 1 ^ 2 := by linarith\n  rwa [sq_le_sq, abs_one] at this", "annotated_tactic": ["suffices 3 * c ^ 2 < 4 by\n    rw [\u2190 <a>Int.cast_pow</a>, \u2190 <a>Int.cast_three</a>, \u2190 <a>Int.cast_four</a>, \u2190 <a>Int.cast_mul</a>, <a>Int.cast_lt</a>] at this\n    replace this : c' ^ 2 \u2264 1 ^ 2 := by linarith\n    rwa [<a>sq_le_sq</a>, <a>abs_one</a>] at this", [{"full_name": "Int.cast_pow", "def_path": "Mathlib/Algebra/Ring/Int.lean", "def_pos": [68, 26], "def_end_pos": [68, 34]}, {"full_name": "Int.cast_three", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [146, 9], "def_end_pos": [146, 19]}, {"full_name": "Int.cast_four", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [149, 9], "def_end_pos": [149, 18]}, {"full_name": "Int.cast_mul", "def_path": "Mathlib/Algebra/Ring/Int.lean", "def_pos": [58, 7], "def_end_pos": [58, 15]}, {"full_name": "Int.cast_lt", "def_path": "Mathlib/Algebra/Order/Ring/Cast.lean", "def_pos": [57, 26], "def_end_pos": [57, 33]}, {"full_name": "sq_le_sq", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [125, 7], "def_end_pos": [125, 15]}, {"full_name": "abs_one", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [42, 15], "def_end_pos": [42, 22]}]], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\n\u22a2 |\u2191g 1 0| \u2264 1", "state_after": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\n\u22a2 3 * c ^ 2 < 4"}, {"tactic": "intro hc", "annotated_tactic": ["intro hc", []], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\n\u22a2 c \u2260 0 \u2192 9 * c ^ 4 < 16", "state_after": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : c \u2260 0\n\u22a2 9 * c ^ 4 < 16"}, {"tactic": "replace hc : 0 < c ^ 4 := by\n  change 0 < c ^ (2 * 2); rw [pow_mul]; apply sq_pos_of_pos (sq_pos_of_ne_zero hc)", "annotated_tactic": ["replace hc : 0 < c ^ 4 := by\n    change 0 < c ^ (2 * 2); rw [<a>pow_mul</a>]; apply <a>sq_pos_of_pos</a> (<a>sq_pos_of_ne_zero</a> hc)", [{"full_name": "pow_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [713, 32], "def_end_pos": [713, 39]}, {"full_name": "sq_pos_of_pos", "def_path": "Mathlib/Algebra/Order/Ring/Basic.lean", "def_pos": [181, 9], "def_end_pos": [181, 22]}, {"full_name": "sq_pos_of_ne_zero", "def_path": "Mathlib/Algebra/Order/Ring/Basic.lean", "def_pos": [387, 11], "def_end_pos": [387, 28]}]], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : c \u2260 0\n\u22a2 9 * c ^ 4 < 16", "state_after": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : 0 < c ^ 4\n\u22a2 9 * c ^ 4 < 16"}, {"tactic": "have h\u2081 :=\n  mul_lt_mul_of_pos_right\n    (mul_lt_mul'' (three_lt_four_mul_im_sq_of_mem_fdo hg) (three_lt_four_mul_im_sq_of_mem_fdo hz)\n      (by linarith) (by linarith))\n    hc", "annotated_tactic": ["have h\u2081 :=\n    <a>mul_lt_mul_of_pos_right</a>\n      (<a>mul_lt_mul''</a> (<a>three_lt_four_mul_im_sq_of_mem_fdo</a> hg) (<a>three_lt_four_mul_im_sq_of_mem_fdo</a> hz)\n        (by linarith) (by linarith))\n      hc", [{"full_name": "mul_lt_mul_of_pos_right", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [220, 9], "def_end_pos": [220, 32]}, {"full_name": "mul_lt_mul''", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [582, 9], "def_end_pos": [582, 21]}, {"full_name": "ModularGroup.three_lt_four_mul_im_sq_of_mem_fdo", "def_path": "Mathlib/NumberTheory/Modular.lean", "def_pos": [414, 9], "def_end_pos": [414, 43]}, {"full_name": "ModularGroup.three_lt_four_mul_im_sq_of_mem_fdo", "def_path": "Mathlib/NumberTheory/Modular.lean", "def_pos": [414, 9], "def_end_pos": [414, 43]}]], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : 0 < c ^ 4\n\u22a2 9 * c ^ 4 < 16", "state_after": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : 0 < c ^ 4\nh\u2081 : 3 * 3 * c ^ 4 < 4 * (g \u2022 z).im ^ 2 * (4 * z.im ^ 2) * c ^ 4\n\u22a2 9 * c ^ 4 < 16"}, {"tactic": "have h\u2082 : (c * z.im) ^ 4 / normSq (denom (\u2191g) z) ^ 2 \u2264 1 :=\n  div_le_one_of_le\n    (pow_four_le_pow_two_of_pow_two_le (UpperHalfPlane.c_mul_im_sq_le_normSq_denom z g))\n    (sq_nonneg _)", "annotated_tactic": ["have h\u2082 : (c * z.im) ^ 4 / <a>normSq</a> (<a>denom</a> (\u2191g) z) ^ 2 \u2264 1 :=\n    <a>div_le_one_of_le</a>\n      (<a>pow_four_le_pow_two_of_pow_two_le</a> (<a>UpperHalfPlane.c_mul_im_sq_le_normSq_denom</a> z g))\n      (<a>sq_nonneg</a> _)", [{"full_name": "Complex.normSq", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [630, 5], "def_end_pos": [630, 11]}, {"full_name": "UpperHalfPlane.denom", "def_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean", "def_pos": [201, 5], "def_end_pos": [201, 10]}, {"full_name": "div_le_one_of_le", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [165, 9], "def_end_pos": [165, 25]}, {"full_name": "pow_four_le_pow_two_of_pow_two_le", "def_path": "Mathlib/Algebra/Order/Ring/Basic.lean", "def_pos": [392, 7], "def_end_pos": [392, 40]}, {"full_name": "UpperHalfPlane.c_mul_im_sq_le_normSq_denom", "def_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean", "def_pos": [401, 9], "def_end_pos": [401, 36]}, {"full_name": "sq_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [1160, 7], "def_end_pos": [1160, 16]}]], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : 0 < c ^ 4\nh\u2081 : 3 * 3 * c ^ 4 < 4 * (g \u2022 z).im ^ 2 * (4 * z.im ^ 2) * c ^ 4\n\u22a2 9 * c ^ 4 < 16", "state_after": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : 0 < c ^ 4\nh\u2081 : 3 * 3 * c ^ 4 < 4 * (g \u2022 z).im ^ 2 * (4 * z.im ^ 2) * c ^ 4\nh\u2082 : (c * z.im) ^ 4 / normSq (denom (\u2191g) z) ^ 2 \u2264 1\n\u22a2 9 * c ^ 4 < 16"}, {"tactic": "let nsq := normSq (denom g z)", "annotated_tactic": ["let nsq := <a>normSq</a> (<a>denom</a> g z)", [{"full_name": "Complex.normSq", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [630, 5], "def_end_pos": [630, 11]}, {"full_name": "UpperHalfPlane.denom", "def_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean", "def_pos": [201, 5], "def_end_pos": [201, 10]}]], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : 0 < c ^ 4\nh\u2081 : 3 * 3 * c ^ 4 < 4 * (g \u2022 z).im ^ 2 * (4 * z.im ^ 2) * c ^ 4\nh\u2082 : (c * z.im) ^ 4 / normSq (denom (\u2191g) z) ^ 2 \u2264 1\n\u22a2 9 * c ^ 4 < 16", "state_after": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : 0 < c ^ 4\nh\u2081 : 3 * 3 * c ^ 4 < 4 * (g \u2022 z).im ^ 2 * (4 * z.im ^ 2) * c ^ 4\nh\u2082 : (c * z.im) ^ 4 / normSq (denom (\u2191g) z) ^ 2 \u2264 1\nnsq : \u211d := normSq (denom (\u2191g) z)\n\u22a2 9 * c ^ 4 < 16"}, {"tactic": "calc\n  9 * c ^ 4 < c ^ 4 * z.im ^ 2 * (g \u2022 z).im ^ 2 * 16 := by linarith\n  _ = c ^ 4 * z.im ^ 4 / nsq ^ 2 * 16 := by\n    rw [ModularGroup.im_smul_eq_div_normSq, div_pow]\n    ring\n  _ \u2264 16 := by rw [\u2190 mul_pow]; linarith", "annotated_tactic": ["calc\n    9 * c ^ 4 < c ^ 4 * z.im ^ 2 * (g \u2022 z).<a>im</a> ^ 2 * 16 := by linarith\n    _ = c ^ 4 * z.im ^ 4 / nsq ^ 2 * 16 := by\n      rw [<a>ModularGroup.im_smul_eq_div_normSq</a>, <a>div_pow</a>]\n      ring\n    _ \u2264 16 := by rw [\u2190 <a>mul_pow</a>]; linarith", [{"full_name": "UpperHalfPlane.im", "def_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean", "def_pos": [76, 5], "def_end_pos": [76, 7]}, {"full_name": "UpperHalfPlane.ModularGroup.im_smul_eq_div_normSq", "def_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean", "def_pos": [444, 16], "def_end_pos": [444, 37]}, {"full_name": "div_pow", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [845, 7], "def_end_pos": [845, 14]}, {"full_name": "mul_pow", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [310, 32], "def_end_pos": [310, 39]}]], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : 0 < c ^ 4\nh\u2081 : 3 * 3 * c ^ 4 < 4 * (g \u2022 z).im ^ 2 * (4 * z.im ^ 2) * c ^ 4\nh\u2082 : (c * z.im) ^ 4 / normSq (denom (\u2191g) z) ^ 2 \u2264 1\nnsq : \u211d := normSq (denom (\u2191g) z)\n\u22a2 9 * c ^ 4 < 16", "state_after": "no goals"}, {"tactic": "rw [\u2190 Int.cast_pow, \u2190 Int.cast_three, \u2190 Int.cast_four, \u2190 Int.cast_mul, Int.cast_lt] at this", "annotated_tactic": ["rw [\u2190 <a>Int.cast_pow</a>, \u2190 <a>Int.cast_three</a>, \u2190 <a>Int.cast_four</a>, \u2190 <a>Int.cast_mul</a>, <a>Int.cast_lt</a>] at this", [{"full_name": "Int.cast_pow", "def_path": "Mathlib/Algebra/Ring/Int.lean", "def_pos": [68, 26], "def_end_pos": [68, 34]}, {"full_name": "Int.cast_three", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [146, 9], "def_end_pos": [146, 19]}, {"full_name": "Int.cast_four", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [149, 9], "def_end_pos": [149, 18]}, {"full_name": "Int.cast_mul", "def_path": "Mathlib/Algebra/Ring/Int.lean", "def_pos": [58, 7], "def_end_pos": [58, 15]}, {"full_name": "Int.cast_lt", "def_path": "Mathlib/Algebra/Order/Ring/Cast.lean", "def_pos": [57, 26], "def_end_pos": [57, 33]}]], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nthis : 3 * c ^ 2 < 4\n\u22a2 |\u2191g 1 0| \u2264 1", "state_after": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nthis : 3 * c' ^ 2 < 4\n\u22a2 |\u2191g 1 0| \u2264 1"}, {"tactic": "replace this : c' ^ 2 \u2264 1 ^ 2 := by linarith", "annotated_tactic": ["replace this : c' ^ 2 \u2264 1 ^ 2 := by linarith", []], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nthis : 3 * c' ^ 2 < 4\n\u22a2 |\u2191g 1 0| \u2264 1", "state_after": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nthis : c' ^ 2 \u2264 1 ^ 2\n\u22a2 |\u2191g 1 0| \u2264 1"}, {"tactic": "rwa [sq_le_sq, abs_one] at this", "annotated_tactic": ["rwa [<a>sq_le_sq</a>, <a>abs_one</a>] at this", [{"full_name": "sq_le_sq", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [125, 7], "def_end_pos": [125, 15]}, {"full_name": "abs_one", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [42, 15], "def_end_pos": [42, 22]}]], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nthis : c' ^ 2 \u2264 1 ^ 2\n\u22a2 |\u2191g 1 0| \u2264 1", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nthis : 3 * c' ^ 2 < 4\n\u22a2 c' ^ 2 \u2264 1 ^ 2", "state_after": "no goals"}, {"tactic": "rcases eq_or_ne c 0 with (hc | hc)", "annotated_tactic": ["rcases <a>eq_or_ne</a> c 0 with (hc | hc)", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 17]}]], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nthis : c \u2260 0 \u2192 9 * c ^ 4 < 16\n\u22a2 3 * c ^ 2 < 4", "state_after": "case inl\ng : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nthis : c \u2260 0 \u2192 9 * c ^ 4 < 16\nhc : c = 0\n\u22a2 3 * c ^ 2 < 4\n\ncase inr\ng : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nthis : c \u2260 0 \u2192 9 * c ^ 4 < 16\nhc : c \u2260 0\n\u22a2 3 * c ^ 2 < 4"}, {"tactic": "rw [hc]", "annotated_tactic": ["rw [hc]", []], "state_before": "case inl\ng : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nthis : c \u2260 0 \u2192 9 * c ^ 4 < 16\nhc : c = 0\n\u22a2 3 * c ^ 2 < 4", "state_after": "case inl\ng : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nthis : c \u2260 0 \u2192 9 * c ^ 4 < 16\nhc : c = 0\n\u22a2 3 * 0 ^ 2 < 4"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "case inl\ng : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nthis : c \u2260 0 \u2192 9 * c ^ 4 < 16\nhc : c = 0\n\u22a2 3 * 0 ^ 2 < 4", "state_after": "no goals"}, {"tactic": "refine (abs_lt_of_sq_lt_sq' ?_ (by norm_num)).2", "annotated_tactic": ["refine (<a>abs_lt_of_sq_lt_sq'</a> ?_ (by norm_num)).2", [{"full_name": "abs_lt_of_sq_lt_sq'", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [138, 7], "def_end_pos": [138, 26]}]], "state_before": "case inr\ng : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nthis : c \u2260 0 \u2192 9 * c ^ 4 < 16\nhc : c \u2260 0\n\u22a2 3 * c ^ 2 < 4", "state_after": "case inr\ng : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nthis : c \u2260 0 \u2192 9 * c ^ 4 < 16\nhc : c \u2260 0\n\u22a2 (3 * c ^ 2) ^ 2 < 4 ^ 2"}, {"tactic": "specialize this hc", "annotated_tactic": ["specialize this hc", []], "state_before": "case inr\ng : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nthis : c \u2260 0 \u2192 9 * c ^ 4 < 16\nhc : c \u2260 0\n\u22a2 (3 * c ^ 2) ^ 2 < 4 ^ 2", "state_after": "case inr\ng : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : c \u2260 0\nthis : 9 * c ^ 4 < 16\n\u22a2 (3 * c ^ 2) ^ 2 < 4 ^ 2"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "case inr\ng : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : c \u2260 0\nthis : 9 * c ^ 4 < 16\n\u22a2 (3 * c ^ 2) ^ 2 < 4 ^ 2", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nthis : c \u2260 0 \u2192 9 * c ^ 4 < 16\nhc : c \u2260 0\n\u22a2 0 \u2264 4", "state_after": "no goals"}, {"tactic": "change 0 < c ^ (2 * 2)", "annotated_tactic": ["change 0 < c ^ (2 * 2)", []], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : c \u2260 0\n\u22a2 0 < c ^ 4", "state_after": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : c \u2260 0\n\u22a2 0 < c ^ (2 * 2)"}, {"tactic": "rw [pow_mul]", "annotated_tactic": ["rw [<a>pow_mul</a>]", [{"full_name": "pow_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [713, 32], "def_end_pos": [713, 39]}]], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : c \u2260 0\n\u22a2 0 < c ^ (2 * 2)", "state_after": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : c \u2260 0\n\u22a2 0 < (c ^ 2) ^ 2"}, {"tactic": "apply sq_pos_of_pos (sq_pos_of_ne_zero hc)", "annotated_tactic": ["apply <a>sq_pos_of_pos</a> (<a>sq_pos_of_ne_zero</a> hc)", [{"full_name": "sq_pos_of_pos", "def_path": "Mathlib/Algebra/Order/Ring/Basic.lean", "def_pos": [181, 9], "def_end_pos": [181, 22]}, {"full_name": "sq_pos_of_ne_zero", "def_path": "Mathlib/Algebra/Order/Ring/Basic.lean", "def_pos": [387, 11], "def_end_pos": [387, 28]}]], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : c \u2260 0\n\u22a2 0 < (c ^ 2) ^ 2", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : 0 < c ^ 4\n\u22a2 0 \u2264 3", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : 0 < c ^ 4\nh\u2081 : 3 * 3 * c ^ 4 < 4 * (g \u2022 z).im ^ 2 * (4 * z.im ^ 2) * c ^ 4\nh\u2082 : (c * z.im) ^ 4 / normSq (denom (\u2191g) z) ^ 2 \u2264 1\nnsq : \u211d := normSq (denom (\u2191g) z)\n\u22a2 9 * c ^ 4 < c ^ 4 * z.im ^ 2 * (g \u2022 z).im ^ 2 * 16", "state_after": "no goals"}, {"tactic": "rw [ModularGroup.im_smul_eq_div_normSq, div_pow]", "annotated_tactic": ["rw [<a>ModularGroup.im_smul_eq_div_normSq</a>, <a>div_pow</a>]", [{"full_name": "UpperHalfPlane.ModularGroup.im_smul_eq_div_normSq", "def_path": "Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean", "def_pos": [444, 16], "def_end_pos": [444, 37]}, {"full_name": "div_pow", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [845, 7], "def_end_pos": [845, 14]}]], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : 0 < c ^ 4\nh\u2081 : 3 * 3 * c ^ 4 < 4 * (g \u2022 z).im ^ 2 * (4 * z.im ^ 2) * c ^ 4\nh\u2082 : (c * z.im) ^ 4 / normSq (denom (\u2191g) z) ^ 2 \u2264 1\nnsq : \u211d := normSq (denom (\u2191g) z)\n\u22a2 c ^ 4 * z.im ^ 2 * (g \u2022 z).im ^ 2 * 16 = c ^ 4 * z.im ^ 4 / nsq ^ 2 * 16", "state_after": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : 0 < c ^ 4\nh\u2081 : 3 * 3 * c ^ 4 < 4 * (g \u2022 z).im ^ 2 * (4 * z.im ^ 2) * c ^ 4\nh\u2082 : (c * z.im) ^ 4 / normSq (denom (\u2191g) z) ^ 2 \u2264 1\nnsq : \u211d := normSq (denom (\u2191g) z)\n\u22a2 c ^ 4 * z.im ^ 2 * (z.im ^ 2 / normSq (denom (\u2191g) z) ^ 2) * 16 = c ^ 4 * z.im ^ 4 / nsq ^ 2 * 16"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : 0 < c ^ 4\nh\u2081 : 3 * 3 * c ^ 4 < 4 * (g \u2022 z).im ^ 2 * (4 * z.im ^ 2) * c ^ 4\nh\u2082 : (c * z.im) ^ 4 / normSq (denom (\u2191g) z) ^ 2 \u2264 1\nnsq : \u211d := normSq (denom (\u2191g) z)\n\u22a2 c ^ 4 * z.im ^ 2 * (z.im ^ 2 / normSq (denom (\u2191g) z) ^ 2) * 16 = c ^ 4 * z.im ^ 4 / nsq ^ 2 * 16", "state_after": "no goals"}, {"tactic": "rw [\u2190 mul_pow]", "annotated_tactic": ["rw [\u2190 <a>mul_pow</a>]", [{"full_name": "mul_pow", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [310, 32], "def_end_pos": [310, 39]}]], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : 0 < c ^ 4\nh\u2081 : 3 * 3 * c ^ 4 < 4 * (g \u2022 z).im ^ 2 * (4 * z.im ^ 2) * c ^ 4\nh\u2082 : (c * z.im) ^ 4 / normSq (denom (\u2191g) z) ^ 2 \u2264 1\nnsq : \u211d := normSq (denom (\u2191g) z)\n\u22a2 c ^ 4 * z.im ^ 4 / nsq ^ 2 * 16 \u2264 16", "state_after": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : 0 < c ^ 4\nh\u2081 : 3 * 3 * c ^ 4 < 4 * (g \u2022 z).im ^ 2 * (4 * z.im ^ 2) * c ^ 4\nh\u2082 : (c * z.im) ^ 4 / normSq (denom (\u2191g) z) ^ 2 \u2264 1\nnsq : \u211d := normSq (denom (\u2191g) z)\n\u22a2 (c * z.im) ^ 4 / nsq ^ 2 * 16 \u2264 16"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "g : SL(2, \u2124)\nz : \u210d\nhz : z \u2208 \ud835\udc9f\u1d52\nhg : g \u2022 z \u2208 \ud835\udc9f\u1d52\nc' : \u2124 := \u2191g 1 0\nc : \u211d := \u2191c'\nhc : 0 < c ^ 4\nh\u2081 : 3 * 3 * c ^ 4 < 4 * (g \u2022 z).im ^ 2 * (4 * z.im ^ 2) * c ^ 4\nh\u2082 : (c * z.im) ^ 4 / normSq (denom (\u2191g) z) ^ 2 \u2264 1\nnsq : \u211d := normSq (denom (\u2191g) z)\n\u22a2 (c * z.im) ^ 4 / nsq ^ 2 * 16 \u2264 16", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/List/Pairwise.lean", "full_name": "List.pairwise_pwFilter", "start": [298, 1], "end": [304, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "full_name": "iteratedFDerivWithin_zero_eq_comp", "start": [795, 1], "end": [797, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/FixedPoint.lean", "full_name": "Ordinal.eq_zero_or_opow_omega_le_of_mul_eq_right", "start": [661, 1], "end": [670, 63], "traced_tactics": [{"tactic": "rcases eq_zero_or_pos a with ha | ha", "annotated_tactic": ["rcases <a>eq_zero_or_pos</a> a with ha | ha", [{"full_name": "Ordinal.eq_zero_or_pos", "def_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "def_pos": [412, 9], "def_end_pos": [412, 23]}]], "state_before": "a b : Ordinal.{u}\nhab : a * b = b\n\u22a2 b = 0 \u2228 a ^ \u03c9 \u2264 b", "state_after": "case inl\na b : Ordinal.{u}\nhab : a * b = b\nha : a = 0\n\u22a2 b = 0 \u2228 a ^ \u03c9 \u2264 b\n\ncase inr\na b : Ordinal.{u}\nhab : a * b = b\nha : 0 < a\n\u22a2 b = 0 \u2228 a ^ \u03c9 \u2264 b"}, {"tactic": "rw [or_iff_not_imp_left]", "annotated_tactic": ["rw [<a>or_iff_not_imp_left</a>]", [{"full_name": "Classical.or_iff_not_imp_left", "def_path": ".lake/packages/lean4/src/lean/Init/Classical.lean", "def_pos": [147, 9], "def_end_pos": [147, 28]}]], "state_before": "case inr\na b : Ordinal.{u}\nhab : a * b = b\nha : 0 < a\n\u22a2 b = 0 \u2228 a ^ \u03c9 \u2264 b", "state_after": "case inr\na b : Ordinal.{u}\nhab : a * b = b\nha : 0 < a\n\u22a2 \u00acb = 0 \u2192 a ^ \u03c9 \u2264 b"}, {"tactic": "intro hb", "annotated_tactic": ["intro hb", []], "state_before": "case inr\na b : Ordinal.{u}\nhab : a * b = b\nha : 0 < a\n\u22a2 \u00acb = 0 \u2192 a ^ \u03c9 \u2264 b", "state_after": "case inr\na b : Ordinal.{u}\nhab : a * b = b\nha : 0 < a\nhb : \u00acb = 0\n\u22a2 a ^ \u03c9 \u2264 b"}, {"tactic": "rw [\u2190 nfp_mul_one ha]", "annotated_tactic": ["rw [\u2190 <a>nfp_mul_one</a> ha]", [{"full_name": "Ordinal.nfp_mul_one", "def_path": "Mathlib/SetTheory/Ordinal/FixedPoint.lean", "def_pos": [612, 9], "def_end_pos": [612, 20]}]], "state_before": "case inr\na b : Ordinal.{u}\nhab : a * b = b\nha : 0 < a\nhb : \u00acb = 0\n\u22a2 a ^ \u03c9 \u2264 b", "state_after": "case inr\na b : Ordinal.{u}\nhab : a * b = b\nha : 0 < a\nhb : \u00acb = 0\n\u22a2 nfp (fun x => a * x) 1 \u2264 b"}, {"tactic": "rw [\u2190 Ne, \u2190 one_le_iff_ne_zero] at hb", "annotated_tactic": ["rw [\u2190 <a>Ne</a>, \u2190 <a>one_le_iff_ne_zero</a>] at hb", [{"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "Ordinal.one_le_iff_ne_zero", "def_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "def_pos": [1065, 9], "def_end_pos": [1065, 27]}]], "state_before": "case inr\na b : Ordinal.{u}\nhab : a * b = b\nha : 0 < a\nhb : \u00acb = 0\n\u22a2 nfp (fun x => a * x) 1 \u2264 b", "state_after": "case inr\na b : Ordinal.{u}\nhab : a * b = b\nha : 0 < a\nhb : 1 \u2264 b\n\u22a2 nfp (fun x => a * x) 1 \u2264 b"}, {"tactic": "exact nfp_le_fp (mul_isNormal ha).monotone hb (le_of_eq hab)", "annotated_tactic": ["exact <a>nfp_le_fp</a> (<a>mul_isNormal</a> ha).<a>monotone</a> hb (<a>le_of_eq</a> hab)", [{"full_name": "Ordinal.nfp_le_fp", "def_path": "Mathlib/SetTheory/Ordinal/FixedPoint.lean", "def_pos": [479, 9], "def_end_pos": [479, 18]}, {"full_name": "Ordinal.mul_isNormal", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [799, 9], "def_end_pos": [799, 21]}, {"full_name": "Ordinal.IsNormal.monotone", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [415, 9], "def_end_pos": [415, 26]}, {"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\na b : Ordinal.{u}\nhab : a * b = b\nha : 0 < a\nhb : 1 \u2264 b\n\u22a2 nfp (fun x => a * x) 1 \u2264 b", "state_after": "no goals"}, {"tactic": "rw [ha, zero_opow omega_ne_zero]", "annotated_tactic": ["rw [ha, <a>zero_opow</a> <a>omega_ne_zero</a>]", [{"full_name": "Ordinal.zero_opow", "def_path": "Mathlib/SetTheory/Ordinal/Exponential.lean", "def_pos": [46, 9], "def_end_pos": [46, 18]}, {"full_name": "Ordinal.omega_ne_zero", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [2471, 9], "def_end_pos": [2471, 22]}]], "state_before": "case inl\na b : Ordinal.{u}\nhab : a * b = b\nha : a = 0\n\u22a2 b = 0 \u2228 a ^ \u03c9 \u2264 b", "state_after": "case inl\na b : Ordinal.{u}\nhab : a * b = b\nha : a = 0\n\u22a2 b = 0 \u2228 0 \u2264 b"}, {"tactic": "exact Or.inr (Ordinal.zero_le b)", "annotated_tactic": ["exact <a>Or.inr</a> (<a>Ordinal.zero_le</a> b)", [{"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}, {"full_name": "Ordinal.zero_le", "def_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "def_pos": [386, 19], "def_end_pos": [386, 26]}]], "state_before": "case inl\na b : Ordinal.{u}\nhab : a * b = b\nha : a = 0\n\u22a2 b = 0 \u2228 0 \u2264 b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.mem_nsmul", "start": [741, 1], "end": [745, 17], "traced_tactics": [{"tactic": "refine \u27e8mem_of_mem_nsmul, fun h => ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>mem_of_mem_nsmul</a>, fun h => ?_\u27e9", [{"full_name": "Multiset.mem_of_mem_nsmul", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [732, 9], "def_end_pos": [732, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\na : \u03b1\ns : Multiset \u03b1\nn : \u2115\nh0 : n \u2260 0\n\u22a2 a \u2208 n \u2022 s \u2194 a \u2208 s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\na : \u03b1\ns : Multiset \u03b1\nn : \u2115\nh0 : n \u2260 0\nh : a \u2208 s\n\u22a2 a \u2208 n \u2022 s"}, {"tactic": "obtain \u27e8n, rfl\u27e9 := exists_eq_succ_of_ne_zero h0", "annotated_tactic": ["obtain \u27e8n, rfl\u27e9 := <a>exists_eq_succ_of_ne_zero</a> h0", [{"full_name": "Nat.exists_eq_succ_of_ne_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [691, 9], "def_end_pos": [691, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\na : \u03b1\ns : Multiset \u03b1\nn : \u2115\nh0 : n \u2260 0\nh : a \u2208 s\n\u22a2 a \u2208 n \u2022 s", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\na : \u03b1\ns : Multiset \u03b1\nh : a \u2208 s\nn : \u2115\nh0 : n.succ \u2260 0\n\u22a2 a \u2208 n.succ \u2022 s"}, {"tactic": "rw [succ_nsmul, mem_add]", "annotated_tactic": ["rw [<a>succ_nsmul</a>, <a>mem_add</a>]", [{"full_name": "succ_nsmul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [656, 15], "def_end_pos": [656, 25]}, {"full_name": "Multiset.mem_add", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [728, 9], "def_end_pos": [728, 16]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\na : \u03b1\ns : Multiset \u03b1\nh : a \u2208 s\nn : \u2115\nh0 : n.succ \u2260 0\n\u22a2 a \u2208 n.succ \u2022 s", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\na : \u03b1\ns : Multiset \u03b1\nh : a \u2208 s\nn : \u2115\nh0 : n.succ \u2260 0\n\u22a2 a \u2208 n \u2022 s \u2228 a \u2208 s"}, {"tactic": "exact Or.inr h", "annotated_tactic": ["exact <a>Or.inr</a> h", [{"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\na : \u03b1\ns : Multiset \u03b1\nh : a \u2208 s\nn : \u2115\nh0 : n.succ \u2260 0\n\u22a2 a \u2208 n \u2022 s \u2228 a \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "IntervalIntegrable.iff_comp_neg", "start": [354, 1], "end": [356, 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": [322, 9], "def_end_pos": [322, 26]}, {"full_name": "neg_ne_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [646, 3], "def_end_pos": [646, 14]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [58, 15], "def_end_pos": [58, 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": [132, 9], "def_end_pos": [132, 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-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean", "full_name": "ContDiff.csin", "start": [416, 1], "end": [417, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/PartrecCode.lean", "full_name": "Nat.Partrec.Code.const_inj", "start": [103, 1], "end": [108, 29], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "x\u271d : Code.const 0 = Code.const 0\n\u22a2 0 = 0", "state_after": "no goals"}, {"tactic": "dsimp [Nat.Partrec.Code.const] at h", "annotated_tactic": ["dsimp [<a>Nat.Partrec.Code.const</a>] at h", [{"full_name": "Nat.Partrec.Code.const", "def_path": "Mathlib/Computability/PartrecCode.lean", "def_pos": [98, 15], "def_end_pos": [98, 20]}]], "state_before": "n\u2081 n\u2082 : \u2115\nh : Code.const (n\u2081 + 1) = Code.const (n\u2082 + 1)\n\u22a2 n\u2081 + 1 = n\u2082 + 1", "state_after": "n\u2081 n\u2082 : \u2115\nh : succ.comp (Code.const n\u2081) = succ.comp (Code.const n\u2082)\n\u22a2 n\u2081 + 1 = n\u2082 + 1"}, {"tactic": "injection h with h\u2081 h\u2082", "annotated_tactic": ["injection h with h\u2081 h\u2082", []], "state_before": "n\u2081 n\u2082 : \u2115\nh : succ.comp (Code.const n\u2081) = succ.comp (Code.const n\u2082)\n\u22a2 n\u2081 + 1 = n\u2082 + 1", "state_after": "n\u2081 n\u2082 : \u2115\nh\u2081 : succ = succ\nh\u2082 : Code.const n\u2081 = Code.const n\u2082\n\u22a2 n\u2081 + 1 = n\u2082 + 1"}, {"tactic": "simp only [const_inj h\u2082]", "annotated_tactic": ["simp only [const_inj h\u2082]", []], "state_before": "n\u2081 n\u2082 : \u2115\nh\u2081 : succ = succ\nh\u2082 : Code.const n\u2081 = Code.const n\u2082\n\u22a2 n\u2081 + 1 = n\u2082 + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Fourier/PoissonSummation.lean", "full_name": "SchwartzMap.tsum_eq_tsum_fourierIntegral", "start": [232, 1], "end": [237, 73], "traced_tactics": [{"tactic": "simp only [\u2190 hfg, Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay f.continuous one_lt_two\n  (f.isBigO_cocompact_rpow (-2)) (hfg \u25b8 g.isBigO_cocompact_rpow (-2))]", "annotated_tactic": ["simp only [\u2190 hfg, <a>Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay</a> f.continuous <a>one_lt_two</a>\n    (f.isBigO_cocompact_rpow (-2)) (hfg \u25b8 g.isBigO_cocompact_rpow (-2))]", [{"full_name": "Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay", "def_path": "Mathlib/Analysis/Fourier/PoissonSummation.lean", "def_pos": [219, 9], "def_end_pos": [219, 56]}, {"full_name": "one_lt_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [106, 7], "def_end_pos": [106, 17]}]], "state_before": "f g : \ud835\udce2(\u211d, \u2102)\nhfg : \ud835\udcd5 \u21d1f = \u21d1g\nx : \u211d\n\u22a2 \u2211' (n : \u2124), f (x + \u2191n) = \u2211' (n : \u2124), g \u2191n * (fourier n) \u2191x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean", "full_name": "MeasureTheory.ae_const_le_iff_forall_lt_measure_zero", "start": [126, 1], "end": [158, 22], "traced_tactics": [{"tactic": "rw [ae_iff]", "annotated_tactic": ["rw [<a>ae_iff</a>]", [{"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean", "def_pos": [75, 9], "def_end_pos": [75, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, c \u2264 f x) \u2194 \u2200 b < c, \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\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 \u03bc {a | \u00acc \u2264 f a} = 0 \u2194 \u2200 b < c, \u03bc {x | f x \u2264 b} = 0"}, {"tactic": "push_neg", "annotated_tactic": ["push_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\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 \u03bc {a | \u00acc \u2264 f a} = 0 \u2194 \u2200 b < c, \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\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 \u03bc {a | f a < c} = 0 \u2194 \u2200 b < c, \u03bc {x | f x \u2264 b} = 0"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 \u03bc {a | f a < c} = 0 \u2194 \u2200 b < c, \u03bc {x | f x \u2264 b} = 0", "state_after": "case mp\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 \u03bc {a | f a < c} = 0 \u2192 \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\n\ncase mpr\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 (\u2200 b < c, \u03bc {x | f x \u2264 b} = 0) \u2192 \u03bc {a | f a < c} = 0"}, {"tactic": "intro hc", "annotated_tactic": ["intro hc", []], "state_before": "case mpr\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 (\u2200 b < c, \u03bc {x | f x \u2264 b} = 0) \u2192 \u03bc {a | f a < c} = 0", "state_after": "case mpr\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\n\u22a2 \u03bc {a | f a < c} = 0"}, {"tactic": "by_cases h : \u2200 b, c \u2264 b", "annotated_tactic": ["by_cases h : \u2200 b, c \u2264 b", []], "state_before": "case mpr\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\n\u22a2 \u03bc {a | f a < c} = 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nh : \u2200 (b : \u03b2), c \u2264 b\n\u22a2 \u03bc {a | f a < c} = 0\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\n\u22a2 \u03bc {a | f a < c} = 0"}, {"tactic": "by_cases H : \u00acIsLUB (Set.Iio c) c", "annotated_tactic": ["by_cases H : \u00ac<a>IsLUB</a> (<a>Set.Iio</a> c) c", [{"full_name": "IsLUB", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [76, 5], "def_end_pos": [76, 10]}, {"full_name": "Set.Iio", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [54, 5], "def_end_pos": [54, 8]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\n\u22a2 \u03bc {a | f a < c} = 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00acIsLUB (Set.Iio c) c\n\u22a2 \u03bc {a | f a < c} = 0\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00ac\u00acIsLUB (Set.Iio c) c\n\u22a2 \u03bc {a | f a < c} = 0"}, {"tactic": "push_neg at H h", "annotated_tactic": ["push_neg at H h", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00ac\u00acIsLUB (Set.Iio c) c\n\u22a2 \u03bc {a | f a < c} = 0", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\n\u22a2 \u03bc {a | f a < c} = 0"}, {"tactic": "obtain \u27e8u, _, u_lt, u_lim, -\u27e9 :\n  \u2203 u : \u2115 \u2192 \u03b2,\n    StrictMono u \u2227 (\u2200 n : \u2115, u n < c) \u2227 Tendsto u atTop (\ud835\udcdd c) \u2227 \u2200 n : \u2115, u n \u2208 Set.Iio c :=\n  H.exists_seq_strictMono_tendsto_of_not_mem (lt_irrefl c) h", "annotated_tactic": ["obtain \u27e8u, _, u_lt, u_lim, -\u27e9 :\n    \u2203 u : \u2115 \u2192 \u03b2,\n      <a>StrictMono</a> u \u2227 (\u2200 n : \u2115, u n < c) \u2227 <a>Tendsto</a> u <a>atTop</a> (\ud835\udcdd c) \u2227 \u2200 n : \u2115, u n \u2208 <a>Set.Iio</a> c :=\n    H.exists_seq_strictMono_tendsto_of_not_mem (<a>lt_irrefl</a> c) h", [{"full_name": "StrictMono", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [97, 5], "def_end_pos": [97, 15]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2991, 5], "def_end_pos": [2991, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Set.Iio", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [54, 5], "def_end_pos": [54, 8]}, {"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\n\u22a2 \u03bc {a | f a < c} = 0", "state_after": "case neg.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\n\u22a2 \u03bc {a | f a < c} = 0"}, {"tactic": "rw [h_Union, measure_iUnion_null_iff]", "annotated_tactic": ["rw [h_Union, <a>measure_iUnion_null_iff</a>]", [{"full_name": "MeasureTheory.measure_iUnion_null_iff", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [116, 9], "def_end_pos": [116, 32]}]], "state_before": "case neg.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nh_Union : {x | f x < c} = \u22c3 n, {x | f x \u2264 u n}\n\u22a2 \u03bc {a | f a < c} = 0", "state_after": "case neg.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nh_Union : {x | f x < c} = \u22c3 n, {x | f x \u2264 u n}\n\u22a2 \u2200 (i : \u2115), \u03bc {x | f x \u2264 u i} = 0"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "case neg.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nh_Union : {x | f x < c} = \u22c3 n, {x | f x \u2264 u n}\n\u22a2 \u2200 (i : \u2115), \u03bc {x | f x \u2264 u i} = 0", "state_after": "case neg.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nh_Union : {x | f x < c} = \u22c3 n, {x | f x \u2264 u n}\nn : \u2115\n\u22a2 \u03bc {x | f x \u2264 u n} = 0"}, {"tactic": "exact hc _ (u_lt n)", "annotated_tactic": ["exact hc _ (u_lt n)", []], "state_before": "case neg.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nh_Union : {x | f x < c} = \u22c3 n, {x | f x \u2264 u n}\nn : \u2115\n\u22a2 \u03bc {x | f x \u2264 u n} = 0", "state_after": "no goals"}, {"tactic": "intro h b hb", "annotated_tactic": ["intro h b hb", []], "state_before": "case mp\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 \u03bc {a | f a < c} = 0 \u2192 \u2200 b < c, \u03bc {x | f x \u2264 b} = 0", "state_after": "case mp\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nh : \u03bc {a | f a < c} = 0\nb : \u03b2\nhb : b < c\n\u22a2 \u03bc {x | f x \u2264 b} = 0"}, {"tactic": "exact measure_mono_null (fun y hy => (lt_of_le_of_lt hy hb : _)) h", "annotated_tactic": ["exact <a>measure_mono_null</a> (fun y hy => (<a>lt_of_le_of_lt</a> hy hb : _)) h", [{"full_name": "MeasureTheory.measure_mono_null", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [56, 9], "def_end_pos": [56, 26]}, {"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 mp\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nh : \u03bc {a | f a < c} = 0\nb : \u03b2\nhb : b < c\n\u22a2 \u03bc {x | f x \u2264 b} = 0", "state_after": "no goals"}, {"tactic": "have : {a : \u03b1 | f a < c} = \u2205 := by\n  apply Set.eq_empty_iff_forall_not_mem.2 fun x hx => ?_\n  exact (lt_irrefl _ (lt_of_lt_of_le hx (h (f x)))).elim", "annotated_tactic": ["have : {a : \u03b1 | f a < c} = \u2205 := by\n      apply <a>Set.eq_empty_iff_forall_not_mem</a>.2 fun x hx => ?_\n      exact (<a>lt_irrefl</a> _ (<a>lt_of_lt_of_le</a> hx (h (f x)))).<a>elim</a>", [{"full_name": "Set.eq_empty_iff_forall_not_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [552, 9], "def_end_pos": [552, 36]}, {"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"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": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nh : \u2200 (b : \u03b2), c \u2264 b\n\u22a2 \u03bc {a | f a < c} = 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nh : \u2200 (b : \u03b2), c \u2264 b\nthis : {a | f a < c} = \u2205\n\u22a2 \u03bc {a | f a < c} = 0"}, {"tactic": "simp [this]", "annotated_tactic": ["simp [this]", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nh : \u2200 (b : \u03b2), c \u2264 b\nthis : {a | f a < c} = \u2205\n\u22a2 \u03bc {a | f a < c} = 0", "state_after": "no goals"}, {"tactic": "apply Set.eq_empty_iff_forall_not_mem.2 fun x hx => ?_", "annotated_tactic": ["apply <a>Set.eq_empty_iff_forall_not_mem</a>.2 fun x hx => ?_", [{"full_name": "Set.eq_empty_iff_forall_not_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [552, 9], "def_end_pos": [552, 36]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nh : \u2200 (b : \u03b2), c \u2264 b\n\u22a2 {a | f a < c} = \u2205", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nh : \u2200 (b : \u03b2), c \u2264 b\nx : \u03b1\nhx : x \u2208 {a | f a < c}\n\u22a2 False"}, {"tactic": "exact (lt_irrefl _ (lt_of_lt_of_le hx (h (f x)))).elim", "annotated_tactic": ["exact (<a>lt_irrefl</a> _ (<a>lt_of_lt_of_le</a> hx (h (f x)))).<a>elim</a>", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nh : \u2200 (b : \u03b2), c \u2264 b\nx : \u03b1\nhx : x \u2208 {a | f a < c}\n\u22a2 False", "state_after": "no goals"}, {"tactic": "have : c \u2208 upperBounds (Set.Iio c) := fun y hy => le_of_lt hy", "annotated_tactic": ["have : c \u2208 <a>upperBounds</a> (<a>Set.Iio</a> c) := fun y hy => <a>le_of_lt</a> hy", [{"full_name": "upperBounds", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [46, 5], "def_end_pos": [46, 16]}, {"full_name": "Set.Iio", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [54, 5], "def_end_pos": [54, 8]}, {"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\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00acIsLUB (Set.Iio c) c\n\u22a2 \u03bc {a | f a < c} = 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00acIsLUB (Set.Iio c) c\nthis : c \u2208 upperBounds (Set.Iio c)\n\u22a2 \u03bc {a | f a < c} = 0"}, {"tactic": "obtain \u27e8b, b_up, bc\u27e9 : \u2203 b : \u03b2, b \u2208 upperBounds (Set.Iio c) \u2227 b < c := by\n  simpa [IsLUB, IsLeast, this, lowerBounds] using H", "annotated_tactic": ["obtain \u27e8b, b_up, bc\u27e9 : \u2203 b : \u03b2, b \u2208 <a>upperBounds</a> (<a>Set.Iio</a> c) \u2227 b < c := by\n      simpa [<a>IsLUB</a>, <a>IsLeast</a>, this, <a>lowerBounds</a>] using H", [{"full_name": "upperBounds", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [46, 5], "def_end_pos": [46, 16]}, {"full_name": "Set.Iio", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [54, 5], "def_end_pos": [54, 8]}, {"full_name": "IsLUB", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [76, 5], "def_end_pos": [76, 10]}, {"full_name": "IsLeast", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [66, 5], "def_end_pos": [66, 12]}, {"full_name": "lowerBounds", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [51, 5], "def_end_pos": [51, 16]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00acIsLUB (Set.Iio c) c\nthis : c \u2208 upperBounds (Set.Iio c)\n\u22a2 \u03bc {a | f a < c} = 0", "state_after": "case pos.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\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00acIsLUB (Set.Iio c) c\nthis : c \u2208 upperBounds (Set.Iio c)\nb : \u03b2\nb_up : b \u2208 upperBounds (Set.Iio c)\nbc : b < c\n\u22a2 \u03bc {a | f a < c} = 0"}, {"tactic": "exact measure_mono_null (fun x hx => b_up hx) (hc b bc)", "annotated_tactic": ["exact <a>measure_mono_null</a> (fun x hx => b_up hx) (hc b bc)", [{"full_name": "MeasureTheory.measure_mono_null", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [56, 9], "def_end_pos": [56, 26]}]], "state_before": "case pos.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\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00acIsLUB (Set.Iio c) c\nthis : c \u2208 upperBounds (Set.Iio c)\nb : \u03b2\nb_up : b \u2208 upperBounds (Set.Iio c)\nbc : b < c\n\u22a2 \u03bc {a | f a < c} = 0", "state_after": "no goals"}, {"tactic": "simpa [IsLUB, IsLeast, this, lowerBounds] using H", "annotated_tactic": ["simpa [<a>IsLUB</a>, <a>IsLeast</a>, this, <a>lowerBounds</a>] using H", [{"full_name": "IsLUB", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [76, 5], "def_end_pos": [76, 10]}, {"full_name": "IsLeast", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [66, 5], "def_end_pos": [66, 12]}, {"full_name": "lowerBounds", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [51, 5], "def_end_pos": [51, 16]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00acIsLUB (Set.Iio c) c\nthis : c \u2208 upperBounds (Set.Iio c)\n\u22a2 \u2203 b \u2208 upperBounds (Set.Iio c), b < c", "state_after": "no goals"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\n\u22a2 {x | f x < c} = \u22c3 n, {x | f x \u2264 u n}", "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\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\n\u22a2 x \u2208 {x | f x < c} \u2194 x \u2208 \u22c3 n, {x | f x \u2264 u n}"}, {"tactic": "simp_rw [Set.mem_iUnion, Set.mem_setOf_eq]", "annotated_tactic": ["simp_rw [<a>Set.mem_iUnion</a>, <a>Set.mem_setOf_eq</a>]", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Order/SetNotation.lean", "def_pos": [268, 9], "def_end_pos": [268, 19]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [79, 29], "def_end_pos": [79, 41]}]], "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\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\n\u22a2 x \u2208 {x | f x < c} \u2194 x \u2208 \u22c3 n, {x | f x \u2264 u n}", "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\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\n\u22a2 f x < c \u2194 \u2203 i, f x \u2264 u i"}, {"tactic": "constructor <;> intro h", "annotated_tactic": ["constructor <;> intro h", []], "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\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\n\u22a2 f x < c \u2194 \u2203 i, f x \u2264 u i", "state_after": "case h.mp\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh\u271d : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\nh : f x < c\n\u22a2 \u2203 i, f x \u2264 u i\n\ncase h.mpr\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh\u271d : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\nh : \u2203 i, f x \u2264 u i\n\u22a2 f x < c"}, {"tactic": "obtain \u27e8n, hn\u27e9 := ((tendsto_order.1 u_lim).1 _ h).exists", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 := ((<a>tendsto_order</a>.1 u_lim).1 _ h).<a>exists</a>", [{"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 22]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1328, 9], "def_end_pos": [1328, 26]}]], "state_before": "case h.mp\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh\u271d : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\nh : f x < c\n\u22a2 \u2203 i, f x \u2264 u i", "state_after": "case h.mp.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh\u271d : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\nh : f x < c\nn : \u2115\nhn : f x < u n\n\u22a2 \u2203 i, f x \u2264 u i"}, {"tactic": "exact \u27e8n, hn.le\u27e9", "annotated_tactic": ["exact \u27e8n, hn.le\u27e9", []], "state_before": "case h.mp.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh\u271d : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\nh : f x < c\nn : \u2115\nhn : f x < u n\n\u22a2 \u2203 i, f x \u2264 u i", "state_after": "no goals"}, {"tactic": "obtain \u27e8n, hn\u27e9 := h", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 := h", []], "state_before": "case h.mpr\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh\u271d : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\nh : \u2203 i, f x \u2264 u i\n\u22a2 f x < c", "state_after": "case h.mpr.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\nn : \u2115\nhn : f x \u2264 u n\n\u22a2 f x < c"}, {"tactic": "exact hn.trans_lt (u_lt _)", "annotated_tactic": ["exact hn.trans_lt (u_lt _)", []], "state_before": "case h.mpr.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 b < c, \u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\nn : \u2115\nhn : f x \u2264 u n\n\u22a2 f x < c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.snorm_lim_le_liminf_snorm", "start": [1412, 1], "end": [1423, 52], "traced_tactics": [{"tactic": "obtain rfl|hp0 := eq_or_ne p 0", "annotated_tactic": ["obtain rfl|hp0 := <a>eq_or_ne</a> p 0", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 17]}]], "state_before": "\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\n\u22a2 snorm f_lim p \u03bc \u2264 liminf (fun n => snorm (f n) p \u03bc) atTop", "state_after": "case inl\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\n\u22a2 snorm f_lim 0 \u03bc \u2264 liminf (fun n => snorm (f n) 0 \u03bc) atTop\n\ncase inr\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\n\u22a2 snorm f_lim p \u03bc \u2264 liminf (fun n => snorm (f n) p \u03bc) atTop"}, {"tactic": "by_cases hp_top : p = \u221e", "annotated_tactic": ["by_cases hp_top : p = \u221e", []], "state_before": "case inr\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\n\u22a2 snorm f_lim p \u03bc \u2264 liminf (fun n => snorm (f n) p \u03bc) atTop", "state_after": "case pos\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\nhp_top : p = \u22a4\n\u22a2 snorm f_lim p \u03bc \u2264 liminf (fun n => snorm (f n) p \u03bc) atTop\n\ncase neg\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm f_lim p \u03bc \u2264 liminf (fun n => snorm (f n) p \u03bc) atTop"}, {"tactic": "simp_rw [snorm_eq_snorm' hp0 hp_top]", "annotated_tactic": ["simp_rw [<a>snorm_eq_snorm'</a> hp0 hp_top]", [{"full_name": "MeasureTheory.snorm_eq_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [86, 9], "def_end_pos": [86, 24]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm f_lim p \u03bc \u2264 liminf (fun n => snorm (f n) p \u03bc) atTop", "state_after": "case neg\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm' f_lim p.toReal \u03bc \u2264 liminf (fun n => snorm' (f n) p.toReal \u03bc) atTop"}, {"tactic": "have hp_pos : 0 < p.toReal := ENNReal.toReal_pos hp0 hp_top", "annotated_tactic": ["have hp_pos : 0 < p.toReal := <a>ENNReal.toReal_pos</a> hp0 hp_top", [{"full_name": "ENNReal.toReal_pos", "def_path": "Mathlib/Data/ENNReal/Real.lean", "def_pos": [171, 9], "def_end_pos": [171, 19]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\n\u22a2 snorm' f_lim p.toReal \u03bc \u2264 liminf (fun n => snorm' (f n) p.toReal \u03bc) atTop", "state_after": "case neg\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < p.toReal\n\u22a2 snorm' f_lim p.toReal \u03bc \u2264 liminf (fun n => snorm' (f n) p.toReal \u03bc) atTop"}, {"tactic": "exact snorm'_lim_le_liminf_snorm' hp_pos hf h_lim", "annotated_tactic": ["exact <a>snorm'_lim_le_liminf_snorm'</a> hp_pos hf h_lim", [{"full_name": "MeasureTheory.Lp.snorm'_lim_le_liminf_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1375, 9], "def_end_pos": [1375, 36]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\nhp_top : \u00acp = \u22a4\nhp_pos : 0 < p.toReal\n\u22a2 snorm' f_lim p.toReal \u03bc \u2264 liminf (fun n => snorm' (f n) p.toReal \u03bc) atTop", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case inl\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\n\u22a2 snorm f_lim 0 \u03bc \u2264 liminf (fun n => snorm (f n) 0 \u03bc) atTop", "state_after": "no goals"}, {"tactic": "simp_rw [hp_top]", "annotated_tactic": ["simp_rw [hp_top]", []], "state_before": "case pos\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\nhp_top : p = \u22a4\n\u22a2 snorm f_lim p \u03bc \u2264 liminf (fun n => snorm (f n) p \u03bc) atTop", "state_after": "case pos\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\nhp_top : p = \u22a4\n\u22a2 snorm f_lim \u22a4 \u03bc \u2264 liminf (fun n => snorm (f n) \u22a4 \u03bc) atTop"}, {"tactic": "exact snorm_exponent_top_lim_le_liminf_snorm_exponent_top h_lim", "annotated_tactic": ["exact <a>snorm_exponent_top_lim_le_liminf_snorm_exponent_top</a> h_lim", [{"full_name": "MeasureTheory.Lp.snorm_exponent_top_lim_le_liminf_snorm_exponent_top", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [1403, 9], "def_end_pos": [1403, 60]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE\u271d : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b3 : NormedAddCommGroup E\u271d\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedAddCommGroup G\nE : Type u_5\ninst\u271d : NormedAddCommGroup E\nf : \u2115 \u2192 \u03b1 \u2192 E\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nf_lim : \u03b1 \u2192 E\nh_lim : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))\nhp0 : p \u2260 0\nhp_top : p = \u22a4\n\u22a2 snorm f_lim \u22a4 \u03bc \u2264 liminf (fun n => snorm (f n) \u22a4 \u03bc) atTop", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/Fixed.lean", "full_name": "FixedPoints.minpoly.irreducible_aux", "start": [225, 1], "end": [241, 85], "traced_tactics": [{"tactic": "have hf2 : f \u2223 minpoly G F x := by rw [\u2190 hfg]; exact dvd_mul_right _ _", "annotated_tactic": ["have hf2 : f \u2223 <a>minpoly</a> G F x := by rw [\u2190 hfg]; exact <a>dvd_mul_right</a> _ _", [{"full_name": "FixedPoints.minpoly", "def_path": "Mathlib/FieldTheory/Fixed.lean", "def_pos": [172, 5], "def_end_pos": [172, 12]}, {"full_name": "dvd_mul_right", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 22]}]], "state_before": "M : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\n\u22a2 f = 1 \u2228 g = 1", "state_after": "M : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\n\u22a2 f = 1 \u2228 g = 1"}, {"tactic": "have hg2 : g \u2223 minpoly G F x := by rw [\u2190 hfg]; exact dvd_mul_left _ _", "annotated_tactic": ["have hg2 : g \u2223 <a>minpoly</a> G F x := by rw [\u2190 hfg]; exact <a>dvd_mul_left</a> _ _", [{"full_name": "FixedPoints.minpoly", "def_path": "Mathlib/FieldTheory/Fixed.lean", "def_pos": [172, 5], "def_end_pos": [172, 12]}, {"full_name": "dvd_mul_left", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [205, 9], "def_end_pos": [205, 21]}]], "state_before": "M : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\n\u22a2 f = 1 \u2228 g = 1", "state_after": "M : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\nhg2 : g \u2223 minpoly G F x\n\u22a2 f = 1 \u2228 g = 1"}, {"tactic": "have := eval\u2082 G F x", "annotated_tactic": ["have := <a>eval\u2082</a> G F x", [{"full_name": "FixedPoints.minpoly.eval\u2082", "def_path": "Mathlib/FieldTheory/Fixed.lean", "def_pos": [185, 9], "def_end_pos": [185, 14]}]], "state_before": "M : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\nhg2 : g \u2223 minpoly G F x\n\u22a2 f = 1 \u2228 g = 1", "state_after": "M : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\nhg2 : g \u2223 minpoly G F x\nthis : Polynomial.eval\u2082 (subfield G F).subtype x (minpoly G F x) = 0\n\u22a2 f = 1 \u2228 g = 1"}, {"tactic": "rw [\u2190 hfg, Polynomial.eval\u2082_mul, mul_eq_zero] at this", "annotated_tactic": ["rw [\u2190 hfg, <a>Polynomial.eval\u2082_mul</a>, <a>mul_eq_zero</a>] at this", [{"full_name": "Polynomial.eval\u2082_mul", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [258, 9], "def_end_pos": [258, 18]}, {"full_name": "mul_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [282, 9], "def_end_pos": [282, 20]}]], "state_before": "M : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\nhg2 : g \u2223 minpoly G F x\nthis : Polynomial.eval\u2082 (subfield G F).subtype x (minpoly G F x) = 0\n\u22a2 f = 1 \u2228 g = 1", "state_after": "M : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\nhg2 : g \u2223 minpoly G F x\nthis : Polynomial.eval\u2082 (subfield G F).subtype x f = 0 \u2228 Polynomial.eval\u2082 (subfield G F).subtype x g = 0\n\u22a2 f = 1 \u2228 g = 1"}, {"tactic": "cases' this with this this", "annotated_tactic": ["cases' this with this this", []], "state_before": "M : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\nhg2 : g \u2223 minpoly G F x\nthis : Polynomial.eval\u2082 (subfield G F).subtype x f = 0 \u2228 Polynomial.eval\u2082 (subfield G F).subtype x g = 0\n\u22a2 f = 1 \u2228 g = 1", "state_after": "case inl\nM : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\nhg2 : g \u2223 minpoly G F x\nthis : Polynomial.eval\u2082 (subfield G F).subtype x f = 0\n\u22a2 f = 1 \u2228 g = 1\n\ncase inr\nM : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\nhg2 : g \u2223 minpoly G F x\nthis : Polynomial.eval\u2082 (subfield G F).subtype x g = 0\n\u22a2 f = 1 \u2228 g = 1"}, {"tactic": "rw [\u2190 hfg]", "annotated_tactic": ["rw [\u2190 hfg]", []], "state_before": "M : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\n\u22a2 f \u2223 minpoly G F x", "state_after": "M : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\n\u22a2 f \u2223 f * g"}, {"tactic": "exact dvd_mul_right _ _", "annotated_tactic": ["exact <a>dvd_mul_right</a> _ _", [{"full_name": "dvd_mul_right", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 22]}]], "state_before": "M : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\n\u22a2 f \u2223 f * g", "state_after": "no goals"}, {"tactic": "rw [\u2190 hfg]", "annotated_tactic": ["rw [\u2190 hfg]", []], "state_before": "M : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\n\u22a2 g \u2223 minpoly G F x", "state_after": "M : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\n\u22a2 g \u2223 f * g"}, {"tactic": "exact dvd_mul_left _ _", "annotated_tactic": ["exact <a>dvd_mul_left</a> _ _", [{"full_name": "dvd_mul_left", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [205, 9], "def_end_pos": [205, 21]}]], "state_before": "M : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\n\u22a2 g \u2223 f * g", "state_after": "no goals"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "case inl\nM : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\nhg2 : g \u2223 minpoly G F x\nthis : Polynomial.eval\u2082 (subfield G F).subtype x f = 0\n\u22a2 f = 1 \u2228 g = 1", "state_after": "case inl.h\nM : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\nhg2 : g \u2223 minpoly G F x\nthis : Polynomial.eval\u2082 (subfield G F).subtype x f = 0\n\u22a2 g = 1"}, {"tactic": "have hf3 : f = minpoly G F x :=\n  Polynomial.eq_of_monic_of_associated hf (monic G F x)\n    (associated_of_dvd_dvd hf2 <| @of_eval\u2082 G _ F _ _ _ x f this)", "annotated_tactic": ["have hf3 : f = <a>minpoly</a> G F x :=\n      <a>Polynomial.eq_of_monic_of_associated</a> hf (<a>monic</a> G F x)\n        (<a>associated_of_dvd_dvd</a> hf2 <| @<a>of_eval\u2082</a> G _ F _ _ _ x f this)", [{"full_name": "FixedPoints.minpoly", "def_path": "Mathlib/FieldTheory/Fixed.lean", "def_pos": [172, 5], "def_end_pos": [172, 12]}, {"full_name": "Polynomial.eq_of_monic_of_associated", "def_path": "Mathlib/Algebra/Polynomial/Monic.lean", "def_pos": [267, 9], "def_end_pos": [267, 34]}, {"full_name": "FixedPoints.minpoly.monic", "def_path": "Mathlib/FieldTheory/Fixed.lean", "def_pos": [180, 9], "def_end_pos": [180, 14]}, {"full_name": "associated_of_dvd_dvd", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [580, 9], "def_end_pos": [580, 30]}, {"full_name": "FixedPoints.minpoly.of_eval\u2082", "def_path": "Mathlib/FieldTheory/Fixed.lean", "def_pos": [202, 9], "def_end_pos": [202, 17]}]], "state_before": "case inl.h\nM : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\nhg2 : g \u2223 minpoly G F x\nthis : Polynomial.eval\u2082 (subfield G F).subtype x f = 0\n\u22a2 g = 1", "state_after": "case inl.h\nM : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\nhg2 : g \u2223 minpoly G F x\nthis : Polynomial.eval\u2082 (subfield G F).subtype x f = 0\nhf3 : f = minpoly G F x\n\u22a2 g = 1"}, {"tactic": "rwa [\u2190 mul_one (minpoly G F x), hf3, mul_right_inj' (monic G F x).ne_zero] at hfg", "annotated_tactic": ["rwa [\u2190 <a>mul_one</a> (<a>minpoly</a> G F x), hf3, <a>mul_right_inj'</a> (<a>monic</a> G F x).<a>ne_zero</a>] at hfg", [{"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "FixedPoints.minpoly", "def_path": "Mathlib/FieldTheory/Fixed.lean", "def_pos": [172, 5], "def_end_pos": [172, 12]}, {"full_name": "mul_right_inj'", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [132, 9], "def_end_pos": [132, 23]}, {"full_name": "FixedPoints.minpoly.monic", "def_path": "Mathlib/FieldTheory/Fixed.lean", "def_pos": [180, 9], "def_end_pos": [180, 14]}, {"full_name": "Polynomial.Monic.ne_zero", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}]], "state_before": "case inl.h\nM : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\nhg2 : g \u2223 minpoly G F x\nthis : Polynomial.eval\u2082 (subfield G F).subtype x f = 0\nhf3 : f = minpoly G F x\n\u22a2 g = 1", "state_after": "no goals"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "case inr\nM : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\nhg2 : g \u2223 minpoly G F x\nthis : Polynomial.eval\u2082 (subfield G F).subtype x g = 0\n\u22a2 f = 1 \u2228 g = 1", "state_after": "case inr.h\nM : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\nhg2 : g \u2223 minpoly G F x\nthis : Polynomial.eval\u2082 (subfield G F).subtype x g = 0\n\u22a2 f = 1"}, {"tactic": "have hg3 : g = minpoly G F x :=\n  Polynomial.eq_of_monic_of_associated hg (monic G F x)\n    (associated_of_dvd_dvd hg2 <| @of_eval\u2082 G _ F _ _ _ x g this)", "annotated_tactic": ["have hg3 : g = <a>minpoly</a> G F x :=\n      <a>Polynomial.eq_of_monic_of_associated</a> hg (<a>monic</a> G F x)\n        (<a>associated_of_dvd_dvd</a> hg2 <| @<a>of_eval\u2082</a> G _ F _ _ _ x g this)", [{"full_name": "FixedPoints.minpoly", "def_path": "Mathlib/FieldTheory/Fixed.lean", "def_pos": [172, 5], "def_end_pos": [172, 12]}, {"full_name": "Polynomial.eq_of_monic_of_associated", "def_path": "Mathlib/Algebra/Polynomial/Monic.lean", "def_pos": [267, 9], "def_end_pos": [267, 34]}, {"full_name": "FixedPoints.minpoly.monic", "def_path": "Mathlib/FieldTheory/Fixed.lean", "def_pos": [180, 9], "def_end_pos": [180, 14]}, {"full_name": "associated_of_dvd_dvd", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [580, 9], "def_end_pos": [580, 30]}, {"full_name": "FixedPoints.minpoly.of_eval\u2082", "def_path": "Mathlib/FieldTheory/Fixed.lean", "def_pos": [202, 9], "def_end_pos": [202, 17]}]], "state_before": "case inr.h\nM : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\nhg2 : g \u2223 minpoly G F x\nthis : Polynomial.eval\u2082 (subfield G F).subtype x g = 0\n\u22a2 f = 1", "state_after": "case inr.h\nM : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\nhg2 : g \u2223 minpoly G F x\nthis : Polynomial.eval\u2082 (subfield G F).subtype x g = 0\nhg3 : g = minpoly G F x\n\u22a2 f = 1"}, {"tactic": "rwa [\u2190 one_mul (minpoly G F x), hg3, mul_left_inj' (monic G F x).ne_zero] at hfg", "annotated_tactic": ["rwa [\u2190 <a>one_mul</a> (<a>minpoly</a> G F x), hg3, <a>mul_left_inj'</a> (<a>monic</a> G F x).<a>ne_zero</a>] at hfg", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "FixedPoints.minpoly", "def_path": "Mathlib/FieldTheory/Fixed.lean", "def_pos": [172, 5], "def_end_pos": [172, 12]}, {"full_name": "mul_left_inj'", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [128, 9], "def_end_pos": [128, 22]}, {"full_name": "FixedPoints.minpoly.monic", "def_path": "Mathlib/FieldTheory/Fixed.lean", "def_pos": [180, 9], "def_end_pos": [180, 14]}, {"full_name": "Polynomial.Monic.ne_zero", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [909, 9], "def_end_pos": [909, 22]}]], "state_before": "case inr.h\nM : Type u\ninst\u271d\u2075 : Monoid M\nG : Type u\ninst\u271d\u2074 : Group G\nF : Type v\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : MulSemiringAction M F\ninst\u271d\u00b9 : MulSemiringAction G F\nm : M\ninst\u271d : Fintype G\nx : F\nf g : Polynomial \u21a5(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f \u2223 minpoly G F x\nhg2 : g \u2223 minpoly G F x\nthis : Polynomial.eval\u2082 (subfield G F).subtype x g = 0\nhg3 : g = minpoly G F x\n\u22a2 f = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean", "full_name": "CategoryTheory.Subgroupoid.isThin_iff", "start": [604, 8], "end": [604, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/NNRat/Defs.lean", "full_name": "NNRat.bddBelow_coe", "start": [226, 1], "end": [227, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Nodup.lean", "full_name": "List.Nodup.insert", "start": [405, 11], "end": [407, 76], "traced_tactics": [{"tactic": "rw [insert_of_mem h']", "annotated_tactic": ["rw [<a>insert_of_mem</a> h']", [{"full_name": "List.insert_of_mem", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [1926, 17], "def_end_pos": [1926, 30]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nl l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : l.Nodup\nh' : a \u2208 l\n\u22a2 (List.insert a l).Nodup", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nl l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : l.Nodup\nh' : a \u2208 l\n\u22a2 l.Nodup"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nl l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : l.Nodup\nh' : a \u2208 l\n\u22a2 l.Nodup", "state_after": "no goals"}, {"tactic": "rw [insert_of_not_mem h', nodup_cons]", "annotated_tactic": ["rw [<a>insert_of_not_mem</a> h', <a>nodup_cons</a>]", [{"full_name": "List.insert_of_not_mem", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [1929, 17], "def_end_pos": [1929, 34]}, {"full_name": "List.nodup_cons", "def_path": "Mathlib/Data/List/Nodup.lean", "def_pos": [39, 9], "def_end_pos": [39, 19]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nl l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : l.Nodup\nh' : a \u2209 l\n\u22a2 (List.insert a l).Nodup", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nl l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : l.Nodup\nh' : a \u2209 l\n\u22a2 a \u2209 l \u2227 l.Nodup"}, {"tactic": "constructor <;> assumption", "annotated_tactic": ["constructor <;> assumption", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nl l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : l.Nodup\nh' : a \u2209 l\n\u22a2 a \u2209 l \u2227 l.Nodup", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/FiberedCategory/BasedCategory.lean", "full_name": "CategoryTheory.BasedFunctor.id_comp", "start": [89, 1], "end": [91, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean", "full_name": "CircleDeg1Lift.tendsto_translationNumber_aux", "start": [680, 1], "end": [681, 101], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Spectrum.lean", "full_name": "spectrum.isUnit_resolvent", "start": [194, 1], "end": [195, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/DedekindDomain/Ideal.lean", "full_name": "singleton_span_mem_normalizedFactors_of_mem_normalizedFactors", "start": [1451, 1], "end": [1468, 83], "traced_tactics": [{"tactic": "by_cases hb : b = 0", "annotated_tactic": ["by_cases hb : b = 0", []], "state_before": "R : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\n\u22a2 span {a} \u2208 normalizedFactors (span {b})", "state_after": "case pos\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\nhb : b = 0\n\u22a2 span {a} \u2208 normalizedFactors (span {b})\n\ncase neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\nhb : \u00acb = 0\n\u22a2 span {a} \u2208 normalizedFactors (span {b})"}, {"tactic": "rw [Ideal.span_singleton_eq_bot.mpr hb, bot_eq_zero, normalizedFactors_zero]", "annotated_tactic": ["rw [Ideal.span_singleton_eq_bot.mpr hb, <a>bot_eq_zero</a>, <a>normalizedFactors_zero</a>]", [{"full_name": "bot_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [210, 3], "def_end_pos": [210, 14]}, {"full_name": "UniqueFactorizationMonoid.normalizedFactors_zero", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [684, 9], "def_end_pos": [684, 31]}]], "state_before": "case pos\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\nhb : b = 0\n\u22a2 span {a} \u2208 normalizedFactors (span {b})", "state_after": "case pos\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\nhb : b = 0\n\u22a2 span {a} \u2208 0"}, {"tactic": "rw [hb, normalizedFactors_zero] at ha", "annotated_tactic": ["rw [hb, <a>normalizedFactors_zero</a>] at ha", [{"full_name": "UniqueFactorizationMonoid.normalizedFactors_zero", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [684, 9], "def_end_pos": [684, 31]}]], "state_before": "case pos\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\nhb : b = 0\n\u22a2 span {a} \u2208 0", "state_after": "case pos\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 0\nhb : b = 0\n\u22a2 span {a} \u2208 0"}, {"tactic": "exact absurd ha (Multiset.not_mem_zero a)", "annotated_tactic": ["exact <a>absurd</a> ha (<a>Multiset.not_mem_zero</a> a)", [{"full_name": "absurd", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [246, 21], "def_end_pos": [246, 27]}, {"full_name": "Multiset.not_mem_zero", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [274, 9], "def_end_pos": [274, 21]}]], "state_before": "case pos\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 0\nhb : b = 0\n\u22a2 span {a} \u2208 0", "state_after": "no goals"}, {"tactic": "suffices Prime (Ideal.span ({a} : Set R)) by\n  obtain \u27e8c, hc, hc'\u27e9 := exists_mem_normalizedFactors_of_dvd ?_ this.irreducible\n      (dvd_iff_le.mpr (span_singleton_le_span_singleton.mpr (dvd_of_mem_normalizedFactors ha)))\n  rwa [associated_iff_eq.mp hc']", "annotated_tactic": ["suffices <a>Prime</a> (<a>Ideal.span</a> ({a} : <a>Set</a> R)) by\n      obtain \u27e8c, hc, hc'\u27e9 := <a>exists_mem_normalizedFactors_of_dvd</a> ?_ this.irreducible\n          (dvd_iff_le.mpr (span_singleton_le_span_singleton.mpr (<a>dvd_of_mem_normalizedFactors</a> ha)))\n      rwa [associated_iff_eq.mp hc']", [{"full_name": "Prime", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "Ideal.span", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [115, 5], "def_end_pos": [115, 9]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "UniqueFactorizationMonoid.exists_mem_normalizedFactors_of_dvd", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [657, 9], "def_end_pos": [657, 44]}, {"full_name": "UniqueFactorizationMonoid.dvd_of_mem_normalizedFactors", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [783, 9], "def_end_pos": [783, 37]}]], "state_before": "case neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\nhb : \u00acb = 0\n\u22a2 span {a} \u2208 normalizedFactors (span {b})", "state_after": "case neg.refine_2\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\nhb : \u00acb = 0\nthis : Prime (span {a})\n\u22a2 span {b} \u2260 0\n\ncase neg.refine_1\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\nhb : \u00acb = 0\n\u22a2 Prime (span {a})"}, {"tactic": "rw [prime_iff_isPrime]", "annotated_tactic": ["rw [<a>prime_iff_isPrime</a>]", [{"full_name": "Ideal.prime_iff_isPrime", "def_path": "Mathlib/RingTheory/DedekindDomain/Ideal.lean", "def_pos": [715, 9], "def_end_pos": [715, 32]}]], "state_before": "case neg.refine_1\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\nhb : \u00acb = 0\n\u22a2 Prime (span {a})", "state_after": "case neg.refine_1\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\nhb : \u00acb = 0\n\u22a2 (span {a}).IsPrime\n\ncase neg.refine_1\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\nhb : \u00acb = 0\n\u22a2 span {a} \u2260 \u22a5"}, {"tactic": "obtain \u27e8c, hc, hc'\u27e9 := exists_mem_normalizedFactors_of_dvd ?_ this.irreducible\n    (dvd_iff_le.mpr (span_singleton_le_span_singleton.mpr (dvd_of_mem_normalizedFactors ha)))", "annotated_tactic": ["obtain \u27e8c, hc, hc'\u27e9 := <a>exists_mem_normalizedFactors_of_dvd</a> ?_ this.irreducible\n          (dvd_iff_le.mpr (span_singleton_le_span_singleton.mpr (<a>dvd_of_mem_normalizedFactors</a> ha)))", [{"full_name": "UniqueFactorizationMonoid.exists_mem_normalizedFactors_of_dvd", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [657, 9], "def_end_pos": [657, 44]}, {"full_name": "UniqueFactorizationMonoid.dvd_of_mem_normalizedFactors", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [783, 9], "def_end_pos": [783, 37]}]], "state_before": "R : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\nhb : \u00acb = 0\nthis : Prime (span {a})\n\u22a2 span {a} \u2208 normalizedFactors (span {b})", "state_after": "case intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\nhb : \u00acb = 0\nthis : Prime (span {a})\nc : Ideal R\nhc : c \u2208 normalizedFactors (span {b})\nhc' : Associated (span {a}) c\n\u22a2 span {a} \u2208 normalizedFactors (span {b})"}, {"tactic": "rwa [associated_iff_eq.mp hc']", "annotated_tactic": ["rwa [associated_iff_eq.mp hc']", []], "state_before": "case intro.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\nhb : \u00acb = 0\nthis : Prime (span {a})\nc : Ideal R\nhc : c \u2208 normalizedFactors (span {b})\nhc' : Associated (span {a}) c\n\u22a2 span {a} \u2208 normalizedFactors (span {b})", "state_after": "no goals"}, {"tactic": "by_contra h", "annotated_tactic": ["by_contra h", []], "state_before": "case neg.refine_2\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\nhb : \u00acb = 0\nthis : Prime (span {a})\n\u22a2 span {b} \u2260 0", "state_after": "case neg.refine_2\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\nhb : \u00acb = 0\nthis : Prime (span {a})\nh : span {b} = 0\n\u22a2 False"}, {"tactic": "exact hb (span_singleton_eq_bot.mp h)", "annotated_tactic": ["exact hb (span_singleton_eq_bot.mp h)", []], "state_before": "case neg.refine_2\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\nhb : \u00acb = 0\nthis : Prime (span {a})\nh : span {b} = 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exact (span_singleton_prime (prime_of_normalized_factor a ha).ne_zero).mpr\n  (prime_of_normalized_factor a ha)", "annotated_tactic": ["exact (<a>span_singleton_prime</a> (<a>prime_of_normalized_factor</a> a ha).<a>ne_zero</a>).<a>mpr</a>\n        (<a>prime_of_normalized_factor</a> a ha)", [{"full_name": "Ideal.span_singleton_prime", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [531, 9], "def_end_pos": [531, 29]}, {"full_name": "UniqueFactorizationMonoid.prime_of_normalized_factor", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [615, 9], "def_end_pos": [615, 35]}, {"full_name": "Prime.ne_zero", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [49, 9], "def_end_pos": [49, 16]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}, {"full_name": "UniqueFactorizationMonoid.prime_of_normalized_factor", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [615, 9], "def_end_pos": [615, 35]}]], "state_before": "case neg.refine_1\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\nhb : \u00acb = 0\n\u22a2 (span {a}).IsPrime", "state_after": "no goals"}, {"tactic": "by_contra h", "annotated_tactic": ["by_contra h", []], "state_before": "case neg.refine_1\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\nhb : \u00acb = 0\n\u22a2 span {a} \u2260 \u22a5", "state_after": "case neg.refine_1\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\nhb : \u00acb = 0\nh : span {a} = \u22a5\n\u22a2 False"}, {"tactic": "exact (prime_of_normalized_factor a ha).ne_zero (span_singleton_eq_bot.mp h)", "annotated_tactic": ["exact (<a>prime_of_normalized_factor</a> a ha).<a>ne_zero</a> (span_singleton_eq_bot.mp h)", [{"full_name": "UniqueFactorizationMonoid.prime_of_normalized_factor", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [615, 9], "def_end_pos": [615, 35]}, {"full_name": "Prime.ne_zero", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [49, 9], "def_end_pos": [49, 16]}]], "state_before": "case neg.refine_1\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDomain R\ninst\u271d\u00b9 : IsPrincipalIdealRing R\ninst\u271d : NormalizationMonoid R\na b : R\nha : a \u2208 normalizedFactors b\nhb : \u00acb = 0\nh : span {a} = \u22a5\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.splitOnP_cons", "start": [2345, 1], "end": [2348, 70], "traced_tactics": [{"tactic": "rw [splitOnP, splitOnP.go]", "annotated_tactic": ["rw [<a>splitOnP</a>, <a>splitOnP.go</a>]", [{"full_name": "List.splitOnP", "def_path": ".lake/packages/batteries/Batteries/Data/List/Basic.lean", "def_pos": [198, 5], "def_end_pos": [198, 13]}, {"full_name": "List.splitOnP.go", "def_path": ".lake/packages/batteries/Batteries/Data/List/Basic.lean", "def_pos": [201, 3], "def_end_pos": [201, 5]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nxs\u271d ys : List \u03b1\nls : List (List \u03b1)\nf : List \u03b1 \u2192 List \u03b1\nx : \u03b1\nxs : List \u03b1\n\u22a2 splitOnP p (x :: xs) = if p x = true then [] :: splitOnP p xs else modifyHead (cons x) (splitOnP p xs)", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nxs\u271d ys : List \u03b1\nls : List (List \u03b1)\nf : List \u03b1 \u2192 List \u03b1\nx : \u03b1\nxs : List \u03b1\n\u22a2 (if p x = true then [].reverse :: splitOnP.go p xs [] else splitOnP.go p xs [x]) =\n    if p x = true then [] :: splitOnP p xs else modifyHead (cons x) (splitOnP p xs)"}, {"tactic": "split <;> [rfl; simp [splitOnP.go_acc]]", "annotated_tactic": ["split <;> [rfl; simp [<a>splitOnP.go_acc</a>]]", [{"full_name": "List.splitOnP.go_acc", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [2331, 9], "def_end_pos": [2331, 24]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nxs\u271d ys : List \u03b1\nls : List (List \u03b1)\nf : List \u03b1 \u2192 List \u03b1\nx : \u03b1\nxs : List \u03b1\n\u22a2 (if p x = true then [].reverse :: splitOnP.go p xs [] else splitOnP.go p xs [x]) =\n    if p x = true then [] :: splitOnP p xs else modifyHead (cons x) (splitOnP p xs)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Equiv/Fin.lean", "full_name": "finSuccEquiv_zero", "start": [170, 1], "end": [171, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Monoid.lean", "full_name": "continuous_multiset_prod", "start": [780, 1], "end": [783, 37], "traced_tactics": [{"tactic": "rcases s with \u27e8l\u27e9", "annotated_tactic": ["rcases s with \u27e8l\u27e9", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\ns : Multiset \u03b9\n\u22a2 (\u2200 i \u2208 s, Continuous (f i)) \u2192 Continuous fun a => (Multiset.map (fun i => f i a) s).prod", "state_after": "case mk\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\ns : Multiset \u03b9\nl : List \u03b9\n\u22a2 (\u2200 i \u2208 Quot.mk Setoid.r l, Continuous (f i)) \u2192\n    Continuous fun a => (Multiset.map (fun i => f i a) (Quot.mk Setoid.r l)).prod"}, {"tactic": "simpa using continuous_list_prod l", "annotated_tactic": ["simpa using <a>continuous_list_prod</a> l", [{"full_name": "continuous_list_prod", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [567, 9], "def_end_pos": [567, 29]}]], "state_before": "case mk\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\ns : Multiset \u03b9\nl : List \u03b9\n\u22a2 (\u2200 i \u2208 Quot.mk Setoid.r l, Continuous (f i)) \u2192\n    Continuous fun a => (Multiset.map (fun i => f i a) (Quot.mk Setoid.r l)).prod", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "full_name": "Real.volume_univ", "start": [100, 1], "end": [104, 54], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b9 : Type u_1\ninst\u271d : Fintype \u03b9\nr : \u211d\u22650\n\u22a2 \u2191r = volume (Icc 0 \u2191r)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/Hermitian.lean", "full_name": "Matrix.IsHermitian.adjugate", "start": [262, 1], "end": [263, 83], "traced_tactics": [{"tactic": "simp [IsHermitian, adjugate_conjTranspose, hA.eq]", "annotated_tactic": ["simp [<a>IsHermitian</a>, <a>adjugate_conjTranspose</a>, hA.eq]", [{"full_name": "Matrix.IsHermitian", "def_path": "Mathlib/LinearAlgebra/Matrix/Hermitian.lean", "def_pos": [42, 5], "def_end_pos": [42, 16]}, {"full_name": "Matrix.adjugate_conjTranspose", "def_path": "Mathlib/LinearAlgebra/Matrix/Adjugate.lean", "def_pos": [462, 9], "def_end_pos": [462, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : Type u_3\nn : Type u_4\nA\u271d : Matrix n n \u03b1\ninst\u271d\u00b3 : CommRing \u03b1\ninst\u271d\u00b2 : StarRing \u03b1\ninst\u271d\u00b9 : Fintype m\ninst\u271d : DecidableEq m\nA : Matrix m m \u03b1\nhA : A.IsHermitian\n\u22a2 A.adjugate.IsHermitian", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Periodic.lean", "full_name": "Nat.filter_multiset_Ico_card_eq_of_periodic", "start": [48, 1], "end": [54, 50], "traced_tactics": [{"tactic": "rw [count_eq_card_filter_range, Finset.card, Finset.filter_val, Finset.range_val, \u2190\n  multiset_Ico_map_mod n, \u2190 map_count_True_eq_filter_card, \u2190 map_count_True_eq_filter_card,\n  map_map]", "annotated_tactic": ["rw [<a>count_eq_card_filter_range</a>, <a>Finset.card</a>, <a>Finset.filter_val</a>, <a>Finset.range_val</a>, \u2190\n    <a>multiset_Ico_map_mod</a> n, \u2190 <a>map_count_True_eq_filter_card</a>, \u2190 <a>map_count_True_eq_filter_card</a>,\n    <a>map_map</a>]", [{"full_name": "Nat.count_eq_card_filter_range", "def_path": "Mathlib/Data/Nat/Count.lean", "def_pos": [54, 9], "def_end_pos": [54, 35]}, {"full_name": "Finset.card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [42, 5], "def_end_pos": [42, 9]}, {"full_name": "Finset.filter_val", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2581, 9], "def_end_pos": [2581, 19]}, {"full_name": "Finset.range_val", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2930, 9], "def_end_pos": [2930, 18]}, {"full_name": "Nat.multiset_Ico_map_mod", "def_path": "Mathlib/Order/Interval/Finset/Nat.lean", "def_pos": [273, 9], "def_end_pos": [273, 29]}, {"full_name": "Multiset.map_count_True_eq_filter_card", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2748, 9], "def_end_pos": [2748, 38]}, {"full_name": "Multiset.map_count_True_eq_filter_card", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2748, 9], "def_end_pos": [2748, 38]}, {"full_name": "Multiset.map_map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1332, 9], "def_end_pos": [1332, 16]}]], "state_before": "n a : \u2115\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\npp : Periodic p a\n\u22a2 card (filter p (Ico n (n + a))) = count p a", "state_after": "n a : \u2115\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\npp : Periodic p a\n\u22a2 Multiset.count True (map p (Ico n (n + a))) = Multiset.count True (map (p \u2218 fun x => x % a) (Ico n (n + a)))"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "n a : \u2115\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\npp : Periodic p a\n\u22a2 Multiset.count True (map p (Ico n (n + a))) = Multiset.count True (map (p \u2218 fun x => x % a) (Ico n (n + a)))", "state_after": "case e_a.e_f\nn a : \u2115\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\npp : Periodic p a\n\u22a2 p = p \u2218 fun x => x % a"}, {"tactic": "funext n", "annotated_tactic": ["funext n", []], "state_before": "case e_a.e_f\nn a : \u2115\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\npp : Periodic p a\n\u22a2 p = p \u2218 fun x => x % a", "state_after": "case e_a.e_f.h\nn\u271d a : \u2115\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\npp : Periodic p a\nn : \u2115\n\u22a2 p n = (p \u2218 fun x => x % a) n"}, {"tactic": "exact (Function.Periodic.map_mod_nat pp n).symm", "annotated_tactic": ["exact (<a>Function.Periodic.map_mod_nat</a> pp n).<a>symm</a>", [{"full_name": "Function.Periodic.map_mod_nat", "def_path": "Mathlib/Data/Nat/Periodic.lean", "def_pos": [37, 9], "def_end_pos": [37, 45]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case e_a.e_f.h\nn\u271d a : \u2115\np : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\npp : Periodic p a\nn : \u2115\n\u22a2 p n = (p \u2218 fun x => x % a) n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "Ideal.IsPrime.radical", "start": [978, 1], "end": [979, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Mirror.lean", "full_name": "Polynomial.coeff_mul_mirror", "start": [161, 1], "end": [169, 85], "traced_tactics": [{"tactic": "rw [coeff_mul, Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk]", "annotated_tactic": ["rw [<a>coeff_mul</a>, <a>Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk</a>]", [{"full_name": "Polynomial.coeff_mul", "def_path": "Mathlib/Algebra/Polynomial/Coeff.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}, {"full_name": "Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk", "def_path": "Mathlib/Algebra/BigOperators/NatAntidiagonal.lean", "def_pos": [60, 3], "def_end_pos": [60, 14]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 (p * p.mirror).coeff (p.natDegree + p.natTrailingDegree) = p.sum fun n x => x ^ 2", "state_after": "R : Type u_1\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 \u2211 k \u2208 Finset.range (p.natDegree + p.natTrailingDegree).succ,\n      p.coeff (k, p.natDegree + p.natTrailingDegree - k).1 *\n        p.mirror.coeff (k, p.natDegree + p.natTrailingDegree - k).2 =\n    p.sum fun n x => x ^ 2"}, {"tactic": "rw [coeff_mirror, \u2190 revAt_le (Finset.mem_range_succ_iff.mp hn), revAt_invol, \u2190 sq]", "annotated_tactic": ["rw [<a>coeff_mirror</a>, \u2190 <a>revAt_le</a> (Finset.mem_range_succ_iff.mp hn), <a>revAt_invol</a>, \u2190 <a>sq</a>]", [{"full_name": "Polynomial.coeff_mirror", "def_path": "Mathlib/Algebra/Polynomial/Mirror.lean", "def_pos": [82, 9], "def_end_pos": [82, 21]}, {"full_name": "Polynomial.revAt_le", "def_path": "Mathlib/Algebra/Polynomial/Reverse.lean", "def_pos": [76, 9], "def_end_pos": [76, 17]}, {"full_name": "Polynomial.revAt_invol", "def_path": "Mathlib/Algebra/Polynomial/Reverse.lean", "def_pos": [71, 9], "def_end_pos": [71, 20]}, {"full_name": "sq", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [684, 41], "def_end_pos": [684, 43]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\np q : R[X]\nn : \u2115\nhn : n \u2208 Finset.range (p.natDegree + p.natTrailingDegree).succ\n\u22a2 p.coeff (n, p.natDegree + p.natTrailingDegree - n).1 * p.mirror.coeff (n, p.natDegree + p.natTrailingDegree - n).2 =\n    p.coeff n ^ 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Antisymmetrization.lean", "full_name": "Antisymmetrization.ind", "start": [112, 11], "end": [114, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/Basic.lean", "full_name": "Set.Ico_subset_Ico_iff", "start": [1167, 1], "end": [1171, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/LiminfLimsup.lean", "full_name": "Filter.isBoundedUnder_ge_inf", "start": [393, 1], "end": [396, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Splits.lean", "full_name": "Polynomial.map_rootOfSplits'", "start": [196, 1], "end": [198, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Factors.lean", "full_name": "Equiv.Perm.cycleOf_self_apply_pow", "start": [149, 1], "end": [151, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Parity.lean", "full_name": "even_two", "start": [94, 1], "end": [94, 73], "traced_tactics": [{"tactic": "rw [one_add_one_eq_two]", "annotated_tactic": ["rw [<a>one_add_one_eq_two</a>]", [{"full_name": "one_add_one_eq_two", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [231, 9], "def_end_pos": [231, 27]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nR : Type u_4\ninst\u271d\u00b9 : Semiring \u03b1\ninst\u271d : Semiring \u03b2\na b : \u03b1\nm n : \u2115\n\u22a2 2 = 1 + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matrix/Basic.lean", "full_name": "Matrix.conjTranspose_smul", "start": [2376, 1], "end": [2378, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Basic.lean", "full_name": "Submodule.convex", "start": [652, 11], "end": [654, 68], "traced_tactics": [{"tactic": "repeat' intro", "annotated_tactic": ["repeat' intro", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\nK : Submodule \ud835\udd5c E\n\u22a2 Convex \ud835\udd5c \u2191K", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\nK : Submodule \ud835\udd5c E\nx\u271d : E\na\u271d\u2075 : x\u271d \u2208 \u2191K\ny\u271d : E\na\u271d\u2074 : y\u271d \u2208 \u2191K\na\u271d\u00b3 b\u271d : \ud835\udd5c\na\u271d\u00b2 : 0 \u2264 a\u271d\u00b3\na\u271d\u00b9 : 0 \u2264 b\u271d\na\u271d : a\u271d\u00b3 + b\u271d = 1\n\u22a2 a\u271d\u00b3 \u2022 x\u271d + b\u271d \u2022 y\u271d \u2208 \u2191K"}, {"tactic": "refine add_mem (smul_mem _ _ ?_) (smul_mem _ _ ?_) <;> assumption", "annotated_tactic": ["refine <a>add_mem</a> (<a>smul_mem</a> _ _ ?_) (<a>smul_mem</a> _ _ ?_) <;> assumption", [{"full_name": "AddMemClass.add_mem", "def_path": "Mathlib/Algebra/Group/Subsemigroup/Basic.lean", "def_pos": [72, 3], "def_end_pos": [72, 10]}, {"full_name": "Submodule.smul_mem", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [224, 9], "def_end_pos": [224, 17]}, {"full_name": "Submodule.smul_mem", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [224, 9], "def_end_pos": [224, 17]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\nK : Submodule \ud835\udd5c E\nx\u271d : E\na\u271d\u2075 : x\u271d \u2208 \u2191K\ny\u271d : E\na\u271d\u2074 : y\u271d \u2208 \u2191K\na\u271d\u00b3 b\u271d : \ud835\udd5c\na\u271d\u00b2 : 0 \u2264 a\u271d\u00b3\na\u271d\u00b9 : 0 \u2264 b\u271d\na\u271d : a\u271d\u00b3 + b\u271d = 1\n\u22a2 a\u271d\u00b3 \u2022 x\u271d + b\u271d \u2022 y\u271d \u2208 \u2191K", "state_after": "no goals"}, {"tactic": "intro", "annotated_tactic": ["intro", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\nK : Submodule \ud835\udd5c E\nx\u271d : E\na\u271d\u2074 : x\u271d \u2208 \u2191K\ny\u271d : E\na\u271d\u00b3 : y\u271d \u2208 \u2191K\na\u271d\u00b2 b\u271d : \ud835\udd5c\na\u271d\u00b9 : 0 \u2264 a\u271d\u00b2\na\u271d : 0 \u2264 b\u271d\n\u22a2 a\u271d\u00b2 + b\u271d = 1 \u2192 a\u271d\u00b2 \u2022 x\u271d + b\u271d \u2022 y\u271d \u2208 \u2191K", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u00b2 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b9 : AddCommMonoid E\ninst\u271d : Module \ud835\udd5c E\nK : Submodule \ud835\udd5c E\nx\u271d : E\na\u271d\u2075 : x\u271d \u2208 \u2191K\ny\u271d : E\na\u271d\u2074 : y\u271d \u2208 \u2191K\na\u271d\u00b3 b\u271d : \ud835\udd5c\na\u271d\u00b2 : 0 \u2264 a\u271d\u00b3\na\u271d\u00b9 : 0 \u2264 b\u271d\na\u271d : a\u271d\u00b3 + b\u271d = 1\n\u22a2 a\u271d\u00b3 \u2022 x\u271d + b\u271d \u2022 y\u271d \u2208 \u2191K"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/NNReal/Basic.lean", "full_name": "Real.toNNReal_eq_natCast", "start": [671, 1], "end": [673, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Notation.lean", "full_name": "ONote.oadd_mul", "start": [585, 1], "end": [588, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Int/Defs.lean", "full_name": "Int.natCast_nonpos_iff", "start": [139, 1], "end": [139, 75], "traced_tactics": [{"tactic": "omega", "annotated_tactic": ["omega", []], "state_before": "a b c d m n\u271d : \u2124\nn : \u2115\n\u22a2 \u2191n \u2264 0 \u2194 n = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean", "full_name": "HurwitzZeta.hurwitzEvenFEPair_neg", "start": [305, 1], "end": [307, 88], "traced_tactics": [{"tactic": "unfold hurwitzEvenFEPair", "annotated_tactic": ["unfold <a>hurwitzEvenFEPair</a>", [{"full_name": "HurwitzZeta.hurwitzEvenFEPair", "def_path": "Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean", "def_pos": [271, 5], "def_end_pos": [271, 22]}]], "state_before": "a : UnitAddCircle\n\u22a2 hurwitzEvenFEPair (-a) = hurwitzEvenFEPair a", "state_after": "a : UnitAddCircle\n\u22a2 { f := ofReal' \u2218 evenKernel (-a), g := ofReal' \u2218 cosKernel (-a), k := 1 / 2, \u03b5 := 1, f\u2080 := if -a = 0 then 1 else 0,\n      g\u2080 := 1, hf_int := \u22ef, hg_int := \u22ef, hk := hurwitzEvenFEPair.proof_4, h\u03b5 := hurwitzEvenFEPair.proof_5, h_feq := \u22ef,\n      hf_top := \u22ef, hg_top := \u22ef } =\n    { f := ofReal' \u2218 evenKernel a, g := ofReal' \u2218 cosKernel a, k := 1 / 2, \u03b5 := 1, f\u2080 := if a = 0 then 1 else 0,\n      g\u2080 := 1, hf_int := \u22ef, hg_int := \u22ef, hk := hurwitzEvenFEPair.proof_4, h\u03b5 := hurwitzEvenFEPair.proof_5, h_feq := \u22ef,\n      hf_top := \u22ef, hg_top := \u22ef }"}, {"tactic": "congr 1 <;> simp only [Function.comp_def, evenKernel_neg, cosKernel_neg, neg_eq_zero]", "annotated_tactic": ["congr 1 <;> simp only [<a>Function.comp_def</a>, <a>evenKernel_neg</a>, <a>cosKernel_neg</a>, <a>neg_eq_zero</a>]", [{"full_name": "Function.comp_def", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [37, 9], "def_end_pos": [37, 26]}, {"full_name": "HurwitzZeta.evenKernel_neg", "def_path": "Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean", "def_pos": [108, 7], "def_end_pos": [108, 21]}, {"full_name": "HurwitzZeta.cosKernel_neg", "def_path": "Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean", "def_pos": [113, 7], "def_end_pos": [113, 20]}, {"full_name": "neg_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [634, 3], "def_end_pos": [634, 14]}]], "state_before": "a : UnitAddCircle\n\u22a2 { f := ofReal' \u2218 evenKernel (-a), g := ofReal' \u2218 cosKernel (-a), k := 1 / 2, \u03b5 := 1, f\u2080 := if -a = 0 then 1 else 0,\n      g\u2080 := 1, hf_int := \u22ef, hg_int := \u22ef, hk := hurwitzEvenFEPair.proof_4, h\u03b5 := hurwitzEvenFEPair.proof_5, h_feq := \u22ef,\n      hf_top := \u22ef, hg_top := \u22ef } =\n    { f := ofReal' \u2218 evenKernel a, g := ofReal' \u2218 cosKernel a, k := 1 / 2, \u03b5 := 1, f\u2080 := if a = 0 then 1 else 0,\n      g\u2080 := 1, hf_int := \u22ef, hg_int := \u22ef, hk := hurwitzEvenFEPair.proof_4, h\u03b5 := hurwitzEvenFEPair.proof_5, h_feq := \u22ef,\n      hf_top := \u22ef, hg_top := \u22ef }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.mul_nonempty", "start": [427, 1], "end": [428, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "full_name": "MeasureTheory.Ico_ae_eq_Ioc", "start": [453, 1], "end": [454, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Decomposition/RadonNikodym.lean", "full_name": "MeasurableEmbedding.singularPart_map", "start": [466, 1], "end": [474, 68], "traced_tactics": [{"tactic": "have h_add : \u03bc.map f = (\u03bc.singularPart \u03bd).map f\n    + (\u03bd.map f).withDensity ((\u03bc.map f).rnDeriv (\u03bd.map f)) := by\n  conv_lhs => rw [\u03bc.haveLebesgueDecomposition_add \u03bd]\n  rw [Measure.map_add _ _ hf.measurable, \u2190 hf.map_withDensity_rnDeriv \u03bc \u03bd]", "annotated_tactic": ["have h_add : \u03bc.map f = (\u03bc.singularPart \u03bd).<a>map</a> f\n      + (\u03bd.map f).<a>withDensity</a> ((\u03bc.map f).<a>rnDeriv</a> (\u03bd.map f)) := by\n    conv_lhs => rw [\u03bc.haveLebesgueDecomposition_add \u03bd]\n    rw [<a>Measure.map_add</a> _ _ hf.measurable, \u2190 hf.map_withDensity_rnDeriv \u03bc \u03bd]", [{"full_name": "MeasureTheory.Measure.map", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1212, 17], "def_end_pos": [1212, 20]}, {"full_name": "MeasureTheory.Measure.withDensity", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [33, 5], "def_end_pos": [33, 24]}, {"full_name": "MeasureTheory.Measure.rnDeriv", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [80, 31], "def_end_pos": [80, 38]}, {"full_name": "MeasureTheory.Measure.map_add", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1227, 9], "def_end_pos": [1227, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\n\u22a2 (map f \u03bc).singularPart (map f \u03bd) = map f (\u03bc.singularPart \u03bd)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nh_add : map f \u03bc = map f (\u03bc.singularPart \u03bd) + (map f \u03bd).withDensity ((map f \u03bc).rnDeriv (map f \u03bd))\n\u22a2 (map f \u03bc).singularPart (map f \u03bd) = map f (\u03bc.singularPart \u03bd)"}, {"tactic": "refine (Measure.eq_singularPart (Measure.measurable_rnDeriv _ _) ?_ h_add).symm", "annotated_tactic": ["refine (<a>Measure.eq_singularPart</a> (<a>Measure.measurable_rnDeriv</a> _ _) ?_ h_add).<a>symm</a>", [{"full_name": "MeasureTheory.Measure.eq_singularPart", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [395, 9], "def_end_pos": [395, 24]}, {"full_name": "MeasureTheory.Measure.measurable_rnDeriv", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [102, 9], "def_end_pos": [102, 27]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nh_add : map f \u03bc = map f (\u03bc.singularPart \u03bd) + (map f \u03bd).withDensity ((map f \u03bc).rnDeriv (map f \u03bd))\n\u22a2 (map f \u03bc).singularPart (map f \u03bd) = map f (\u03bc.singularPart \u03bd)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nh_add : map f \u03bc = map f (\u03bc.singularPart \u03bd) + (map f \u03bd).withDensity ((map f \u03bc).rnDeriv (map f \u03bd))\n\u22a2 map f (\u03bc.singularPart \u03bd) \u27c2\u2098 map f \u03bd"}, {"tactic": "exact hf.mutuallySingular_map (\u03bc.mutuallySingular_singularPart \u03bd)", "annotated_tactic": ["exact hf.mutuallySingular_map (\u03bc.mutuallySingular_singularPart \u03bd)", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\nh_add : map f \u03bc = map f (\u03bc.singularPart \u03bd) + (map f \u03bd).withDensity ((map f \u03bc).rnDeriv (map f \u03bd))\n\u22a2 map f (\u03bc.singularPart \u03bd) \u27c2\u2098 map f \u03bd", "state_after": "no goals"}, {"tactic": "conv_lhs => rw [\u03bc.haveLebesgueDecomposition_add \u03bd]", "annotated_tactic": ["conv_lhs => rw [\u03bc.haveLebesgueDecomposition_add \u03bd]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\n\u22a2 map f \u03bc = map f (\u03bc.singularPart \u03bd) + (map f \u03bd).withDensity ((map f \u03bc).rnDeriv (map f \u03bd))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\n\u22a2 map f (\u03bc.singularPart \u03bd + \u03bd.withDensity (\u03bc.rnDeriv \u03bd)) =\n    map f (\u03bc.singularPart \u03bd) + (map f \u03bd).withDensity ((map f \u03bc).rnDeriv (map f \u03bd))"}, {"tactic": "rw [Measure.map_add _ _ hf.measurable, \u2190 hf.map_withDensity_rnDeriv \u03bc \u03bd]", "annotated_tactic": ["rw [<a>Measure.map_add</a> _ _ hf.measurable, \u2190 hf.map_withDensity_rnDeriv \u03bc \u03bd]", [{"full_name": "MeasureTheory.Measure.map_add", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1227, 9], "def_end_pos": [1227, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc\u271d \u03bd\u271d : Measure \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : MeasurableEmbedding f\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u00b9 : SigmaFinite \u03bc\ninst\u271d : SigmaFinite \u03bd\n\u22a2 map f (\u03bc.singularPart \u03bd + \u03bd.withDensity (\u03bc.rnDeriv \u03bd)) =\n    map f (\u03bc.singularPart \u03bd) + (map f \u03bd).withDensity ((map f \u03bc).rnDeriv (map f \u03bd))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Invariants.lean", "full_name": "MeasurableSpace.measurable_invariants_dom", "start": [58, 1], "end": [60, 44], "traced_tactics": [{"tactic": "simp only [Measurable, \u2190 forall_and]", "annotated_tactic": ["simp only [<a>Measurable</a>, \u2190 <a>forall_and</a>]", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [551, 5], "def_end_pos": [551, 15]}, {"full_name": "forall_and", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [256, 9], "def_end_pos": [256, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b1\ng : \u03b1 \u2192 \u03b2\n\u22a2 Measurable g \u2194 Measurable g \u2227 \u2200 (s : Set \u03b2), MeasurableSet s \u2192 g \u2218 f \u207b\u00b9' s = g \u207b\u00b9' s", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b1\ng : \u03b1 \u2192 \u03b2\n\u22a2 (\u2200 \u2983t : Set \u03b2\u2984, MeasurableSet t \u2192 MeasurableSet (g \u207b\u00b9' t)) \u2194\n    \u2200 (x : Set \u03b2), MeasurableSet x \u2192 MeasurableSet (g \u207b\u00b9' x) \u2227 g \u2218 f \u207b\u00b9' x = g \u207b\u00b9' x"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b2\nf : \u03b1 \u2192 \u03b1\ng : \u03b1 \u2192 \u03b2\n\u22a2 (\u2200 \u2983t : Set \u03b2\u2984, MeasurableSet t \u2192 MeasurableSet (g \u207b\u00b9' t)) \u2194\n    \u2200 (x : Set \u03b2), MeasurableSet x \u2192 MeasurableSet (g \u207b\u00b9' x) \u2227 g \u2218 f \u207b\u00b9' x = g \u207b\u00b9' x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean", "full_name": "not_integrableOn_Ioi_rpow", "start": [93, 1], "end": [101, 25], "traced_tactics": [{"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "s : \u211d\n\u22a2 \u00acIntegrableOn (fun x => x ^ s) (Ioi 0) volume", "state_after": "s : \u211d\nh : IntegrableOn (fun x => x ^ s) (Ioi 0) volume\n\u22a2 False"}, {"tactic": "rcases le_or_lt s (-1) with hs|hs", "annotated_tactic": ["rcases <a>le_or_lt</a> s (-1) with hs|hs", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [342, 9], "def_end_pos": [342, 17]}]], "state_before": "s : \u211d\nh : IntegrableOn (fun x => x ^ s) (Ioi 0) volume\n\u22a2 False", "state_after": "case inl\ns : \u211d\nh : IntegrableOn (fun x => x ^ s) (Ioi 0) volume\nhs : s \u2264 -1\n\u22a2 False\n\ncase inr\ns : \u211d\nh : IntegrableOn (fun x => x ^ s) (Ioi 0) volume\nhs : -1 < s\n\u22a2 False"}, {"tactic": "have : IntegrableOn (fun x \u21a6 x ^ s) (Ioo (0 : \u211d) 1) := h.mono Ioo_subset_Ioi_self le_rfl", "annotated_tactic": ["have : <a>IntegrableOn</a> (fun x \u21a6 x ^ s) (<a>Ioo</a> (0 : \u211d) 1) := h.mono <a>Ioo_subset_Ioi_self</a> <a>le_rfl</a>", [{"full_name": "MeasureTheory.IntegrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [90, 5], "def_end_pos": [90, 17]}, {"full_name": "Set.Ioo", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "Set.Ioo_subset_Ioi_self", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [545, 9], "def_end_pos": [545, 28]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}]], "state_before": "case inl\ns : \u211d\nh : IntegrableOn (fun x => x ^ s) (Ioi 0) volume\nhs : s \u2264 -1\n\u22a2 False", "state_after": "case inl\ns : \u211d\nh : IntegrableOn (fun x => x ^ s) (Ioi 0) volume\nhs : s \u2264 -1\nthis : IntegrableOn (fun x => x ^ s) (Ioo 0 1) volume\n\u22a2 False"}, {"tactic": "rw [integrableOn_Ioo_rpow_iff zero_lt_one] at this", "annotated_tactic": ["rw [<a>integrableOn_Ioo_rpow_iff</a> <a>zero_lt_one</a>] at this", [{"full_name": "intervalIntegral.integrableOn_Ioo_rpow_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Integrals.lean", "def_pos": [99, 7], "def_end_pos": [99, 32]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "case inl\ns : \u211d\nh : IntegrableOn (fun x => x ^ s) (Ioi 0) volume\nhs : s \u2264 -1\nthis : IntegrableOn (fun x => x ^ s) (Ioo 0 1) volume\n\u22a2 False", "state_after": "case inl\ns : \u211d\nh : IntegrableOn (fun x => x ^ s) (Ioi 0) volume\nhs : s \u2264 -1\nthis : -1 < s\n\u22a2 False"}, {"tactic": "exact hs.not_lt this", "annotated_tactic": ["exact hs.not_lt this", []], "state_before": "case inl\ns : \u211d\nh : IntegrableOn (fun x => x ^ s) (Ioi 0) volume\nhs : s \u2264 -1\nthis : -1 < s\n\u22a2 False", "state_after": "no goals"}, {"tactic": "have : IntegrableOn (fun x \u21a6 x ^ s) (Ioi (1 : \u211d)) := h.mono (Ioi_subset_Ioi zero_le_one) le_rfl", "annotated_tactic": ["have : <a>IntegrableOn</a> (fun x \u21a6 x ^ s) (<a>Ioi</a> (1 : \u211d)) := h.mono (<a>Ioi_subset_Ioi</a> <a>zero_le_one</a>) <a>le_rfl</a>", [{"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/Order/Interval/Set/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "Set.Ioi_subset_Ioi", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [615, 9], "def_end_pos": [615, 23]}, {"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/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}]], "state_before": "case inr\ns : \u211d\nh : IntegrableOn (fun x => x ^ s) (Ioi 0) volume\nhs : -1 < s\n\u22a2 False", "state_after": "case inr\ns : \u211d\nh : IntegrableOn (fun x => x ^ s) (Ioi 0) volume\nhs : -1 < s\nthis : IntegrableOn (fun x => x ^ s) (Ioi 1) volume\n\u22a2 False"}, {"tactic": "rw [integrableOn_Ioi_rpow_iff zero_lt_one] at this", "annotated_tactic": ["rw [<a>integrableOn_Ioi_rpow_iff</a> <a>zero_lt_one</a>] at this", [{"full_name": "integrableOn_Ioi_rpow_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/ImproperIntegrals.lean", "def_pos": [76, 9], "def_end_pos": [76, 34]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "case inr\ns : \u211d\nh : IntegrableOn (fun x => x ^ s) (Ioi 0) volume\nhs : -1 < s\nthis : IntegrableOn (fun x => x ^ s) (Ioi 1) volume\n\u22a2 False", "state_after": "case inr\ns : \u211d\nh : IntegrableOn (fun x => x ^ s) (Ioi 0) volume\nhs : -1 < s\nthis : s < -1\n\u22a2 False"}, {"tactic": "exact hs.not_lt this", "annotated_tactic": ["exact hs.not_lt this", []], "state_before": "case inr\ns : \u211d\nh : IntegrableOn (fun x => x ^ s) (Ioi 0) volume\nhs : -1 < s\nthis : s < -1\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monoidal/Functor.lean", "full_name": "CategoryTheory.MonoidalFunctor.map_rightUnitor", "start": [390, 1], "end": [396, 7], "traced_tactics": [{"tactic": "simp only [LaxMonoidalFunctor.right_unitality]", "annotated_tactic": ["simp only [<a>LaxMonoidalFunctor.right_unitality</a>]", [{"full_name": "CategoryTheory.LaxMonoidalFunctor.right_unitality", "def_path": "Mathlib/CategoryTheory/Monoidal/Functor.lean", "def_pos": [86, 3], "def_end_pos": [86, 18]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : MonoidalFunctor C D\nX : C\n\u22a2 F.map (\u03c1_ X).hom = inv (F.\u03bc X (\ud835\udfd9_ C)) \u226b F.obj X \u25c1 inv F.\u03b5 \u226b (\u03c1_ (F.obj X)).hom", "state_after": "C : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : MonoidalFunctor C D\nX : C\n\u22a2 F.map (\u03c1_ X).hom = inv (F.\u03bc X (\ud835\udfd9_ C)) \u226b F.obj X \u25c1 inv F.\u03b5 \u226b F.obj X \u25c1 F.\u03b5 \u226b F.\u03bc X (\ud835\udfd9_ C) \u226b F.map (\u03c1_ X).hom"}, {"tactic": "slice_rhs 2 3 =>\n  rw [\u2190 MonoidalCategory.whiskerLeft_comp]\n  simp", "annotated_tactic": ["slice_rhs 2 3 =>\n    rw [\u2190 <a>MonoidalCategory.whiskerLeft_comp</a>]\n    simp", [{"full_name": "CategoryTheory.MonoidalCategory.whiskerLeft_comp", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [235, 9], "def_end_pos": [235, 25]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : MonoidalFunctor C D\nX : C\n\u22a2 F.map (\u03c1_ X).hom = inv (F.\u03bc X (\ud835\udfd9_ C)) \u226b F.obj X \u25c1 inv F.\u03b5 \u226b F.obj X \u25c1 F.\u03b5 \u226b F.\u03bc X (\ud835\udfd9_ C) \u226b F.map (\u03c1_ X).hom", "state_after": "C : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : MonoidalFunctor C D\nX : C\n\u22a2 F.map (\u03c1_ X).hom = inv (F.\u03bc X (\ud835\udfd9_ C)) \u226b (\ud835\udfd9 (F.obj X \u2297 F.obj (\ud835\udfd9_ C)) \u226b F.\u03bc X (\ud835\udfd9_ C)) \u226b F.map (\u03c1_ X).hom"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b2 : MonoidalCategory C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : MonoidalCategory D\nF : MonoidalFunctor C D\nX : C\n\u22a2 F.map (\u03c1_ X).hom = inv (F.\u03bc X (\ud835\udfd9_ C)) \u226b (\ud835\udfd9 (F.obj X \u2297 F.obj (\ud835\udfd9_ C)) \u226b F.\u03bc X (\ud835\udfd9_ C)) \u226b F.map (\u03c1_ X).hom", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monoidal/FunctorCategory.lean", "full_name": "CategoryTheory.Monoidal.tensorObj_obj", "start": [103, 1], "end": [104, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/HahnSeries/Summable.lean", "full_name": "HahnSeries.SummableFamily.finite_co_support", "start": [138, 1], "end": [140, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Constructions.lean", "full_name": 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M\ninst\u271d\u00b2 : Module R M\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\ng : TensorAlgebra R M \u2192\u2090[R] A\n\u22a2 (lift R) (g.toLinearMap \u2218\u2097 \u03b9 R) = g", "state_after": "R : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Type u_2\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\ng : TensorAlgebra R M \u2192\u2090[R] A\n\u22a2 (lift R) ((lift R).symm g) = g"}, {"tactic": "exact (lift R).apply_symm_apply g", "annotated_tactic": ["exact (<a>lift</a> R).<a>apply_symm_apply</a> g", [{"full_name": "TensorAlgebra.lift", "def_path": "Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean", "def_pos": [129, 5], "def_end_pos": [129, 9]}, {"full_name": "Equiv.apply_symm_apply", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [279, 17], "def_end_pos": [279, 33]}]], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommSemiring R\nM : Type u_2\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\nA : Type u_3\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\ng : TensorAlgebra R M \u2192\u2090[R] A\n\u22a2 (lift R) ((lift R).symm g) = g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Periodic.lean", "full_name": "Function.Periodic.comp_addHom", "start": [59, 1], "end": [61, 49], "traced_tactics": [{"tactic": "simp only [hg c, h (g x), map_add, comp_apply]", "annotated_tactic": ["simp only [hg c, h (g x), <a>map_add</a>, <a>comp_apply</a>]", [{"full_name": "map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [308, 3], "def_end_pos": [308, 14]}, {"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf g\u271d : \u03b1 \u2192 \u03b2\nc c\u2081 c\u2082 x\u271d : \u03b1\ninst\u271d\u00b9 : Add \u03b1\ninst\u271d : Add \u03b3\nh : Periodic f c\ng : AddHom \u03b3 \u03b1\ng_inv : \u03b1 \u2192 \u03b3\nhg : RightInverse g_inv \u21d1g\nx : \u03b3\n\u22a2 (f \u2218 \u21d1g) (x + g_inv c) = (f \u2218 \u21d1g) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean", "full_name": "Real.Angle.expMapCircle_add", "start": [142, 1], "end": [146, 36], "traced_tactics": [{"tactic": "induction \u03b8\u2081 using Real.Angle.induction_on", "annotated_tactic": ["induction \u03b8\u2081 using <a>Real.Angle.induction_on</a>", [{"full_name": "Real.Angle.induction_on", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [74, 19], "def_end_pos": [74, 31]}]], "state_before": "\u03b8\u2081 \u03b8\u2082 : Angle\n\u22a2 (\u03b8\u2081 + \u03b8\u2082).expMapCircle = \u03b8\u2081.expMapCircle * \u03b8\u2082.expMapCircle", "state_after": "case h\n\u03b8\u2082 : Angle\nx\u271d : \u211d\n\u22a2 (\u2191x\u271d + \u03b8\u2082).expMapCircle = (\u2191x\u271d).expMapCircle * \u03b8\u2082.expMapCircle"}, {"tactic": "induction \u03b8\u2082 using Real.Angle.induction_on", "annotated_tactic": ["induction \u03b8\u2082 using <a>Real.Angle.induction_on</a>", [{"full_name": "Real.Angle.induction_on", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [74, 19], "def_end_pos": [74, 31]}]], "state_before": "case h\n\u03b8\u2082 : Angle\nx\u271d : \u211d\n\u22a2 (\u2191x\u271d + \u03b8\u2082).expMapCircle = (\u2191x\u271d).expMapCircle * \u03b8\u2082.expMapCircle", "state_after": "case h.h\nx\u271d\u00b9 x\u271d : \u211d\n\u22a2 (\u2191x\u271d\u00b9 + \u2191x\u271d).expMapCircle = (\u2191x\u271d\u00b9).expMapCircle * (\u2191x\u271d).expMapCircle"}, {"tactic": "exact _root_.expMapCircle_add _ _", "annotated_tactic": ["exact <a>_root_.expMapCircle_add</a> _ _", [{"full_name": "expMapCircle_add", "def_path": "Mathlib/Analysis/Complex/Circle.lean", "def_pos": [136, 9], "def_end_pos": [136, 25]}]], "state_before": "case h.h\nx\u271d\u00b9 x\u271d : \u211d\n\u22a2 (\u2191x\u271d\u00b9 + \u2191x\u271d).expMapCircle = (\u2191x\u271d\u00b9).expMapCircle * (\u2191x\u271d).expMapCircle", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.card_attach", "start": [1623, 1], "end": [1624, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Rat/Cast/Order.lean", "full_name": "Rat.preimage_cast_Iic", "start": [110, 1], "end": [111, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Cover.lean", "full_name": "CovBy.of_le_of_lt", "start": [302, 1], "end": [303, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.union_diff_left", "start": [1805, 1], "end": [1806, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Basic.lean", "full_name": "SimpleGraph.commonNeighbors_top_eq", "start": [863, 1], "end": [866, 42], "traced_tactics": [{"tactic": "ext u", "annotated_tactic": ["ext u", []], "state_before": "\u03b9 : Sort u_1\nV : Type u\nG : SimpleGraph V\na b c u v\u271d w\u271d : V\ne : Sym2 V\nv w : V\n\u22a2 \u22a4.commonNeighbors v w = Set.univ \\ {v, w}", "state_after": "case h\n\u03b9 : Sort u_1\nV : Type u\nG : SimpleGraph V\na b c u\u271d v\u271d w\u271d : V\ne : Sym2 V\nv w u : V\n\u22a2 u \u2208 \u22a4.commonNeighbors v w \u2194 u \u2208 Set.univ \\ {v, w}"}, {"tactic": "simp [commonNeighbors, eq_comm, not_or]", "annotated_tactic": ["simp [<a>commonNeighbors</a>, <a>eq_comm</a>, <a>not_or</a>]", [{"full_name": "SimpleGraph.commonNeighbors", "def_path": "Mathlib/Combinatorics/SimpleGraph/Basic.lean", "def_pos": [824, 5], "def_end_pos": [824, 20]}, {"full_name": "eq_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}, {"full_name": "not_or", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [134, 17], "def_end_pos": [134, 23]}]], "state_before": "case h\n\u03b9 : Sort u_1\nV : Type u\nG : SimpleGraph V\na b c u\u271d v\u271d w\u271d : V\ne : Sym2 V\nv w u : V\n\u22a2 u \u2208 \u22a4.commonNeighbors v w \u2194 u \u2208 Set.univ \\ {v, w}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean", "full_name": "MeasureTheory.snorm_sum_le", "start": [168, 1], "end": [173, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Defs.lean", "full_name": "Nat.div_pos", "start": [553, 1], "end": [558, 19], "traced_tactics": [{"tactic": "simpa [h] using (mod_add_div a b).symm", "annotated_tactic": ["simpa [h] using (<a>mod_add_div</a> a b).<a>symm</a>", [{"full_name": "Nat.mod_add_div", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [201, 9], "def_end_pos": [201, 20]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "a b c d m n k : \u2115\np q : \u2115 \u2192 Prop\nhba : b \u2264 a\nhb : 0 < b\nh : a / b = 0\n\u22a2 a = a % b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/GromovHausdorff.lean", "full_name": "GromovHausdorff.ghDist_eq_hausdorffDist", "start": [398, 1], "end": [411, 70], "traced_tactics": [{"tactic": "let F := kuratowskiEmbedding (OptimalGHCoupling X Y)", "annotated_tactic": ["let F := <a>kuratowskiEmbedding</a> (<a>OptimalGHCoupling</a> X Y)", [{"full_name": "kuratowskiEmbedding", "def_path": "Mathlib/Topology/MetricSpace/Kuratowski.lean", "def_pos": [111, 5], "def_end_pos": [111, 24]}, {"full_name": "GromovHausdorff.OptimalGHCoupling", "def_path": "Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean", "def_pos": [457, 5], "def_end_pos": [457, 22]}]], "state_before": "X : Type u\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\nY : Type v\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\n\u22a2 \u2203 \u03a6 \u03a8, Isometry \u03a6 \u2227 Isometry \u03a8 \u2227 ghDist X Y = hausdorffDist (range \u03a6) (range \u03a8)", "state_after": "X : Type u\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\nY : Type v\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nF : OptimalGHCoupling X Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := kuratowskiEmbedding (OptimalGHCoupling X Y)\n\u22a2 \u2203 \u03a6 \u03a8, Isometry \u03a6 \u2227 Isometry \u03a8 \u2227 ghDist X Y = hausdorffDist (range \u03a6) (range \u03a8)"}, {"tactic": "let \u03a6 := F \u2218 optimalGHInjl X Y", "annotated_tactic": ["let \u03a6 := F \u2218 <a>optimalGHInjl</a> X Y", [{"full_name": "GromovHausdorff.optimalGHInjl", "def_path": "Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean", "def_pos": [466, 5], "def_end_pos": [466, 18]}]], "state_before": "X : Type u\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\nY : Type v\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nF : OptimalGHCoupling X Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := kuratowskiEmbedding (OptimalGHCoupling X Y)\n\u22a2 \u2203 \u03a6 \u03a8, Isometry \u03a6 \u2227 Isometry \u03a8 \u2227 ghDist X Y = hausdorffDist (range \u03a6) (range \u03a8)", "state_after": "X : Type u\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\nY : Type v\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nF : OptimalGHCoupling X Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := kuratowskiEmbedding (OptimalGHCoupling X Y)\n\u03a6 : X \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjl X Y\n\u22a2 \u2203 \u03a6 \u03a8, Isometry \u03a6 \u2227 Isometry \u03a8 \u2227 ghDist X Y = hausdorffDist (range \u03a6) (range \u03a8)"}, {"tactic": "let \u03a8 := F \u2218 optimalGHInjr X Y", "annotated_tactic": ["let \u03a8 := F \u2218 <a>optimalGHInjr</a> X Y", [{"full_name": "GromovHausdorff.optimalGHInjr", "def_path": "Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean", "def_pos": [476, 5], "def_end_pos": [476, 18]}]], "state_before": "X : Type u\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\nY : Type v\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nF : OptimalGHCoupling X Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := kuratowskiEmbedding (OptimalGHCoupling X Y)\n\u03a6 : X \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjl X Y\n\u22a2 \u2203 \u03a6 \u03a8, Isometry \u03a6 \u2227 Isometry \u03a8 \u2227 ghDist X Y = hausdorffDist (range \u03a6) (range \u03a8)", "state_after": "X : Type u\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\nY : Type v\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nF : OptimalGHCoupling X Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := kuratowskiEmbedding (OptimalGHCoupling X Y)\n\u03a6 : X \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjl X Y\n\u03a8 : Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjr X Y\n\u22a2 \u2203 \u03a6 \u03a8, Isometry \u03a6 \u2227 Isometry \u03a8 \u2227 ghDist X Y = hausdorffDist (range \u03a6) (range \u03a8)"}, {"tactic": "refine \u27e8\u03a6, \u03a8, ?_, ?_, ?_\u27e9", "annotated_tactic": ["refine \u27e8\u03a6, \u03a8, ?_, ?_, ?_\u27e9", []], "state_before": "X : Type u\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\nY : Type v\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nF : OptimalGHCoupling X Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := kuratowskiEmbedding (OptimalGHCoupling X Y)\n\u03a6 : X \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjl X Y\n\u03a8 : Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjr X Y\n\u22a2 \u2203 \u03a6 \u03a8, Isometry \u03a6 \u2227 Isometry \u03a8 \u2227 ghDist X Y = hausdorffDist (range \u03a6) (range \u03a8)", "state_after": "case refine_1\nX : Type u\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\nY : Type v\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nF : OptimalGHCoupling X Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := kuratowskiEmbedding (OptimalGHCoupling X Y)\n\u03a6 : X \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjl X Y\n\u03a8 : Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjr X Y\n\u22a2 Isometry \u03a6\n\ncase refine_2\nX : Type u\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\nY : Type v\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nF : OptimalGHCoupling X Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := kuratowskiEmbedding (OptimalGHCoupling X Y)\n\u03a6 : X \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjl X Y\n\u03a8 : Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjr X Y\n\u22a2 Isometry \u03a8\n\ncase refine_3\nX : Type u\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\nY : Type v\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nF : OptimalGHCoupling X Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := kuratowskiEmbedding (OptimalGHCoupling X Y)\n\u03a6 : X \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjl X Y\n\u03a8 : Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjr X Y\n\u22a2 ghDist X Y = hausdorffDist (range \u03a6) (range \u03a8)"}, {"tactic": "exact (kuratowskiEmbedding.isometry _).comp (isometry_optimalGHInjl X Y)", "annotated_tactic": ["exact (<a>kuratowskiEmbedding.isometry</a> _).<a>comp</a> (<a>isometry_optimalGHInjl</a> X Y)", [{"full_name": "kuratowskiEmbedding.isometry", "def_path": "Mathlib/Topology/MetricSpace/Kuratowski.lean", "def_pos": [118, 19], "def_end_pos": [118, 47]}, {"full_name": "Isometry.comp", "def_path": "Mathlib/Topology/MetricSpace/Isometry.lean", "def_pos": [109, 9], "def_end_pos": [109, 13]}, {"full_name": "GromovHausdorff.isometry_optimalGHInjl", "def_path": "Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean", "def_pos": [471, 9], "def_end_pos": [471, 31]}]], "state_before": "case refine_1\nX : Type u\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\nY : Type v\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nF : OptimalGHCoupling X Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := kuratowskiEmbedding (OptimalGHCoupling X Y)\n\u03a6 : X \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjl X Y\n\u03a8 : Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjr X Y\n\u22a2 Isometry \u03a6", "state_after": "no goals"}, {"tactic": "exact (kuratowskiEmbedding.isometry _).comp (isometry_optimalGHInjr X Y)", "annotated_tactic": ["exact (<a>kuratowskiEmbedding.isometry</a> _).<a>comp</a> (<a>isometry_optimalGHInjr</a> X Y)", [{"full_name": "kuratowskiEmbedding.isometry", "def_path": "Mathlib/Topology/MetricSpace/Kuratowski.lean", "def_pos": [118, 19], "def_end_pos": [118, 47]}, {"full_name": "Isometry.comp", "def_path": "Mathlib/Topology/MetricSpace/Isometry.lean", "def_pos": [109, 9], "def_end_pos": [109, 13]}, {"full_name": "GromovHausdorff.isometry_optimalGHInjr", "def_path": "Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean", "def_pos": [481, 9], "def_end_pos": [481, 31]}]], "state_before": "case refine_2\nX : Type u\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\nY : Type v\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nF : OptimalGHCoupling X Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := kuratowskiEmbedding (OptimalGHCoupling X Y)\n\u03a6 : X \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjl X Y\n\u03a8 : Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjr X Y\n\u22a2 Isometry \u03a8", "state_after": "no goals"}, {"tactic": "rw [\u2190 image_univ, \u2190 image_univ, image_comp F, image_univ, image_comp F (optimalGHInjr X Y),\n  image_univ, \u2190 hausdorffDist_optimal]", "annotated_tactic": ["rw [\u2190 <a>image_univ</a>, \u2190 <a>image_univ</a>, <a>image_comp</a> F, <a>image_univ</a>, <a>image_comp</a> F (<a>optimalGHInjr</a> X Y),\n      <a>image_univ</a>, \u2190 <a>hausdorffDist_optimal</a>]", [{"full_name": "Set.image_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [701, 9], "def_end_pos": [701, 19]}, {"full_name": "Set.image_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [701, 9], "def_end_pos": [701, 19]}, {"full_name": "Set.image_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [263, 9], "def_end_pos": [263, 19]}, {"full_name": "Set.image_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [701, 9], "def_end_pos": [701, 19]}, {"full_name": "Set.image_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [263, 9], "def_end_pos": [263, 19]}, {"full_name": "GromovHausdorff.optimalGHInjr", "def_path": "Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean", "def_pos": [476, 5], "def_end_pos": [476, 18]}, {"full_name": "Set.image_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [701, 9], "def_end_pos": [701, 19]}, {"full_name": "GromovHausdorff.hausdorffDist_optimal", "def_path": "Mathlib/Topology/MetricSpace/GromovHausdorff.lean", "def_pos": [253, 9], "def_end_pos": [253, 30]}]], "state_before": "case refine_3\nX : Type u\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\nY : Type v\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nF : OptimalGHCoupling X Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := kuratowskiEmbedding (OptimalGHCoupling X Y)\n\u03a6 : X \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjl X Y\n\u03a8 : Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjr X Y\n\u22a2 ghDist X Y = hausdorffDist (range \u03a6) (range \u03a8)", "state_after": "case refine_3\nX : Type u\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\nY : Type v\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nF : OptimalGHCoupling X Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := kuratowskiEmbedding (OptimalGHCoupling X Y)\n\u03a6 : X \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjl X Y\n\u03a8 : Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjr X Y\n\u22a2 hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) =\n    hausdorffDist (F '' range (optimalGHInjl X Y)) (F '' range (optimalGHInjr X Y))"}, {"tactic": "exact (hausdorffDist_image (kuratowskiEmbedding.isometry _)).symm", "annotated_tactic": ["exact (<a>hausdorffDist_image</a> (<a>kuratowskiEmbedding.isometry</a> _)).<a>symm</a>", [{"full_name": "Metric.hausdorffDist_image", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "def_pos": [846, 9], "def_end_pos": [846, 28]}, {"full_name": "kuratowskiEmbedding.isometry", "def_path": "Mathlib/Topology/MetricSpace/Kuratowski.lean", "def_pos": [118, 19], "def_end_pos": [118, 47]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case refine_3\nX : Type u\ninst\u271d\u2075 : MetricSpace X\ninst\u271d\u2074 : CompactSpace X\ninst\u271d\u00b3 : Nonempty X\nY : Type v\ninst\u271d\u00b2 : MetricSpace Y\ninst\u271d\u00b9 : CompactSpace Y\ninst\u271d : Nonempty Y\nF : OptimalGHCoupling X Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := kuratowskiEmbedding (OptimalGHCoupling X Y)\n\u03a6 : X \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjl X Y\n\u03a8 : Y \u2192 \u21a5(lp (fun i => \u211d) \u22a4) := F \u2218 optimalGHInjr X Y\n\u22a2 hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHInjr X Y)) =\n    hausdorffDist (F '' range (optimalGHInjl X Y)) (F '' range (optimalGHInjr X Y))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/OrderClosed.lean", "full_name": "continuousWithinAt_Ioc_iff_Ioi", "start": [554, 1], "end": [556, 69], "traced_tactics": [{"tactic": "simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsWithin_Ioi h]", "annotated_tactic": ["simp only [<a>ContinuousWithinAt</a>, <a>nhdsWithin_Ioc_eq_nhdsWithin_Ioi</a> h]", [{"full_name": "ContinuousWithinAt", "def_path": "Mathlib/Topology/Defs/Filter.lean", "def_pos": [163, 5], "def_end_pos": [163, 23]}, {"full_name": "nhdsWithin_Ioc_eq_nhdsWithin_Ioi", "def_path": "Mathlib/Topology/Order/OrderClosed.lean", "def_pos": [544, 9], "def_end_pos": [544, 41]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : ClosedIciTopology \u03b1\ninst\u271d : TopologicalSpace \u03b2\na b c : \u03b1\nf : \u03b1 \u2192 \u03b2\nh : a < b\n\u22a2 ContinuousWithinAt f (Ioc a b) a \u2194 ContinuousWithinAt f (Ioi a) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Hom/Monoid.lean", "full_name": "OrderMonoidHom.coe_copy", "start": [329, 1], "end": [330, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "IsClosed.mul_left_of_isCompact", "start": [1513, 1], "end": [1514, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Localization/Basic.lean", "full_name": "IsLocalization.lift_surjective_iff", "start": [604, 1], "end": [606, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Semicontinuous.lean", "full_name": "lowerSemicontinuous_iSup", "start": [707, 1], "end": [709, 40], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Sort u_3\n\u03b4 : Type u_4\n\u03b4' : Type u_5\ninst\u271d\u00b9 : CompleteLinearOrder \u03b4\ninst\u271d : ConditionallyCompleteLinearOrder \u03b4'\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b4\nh : \u2200 (i : \u03b9), LowerSemicontinuous (f i)\n\u22a2 \u2200 (x : \u03b1), BddAbove (range fun i => f i x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Dynamics/Ergodic/Ergodic.lean", "full_name": "PreErgodic.measure_self_or_compl_eq_zero", "start": [64, 1], "end": [66, 41], "traced_tactics": [{"tactic": "simpa using hf.ae_empty_or_univ hs hs'", "annotated_tactic": ["simpa using hf.ae_empty_or_univ hs hs'", []], "state_before": "\u03b1 : Type u_1\nm : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \u03b1\ns : Set \u03b1\n\u03bc : Measure \u03b1\nhf : PreErgodic f \u03bc\nhs : MeasurableSet s\nhs' : f \u207b\u00b9' s = s\n\u22a2 \u03bc s = 0 \u2228 \u03bc s\u1d9c = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.nat_div2", "start": [843, 1], "end": [844, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Sign.lean", "full_name": "SignType.univ_eq", "start": [300, 1], "end": [301, 9], "traced_tactics": [{"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "\u22a2 Finset.univ = {0, -1, 1}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/Irrational.lean", "full_name": "irrational_int_add_iff", "start": [571, 1], "end": [572, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finsupp/Defs.lean", "full_name": "Finsupp.unique_single", "start": [432, 1], "end": [433, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "full_name": "TensorProduct.tmul_sum", "start": [473, 1], "end": [478, 49], "traced_tactics": [{"tactic": "induction' s using Finset.induction with a s has ih h", "annotated_tactic": ["induction' s using <a>Finset.induction</a> with a s has ih h", [{"full_name": "Finset.induction", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1267, 19], "def_end_pos": [1267, 28]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9\u2078 : CommSemiring R\nR' : Type u_2\ninst\u271d\u00b9\u2077 : Monoid R'\nR'' : Type u_3\ninst\u271d\u00b9\u2076 : Semiring R''\nM : Type u_4\nN : Type u_5\nP : Type u_6\nQ : Type u_7\nS : Type u_8\nT : Type u_9\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : AddCommMonoid N\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\ninst\u271d\u00b9\u00b9 : AddCommMonoid S\ninst\u271d\u00b9\u2070 : AddCommMonoid T\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R N\ninst\u271d\u2077 : Module R P\ninst\u271d\u2076 : Module R Q\ninst\u271d\u2075 : Module R S\ninst\u271d\u2074 : Module R T\ninst\u271d\u00b3 : DistribMulAction R' M\ninst\u271d\u00b2 : Module R'' M\ninst\u271d\u00b9 : SMulCommClass R R' M\ninst\u271d : SMulCommClass R R'' M\nm : M\n\u03b1 : Type u_10\ns : Finset \u03b1\nn : \u03b1 \u2192 N\n\u22a2 m \u2297\u209c[R] \u2211 a \u2208 s, n a = \u2211 a \u2208 s, m \u2297\u209c[R] n a", "state_after": "case empty\nR : Type u_1\ninst\u271d\u00b9\u2078 : CommSemiring R\nR' : Type u_2\ninst\u271d\u00b9\u2077 : Monoid R'\nR'' : Type u_3\ninst\u271d\u00b9\u2076 : Semiring R''\nM : Type u_4\nN : Type u_5\nP : Type u_6\nQ : Type u_7\nS : Type u_8\nT : Type u_9\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : AddCommMonoid N\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\ninst\u271d\u00b9\u00b9 : AddCommMonoid S\ninst\u271d\u00b9\u2070 : AddCommMonoid T\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R N\ninst\u271d\u2077 : Module R P\ninst\u271d\u2076 : Module R Q\ninst\u271d\u2075 : Module R S\ninst\u271d\u2074 : Module R T\ninst\u271d\u00b3 : DistribMulAction R' M\ninst\u271d\u00b2 : Module R'' M\ninst\u271d\u00b9 : SMulCommClass R R' M\ninst\u271d : SMulCommClass R R'' M\nm : M\n\u03b1 : Type u_10\nn : \u03b1 \u2192 N\n\u22a2 m \u2297\u209c[R] \u2211 a \u2208 \u2205, n a = \u2211 a \u2208 \u2205, m \u2297\u209c[R] n a\n\ncase insert\nR : Type u_1\ninst\u271d\u00b9\u2078 : CommSemiring R\nR' : Type u_2\ninst\u271d\u00b9\u2077 : Monoid R'\nR'' : Type u_3\ninst\u271d\u00b9\u2076 : Semiring R''\nM : Type u_4\nN : Type u_5\nP : Type u_6\nQ : Type u_7\nS : Type u_8\nT : Type u_9\ninst\u271d\u00b9\u2075 : AddCommMonoid M\ninst\u271d\u00b9\u2074 : AddCommMonoid N\ninst\u271d\u00b9\u00b3 : AddCommMonoid P\ninst\u271d\u00b9\u00b2 : AddCommMonoid Q\ninst\u271d\u00b9\u00b9 : AddCommMonoid S\ninst\u271d\u00b9\u2070 : AddCommMonoid T\ninst\u271d\u2079 : Module R M\ninst\u271d\u2078 : Module R N\ninst\u271d\u2077 : Module R P\ninst\u271d\u2076 : Module R Q\ninst\u271d\u2075 : Module R 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"Mathlib/Algebra/Order/Group/Defs.lean", "full_name": "le_inv_iff_mul_le_one_right", "start": [236, 1], "end": [237, 62], "traced_tactics": [{"tactic": "rw [inv_mul_self]", "annotated_tactic": ["rw [<a>inv_mul_self</a>]", [{"full_name": "inv_mul_self", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1228, 9], "def_end_pos": [1228, 21]}]], "state_before": "\u03b1 : Type u\ninst\u271d\u00b2 : Group \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c : \u03b1\n\u22a2 a * b \u2264 b\u207b\u00b9 * b \u2194 a * b \u2264 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Log.lean", "full_name": "Nat.log_div_base", "start": [210, 1], "end": [215, 58], "traced_tactics": [{"tactic": "rcases le_or_lt b 1 with hb | hb", "annotated_tactic": ["rcases <a>le_or_lt</a> b 1 with 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(p.coeff \u2191i) = 0"}, {"tactic": "simp_rw [aeval_eq_sum_range' hlt, Finset.sum_range, \u2190 pb.basis_eq_pow] at root", "annotated_tactic": ["simp_rw [<a>aeval_eq_sum_range'</a> hlt, <a>Finset.sum_range</a>, \u2190 pb.basis_eq_pow] at root", [{"full_name": "Polynomial.aeval_eq_sum_range'", "def_path": "Mathlib/Algebra/Polynomial/AlgebraMap.lean", "def_pos": [398, 9], "def_end_pos": [398, 28]}, {"full_name": "Finset.sum_range", "def_path": "Mathlib/Algebra/BigOperators/Fin.lean", "def_pos": [34, 3], "def_end_pos": [34, 14]}]], "state_before": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nA : Type u_4\nB : Type u_5\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : IsDomain B\ninst\u271d\u00b2 : Algebra A B\nK : Type u_6\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A S\npb : PowerBasis A S\np : A[X]\nne_zero : p \u2260 0\nroot : (aeval pb.gen) p = 0\nhlt : p.natDegree < pb.dim\ni : Fin pb.dim\n\u22a2 (monomial \u2191i) (p.coeff \u2191i) = 0", "state_after": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nA : Type u_4\nB : Type u_5\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : IsDomain B\ninst\u271d\u00b2 : Algebra A B\nK : Type u_6\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A S\npb : PowerBasis A S\np : A[X]\nne_zero : p \u2260 0\nhlt : p.natDegree < pb.dim\ni : Fin pb.dim\nroot : \u2211 x : Fin pb.dim, p.coeff \u2191x \u2022 pb.basis x = 0\n\u22a2 (monomial \u2191i) (p.coeff \u2191i) = 0"}, {"tactic": "have := Fintype.linearIndependent_iff.1 pb.basis.linearIndependent _ root", "annotated_tactic": ["have := <a>Fintype.linearIndependent_iff</a>.1 pb.basis.linearIndependent _ root", [{"full_name": "Fintype.linearIndependent_iff", "def_path": "Mathlib/LinearAlgebra/LinearIndependent.lean", "def_pos": [175, 9], "def_end_pos": [175, 38]}]], "state_before": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nA : Type u_4\nB : Type u_5\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : IsDomain B\ninst\u271d\u00b2 : Algebra A B\nK : Type u_6\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A S\npb : PowerBasis A S\np : A[X]\nne_zero : p \u2260 0\nhlt : p.natDegree < pb.dim\ni : Fin pb.dim\nroot : \u2211 x : Fin pb.dim, p.coeff \u2191x \u2022 pb.basis x = 0\n\u22a2 (monomial \u2191i) (p.coeff \u2191i) = 0", "state_after": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nA : Type u_4\nB : Type u_5\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : IsDomain B\ninst\u271d\u00b2 : Algebra A B\nK : Type u_6\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A S\npb : PowerBasis A S\np : A[X]\nne_zero : p \u2260 0\nhlt : p.natDegree < pb.dim\ni : Fin pb.dim\nroot : \u2211 x : Fin pb.dim, p.coeff \u2191x \u2022 pb.basis x = 0\nthis : \u2200 (i : Fin pb.dim), p.coeff \u2191i = 0\n\u22a2 (monomial \u2191i) (p.coeff \u2191i) = 0"}, {"tactic": "rw [this, monomial_zero_right]", "annotated_tactic": ["rw [this, <a>monomial_zero_right</a>]", [{"full_name": "Polynomial.monomial_zero_right", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [453, 9], "def_end_pos": [453, 28]}]], "state_before": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : Ring S\ninst\u271d\u2076 : Algebra R S\nA : Type u_4\nB : Type u_5\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing B\ninst\u271d\u00b3 : IsDomain B\ninst\u271d\u00b2 : Algebra A B\nK : Type u_6\ninst\u271d\u00b9 : Field K\ninst\u271d : Algebra A S\npb : PowerBasis A S\np : A[X]\nne_zero : p \u2260 0\nhlt : p.natDegree < pb.dim\ni : Fin pb.dim\nroot : \u2211 x : Fin pb.dim, p.coeff \u2191x \u2022 pb.basis x = 0\nthis : \u2200 (i : Fin pb.dim), p.coeff \u2191i = 0\n\u22a2 (monomial \u2191i) (p.coeff \u2191i) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Count.lean", "full_name": "MeasureTheory.Measure.count_apply_eq_top", "start": [90, 1], "end": [94, 36], "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\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns : Set \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\n\u22a2 count s = \u22a4 \u2194 s.Infinite", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns : Set \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\nhs : s.Finite\n\u22a2 count s = \u22a4 \u2194 s.Infinite\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns : Set \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\nhs : \u00acs.Finite\n\u22a2 count s = \u22a4 \u2194 s.Infinite"}, {"tactic": "exact count_apply_eq_top' hs.measurableSet", "annotated_tactic": ["exact <a>count_apply_eq_top'</a> hs.measurableSet", [{"full_name": "MeasureTheory.Measure.count_apply_eq_top'", "def_path": "Mathlib/MeasureTheory/Measure/Count.lean", "def_pos": [82, 9], "def_end_pos": [82, 28]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns : Set \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\nhs : s.Finite\n\u22a2 count s = \u22a4 \u2194 s.Infinite", "state_after": "no goals"}, {"tactic": "change s.Infinite at hs", "annotated_tactic": ["change s.Infinite at hs", []], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns : Set \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\nhs : \u00acs.Finite\n\u22a2 count s = \u22a4 \u2194 s.Infinite", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns : Set \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\nhs : s.Infinite\n\u22a2 count s = \u22a4 \u2194 s.Infinite"}, {"tactic": "simp [hs, count_apply_infinite]", "annotated_tactic": ["simp [hs, <a>count_apply_infinite</a>]", [{"full_name": "MeasureTheory.Measure.count_apply_infinite", "def_path": "Mathlib/MeasureTheory/Measure/Count.lean", "def_pos": [71, 9], "def_end_pos": [71, 29]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ns : Set \u03b1\ninst\u271d : MeasurableSingletonClass \u03b1\nhs : s.Infinite\n\u22a2 count s = \u22a4 \u2194 s.Infinite", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean", "full_name": "Rat.normalize_mul_normalize", "start": [285, 1], "end": [292, 43], "traced_tactics": [{"tactic": "cases e\u2081 : normalize n\u2081 d\u2081 z\u2081", "annotated_tactic": ["cases e\u2081 : <a>normalize</a> n\u2081 d\u2081 z\u2081", [{"full_name": "Rat.normalize", "def_path": ".lake/packages/batteries/Batteries/Data/Rat/Basic.lean", "def_pos": [73, 15], "def_end_pos": [73, 28]}]], "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 normalize n\u2081 d\u2081 z\u2081 * normalize n\u2082 d\u2082 z\u2082 = normalize (n\u2081 * n\u2082) (d\u2081 * d\u2082) \u22ef", "state_after": "case mk'\nn\u2081 n\u2082 : Int\nd\u2081 d\u2082 : Nat\nz\u2081 : d\u2081 \u2260 0\nz\u2082 : d\u2082 \u2260 0\nnum\u271d : Int\nden\u271d : Nat\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : num\u271d.natAbs.Coprime den\u271d\ne\u2081 : normalize n\u2081 d\u2081 z\u2081 = { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d }\n\u22a2 { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d } * normalize n\u2082 d\u2082 z\u2082 =\n    normalize (n\u2081 * n\u2082) (d\u2081 * d\u2082) \u22ef"}, {"tactic": "rcases normalize_num_den e\u2081 with \u27e8g\u2081, zg\u2081, rfl, rfl\u27e9", "annotated_tactic": ["rcases <a>normalize_num_den</a> e\u2081 with \u27e8g\u2081, zg\u2081, rfl, rfl\u27e9", [{"full_name": "Rat.normalize_num_den", "def_path": ".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean", "def_pos": [94, 9], "def_end_pos": [94, 26]}]], "state_before": "case mk'\nn\u2081 n\u2082 : Int\nd\u2081 d\u2082 : Nat\nz\u2081 : d\u2081 \u2260 0\nz\u2082 : d\u2082 \u2260 0\nnum\u271d : Int\nden\u271d : Nat\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : num\u271d.natAbs.Coprime den\u271d\ne\u2081 : normalize n\u2081 d\u2081 z\u2081 = { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d }\n\u22a2 { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d } * normalize n\u2082 d\u2082 z\u2082 =\n    normalize (n\u2081 * n\u2082) (d\u2081 * d\u2082) \u22ef", "state_after": "case mk'.intro.intro.intro\nn\u2082 : Int\nd\u2082 : Nat\nz\u2082 : d\u2082 \u2260 0\nnum\u271d : Int\nden\u271d : Nat\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : num\u271d.natAbs.Coprime den\u271d\ng\u2081 : Nat\nzg\u2081 : g\u2081 \u2260 0\nz\u2081 : den\u271d * g\u2081 \u2260 0\ne\u2081 : normalize (num\u271d * \u2191g\u2081) (den\u271d * g\u2081) z\u2081 = { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d }\n\u22a2 { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d } * normalize n\u2082 d\u2082 z\u2082 =\n    normalize (num\u271d * \u2191g\u2081 * n\u2082) (den\u271d * g\u2081 * d\u2082) \u22ef"}, {"tactic": "cases e\u2082 : normalize n\u2082 d\u2082 z\u2082", "annotated_tactic": ["cases e\u2082 : <a>normalize</a> n\u2082 d\u2082 z\u2082", [{"full_name": "Rat.normalize", "def_path": ".lake/packages/batteries/Batteries/Data/Rat/Basic.lean", "def_pos": [73, 15], "def_end_pos": [73, 28]}]], "state_before": "case mk'.intro.intro.intro\nn\u2082 : Int\nd\u2082 : Nat\nz\u2082 : d\u2082 \u2260 0\nnum\u271d : Int\nden\u271d : Nat\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : num\u271d.natAbs.Coprime den\u271d\ng\u2081 : Nat\nzg\u2081 : g\u2081 \u2260 0\nz\u2081 : den\u271d * g\u2081 \u2260 0\ne\u2081 : normalize (num\u271d * \u2191g\u2081) (den\u271d * g\u2081) z\u2081 = { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d }\n\u22a2 { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d } * normalize n\u2082 d\u2082 z\u2082 =\n    normalize (num\u271d * \u2191g\u2081 * n\u2082) (den\u271d * g\u2081 * d\u2082) \u22ef", "state_after": "case mk'.intro.intro.intro.mk'\nn\u2082 : Int\nd\u2082 : Nat\nz\u2082 : d\u2082 \u2260 0\nnum\u271d\u00b9 : Int\nden\u271d\u00b9 : Nat\nden_nz\u271d\u00b9 : den\u271d\u00b9 \u2260 0\nreduced\u271d\u00b9 : num\u271d\u00b9.natAbs.Coprime den\u271d\u00b9\ng\u2081 : Nat\nzg\u2081 : g\u2081 \u2260 0\nz\u2081 : den\u271d\u00b9 * g\u2081 \u2260 0\ne\u2081 : normalize (num\u271d\u00b9 * \u2191g\u2081) (den\u271d\u00b9 * g\u2081) z\u2081 = { num := num\u271d\u00b9, den := den\u271d\u00b9, den_nz := den_nz\u271d\u00b9, reduced := reduced\u271d\u00b9 }\nnum\u271d : Int\nden\u271d : Nat\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : num\u271d.natAbs.Coprime den\u271d\ne\u2082 : normalize n\u2082 d\u2082 z\u2082 = { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d }\n\u22a2 { num := num\u271d\u00b9, den := den\u271d\u00b9, den_nz := den_nz\u271d\u00b9, reduced := reduced\u271d\u00b9 } *\n      { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d } =\n    normalize (num\u271d\u00b9 * \u2191g\u2081 * n\u2082) (den\u271d\u00b9 * g\u2081 * d\u2082) \u22ef"}, {"tactic": "rcases normalize_num_den e\u2082 with \u27e8g\u2082, zg\u2082, rfl, rfl\u27e9", "annotated_tactic": ["rcases <a>normalize_num_den</a> e\u2082 with \u27e8g\u2082, zg\u2082, rfl, rfl\u27e9", [{"full_name": "Rat.normalize_num_den", "def_path": ".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean", "def_pos": [94, 9], "def_end_pos": [94, 26]}]], "state_before": "case mk'.intro.intro.intro.mk'\nn\u2082 : Int\nd\u2082 : Nat\nz\u2082 : d\u2082 \u2260 0\nnum\u271d\u00b9 : Int\nden\u271d\u00b9 : Nat\nden_nz\u271d\u00b9 : den\u271d\u00b9 \u2260 0\nreduced\u271d\u00b9 : num\u271d\u00b9.natAbs.Coprime den\u271d\u00b9\ng\u2081 : Nat\nzg\u2081 : g\u2081 \u2260 0\nz\u2081 : den\u271d\u00b9 * g\u2081 \u2260 0\ne\u2081 : normalize (num\u271d\u00b9 * \u2191g\u2081) (den\u271d\u00b9 * g\u2081) z\u2081 = { num := num\u271d\u00b9, den := den\u271d\u00b9, den_nz := den_nz\u271d\u00b9, reduced := reduced\u271d\u00b9 }\nnum\u271d : Int\nden\u271d : Nat\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : num\u271d.natAbs.Coprime den\u271d\ne\u2082 : normalize n\u2082 d\u2082 z\u2082 = { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d }\n\u22a2 { num := num\u271d\u00b9, den := den\u271d\u00b9, den_nz := den_nz\u271d\u00b9, reduced := reduced\u271d\u00b9 } *\n      { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d } =\n    normalize (num\u271d\u00b9 * \u2191g\u2081 * n\u2082) (den\u271d\u00b9 * g\u2081 * d\u2082) \u22ef", "state_after": "case mk'.intro.intro.intro.mk'.intro.intro.intro\nnum\u271d\u00b9 : Int\nden\u271d\u00b9 : Nat\nden_nz\u271d\u00b9 : den\u271d\u00b9 \u2260 0\nreduced\u271d\u00b9 : num\u271d\u00b9.natAbs.Coprime den\u271d\u00b9\ng\u2081 : Nat\nzg\u2081 : g\u2081 \u2260 0\nz\u2081 : den\u271d\u00b9 * g\u2081 \u2260 0\ne\u2081 : normalize (num\u271d\u00b9 * \u2191g\u2081) (den\u271d\u00b9 * g\u2081) z\u2081 = { num := num\u271d\u00b9, den := den\u271d\u00b9, den_nz := den_nz\u271d\u00b9, reduced := reduced\u271d\u00b9 }\nnum\u271d : Int\nden\u271d : Nat\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : num\u271d.natAbs.Coprime den\u271d\ng\u2082 : Nat\nzg\u2082 : g\u2082 \u2260 0\nz\u2082 : den\u271d * g\u2082 \u2260 0\ne\u2082 : normalize (num\u271d * \u2191g\u2082) (den\u271d * g\u2082) z\u2082 = { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d }\n\u22a2 { num := num\u271d\u00b9, den := den\u271d\u00b9, den_nz := den_nz\u271d\u00b9, reduced := reduced\u271d\u00b9 } *\n      { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d } =\n    normalize (num\u271d\u00b9 * \u2191g\u2081 * (num\u271d * \u2191g\u2082)) (den\u271d\u00b9 * g\u2081 * (den\u271d * g\u2082)) \u22ef"}, {"tactic": "simp only [mul_def]", "annotated_tactic": ["simp only [<a>mul_def</a>]", [{"full_name": "Rat.mul_def", "def_path": ".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean", "def_pos": [264, 9], "def_end_pos": [264, 16]}]], "state_before": "case mk'.intro.intro.intro.mk'.intro.intro.intro\nnum\u271d\u00b9 : Int\nden\u271d\u00b9 : Nat\nden_nz\u271d\u00b9 : den\u271d\u00b9 \u2260 0\nreduced\u271d\u00b9 : num\u271d\u00b9.natAbs.Coprime den\u271d\u00b9\ng\u2081 : Nat\nzg\u2081 : g\u2081 \u2260 0\nz\u2081 : den\u271d\u00b9 * g\u2081 \u2260 0\ne\u2081 : normalize (num\u271d\u00b9 * \u2191g\u2081) (den\u271d\u00b9 * g\u2081) z\u2081 = { num := num\u271d\u00b9, den := den\u271d\u00b9, den_nz := den_nz\u271d\u00b9, reduced := reduced\u271d\u00b9 }\nnum\u271d : Int\nden\u271d : Nat\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : num\u271d.natAbs.Coprime den\u271d\ng\u2082 : Nat\nzg\u2082 : g\u2082 \u2260 0\nz\u2082 : den\u271d * g\u2082 \u2260 0\ne\u2082 : normalize (num\u271d * \u2191g\u2082) (den\u271d * g\u2082) z\u2082 = { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d }\n\u22a2 { num := num\u271d\u00b9, den := den\u271d\u00b9, den_nz := den_nz\u271d\u00b9, reduced := reduced\u271d\u00b9 } *\n      { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d } =\n    normalize (num\u271d\u00b9 * \u2191g\u2081 * (num\u271d * \u2191g\u2082)) (den\u271d\u00b9 * g\u2081 * (den\u271d * g\u2082)) \u22ef", "state_after": "case mk'.intro.intro.intro.mk'.intro.intro.intro\nnum\u271d\u00b9 : Int\nden\u271d\u00b9 : Nat\nden_nz\u271d\u00b9 : den\u271d\u00b9 \u2260 0\nreduced\u271d\u00b9 : num\u271d\u00b9.natAbs.Coprime den\u271d\u00b9\ng\u2081 : Nat\nzg\u2081 : g\u2081 \u2260 0\nz\u2081 : den\u271d\u00b9 * g\u2081 \u2260 0\ne\u2081 : normalize (num\u271d\u00b9 * \u2191g\u2081) (den\u271d\u00b9 * g\u2081) z\u2081 = { num := num\u271d\u00b9, den := den\u271d\u00b9, den_nz := den_nz\u271d\u00b9, reduced := reduced\u271d\u00b9 }\nnum\u271d : Int\nden\u271d : Nat\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : num\u271d.natAbs.Coprime den\u271d\ng\u2082 : Nat\nzg\u2082 : g\u2082 \u2260 0\nz\u2082 : den\u271d * g\u2082 \u2260 0\ne\u2082 : normalize (num\u271d * \u2191g\u2082) (den\u271d * g\u2082) z\u2082 = { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d }\n\u22a2 normalize (num\u271d\u00b9 * num\u271d) (den\u271d\u00b9 * den\u271d) \u22ef = normalize (num\u271d\u00b9 * \u2191g\u2081 * (num\u271d * \u2191g\u2082)) (den\u271d\u00b9 * g\u2081 * (den\u271d * g\u2082)) \u22ef"}, {"tactic": "rw [\u2190 normalize_mul_right _ (Nat.mul_ne_zero zg\u2081 zg\u2082)]", "annotated_tactic": ["rw [\u2190 <a>normalize_mul_right</a> _ (<a>Nat.mul_ne_zero</a> zg\u2081 zg\u2082)]", [{"full_name": "Rat.normalize_mul_right", "def_path": ".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean", "def_pos": [58, 9], "def_end_pos": [58, 28]}, {"full_name": "Nat.mul_ne_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean", "def_pos": [437, 19], "def_end_pos": [437, 30]}]], "state_before": "case mk'.intro.intro.intro.mk'.intro.intro.intro\nnum\u271d\u00b9 : Int\nden\u271d\u00b9 : Nat\nden_nz\u271d\u00b9 : den\u271d\u00b9 \u2260 0\nreduced\u271d\u00b9 : num\u271d\u00b9.natAbs.Coprime den\u271d\u00b9\ng\u2081 : Nat\nzg\u2081 : g\u2081 \u2260 0\nz\u2081 : den\u271d\u00b9 * g\u2081 \u2260 0\ne\u2081 : normalize (num\u271d\u00b9 * \u2191g\u2081) (den\u271d\u00b9 * g\u2081) z\u2081 = { num := num\u271d\u00b9, den := den\u271d\u00b9, den_nz := den_nz\u271d\u00b9, reduced := reduced\u271d\u00b9 }\nnum\u271d : Int\nden\u271d : Nat\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : num\u271d.natAbs.Coprime den\u271d\ng\u2082 : Nat\nzg\u2082 : g\u2082 \u2260 0\nz\u2082 : den\u271d * g\u2082 \u2260 0\ne\u2082 : normalize (num\u271d * \u2191g\u2082) (den\u271d * g\u2082) z\u2082 = { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d }\n\u22a2 normalize (num\u271d\u00b9 * num\u271d) (den\u271d\u00b9 * den\u271d) \u22ef = normalize (num\u271d\u00b9 * \u2191g\u2081 * (num\u271d * \u2191g\u2082)) (den\u271d\u00b9 * g\u2081 * (den\u271d * g\u2082)) \u22ef", "state_after": "case mk'.intro.intro.intro.mk'.intro.intro.intro\nnum\u271d\u00b9 : Int\nden\u271d\u00b9 : Nat\nden_nz\u271d\u00b9 : den\u271d\u00b9 \u2260 0\nreduced\u271d\u00b9 : num\u271d\u00b9.natAbs.Coprime den\u271d\u00b9\ng\u2081 : Nat\nzg\u2081 : g\u2081 \u2260 0\nz\u2081 : den\u271d\u00b9 * g\u2081 \u2260 0\ne\u2081 : normalize (num\u271d\u00b9 * \u2191g\u2081) (den\u271d\u00b9 * g\u2081) z\u2081 = { num := num\u271d\u00b9, den := den\u271d\u00b9, den_nz := den_nz\u271d\u00b9, reduced := reduced\u271d\u00b9 }\nnum\u271d : Int\nden\u271d : Nat\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : num\u271d.natAbs.Coprime den\u271d\ng\u2082 : Nat\nzg\u2082 : g\u2082 \u2260 0\nz\u2082 : den\u271d * g\u2082 \u2260 0\ne\u2082 : normalize (num\u271d * \u2191g\u2082) (den\u271d * g\u2082) z\u2082 = { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d }\n\u22a2 normalize (num\u271d\u00b9 * num\u271d * \u2191(g\u2081 * g\u2082)) (den\u271d\u00b9 * den\u271d * (g\u2081 * g\u2082)) \u22ef =\n    normalize (num\u271d\u00b9 * \u2191g\u2081 * (num\u271d * \u2191g\u2082)) (den\u271d\u00b9 * g\u2081 * (den\u271d * g\u2082)) \u22ef"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case mk'.intro.intro.intro.mk'.intro.intro.intro\nnum\u271d\u00b9 : Int\nden\u271d\u00b9 : Nat\nden_nz\u271d\u00b9 : den\u271d\u00b9 \u2260 0\nreduced\u271d\u00b9 : num\u271d\u00b9.natAbs.Coprime den\u271d\u00b9\ng\u2081 : Nat\nzg\u2081 : g\u2081 \u2260 0\nz\u2081 : den\u271d\u00b9 * g\u2081 \u2260 0\ne\u2081 : normalize (num\u271d\u00b9 * \u2191g\u2081) (den\u271d\u00b9 * g\u2081) z\u2081 = { num := num\u271d\u00b9, den := den\u271d\u00b9, den_nz := den_nz\u271d\u00b9, reduced := reduced\u271d\u00b9 }\nnum\u271d : Int\nden\u271d : Nat\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : num\u271d.natAbs.Coprime den\u271d\ng\u2082 : Nat\nzg\u2082 : g\u2082 \u2260 0\nz\u2082 : den\u271d * g\u2082 \u2260 0\ne\u2082 : normalize (num\u271d * \u2191g\u2082) (den\u271d * g\u2082) z\u2082 = { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d }\n\u22a2 normalize (num\u271d\u00b9 * num\u271d * \u2191(g\u2081 * g\u2082)) (den\u271d\u00b9 * den\u271d * (g\u2081 * g\u2082)) \u22ef =\n    normalize (num\u271d\u00b9 * \u2191g\u2081 * (num\u271d * \u2191g\u2082)) (den\u271d\u00b9 * g\u2081 * (den\u271d * g\u2082)) \u22ef", "state_after": "case mk'.intro.intro.intro.mk'.intro.intro.intro.e_num\nnum\u271d\u00b9 : Int\nden\u271d\u00b9 : Nat\nden_nz\u271d\u00b9 : den\u271d\u00b9 \u2260 0\nreduced\u271d\u00b9 : num\u271d\u00b9.natAbs.Coprime den\u271d\u00b9\ng\u2081 : Nat\nzg\u2081 : g\u2081 \u2260 0\nz\u2081 : den\u271d\u00b9 * g\u2081 \u2260 0\ne\u2081 : normalize (num\u271d\u00b9 * \u2191g\u2081) (den\u271d\u00b9 * g\u2081) z\u2081 = { num := num\u271d\u00b9, den := den\u271d\u00b9, den_nz := den_nz\u271d\u00b9, reduced := reduced\u271d\u00b9 }\nnum\u271d : Int\nden\u271d : Nat\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : num\u271d.natAbs.Coprime den\u271d\ng\u2082 : Nat\nzg\u2082 : g\u2082 \u2260 0\nz\u2082 : den\u271d * g\u2082 \u2260 0\ne\u2082 : normalize (num\u271d * \u2191g\u2082) (den\u271d * g\u2082) z\u2082 = { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d }\n\u22a2 num\u271d\u00b9 * num\u271d * \u2191(g\u2081 * g\u2082) = num\u271d\u00b9 * \u2191g\u2081 * (num\u271d * \u2191g\u2082)\n\ncase mk'.intro.intro.intro.mk'.intro.intro.intro.e_den\nnum\u271d\u00b9 : Int\nden\u271d\u00b9 : Nat\nden_nz\u271d\u00b9 : den\u271d\u00b9 \u2260 0\nreduced\u271d\u00b9 : num\u271d\u00b9.natAbs.Coprime den\u271d\u00b9\ng\u2081 : Nat\nzg\u2081 : g\u2081 \u2260 0\nz\u2081 : den\u271d\u00b9 * g\u2081 \u2260 0\ne\u2081 : normalize (num\u271d\u00b9 * \u2191g\u2081) (den\u271d\u00b9 * g\u2081) z\u2081 = { num := num\u271d\u00b9, den := den\u271d\u00b9, den_nz := den_nz\u271d\u00b9, reduced := reduced\u271d\u00b9 }\nnum\u271d : Int\nden\u271d : Nat\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : num\u271d.natAbs.Coprime den\u271d\ng\u2082 : Nat\nzg\u2082 : g\u2082 \u2260 0\nz\u2082 : den\u271d * g\u2082 \u2260 0\ne\u2082 : normalize (num\u271d * \u2191g\u2082) (den\u271d * g\u2082) z\u2082 = { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d }\n\u22a2 den\u271d\u00b9 * den\u271d * (g\u2081 * g\u2082) = den\u271d\u00b9 * g\u2081 * (den\u271d * g\u2082)"}, {"tactic": "simp [Int.ofNat_mul, Int.mul_assoc, Int.mul_left_comm]", "annotated_tactic": ["simp [<a>Int.ofNat_mul</a>, <a>Int.mul_assoc</a>, <a>Int.mul_left_comm</a>]", [{"full_name": "Int.ofNat_mul", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean", "def_pos": [26, 22], "def_end_pos": [26, 31]}, {"full_name": "Int.mul_assoc", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean", "def_pos": [376, 19], "def_end_pos": [376, 28]}, {"full_name": "Int.mul_left_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean", "def_pos": [380, 19], "def_end_pos": [380, 32]}]], "state_before": "case mk'.intro.intro.intro.mk'.intro.intro.intro.e_num\nnum\u271d\u00b9 : Int\nden\u271d\u00b9 : Nat\nden_nz\u271d\u00b9 : den\u271d\u00b9 \u2260 0\nreduced\u271d\u00b9 : num\u271d\u00b9.natAbs.Coprime den\u271d\u00b9\ng\u2081 : Nat\nzg\u2081 : g\u2081 \u2260 0\nz\u2081 : den\u271d\u00b9 * g\u2081 \u2260 0\ne\u2081 : normalize (num\u271d\u00b9 * \u2191g\u2081) (den\u271d\u00b9 * g\u2081) z\u2081 = { num := num\u271d\u00b9, den := den\u271d\u00b9, den_nz := den_nz\u271d\u00b9, reduced := reduced\u271d\u00b9 }\nnum\u271d : Int\nden\u271d : Nat\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : num\u271d.natAbs.Coprime den\u271d\ng\u2082 : Nat\nzg\u2082 : g\u2082 \u2260 0\nz\u2082 : den\u271d * g\u2082 \u2260 0\ne\u2082 : normalize (num\u271d * \u2191g\u2082) (den\u271d * g\u2082) z\u2082 = { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d }\n\u22a2 num\u271d\u00b9 * num\u271d * \u2191(g\u2081 * g\u2082) = num\u271d\u00b9 * \u2191g\u2081 * (num\u271d * \u2191g\u2082)", "state_after": "no goals"}, {"tactic": "simp [Nat.mul_left_comm, Nat.mul_comm]", "annotated_tactic": ["simp [<a>Nat.mul_left_comm</a>, <a>Nat.mul_comm</a>]", [{"full_name": "Nat.mul_left_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [250, 19], "def_end_pos": [250, 32]}, {"full_name": "Nat.mul_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [217, 19], "def_end_pos": [217, 27]}]], "state_before": "case mk'.intro.intro.intro.mk'.intro.intro.intro.e_den\nnum\u271d\u00b9 : Int\nden\u271d\u00b9 : Nat\nden_nz\u271d\u00b9 : den\u271d\u00b9 \u2260 0\nreduced\u271d\u00b9 : num\u271d\u00b9.natAbs.Coprime den\u271d\u00b9\ng\u2081 : Nat\nzg\u2081 : g\u2081 \u2260 0\nz\u2081 : den\u271d\u00b9 * g\u2081 \u2260 0\ne\u2081 : normalize (num\u271d\u00b9 * \u2191g\u2081) (den\u271d\u00b9 * g\u2081) z\u2081 = { num := num\u271d\u00b9, den := den\u271d\u00b9, den_nz := den_nz\u271d\u00b9, reduced := reduced\u271d\u00b9 }\nnum\u271d : Int\nden\u271d : Nat\nden_nz\u271d : den\u271d \u2260 0\nreduced\u271d : num\u271d.natAbs.Coprime den\u271d\ng\u2082 : Nat\nzg\u2082 : g\u2082 \u2260 0\nz\u2082 : den\u271d * g\u2082 \u2260 0\ne\u2082 : normalize (num\u271d * \u2191g\u2082) (den\u271d * g\u2082) z\u2082 = { num := num\u271d, den := den\u271d, den_nz := den_nz\u271d, reduced := reduced\u271d }\n\u22a2 den\u271d\u00b9 * den\u271d * (g\u2081 * g\u2082) = den\u271d\u00b9 * g\u2081 * (den\u271d * g\u2082)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean", "full_name": "NumberField.canonicalEmbedding.integerLattice.inter_ball_finite", "start": [93, 1], "end": [105, 57], "traced_tactics": [{"tactic": "obtain hr | _ := lt_or_le r 0", "annotated_tactic": ["obtain hr | _ := <a>lt_or_le</a> r 0", [{"full_name": "lt_or_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [338, 9], "def_end_pos": [338, 17]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\n\u22a2 (\u2191(integerLattice K) \u2229 Metric.closedBall 0 r).Finite", "state_after": "case inl\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nhr : r < 0\n\u22a2 (\u2191(integerLattice K) \u2229 Metric.closedBall 0 r).Finite\n\ncase inr\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\n\u22a2 (\u2191(integerLattice K) \u2229 Metric.closedBall 0 r).Finite"}, {"tactic": "simp [Metric.closedBall_eq_empty.2 hr]", "annotated_tactic": ["simp [<a>Metric.closedBall_eq_empty</a>.2 hr]", [{"full_name": "Metric.closedBall_eq_empty", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [515, 9], "def_end_pos": [515, 28]}]], "state_before": "case inl\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nhr : r < 0\n\u22a2 (\u2191(integerLattice K) \u2229 Metric.closedBall 0 r).Finite", "state_after": "no goals"}, {"tactic": "have heq : \u2200 x, canonicalEmbedding K x \u2208 Metric.closedBall 0 r \u2194\n    \u2200 \u03c6 : K \u2192+* \u2102, \u2016\u03c6 x\u2016 \u2264 r := by\n  intro x; rw [\u2190 norm_le_iff, mem_closedBall_zero_iff]", "annotated_tactic": ["have heq : \u2200 x, <a>canonicalEmbedding</a> K x \u2208 <a>Metric.closedBall</a> 0 r \u2194\n        \u2200 \u03c6 : K \u2192+* \u2102, \u2016\u03c6 x\u2016 \u2264 r := by\n      intro x; rw [\u2190 <a>norm_le_iff</a>, <a>mem_closedBall_zero_iff</a>]", [{"full_name": "NumberField.canonicalEmbedding", "def_path": "Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean", "def_pos": [47, 5], "def_end_pos": [47, 42]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "NumberField.canonicalEmbedding.norm_le_iff", "def_path": "Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean", "def_pos": [76, 9], "def_end_pos": [76, 20]}, {"full_name": "mem_closedBall_zero_iff", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [631, 3], "def_end_pos": [631, 14]}]], "state_before": "case inr\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\n\u22a2 (\u2191(integerLattice K) \u2229 Metric.closedBall 0 r).Finite", "state_after": "case inr\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nheq : \u2200 (x : K), (canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r\n\u22a2 (\u2191(integerLattice K) \u2229 Metric.closedBall 0 r).Finite"}, {"tactic": "convert (Embeddings.finite_of_norm_le K \u2102 r).image (canonicalEmbedding K)", "annotated_tactic": ["convert (<a>Embeddings.finite_of_norm_le</a> K \u2102 r).<a>image</a> (<a>canonicalEmbedding</a> K)", [{"full_name": "NumberField.Embeddings.finite_of_norm_le", "def_path": "Mathlib/NumberTheory/NumberField/Embeddings.lean", "def_pos": [105, 9], "def_end_pos": [105, 26]}, {"full_name": "Set.Finite.image", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [863, 9], "def_end_pos": [863, 21]}, {"full_name": "NumberField.canonicalEmbedding", "def_path": "Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean", "def_pos": [47, 5], "def_end_pos": [47, 42]}]], "state_before": "case inr\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nheq : \u2200 (x : K), (canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r\n\u22a2 (\u2191(integerLattice K) \u2229 Metric.closedBall 0 r).Finite", "state_after": "case h.e'_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nheq : \u2200 (x : K), (canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r\n\u22a2 \u2191(integerLattice K) \u2229 Metric.closedBall 0 r =\n    \u21d1(canonicalEmbedding K) '' {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r}"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "case h.e'_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nheq : \u2200 (x : K), (canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r\n\u22a2 \u2191(integerLattice K) \u2229 Metric.closedBall 0 r =\n    \u21d1(canonicalEmbedding K) '' {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r}", "state_after": "case h.e'_2.h\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nheq : \u2200 (x : K), (canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r\nx\u271d : (K \u2192+* \u2102) \u2192 \u2102\n\u22a2 x\u271d \u2208 \u2191(integerLattice K) \u2229 Metric.closedBall 0 r \u2194\n    x\u271d \u2208 \u21d1(canonicalEmbedding K) '' {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r}"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h.e'_2.h\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nheq : \u2200 (x : K), (canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r\nx\u271d : (K \u2192+* \u2102) \u2192 \u2102\n\u22a2 x\u271d \u2208 \u2191(integerLattice K) \u2229 Metric.closedBall 0 r \u2194\n    x\u271d \u2208 \u21d1(canonicalEmbedding K) '' {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r}", "state_after": "case h.e'_2.h.mp\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nheq : \u2200 (x : K), (canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r\nx\u271d : (K \u2192+* \u2102) \u2192 \u2102\n\u22a2 x\u271d \u2208 \u2191(integerLattice K) \u2229 Metric.closedBall 0 r \u2192\n    x\u271d \u2208 \u21d1(canonicalEmbedding K) '' {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r}\n\ncase h.e'_2.h.mpr\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nheq : \u2200 (x : K), (canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r\nx\u271d : (K \u2192+* \u2102) \u2192 \u2102\n\u22a2 x\u271d \u2208 \u21d1(canonicalEmbedding K) '' {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r} \u2192\n    x\u271d \u2208 \u2191(integerLattice K) \u2229 Metric.closedBall 0 r"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "K : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\n\u22a2 \u2200 (x : K), (canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r", "state_after": "K : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nx : K\n\u22a2 (canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r"}, {"tactic": "rw [\u2190 norm_le_iff, mem_closedBall_zero_iff]", "annotated_tactic": ["rw [\u2190 <a>norm_le_iff</a>, <a>mem_closedBall_zero_iff</a>]", [{"full_name": "NumberField.canonicalEmbedding.norm_le_iff", "def_path": "Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean", "def_pos": [76, 9], "def_end_pos": [76, 20]}, {"full_name": "mem_closedBall_zero_iff", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [631, 3], "def_end_pos": [631, 14]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nx : K\n\u22a2 (canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r", "state_after": "no goals"}, {"tactic": "rintro \u27e8\u27e8_, \u27e8x, rfl\u27e9, rfl\u27e9, hx\u27e9", "annotated_tactic": ["rintro \u27e8\u27e8_, \u27e8x, rfl\u27e9, rfl\u27e9, hx\u27e9", []], "state_before": "case h.e'_2.h.mp\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nheq : \u2200 (x : K), (canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r\nx\u271d : (K \u2192+* \u2102) \u2192 \u2102\n\u22a2 x\u271d \u2208 \u2191(integerLattice K) \u2229 Metric.closedBall 0 r \u2192\n    x\u271d \u2208 \u21d1(canonicalEmbedding K) '' {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r}", "state_after": "case h.e'_2.h.mp.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nheq : \u2200 (x : K), (canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r\nx : \ud835\udcde K\nhx : (canonicalEmbedding K) ((algebraMap (\ud835\udcde K) K) x) \u2208 Metric.closedBall 0 r\n\u22a2 (canonicalEmbedding K) ((algebraMap (\ud835\udcde K) K) x) \u2208\n    \u21d1(canonicalEmbedding K) '' {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r}"}, {"tactic": "exact \u27e8x, \u27e8SetLike.coe_mem x, fun \u03c6 => (heq _).mp hx \u03c6\u27e9, rfl\u27e9", "annotated_tactic": ["exact \u27e8x, \u27e8<a>SetLike.coe_mem</a> x, fun \u03c6 => (heq _).<a>mp</a> hx \u03c6\u27e9, <a>rfl</a>\u27e9", [{"full_name": "SetLike.coe_mem", "def_path": "Mathlib/Data/SetLike/Basic.lean", "def_pos": [193, 9], "def_end_pos": [193, 16]}, {"full_name": "Iff.mp", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [118, 3], "def_end_pos": [118, 5]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case h.e'_2.h.mp.intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nheq : \u2200 (x : K), (canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r\nx : \ud835\udcde K\nhx : (canonicalEmbedding K) ((algebraMap (\ud835\udcde K) K) x) \u2208 Metric.closedBall 0 r\n\u22a2 (canonicalEmbedding K) ((algebraMap (\ud835\udcde K) K) x) \u2208\n    \u21d1(canonicalEmbedding K) '' {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r}", "state_after": "no goals"}, {"tactic": "rintro \u27e8x, \u27e8hx1, hx2\u27e9, rfl\u27e9", "annotated_tactic": ["rintro \u27e8x, \u27e8hx1, hx2\u27e9, rfl\u27e9", []], "state_before": "case h.e'_2.h.mpr\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nheq : \u2200 (x : K), (canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r\nx\u271d : (K \u2192+* \u2102) \u2192 \u2102\n\u22a2 x\u271d \u2208 \u21d1(canonicalEmbedding K) '' {x | IsIntegral \u2124 x \u2227 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r} \u2192\n    x\u271d \u2208 \u2191(integerLattice K) \u2229 Metric.closedBall 0 r", "state_after": "case h.e'_2.h.mpr.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nheq : \u2200 (x : K), (canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r\nx : K\nhx1 : IsIntegral \u2124 x\nhx2 : \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r\n\u22a2 (canonicalEmbedding K) x \u2208 \u2191(integerLattice K) \u2229 Metric.closedBall 0 r"}, {"tactic": "exact \u27e8\u27e8x, \u27e8\u27e8x, hx1\u27e9, rfl\u27e9, rfl\u27e9, (heq x).mpr hx2\u27e9", "annotated_tactic": ["exact \u27e8\u27e8x, \u27e8\u27e8x, hx1\u27e9, <a>rfl</a>\u27e9, <a>rfl</a>\u27e9, (heq x).<a>mpr</a> hx2\u27e9", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}]], "state_before": "case h.e'_2.h.mpr.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nr : \u211d\nh\u271d : 0 \u2264 r\nheq : \u2200 (x : K), (canonicalEmbedding K) x \u2208 Metric.closedBall 0 r \u2194 \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r\nx : K\nhx1 : IsIntegral \u2124 x\nhx2 : \u2200 (\u03c6 : K \u2192+* \u2102), \u2016\u03c6 x\u2016 \u2264 r\n\u22a2 (canonicalEmbedding K) x \u2208 \u2191(integerLattice K) \u2229 Metric.closedBall 0 r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean", "full_name": "HurwitzZeta.hasSum_nat_cosKernel\u2080", "start": [218, 1], "end": [231, 19], "traced_tactics": [{"tactic": "rw [\u2190 hasSum_ofReal, ofReal_sub, ofReal_one]", "annotated_tactic": ["rw [\u2190 <a>hasSum_ofReal</a>, <a>ofReal_sub</a>, <a>ofReal_one</a>]", [{"full_name": "Complex.hasSum_ofReal", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [664, 9], "def_end_pos": [664, 22]}, {"full_name": "Complex.ofReal_sub", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [789, 9], "def_end_pos": [789, 19]}, {"full_name": "Complex.ofReal_one", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [180, 9], "def_end_pos": [180, 19]}]], "state_before": "a t : \u211d\nht : 0 < t\n\u22a2 HasSum (fun n => 2 * Real.cos (2 * \u03c0 * a * (\u2191n + 1)) * rexp (-\u03c0 * (\u2191n + 1) ^ 2 * t)) (cosKernel (\u2191a) t - 1)", "state_after": "a t : \u211d\nht : 0 < t\n\u22a2 HasSum (fun x => \u2191(2 * Real.cos (2 * \u03c0 * a * (\u2191x + 1)) * rexp (-\u03c0 * (\u2191x + 1) ^ 2 * t))) (\u2191(cosKernel (\u2191a) t) - 1)"}, {"tactic": "have := (hasSum_int_cosKernel a ht).nat_add_neg", "annotated_tactic": ["have := (<a>hasSum_int_cosKernel</a> a ht).<a>nat_add_neg</a>", [{"full_name": "HurwitzZeta.hasSum_int_cosKernel", "def_path": "Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean", "def_pos": [180, 7], "def_end_pos": [180, 27]}, {"full_name": "HasSum.nat_add_neg", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean", "def_pos": [417, 3], "def_end_pos": [417, 14]}]], "state_before": "a t : \u211d\nht : 0 < t\n\u22a2 HasSum (fun x => \u2191(2 * Real.cos (2 * \u03c0 * a * (\u2191x + 1)) * rexp (-\u03c0 * (\u2191x + 1) ^ 2 * t))) (\u2191(cosKernel (\u2191a) t) - 1)", "state_after": "a t : \u211d\nht : 0 < t\nthis :\n  HasSum\n    (fun n =>\n      cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191\u2191n) * \u2191(rexp (-\u03c0 * \u2191\u2191n ^ 2 * t)) +\n        cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191(-\u2191n)) * \u2191(rexp (-\u03c0 * \u2191(-\u2191n) ^ 2 * t)))\n    (\u2191(cosKernel (\u2191a) t) + cexp (2 * \u2191\u03c0 * I * \u2191a * \u21910) * \u2191(rexp (-\u03c0 * \u21910 ^ 2 * t)))\n\u22a2 HasSum (fun x => \u2191(2 * Real.cos (2 * \u03c0 * a * (\u2191x + 1)) * rexp (-\u03c0 * (\u2191x + 1) ^ 2 * t))) (\u2191(cosKernel (\u2191a) t) - 1)"}, {"tactic": "rw [\u2190 hasSum_nat_add_iff' 1] at this", "annotated_tactic": ["rw [\u2190 <a>hasSum_nat_add_iff'</a> 1] at this", [{"full_name": "hasSum_nat_add_iff'", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean", "def_pos": [232, 3], "def_end_pos": [232, 14]}]], "state_before": "a t : \u211d\nht : 0 < t\nthis :\n  HasSum\n    (fun n =>\n      cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191\u2191n) * \u2191(rexp (-\u03c0 * \u2191\u2191n ^ 2 * t)) +\n        cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191(-\u2191n)) * \u2191(rexp (-\u03c0 * \u2191(-\u2191n) ^ 2 * t)))\n    (\u2191(cosKernel (\u2191a) t) + cexp (2 * \u2191\u03c0 * I * \u2191a * \u21910) * \u2191(rexp (-\u03c0 * \u21910 ^ 2 * t)))\n\u22a2 HasSum (fun x => \u2191(2 * Real.cos (2 * \u03c0 * a * (\u2191x + 1)) * rexp (-\u03c0 * (\u2191x + 1) ^ 2 * t))) (\u2191(cosKernel (\u2191a) t) - 1)", "state_after": "a t : \u211d\nht : 0 < t\nthis :\n  HasSum\n    (fun n =>\n      cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191\u2191(n + 1)) * \u2191(rexp (-\u03c0 * \u2191\u2191(n + 1) ^ 2 * t)) +\n        cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191(-\u2191(n + 1))) * \u2191(rexp (-\u03c0 * \u2191(-\u2191(n + 1)) ^ 2 * t)))\n    (\u2191(cosKernel (\u2191a) t) + cexp (2 * \u2191\u03c0 * I * \u2191a * \u21910) * \u2191(rexp (-\u03c0 * \u21910 ^ 2 * t)) -\n      \u2211 i \u2208 Finset.range 1,\n        (cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191\u2191i) * \u2191(rexp (-\u03c0 * \u2191\u2191i ^ 2 * t)) +\n          cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191(-\u2191i)) * \u2191(rexp (-\u03c0 * \u2191(-\u2191i) ^ 2 * t))))\n\u22a2 HasSum (fun x => \u2191(2 * Real.cos (2 * \u03c0 * a * (\u2191x + 1)) * rexp (-\u03c0 * (\u2191x + 1) ^ 2 * t))) (\u2191(cosKernel (\u2191a) t) - 1)"}, {"tactic": "simp_rw [Finset.sum_range_one, Nat.cast_zero, neg_zero, Int.cast_zero, zero_pow two_ne_zero,\n  mul_zero, zero_mul, Complex.exp_zero, Real.exp_zero, ofReal_one, mul_one, Int.cast_neg,\n  Int.cast_natCast, neg_sq, \u2190 add_mul, add_sub_assoc, \u2190 sub_sub, sub_self, zero_sub,\n  \u2190 sub_eq_add_neg, mul_neg] at this", "annotated_tactic": ["simp_rw [<a>Finset.sum_range_one</a>, <a>Nat.cast_zero</a>, <a>neg_zero</a>, <a>Int.cast_zero</a>, <a>zero_pow</a> <a>two_ne_zero</a>,\n    <a>mul_zero</a>, <a>zero_mul</a>, <a>Complex.exp_zero</a>, <a>Real.exp_zero</a>, <a>ofReal_one</a>, <a>mul_one</a>, <a>Int.cast_neg</a>,\n    <a>Int.cast_natCast</a>, <a>neg_sq</a>, \u2190 <a>add_mul</a>, <a>add_sub_assoc</a>, \u2190 <a>sub_sub</a>, <a>sub_self</a>, <a>zero_sub</a>,\n    \u2190 <a>sub_eq_add_neg</a>, <a>mul_neg</a>] at this", [{"full_name": "Finset.sum_range_one", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [1566, 15], "def_end_pos": [1566, 28]}, {"full_name": "Nat.cast_zero", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}, {"full_name": "neg_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1117, 3], "def_end_pos": [1117, 14]}, {"full_name": "Int.cast_zero", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [59, 9], "def_end_pos": [59, 18]}, {"full_name": "zero_pow", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [160, 15], "def_end_pos": [160, 23]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [65, 7], "def_end_pos": [65, 18]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}, {"full_name": "Complex.exp_zero", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [171, 9], "def_end_pos": [171, 17]}, {"full_name": "Real.exp_zero", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [825, 9], "def_end_pos": [825, 17]}, {"full_name": "Complex.ofReal_one", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [180, 9], "def_end_pos": [180, 19]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "Int.cast_neg", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [85, 9], "def_end_pos": [85, 17]}, {"full_name": "Int.cast_natCast", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [66, 9], "def_end_pos": [66, 21]}, {"full_name": "neg_sq", "def_path": "Mathlib/Algebra/Ring/Commute.lean", "def_pos": [196, 7], "def_end_pos": [196, 13]}, {"full_name": "add_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [91, 7], "def_end_pos": [91, 14]}, {"full_name": "add_sub_assoc", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [455, 3], "def_end_pos": [455, 14]}, {"full_name": "sub_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [770, 3], "def_end_pos": [770, 14]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1003, 30], "def_end_pos": [1003, 38]}, {"full_name": "zero_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [467, 3], "def_end_pos": [467, 14]}, {"full_name": "sub_eq_add_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1057, 3], "def_end_pos": [1057, 14]}, {"full_name": "mul_neg", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "a t : \u211d\nht : 0 < t\nthis :\n  HasSum\n    (fun n =>\n      cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191\u2191(n + 1)) * \u2191(rexp (-\u03c0 * \u2191\u2191(n + 1) ^ 2 * t)) +\n        cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191(-\u2191(n + 1))) * \u2191(rexp (-\u03c0 * \u2191(-\u2191(n + 1)) ^ 2 * t)))\n    (\u2191(cosKernel (\u2191a) t) + cexp (2 * \u2191\u03c0 * I * \u2191a * \u21910) * \u2191(rexp (-\u03c0 * \u21910 ^ 2 * t)) -\n      \u2211 i \u2208 Finset.range 1,\n        (cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191\u2191i) * \u2191(rexp (-\u03c0 * \u2191\u2191i ^ 2 * t)) +\n          cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191(-\u2191i)) * \u2191(rexp (-\u03c0 * \u2191(-\u2191i) ^ 2 * t))))\n\u22a2 HasSum (fun x => \u2191(2 * Real.cos (2 * \u03c0 * a * (\u2191x + 1)) * rexp (-\u03c0 * (\u2191x + 1) ^ 2 * t))) (\u2191(cosKernel (\u2191a) t) - 1)", "state_after": "a t : \u211d\nht : 0 < t\nthis :\n  HasSum\n    (fun n =>\n      (cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191(n + 1)) + cexp (-(2 * \u2191\u03c0 * I * \u2191a * \u2191(n + 1)))) * \u2191(rexp (-\u03c0 * \u2191(n + 1) ^ 2 * t)))\n    (\u2191(cosKernel (\u2191a) t) - 1)\n\u22a2 HasSum (fun x => \u2191(2 * Real.cos (2 * \u03c0 * a * (\u2191x + 1)) * rexp (-\u03c0 * (\u2191x + 1) ^ 2 * t))) (\u2191(cosKernel (\u2191a) t) - 1)"}, {"tactic": "refine this.congr_fun fun n \u21a6 ?_", "annotated_tactic": ["refine this.congr_fun fun n \u21a6 ?_", []], "state_before": "a t : \u211d\nht : 0 < t\nthis :\n  HasSum\n    (fun n =>\n      (cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191(n + 1)) + cexp (-(2 * \u2191\u03c0 * I * \u2191a * \u2191(n + 1)))) * \u2191(rexp (-\u03c0 * \u2191(n + 1) ^ 2 * t)))\n    (\u2191(cosKernel (\u2191a) t) - 1)\n\u22a2 HasSum (fun x => \u2191(2 * Real.cos (2 * \u03c0 * a * (\u2191x + 1)) * rexp (-\u03c0 * (\u2191x + 1) ^ 2 * t))) (\u2191(cosKernel (\u2191a) t) - 1)", "state_after": "a t : \u211d\nht : 0 < t\nthis :\n  HasSum\n    (fun n =>\n      (cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191(n + 1)) + cexp (-(2 * \u2191\u03c0 * I * \u2191a * \u2191(n + 1)))) * \u2191(rexp (-\u03c0 * \u2191(n + 1) ^ 2 * t)))\n    (\u2191(cosKernel (\u2191a) t) - 1)\nn : \u2115\n\u22a2 \u2191(2 * Real.cos (2 * \u03c0 * a * (\u2191n + 1)) * rexp (-\u03c0 * (\u2191n + 1) ^ 2 * t)) =\n    (cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191(n + 1)) + cexp (-(2 * \u2191\u03c0 * I * \u2191a * \u2191(n + 1)))) * \u2191(rexp (-\u03c0 * \u2191(n + 1) ^ 2 * t))"}, {"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "a t : \u211d\nht : 0 < t\nthis :\n  HasSum\n    (fun n =>\n      (cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191(n + 1)) + cexp (-(2 * \u2191\u03c0 * I * \u2191a * \u2191(n + 1)))) * \u2191(rexp (-\u03c0 * \u2191(n + 1) ^ 2 * t)))\n    (\u2191(cosKernel (\u2191a) t) - 1)\nn : \u2115\n\u22a2 \u2191(2 * Real.cos (2 * \u03c0 * a * (\u2191n + 1)) * rexp (-\u03c0 * (\u2191n + 1) ^ 2 * t)) =\n    (cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191(n + 1)) + cexp (-(2 * \u2191\u03c0 * I * \u2191a * \u2191(n + 1)))) * \u2191(rexp (-\u03c0 * \u2191(n + 1) ^ 2 * t))", "state_after": "a t : \u211d\nht : 0 < t\nthis :\n  HasSum\n    (fun n =>\n      (cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191(n + 1)) + cexp (-(2 * \u2191\u03c0 * I * \u2191a * \u2191(n + 1)))) * \u2191(rexp (-\u03c0 * \u2191(n + 1) ^ 2 * t)))\n    (\u2191(cosKernel (\u2191a) t) - 1)\nn : \u2115\n\u22a2 2 * Complex.cos (2 * \u2191\u03c0 * \u2191a * (\u2191n + 1)) * cexp (-\u2191\u03c0 * (\u2191n + 1) ^ 2 * \u2191t) =\n    (cexp (2 * \u2191\u03c0 * I * \u2191a * (\u2191n + 1)) + cexp (-(2 * \u2191\u03c0 * I * \u2191a * (\u2191n + 1)))) * cexp (-\u2191\u03c0 * (\u2191n + 1) ^ 2 * \u2191t)"}, {"tactic": "rw [Complex.cos, mul_div_cancel\u2080 _ two_ne_zero]", "annotated_tactic": ["rw [<a>Complex.cos</a>, <a>mul_div_cancel\u2080</a> _ <a>two_ne_zero</a>]", [{"full_name": "Complex.cos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [67, 5], "def_end_pos": [67, 8]}, {"full_name": "mul_div_cancel\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [516, 7], "def_end_pos": [516, 22]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [65, 7], "def_end_pos": [65, 18]}]], "state_before": "a t : \u211d\nht : 0 < t\nthis :\n  HasSum\n    (fun n =>\n      (cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191(n + 1)) + cexp (-(2 * \u2191\u03c0 * I * \u2191a * \u2191(n + 1)))) * \u2191(rexp (-\u03c0 * \u2191(n + 1) ^ 2 * t)))\n    (\u2191(cosKernel (\u2191a) t) - 1)\nn : \u2115\n\u22a2 2 * Complex.cos (2 * \u2191\u03c0 * \u2191a * (\u2191n + 1)) * cexp (-\u2191\u03c0 * (\u2191n + 1) ^ 2 * \u2191t) =\n    (cexp (2 * \u2191\u03c0 * I * \u2191a * (\u2191n + 1)) + cexp (-(2 * \u2191\u03c0 * I * \u2191a * (\u2191n + 1)))) * cexp (-\u2191\u03c0 * (\u2191n + 1) ^ 2 * \u2191t)", "state_after": "a t : \u211d\nht : 0 < t\nthis :\n  HasSum\n    (fun n =>\n      (cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191(n + 1)) + cexp (-(2 * \u2191\u03c0 * I * \u2191a * \u2191(n + 1)))) * \u2191(rexp (-\u03c0 * \u2191(n + 1) ^ 2 * t)))\n    (\u2191(cosKernel (\u2191a) t) - 1)\nn : \u2115\n\u22a2 (cexp (2 * \u2191\u03c0 * \u2191a * (\u2191n + 1) * I) + cexp (-(2 * \u2191\u03c0 * \u2191a * (\u2191n + 1)) * I)) * cexp (-\u2191\u03c0 * (\u2191n + 1) ^ 2 * \u2191t) =\n    (cexp (2 * \u2191\u03c0 * I * \u2191a * (\u2191n + 1)) + cexp (-(2 * \u2191\u03c0 * I * \u2191a * (\u2191n + 1)))) * cexp (-\u2191\u03c0 * (\u2191n + 1) ^ 2 * \u2191t)"}, {"tactic": "congr 3 <;> ring", "annotated_tactic": ["congr 3 <;> ring", []], "state_before": "a t : \u211d\nht : 0 < t\nthis :\n  HasSum\n    (fun n =>\n      (cexp (2 * \u2191\u03c0 * I * \u2191a * \u2191(n + 1)) + cexp (-(2 * \u2191\u03c0 * I * \u2191a * \u2191(n + 1)))) * \u2191(rexp (-\u03c0 * \u2191(n + 1) ^ 2 * t)))\n    (\u2191(cosKernel (\u2191a) t) - 1)\nn : \u2115\n\u22a2 (cexp (2 * \u2191\u03c0 * \u2191a * (\u2191n + 1) * I) + cexp (-(2 * \u2191\u03c0 * \u2191a * (\u2191n + 1)) * I)) * cexp (-\u2191\u03c0 * (\u2191n + 1) ^ 2 * \u2191t) =\n    (cexp (2 * \u2191\u03c0 * I * \u2191a * (\u2191n + 1)) + cexp (-(2 * \u2191\u03c0 * I * \u2191a * (\u2191n + 1)))) * cexp (-\u2191\u03c0 * (\u2191n + 1) ^ 2 * \u2191t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/FiberedCategory/HomLift.lean", "full_name": "CategoryTheory.IsHomLift.fac'", "start": [85, 1], "end": [86, 29], "traced_tactics": [{"tactic": "subst_hom_lift p f \u03c6", "annotated_tactic": ["subst_hom_lift p f \u03c6", []], "state_before": "\ud835\udcae : Type u\u2081\n\ud835\udcb3 : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} \ud835\udcb3\ninst\u271d\u00b9 : Category.{v\u2082, u\u2081} \ud835\udcae\np : \ud835\udcb3 \u2964 \ud835\udcae\nR S : \ud835\udcae\na b : \ud835\udcb3\nf : R \u27f6 S\n\u03c6 : a \u27f6 b\ninst\u271d : p.IsHomLift f \u03c6\n\u22a2 p.map \u03c6 = eqToHom \u22ef \u226b f \u226b eqToHom \u22ef", "state_after": "case map\n\ud835\udcae : Type u\u2081\n\ud835\udcb3 : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} \ud835\udcb3\ninst\u271d\u00b9 : Category.{v\u2082, u\u2081} \ud835\udcae\np : \ud835\udcb3 \u2964 \ud835\udcae\na\u271d b\u271d : \ud835\udcb3\n\u03c6 : a\u271d \u27f6 b\u271d\nR S : \ud835\udcae\na b : \ud835\udcb3\ninst\u271d : p.IsHomLift (p.map \u03c6) \u03c6\n\u22a2 p.map \u03c6 = eqToHom \u22ef \u226b p.map \u03c6 \u226b eqToHom \u22ef"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case map\n\ud835\udcae : Type u\u2081\n\ud835\udcb3 : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2081, u\u2082} \ud835\udcb3\ninst\u271d\u00b9 : Category.{v\u2082, u\u2081} \ud835\udcae\np : \ud835\udcb3 \u2964 \ud835\udcae\na\u271d b\u271d : \ud835\udcb3\n\u03c6 : a\u271d \u27f6 b\u271d\nR S : \ud835\udcae\na b : \ud835\udcb3\ninst\u271d : p.IsHomLift (p.map \u03c6) \u03c6\n\u22a2 p.map \u03c6 = eqToHom \u22ef \u226b p.map \u03c6 \u226b eqToHom \u22ef", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Module/AEval.lean", "full_name": "Module.AEval.injective_comapSubmodule", "start": [157, 1], "end": [160, 15], "traced_tactics": [{"tactic": "intro q\u2081 q\u2082 hq", "annotated_tactic": ["intro q\u2081 q\u2082 hq", []], "state_before": "R : Type u_2\nA : Type u_3\nM : Type u_1\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\na : A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsScalarTower R A M\np : Submodule R M\nhp : p \u2264 Submodule.comap ((Algebra.lsmul R R M) a) p\nq : Submodule R[X] (AEval R M a)\n\u22a2 Injective \u21d1(comapSubmodule R M a)", "state_after": "R : Type u_2\nA : Type u_3\nM : Type u_1\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\na : A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsScalarTower R A M\np : Submodule R M\nhp : p \u2264 Submodule.comap ((Algebra.lsmul R R M) a) p\nq q\u2081 q\u2082 : Submodule R[X] (AEval R M a)\nhq : (comapSubmodule R M a) q\u2081 = (comapSubmodule R M a) q\u2082\n\u22a2 q\u2081 = q\u2082"}, {"tactic": "rw [\u2190 mapSubmodule_comapSubmodule (q := q\u2081), \u2190 mapSubmodule_comapSubmodule (q := q\u2082)]", "annotated_tactic": ["rw [\u2190 <a>mapSubmodule_comapSubmodule</a> (q := q\u2081), \u2190 <a>mapSubmodule_comapSubmodule</a> (q := q\u2082)]", [{"full_name": "Module.AEval.mapSubmodule_comapSubmodule", "def_path": "Mathlib/Algebra/Polynomial/Module/AEval.lean", "def_pos": [147, 15], "def_end_pos": [147, 42]}, {"full_name": "Module.AEval.mapSubmodule_comapSubmodule", "def_path": "Mathlib/Algebra/Polynomial/Module/AEval.lean", "def_pos": [147, 15], "def_end_pos": [147, 42]}]], "state_before": "R : Type u_2\nA : Type u_3\nM : Type u_1\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\na : A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsScalarTower R A M\np : Submodule R M\nhp : p \u2264 Submodule.comap ((Algebra.lsmul R R M) a) p\nq q\u2081 q\u2082 : Submodule R[X] (AEval R M a)\nhq : (comapSubmodule R M a) q\u2081 = (comapSubmodule R M a) q\u2082\n\u22a2 q\u2081 = q\u2082", "state_after": "R : Type u_2\nA : Type u_3\nM : Type u_1\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\na : A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsScalarTower R A M\np : Submodule R M\nhp : p \u2264 Submodule.comap ((Algebra.lsmul R R M) a) p\nq q\u2081 q\u2082 : Submodule R[X] (AEval R M a)\nhq : (comapSubmodule R M a) q\u2081 = (comapSubmodule R M a) q\u2082\n\u22a2 mapSubmodule a \u22ef = mapSubmodule a \u22ef"}, {"tactic": "simp_rw [hq]", "annotated_tactic": ["simp_rw [hq]", []], "state_before": "R : Type u_2\nA : Type u_3\nM : Type u_1\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : Semiring A\na : A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module A M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsScalarTower R A M\np : Submodule R M\nhp : p \u2264 Submodule.comap ((Algebra.lsmul R R M) a) p\nq q\u2081 q\u2082 : Submodule R[X] (AEval R M a)\nhq : (comapSubmodule R M a) q\u2081 = (comapSubmodule R M a) q\u2082\n\u22a2 mapSubmodule a \u22ef = mapSubmodule a \u22ef", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean", "full_name": "Fin.foldr_loop", "start": [108, 1], "end": [113, 69], "traced_tactics": [{"tactic": "induction m generalizing x with\n| zero => simp [foldr_loop_zero, foldr_loop_succ]\n| succ m ih => rw [foldr_loop_succ, ih, foldr_loop_succ, Fin.succ]", "annotated_tactic": ["induction m generalizing x with\n  | zero => simp [<a>foldr_loop_zero</a>, <a>foldr_loop_succ</a>]\n  | succ m ih => rw [<a>foldr_loop_succ</a>, ih, <a>foldr_loop_succ</a>, <a>Fin.succ</a>]", [{"full_name": "Fin.foldr_loop_zero", "def_path": ".lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean", "def_pos": [100, 9], "def_end_pos": [100, 24]}, {"full_name": "Fin.foldr_loop_succ", "def_path": ".lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean", "def_pos": [104, 9], "def_end_pos": [104, 24]}, {"full_name": "Fin.foldr_loop_succ", "def_path": ".lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean", "def_pos": [104, 9], "def_end_pos": [104, 24]}, {"full_name": "Fin.foldr_loop_succ", "def_path": ".lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean", "def_pos": [104, 9], "def_end_pos": [104, 24]}, {"full_name": "Fin.succ", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Fin/Basic.lean", "def_pos": [34, 5], "def_end_pos": [34, 9]}]], "state_before": "n : Nat\n\u03b1 : Sort u_1\nm : Nat\nf : Fin (n + 1) \u2192 \u03b1 \u2192 \u03b1\nx : \u03b1\nh : m + 1 \u2264 n + 1\n\u22a2 foldr.loop (n + 1) f \u27e8m + 1, h\u27e9 x = f 0 (foldr.loop n (fun i => f i.succ) \u27e8m, \u22ef\u27e9 x)", "state_after": "no goals"}, {"tactic": "simp [foldr_loop_zero, foldr_loop_succ]", "annotated_tactic": ["simp [<a>foldr_loop_zero</a>, <a>foldr_loop_succ</a>]", [{"full_name": "Fin.foldr_loop_zero", "def_path": ".lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean", "def_pos": [100, 9], "def_end_pos": [100, 24]}, {"full_name": "Fin.foldr_loop_succ", "def_path": ".lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean", "def_pos": [104, 9], "def_end_pos": [104, 24]}]], "state_before": "case zero\nn : Nat\n\u03b1 : Sort u_1\nf : Fin (n + 1) \u2192 \u03b1 \u2192 \u03b1\nx : \u03b1\nh : 0 + 1 \u2264 n + 1\n\u22a2 foldr.loop (n + 1) f \u27e80 + 1, h\u27e9 x = f 0 (foldr.loop n (fun i => f i.succ) \u27e80, \u22ef\u27e9 x)", "state_after": "no goals"}, {"tactic": "rw [foldr_loop_succ, ih, foldr_loop_succ, Fin.succ]", "annotated_tactic": ["rw [<a>foldr_loop_succ</a>, ih, <a>foldr_loop_succ</a>, <a>Fin.succ</a>]", [{"full_name": "Fin.foldr_loop_succ", "def_path": ".lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean", "def_pos": [104, 9], "def_end_pos": [104, 24]}, {"full_name": "Fin.foldr_loop_succ", "def_path": ".lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean", "def_pos": [104, 9], "def_end_pos": [104, 24]}, {"full_name": "Fin.succ", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Fin/Basic.lean", "def_pos": [34, 5], "def_end_pos": [34, 9]}]], "state_before": "case succ\nn : Nat\n\u03b1 : Sort u_1\nf : Fin (n + 1) \u2192 \u03b1 \u2192 \u03b1\nm : Nat\nih : \u2200 (x : \u03b1) (h : m + 1 \u2264 n + 1), foldr.loop (n + 1) f \u27e8m + 1, h\u27e9 x = f 0 (foldr.loop n (fun i => f i.succ) \u27e8m, \u22ef\u27e9 x)\nx : \u03b1\nh : m + 1 + 1 \u2264 n + 1\n\u22a2 foldr.loop (n + 1) f \u27e8m + 1 + 1, h\u27e9 x = f 0 (foldr.loop n (fun i => f i.succ) \u27e8m + 1, \u22ef\u27e9 x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/IsAdjoinRoot.lean", "full_name": "IsAdjoinRoot.aeval_root", "start": [158, 1], "end": [158, 93], "traced_tactics": [{"tactic": "rw [aeval_eq, map_self]", "annotated_tactic": ["rw [<a>aeval_eq</a>, <a>map_self</a>]", [{"full_name": "IsAdjoinRoot.aeval_eq", "def_path": "Mathlib/RingTheory/IsAdjoinRoot.lean", "def_pos": [150, 9], "def_end_pos": [150, 17]}, {"full_name": "IsAdjoinRoot.map_self", "def_path": "Mathlib/RingTheory/IsAdjoinRoot.lean", "def_pos": [146, 9], "def_end_pos": [146, 17]}]], "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Ring S\nf : R[X]\ninst\u271d : Algebra R S\nh : IsAdjoinRoot S f\n\u22a2 (aeval h.root) f = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "full_name": "Polynomial.degree_zero_le", "start": [1164, 1], "end": [1164, 94], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Category/Grp/Basic.lean", "full_name": "CommGrp.coe_id'", "start": [288, 1], "end": [291, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "full_name": "Cardinal.aleph_lt", "start": [252, 1], "end": [253, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.le_lintegral_add", "start": [541, 1], "end": [546, 77], "traced_tactics": [{"tactic": "simp only [lintegral]", "annotated_tactic": ["simp only [<a>lintegral</a>]", [{"full_name": "MeasureTheory.lintegral", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [61, 17], "def_end_pos": [61, 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 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), f a \u2202\u03bc + \u222b\u207b (a : \u03b1), g a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a + g a \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 (\u2a06 g, \u2a06 (_ : \u2191g \u2264 fun a => f a), g.lintegral \u03bc) + \u2a06 g_1, \u2a06 (_ : \u2191g_1 \u2264 fun a => g a), g_1.lintegral \u03bc \u2264\n    \u2a06 g_1, \u2a06 (_ : \u2191g_1 \u2264 fun a => f a + g a), g_1.lintegral \u03bc"}, {"tactic": "refine ENNReal.biSup_add_biSup_le' (p := fun h : \u03b1 \u2192\u209b \u211d\u22650\u221e => h \u2264 f)\n  (q := fun h : \u03b1 \u2192\u209b \u211d\u22650\u221e => h \u2264 g) \u27e80, zero_le f\u27e9 \u27e80, zero_le g\u27e9 fun f' hf' g' hg' => ?_", "annotated_tactic": ["refine <a>ENNReal.biSup_add_biSup_le'</a> (p := fun h : \u03b1 \u2192\u209b \u211d\u22650\u221e => h \u2264 f)\n    (q := fun h : \u03b1 \u2192\u209b \u211d\u22650\u221e => h \u2264 g) \u27e80, <a>zero_le</a> f\u27e9 \u27e80, <a>zero_le</a> g\u27e9 fun f' hf' g' hg' => ?_", [{"full_name": "ENNReal.biSup_add_biSup_le'", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [613, 9], "def_end_pos": [613, 28]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [204, 30], "def_end_pos": [204, 37]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [204, 30], "def_end_pos": [204, 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 \u211d\u22650\u221e\n\u22a2 (\u2a06 g, \u2a06 (_ : \u2191g \u2264 fun a => f a), g.lintegral \u03bc) + \u2a06 g_1, \u2a06 (_ : \u2191g_1 \u2264 fun a => g a), g_1.lintegral \u03bc \u2264\n    \u2a06 g_1, \u2a06 (_ : \u2191g_1 \u2264 fun a => f a + g a), g_1.lintegral \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhf' : (fun h => \u2191h \u2264 f) f'\ng' : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhg' : (fun h => \u2191h \u2264 g) g'\n\u22a2 f'.lintegral \u03bc + g'.lintegral \u03bc \u2264 \u2a06 g_1, \u2a06 (_ : \u2191g_1 \u2264 fun a => f a + g a), g_1.lintegral \u03bc"}, {"tactic": "exact le_iSup\u2082_of_le (f' + g') (add_le_add hf' hg') (add_lintegral _ _).ge", "annotated_tactic": ["exact <a>le_iSup\u2082_of_le</a> (f' + g') (<a>add_le_add</a> hf' hg') (<a>add_lintegral</a> _ _).<a>ge</a>", [{"full_name": "le_iSup\u2082_of_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [750, 9], "def_end_pos": [750, 23]}, {"full_name": "add_le_add", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [205, 32], "def_end_pos": [205, 42]}, {"full_name": "MeasureTheory.SimpleFunc.add_lintegral", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [1005, 9], "def_end_pos": [1005, 22]}, {"full_name": "Eq.ge", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [218, 19], "def_end_pos": [218, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nf' : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhf' : (fun h => \u2191h \u2264 f) f'\ng' : \u03b1 \u2192\u209b \u211d\u22650\u221e\nhg' : (fun h => \u2191h \u2264 g) g'\n\u22a2 f'.lintegral \u03bc + g'.lintegral \u03bc \u2264 \u2a06 g_1, \u2a06 (_ : \u2191g_1 \u2264 fun a => f a + g a), g_1.lintegral \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "full_name": "Ideal.Quotient.lift_surjective_of_surjective", "start": [270, 1], "end": [275, 37], "traced_tactics": [{"tactic": "intro y", "annotated_tactic": ["intro y", []], "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx y : R\ninst\u271d : Semiring S\nI : Ideal R\nf : R \u2192+* S\nH : \u2200 a \u2208 I, f a = 0\nhf : Function.Surjective \u21d1f\n\u22a2 Function.Surjective \u21d1(lift I f H)", "state_after": "R : Type u\ninst\u271d\u00b9 : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx y\u271d : R\ninst\u271d : Semiring S\nI : Ideal R\nf : R \u2192+* S\nH : \u2200 a \u2208 I, f a = 0\nhf : Function.Surjective \u21d1f\ny : S\n\u22a2 \u2203 a, (lift I f H) a = y"}, {"tactic": "obtain \u27e8x, rfl\u27e9 := hf y", "annotated_tactic": ["obtain \u27e8x, rfl\u27e9 := hf y", []], "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx y\u271d : R\ninst\u271d : Semiring S\nI : Ideal R\nf : R \u2192+* S\nH : \u2200 a \u2208 I, f a = 0\nhf : Function.Surjective \u21d1f\ny : S\n\u22a2 \u2203 a, (lift I f H) a = y", "state_after": "case intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx\u271d y : R\ninst\u271d : Semiring S\nI : Ideal R\nf : R \u2192+* S\nH : \u2200 a \u2208 I, f a = 0\nhf : Function.Surjective \u21d1f\nx : R\n\u22a2 \u2203 a, (lift I f H) a = f x"}, {"tactic": "use Ideal.Quotient.mk I x", "annotated_tactic": ["use <a>Ideal.Quotient.mk</a> I x", [{"full_name": "Ideal.Quotient.mk", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [92, 5], "def_end_pos": [92, 7]}]], "state_before": "case intro\nR : Type u\ninst\u271d\u00b9 : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx\u271d y : R\ninst\u271d : Semiring S\nI : Ideal R\nf : R \u2192+* S\nH : \u2200 a \u2208 I, f a = 0\nhf : Function.Surjective \u21d1f\nx : R\n\u22a2 \u2203 a, (lift I f H) a = f x", "state_after": "case h\nR : Type u\ninst\u271d\u00b9 : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx\u271d y : R\ninst\u271d : Semiring S\nI : Ideal R\nf : R \u2192+* S\nH : \u2200 a \u2208 I, f a = 0\nhf : Function.Surjective \u21d1f\nx : R\n\u22a2 (lift I f H) ((mk I) x) = f x"}, {"tactic": "simp only [Ideal.Quotient.lift_mk]", "annotated_tactic": ["simp only [<a>Ideal.Quotient.lift_mk</a>]", [{"full_name": "Ideal.Quotient.lift_mk", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [265, 9], "def_end_pos": [265, 16]}]], "state_before": "case h\nR : Type u\ninst\u271d\u00b9 : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx\u271d y : R\ninst\u271d : Semiring S\nI : Ideal R\nf : R \u2192+* S\nH : \u2200 a \u2208 I, f a = 0\nhf : Function.Surjective \u21d1f\nx : R\n\u22a2 (lift I f H) ((mk I) x) = f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Invertible.lean", "full_name": "invOf_pos", "start": [19, 1], "end": [21, 84], "traced_tactics": [{"tactic": "simp only [mul_invOf_self, zero_lt_one]", "annotated_tactic": ["simp only [<a>mul_invOf_self</a>, <a>zero_lt_one</a>]", [{"full_name": "mul_invOf_self", "def_path": "Mathlib/Algebra/Group/Invertible/Defs.lean", "def_pos": [112, 9], "def_end_pos": [112, 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\u00b9 : LinearOrderedSemiring \u03b1\na : \u03b1\ninst\u271d : Invertible a\n\u22a2 0 < a * \u215fa", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean", "full_name": "TensorProduct.gradedComm_of_zero_tmul", "start": [137, 1], "end": [145, 61], "traced_tactics": [{"tactic": "suffices\n  (gradedComm R \ud835\udc9c \u212c).toLinearMap \u2218\u2097 (TensorProduct.mk R (\u2a01 i, \ud835\udc9c i) (\u2a01 i, \u212c i)) (lof R _ \ud835\udc9c 0 a) =\n    (TensorProduct.mk R _ _).flip (lof R _ \ud835\udc9c 0 a) from\n  DFunLike.congr_fun this b", "annotated_tactic": ["suffices\n    (<a>gradedComm</a> R \ud835\udc9c \u212c).<a>toLinearMap</a> \u2218\u2097 (<a>TensorProduct.mk</a> R (\u2a01 i, \ud835\udc9c i) (\u2a01 i, \u212c i)) (<a>lof</a> R _ \ud835\udc9c 0 a) =\n      (<a>TensorProduct.mk</a> R _ _).<a>flip</a> (<a>lof</a> R _ \ud835\udc9c 0 a) from\n    <a>DFunLike.congr_fun</a> this b", [{"full_name": "TensorProduct.gradedComm", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean", "def_pos": [103, 5], "def_end_pos": [103, 15]}, {"full_name": "LinearEquiv.toLinearMap", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [61, 14], "def_end_pos": [61, 37]}, {"full_name": "TensorProduct.mk", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [443, 5], "def_end_pos": [443, 7]}, {"full_name": "DirectSum.lof", "def_path": "Mathlib/Algebra/DirectSum/Module.lean", "def_pos": [67, 5], "def_end_pos": [67, 8]}, {"full_name": "TensorProduct.mk", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [443, 5], "def_end_pos": [443, 7]}, {"full_name": "LinearMap.flip", "def_path": "Mathlib/LinearAlgebra/BilinearMap.lean", "def_pos": [113, 5], "def_end_pos": [113, 9]}, {"full_name": "DirectSum.lof", "def_path": "Mathlib/Algebra/DirectSum/Module.lean", "def_pos": [67, 5], "def_end_pos": [67, 8]}, {"full_name": "DFunLike.congr_fun", "def_path": "Mathlib/Data/FunLike/Basic.lean", "def_pos": [209, 19], "def_end_pos": [209, 28]}]], "state_before": "R : Type u_1\n\u03b9 : Type u_2\nA : Type u_3\nB : Type u_4\ninst\u271d\u00b9\u00b9 : CommSemiring \u03b9\ninst\u271d\u00b9\u2070 : Module \u03b9 (Additive \u2124\u02e3)\ninst\u271d\u2079 : DecidableEq \u03b9\n\ud835\udc9c : \u03b9 \u2192 Type u_5\n\u212c : \u03b9 \u2192 Type u_6\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommGroup (\ud835\udc9c i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommGroup (\u212c i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (\ud835\udc9c i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R (\u212c i)\ninst\u271d\u00b3 : DirectSum.GRing \ud835\udc9c\ninst\u271d\u00b2 : DirectSum.GRing \u212c\ninst\u271d\u00b9 : DirectSum.GAlgebra R \ud835\udc9c\ninst\u271d : DirectSum.GAlgebra R \u212c\na : \ud835\udc9c 0\nb : \u2a01 (i : \u03b9), \u212c i\n\u22a2 (gradedComm R \ud835\udc9c \u212c) ((lof R \u03b9 \ud835\udc9c 0) a \u2297\u209c[R] b) = b \u2297\u209c[R] (lof R \u03b9 \ud835\udc9c 0) a", "state_after": "R : Type u_1\n\u03b9 : Type u_2\nA : Type u_3\nB : Type u_4\ninst\u271d\u00b9\u00b9 : CommSemiring \u03b9\ninst\u271d\u00b9\u2070 : Module \u03b9 (Additive \u2124\u02e3)\ninst\u271d\u2079 : DecidableEq \u03b9\n\ud835\udc9c : \u03b9 \u2192 Type u_5\n\u212c : \u03b9 \u2192 Type u_6\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommGroup (\ud835\udc9c i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommGroup (\u212c i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (\ud835\udc9c i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R (\u212c i)\ninst\u271d\u00b3 : DirectSum.GRing \ud835\udc9c\ninst\u271d\u00b2 : DirectSum.GRing \u212c\ninst\u271d\u00b9 : DirectSum.GAlgebra R \ud835\udc9c\ninst\u271d : DirectSum.GAlgebra R \u212c\na : \ud835\udc9c 0\nb : \u2a01 (i : \u03b9), \u212c i\n\u22a2 \u2191(gradedComm R \ud835\udc9c \u212c) \u2218\u2097 (mk R (\u2a01 (i : \u03b9), \ud835\udc9c i) (\u2a01 (i : \u03b9), \u212c i)) ((lof R \u03b9 \ud835\udc9c 0) a) =\n    (mk R (\u2a01 (i : \u03b9), \u212c i) (\u2a01 (i : \u03b9), \ud835\udc9c i)).flip ((lof R \u03b9 \ud835\udc9c 0) a)"}, {"tactic": "ext i b", "annotated_tactic": ["ext i b", []], "state_before": "R : Type u_1\n\u03b9 : Type u_2\nA : Type u_3\nB : Type u_4\ninst\u271d\u00b9\u00b9 : CommSemiring \u03b9\ninst\u271d\u00b9\u2070 : Module \u03b9 (Additive \u2124\u02e3)\ninst\u271d\u2079 : DecidableEq \u03b9\n\ud835\udc9c : \u03b9 \u2192 Type u_5\n\u212c : \u03b9 \u2192 Type u_6\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommGroup (\ud835\udc9c i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommGroup (\u212c i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (\ud835\udc9c i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R (\u212c i)\ninst\u271d\u00b3 : DirectSum.GRing \ud835\udc9c\ninst\u271d\u00b2 : DirectSum.GRing \u212c\ninst\u271d\u00b9 : DirectSum.GAlgebra R \ud835\udc9c\ninst\u271d : DirectSum.GAlgebra R \u212c\na : \ud835\udc9c 0\nb : \u2a01 (i : \u03b9), \u212c i\n\u22a2 \u2191(gradedComm R \ud835\udc9c \u212c) \u2218\u2097 (mk R (\u2a01 (i : \u03b9), \ud835\udc9c i) (\u2a01 (i : \u03b9), \u212c i)) ((lof R \u03b9 \ud835\udc9c 0) a) =\n    (mk R (\u2a01 (i : \u03b9), \u212c i) (\u2a01 (i : \u03b9), \ud835\udc9c i)).flip ((lof R \u03b9 \ud835\udc9c 0) a)", "state_after": "case H.h\nR : Type u_1\n\u03b9 : Type u_2\nA : Type u_3\nB : Type u_4\ninst\u271d\u00b9\u00b9 : CommSemiring \u03b9\ninst\u271d\u00b9\u2070 : Module \u03b9 (Additive \u2124\u02e3)\ninst\u271d\u2079 : DecidableEq \u03b9\n\ud835\udc9c : \u03b9 \u2192 Type u_5\n\u212c : \u03b9 \u2192 Type u_6\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommGroup (\ud835\udc9c i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommGroup (\u212c i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (\ud835\udc9c i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R (\u212c i)\ninst\u271d\u00b3 : DirectSum.GRing \ud835\udc9c\ninst\u271d\u00b2 : DirectSum.GRing \u212c\ninst\u271d\u00b9 : DirectSum.GAlgebra R \ud835\udc9c\ninst\u271d : DirectSum.GAlgebra R \u212c\na : \ud835\udc9c 0\nb\u271d : \u2a01 (i : \u03b9), \u212c i\ni : \u03b9\nb : \u212c i\n\u22a2 ((\u2191(gradedComm R \ud835\udc9c \u212c) \u2218\u2097 (mk R (\u2a01 (i : \u03b9), \ud835\udc9c i) (\u2a01 (i : \u03b9), \u212c i)) ((lof R \u03b9 \ud835\udc9c 0) a)) \u2218\u2097 lof R \u03b9 \u212c i) b =\n    ((mk R (\u2a01 (i : \u03b9), \u212c i) (\u2a01 (i : \u03b9), \ud835\udc9c i)).flip ((lof R \u03b9 \ud835\udc9c 0) a) \u2218\u2097 lof R \u03b9 \u212c i) b"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case H.h\nR : Type u_1\n\u03b9 : Type u_2\nA : Type u_3\nB : Type u_4\ninst\u271d\u00b9\u00b9 : CommSemiring \u03b9\ninst\u271d\u00b9\u2070 : Module \u03b9 (Additive \u2124\u02e3)\ninst\u271d\u2079 : DecidableEq \u03b9\n\ud835\udc9c : \u03b9 \u2192 Type u_5\n\u212c : \u03b9 \u2192 Type u_6\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommGroup (\ud835\udc9c i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommGroup (\u212c i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (\ud835\udc9c i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R (\u212c i)\ninst\u271d\u00b3 : DirectSum.GRing \ud835\udc9c\ninst\u271d\u00b2 : DirectSum.GRing \u212c\ninst\u271d\u00b9 : DirectSum.GAlgebra R \ud835\udc9c\ninst\u271d : DirectSum.GAlgebra R \u212c\na : \ud835\udc9c 0\nb\u271d : \u2a01 (i : \u03b9), \u212c i\ni : \u03b9\nb : \u212c i\n\u22a2 ((\u2191(gradedComm R \ud835\udc9c \u212c) \u2218\u2097 (mk R (\u2a01 (i : \u03b9), \ud835\udc9c i) (\u2a01 (i : \u03b9), \u212c i)) ((lof R \u03b9 \ud835\udc9c 0) a)) \u2218\u2097 lof R \u03b9 \u212c i) b =\n    ((mk R (\u2a01 (i : \u03b9), \u212c i) (\u2a01 (i : \u03b9), \ud835\udc9c i)).flip ((lof R \u03b9 \ud835\udc9c 0) a) \u2218\u2097 lof R \u03b9 \u212c i) b", "state_after": "case H.h\nR : Type u_1\n\u03b9 : Type u_2\nA : Type u_3\nB : Type u_4\ninst\u271d\u00b9\u00b9 : CommSemiring \u03b9\ninst\u271d\u00b9\u2070 : Module \u03b9 (Additive \u2124\u02e3)\ninst\u271d\u2079 : DecidableEq \u03b9\n\ud835\udc9c : \u03b9 \u2192 Type u_5\n\u212c : \u03b9 \u2192 Type u_6\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommGroup (\ud835\udc9c i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommGroup (\u212c i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (\ud835\udc9c i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R (\u212c i)\ninst\u271d\u00b3 : DirectSum.GRing \ud835\udc9c\ninst\u271d\u00b2 : DirectSum.GRing \u212c\ninst\u271d\u00b9 : DirectSum.GAlgebra R \ud835\udc9c\ninst\u271d : DirectSum.GAlgebra R \u212c\na : \ud835\udc9c 0\nb\u271d : \u2a01 (i : \u03b9), \u212c i\ni : \u03b9\nb : \u212c i\n\u22a2 (gradedComm R \ud835\udc9c \u212c) ((lof R \u03b9 \ud835\udc9c 0) a \u2297\u209c[R] (lof R \u03b9 \u212c i) b) = (lof R \u03b9 \u212c i) b \u2297\u209c[R] (lof R \u03b9 \ud835\udc9c 0) a"}, {"tactic": "rw [gradedComm_of_tmul_of, mul_zero, uzpow_zero, one_smul]", "annotated_tactic": ["rw [<a>gradedComm_of_tmul_of</a>, <a>mul_zero</a>, <a>uzpow_zero</a>, <a>one_smul</a>]", [{"full_name": "TensorProduct.gradedComm_of_tmul_of", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean", "def_pos": [116, 9], "def_end_pos": [116, 30]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}, {"full_name": "uzpow_zero", "def_path": "Mathlib/Data/ZMod/IntUnitsPower.lean", "def_pos": [85, 15], "def_end_pos": [85, 25]}, {"full_name": "one_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [453, 7], "def_end_pos": [453, 15]}]], "state_before": "case H.h\nR : Type u_1\n\u03b9 : Type u_2\nA : Type u_3\nB : Type u_4\ninst\u271d\u00b9\u00b9 : CommSemiring \u03b9\ninst\u271d\u00b9\u2070 : Module \u03b9 (Additive \u2124\u02e3)\ninst\u271d\u2079 : DecidableEq \u03b9\n\ud835\udc9c : \u03b9 \u2192 Type u_5\n\u212c : \u03b9 \u2192 Type u_6\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : (i : \u03b9) \u2192 AddCommGroup (\ud835\udc9c i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommGroup (\u212c i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (\ud835\udc9c i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module R (\u212c i)\ninst\u271d\u00b3 : DirectSum.GRing \ud835\udc9c\ninst\u271d\u00b2 : DirectSum.GRing \u212c\ninst\u271d\u00b9 : DirectSum.GAlgebra R \ud835\udc9c\ninst\u271d : DirectSum.GAlgebra R \u212c\na : \ud835\udc9c 0\nb\u271d : \u2a01 (i : \u03b9), \u212c i\ni : \u03b9\nb : \u212c i\n\u22a2 (gradedComm R \ud835\udc9c \u212c) ((lof R \u03b9 \ud835\udc9c 0) a \u2297\u209c[R] (lof R \u03b9 \u212c i) b) = (lof R \u03b9 \u212c i) b \u2297\u209c[R] (lof R \u03b9 \ud835\udc9c 0) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.tendsto_iff_eventually", "start": [3000, 1], "end": [3002, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/LSeries/Convergence.lean", "full_name": "LSeries.abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable'", "start": [74, 1], "end": [84, 17], "traced_tactics": [{"tactic": "induction' x with y", "annotated_tactic": ["induction' x with y", []], "state_before": "f : \u2115 \u2192 \u2102\nx : EReal\nh : \u2200 (y : \u211d), x < \u2191y \u2192 LSeriesSummable f \u2191y\n\u22a2 abscissaOfAbsConv f \u2264 x", "state_after": "case h_bot\nf : \u2115 \u2192 \u2102\nh : \u2200 (y : \u211d), \u22a5 < \u2191y \u2192 LSeriesSummable f \u2191y\n\u22a2 abscissaOfAbsConv f \u2264 \u22a5\n\ncase h_real\nf : \u2115 \u2192 \u2102\ny : \u211d\nh : \u2200 (y_1 : \u211d), \u2191y < \u2191y_1 \u2192 LSeriesSummable f \u2191y_1\n\u22a2 abscissaOfAbsConv f \u2264 \u2191y\n\ncase h_top\nf : \u2115 \u2192 \u2102\nh : \u2200 (y : \u211d), \u22a4 < \u2191y \u2192 LSeriesSummable f \u2191y\n\u22a2 abscissaOfAbsConv f \u2264 \u22a4"}, {"tactic": "refine le_of_eq <| sInf_eq_bot.mpr fun y hy \u21a6 ?_", "annotated_tactic": ["refine <a>le_of_eq</a> <| sInf_eq_bot.mpr fun y hy \u21a6 ?_", [{"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "case h_bot\nf : \u2115 \u2192 \u2102\nh : \u2200 (y : \u211d), \u22a5 < \u2191y \u2192 LSeriesSummable f \u2191y\n\u22a2 abscissaOfAbsConv f \u2264 \u22a5", "state_after": "case h_bot\nf : \u2115 \u2192 \u2102\nh : \u2200 (y : \u211d), \u22a5 < \u2191y \u2192 LSeriesSummable f \u2191y\ny : EReal\nhy : y > \u22a5\n\u22a2 \u2203 a \u2208 Real.toEReal '' {x | LSeriesSummable f \u2191x}, a < y"}, {"tactic": "induction' y with z", "annotated_tactic": ["induction' y with z", []], "state_before": "case h_bot\nf : \u2115 \u2192 \u2102\nh : \u2200 (y : \u211d), \u22a5 < \u2191y \u2192 LSeriesSummable f \u2191y\ny : EReal\nhy : y > \u22a5\n\u22a2 \u2203 a \u2208 Real.toEReal '' {x | LSeriesSummable f \u2191x}, a < y", "state_after": "case h_bot.h_bot\nf : \u2115 \u2192 \u2102\nh : \u2200 (y : \u211d), \u22a5 < \u2191y \u2192 LSeriesSummable f \u2191y\nhy : \u22a5 > \u22a5\n\u22a2 \u2203 a \u2208 Real.toEReal '' {x | LSeriesSummable f \u2191x}, a < \u22a5\n\ncase h_bot.h_real\nf : \u2115 \u2192 \u2102\nh : \u2200 (y : \u211d), \u22a5 < \u2191y \u2192 LSeriesSummable f \u2191y\nz : \u211d\nhy : \u2191z > \u22a5\n\u22a2 \u2203 a \u2208 Real.toEReal '' {x | LSeriesSummable f \u2191x}, a < \u2191z\n\ncase h_bot.h_top\nf : \u2115 \u2192 \u2102\nh : \u2200 (y : \u211d), \u22a5 < \u2191y \u2192 LSeriesSummable f \u2191y\nhy : \u22a4 > \u22a5\n\u22a2 \u2203 a \u2208 Real.toEReal '' {x | LSeriesSummable f \u2191x}, a < \u22a4"}, {"tactic": "simp only [gt_iff_lt, lt_self_iff_false] at hy", "annotated_tactic": ["simp only [<a>gt_iff_lt</a>, <a>lt_self_iff_false</a>] at hy", [{"full_name": "gt_iff_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1949, 17], "def_end_pos": [1949, 26]}, {"full_name": "lt_self_iff_false", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [165, 9], "def_end_pos": [165, 26]}]], "state_before": "case h_bot.h_bot\nf : \u2115 \u2192 \u2102\nh : \u2200 (y : \u211d), \u22a5 < \u2191y \u2192 LSeriesSummable f \u2191y\nhy : \u22a5 > \u22a5\n\u22a2 \u2203 a \u2208 Real.toEReal '' {x | LSeriesSummable f \u2191x}, a < \u22a5", "state_after": "no goals"}, {"tactic": "exact \u27e8z - 1,  \u27e8z-1, h (z - 1) <| EReal.bot_lt_coe _, rfl\u27e9, by norm_cast; exact sub_one_lt z\u27e9", "annotated_tactic": ["exact \u27e8z - 1,  \u27e8z-1, h (z - 1) <| <a>EReal.bot_lt_coe</a> _, <a>rfl</a>\u27e9, by norm_cast; exact <a>sub_one_lt</a> z\u27e9", [{"full_name": "EReal.bot_lt_coe", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [284, 9], "def_end_pos": [284, 19]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "sub_one_lt", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [1261, 7], "def_end_pos": [1261, 17]}]], "state_before": "case h_bot.h_real\nf : \u2115 \u2192 \u2102\nh : \u2200 (y : \u211d), \u22a5 < \u2191y \u2192 LSeriesSummable f \u2191y\nz : \u211d\nhy : \u2191z > \u22a5\n\u22a2 \u2203 a \u2208 Real.toEReal '' {x | LSeriesSummable f \u2191x}, a < \u2191z", "state_after": "no goals"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "f : \u2115 \u2192 \u2102\nh : \u2200 (y : \u211d), \u22a5 < \u2191y \u2192 LSeriesSummable f \u2191y\nz : \u211d\nhy : \u2191z > \u22a5\n\u22a2 \u2191z - 1 < \u2191z", "state_after": "f : \u2115 \u2192 \u2102\nh : \u2200 (y : \u211d), \u22a5 < \u2191y \u2192 LSeriesSummable f \u2191y\nz : \u211d\nhy : \u2191z > \u22a5\n\u22a2 z - 1 < z"}, {"tactic": "exact sub_one_lt z", "annotated_tactic": ["exact <a>sub_one_lt</a> z", [{"full_name": "sub_one_lt", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [1261, 7], "def_end_pos": [1261, 17]}]], "state_before": "f : \u2115 \u2192 \u2102\nh : \u2200 (y : \u211d), \u22a5 < \u2191y \u2192 LSeriesSummable f \u2191y\nz : \u211d\nhy : \u2191z > \u22a5\n\u22a2 z - 1 < z", "state_after": "no goals"}, {"tactic": "exact \u27e80, \u27e80, h 0 <| EReal.bot_lt_coe 0, rfl\u27e9, EReal.zero_lt_top\u27e9", "annotated_tactic": ["exact \u27e80, \u27e80, h 0 <| <a>EReal.bot_lt_coe</a> 0, <a>rfl</a>\u27e9, <a>EReal.zero_lt_top</a>\u27e9", [{"full_name": "EReal.bot_lt_coe", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [284, 9], "def_end_pos": [284, 19]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "EReal.zero_lt_top", "def_path": "Mathlib/Data/Real/EReal.lean", "def_pos": [329, 9], "def_end_pos": [329, 20]}]], "state_before": "case h_bot.h_top\nf : \u2115 \u2192 \u2102\nh : \u2200 (y : \u211d), \u22a5 < \u2191y \u2192 LSeriesSummable f \u2191y\nhy : \u22a4 > \u22a5\n\u22a2 \u2203 a \u2208 Real.toEReal '' {x | LSeriesSummable f \u2191x}, a < \u22a4", "state_after": "no goals"}, {"tactic": "exact abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable <| by exact_mod_cast h", "annotated_tactic": ["exact <a>abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable</a> <| by exact_mod_cast h", [{"full_name": "LSeries.abscissaOfAbsConv_le_of_forall_lt_LSeriesSummable", "def_path": "Mathlib/NumberTheory/LSeries/Convergence.lean", "def_pos": [61, 7], "def_end_pos": [61, 64]}]], "state_before": "case h_real\nf : \u2115 \u2192 \u2102\ny : \u211d\nh : \u2200 (y_1 : \u211d), \u2191y < \u2191y_1 \u2192 LSeriesSummable f \u2191y_1\n\u22a2 abscissaOfAbsConv f \u2264 \u2191y", "state_after": "no goals"}, {"tactic": "exact_mod_cast h", "annotated_tactic": ["exact_mod_cast h", []], "state_before": "f : \u2115 \u2192 \u2102\ny : \u211d\nh : \u2200 (y_1 : \u211d), \u2191y < \u2191y_1 \u2192 LSeriesSummable f \u2191y_1\n\u22a2 \u2200 (y_1 : \u211d), y < y_1 \u2192 LSeriesSummable f \u2191y_1", "state_after": "no goals"}, {"tactic": "exact le_top", "annotated_tactic": ["exact <a>le_top</a>", [{"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [66, 9], "def_end_pos": [66, 15]}]], "state_before": "case h_top\nf : \u2115 \u2192 \u2102\nh : \u2200 (y : \u211d), \u22a4 < \u2191y \u2192 LSeriesSummable f \u2191y\n\u22a2 abscissaOfAbsConv f \u2264 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/NNReal/Basic.lean", "full_name": "NNReal.coe_inv", "start": [190, 11], "end": [191, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Divisibility/Basic.lean", "full_name": "dvd_of_mul_left_dvd", "start": [221, 1], "end": [222, 60], "traced_tactics": [{"tactic": "simp [ceq]", "annotated_tactic": ["simp [ceq]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : CommSemigroup \u03b1\na b c : \u03b1\nh : a * b \u2223 c\nd : \u03b1\nceq : c = a * b * d\n\u22a2 b * (a * d) = c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Basic.lean", "full_name": "Equiv.Perm.SameCycle.of_zpow", "start": [202, 1], "end": [203, 33], "traced_tactics": [{"tactic": "simp [zpow_mul, h]", "annotated_tactic": ["simp [<a>zpow_mul</a>, h]", [{"full_name": "zpow_mul", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [661, 33], "def_end_pos": [661, 41]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nf g : Perm \u03b1\np : \u03b1 \u2192 Prop\nx y z : \u03b1\nn : \u2124\nx\u271d : (f ^ n).SameCycle x y\nm : \u2124\nh : ((f ^ n) ^ m) x = y\n\u22a2 (f ^ (n * m)) x = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "full_name": "add_tsub_cancel_left", "start": [361, 1], "end": [362, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/ContinuousOn.lean", "full_name": "Filter.Tendsto.piecewise_nhdsWithin", "start": [354, 1], "end": [357, 56], "traced_tactics": [{"tactic": "apply Tendsto.piecewise <;> rwa [\u2190 nhdsWithin_inter']", "annotated_tactic": ["apply <a>Tendsto.piecewise</a> <;> rwa [\u2190 <a>nhdsWithin_inter'</a>]", [{"full_name": "Filter.Tendsto.piecewise", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3295, 19], "def_end_pos": [3295, 36]}, {"full_name": "nhdsWithin_inter'", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [257, 9], "def_end_pos": [257, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : TopologicalSpace \u03b1\n\u03b9 : Type u_5\n\u03c0 : \u03b9 \u2192 Type u_6\ninst\u271d\u00b9 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\nf g : \u03b1 \u2192 \u03b2\nt : Set \u03b1\ninst\u271d : (x : \u03b1) \u2192 Decidable (x \u2208 t)\na : \u03b1\ns : Set \u03b1\nl : Filter \u03b2\nh\u2080 : Tendsto f (\ud835\udcdd[s \u2229 t] a) l\nh\u2081 : Tendsto g (\ud835\udcdd[s \u2229 t\u1d9c] a) l\n\u22a2 Tendsto (t.piecewise f g) (\ud835\udcdd[s] a) l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Int/Bitwise.lean", "full_name": "Int.bodd_two", "start": [135, 1], "end": [136, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/Star/Multiplier.lean", "full_name": "DoubleCentralizer.add_fst", "start": [256, 1], "end": [257, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Adjoin/FG.lean", "full_name": "Subalgebra.fg_bot", "start": [104, 1], "end": [105, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matrix/ColumnRowPartitioned.lean", "full_name": "Matrix.toRows\u2082_fromRows", "start": [84, 1], "end": [86, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.map_map", "start": [558, 1], "end": [559, 55], "traced_tactics": [{"tactic": "erw [\u2190 bind_some_eq_map, bind_map, bind_some_eq_map]", "annotated_tactic": ["erw [\u2190 <a>bind_some_eq_map</a>, <a>bind_map</a>, <a>bind_some_eq_map</a>]", [{"full_name": "Part.bind_some_eq_map", "def_path": "Mathlib/Data/Part.lean", "def_pos": [525, 9], "def_end_pos": [525, 25]}, {"full_name": "Part.bind_map", "def_path": "Mathlib/Data/Part.lean", "def_pos": [548, 9], "def_end_pos": [548, 17]}, {"full_name": "Part.bind_some_eq_map", "def_path": "Mathlib/Data/Part.lean", "def_pos": [525, 9], "def_end_pos": [525, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ng : \u03b2 \u2192 \u03b3\nf : \u03b1 \u2192 \u03b2\no : Part \u03b1\n\u22a2 map g (map f o) = map (g \u2218 f) o", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/TwoDim.lean", "full_name": "Orientation.abs_areaForm_le", "start": [143, 1], "end": [144, 93], "traced_tactics": [{"tactic": "simpa [areaForm_to_volumeForm, Fin.prod_univ_succ] using o.abs_volumeForm_apply_le ![x, y]", "annotated_tactic": ["simpa [<a>areaForm_to_volumeForm</a>, <a>Fin.prod_univ_succ</a>] using o.abs_volumeForm_apply_le ![x, y]", [{"full_name": "Orientation.areaForm_to_volumeForm", "def_path": "Mathlib/Analysis/InnerProductSpace/TwoDim.lean", "def_pos": [106, 9], "def_end_pos": [106, 31]}, {"full_name": "Fin.prod_univ_succ", "def_path": "Mathlib/Algebra/BigOperators/Fin.lean", "def_pos": [80, 9], "def_end_pos": [80, 23]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 |(o.areaForm x) y| \u2264 \u2016x\u2016 * \u2016y\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.mem_ae_map_iff", "start": [2012, 1], "end": [2014, 80], "traced_tactics": [{"tactic": "simp only [mem_ae_iff, map_apply_of_aemeasurable hf hs.compl, preimage_compl]", "annotated_tactic": ["simp only [<a>mem_ae_iff</a>, <a>map_apply_of_aemeasurable</a> hf hs.compl, <a>preimage_compl</a>]", [{"full_name": "MeasureTheory.mem_ae_iff", "def_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean", "def_pos": [71, 9], "def_end_pos": [71, 19]}, {"full_name": "MeasureTheory.Measure.map_apply_of_aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1286, 9], "def_end_pos": [1286, 34]}, {"full_name": "Set.preimage_compl", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [82, 9], "def_end_pos": [82, 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\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 : \u03b1 \u2192 \u03b2\nhf : AEMeasurable f \u03bc\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 s \u2208 ae (Measure.map f \u03bc) \u2194 f \u207b\u00b9' s \u2208 ae \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Instances/Matrix.lean", "full_name": "Continuous.matrix_cramer", "start": [219, 1], "end": [222, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Abelian/Homology.lean", "full_name": "homology'.map_eq_desc'_lift_left", "start": [239, 1], "end": [253, 11], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "A : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u226b cokernel.desc f' g' w' = 0", "state_after": "no goals"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "A : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 kernel.lift g f w \u226b lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u22ef = 0", "state_after": "case h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 (kernel.lift g f w \u226b lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u22ef) \u226b \u03b9 f' g' w' = 0 \u226b \u03b9 f' g' w'"}, {"tactic": "simp only [\u2190 h, Category.assoc, zero_comp, lift_\u03b9, kernel.lift_\u03b9_assoc]", "annotated_tactic": ["simp only [\u2190 h, <a>Category.assoc</a>, <a>zero_comp</a>, <a>lift_\u03b9</a>, <a>kernel.lift_\u03b9_assoc</a>]", [{"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.Limits.zero_comp", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean", "def_pos": [72, 9], "def_end_pos": [72, 18]}, {"full_name": "homology'.lift_\u03b9", "def_path": "Mathlib/CategoryTheory/Abelian/Homology.lean", "def_pos": [144, 9], "def_end_pos": [144, 15]}, {"full_name": "CategoryTheory.Limits.kernel.lift_\u03b9_assoc", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean", "def_pos": [287, 3], "def_end_pos": [287, 25]}]], "state_before": "case h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 (kernel.lift g f w \u226b lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u22ef) \u226b \u03b9 f' g' w' = 0 \u226b \u03b9 f' g' w'", "state_after": "case h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 f \u226b \u03b1.right \u226b cokernel.\u03c0 f' = 0"}, {"tactic": "erw [\u2190 reassoc_of% \u03b1.w]", "annotated_tactic": ["erw [\u2190 reassoc_of% \u03b1.w]", []], "state_before": "case h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 f \u226b \u03b1.right \u226b cokernel.\u03c0 f' = 0", "state_after": "case h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 \u03b1.left \u226b (Arrow.mk f').hom \u226b cokernel.\u03c0 f' = 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 \u03b1.left \u226b (Arrow.mk f').hom \u226b cokernel.\u03c0 f' = 0", "state_after": "no goals"}, {"tactic": "apply homology'.hom_from_ext", "annotated_tactic": ["apply <a>homology'.hom_from_ext</a>", [{"full_name": "homology'.hom_from_ext", "def_path": "Mathlib/CategoryTheory/Abelian/Homology.lean", "def_pos": [163, 9], "def_end_pos": [163, 21]}]], "state_before": "A : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 map w w' \u03b1 \u03b2 h = desc' f g w (lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u22ef) \u22ef", "state_after": "case h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 \u03c0' f g w \u226b map w w' \u03b1 \u03b2 h = \u03c0' f g w \u226b desc' f g w (lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u22ef) \u22ef"}, {"tactic": "simp only [\u03c0'_map, \u03c0'_desc']", "annotated_tactic": ["simp only [<a>\u03c0'_map</a>, <a>\u03c0'_desc'</a>]", [{"full_name": "homology'.\u03c0'_map", "def_path": "Mathlib/CategoryTheory/Abelian/Homology.lean", "def_pos": [207, 9], "def_end_pos": [207, 15]}, {"full_name": "homology'.\u03c0'_desc'", "def_path": "Mathlib/CategoryTheory/Abelian/Homology.lean", "def_pos": [137, 9], "def_end_pos": [137, 17]}]], "state_before": "case h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 \u03c0' f g w \u226b map w w' \u03b1 \u03b2 h = \u03c0' f g w \u226b desc' f g w (lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u22ef) \u22ef", "state_after": "case h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 kernel.map g g' \u03b1.right \u03b2.right \u22ef \u226b \u03c0' f' g' w' = lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u22ef"}, {"tactic": "dsimp [\u03c0', lift]", "annotated_tactic": ["dsimp [<a>\u03c0'</a>, <a>lift</a>]", [{"full_name": "homology'.\u03c0'", "def_path": "Mathlib/CategoryTheory/Abelian/Homology.lean", "def_pos": [117, 5], "def_end_pos": [117, 7]}, {"full_name": "homology'.lift", "def_path": "Mathlib/CategoryTheory/Abelian/Homology.lean", "def_pos": [132, 5], "def_end_pos": [132, 9]}]], "state_before": "case h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 kernel.map g g' \u03b1.right \u03b2.right \u22ef \u226b \u03c0' f' g' w' = lift f' g' w' (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u22ef", "state_after": "case h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 kernel.map g g' \u03b1.right \u03b2.right \u22ef \u226b cokernel.\u03c0 (kernel.lift g' f' w') \u226b (homology'IsoCokernelLift f' g' w').inv =\n    kernel.lift (cokernel.desc f' g' w') (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u22ef \u226b (homology'IsoKernelDesc f' g' w').inv"}, {"tactic": "rw [Iso.eq_comp_inv]", "annotated_tactic": ["rw [<a>Iso.eq_comp_inv</a>]", [{"full_name": "CategoryTheory.Iso.eq_comp_inv", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [223, 9], "def_end_pos": [223, 20]}]], "state_before": "case h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 kernel.map g g' \u03b1.right \u03b2.right \u22ef \u226b cokernel.\u03c0 (kernel.lift g' f' w') \u226b (homology'IsoCokernelLift f' g' w').inv =\n    kernel.lift (cokernel.desc f' g' w') (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u22ef \u226b (homology'IsoKernelDesc f' g' w').inv", "state_after": "case h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 (kernel.map g g' \u03b1.right \u03b2.right \u22ef \u226b cokernel.\u03c0 (kernel.lift g' f' w') \u226b (homology'IsoCokernelLift f' g' w').inv) \u226b\n      (homology'IsoKernelDesc f' g' w').hom =\n    kernel.lift (cokernel.desc f' g' w') (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u22ef"}, {"tactic": "dsimp [homology'IsoKernelDesc]", "annotated_tactic": ["dsimp [<a>homology'IsoKernelDesc</a>]", [{"full_name": "homology'IsoKernelDesc", "def_path": "Mathlib/CategoryTheory/Abelian/Homology.lean", "def_pos": [109, 5], "def_end_pos": [109, 27]}]], "state_before": "case h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 (kernel.map g g' \u03b1.right \u03b2.right \u22ef \u226b cokernel.\u03c0 (kernel.lift g' f' w') \u226b (homology'IsoCokernelLift f' g' w').inv) \u226b\n      (homology'IsoKernelDesc f' g' w').hom =\n    kernel.lift (cokernel.desc f' g' w') (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u22ef", "state_after": "case h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 (kernel.map g g' \u03b1.right \u03b2.right \u22ef \u226b cokernel.\u03c0 (kernel.lift g' f' w') \u226b (homology'IsoCokernelLift f' g' w').inv) \u226b\n      (homology'IsoCokernelLift f' g' w').hom \u226b Abelian.homologyCToK f' g' w' =\n    kernel.lift (cokernel.desc f' g' w') (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u22ef"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "case h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 (kernel.map g g' \u03b1.right \u03b2.right \u22ef \u226b cokernel.\u03c0 (kernel.lift g' f' w') \u226b (homology'IsoCokernelLift f' g' w').inv) \u226b\n      (homology'IsoCokernelLift f' g' w').hom \u226b Abelian.homologyCToK f' g' w' =\n    kernel.lift (cokernel.desc f' g' w') (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u22ef", "state_after": "case h.h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 ((kernel.map g g' \u03b1.right \u03b2.right \u22ef \u226b cokernel.\u03c0 (kernel.lift g' f' w') \u226b (homology'IsoCokernelLift f' g' w').inv) \u226b\n        (homology'IsoCokernelLift f' g' w').hom \u226b Abelian.homologyCToK f' g' w') \u226b\n      equalizer.\u03b9 (cokernel.desc f' g' w') 0 =\n    kernel.lift (cokernel.desc f' g' w') (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u22ef \u226b\n      equalizer.\u03b9 (cokernel.desc f' g' w') 0"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "case h.h\nA : Type u\ninst\u271d\u00b9 : Category.{v, u} A\ninst\u271d : Abelian A\nX Y Z : A\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\nX' Y' Z' : A\nf' : X' \u27f6 Y'\ng' : Y' \u27f6 Z'\nw' : f' \u226b g' = 0\n\u03b1 : Arrow.mk f \u27f6 Arrow.mk f'\n\u03b2 : Arrow.mk g \u27f6 Arrow.mk g'\nh : \u03b1.right = \u03b2.left\n\u22a2 ((kernel.map g g' \u03b1.right \u03b2.right \u22ef \u226b cokernel.\u03c0 (kernel.lift g' f' w') \u226b (homology'IsoCokernelLift f' g' w').inv) \u226b\n        (homology'IsoCokernelLift f' g' w').hom \u226b Abelian.homologyCToK f' g' w') \u226b\n      equalizer.\u03b9 (cokernel.desc f' g' w') 0 =\n    kernel.lift (cokernel.desc f' g' w') (kernel.\u03b9 g \u226b \u03b2.left \u226b cokernel.\u03c0 f') \u22ef \u226b\n      equalizer.\u03b9 (cokernel.desc f' g' w') 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Sets/Compacts.lean", "full_name": "TopologicalSpace.NonemptyCompacts.coe_toCompacts", "start": [255, 1], "end": [255, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Hom/Defs.lean", "full_name": "RingHom.coe_coe", "start": [446, 1], "end": [448, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subsemigroup/Basic.lean", "full_name": "Subsemigroup.mem_mk", "start": [118, 1], "end": [119, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Commutator.lean", "full_name": "Subgroup.commutator_le_right", "start": [149, 1], "end": [150, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Projection.lean", "full_name": "Submodule.linearProjOfIsCompl_comp_subtype", "start": [190, 1], "end": [192, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/Star/Multiplier.lean", "full_name": "DoubleCentralizer.natCast_snd", "start": [313, 1], "end": [314, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "full_name": "Submodule.disjoint_iff_comap_eq_bot", "start": [443, 1], "end": [445, 46], "traced_tactics": [{"tactic": "rw [\u2190 (map_injective_of_injective (show Injective p.subtype from Subtype.coe_injective)).eq_iff,\n  map_comap_subtype, map_bot, disjoint_iff]", "annotated_tactic": ["rw [\u2190 (<a>map_injective_of_injective</a> (show <a>Injective</a> p.subtype from <a>Subtype.coe_injective</a>)).<a>eq_iff</a>,\n    <a>map_comap_subtype</a>, <a>map_bot</a>, <a>disjoint_iff</a>]", [{"full_name": "Submodule.map_injective_of_injective", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [357, 9], "def_end_pos": [357, 35]}, {"full_name": "Function.Injective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [123, 5], "def_end_pos": [123, 14]}, {"full_name": "Subtype.coe_injective", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [128, 9], "def_end_pos": [128, 22]}, {"full_name": "Function.Injective.eq_iff", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [85, 9], "def_end_pos": [85, 25]}, {"full_name": "Submodule.map_comap_subtype", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [423, 9], "def_end_pos": [423, 26]}, {"full_name": "Submodule.map_bot", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [234, 9], "def_end_pos": [234, 16]}, {"full_name": "disjoint_iff", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [135, 9], "def_end_pos": [135, 21]}]], "state_before": "R : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nM : Type u_5\nM\u2081 : Type u_6\nM\u2082 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u00b9\u00b9 : Semiring R\ninst\u271d\u00b9\u2070 : Semiring R\u2082\ninst\u271d\u2079 : Semiring R\u2083\ninst\u271d\u2078 : AddCommMonoid M\ninst\u271d\u2077 : AddCommMonoid M\u2082\ninst\u271d\u2076 : AddCommMonoid M\u2083\ninst\u271d\u2075 : Module R M\ninst\u271d\u2074 : Module R\u2082 M\u2082\ninst\u271d\u00b3 : Module R\u2083 M\u2083\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2081\u2083 : R \u2192+* R\u2083\ninst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\np\u271d p' : Submodule R M\nq\u271d q' : Submodule R\u2082 M\u2082\nx : M\nF : Type u_9\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\np q : Submodule R M\n\u22a2 Disjoint p q \u2194 comap p.subtype q = \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Rotate.lean", "full_name": "List.isRotated_concat", "start": [503, 1], "end": [504, 30], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u\nl l' : List \u03b1\nhd : \u03b1\ntl : List \u03b1\n\u22a2 (hd :: tl).rotate 1 = tl ++ [hd]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/Adjoin.lean", "full_name": "IntermediateField.restrictScalars_adjoin_of_algEquiv", "start": [499, 1], "end": [511, 94], "traced_tactics": [{"tactic": "apply_fun toSubfield using (fun K K' h \u21a6 by\n  ext x; change x \u2208 K.toSubfield \u2194 x \u2208 K'.toSubfield; rw [h])", "annotated_tactic": ["apply_fun <a>toSubfield</a> using (fun K K' h \u21a6 by\n    ext x; change x \u2208 K.toSubfield \u2194 x \u2208 K'.toSubfield; rw [h])", [{"full_name": "IntermediateField.toSubfield", "def_path": "Mathlib/FieldTheory/IntermediateField.lean", "def_pos": [71, 5], "def_end_pos": [71, 15]}]], "state_before": "F : Type u_1\ninst\u271d\u00b9\u2070 : Field F\nE : Type u_2\ninst\u271d\u2079 : Field E\ninst\u271d\u2078 : Algebra F E\nS\u271d : Set E\nL : Type u_3\nL' : Type u_4\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra F L\ninst\u271d\u2074 : Algebra L E\ninst\u271d\u00b3 : Algebra F L'\ninst\u271d\u00b2 : Algebra L' E\ninst\u271d\u00b9 : IsScalarTower F L E\ninst\u271d : IsScalarTower F L' E\ni : L \u2243\u2090[F] L'\nhi : \u21d1(algebraMap L E) = \u21d1(algebraMap L' E) \u2218 \u21d1i\nS : Set E\n\u22a2 restrictScalars F (adjoin L S) = restrictScalars F (adjoin L' S)", "state_after": "F : Type u_1\ninst\u271d\u00b9\u2070 : Field F\nE : Type u_2\ninst\u271d\u2079 : Field E\ninst\u271d\u2078 : Algebra F E\nS\u271d : Set E\nL : Type u_3\nL' : Type u_4\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra F L\ninst\u271d\u2074 : Algebra L E\ninst\u271d\u00b3 : Algebra F L'\ninst\u271d\u00b2 : Algebra L' E\ninst\u271d\u00b9 : IsScalarTower F L E\ninst\u271d : IsScalarTower F L' E\ni : L \u2243\u2090[F] L'\nhi : \u21d1(algebraMap L E) = \u21d1(algebraMap L' E) \u2218 \u21d1i\nS : Set E\n\u22a2 (restrictScalars F (adjoin L S)).toSubfield = (restrictScalars F (adjoin L' S)).toSubfield"}, {"tactic": "change Subfield.closure _ = Subfield.closure _", "annotated_tactic": ["change <a>Subfield.closure</a> _ = <a>Subfield.closure</a> _", [{"full_name": "Subfield.closure", "def_path": "Mathlib/Algebra/Field/Subfield.lean", "def_pos": [660, 5], "def_end_pos": [660, 12]}, {"full_name": "Subfield.closure", "def_path": "Mathlib/Algebra/Field/Subfield.lean", "def_pos": [660, 5], "def_end_pos": [660, 12]}]], "state_before": "F : Type u_1\ninst\u271d\u00b9\u2070 : Field F\nE : Type u_2\ninst\u271d\u2079 : Field E\ninst\u271d\u2078 : Algebra F E\nS\u271d : Set E\nL : Type u_3\nL' : Type u_4\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra F L\ninst\u271d\u2074 : Algebra L E\ninst\u271d\u00b3 : Algebra F L'\ninst\u271d\u00b2 : Algebra L' E\ninst\u271d\u00b9 : IsScalarTower F L E\ninst\u271d : IsScalarTower F L' E\ni : L \u2243\u2090[F] L'\nhi : \u21d1(algebraMap L E) = \u21d1(algebraMap L' E) \u2218 \u21d1i\nS : Set E\n\u22a2 (restrictScalars F (adjoin L S)).toSubfield = (restrictScalars F (adjoin L' S)).toSubfield", "state_after": "F : Type u_1\ninst\u271d\u00b9\u2070 : Field F\nE : Type u_2\ninst\u271d\u2079 : Field E\ninst\u271d\u2078 : Algebra F E\nS\u271d : Set E\nL : Type u_3\nL' : Type u_4\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra F L\ninst\u271d\u2074 : Algebra L E\ninst\u271d\u00b3 : Algebra F L'\ninst\u271d\u00b2 : Algebra L' E\ninst\u271d\u00b9 : IsScalarTower F L E\ninst\u271d : IsScalarTower F L' E\ni : L \u2243\u2090[F] L'\nhi : \u21d1(algebraMap L E) = \u21d1(algebraMap L' E) \u2218 \u21d1i\nS : Set E\n\u22a2 Subfield.closure (Set.range \u21d1(algebraMap L E) \u222a S) = Subfield.closure (Set.range \u21d1(algebraMap L' E) \u222a S)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "F : Type u_1\ninst\u271d\u00b9\u2070 : Field F\nE : Type u_2\ninst\u271d\u2079 : Field E\ninst\u271d\u2078 : Algebra F E\nS\u271d : Set E\nL : Type u_3\nL' : Type u_4\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra F L\ninst\u271d\u2074 : Algebra L E\ninst\u271d\u00b3 : Algebra F L'\ninst\u271d\u00b2 : Algebra L' E\ninst\u271d\u00b9 : IsScalarTower F L E\ninst\u271d : IsScalarTower F L' E\ni : L \u2243\u2090[F] L'\nhi : \u21d1(algebraMap L E) = \u21d1(algebraMap L' E) \u2218 \u21d1i\nS : Set E\n\u22a2 Subfield.closure (Set.range \u21d1(algebraMap L E) \u222a S) = Subfield.closure (Set.range \u21d1(algebraMap L' E) \u222a S)", "state_after": "case e_s.e_a\nF : Type u_1\ninst\u271d\u00b9\u2070 : Field F\nE : Type u_2\ninst\u271d\u2079 : Field E\ninst\u271d\u2078 : Algebra F E\nS\u271d : Set E\nL : Type u_3\nL' : Type u_4\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra F L\ninst\u271d\u2074 : Algebra L E\ninst\u271d\u00b3 : Algebra F L'\ninst\u271d\u00b2 : Algebra L' E\ninst\u271d\u00b9 : IsScalarTower F L E\ninst\u271d : IsScalarTower F L' E\ni : L \u2243\u2090[F] L'\nhi : \u21d1(algebraMap L E) = \u21d1(algebraMap L' E) \u2218 \u21d1i\nS : Set E\n\u22a2 Set.range \u21d1(algebraMap L E) = Set.range \u21d1(algebraMap L' E)"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case e_s.e_a\nF : Type u_1\ninst\u271d\u00b9\u2070 : Field F\nE : Type u_2\ninst\u271d\u2079 : Field E\ninst\u271d\u2078 : Algebra F E\nS\u271d : Set E\nL : Type u_3\nL' : Type u_4\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra F L\ninst\u271d\u2074 : Algebra L E\ninst\u271d\u00b3 : Algebra F L'\ninst\u271d\u00b2 : Algebra L' E\ninst\u271d\u00b9 : IsScalarTower F L E\ninst\u271d : IsScalarTower F L' E\ni : L \u2243\u2090[F] L'\nhi : \u21d1(algebraMap L E) = \u21d1(algebraMap L' E) \u2218 \u21d1i\nS : Set E\n\u22a2 Set.range \u21d1(algebraMap L E) = Set.range \u21d1(algebraMap L' E)", "state_after": "case e_s.e_a.h\nF : Type u_1\ninst\u271d\u00b9\u2070 : Field F\nE : Type u_2\ninst\u271d\u2079 : Field E\ninst\u271d\u2078 : Algebra F E\nS\u271d : Set E\nL : Type u_3\nL' : Type u_4\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra F L\ninst\u271d\u2074 : Algebra L E\ninst\u271d\u00b3 : Algebra F L'\ninst\u271d\u00b2 : Algebra L' E\ninst\u271d\u00b9 : IsScalarTower F L E\ninst\u271d : IsScalarTower F L' E\ni : L \u2243\u2090[F] L'\nhi : \u21d1(algebraMap L E) = \u21d1(algebraMap L' E) \u2218 \u21d1i\nS : Set E\nx : E\n\u22a2 x \u2208 Set.range \u21d1(algebraMap L E) \u2194 x \u2208 Set.range \u21d1(algebraMap L' E)"}, {"tactic": "exact \u27e8fun \u27e8y, h\u27e9 \u21a6 \u27e8i y, by rw [\u2190 h, hi]; rfl\u27e9,\n  fun \u27e8y, h\u27e9 \u21a6 \u27e8i.symm y, by rw [\u2190 h, hi, Function.comp_apply, AlgEquiv.apply_symm_apply]\u27e9\u27e9", "annotated_tactic": ["exact \u27e8fun \u27e8y, h\u27e9 \u21a6 \u27e8i y, by rw [\u2190 h, hi]; rfl\u27e9,\n    fun \u27e8y, h\u27e9 \u21a6 \u27e8i.symm y, by rw [\u2190 h, hi, <a>Function.comp_apply</a>, <a>AlgEquiv.apply_symm_apply</a>]\u27e9\u27e9", [{"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}, {"full_name": "AlgEquiv.apply_symm_apply", "def_path": "Mathlib/Algebra/Algebra/Equiv.lean", "def_pos": [407, 9], "def_end_pos": [407, 25]}]], "state_before": "case e_s.e_a.h\nF : Type u_1\ninst\u271d\u00b9\u2070 : Field F\nE : Type u_2\ninst\u271d\u2079 : Field E\ninst\u271d\u2078 : Algebra F E\nS\u271d : Set E\nL : Type u_3\nL' : Type u_4\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra F L\ninst\u271d\u2074 : Algebra L E\ninst\u271d\u00b3 : Algebra F L'\ninst\u271d\u00b2 : Algebra L' E\ninst\u271d\u00b9 : IsScalarTower F L E\ninst\u271d : IsScalarTower F L' E\ni : L \u2243\u2090[F] L'\nhi : \u21d1(algebraMap L E) = \u21d1(algebraMap L' E) \u2218 \u21d1i\nS : Set E\nx : E\n\u22a2 x \u2208 Set.range \u21d1(algebraMap L E) \u2194 x \u2208 Set.range \u21d1(algebraMap L' E)", "state_after": "no goals"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "F : Type u_1\ninst\u271d\u00b9\u2070 : Field F\nE : Type u_2\ninst\u271d\u2079 : Field E\ninst\u271d\u2078 : Algebra F E\nS\u271d : Set E\nL : Type u_3\nL' : Type u_4\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra F L\ninst\u271d\u2074 : Algebra L E\ninst\u271d\u00b3 : Algebra F L'\ninst\u271d\u00b2 : Algebra L' E\ninst\u271d\u00b9 : IsScalarTower F L E\ninst\u271d : IsScalarTower F L' E\ni : L \u2243\u2090[F] L'\nhi : \u21d1(algebraMap L E) = \u21d1(algebraMap L' E) \u2218 \u21d1i\nS : Set E\nK K' : IntermediateField F E\nh : K.toSubfield = K'.toSubfield\n\u22a2 K = K'", "state_after": "case h\nF : Type u_1\ninst\u271d\u00b9\u2070 : Field F\nE : Type u_2\ninst\u271d\u2079 : Field E\ninst\u271d\u2078 : Algebra F E\nS\u271d : Set E\nL : Type u_3\nL' : Type u_4\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra F L\ninst\u271d\u2074 : Algebra L E\ninst\u271d\u00b3 : Algebra F L'\ninst\u271d\u00b2 : Algebra L' E\ninst\u271d\u00b9 : IsScalarTower F L E\ninst\u271d : IsScalarTower F L' E\ni : L \u2243\u2090[F] L'\nhi : \u21d1(algebraMap L E) = \u21d1(algebraMap L' E) \u2218 \u21d1i\nS : Set E\nK K' : IntermediateField F E\nh : K.toSubfield = K'.toSubfield\nx : E\n\u22a2 x \u2208 K \u2194 x \u2208 K'"}, {"tactic": "change x \u2208 K.toSubfield \u2194 x \u2208 K'.toSubfield", "annotated_tactic": ["change x \u2208 K.toSubfield \u2194 x \u2208 K'.toSubfield", []], "state_before": "case h\nF : Type u_1\ninst\u271d\u00b9\u2070 : Field F\nE : Type u_2\ninst\u271d\u2079 : Field E\ninst\u271d\u2078 : Algebra F E\nS\u271d : Set E\nL : Type u_3\nL' : Type u_4\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra F L\ninst\u271d\u2074 : Algebra L E\ninst\u271d\u00b3 : Algebra F L'\ninst\u271d\u00b2 : Algebra L' E\ninst\u271d\u00b9 : IsScalarTower F L E\ninst\u271d : IsScalarTower F L' E\ni : L \u2243\u2090[F] L'\nhi : \u21d1(algebraMap L E) = \u21d1(algebraMap L' E) \u2218 \u21d1i\nS : Set E\nK K' : IntermediateField F E\nh : K.toSubfield = K'.toSubfield\nx : E\n\u22a2 x \u2208 K \u2194 x \u2208 K'", "state_after": "case h\nF : Type u_1\ninst\u271d\u00b9\u2070 : Field F\nE : Type u_2\ninst\u271d\u2079 : Field E\ninst\u271d\u2078 : Algebra F E\nS\u271d : Set E\nL : Type u_3\nL' : Type u_4\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra F L\ninst\u271d\u2074 : Algebra L E\ninst\u271d\u00b3 : Algebra F L'\ninst\u271d\u00b2 : Algebra L' E\ninst\u271d\u00b9 : IsScalarTower F L E\ninst\u271d : IsScalarTower F L' E\ni : L \u2243\u2090[F] L'\nhi : \u21d1(algebraMap L E) = \u21d1(algebraMap L' E) \u2218 \u21d1i\nS : Set E\nK K' : IntermediateField F E\nh : K.toSubfield = K'.toSubfield\nx : E\n\u22a2 x \u2208 K.toSubfield \u2194 x \u2208 K'.toSubfield"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "case h\nF : Type u_1\ninst\u271d\u00b9\u2070 : Field F\nE : Type u_2\ninst\u271d\u2079 : Field E\ninst\u271d\u2078 : Algebra F E\nS\u271d : Set E\nL : Type u_3\nL' : Type u_4\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra F L\ninst\u271d\u2074 : Algebra L E\ninst\u271d\u00b3 : Algebra F L'\ninst\u271d\u00b2 : Algebra L' E\ninst\u271d\u00b9 : IsScalarTower F L E\ninst\u271d : IsScalarTower F L' E\ni : L \u2243\u2090[F] L'\nhi : \u21d1(algebraMap L E) = \u21d1(algebraMap L' E) \u2218 \u21d1i\nS : Set E\nK K' : IntermediateField F E\nh : K.toSubfield = K'.toSubfield\nx : E\n\u22a2 x \u2208 K.toSubfield \u2194 x \u2208 K'.toSubfield", "state_after": "no goals"}, {"tactic": "rw [\u2190 h, hi]", "annotated_tactic": ["rw [\u2190 h, hi]", []], "state_before": "F : Type u_1\ninst\u271d\u00b9\u2070 : Field F\nE : Type u_2\ninst\u271d\u2079 : Field E\ninst\u271d\u2078 : Algebra F E\nS\u271d : Set E\nL : Type u_3\nL' : Type u_4\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra F L\ninst\u271d\u2074 : Algebra L E\ninst\u271d\u00b3 : Algebra F L'\ninst\u271d\u00b2 : Algebra L' E\ninst\u271d\u00b9 : IsScalarTower F L E\ninst\u271d : IsScalarTower F L' E\ni : L \u2243\u2090[F] L'\nhi : \u21d1(algebraMap L E) = \u21d1(algebraMap L' E) \u2218 \u21d1i\nS : Set E\nx : E\nx\u271d : x \u2208 Set.range \u21d1(algebraMap L E)\ny : L\nh : (algebraMap L E) y = x\n\u22a2 (algebraMap L' E) (i y) = x", "state_after": "F : Type u_1\ninst\u271d\u00b9\u2070 : Field F\nE : Type u_2\ninst\u271d\u2079 : Field E\ninst\u271d\u2078 : Algebra F E\nS\u271d : Set E\nL : Type u_3\nL' : Type u_4\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra F L\ninst\u271d\u2074 : Algebra L E\ninst\u271d\u00b3 : Algebra F L'\ninst\u271d\u00b2 : Algebra L' E\ninst\u271d\u00b9 : IsScalarTower F L E\ninst\u271d : IsScalarTower F L' E\ni : L \u2243\u2090[F] L'\nhi : \u21d1(algebraMap L E) = \u21d1(algebraMap L' E) \u2218 \u21d1i\nS : Set E\nx : E\nx\u271d : x \u2208 Set.range \u21d1(algebraMap L E)\ny : L\nh : (algebraMap L E) y = x\n\u22a2 (algebraMap L' E) (i y) = (\u21d1(algebraMap L' E) \u2218 \u21d1i) y"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "F : Type u_1\ninst\u271d\u00b9\u2070 : Field F\nE : Type u_2\ninst\u271d\u2079 : Field E\ninst\u271d\u2078 : Algebra F E\nS\u271d : Set E\nL : Type u_3\nL' : Type u_4\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra F L\ninst\u271d\u2074 : Algebra L E\ninst\u271d\u00b3 : Algebra F L'\ninst\u271d\u00b2 : Algebra L' E\ninst\u271d\u00b9 : IsScalarTower F L E\ninst\u271d : IsScalarTower F L' E\ni : L \u2243\u2090[F] L'\nhi : \u21d1(algebraMap L E) = \u21d1(algebraMap L' E) \u2218 \u21d1i\nS : Set E\nx : E\nx\u271d : x \u2208 Set.range \u21d1(algebraMap L E)\ny : L\nh : (algebraMap L E) y = x\n\u22a2 (algebraMap L' E) (i y) = (\u21d1(algebraMap L' E) \u2218 \u21d1i) y", "state_after": "no goals"}, {"tactic": "rw [\u2190 h, hi, Function.comp_apply, AlgEquiv.apply_symm_apply]", "annotated_tactic": ["rw [\u2190 h, hi, <a>Function.comp_apply</a>, <a>AlgEquiv.apply_symm_apply</a>]", [{"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}, {"full_name": "AlgEquiv.apply_symm_apply", "def_path": "Mathlib/Algebra/Algebra/Equiv.lean", "def_pos": [407, 9], "def_end_pos": [407, 25]}]], "state_before": "F : Type u_1\ninst\u271d\u00b9\u2070 : Field F\nE : Type u_2\ninst\u271d\u2079 : Field E\ninst\u271d\u2078 : Algebra F E\nS\u271d : Set E\nL : Type u_3\nL' : Type u_4\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : Field L'\ninst\u271d\u2075 : Algebra F L\ninst\u271d\u2074 : Algebra L E\ninst\u271d\u00b3 : Algebra F L'\ninst\u271d\u00b2 : Algebra L' E\ninst\u271d\u00b9 : IsScalarTower F L E\ninst\u271d : IsScalarTower F L' E\ni : L \u2243\u2090[F] L'\nhi : \u21d1(algebraMap L E) = \u21d1(algebraMap L' E) \u2218 \u21d1i\nS : Set E\nx : E\nx\u271d : x \u2208 Set.range \u21d1(algebraMap L' E)\ny : L'\nh : (algebraMap L' E) y = x\n\u22a2 (algebraMap L E) (i.symm y) = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.iIndep.iIndepSets", "start": [401, 1], "end": [404, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Polynomial/Dickson.lean", "full_name": "Polynomial.dickson_one_one_eq_chebyshev_T", "start": [149, 1], "end": [162, 9], "traced_tactics": [{"tactic": "simp only [Chebyshev.T_zero, mul_one, one_comp, dickson_zero]", "annotated_tactic": ["simp only [<a>Chebyshev.T_zero</a>, <a>mul_one</a>, <a>one_comp</a>, <a>dickson_zero</a>]", [{"full_name": "Polynomial.Chebyshev.T_zero", "def_path": "Mathlib/RingTheory/Polynomial/Chebyshev.lean", "def_pos": [104, 9], "def_end_pos": [104, 15]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "Polynomial.one_comp", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [585, 9], "def_end_pos": [585, 17]}, {"full_name": "Polynomial.dickson_zero", "def_path": "Mathlib/RingTheory/Polynomial/Dickson.lean", "def_pos": [68, 9], "def_end_pos": [68, 21]}]], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\ninst\u271d : Invertible 2\n\u22a2 dickson 1 1 0 = 2 * (Chebyshev.T R \u21910).comp (C \u215f2 * X)", "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\ninst\u271d : Invertible 2\n\u22a2 3 - \u21911 = 2 * (Chebyshev.T R \u21910).comp (C \u215f2 * X)"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\ninst\u271d : Invertible 2\n\u22a2 3 - \u21911 = 2 * (Chebyshev.T R \u21910).comp (C \u215f2 * X)", "state_after": "no goals"}, {"tactic": "rw [dickson_one, Nat.cast_one, Chebyshev.T_one, X_comp, \u2190 mul_assoc, two_mul_C_half_eq_one,\n  one_mul]", "annotated_tactic": ["rw [<a>dickson_one</a>, <a>Nat.cast_one</a>, <a>Chebyshev.T_one</a>, <a>X_comp</a>, \u2190 <a>mul_assoc</a>, <a>two_mul_C_half_eq_one</a>,\n      <a>one_mul</a>]", [{"full_name": "Polynomial.dickson_one", "def_path": "Mathlib/RingTheory/Polynomial/Dickson.lean", "def_pos": [73, 9], "def_end_pos": [73, 20]}, {"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [154, 9], "def_end_pos": [154, 17]}, {"full_name": "Polynomial.Chebyshev.T_one", "def_path": "Mathlib/RingTheory/Polynomial/Chebyshev.lean", "def_pos": [108, 9], "def_end_pos": [108, 14]}, {"full_name": "Polynomial.X_comp", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [548, 9], "def_end_pos": [548, 15]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "_private.Mathlib.RingTheory.Polynomial.Dickson.0.Polynomial.two_mul_C_half_eq_one", "def_path": "Mathlib/RingTheory/Polynomial/Dickson.lean", "def_pos": [143, 17], "def_end_pos": [143, 38]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\ninst\u271d : Invertible 2\n\u22a2 dickson 1 1 1 = 2 * (Chebyshev.T R \u21911).comp (C \u215f2 * X)", "state_after": "no goals"}, {"tactic": "rw [dickson_add_two, C_1, Nat.cast_add, Nat.cast_two, Chebyshev.T_add_two,\n  dickson_one_one_eq_chebyshev_T (n + 1), dickson_one_one_eq_chebyshev_T n, sub_comp, mul_comp,\n  mul_comp, X_comp, ofNat_comp]", "annotated_tactic": ["rw [<a>dickson_add_two</a>, <a>C_1</a>, <a>Nat.cast_add</a>, <a>Nat.cast_two</a>, <a>Chebyshev.T_add_two</a>,\n      dickson_one_one_eq_chebyshev_T (n + 1), dickson_one_one_eq_chebyshev_T n, <a>sub_comp</a>, <a>mul_comp</a>,\n      <a>mul_comp</a>, <a>X_comp</a>, <a>ofNat_comp</a>]", [{"full_name": "Polynomial.dickson_add_two", "def_path": "Mathlib/RingTheory/Polynomial/Dickson.lean", "def_pos": [82, 9], "def_end_pos": [82, 24]}, {"full_name": "Polynomial.C_1", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [521, 9], "def_end_pos": [521, 12]}, {"full_name": "Nat.cast_add", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 17]}, {"full_name": "Nat.cast_two", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [208, 9], "def_end_pos": [208, 17]}, {"full_name": "Polynomial.Chebyshev.T_add_two", "def_path": "Mathlib/RingTheory/Polynomial/Chebyshev.lean", "def_pos": [85, 9], "def_end_pos": [85, 18]}, {"full_name": "Polynomial.sub_comp", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [1339, 9], "def_end_pos": [1339, 17]}, {"full_name": "Polynomial.mul_comp", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [648, 9], "def_end_pos": [648, 17]}, {"full_name": "Polynomial.mul_comp", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [648, 9], "def_end_pos": [648, 17]}, {"full_name": "Polynomial.X_comp", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [548, 9], "def_end_pos": [548, 15]}, {"full_name": "Polynomial.ofNat_comp", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [569, 9], "def_end_pos": [569, 19]}]], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\ninst\u271d : Invertible 2\nn : \u2115\n\u22a2 dickson 1 1 (n + 2) = 2 * (Chebyshev.T R \u2191(n + 2)).comp (C \u215f2 * X)", "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\ninst\u271d : Invertible 2\nn : \u2115\n\u22a2 X * (2 * (Chebyshev.T R \u2191(n + 1)).comp (C \u215f2 * X)) - 1 * (2 * (Chebyshev.T R \u2191n).comp (C \u215f2 * X)) =\n    2 * (\u21912 * (C \u215f2 * X) * (Chebyshev.T R (\u2191n + 1)).comp (C \u215f2 * X) - (Chebyshev.T R \u2191n).comp (C \u215f2 * X))"}, {"tactic": "simp_rw [\u2190 mul_assoc, Nat.cast_ofNat, two_mul_C_half_eq_one, Nat.cast_add, Nat.cast_one]", "annotated_tactic": ["simp_rw [\u2190 <a>mul_assoc</a>, <a>Nat.cast_ofNat</a>, <a>two_mul_C_half_eq_one</a>, <a>Nat.cast_add</a>, <a>Nat.cast_one</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "Nat.cast_ofNat", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [76, 28], "def_end_pos": [76, 42]}, {"full_name": "_private.Mathlib.RingTheory.Polynomial.Dickson.0.Polynomial.two_mul_C_half_eq_one", "def_path": "Mathlib/RingTheory/Polynomial/Dickson.lean", "def_pos": [143, 17], "def_end_pos": [143, 38]}, {"full_name": "Nat.cast_add", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 17]}, {"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [154, 9], "def_end_pos": [154, 17]}]], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\ninst\u271d : Invertible 2\nn : \u2115\n\u22a2 X * (2 * (Chebyshev.T R \u2191(n + 1)).comp (C \u215f2 * X)) - 1 * (2 * (Chebyshev.T R \u2191n).comp (C \u215f2 * X)) =\n    2 * (\u21912 * (C \u215f2 * X) * (Chebyshev.T R (\u2191n + 1)).comp (C \u215f2 * X) - (Chebyshev.T R \u2191n).comp (C \u215f2 * X))", "state_after": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\ninst\u271d : Invertible 2\nn : \u2115\n\u22a2 X * 2 * (Chebyshev.T R (\u2191n + 1)).comp (C \u215f2 * X) - 1 * 2 * (Chebyshev.T R \u2191n).comp (C \u215f2 * X) =\n    2 * (1 * X * (Chebyshev.T R (\u2191n + 1)).comp (C \u215f2 * X) - (Chebyshev.T R \u2191n).comp (C \u215f2 * X))"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\nk : \u2115\na : R\ninst\u271d : Invertible 2\nn : \u2115\n\u22a2 X * 2 * (Chebyshev.T R (\u2191n + 1)).comp (C \u215f2 * X) - 1 * 2 * (Chebyshev.T R \u2191n).comp (C \u215f2 * X) =\n    2 * (1 * X * (Chebyshev.T R (\u2191n + 1)).comp (C \u215f2 * X) - (Chebyshev.T R \u2191n).comp (C \u215f2 * X))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Cycle.lean", "full_name": "Cycle.next_prev", "start": [881, 8], "end": [883, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/BilinearMap.lean", "full_name": "LinearMap.compl\u2081\u2082_apply", "start": [358, 1], "end": [359, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Finset/Basic.lean", "full_name": "Finset.Ioo_subset_Ioo_left", "start": [200, 1], "end": [201, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Ring/Int.lean", "full_name": "Int.add_two_le_iff_lt_of_even_sub", "start": [73, 1], "end": [74, 73], "traced_tactics": [{"tactic": "rw [add_comm]", "annotated_tactic": ["rw [<a>add_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "m n : \u2124\neven : Even (n - m)\n\u22a2 m + 2 \u2264 n \u2194 m < n", "state_after": "m n : \u2124\neven : Even (n - m)\n\u22a2 2 + m \u2264 n \u2194 m < n"}, {"tactic": "exact le_add_iff_lt_of_dvd_sub (by decide) even.two_dvd", "annotated_tactic": ["exact <a>le_add_iff_lt_of_dvd_sub</a> (by decide) even.two_dvd", [{"full_name": "Int.le_add_iff_lt_of_dvd_sub", "def_path": "Mathlib/Data/Int/Defs.lean", "def_pos": [769, 7], "def_end_pos": [769, 31]}]], "state_before": "m n : \u2124\neven : Even (n - m)\n\u22a2 2 + m \u2264 n \u2194 m < n", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "m n : \u2124\neven : Even (n - m)\n\u22a2 0 < 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "full_name": "Commute.pow_self", "start": [207, 1], "end": [208, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Complex/Schwarz.lean", "full_name": "Complex.norm_deriv_le_div_of_mapsTo_ball", "start": [146, 1], "end": [148, 96], "traced_tactics": [{"tactic": "simpa only [dslope_same] using norm_dslope_le_div_of_mapsTo_ball hd h_maps (mem_ball_self h\u2080)", "annotated_tactic": ["simpa only [<a>dslope_same</a>] using <a>norm_dslope_le_div_of_mapsTo_ball</a> hd h_maps (<a>mem_ball_self</a> h\u2080)", [{"full_name": "dslope_same", "def_path": "Mathlib/Analysis/Calculus/Dslope.lean", "def_pos": [36, 9], "def_end_pos": [36, 20]}, {"full_name": "Complex.norm_dslope_le_div_of_mapsTo_ball", "def_path": "Mathlib/Analysis/Complex/Schwarz.lean", "def_pos": [92, 9], "def_end_pos": [92, 42]}, {"full_name": "Metric.mem_ball_self", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [415, 9], "def_end_pos": [415, 22]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nR R\u2081 R\u2082 : \u211d\nf : \u2102 \u2192 E\nc z z\u2080 : \u2102\nhd : DifferentiableOn \u2102 f (ball c R\u2081)\nh_maps : MapsTo f (ball c R\u2081) (ball (f c) R\u2082)\nh\u2080 : 0 < R\u2081\n\u22a2 \u2016deriv f c\u2016 \u2264 R\u2082 / R\u2081", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/Compact.lean", "full_name": "Continuous.tendstoUniformly", "start": [234, 1], "end": [239, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Normed/Group/SemiNormedGrp/Kernels.lean", "full_name": "SemiNormedGrp.explicitCokernelIso_hom_desc", "start": [394, 1], "end": [399, 40], "traced_tactics": [{"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "X Y Z : SemiNormedGrp\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 (explicitCokernelIso f).hom \u226b cokernel.desc f g w = explicitCokernelDesc w", "state_after": "case h\nX Y Z : SemiNormedGrp\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 explicitCokernel\u03c0 f \u226b (explicitCokernelIso f).hom \u226b cokernel.desc f g w = explicitCokernel\u03c0 f \u226b explicitCokernelDesc w"}, {"tactic": "simp [explicitCokernelDesc, explicitCokernel\u03c0, explicitCokernelIso,\n  IsColimit.coconePointUniqueUpToIso]", "annotated_tactic": ["simp [<a>explicitCokernelDesc</a>, <a>explicitCokernel\u03c0</a>, <a>explicitCokernelIso</a>,\n    <a>IsColimit.coconePointUniqueUpToIso</a>]", [{"full_name": "SemiNormedGrp.explicitCokernelDesc", "def_path": "Mathlib/Analysis/Normed/Group/SemiNormedGrp/Kernels.lean", "def_pos": [222, 5], "def_end_pos": [222, 25]}, {"full_name": "SemiNormedGrp.explicitCokernel\u03c0", "def_path": "Mathlib/Analysis/Normed/Group/SemiNormedGrp/Kernels.lean", "def_pos": [230, 5], "def_end_pos": [230, 22]}, {"full_name": "SemiNormedGrp.explicitCokernelIso", "def_path": "Mathlib/Analysis/Normed/Group/SemiNormedGrp/Kernels.lean", "def_pos": [373, 5], "def_end_pos": [373, 24]}, {"full_name": "CategoryTheory.Limits.IsColimit.coconePointUniqueUpToIso", "def_path": "Mathlib/CategoryTheory/Limits/IsLimit.lean", "def_pos": [660, 5], "def_end_pos": [660, 29]}]], "state_before": "case h\nX Y Z : SemiNormedGrp\nf : X \u27f6 Y\ng : Y \u27f6 Z\nw : f \u226b g = 0\n\u22a2 explicitCokernel\u03c0 f \u226b (explicitCokernelIso f).hom \u226b cokernel.desc f g w = explicitCokernel\u03c0 f \u226b explicitCokernelDesc w", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/HausdorffDimension.lean", "full_name": "HolderOnWith.dimH_image_le", "start": [318, 1], "end": [329, 31], "traced_tactics": [{"tactic": "borelize X Y", "annotated_tactic": ["borelize X Y", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nC K r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\n\u22a2 dimH (f '' s) \u2264 dimH s / \u2191r", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nC K r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis\u271d\u00b3 : MeasurableSpace X := borel X\nthis\u271d\u00b2 : BorelSpace X\nthis\u271d\u00b9 : MeasurableSpace Y := borel Y\nthis\u271d : BorelSpace Y\n\u22a2 dimH (f '' s) \u2264 dimH s / \u2191r"}, {"tactic": "refine dimH_le fun d hd => ?_", "annotated_tactic": ["refine <a>dimH_le</a> fun d hd => ?_", [{"full_name": "dimH_le", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDimension.lean", "def_pos": [123, 9], "def_end_pos": [123, 16]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nC K r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis\u271d\u00b3 : MeasurableSpace X := borel X\nthis\u271d\u00b2 : BorelSpace X\nthis\u271d\u00b9 : MeasurableSpace Y := borel Y\nthis\u271d : BorelSpace Y\n\u22a2 dimH (f '' s) \u2264 dimH s / \u2191r", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nC K r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis\u271d\u00b3 : MeasurableSpace X := borel X\nthis\u271d\u00b2 : BorelSpace X\nthis\u271d\u00b9 : MeasurableSpace Y := borel Y\nthis\u271d : BorelSpace Y\nd : \u211d\u22650\nhd : \u03bcH[\u2191d] (f '' s) = \u22a4\n\u22a2 \u2191d \u2264 dimH s / \u2191r"}, {"tactic": "have := h.hausdorffMeasure_image_le hr d.coe_nonneg", "annotated_tactic": ["have := h.hausdorffMeasure_image_le hr d.coe_nonneg", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nC K r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis\u271d\u00b3 : MeasurableSpace X := borel X\nthis\u271d\u00b2 : BorelSpace X\nthis\u271d\u00b9 : MeasurableSpace Y := borel Y\nthis\u271d : BorelSpace Y\nd : \u211d\u22650\nhd : \u03bcH[\u2191d] (f '' s) = \u22a4\n\u22a2 \u2191d \u2264 dimH s / \u2191r", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nC K r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis\u271d\u00b3 : MeasurableSpace X := borel X\nthis\u271d\u00b2 : BorelSpace X\nthis\u271d\u00b9 : MeasurableSpace Y := borel Y\nthis\u271d : BorelSpace Y\nd : \u211d\u22650\nhd : \u03bcH[\u2191d] (f '' s) = \u22a4\nthis : \u03bcH[\u2191d] (f '' s) \u2264 \u2191C ^ \u2191d * \u03bcH[\u2191r * \u2191d] s\n\u22a2 \u2191d \u2264 dimH s / \u2191r"}, {"tactic": "rw [hd, ENNReal.coe_rpow_of_nonneg _ d.coe_nonneg, top_le_iff] at this", "annotated_tactic": ["rw [hd, <a>ENNReal.coe_rpow_of_nonneg</a> _ d.coe_nonneg, <a>top_le_iff</a>] at this", [{"full_name": "ENNReal.coe_rpow_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [459, 9], "def_end_pos": [459, 27]}, {"full_name": "top_le_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [125, 9], "def_end_pos": [125, 19]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nC K r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis\u271d\u00b3 : MeasurableSpace X := borel X\nthis\u271d\u00b2 : BorelSpace X\nthis\u271d\u00b9 : MeasurableSpace Y := borel Y\nthis\u271d : BorelSpace Y\nd : \u211d\u22650\nhd : \u03bcH[\u2191d] (f '' s) = \u22a4\nthis : \u03bcH[\u2191d] (f '' s) \u2264 \u2191C ^ \u2191d * \u03bcH[\u2191r * \u2191d] s\n\u22a2 \u2191d \u2264 dimH s / \u2191r", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nC K r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis\u271d\u00b3 : MeasurableSpace X := borel X\nthis\u271d\u00b2 : BorelSpace X\nthis\u271d\u00b9 : MeasurableSpace Y := borel Y\nthis\u271d : BorelSpace Y\nd : \u211d\u22650\nhd : \u03bcH[\u2191d] (f '' s) = \u22a4\nthis : \u2191(C ^ \u2191d) * \u03bcH[\u2191r * \u2191d] s = \u22a4\n\u22a2 \u2191d \u2264 dimH s / \u2191r"}, {"tactic": "have Hrd : \u03bcH[(r * d : \u211d\u22650)] s = \u22a4 := by\n  contrapose this\n  exact ENNReal.mul_ne_top ENNReal.coe_ne_top this", "annotated_tactic": ["have Hrd : \u03bcH[(r * d : \u211d\u22650)] s = \u22a4 := by\n    contrapose this\n    exact <a>ENNReal.mul_ne_top</a> <a>ENNReal.coe_ne_top</a> this", [{"full_name": "ENNReal.mul_ne_top", "def_path": "Mathlib/Data/ENNReal/Operations.lean", "def_pos": [232, 9], "def_end_pos": [232, 19]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [325, 17], "def_end_pos": [325, 27]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nC K r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis\u271d\u00b3 : MeasurableSpace X := borel X\nthis\u271d\u00b2 : BorelSpace X\nthis\u271d\u00b9 : MeasurableSpace Y := borel Y\nthis\u271d : BorelSpace Y\nd : \u211d\u22650\nhd : \u03bcH[\u2191d] (f '' s) = \u22a4\nthis : \u2191(C ^ \u2191d) * \u03bcH[\u2191r * \u2191d] s = \u22a4\n\u22a2 \u2191d \u2264 dimH s / \u2191r", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nC K r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis\u271d\u00b3 : MeasurableSpace X := borel X\nthis\u271d\u00b2 : BorelSpace X\nthis\u271d\u00b9 : MeasurableSpace Y := borel Y\nthis\u271d : BorelSpace Y\nd : \u211d\u22650\nhd : \u03bcH[\u2191d] (f '' s) = \u22a4\nthis : \u2191(C ^ \u2191d) * \u03bcH[\u2191r * \u2191d] s = \u22a4\nHrd : \u03bcH[\u2191(r * d)] s = \u22a4\n\u22a2 \u2191d \u2264 dimH s / \u2191r"}, {"tactic": "rw [ENNReal.le_div_iff_mul_le, mul_comm, \u2190 ENNReal.coe_mul]", "annotated_tactic": ["rw [<a>ENNReal.le_div_iff_mul_le</a>, <a>mul_comm</a>, \u2190 <a>ENNReal.coe_mul</a>]", [{"full_name": "ENNReal.le_div_iff_mul_le", "def_path": "Mathlib/Data/ENNReal/Inv.lean", "def_pos": [308, 19], "def_end_pos": [308, 36]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "ENNReal.coe_mul", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [413, 26], "def_end_pos": [413, 33]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nC K r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis\u271d\u00b3 : MeasurableSpace X := borel X\nthis\u271d\u00b2 : BorelSpace X\nthis\u271d\u00b9 : MeasurableSpace Y := borel Y\nthis\u271d : BorelSpace Y\nd : \u211d\u22650\nhd : \u03bcH[\u2191d] (f '' s) = \u22a4\nthis : \u2191(C ^ \u2191d) * \u03bcH[\u2191r * \u2191d] s = \u22a4\nHrd : \u03bcH[\u2191(r * d)] s = \u22a4\n\u22a2 \u2191d \u2264 dimH s / \u2191r", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nC K r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis\u271d\u00b3 : MeasurableSpace X := borel X\nthis\u271d\u00b2 : BorelSpace X\nthis\u271d\u00b9 : MeasurableSpace Y := borel Y\nthis\u271d : BorelSpace Y\nd : \u211d\u22650\nhd : \u03bcH[\u2191d] (f '' s) = \u22a4\nthis : \u2191(C ^ \u2191d) * \u03bcH[\u2191r * \u2191d] s = \u22a4\nHrd : \u03bcH[\u2191(r * d)] s = \u22a4\n\u22a2 \u2191(r * d) \u2264 dimH s\n\ncase h0\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nC K r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis\u271d\u00b3 : MeasurableSpace X := borel X\nthis\u271d\u00b2 : BorelSpace X\nthis\u271d\u00b9 : MeasurableSpace Y := borel Y\nthis\u271d : BorelSpace Y\nd : \u211d\u22650\nhd : \u03bcH[\u2191d] (f '' s) = \u22a4\nthis : \u2191(C ^ \u2191d) * \u03bcH[\u2191r * \u2191d] s = \u22a4\nHrd : \u03bcH[\u2191(r * d)] s = \u22a4\n\u22a2 \u2191r \u2260 0 \u2228 dimH s \u2260 0\n\ncase ht\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nC K r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis\u271d\u00b3 : MeasurableSpace X := borel X\nthis\u271d\u00b2 : BorelSpace X\nthis\u271d\u00b9 : MeasurableSpace Y := borel Y\nthis\u271d : BorelSpace Y\nd : \u211d\u22650\nhd : \u03bcH[\u2191d] (f '' s) = \u22a4\nthis : \u2191(C ^ \u2191d) * \u03bcH[\u2191r * \u2191d] s = \u22a4\nHrd : \u03bcH[\u2191(r * d)] s = \u22a4\n\u22a2 \u2191r \u2260 \u22a4 \u2228 dimH s \u2260 \u22a4"}, {"tactic": "exacts [le_dimH_of_hausdorffMeasure_eq_top Hrd, Or.inl (mt ENNReal.coe_eq_zero.1 hr.ne'),\n  Or.inl ENNReal.coe_ne_top]", "annotated_tactic": ["exacts [<a>le_dimH_of_hausdorffMeasure_eq_top</a> Hrd, <a>Or.inl</a> (<a>mt</a> <a>ENNReal.coe_eq_zero</a>.1 hr.ne'),\n    <a>Or.inl</a> <a>ENNReal.coe_ne_top</a>]", [{"full_name": "le_dimH_of_hausdorffMeasure_eq_top", "def_path": "Mathlib/Topology/MetricSpace/HausdorffDimension.lean", "def_pos": [133, 9], "def_end_pos": [133, 43]}, {"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "mt", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [647, 9], "def_end_pos": [647, 11]}, {"full_name": "ENNReal.coe_eq_zero", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [390, 28], "def_end_pos": [390, 39]}, {"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [325, 17], "def_end_pos": [325, 27]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nC K r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis\u271d\u00b3 : MeasurableSpace X := borel X\nthis\u271d\u00b2 : BorelSpace X\nthis\u271d\u00b9 : MeasurableSpace Y := borel Y\nthis\u271d : BorelSpace Y\nd : \u211d\u22650\nhd : \u03bcH[\u2191d] (f '' s) = \u22a4\nthis : \u2191(C ^ \u2191d) * \u03bcH[\u2191r * \u2191d] s = \u22a4\nHrd : \u03bcH[\u2191(r * d)] s = \u22a4\n\u22a2 \u2191(r * d) \u2264 dimH s\n\ncase h0\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nC K r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis\u271d\u00b3 : MeasurableSpace X := borel X\nthis\u271d\u00b2 : BorelSpace X\nthis\u271d\u00b9 : MeasurableSpace Y := borel Y\nthis\u271d : BorelSpace Y\nd : \u211d\u22650\nhd : \u03bcH[\u2191d] (f '' s) = \u22a4\nthis : \u2191(C ^ \u2191d) * \u03bcH[\u2191r * \u2191d] s = \u22a4\nHrd : \u03bcH[\u2191(r * d)] s = \u22a4\n\u22a2 \u2191r \u2260 0 \u2228 dimH s \u2260 0\n\ncase ht\n\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nC K r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis\u271d\u00b3 : MeasurableSpace X := borel X\nthis\u271d\u00b2 : BorelSpace X\nthis\u271d\u00b9 : MeasurableSpace Y := borel Y\nthis\u271d : BorelSpace Y\nd : \u211d\u22650\nhd : \u03bcH[\u2191d] (f '' s) = \u22a4\nthis : \u2191(C ^ \u2191d) * \u03bcH[\u2191r * \u2191d] s = \u22a4\nHrd : \u03bcH[\u2191(r * d)] s = \u22a4\n\u22a2 \u2191r \u2260 \u22a4 \u2228 dimH s \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "contrapose this", "annotated_tactic": ["contrapose this", []], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nC K r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis\u271d\u00b3 : MeasurableSpace X := borel X\nthis\u271d\u00b2 : BorelSpace X\nthis\u271d\u00b9 : MeasurableSpace Y := borel Y\nthis\u271d : BorelSpace Y\nd : \u211d\u22650\nhd : \u03bcH[\u2191d] (f '' s) = \u22a4\nthis : \u2191(C ^ \u2191d) * \u03bcH[\u2191r * \u2191d] s = \u22a4\n\u22a2 \u03bcH[\u2191(r * d)] s = \u22a4", "state_after": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nC K r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis\u271d\u00b3 : MeasurableSpace X := borel X\nthis\u271d\u00b2 : BorelSpace X\nthis\u271d\u00b9 : MeasurableSpace Y := borel Y\nthis\u271d : BorelSpace Y\nd : \u211d\u22650\nhd : \u03bcH[\u2191d] (f '' s) = \u22a4\nthis : \u00ac\u03bcH[\u2191(r * d)] s = \u22a4\n\u22a2 \u00ac\u2191(C ^ \u2191d) * \u03bcH[\u2191r * \u2191d] s = \u22a4"}, {"tactic": "exact ENNReal.mul_ne_top ENNReal.coe_ne_top this", "annotated_tactic": ["exact <a>ENNReal.mul_ne_top</a> <a>ENNReal.coe_ne_top</a> this", [{"full_name": "ENNReal.mul_ne_top", "def_path": "Mathlib/Data/ENNReal/Operations.lean", "def_pos": [232, 9], "def_end_pos": [232, 19]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [325, 17], "def_end_pos": [325, 27]}]], "state_before": "\u03b9 : Type u_1\nX : Type u_2\nY : Type u_3\ninst\u271d\u00b9 : EMetricSpace X\ninst\u271d : EMetricSpace Y\nC K r : \u211d\u22650\nf : X \u2192 Y\ns t : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nthis\u271d\u00b3 : MeasurableSpace X := borel X\nthis\u271d\u00b2 : BorelSpace X\nthis\u271d\u00b9 : MeasurableSpace Y := borel Y\nthis\u271d : BorelSpace Y\nd : \u211d\u22650\nhd : \u03bcH[\u2191d] (f '' s) = \u22a4\nthis : \u00ac\u03bcH[\u2191(r * d)] s = \u22a4\n\u22a2 \u00ac\u2191(C ^ \u2191d) * \u03bcH[\u2191r * \u2191d] s = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "full_name": "Metric.continuousOn_iff", "start": [1067, 1], "end": [1069, 46], "traced_tactics": [{"tactic": "simp [ContinuousOn, continuousWithinAt_iff]", "annotated_tactic": ["simp [<a>ContinuousOn</a>, <a>continuousWithinAt_iff</a>]", [{"full_name": "ContinuousOn", "def_path": "Mathlib/Topology/Defs/Filter.lean", "def_pos": [170, 5], "def_end_pos": [170, 17]}, {"full_name": "Metric.continuousWithinAt_iff", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [1061, 9], "def_end_pos": [1061, 31]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\nx y z : \u03b1\n\u03b4 \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\ns\u271d : Set \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Set \u03b1\n\u22a2 ContinuousOn f s \u2194 \u2200 b \u2208 s, \u2200 \u03b5 > 0, \u2203 \u03b4 > 0, \u2200 a \u2208 s, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Indicator.lean", "full_name": "Set.mulIndicator_empty'", "start": [201, 1], "end": [202, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/Completion.lean", "full_name": "CauchyFilter.denseInducing_pureCauchy", "start": [205, 1], "end": [206, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/SeparatedMap.lean", "full_name": "T2Space.isSeparatedMap", "start": [64, 1], "end": [64, 101], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/WithZeroTopology.lean", "full_name": "WithZeroTopology.nhds_eq_update", "start": [47, 1], "end": [49, 68], "traced_tactics": [{"tactic": "rw [nhds_nhdsAdjoint, sup_of_le_right]", "annotated_tactic": ["rw [<a>nhds_nhdsAdjoint</a>, <a>sup_of_le_right</a>]", [{"full_name": "nhds_nhdsAdjoint", "def_path": "Mathlib/Topology/Order.lean", "def_pos": [634, 9], "def_end_pos": [634, 25]}, {"full_name": "sup_of_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [176, 22], "def_end_pos": [176, 37]}]], "state_before": "\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u22a2 \ud835\udcdd = update pure 0 (\u2a05 \u03b3, \u2a05 (_ : \u03b3 \u2260 0), \ud835\udcdf (Iio \u03b3))", "state_after": "\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u22a2 pure 0 \u2264 \u2a05 \u03b3, \u2a05 (_ : \u03b3 \u2260 0), \ud835\udcdf (Iio \u03b3)"}, {"tactic": "exact le_iInf\u2082 fun \u03b3 h\u03b3 \u21a6 le_principal_iff.2 <| zero_lt_iff.2 h\u03b3", "annotated_tactic": ["exact <a>le_iInf\u2082</a> fun \u03b3 h\u03b3 \u21a6 <a>le_principal_iff</a>.2 <| <a>zero_lt_iff</a>.2 h\u03b3", [{"full_name": "le_iInf\u2082", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [772, 9], "def_end_pos": [772, 17]}, {"full_name": "Filter.le_principal_iff", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [668, 9], "def_end_pos": [668, 25]}, {"full_name": "zero_lt_iff", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Canonical.lean", "def_pos": [92, 9], "def_end_pos": [92, 20]}]], "state_before": "\u03b1 : Type u_1\n\u0393\u2080 : Type u_2\ninst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080\n\u03b3 \u03b3\u2081 \u03b3\u2082 : \u0393\u2080\nl : Filter \u03b1\nf : \u03b1 \u2192 \u0393\u2080\n\u22a2 pure 0 \u2264 \u2a05 \u03b3, \u2a05 (_ : \u03b3 \u2260 0), \ud835\udcdf (Iio \u03b3)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/EulerProduct/Basic.lean", "full_name": "EulerProduct.prod_primesBelow_geometric_eq_tsum_smoothNumbers", "start": [317, 1], "end": [322, 67], "traced_tactics": [{"tactic": "rw [smoothNumbers_eq_factoredNumbers, primesBelow]", "annotated_tactic": ["rw [<a>smoothNumbers_eq_factoredNumbers</a>, <a>primesBelow</a>]", [{"full_name": "Nat.smoothNumbers_eq_factoredNumbers", "def_path": "Mathlib/NumberTheory/SmoothNumbers.lean", "def_pos": [275, 7], "def_end_pos": [275, 39]}, {"full_name": "Nat.primesBelow", "def_path": "Mathlib/NumberTheory/SmoothNumbers.lean", "def_pos": [36, 5], "def_end_pos": [36, 16]}]], "state_before": "F : Type u_1\ninst\u271d\u00b9 : NormedField F\ninst\u271d : CompleteSpace F\nf : \u2115 \u2192* F\nhsum : Summable \u21d1f\nN : \u2115\n\u22a2 \u220f p \u2208 N.primesBelow, (1 - f p)\u207b\u00b9 = \u2211' (m : \u2191N.smoothNumbers), f \u2191m", "state_after": "F : Type u_1\ninst\u271d\u00b9 : NormedField F\ninst\u271d : CompleteSpace F\nf : \u2115 \u2192* F\nhsum : Summable \u21d1f\nN : \u2115\n\u22a2 \u220f p \u2208 filter (fun p => Nat.Prime p) (range N), (1 - f p)\u207b\u00b9 = \u2211' (m : \u2191(factoredNumbers (range N))), f \u2191m"}, {"tactic": "exact prod_filter_prime_geometric_eq_tsum_factoredNumbers hsum _", "annotated_tactic": ["exact <a>prod_filter_prime_geometric_eq_tsum_factoredNumbers</a> hsum _", [{"full_name": "EulerProduct.prod_filter_prime_geometric_eq_tsum_factoredNumbers", "def_path": "Mathlib/NumberTheory/EulerProduct/Basic.lean", "def_pos": [300, 7], "def_end_pos": [300, 58]}]], "state_before": "F : Type u_1\ninst\u271d\u00b9 : NormedField F\ninst\u271d : CompleteSpace F\nf : \u2115 \u2192* F\nhsum : Summable \u21d1f\nN : \u2115\n\u22a2 \u220f p \u2208 filter (fun p => Nat.Prime p) (range N), (1 - f p)\u207b\u00b9 = \u2211' (m : \u2191(factoredNumbers (range N))), f \u2191m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/WithBot.lean", "full_name": "WithBot.map\u2082_bot_right", "start": [177, 1], "end": [177, 88], "traced_tactics": [{"tactic": "cases a <;> rfl", "annotated_tactic": ["cases a <;> rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\na\u271d b : \u03b1\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\na : WithBot \u03b1\n\u22a2 map\u2082 f a \u22a5 = \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Span.lean", "full_name": "Submodule.lt_sup_iff_not_mem", "start": [780, 1], "end": [780, 92], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "R : Type u_1\nR\u2082 : Type u_2\nK : Type u_3\nM : Type u_4\nM\u2082 : Type u_5\nV : Type u_6\nS : Type u_7\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : Module R M\nx : M\np p' : Submodule R M\ninst\u271d\u2074 : Semiring R\u2082\n\u03c3\u2081\u2082 : R \u2192+* R\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b2 : Module R\u2082 M\u2082\nF : Type u_8\ninst\u271d\u00b9 : FunLike F M M\u2082\ninst\u271d : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082\ns t : Set M\nI : Submodule R M\na : M\n\u22a2 I < I \u2294 span R {a} \u2194 a \u2209 I", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "Isometry.hausdorffMeasure_preimage", "start": [877, 1], "end": [879, 69], "traced_tactics": [{"tactic": "rw [\u2190 hf.hausdorffMeasure_image hd, image_preimage_eq_inter_range]", "annotated_tactic": ["rw [\u2190 hf.hausdorffMeasure_image hd, <a>image_preimage_eq_inter_range</a>]", [{"full_name": "Set.image_preimage_eq_inter_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [786, 9], "def_end_pos": [786, 38]}]], "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 Y\n\u22a2 \u03bcH[d] (f \u207b\u00b9' s) = \u03bcH[d] (s \u2229 range f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Semicontinuous.lean", "full_name": "lowerSemicontinuousWithinAt_tsum", "start": [727, 1], "end": [732, 57], "traced_tactics": [{"tactic": "simp_rw [ENNReal.tsum_eq_iSup_sum]", "annotated_tactic": ["simp_rw [<a>ENNReal.tsum_eq_iSup_sum</a>]", [{"full_name": "ENNReal.tsum_eq_iSup_sum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [816, 19], "def_end_pos": [816, 35]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200 (i : \u03b9), LowerSemicontinuousWithinAt (f i) s x\n\u22a2 LowerSemicontinuousWithinAt (fun x' => \u2211' (i : \u03b9), f i x') s x", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200 (i : \u03b9), LowerSemicontinuousWithinAt (f i) s x\n\u22a2 LowerSemicontinuousWithinAt (fun x' => \u2a06 s, \u2211 i \u2208 s, f i x') s x"}, {"tactic": "refine lowerSemicontinuousWithinAt_iSup fun b => ?_", "annotated_tactic": ["refine <a>lowerSemicontinuousWithinAt_iSup</a> fun b => ?_", [{"full_name": "lowerSemicontinuousWithinAt_iSup", "def_path": "Mathlib/Topology/Semicontinuous.lean", "def_pos": [654, 9], "def_end_pos": [654, 41]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200 (i : \u03b9), LowerSemicontinuousWithinAt (f i) s x\n\u22a2 LowerSemicontinuousWithinAt (fun x' => \u2a06 s, \u2211 i \u2208 s, f i x') s x", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200 (i : \u03b9), LowerSemicontinuousWithinAt (f i) s x\nb : Finset \u03b9\n\u22a2 LowerSemicontinuousWithinAt (fun x' => \u2211 i \u2208 b, f i x') s x"}, {"tactic": "exact lowerSemicontinuousWithinAt_sum fun i _hi => h i", "annotated_tactic": ["exact <a>lowerSemicontinuousWithinAt_sum</a> fun i _hi => h i", [{"full_name": "lowerSemicontinuousWithinAt_sum", "def_path": "Mathlib/Topology/Semicontinuous.lean", "def_pos": [604, 9], "def_end_pos": [604, 40]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b9 : Type u_3\nf : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e\nh : \u2200 (i : \u03b9), LowerSemicontinuousWithinAt (f i) s x\nb : Finset \u03b9\n\u22a2 LowerSemicontinuousWithinAt (fun x' => \u2211 i \u2208 b, f i x') s x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.StronglyMeasurable.div", "start": [447, 11], "end": [449, 98], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.count_replicate", "start": [2542, 1], "end": [2544, 34], "traced_tactics": [{"tactic": "convert List.count_replicate a b n", "annotated_tactic": ["convert <a>List.count_replicate</a> a b n", [{"full_name": "List.count_replicate", "def_path": ".lake/packages/batteries/Batteries/Data/List/Count.lean", "def_pos": [188, 9], "def_end_pos": [188, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns : Multiset \u03b1\na b : \u03b1\nn : \u2115\n\u22a2 count a (replicate n b) = if a = b then n else 0", "state_after": "case h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns : Multiset \u03b1\na b : \u03b1\nn : \u2115\n\u22a2 count a (replicate n b) = List.count a (List.replicate n b)"}, {"tactic": "rw [\u2190 coe_count, coe_replicate]", "annotated_tactic": ["rw [\u2190 <a>coe_count</a>, <a>coe_replicate</a>]", [{"full_name": "Multiset.coe_count", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2444, 9], "def_end_pos": [2444, 18]}, {"full_name": "Multiset.coe_replicate", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [933, 9], "def_end_pos": [933, 22]}]], "state_before": "case h.e'_2\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\ninst\u271d : DecidableEq \u03b1\ns : Multiset \u03b1\na b : \u03b1\nn : \u2115\n\u22a2 count a (replicate n b) = List.count a (List.replicate n b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/NNReal/Basic.lean", "full_name": "NNReal.ne_iff", "start": [105, 1], "end": [106, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "full_name": "MeasureTheory.JordanDecomposition.toSignedMeasure_injective", "start": [375, 1], "end": [425, 85], "traced_tactics": [{"tactic": "intro j\u2081 j\u2082 hj", "annotated_tactic": ["intro j\u2081 j\u2082 hj", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\n\u22a2 Injective toSignedMeasure", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\n\u22a2 j\u2081 = j\u2082"}, {"tactic": "obtain \u27e8S, hS\u2081, hS\u2082, hS\u2083, hS\u2084, hS\u2085\u27e9 := j\u2081.exists_compl_positive_negative", "annotated_tactic": ["obtain \u27e8S, hS\u2081, hS\u2082, hS\u2083, hS\u2084, hS\u2085\u27e9 := j\u2081.exists_compl_positive_negative", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\n\u22a2 j\u2081 = j\u2082", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\n\u22a2 j\u2081 = j\u2082"}, {"tactic": "obtain \u27e8T, hT\u2081, hT\u2082, hT\u2083, hT\u2084, hT\u2085\u27e9 := j\u2082.exists_compl_positive_negative", "annotated_tactic": ["obtain \u27e8T, hT\u2081, hT\u2082, hT\u2083, hT\u2084, hT\u2085\u27e9 := j\u2082.exists_compl_positive_negative", []], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\n\u22a2 j\u2081 = j\u2082", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2082.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2082.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\n\u22a2 j\u2081 = j\u2082"}, {"tactic": "rw [\u2190 hj] at hT\u2082 hT\u2083", "annotated_tactic": ["rw [\u2190 hj] at hT\u2082 hT\u2083", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2082.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2082.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\n\u22a2 j\u2081 = j\u2082", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\n\u22a2 j\u2081 = j\u2082"}, {"tactic": "obtain \u27e8hST\u2081, -\u27e9 :=\n  of_symmDiff_compl_positive_negative hS\u2081.compl hT\u2081.compl \u27e8hS\u2083, (compl_compl S).symm \u25b8 hS\u2082\u27e9\n    \u27e8hT\u2083, (compl_compl T).symm \u25b8 hT\u2082\u27e9", "annotated_tactic": ["obtain \u27e8hST\u2081, -\u27e9 :=\n    <a>of_symmDiff_compl_positive_negative</a> hS\u2081.compl hT\u2081.compl \u27e8hS\u2083, (<a>compl_compl</a> S).<a>symm</a> \u25b8 hS\u2082\u27e9\n      \u27e8hT\u2083, (<a>compl_compl</a> T).<a>symm</a> \u25b8 hT\u2082\u27e9", [{"full_name": "MeasureTheory.SignedMeasure.of_symmDiff_compl_positive_negative", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedHahn.lean", "def_pos": [420, 9], "def_end_pos": [420, 44]}, {"full_name": "compl_compl", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [655, 9], "def_end_pos": [655, 20]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}, {"full_name": "compl_compl", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [655, 9], "def_end_pos": [655, 20]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\n\u22a2 j\u2081 = j\u2082", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\n\u22a2 j\u2081 = j\u2082"}, {"tactic": "refine eq_of_posPart_eq_posPart ?_ hj", "annotated_tactic": ["refine <a>eq_of_posPart_eq_posPart</a> ?_ hj", [{"full_name": "_private.Mathlib.MeasureTheory.Decomposition.Jordan.0.MeasureTheory.JordanDecomposition.eq_of_posPart_eq_posPart", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [364, 17], "def_end_pos": [364, 41]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\n\u22a2 j\u2081 = j\u2082", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\n\u22a2 j\u2081.posPart = j\u2082.posPart"}, {"tactic": "ext1 i hi", "annotated_tactic": ["ext1 i hi", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\n\u22a2 j\u2081.posPart = j\u2082.posPart", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 j\u2081.posPart i = j\u2082.posPart i"}, {"tactic": "rw [\u2190 ENNReal.toReal_eq_toReal (measure_ne_top _ _) (measure_ne_top _ _), h\u03bc\u2081, h\u03bc\u2082, \u2190 hj]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.toReal_eq_toReal</a> (<a>measure_ne_top</a> _ _) (<a>measure_ne_top</a> _ _), h\u03bc\u2081, h\u03bc\u2082, \u2190 hj]", [{"full_name": "ENNReal.toReal_eq_toReal", "def_path": "Mathlib/Data/ENNReal/Real.lean", "def_pos": [456, 9], "def_end_pos": [456, 25]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "def_pos": [61, 9], "def_end_pos": [61, 23]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "def_pos": [61, 9], "def_end_pos": [61, 23]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : (j\u2081.posPart i).toReal = \u2191j\u2081.toSignedMeasure (i \u2229 S\u1d9c)\nh\u03bc\u2082 : (j\u2082.posPart i).toReal = \u2191j\u2082.toSignedMeasure (i \u2229 T\u1d9c)\n\u22a2 j\u2081.posPart i = j\u2082.posPart i", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : (j\u2081.posPart i).toReal = \u2191j\u2081.toSignedMeasure (i \u2229 S\u1d9c)\nh\u03bc\u2082 : (j\u2082.posPart i).toReal = \u2191j\u2082.toSignedMeasure (i \u2229 T\u1d9c)\n\u22a2 \u2191j\u2081.toSignedMeasure (i \u2229 S\u1d9c) = \u2191j\u2081.toSignedMeasure (i \u2229 T\u1d9c)"}, {"tactic": "exact of_inter_eq_of_symmDiff_eq_zero_positive hS\u2081.compl hT\u2081.compl hi hS\u2083 hT\u2083 hST\u2081", "annotated_tactic": ["exact <a>of_inter_eq_of_symmDiff_eq_zero_positive</a> hS\u2081.compl hT\u2081.compl hi hS\u2083 hT\u2083 hST\u2081", [{"full_name": "MeasureTheory.SignedMeasure.of_inter_eq_of_symmDiff_eq_zero_positive", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [329, 9], "def_end_pos": [329, 49]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : (j\u2081.posPart i).toReal = \u2191j\u2081.toSignedMeasure (i \u2229 S\u1d9c)\nh\u03bc\u2082 : (j\u2082.posPart i).toReal = \u2191j\u2082.toSignedMeasure (i \u2229 T\u1d9c)\n\u22a2 \u2191j\u2081.toSignedMeasure (i \u2229 S\u1d9c) = \u2191j\u2081.toSignedMeasure (i \u2229 T\u1d9c)", "state_after": "no goals"}, {"tactic": "rw [toSignedMeasure, toSignedMeasure_sub_apply (hi.inter hS\u2081.compl),\n  show j\u2081.negPart (i \u2229 S\u1d9c) = 0 from\n    nonpos_iff_eq_zero.1 (hS\u2085 \u25b8 measure_mono Set.inter_subset_right),\n  ENNReal.zero_toReal, sub_zero]", "annotated_tactic": ["rw [<a>toSignedMeasure</a>, <a>toSignedMeasure_sub_apply</a> (hi.inter hS\u2081.compl),\n      show j\u2081.negPart (i \u2229 S\u1d9c) = 0 from\n        <a>nonpos_iff_eq_zero</a>.1 (hS\u2085 \u25b8 <a>measure_mono</a> <a>Set.inter_subset_right</a>),\n      <a>ENNReal.zero_toReal</a>, <a>sub_zero</a>]", [{"full_name": "MeasureTheory.JordanDecomposition.toSignedMeasure", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [169, 5], "def_end_pos": [169, 20]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_sub_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [497, 9], "def_end_pos": [497, 34]}, {"full_name": "nonpos_iff_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [224, 3], "def_end_pos": [224, 14]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 21]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [934, 9], "def_end_pos": [934, 27]}, {"full_name": "ENNReal.zero_toReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [268, 17], "def_end_pos": [268, 28]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [489, 3], "def_end_pos": [489, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 (j\u2081.posPart i).toReal = \u2191j\u2081.toSignedMeasure (i \u2229 S\u1d9c)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 (j\u2081.posPart i).toReal = (j\u2081.posPart (i \u2229 S\u1d9c)).toReal"}, {"tactic": "conv_lhs => rw [\u2190 Set.inter_union_compl i S]", "annotated_tactic": ["conv_lhs => rw [\u2190 <a>Set.inter_union_compl</a> i S]", [{"full_name": "Set.inter_union_compl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1839, 9], "def_end_pos": [1839, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 (j\u2081.posPart i).toReal = (j\u2081.posPart (i \u2229 S\u1d9c)).toReal", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 (j\u2081.posPart (i \u2229 S \u222a i \u2229 S\u1d9c)).toReal = (j\u2081.posPart (i \u2229 S\u1d9c)).toReal"}, {"tactic": "rw [measure_union,\n  show j\u2081.posPart (i \u2229 S) = 0 from\n    nonpos_iff_eq_zero.1 (hS\u2084 \u25b8 measure_mono Set.inter_subset_right),\n  zero_add]", "annotated_tactic": ["rw [<a>measure_union</a>,\n      show j\u2081.posPart (i \u2229 S) = 0 from\n        <a>nonpos_iff_eq_zero</a>.1 (hS\u2084 \u25b8 <a>measure_mono</a> <a>Set.inter_subset_right</a>),\n      <a>zero_add</a>]", [{"full_name": "MeasureTheory.measure_union", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [112, 9], "def_end_pos": [112, 22]}, {"full_name": "nonpos_iff_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [224, 3], "def_end_pos": [224, 14]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 21]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [934, 9], "def_end_pos": [934, 27]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [476, 3], "def_end_pos": [476, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 (j\u2081.posPart (i \u2229 S \u222a i \u2229 S\u1d9c)).toReal = (j\u2081.posPart (i \u2229 S\u1d9c)).toReal", "state_after": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 Disjoint (i \u2229 S) (i \u2229 S\u1d9c)\n\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 MeasurableSet (i \u2229 S\u1d9c)"}, {"tactic": "refine\n  Set.disjoint_of_subset_left Set.inter_subset_right\n    (Set.disjoint_of_subset_right Set.inter_subset_right disjoint_compl_right)", "annotated_tactic": ["refine\n        <a>Set.disjoint_of_subset_left</a> <a>Set.inter_subset_right</a>\n          (<a>Set.disjoint_of_subset_right</a> <a>Set.inter_subset_right</a> <a>disjoint_compl_right</a>)", [{"full_name": "Set.disjoint_of_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1523, 7], "def_end_pos": [1523, 30]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [934, 9], "def_end_pos": [934, 27]}, {"full_name": "Set.disjoint_of_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1525, 7], "def_end_pos": [1525, 31]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [934, 9], "def_end_pos": [934, 27]}, {"full_name": "disjoint_compl_right", "def_path": "Mathlib/Order/Heyting/Basic.lean", "def_pos": [785, 9], "def_end_pos": [785, 29]}]], "state_before": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 Disjoint (i \u2229 S) (i \u2229 S\u1d9c)", "state_after": "no goals"}, {"tactic": "exact hi.inter hS\u2081.compl", "annotated_tactic": ["exact hi.inter hS\u2081.compl", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\n\u22a2 MeasurableSet (i \u2229 S\u1d9c)", "state_after": "no goals"}, {"tactic": "rw [toSignedMeasure, toSignedMeasure_sub_apply (hi.inter hT\u2081.compl),\n  show j\u2082.negPart (i \u2229 T\u1d9c) = 0 from\n    nonpos_iff_eq_zero.1 (hT\u2085 \u25b8 measure_mono Set.inter_subset_right),\n  ENNReal.zero_toReal, sub_zero]", "annotated_tactic": ["rw [<a>toSignedMeasure</a>, <a>toSignedMeasure_sub_apply</a> (hi.inter hT\u2081.compl),\n      show j\u2082.negPart (i \u2229 T\u1d9c) = 0 from\n        <a>nonpos_iff_eq_zero</a>.1 (hT\u2085 \u25b8 <a>measure_mono</a> <a>Set.inter_subset_right</a>),\n      <a>ENNReal.zero_toReal</a>, <a>sub_zero</a>]", [{"full_name": "MeasureTheory.JordanDecomposition.toSignedMeasure", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [169, 5], "def_end_pos": [169, 20]}, {"full_name": "MeasureTheory.Measure.toSignedMeasure_sub_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [497, 9], "def_end_pos": [497, 34]}, {"full_name": "nonpos_iff_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [224, 3], "def_end_pos": [224, 14]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 21]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [934, 9], "def_end_pos": [934, 27]}, {"full_name": "ENNReal.zero_toReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [268, 17], "def_end_pos": [268, 28]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [489, 3], "def_end_pos": [489, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : (j\u2081.posPart i).toReal = \u2191j\u2081.toSignedMeasure (i \u2229 S\u1d9c)\n\u22a2 (j\u2082.posPart i).toReal = \u2191j\u2082.toSignedMeasure (i \u2229 T\u1d9c)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : (j\u2081.posPart i).toReal = \u2191j\u2081.toSignedMeasure (i \u2229 S\u1d9c)\n\u22a2 (j\u2082.posPart i).toReal = (j\u2082.posPart (i \u2229 T\u1d9c)).toReal"}, {"tactic": "conv_lhs => rw [\u2190 Set.inter_union_compl i T]", "annotated_tactic": ["conv_lhs => rw [\u2190 <a>Set.inter_union_compl</a> i T]", [{"full_name": "Set.inter_union_compl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1839, 9], "def_end_pos": [1839, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : (j\u2081.posPart i).toReal = \u2191j\u2081.toSignedMeasure (i \u2229 S\u1d9c)\n\u22a2 (j\u2082.posPart i).toReal = (j\u2082.posPart (i \u2229 T\u1d9c)).toReal", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : (j\u2081.posPart i).toReal = \u2191j\u2081.toSignedMeasure (i \u2229 S\u1d9c)\n\u22a2 (j\u2082.posPart (i \u2229 T \u222a i \u2229 T\u1d9c)).toReal = (j\u2082.posPart (i \u2229 T\u1d9c)).toReal"}, {"tactic": "rw [measure_union,\n  show j\u2082.posPart (i \u2229 T) = 0 from\n    nonpos_iff_eq_zero.1 (hT\u2084 \u25b8 measure_mono Set.inter_subset_right),\n  zero_add]", "annotated_tactic": ["rw [<a>measure_union</a>,\n      show j\u2082.posPart (i \u2229 T) = 0 from\n        <a>nonpos_iff_eq_zero</a>.1 (hT\u2084 \u25b8 <a>measure_mono</a> <a>Set.inter_subset_right</a>),\n      <a>zero_add</a>]", [{"full_name": "MeasureTheory.measure_union", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [112, 9], "def_end_pos": [112, 22]}, {"full_name": "nonpos_iff_eq_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [224, 3], "def_end_pos": [224, 14]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 21]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [934, 9], "def_end_pos": [934, 27]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [476, 3], "def_end_pos": [476, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : (j\u2081.posPart i).toReal = \u2191j\u2081.toSignedMeasure (i \u2229 S\u1d9c)\n\u22a2 (j\u2082.posPart (i \u2229 T \u222a i \u2229 T\u1d9c)).toReal = (j\u2082.posPart (i \u2229 T\u1d9c)).toReal", "state_after": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : (j\u2081.posPart i).toReal = \u2191j\u2081.toSignedMeasure (i \u2229 S\u1d9c)\n\u22a2 Disjoint (i \u2229 T) (i \u2229 T\u1d9c)\n\ncase h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : (j\u2081.posPart i).toReal = \u2191j\u2081.toSignedMeasure (i \u2229 S\u1d9c)\n\u22a2 MeasurableSet (i \u2229 T\u1d9c)"}, {"tactic": "exact\n  Set.disjoint_of_subset_left Set.inter_subset_right\n    (Set.disjoint_of_subset_right Set.inter_subset_right disjoint_compl_right)", "annotated_tactic": ["exact\n        <a>Set.disjoint_of_subset_left</a> <a>Set.inter_subset_right</a>\n          (<a>Set.disjoint_of_subset_right</a> <a>Set.inter_subset_right</a> <a>disjoint_compl_right</a>)", [{"full_name": "Set.disjoint_of_subset_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1523, 7], "def_end_pos": [1523, 30]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [934, 9], "def_end_pos": [934, 27]}, {"full_name": "Set.disjoint_of_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1525, 7], "def_end_pos": [1525, 31]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [934, 9], "def_end_pos": [934, 27]}, {"full_name": "disjoint_compl_right", "def_path": "Mathlib/Order/Heyting/Basic.lean", "def_pos": [785, 9], "def_end_pos": [785, 29]}]], "state_before": "case hd\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : (j\u2081.posPart i).toReal = \u2191j\u2081.toSignedMeasure (i \u2229 S\u1d9c)\n\u22a2 Disjoint (i \u2229 T) (i \u2229 T\u1d9c)", "state_after": "no goals"}, {"tactic": "exact hi.inter hT\u2081.compl", "annotated_tactic": ["exact hi.inter hT\u2081.compl", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\nj\u2081 j\u2082 : JordanDecomposition \u03b1\nhj : j\u2081.toSignedMeasure = j\u2082.toSignedMeasure\nS : Set \u03b1\nhS\u2081 : MeasurableSet S\nhS\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure S \u2264 VectorMeasure.restrict 0 S\nhS\u2083 : VectorMeasure.restrict 0 S\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure S\u1d9c\nhS\u2084 : j\u2081.posPart S = 0\nhS\u2085 : j\u2081.negPart S\u1d9c = 0\nT : Set \u03b1\nhT\u2081 : MeasurableSet T\nhT\u2082 : VectorMeasure.restrict j\u2081.toSignedMeasure T \u2264 VectorMeasure.restrict 0 T\nhT\u2083 : VectorMeasure.restrict 0 T\u1d9c \u2264 VectorMeasure.restrict j\u2081.toSignedMeasure T\u1d9c\nhT\u2084 : j\u2082.posPart T = 0\nhT\u2085 : j\u2082.negPart T\u1d9c = 0\nhST\u2081 : \u2191j\u2081.toSignedMeasure (symmDiff S\u1d9c T\u1d9c) = 0\ni : Set \u03b1\nhi : MeasurableSet i\nh\u03bc\u2081 : (j\u2081.posPart i).toReal = \u2191j\u2081.toSignedMeasure (i \u2229 S\u1d9c)\n\u22a2 MeasurableSet (i \u2229 T\u1d9c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "full_name": "PartialHomeomorph.tendsto_extend_comp_iff", "start": [976, 1], "end": [983, 71], "traced_tactics": [{"tactic": "refine \u27e8fun h u hu \u21a6 mem_map.2 ?_, (continuousAt_extend _ _ hy).tendsto.comp\u27e9", "annotated_tactic": ["refine \u27e8fun h u hu \u21a6 <a>mem_map</a>.2 ?_, (<a>continuousAt_extend</a> _ _ hy).tendsto.comp\u27e9", [{"full_name": "Filter.mem_map", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1930, 9], "def_end_pos": [1930, 16]}, {"full_name": "PartialHomeomorph.continuousAt_extend", "def_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "def_pos": [863, 9], "def_end_pos": [863, 28]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : TopologicalSpace H\ninst\u271d\u2074 : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners \ud835\udd5c E H\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9 : TopologicalSpace H'\ninst\u271d : TopologicalSpace M'\nI' : ModelWithCorners \ud835\udd5c E' H'\ns t : Set M\n\u03b1 : Type u_8\nl : Filter \u03b1\ng : \u03b1 \u2192 M\nhg : \u2200\u1da0 (z : \u03b1) in l, g z \u2208 f.source\ny : M\nhy : y \u2208 f.source\n\u22a2 Tendsto (\u2191(f.extend I) \u2218 g) l (\ud835\udcdd (\u2191(f.extend I) y)) \u2194 Tendsto g l (\ud835\udcdd y)", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : TopologicalSpace H\ninst\u271d\u2074 : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners \ud835\udd5c E H\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9 : TopologicalSpace H'\ninst\u271d : TopologicalSpace M'\nI' : ModelWithCorners \ud835\udd5c E' H'\ns t : Set M\n\u03b1 : Type u_8\nl : Filter \u03b1\ng : \u03b1 \u2192 M\nhg : \u2200\u1da0 (z : \u03b1) in l, g z \u2208 f.source\ny : M\nhy : y \u2208 f.source\nh : Tendsto (\u2191(f.extend I) \u2218 g) l (\ud835\udcdd (\u2191(f.extend I) y))\nu : Set M\nhu : u \u2208 \ud835\udcdd y\n\u22a2 g \u207b\u00b9' u \u2208 l"}, {"tactic": "have := (f.continuousAt_extend_symm I hy).tendsto.comp h", "annotated_tactic": ["have := (f.continuousAt_extend_symm I hy).tendsto.comp h", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : TopologicalSpace H\ninst\u271d\u2074 : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners \ud835\udd5c E H\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9 : TopologicalSpace H'\ninst\u271d : TopologicalSpace M'\nI' : ModelWithCorners \ud835\udd5c E' H'\ns t : Set M\n\u03b1 : Type u_8\nl : Filter \u03b1\ng : \u03b1 \u2192 M\nhg : \u2200\u1da0 (z : \u03b1) in l, g z \u2208 f.source\ny : M\nhy : y \u2208 f.source\nh : Tendsto (\u2191(f.extend I) \u2218 g) l (\ud835\udcdd (\u2191(f.extend I) y))\nu : Set M\nhu : u \u2208 \ud835\udcdd y\n\u22a2 g \u207b\u00b9' u \u2208 l", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : TopologicalSpace H\ninst\u271d\u2074 : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners \ud835\udd5c E H\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9 : TopologicalSpace H'\ninst\u271d : TopologicalSpace M'\nI' : ModelWithCorners \ud835\udd5c E' H'\ns t : Set M\n\u03b1 : Type u_8\nl : Filter \u03b1\ng : \u03b1 \u2192 M\nhg : \u2200\u1da0 (z : \u03b1) in l, g z \u2208 f.source\ny : M\nhy : y \u2208 f.source\nh : Tendsto (\u2191(f.extend I) \u2218 g) l (\ud835\udcdd (\u2191(f.extend I) y))\nu : Set M\nhu : u \u2208 \ud835\udcdd y\nthis : Tendsto (\u2191(f.extend I).symm \u2218 \u2191(f.extend I) \u2218 g) l (\ud835\udcdd (\u2191(f.extend I).symm (\u2191(f.extend I) y)))\n\u22a2 g \u207b\u00b9' u \u2208 l"}, {"tactic": "rw [extend_left_inv _ _ hy] at this", "annotated_tactic": ["rw [<a>extend_left_inv</a> _ _ hy] at this", [{"full_name": "PartialHomeomorph.extend_left_inv", "def_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "def_pos": [841, 9], "def_end_pos": [841, 24]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : TopologicalSpace H\ninst\u271d\u2074 : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners \ud835\udd5c E H\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9 : TopologicalSpace H'\ninst\u271d : TopologicalSpace M'\nI' : ModelWithCorners \ud835\udd5c E' H'\ns t : Set M\n\u03b1 : Type u_8\nl : Filter \u03b1\ng : \u03b1 \u2192 M\nhg : \u2200\u1da0 (z : \u03b1) in l, g z \u2208 f.source\ny : M\nhy : y \u2208 f.source\nh : Tendsto (\u2191(f.extend I) \u2218 g) l (\ud835\udcdd (\u2191(f.extend I) y))\nu : Set M\nhu : u \u2208 \ud835\udcdd y\nthis : Tendsto (\u2191(f.extend I).symm \u2218 \u2191(f.extend I) \u2218 g) l (\ud835\udcdd (\u2191(f.extend I).symm (\u2191(f.extend I) y)))\n\u22a2 g \u207b\u00b9' u \u2208 l", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : TopologicalSpace H\ninst\u271d\u2074 : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners \ud835\udd5c E H\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9 : TopologicalSpace H'\ninst\u271d : TopologicalSpace M'\nI' : ModelWithCorners \ud835\udd5c E' H'\ns t : Set M\n\u03b1 : Type u_8\nl : Filter \u03b1\ng : \u03b1 \u2192 M\nhg : \u2200\u1da0 (z : \u03b1) in l, g z \u2208 f.source\ny : M\nhy : y \u2208 f.source\nh : Tendsto (\u2191(f.extend I) \u2218 g) l (\ud835\udcdd (\u2191(f.extend I) y))\nu : Set M\nhu : u \u2208 \ud835\udcdd y\nthis : Tendsto (\u2191(f.extend I).symm \u2218 \u2191(f.extend I) \u2218 g) l (\ud835\udcdd y)\n\u22a2 g \u207b\u00b9' u \u2208 l"}, {"tactic": "filter_upwards [hg, mem_map.1 (this hu)] with z hz hzu", "annotated_tactic": ["filter_upwards [hg, <a>mem_map</a>.1 (this hu)] with z hz hzu", [{"full_name": "Filter.mem_map", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1930, 9], "def_end_pos": [1930, 16]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : TopologicalSpace H\ninst\u271d\u2074 : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners \ud835\udd5c E H\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9 : TopologicalSpace H'\ninst\u271d : TopologicalSpace M'\nI' : ModelWithCorners \ud835\udd5c E' H'\ns t : Set M\n\u03b1 : Type u_8\nl : Filter \u03b1\ng : \u03b1 \u2192 M\nhg : \u2200\u1da0 (z : \u03b1) in l, g z \u2208 f.source\ny : M\nhy : y \u2208 f.source\nh : Tendsto (\u2191(f.extend I) \u2218 g) l (\ud835\udcdd (\u2191(f.extend I) y))\nu : Set M\nhu : u \u2208 \ud835\udcdd y\nthis : Tendsto (\u2191(f.extend I).symm \u2218 \u2191(f.extend I) \u2218 g) l (\ud835\udcdd y)\n\u22a2 g \u207b\u00b9' u \u2208 l", "state_after": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : TopologicalSpace H\ninst\u271d\u2074 : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners \ud835\udd5c E H\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9 : TopologicalSpace H'\ninst\u271d : TopologicalSpace M'\nI' : ModelWithCorners \ud835\udd5c E' H'\ns t : Set M\n\u03b1 : Type u_8\nl : Filter \u03b1\ng : \u03b1 \u2192 M\nhg : \u2200\u1da0 (z : \u03b1) in l, g z \u2208 f.source\ny : M\nhy : y \u2208 f.source\nh : Tendsto (\u2191(f.extend I) \u2218 g) l (\ud835\udcdd (\u2191(f.extend I) y))\nu : Set M\nhu : u \u2208 \ud835\udcdd y\nthis : Tendsto (\u2191(f.extend I).symm \u2218 \u2191(f.extend I) \u2218 g) l (\ud835\udcdd y)\nz : \u03b1\nhz : g z \u2208 f.source\nhzu : z \u2208 \u2191(f.extend I).symm \u2218 \u2191(f.extend I) \u2218 g \u207b\u00b9' u\n\u22a2 z \u2208 g \u207b\u00b9' u"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/AddChar.lean", "full_name": "AddChar.coe_toAddMonoidHomEquiv_symm", "start": [177, 1], "end": [178, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/List/Perm.lean", "full_name": "List.Subperm.refl", "start": [229, 9], "end": [229, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/MvPolynomial/Degrees.lean", "full_name": "MvPolynomial.totalDegree_le_degrees_card", "start": [369, 1], "end": [373, 76], "traced_tactics": [{"tactic": "classical\nrw [totalDegree_eq]\nexact Finset.sup_le fun s hs => Multiset.card_le_card <| Finset.le_sup hs", "annotated_tactic": ["classical\n  rw [<a>totalDegree_eq</a>]\n  exact <a>Finset.sup_le</a> fun s hs => <a>Multiset.card_le_card</a> <| <a>Finset.le_sup</a> hs", [{"full_name": "MvPolynomial.totalDegree_eq", "def_path": "Mathlib/Algebra/MvPolynomial/Degrees.lean", "def_pos": [358, 9], "def_end_pos": [358, 23]}, {"full_name": "Finset.sup_le", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [112, 21], "def_end_pos": [112, 27]}, {"full_name": "Multiset.card_le_card", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [806, 9], "def_end_pos": [806, 21]}, {"full_name": "Finset.le_sup", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [119, 9], "def_end_pos": [119, 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\u271d q p : MvPolynomial \u03c3 R\n\u22a2 p.totalDegree \u2264 Multiset.card p.degrees", "state_after": "no goals"}, {"tactic": "rw [totalDegree_eq]", "annotated_tactic": ["rw [<a>totalDegree_eq</a>]", [{"full_name": "MvPolynomial.totalDegree_eq", "def_path": "Mathlib/Algebra/MvPolynomial/Degrees.lean", "def_pos": [358, 9], "def_end_pos": [358, 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 : CommSemiring R\np\u271d q p : MvPolynomial \u03c3 R\n\u22a2 p.totalDegree \u2264 Multiset.card p.degrees", "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\n\u22a2 (p.support.sup fun m => Multiset.card (toMultiset m)) \u2264 Multiset.card p.degrees"}, {"tactic": "exact Finset.sup_le fun s hs => Multiset.card_le_card <| Finset.le_sup hs", "annotated_tactic": ["exact <a>Finset.sup_le</a> fun s hs => <a>Multiset.card_le_card</a> <| <a>Finset.le_sup</a> hs", [{"full_name": "Finset.sup_le", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [112, 21], "def_end_pos": [112, 27]}, {"full_name": "Multiset.card_le_card", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [806, 9], "def_end_pos": [806, 21]}, {"full_name": "Finset.le_sup", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [119, 9], "def_end_pos": [119, 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\u271d q p : MvPolynomial \u03c3 R\n\u22a2 (p.support.sup fun m => Multiset.card (toMultiset m)) \u2264 Multiset.card p.degrees", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Operations.lean", "full_name": "SimpleGraph.adj_replaceVertex_iff_of_ne", "start": [70, 1], "end": [72, 81], "traced_tactics": [{"tactic": "simp [replaceVertex, hv, hw]", "annotated_tactic": ["simp [<a>replaceVertex</a>, hv, hw]", [{"full_name": "SimpleGraph.replaceVertex", "def_path": "Mathlib/Combinatorics/SimpleGraph/Operations.lean", "def_pos": [47, 5], "def_end_pos": [47, 18]}]], "state_before": "V : Type u_1\ninst\u271d : DecidableEq V\nG : SimpleGraph V\ns t v w : V\nhv : v \u2260 t\nhw : w \u2260 t\n\u22a2 (G.replaceVertex s t).Adj v w \u2194 G.Adj v w", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/NumberField/Embeddings.lean", "full_name": "NumberField.ComplexEmbedding.place_conjugate", "start": [168, 1], "end": [169, 72], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "K : Type u_1\ninst\u271d\u00b9 : Field K\nk : Type u_2\ninst\u271d : Field k\n\u03c6 : K \u2192+* \u2102\n\u22a2 place (conjugate \u03c6) = place \u03c6", "state_after": "case a\nK : Type u_1\ninst\u271d\u00b9 : Field K\nk : Type u_2\ninst\u271d : Field k\n\u03c6 : K \u2192+* \u2102\nx\u271d : K\n\u22a2 (place (conjugate \u03c6)) x\u271d = (place \u03c6) x\u271d"}, {"tactic": "simp only [place_apply, norm_eq_abs, abs_conj, conjugate_coe_eq]", "annotated_tactic": ["simp only [<a>place_apply</a>, <a>norm_eq_abs</a>, <a>abs_conj</a>, <a>conjugate_coe_eq</a>]", [{"full_name": "NumberField.place_apply", "def_path": "Mathlib/NumberTheory/NumberField/Embeddings.lean", "def_pos": [147, 9], "def_end_pos": [147, 32]}, {"full_name": "Complex.norm_eq_abs", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [55, 9], "def_end_pos": [55, 20]}, {"full_name": "Complex.abs_conj", "def_path": "Mathlib/Data/Complex/Abs.lean", "def_pos": [135, 9], "def_end_pos": [135, 17]}, {"full_name": "NumberField.ComplexEmbedding.conjugate_coe_eq", "def_path": "Mathlib/NumberTheory/NumberField/Embeddings.lean", "def_pos": [165, 9], "def_end_pos": [165, 25]}]], "state_before": "case a\nK : Type u_1\ninst\u271d\u00b9 : Field K\nk : Type u_2\ninst\u271d : Field k\n\u03c6 : K \u2192+* \u2102\nx\u271d : K\n\u22a2 (place (conjugate \u03c6)) x\u271d = (place \u03c6) x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "full_name": "Real.one_le_rpow_of_pos_of_le_one_of_nonpos", "start": [733, 1], "end": [736, 27], "traced_tactics": [{"tactic": "convert rpow_le_rpow_of_exponent_ge hx1 hx2 hz", "annotated_tactic": ["convert <a>rpow_le_rpow_of_exponent_ge</a> hx1 hx2 hz", [{"full_name": "Real.rpow_le_rpow_of_exponent_ge", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [679, 9], "def_end_pos": [679, 36]}]], "state_before": "x y z : \u211d\nn : \u2115\nhx1 : 0 < x\nhx2 : x \u2264 1\nhz : z \u2264 0\n\u22a2 1 \u2264 x ^ z", "state_after": "case h.e'_3\nx y z : \u211d\nn : \u2115\nhx1 : 0 < x\nhx2 : x \u2264 1\nhz : z \u2264 0\n\u22a2 1 = x ^ 0"}, {"tactic": "exact (rpow_zero x).symm", "annotated_tactic": ["exact (<a>rpow_zero</a> x).<a>symm</a>", [{"full_name": "Real.rpow_zero", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [125, 9], "def_end_pos": [125, 18]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case h.e'_3\nx y z : \u211d\nn : \u2115\nhx1 : 0 < x\nhx2 : x \u2264 1\nhz : z \u2264 0\n\u22a2 1 = x ^ 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/RCLike/Basic.lean", "full_name": "RCLike.inv_im", "start": [547, 1], "end": [548, 81], "traced_tactics": [{"tactic": "rw [inv_def, normSq_eq_def', mul_comm, im_ofReal_mul, conj_im, div_eq_inv_mul]", "annotated_tactic": ["rw [<a>inv_def</a>, <a>normSq_eq_def'</a>, <a>mul_comm</a>, <a>im_ofReal_mul</a>, <a>conj_im</a>, <a>div_eq_inv_mul</a>]", [{"full_name": "RCLike.inv_def", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [533, 9], "def_end_pos": [533, 16]}, {"full_name": "RCLike.normSq_eq_def'", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [452, 9], "def_end_pos": [452, 23]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "RCLike.im_ofReal_mul", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [258, 9], "def_end_pos": [258, 22]}, {"full_name": "RCLike.conj_im", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [327, 9], "def_end_pos": [327, 16]}, {"full_name": "div_eq_inv_mul", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 23]}]], "state_before": "K : Type u_1\nE : Type u_2\ninst\u271d : RCLike K\nz\u271d z : K\n\u22a2 im z\u207b\u00b9 = -im z / normSq z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Independence/Kernel.lean", "full_name": "ProbabilityTheory.kernel.iIndepFun.meas_biInter", "start": [146, 1], "end": [148, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.MapsTo.piecewise_ite", "start": [1655, 1], "end": [1661, 65], "traced_tactics": [{"tactic": "refine (h\u2081.congr ?_).union_union (h\u2082.congr ?_)", "annotated_tactic": ["refine (h\u2081.congr ?_).<a>union_union</a> (h\u2082.congr ?_)", [{"full_name": "Set.MapsTo.union_union", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [481, 9], "def_end_pos": [481, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03c0 : \u03b1 \u2192 Type u_5\n\u03b4 : \u03b1 \u2192 Sort u_6\ns\u271d : Set \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s\u271d)\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nf\u2081 f\u2082 : \u03b1 \u2192 \u03b2\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 s)\nh\u2081 : MapsTo f\u2081 (s\u2081 \u2229 s) (t\u2081 \u2229 t)\nh\u2082 : MapsTo f\u2082 (s\u2082 \u2229 s\u1d9c) (t\u2082 \u2229 t\u1d9c)\n\u22a2 MapsTo (s.piecewise f\u2081 f\u2082) (s.ite s\u2081 s\u2082) (t.ite t\u2081 t\u2082)", "state_after": "case refine_1\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\n\u03b4 : \u03b1 \u2192 Sort u_6\ns\u271d : Set \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s\u271d)\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nf\u2081 f\u2082 : \u03b1 \u2192 \u03b2\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 s)\nh\u2081 : MapsTo f\u2081 (s\u2081 \u2229 s) (t\u2081 \u2229 t)\nh\u2082 : MapsTo f\u2082 (s\u2082 \u2229 s\u1d9c) (t\u2082 \u2229 t\u1d9c)\n\u22a2 EqOn f\u2081 (s.piecewise f\u2081 f\u2082) (s\u2081 \u2229 s)\n\ncase refine_2\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\n\u03b4 : \u03b1 \u2192 Sort u_6\ns\u271d : Set \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s\u271d)\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nf\u2081 f\u2082 : \u03b1 \u2192 \u03b2\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 s)\nh\u2081 : MapsTo f\u2081 (s\u2081 \u2229 s) (t\u2081 \u2229 t)\nh\u2082 : MapsTo f\u2082 (s\u2082 \u2229 s\u1d9c) (t\u2082 \u2229 t\u1d9c)\n\u22a2 EqOn f\u2082 (s.piecewise f\u2081 f\u2082) (s\u2082 \u2229 s\u1d9c)"}, {"tactic": "exacts [(piecewise_eqOn s f\u2081 f\u2082).symm.mono inter_subset_right,\n  (piecewise_eqOn_compl s f\u2081 f\u2082).symm.mono inter_subset_right]", "annotated_tactic": ["exacts [(<a>piecewise_eqOn</a> s f\u2081 f\u2082).symm.mono <a>inter_subset_right</a>,\n    (<a>piecewise_eqOn_compl</a> s f\u2081 f\u2082).symm.mono <a>inter_subset_right</a>]", [{"full_name": "Set.piecewise_eqOn", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1614, 9], "def_end_pos": [1614, 23]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [934, 9], "def_end_pos": [934, 27]}, {"full_name": "Set.piecewise_eqOn_compl", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [1618, 9], "def_end_pos": [1618, 29]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [934, 9], "def_end_pos": [934, 27]}]], "state_before": "case refine_1\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\n\u03b4 : \u03b1 \u2192 Sort u_6\ns\u271d : Set \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s\u271d)\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nf\u2081 f\u2082 : \u03b1 \u2192 \u03b2\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 s)\nh\u2081 : MapsTo f\u2081 (s\u2081 \u2229 s) (t\u2081 \u2229 t)\nh\u2082 : MapsTo f\u2082 (s\u2082 \u2229 s\u1d9c) (t\u2082 \u2229 t\u1d9c)\n\u22a2 EqOn f\u2081 (s.piecewise f\u2081 f\u2082) (s\u2081 \u2229 s)\n\ncase refine_2\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\n\u03b4 : \u03b1 \u2192 Sort u_6\ns\u271d : Set \u03b1\nf g : (i : \u03b1) \u2192 \u03b4 i\ninst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s\u271d)\ns s\u2081 s\u2082 : Set \u03b1\nt t\u2081 t\u2082 : Set \u03b2\nf\u2081 f\u2082 : \u03b1 \u2192 \u03b2\ninst\u271d : (i : \u03b1) \u2192 Decidable (i \u2208 s)\nh\u2081 : MapsTo f\u2081 (s\u2081 \u2229 s) (t\u2081 \u2229 t)\nh\u2082 : MapsTo f\u2082 (s\u2082 \u2229 s\u1d9c) (t\u2082 \u2229 t\u1d9c)\n\u22a2 EqOn f\u2082 (s.piecewise f\u2081 f\u2082) (s\u2082 \u2229 s\u1d9c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "full_name": "MeasureTheory.Integrable.integrableAtFilter", "start": [422, 1], "end": [424, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GroupWithZero/WithZero.lean", "full_name": "WithZero.coe_one", "start": [36, 1], "end": [36, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Normed/Order/UpperLower.lean", "full_name": "IsLowerSet.mem_interior_of_forall_lt", "start": [111, 1], "end": [127, 51], "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": [443, 9], "def_end_pos": [443, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\n\u22a2 y \u2208 interior s", "state_after": "case intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u22a2 y \u2208 interior s"}, {"tactic": "obtain \u27e8\u03b5, h\u03b5, hxy\u27e9 := Pi.exists_forall_pos_add_lt h", "annotated_tactic": ["obtain \u27e8\u03b5, h\u03b5, hxy\u27e9 := <a>Pi.exists_forall_pos_add_lt</a> h", [{"full_name": "Pi.exists_forall_pos_add_lt", "def_path": "Mathlib/Algebra/Order/Field/Pi.lean", "def_pos": [21, 9], "def_end_pos": [21, 36]}]], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u22a2 y \u2208 interior s", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\n\u22a2 y \u2208 interior s"}, {"tactic": "obtain \u27e8z, hz, hxz\u27e9 := Metric.mem_closure_iff.1 hx _ h\u03b5", "annotated_tactic": ["obtain \u27e8z, hz, hxz\u27e9 := <a>Metric.mem_closure_iff</a>.1 hx _ h\u03b5", [{"full_name": "Metric.mem_closure_iff", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [1451, 9], "def_end_pos": [1451, 24]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\n\u22a2 y \u2208 interior s", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 s\nhxz : dist x z < \u03b5\n\u22a2 y \u2208 interior s"}, {"tactic": "rw [dist_pi_lt_iff h\u03b5] at hxz", "annotated_tactic": ["rw [<a>dist_pi_lt_iff</a> h\u03b5] at hxz", [{"full_name": "dist_pi_lt_iff", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean", "def_pos": [328, 7], "def_end_pos": [328, 21]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 s\nhxz : dist x z < \u03b5\n\u22a2 y \u2208 interior s", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 s\nhxz : \u2200 (b : \u03b9), dist (x b) (z b) < \u03b5\n\u22a2 y \u2208 interior s"}, {"tactic": "have hyz : \u2200 i, y i < z i := by\n  refine fun i =>\n    (lt_sub_iff_add_lt.2 <| hxy _).trans_le (sub_le_comm.1 <| (le_abs_self _).trans ?_)\n  rw [\u2190 Real.norm_eq_abs, \u2190 dist_eq_norm]\n  exact (hxz _).le", "annotated_tactic": ["have hyz : \u2200 i, y i < z i := by\n    refine fun i =>\n      (<a>lt_sub_iff_add_lt</a>.2 <| hxy _).<a>trans_le</a> (<a>sub_le_comm</a>.1 <| (<a>le_abs_self</a> _).<a>trans</a> ?_)\n    rw [\u2190 <a>Real.norm_eq_abs</a>, \u2190 <a>dist_eq_norm</a>]\n    exact (hxz _).<a>le</a>", [{"full_name": "lt_sub_iff_add_lt", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [909, 3], "def_end_pos": [909, 14]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [143, 7], "def_end_pos": [143, 21]}, {"full_name": "sub_le_comm", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [835, 3], "def_end_pos": [835, 14]}, {"full_name": "le_abs_self", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [73, 3], "def_end_pos": [73, 14]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}, {"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1434, 9], "def_end_pos": [1434, 20]}, {"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [401, 7], "def_end_pos": [401, 19]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 s\nhxz : \u2200 (b : \u03b9), dist (x b) (z b) < \u03b5\n\u22a2 y \u2208 interior s", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 s\nhxz : \u2200 (b : \u03b9), dist (x b) (z b) < \u03b5\nhyz : \u2200 (i : \u03b9), y i < z i\n\u22a2 y \u2208 interior s"}, {"tactic": "obtain \u27e8\u03b4, h\u03b4, hyz\u27e9 := Pi.exists_forall_pos_add_lt hyz", "annotated_tactic": ["obtain \u27e8\u03b4, h\u03b4, hyz\u27e9 := <a>Pi.exists_forall_pos_add_lt</a> hyz", [{"full_name": "Pi.exists_forall_pos_add_lt", "def_path": "Mathlib/Algebra/Order/Field/Pi.lean", "def_pos": [21, 9], "def_end_pos": [21, 36]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 s\nhxz : \u2200 (b : \u03b9), dist (x b) (z b) < \u03b5\nhyz : \u2200 (i : \u03b9), y i < z i\n\u22a2 y \u2208 interior s", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 s\nhxz : \u2200 (b : \u03b9), dist (x b) (z b) < \u03b5\nhyz\u271d : \u2200 (i : \u03b9), y i < z i\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhyz : \u2200 (i : \u03b9), y i + \u03b4 < z i\n\u22a2 y \u2208 interior s"}, {"tactic": "refine mem_interior.2 \u27e8ball y \u03b4, ?_, isOpen_ball, mem_ball_self h\u03b4\u27e9", "annotated_tactic": ["refine <a>mem_interior</a>.2 \u27e8<a>ball</a> y \u03b4, ?_, <a>isOpen_ball</a>, <a>mem_ball_self</a> h\u03b4\u27e9", [{"full_name": "mem_interior", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [248, 9], "def_end_pos": [248, 21]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [399, 5], "def_end_pos": [399, 9]}, {"full_name": "Metric.isOpen_ball", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [1012, 9], "def_end_pos": [1012, 20]}, {"full_name": "Metric.mem_ball_self", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [415, 9], "def_end_pos": [415, 22]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 s\nhxz : \u2200 (b : \u03b9), dist (x b) (z b) < \u03b5\nhyz\u271d : \u2200 (i : \u03b9), y i < z i\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhyz : \u2200 (i : \u03b9), y i + \u03b4 < z i\n\u22a2 y \u2208 interior s", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 s\nhxz : \u2200 (b : \u03b9), dist (x b) (z b) < \u03b5\nhyz\u271d : \u2200 (i : \u03b9), y i < z i\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhyz : \u2200 (i : \u03b9), y i + \u03b4 < z i\n\u22a2 ball y \u03b4 \u2286 s"}, {"tactic": "rintro w hw", "annotated_tactic": ["rintro w hw", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 s\nhxz : \u2200 (b : \u03b9), dist (x b) (z b) < \u03b5\nhyz\u271d : \u2200 (i : \u03b9), y i < z i\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhyz : \u2200 (i : \u03b9), y i + \u03b4 < z i\n\u22a2 ball y \u03b4 \u2286 s", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 s\nhxz : \u2200 (b : \u03b9), dist (x b) (z b) < \u03b5\nhyz\u271d : \u2200 (i : \u03b9), y i < z i\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhyz : \u2200 (i : \u03b9), y i + \u03b4 < z i\nw : \u03b9 \u2192 \u211d\nhw : w \u2208 ball y \u03b4\n\u22a2 w \u2208 s"}, {"tactic": "refine hs (fun i => ?_) hz", "annotated_tactic": ["refine hs (fun i => ?_) hz", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 s\nhxz : \u2200 (b : \u03b9), dist (x b) (z b) < \u03b5\nhyz\u271d : \u2200 (i : \u03b9), y i < z i\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhyz : \u2200 (i : \u03b9), y i + \u03b4 < z i\nw : \u03b9 \u2192 \u211d\nhw : w \u2208 ball y \u03b4\n\u22a2 w \u2208 s", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 s\nhxz : \u2200 (b : \u03b9), dist (x b) (z b) < \u03b5\nhyz\u271d : \u2200 (i : \u03b9), y i < z i\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhyz : \u2200 (i : \u03b9), y i + \u03b4 < z i\nw : \u03b9 \u2192 \u211d\nhw : w \u2208 ball y \u03b4\ni : \u03b9\n\u22a2 w i \u2264 z i"}, {"tactic": "simp_rw [ball_pi _ h\u03b4, Real.ball_eq_Ioo] at hw", "annotated_tactic": ["simp_rw [<a>ball_pi</a> _ h\u03b4, <a>Real.ball_eq_Ioo</a>] at hw", [{"full_name": "ball_pi", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean", "def_pos": [383, 7], "def_end_pos": [383, 14]}, {"full_name": "Real.ball_eq_Ioo", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [1364, 9], "def_end_pos": [1364, 25]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 s\nhxz : \u2200 (b : \u03b9), dist (x b) (z b) < \u03b5\nhyz\u271d : \u2200 (i : \u03b9), y i < z i\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhyz : \u2200 (i : \u03b9), y i + \u03b4 < z i\nw : \u03b9 \u2192 \u211d\nhw : w \u2208 ball y \u03b4\ni : \u03b9\n\u22a2 w i \u2264 z i", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 s\nhxz : \u2200 (b : \u03b9), dist (x b) (z b) < \u03b5\nhyz\u271d : \u2200 (i : \u03b9), y i < z i\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhyz : \u2200 (i : \u03b9), y i + \u03b4 < z i\nw : \u03b9 \u2192 \u211d\ni : \u03b9\nhw : w \u2208 univ.pi fun b => Ioo (y b - \u03b4) (y b + \u03b4)\n\u22a2 w i \u2264 z i"}, {"tactic": "exact ((hw _ <| mem_univ _).2.trans <| hyz _).le", "annotated_tactic": ["exact ((hw _ <| <a>mem_univ</a> _).2.<a>trans</a> <| hyz _).<a>le</a>", [{"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [82, 29], "def_end_pos": [82, 37]}, {"full_name": "LT.lt.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [139, 7], "def_end_pos": [139, 18]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}]], "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 s\nhxz : \u2200 (b : \u03b9), dist (x b) (z b) < \u03b5\nhyz\u271d : \u2200 (i : \u03b9), y i < z i\n\u03b4 : \u211d\nh\u03b4 : 0 < \u03b4\nhyz : \u2200 (i : \u03b9), y i + \u03b4 < z i\nw : \u03b9 \u2192 \u211d\ni : \u03b9\nhw : w \u2208 univ.pi fun b => Ioo (y b - \u03b4) (y b + \u03b4)\n\u22a2 w i \u2264 z i", "state_after": "no goals"}, {"tactic": "refine fun i =>\n  (lt_sub_iff_add_lt.2 <| hxy _).trans_le (sub_le_comm.1 <| (le_abs_self _).trans ?_)", "annotated_tactic": ["refine fun i =>\n      (<a>lt_sub_iff_add_lt</a>.2 <| hxy _).<a>trans_le</a> (<a>sub_le_comm</a>.1 <| (<a>le_abs_self</a> _).<a>trans</a> ?_)", [{"full_name": "lt_sub_iff_add_lt", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [909, 3], "def_end_pos": [909, 14]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [143, 7], "def_end_pos": [143, 21]}, {"full_name": "sub_le_comm", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [835, 3], "def_end_pos": [835, 14]}, {"full_name": "le_abs_self", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [73, 3], "def_end_pos": [73, 14]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 s\nhxz : \u2200 (b : \u03b9), dist (x b) (z b) < \u03b5\n\u22a2 \u2200 (i : \u03b9), y i < z i", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 s\nhxz : \u2200 (b : \u03b9), dist (x b) (z b) < \u03b5\ni : \u03b9\n\u22a2 |x i - z i| \u2264 \u03b5"}, {"tactic": "rw [\u2190 Real.norm_eq_abs, \u2190 dist_eq_norm]", "annotated_tactic": ["rw [\u2190 <a>Real.norm_eq_abs</a>, \u2190 <a>dist_eq_norm</a>]", [{"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1434, 9], "def_end_pos": [1434, 20]}, {"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [401, 7], "def_end_pos": [401, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 s\nhxz : \u2200 (b : \u03b9), dist (x b) (z b) < \u03b5\ni : \u03b9\n\u22a2 |x i - z i| \u2264 \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 s\nhxz : \u2200 (b : \u03b9), dist (x b) (z b) < \u03b5\ni : \u03b9\n\u22a2 dist (x i) (z i) \u2264 \u03b5"}, {"tactic": "exact (hxz _).le", "annotated_tactic": ["exact (hxz _).<a>le</a>", [{"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\ns : Set (\u03b9 \u2192 \u211d)\nx y : \u03b9 \u2192 \u211d\nhs : IsLowerSet s\nhx : x \u2208 closure s\nh : \u2200 (i : \u03b9), y i < x i\nval\u271d : Fintype \u03b9\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nhxy : \u2200 (i : \u03b9), y i + \u03b5 < x i\nz : \u03b9 \u2192 \u211d\nhz : z \u2208 s\nhxz : \u2200 (b : \u03b9), dist (x b) (z b) < \u03b5\ni : \u03b9\n\u22a2 dist (x i) (z i) \u2264 \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Galois/Prorepresentability.lean", "full_name": "CategoryTheory.PreGaloisCategory.PointedGaloisObject.id_val", "start": [97, 1], "end": [99, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.InjOn.preimage_image_inter", "start": [742, 1], "end": [743, 101], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "full_name": "MeasureTheory.Measure.singularPart_eq_zero", "start": [243, 1], "end": [249, 45], "traced_tactics": [{"tactic": "have h_dec := haveLebesgueDecomposition_add \u03bc \u03bd", "annotated_tactic": ["have h_dec := <a>haveLebesgueDecomposition_add</a> \u03bc \u03bd", [{"full_name": "MeasureTheory.Measure.haveLebesgueDecomposition_add", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [116, 9], "def_end_pos": [116, 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 : \u03bc.HaveLebesgueDecomposition \u03bd\n\u22a2 \u03bc.singularPart \u03bd = 0 \u2194 \u03bc \u226a \u03bd", "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 : \u03bc.HaveLebesgueDecomposition \u03bd\nh_dec : \u03bc = \u03bc.singularPart \u03bd + \u03bd.withDensity (\u03bc.rnDeriv \u03bd)\n\u22a2 \u03bc.singularPart \u03bd = 0 \u2194 \u03bc \u226a \u03bd"}, {"tactic": "refine \u27e8fun h \u21a6 ?_, singularPart_eq_zero_of_ac\u27e9", "annotated_tactic": ["refine \u27e8fun h \u21a6 ?_, <a>singularPart_eq_zero_of_ac</a>\u27e9", [{"full_name": "MeasureTheory.Measure.singularPart_eq_zero_of_ac", "def_path": "Mathlib/MeasureTheory/Decomposition/Lebesgue.lean", "def_pos": [228, 7], "def_end_pos": [228, 33]}]], "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 : \u03bc.HaveLebesgueDecomposition \u03bd\nh_dec : \u03bc = \u03bc.singularPart \u03bd + \u03bd.withDensity (\u03bc.rnDeriv \u03bd)\n\u22a2 \u03bc.singularPart \u03bd = 0 \u2194 \u03bc \u226a \u03bd", "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 : \u03bc.HaveLebesgueDecomposition \u03bd\nh_dec : \u03bc = \u03bc.singularPart \u03bd + \u03bd.withDensity (\u03bc.rnDeriv \u03bd)\nh : \u03bc.singularPart \u03bd = 0\n\u22a2 \u03bc \u226a \u03bd"}, {"tactic": "rw [h, zero_add] at h_dec", "annotated_tactic": ["rw [h, <a>zero_add</a>] at h_dec", [{"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [476, 3], "def_end_pos": [476, 14]}]], "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 : \u03bc.HaveLebesgueDecomposition \u03bd\nh_dec : \u03bc = \u03bc.singularPart \u03bd + \u03bd.withDensity (\u03bc.rnDeriv \u03bd)\nh : \u03bc.singularPart \u03bd = 0\n\u22a2 \u03bc \u226a \u03bd", "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 : \u03bc.HaveLebesgueDecomposition \u03bd\nh_dec : \u03bc = \u03bd.withDensity (\u03bc.rnDeriv \u03bd)\nh : \u03bc.singularPart \u03bd = 0\n\u22a2 \u03bc \u226a \u03bd"}, {"tactic": "rw [h_dec]", "annotated_tactic": ["rw [h_dec]", []], "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 : \u03bc.HaveLebesgueDecomposition \u03bd\nh_dec : \u03bc = \u03bd.withDensity (\u03bc.rnDeriv \u03bd)\nh : \u03bc.singularPart \u03bd = 0\n\u22a2 \u03bc \u226a \u03bd", "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 : \u03bc.HaveLebesgueDecomposition \u03bd\nh_dec : \u03bc = \u03bd.withDensity (\u03bc.rnDeriv \u03bd)\nh : \u03bc.singularPart \u03bd = 0\n\u22a2 \u03bd.withDensity (\u03bc.rnDeriv \u03bd) \u226a \u03bd"}, {"tactic": "exact withDensity_absolutelyContinuous \u03bd _", "annotated_tactic": ["exact <a>withDensity_absolutelyContinuous</a> \u03bd _", [{"full_name": "MeasureTheory.withDensity_absolutelyContinuous", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [150, 9], "def_end_pos": [150, 41]}]], "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 : \u03bc.HaveLebesgueDecomposition \u03bd\nh_dec : \u03bc = \u03bd.withDensity (\u03bc.rnDeriv \u03bd)\nh : \u03bc.singularPart \u03bd = 0\n\u22a2 \u03bd.withDensity (\u03bc.rnDeriv \u03bd) \u226a \u03bd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Classes/SatisfiesM.lean", "full_name": "SatisfiesM.seq_pre", "start": [108, 11], "end": [111, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "MeasureTheory.lintegral_prod_symm", "start": [980, 1], "end": [983, 38], "traced_tactics": [{"tactic": "simp_rw [\u2190 lintegral_prod_swap f]", "annotated_tactic": ["simp_rw [\u2190 <a>lintegral_prod_swap</a> f]", [{"full_name": "MeasureTheory.lintegral_prod_swap", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [932, 9], "def_end_pos": [932, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SFinite \u03bd\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f (\u03bc.prod \u03bd)\n\u22a2 \u222b\u207b (z : \u03b1 \u00d7 \u03b2), f z \u2202\u03bc.prod \u03bd = \u222b\u207b (y : \u03b2), \u222b\u207b (x : \u03b1), f (x, y) \u2202\u03bc \u2202\u03bd", "state_after": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SFinite \u03bd\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f (\u03bc.prod \u03bd)\n\u22a2 \u222b\u207b (z : \u03b2 \u00d7 \u03b1), f z.swap \u2202\u03bd.prod \u03bc = \u222b\u207b (y : \u03b2), \u222b\u207b (x : \u03b1), f (x, y) \u2202\u03bc \u2202\u03bd"}, {"tactic": "exact lintegral_prod _ hf.prod_swap", "annotated_tactic": ["exact <a>lintegral_prod</a> _ hf.prod_swap", [{"full_name": "MeasureTheory.lintegral_prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [969, 9], "def_end_pos": [969, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\u2077 : MeasurableSpace \u03b1\ninst\u271d\u2076 : MeasurableSpace \u03b1'\ninst\u271d\u2075 : MeasurableSpace \u03b2\ninst\u271d\u2074 : MeasurableSpace \u03b2'\ninst\u271d\u00b3 : MeasurableSpace \u03b3\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\n\u03c4 : Measure \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : SFinite \u03bd\ninst\u271d : SFinite \u03bc\nf : \u03b1 \u00d7 \u03b2 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f (\u03bc.prod \u03bd)\n\u22a2 \u222b\u207b (z : \u03b2 \u00d7 \u03b1), f z.swap \u2202\u03bd.prod \u03bc = \u222b\u207b (y : \u03b2), \u222b\u207b (x : \u03b1), f (x, y) \u2202\u03bc \u2202\u03bd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.InjOn.image_subset_image_iff", "start": [773, 1], "end": [777, 53], "traced_tactics": [{"tactic": "refine' \u27e8fun h' \u21a6 _, image_subset _\u27e9", "annotated_tactic": ["refine' \u27e8fun h' \u21a6 _, <a>image_subset</a> _\u27e9", [{"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [291, 9], "def_end_pos": [291, 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\nh : InjOn f s\nh\u2081 : s\u2081 \u2286 s\nh\u2082 : s\u2082 \u2286 s\n\u22a2 f '' s\u2081 \u2286 f '' s\u2082 \u2194 s\u2081 \u2286 s\u2082", "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\nh : InjOn f s\nh\u2081 : s\u2081 \u2286 s\nh\u2082 : s\u2082 \u2286 s\nh' : f '' s\u2081 \u2286 f '' s\u2082\n\u22a2 s\u2081 \u2286 s\u2082"}, {"tactic": "rw [\u2190 h.preimage_image_inter h\u2081, \u2190 h.preimage_image_inter h\u2082]", "annotated_tactic": ["rw [\u2190 h.preimage_image_inter h\u2081, \u2190 h.preimage_image_inter h\u2082]", []], "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\nh : InjOn f s\nh\u2081 : s\u2081 \u2286 s\nh\u2082 : s\u2082 \u2286 s\nh' : f '' s\u2081 \u2286 f '' s\u2082\n\u22a2 s\u2081 \u2286 s\u2082", "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\nh : InjOn f s\nh\u2081 : s\u2081 \u2286 s\nh\u2082 : s\u2082 \u2286 s\nh' : f '' s\u2081 \u2286 f '' s\u2082\n\u22a2 f \u207b\u00b9' (f '' s\u2081) \u2229 s \u2286 f \u207b\u00b9' (f '' s\u2082) \u2229 s"}, {"tactic": "exact inter_subset_inter_left _ (preimage_mono h')", "annotated_tactic": ["exact <a>inter_subset_inter_left</a> _ (<a>preimage_mono</a> h')", [{"full_name": "Set.inter_subset_inter_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [994, 9], "def_end_pos": [994, 32]}, {"full_name": "Set.preimage_mono", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [59, 9], "def_end_pos": [59, 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 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\nh : InjOn f s\nh\u2081 : s\u2081 \u2286 s\nh\u2082 : s\u2082 \u2286 s\nh' : f '' s\u2081 \u2286 f '' s\u2082\n\u22a2 f \u207b\u00b9' (f '' s\u2081) \u2229 s \u2286 f \u207b\u00b9' (f '' s\u2082) \u2229 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Complex/Arg.lean", "full_name": "Complex.abs_add_eq_iff", "start": [49, 1], "end": [50, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/BigOperators/Group/Finset.lean", "full_name": "Finset.card_le_mul_card_image_of_maps_to", "start": [281, 1], "end": [287, 32], "traced_tactics": [{"tactic": "simp [mul_comm]", "annotated_tactic": ["simp [<a>mul_comm</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nG : Type u_6\nk : Type u_7\nR : Type u_8\ninst\u271d : DecidableEq \u03b2\nf : \u03b1 \u2192 \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\nHf : \u2200 a \u2208 s, f a \u2208 t\nn : \u2115\nhn : \u2200 a \u2208 t, (filter (fun x => f x = a) s).card \u2264 n\n\u22a2 \u2211 _a \u2208 t, n = n * t.card", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "full_name": "Subgroup.coe_copy", "start": [540, 1], "end": [541, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Contraction.lean", "full_name": "rTensorHomEquivHomRTensor_toLinearMap", "start": [260, 1], "end": [271, 67], "traced_tactics": [{"tactic": "classical let e := congr (dualTensorHomEquiv R M P) (LinearEquiv.refl R Q)\nhave h : Function.Surjective e.toLinearMap := e.surjective\nrefine (cancel_right h).1 ?_\next f p q m\nsimp only [e, rTensorHomEquivHomRTensor, dualTensorHomEquiv, compr\u2082_apply, mk_apply, coe_comp,\n  LinearEquiv.coe_toLinearMap, Function.comp_apply, map_tmul, LinearEquiv.coe_coe,\n  dualTensorHomEquivOfBasis_apply, LinearEquiv.trans_apply, congr_tmul,\n  dualTensorHomEquivOfBasis_symm_cancel_left, LinearEquiv.refl_apply, assoc_tmul,\n  dualTensorHom_apply, rTensorHomToHomRTensor_apply, smul_tmul']", "annotated_tactic": ["classical -- Porting note: missing decidable for choosing basis\n  let e := <a>congr</a> (<a>dualTensorHomEquiv</a> R M P) (<a>LinearEquiv.refl</a> R Q)\n  have h : <a>Function.Surjective</a> e.toLinearMap := e.surjective\n  refine (<a>cancel_right</a> h).1 ?_\n  ext f p q m\n  simp only [e, <a>rTensorHomEquivHomRTensor</a>, <a>dualTensorHomEquiv</a>, <a>compr\u2082_apply</a>, <a>mk_apply</a>, <a>coe_comp</a>,\n    <a>LinearEquiv.coe_toLinearMap</a>, <a>Function.comp_apply</a>, <a>map_tmul</a>, <a>LinearEquiv.coe_coe</a>,\n    <a>dualTensorHomEquivOfBasis_apply</a>, <a>LinearEquiv.trans_apply</a>, <a>congr_tmul</a>,\n    <a>dualTensorHomEquivOfBasis_symm_cancel_left</a>, <a>LinearEquiv.refl_apply</a>, <a>assoc_tmul</a>,\n    <a>dualTensorHom_apply</a>, <a>rTensorHomToHomRTensor_apply</a>, <a>smul_tmul'</a>]", [{"full_name": "TensorProduct.congr", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [1025, 5], "def_end_pos": [1025, 10]}, {"full_name": "dualTensorHomEquiv", "def_path": "Mathlib/LinearAlgebra/Contraction.lean", "def_pos": [204, 19], "def_end_pos": [204, 37]}, {"full_name": "LinearEquiv.refl", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [269, 5], "def_end_pos": [269, 9]}, {"full_name": "Function.Surjective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [133, 5], "def_end_pos": [133, 15]}, {"full_name": "LinearMap.cancel_right", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [588, 9], "def_end_pos": [588, 21]}, {"full_name": "rTensorHomEquivHomRTensor", "def_path": "Mathlib/LinearAlgebra/Contraction.lean", "def_pos": [239, 19], "def_end_pos": [239, 44]}, {"full_name": "dualTensorHomEquiv", "def_path": "Mathlib/LinearAlgebra/Contraction.lean", "def_pos": [204, 19], "def_end_pos": [204, 37]}, {"full_name": "LinearMap.compr\u2082_apply", "def_path": "Mathlib/LinearAlgebra/BilinearMap.lean", "def_pos": [390, 9], "def_end_pos": [390, 21]}, {"full_name": "TensorProduct.mk_apply", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [451, 9], "def_end_pos": [451, 17]}, {"full_name": "LinearMap.coe_comp", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [557, 9], "def_end_pos": [557, 17]}, {"full_name": "LinearEquiv.coe_toLinearMap", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [232, 9], "def_end_pos": [232, 24]}, {"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}, {"full_name": "TensorProduct.map_tmul", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [808, 9], "def_end_pos": [808, 17]}, {"full_name": "LinearEquiv.coe_coe", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [222, 9], "def_end_pos": [222, 16]}, {"full_name": "dualTensorHomEquivOfBasis_apply", "def_path": "Mathlib/LinearAlgebra/Contraction.lean", "def_pos": [172, 9], "def_end_pos": [172, 40]}, {"full_name": "LinearEquiv.trans_apply", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [366, 9], "def_end_pos": [366, 20]}, {"full_name": "TensorProduct.congr_tmul", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [1032, 9], "def_end_pos": [1032, 19]}, {"full_name": "dualTensorHomEquivOfBasis_symm_cancel_left", "def_path": "Mathlib/LinearAlgebra/Contraction.lean", "def_pos": [186, 9], "def_end_pos": [186, 51]}, {"full_name": "LinearEquiv.refl_apply", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [276, 9], "def_end_pos": [276, 19]}, {"full_name": "TensorProduct.assoc_tmul", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [791, 9], "def_end_pos": [791, 19]}, {"full_name": "dualTensorHom_apply", "def_path": "Mathlib/LinearAlgebra/Contraction.lean", "def_pos": [79, 9], "def_end_pos": [79, 28]}, {"full_name": "TensorProduct.rTensorHomToHomRTensor_apply", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [998, 9], "def_end_pos": [998, 37]}, {"full_name": "TensorProduct.smul_tmul'", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [368, 9], "def_end_pos": [368, 19]}]], "state_before": "\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\n\u22a2 \u2191(rTensorHomEquivHomRTensor R M P Q) = rTensorHomToHomRTensor R M P Q", "state_after": "no goals"}, {"tactic": "let e := congr (dualTensorHomEquiv R M P) (LinearEquiv.refl R Q)", "annotated_tactic": ["let e := <a>congr</a> (<a>dualTensorHomEquiv</a> R M P) (<a>LinearEquiv.refl</a> R Q)", [{"full_name": "TensorProduct.congr", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [1025, 5], "def_end_pos": [1025, 10]}, {"full_name": "dualTensorHomEquiv", "def_path": "Mathlib/LinearAlgebra/Contraction.lean", "def_pos": [204, 19], "def_end_pos": [204, 37]}, {"full_name": "LinearEquiv.refl", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [269, 5], "def_end_pos": [269, 9]}]], "state_before": "\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\n\u22a2 \u2191(rTensorHomEquivHomRTensor R M P Q) = rTensorHomToHomRTensor R M P Q", "state_after": "\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\ne : (Dual R M \u2297[R] P) \u2297[R] Q \u2243\u2097[R] (M \u2192\u2097[R] P) \u2297[R] Q :=\n  TensorProduct.congr (dualTensorHomEquiv R M P) (LinearEquiv.refl R Q)\n\u22a2 \u2191(rTensorHomEquivHomRTensor R M P Q) = rTensorHomToHomRTensor R M P Q"}, {"tactic": "have h : Function.Surjective e.toLinearMap := e.surjective", "annotated_tactic": ["have h : <a>Function.Surjective</a> e.toLinearMap := e.surjective", [{"full_name": "Function.Surjective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [133, 5], "def_end_pos": [133, 15]}]], "state_before": "\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\ne : (Dual R M \u2297[R] P) \u2297[R] Q \u2243\u2097[R] (M \u2192\u2097[R] P) \u2297[R] Q :=\n  TensorProduct.congr (dualTensorHomEquiv R M P) (LinearEquiv.refl R Q)\n\u22a2 \u2191(rTensorHomEquivHomRTensor R M P Q) = rTensorHomToHomRTensor R M P Q", "state_after": "\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\ne : (Dual R M \u2297[R] P) \u2297[R] Q \u2243\u2097[R] (M \u2192\u2097[R] P) \u2297[R] Q :=\n  TensorProduct.congr (dualTensorHomEquiv R M P) (LinearEquiv.refl R Q)\nh : Function.Surjective \u21d1\u2191e\n\u22a2 \u2191(rTensorHomEquivHomRTensor R M P Q) = rTensorHomToHomRTensor R M P Q"}, {"tactic": "refine (cancel_right h).1 ?_", "annotated_tactic": ["refine (<a>cancel_right</a> h).1 ?_", [{"full_name": "LinearMap.cancel_right", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [588, 9], "def_end_pos": [588, 21]}]], "state_before": "\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\ne : (Dual R M \u2297[R] P) \u2297[R] Q \u2243\u2097[R] (M \u2192\u2097[R] P) \u2297[R] Q :=\n  TensorProduct.congr (dualTensorHomEquiv R M P) (LinearEquiv.refl R Q)\nh : Function.Surjective \u21d1\u2191e\n\u22a2 \u2191(rTensorHomEquivHomRTensor R M P Q) = rTensorHomToHomRTensor R M P Q", "state_after": "\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\ne : (Dual R M \u2297[R] P) \u2297[R] Q \u2243\u2097[R] (M \u2192\u2097[R] P) \u2297[R] Q :=\n  TensorProduct.congr (dualTensorHomEquiv R M P) (LinearEquiv.refl R Q)\nh : Function.Surjective \u21d1\u2191e\n\u22a2 \u2191(rTensorHomEquivHomRTensor R M P Q) \u2218\u2097 \u2191e = rTensorHomToHomRTensor R M P Q \u2218\u2097 \u2191e"}, {"tactic": "ext f p q m", "annotated_tactic": ["ext f p q m", []], "state_before": "\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\ne : (Dual R M \u2297[R] P) \u2297[R] Q \u2243\u2097[R] (M \u2192\u2097[R] P) \u2297[R] Q :=\n  TensorProduct.congr (dualTensorHomEquiv R M P) (LinearEquiv.refl R Q)\nh : Function.Surjective \u21d1\u2191e\n\u22a2 \u2191(rTensorHomEquivHomRTensor R M P Q) \u2218\u2097 \u2191e = rTensorHomToHomRTensor R M P Q \u2218\u2097 \u2191e", "state_after": "case H.H.h.h.h.h\n\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\ne : (Dual R M \u2297[R] P) \u2297[R] Q \u2243\u2097[R] (M \u2192\u2097[R] P) \u2297[R] Q :=\n  TensorProduct.congr (dualTensorHomEquiv R M P) (LinearEquiv.refl R Q)\nh : Function.Surjective \u21d1\u2191e\nf : Dual R M\np : P\nq : Q\nm : M\n\u22a2 (((((TensorProduct.mk R (Dual R M) P).compr\u2082\n              ((TensorProduct.mk R (Dual R M \u2297[R] P) Q).compr\u2082 (\u2191(rTensorHomEquivHomRTensor R M P Q) \u2218\u2097 \u2191e)))\n            f)\n          p)\n        q)\n      m =\n    (((((TensorProduct.mk R (Dual R M) P).compr\u2082\n              ((TensorProduct.mk R (Dual R M \u2297[R] P) Q).compr\u2082 (rTensorHomToHomRTensor R M P Q \u2218\u2097 \u2191e)))\n            f)\n          p)\n        q)\n      m"}, {"tactic": "simp only [e, rTensorHomEquivHomRTensor, dualTensorHomEquiv, compr\u2082_apply, mk_apply, coe_comp,\n  LinearEquiv.coe_toLinearMap, Function.comp_apply, map_tmul, LinearEquiv.coe_coe,\n  dualTensorHomEquivOfBasis_apply, LinearEquiv.trans_apply, congr_tmul,\n  dualTensorHomEquivOfBasis_symm_cancel_left, LinearEquiv.refl_apply, assoc_tmul,\n  dualTensorHom_apply, rTensorHomToHomRTensor_apply, smul_tmul']", "annotated_tactic": ["simp only [e, <a>rTensorHomEquivHomRTensor</a>, <a>dualTensorHomEquiv</a>, <a>compr\u2082_apply</a>, <a>mk_apply</a>, <a>coe_comp</a>,\n    <a>LinearEquiv.coe_toLinearMap</a>, <a>Function.comp_apply</a>, <a>map_tmul</a>, <a>LinearEquiv.coe_coe</a>,\n    <a>dualTensorHomEquivOfBasis_apply</a>, <a>LinearEquiv.trans_apply</a>, <a>congr_tmul</a>,\n    <a>dualTensorHomEquivOfBasis_symm_cancel_left</a>, <a>LinearEquiv.refl_apply</a>, <a>assoc_tmul</a>,\n    <a>dualTensorHom_apply</a>, <a>rTensorHomToHomRTensor_apply</a>, <a>smul_tmul'</a>]", [{"full_name": "rTensorHomEquivHomRTensor", "def_path": "Mathlib/LinearAlgebra/Contraction.lean", "def_pos": [239, 19], "def_end_pos": [239, 44]}, {"full_name": "dualTensorHomEquiv", "def_path": "Mathlib/LinearAlgebra/Contraction.lean", "def_pos": [204, 19], "def_end_pos": [204, 37]}, {"full_name": "LinearMap.compr\u2082_apply", "def_path": "Mathlib/LinearAlgebra/BilinearMap.lean", "def_pos": [390, 9], "def_end_pos": [390, 21]}, {"full_name": "TensorProduct.mk_apply", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [451, 9], "def_end_pos": [451, 17]}, {"full_name": "LinearMap.coe_comp", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [557, 9], "def_end_pos": [557, 17]}, {"full_name": "LinearEquiv.coe_toLinearMap", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [232, 9], "def_end_pos": [232, 24]}, {"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}, {"full_name": "TensorProduct.map_tmul", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [808, 9], "def_end_pos": [808, 17]}, {"full_name": "LinearEquiv.coe_coe", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [222, 9], "def_end_pos": [222, 16]}, {"full_name": "dualTensorHomEquivOfBasis_apply", "def_path": "Mathlib/LinearAlgebra/Contraction.lean", "def_pos": [172, 9], "def_end_pos": [172, 40]}, {"full_name": "LinearEquiv.trans_apply", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [366, 9], "def_end_pos": [366, 20]}, {"full_name": "TensorProduct.congr_tmul", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [1032, 9], "def_end_pos": [1032, 19]}, {"full_name": "dualTensorHomEquivOfBasis_symm_cancel_left", "def_path": "Mathlib/LinearAlgebra/Contraction.lean", "def_pos": [186, 9], "def_end_pos": [186, 51]}, {"full_name": "LinearEquiv.refl_apply", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [276, 9], "def_end_pos": [276, 19]}, {"full_name": "TensorProduct.assoc_tmul", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [791, 9], "def_end_pos": [791, 19]}, {"full_name": "dualTensorHom_apply", "def_path": "Mathlib/LinearAlgebra/Contraction.lean", "def_pos": [79, 9], "def_end_pos": [79, 28]}, {"full_name": "TensorProduct.rTensorHomToHomRTensor_apply", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [998, 9], "def_end_pos": [998, 37]}, {"full_name": "TensorProduct.smul_tmul'", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [368, 9], "def_end_pos": [368, 19]}]], "state_before": "case H.H.h.h.h.h\n\u03b9 : Type w\nR : Type u\nM : Type v\u2081\nN : Type v\u2082\nP : Type v\u2083\nQ : Type v\u2084\ninst\u271d\u00b9\u00b3 : CommRing R\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : AddCommGroup N\ninst\u271d\u00b9\u2070 : AddCommGroup P\ninst\u271d\u2079 : AddCommGroup Q\ninst\u271d\u2078 : Module R M\ninst\u271d\u2077 : Module R N\ninst\u271d\u2076 : Module R P\ninst\u271d\u2075 : Module R Q\ninst\u271d\u2074 : Free R M\ninst\u271d\u00b3 : Module.Finite R M\ninst\u271d\u00b2 : Free R N\ninst\u271d\u00b9 : Module.Finite R N\ninst\u271d : Nontrivial R\ne : (Dual R M \u2297[R] P) \u2297[R] Q \u2243\u2097[R] (M \u2192\u2097[R] P) \u2297[R] Q :=\n  TensorProduct.congr (dualTensorHomEquiv R M P) (LinearEquiv.refl R Q)\nh : Function.Surjective \u21d1\u2191e\nf : Dual R M\np : P\nq : Q\nm : M\n\u22a2 (((((TensorProduct.mk R (Dual R M) P).compr\u2082\n              ((TensorProduct.mk R (Dual R M \u2297[R] P) Q).compr\u2082 (\u2191(rTensorHomEquivHomRTensor R M P Q) \u2218\u2097 \u2191e)))\n            f)\n          p)\n        q)\n      m =\n    (((((TensorProduct.mk R (Dual R M) P).compr\u2082\n              ((TensorProduct.mk R (Dual R M \u2297[R] P) Q).compr\u2082 (rTensorHomToHomRTensor R M P Q \u2218\u2097 \u2191e)))\n            f)\n          p)\n        q)\n      m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Normed/Ring/Seminorm.lean", "full_name": "MulRingNorm.equiv_symm", "start": [336, 1], "end": [343, 92], "traced_tactics": [{"tactic": "rcases hfg with \u27e8c, hcpos, h\u27e9", "annotated_tactic": ["rcases hfg with \u27e8c, hcpos, h\u27e9", []], "state_before": "F : Type u_1\nR\u271d : Type u_2\nS : Type u_3\nx y : R\u271d\nr : \u211d\ninst\u271d\u2074 : NonAssocRing R\u271d\ninst\u271d\u00b3 : DecidableEq R\u271d\ninst\u271d\u00b2 : NoZeroDivisors R\u271d\ninst\u271d\u00b9 : Nontrivial R\u271d\nR : Type u_4\ninst\u271d : Ring R\nf g : MulRingNorm R\nhfg : f.equiv g\n\u22a2 g.equiv f", "state_after": "case intro.intro\nF : Type u_1\nR\u271d : Type u_2\nS : Type u_3\nx y : R\u271d\nr : \u211d\ninst\u271d\u2074 : NonAssocRing R\u271d\ninst\u271d\u00b3 : DecidableEq R\u271d\ninst\u271d\u00b2 : NoZeroDivisors R\u271d\ninst\u271d\u00b9 : Nontrivial R\u271d\nR : Type u_4\ninst\u271d : Ring R\nf g : MulRingNorm R\nc : \u211d\nhcpos : 0 < c\nh : (fun x => f x ^ c) = \u21d1g\n\u22a2 g.equiv f"}, {"tactic": "use 1/c", "annotated_tactic": ["use 1/c", []], "state_before": "case intro.intro\nF : Type u_1\nR\u271d : Type u_2\nS : Type u_3\nx y : R\u271d\nr : \u211d\ninst\u271d\u2074 : NonAssocRing R\u271d\ninst\u271d\u00b3 : DecidableEq R\u271d\ninst\u271d\u00b2 : NoZeroDivisors R\u271d\ninst\u271d\u00b9 : Nontrivial R\u271d\nR : Type u_4\ninst\u271d : Ring R\nf g : MulRingNorm R\nc : \u211d\nhcpos : 0 < c\nh : (fun x => f x ^ c) = \u21d1g\n\u22a2 g.equiv f", "state_after": "case h\nF : Type u_1\nR\u271d : Type u_2\nS : Type u_3\nx y : R\u271d\nr : \u211d\ninst\u271d\u2074 : NonAssocRing R\u271d\ninst\u271d\u00b3 : DecidableEq R\u271d\ninst\u271d\u00b2 : NoZeroDivisors R\u271d\ninst\u271d\u00b9 : Nontrivial R\u271d\nR : Type u_4\ninst\u271d : Ring R\nf g : MulRingNorm R\nc : \u211d\nhcpos : 0 < c\nh : (fun x => f x ^ c) = \u21d1g\n\u22a2 0 < 1 / c \u2227 (fun x => g x ^ (1 / c)) = \u21d1f"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\nF : Type u_1\nR\u271d : Type u_2\nS : Type u_3\nx y : R\u271d\nr : \u211d\ninst\u271d\u2074 : NonAssocRing R\u271d\ninst\u271d\u00b3 : DecidableEq R\u271d\ninst\u271d\u00b2 : NoZeroDivisors R\u271d\ninst\u271d\u00b9 : Nontrivial R\u271d\nR : Type u_4\ninst\u271d : Ring R\nf g : MulRingNorm R\nc : \u211d\nhcpos : 0 < c\nh : (fun x => f x ^ c) = \u21d1g\n\u22a2 0 < 1 / c \u2227 (fun x => g x ^ (1 / c)) = \u21d1f", "state_after": "case h.left\nF : Type u_1\nR\u271d : Type u_2\nS : Type u_3\nx y : R\u271d\nr : \u211d\ninst\u271d\u2074 : NonAssocRing R\u271d\ninst\u271d\u00b3 : DecidableEq R\u271d\ninst\u271d\u00b2 : NoZeroDivisors R\u271d\ninst\u271d\u00b9 : Nontrivial R\u271d\nR : Type u_4\ninst\u271d : Ring R\nf g : MulRingNorm R\nc : \u211d\nhcpos : 0 < c\nh : (fun x => f x ^ c) = \u21d1g\n\u22a2 0 < 1 / c\n\ncase h.right\nF : Type u_1\nR\u271d : Type u_2\nS : Type u_3\nx y : R\u271d\nr : \u211d\ninst\u271d\u2074 : NonAssocRing R\u271d\ninst\u271d\u00b3 : DecidableEq R\u271d\ninst\u271d\u00b2 : NoZeroDivisors R\u271d\ninst\u271d\u00b9 : Nontrivial R\u271d\nR : Type u_4\ninst\u271d : Ring R\nf g : MulRingNorm R\nc : \u211d\nhcpos : 0 < c\nh : (fun x => f x ^ c) = \u21d1g\n\u22a2 (fun x => g x ^ (1 / c)) = \u21d1f"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case h.right\nF : Type u_1\nR\u271d : Type u_2\nS : Type u_3\nx y : R\u271d\nr : \u211d\ninst\u271d\u2074 : NonAssocRing R\u271d\ninst\u271d\u00b3 : DecidableEq R\u271d\ninst\u271d\u00b2 : NoZeroDivisors R\u271d\ninst\u271d\u00b9 : Nontrivial R\u271d\nR : Type u_4\ninst\u271d : Ring R\nf g : MulRingNorm R\nc : \u211d\nhcpos : 0 < c\nh : (fun x => f x ^ c) = \u21d1g\n\u22a2 (fun x => g x ^ (1 / c)) = \u21d1f", "state_after": "case h.right.h\nF : Type u_1\nR\u271d : Type u_2\nS : Type u_3\nx\u271d y : R\u271d\nr : \u211d\ninst\u271d\u2074 : NonAssocRing R\u271d\ninst\u271d\u00b3 : DecidableEq R\u271d\ninst\u271d\u00b2 : NoZeroDivisors R\u271d\ninst\u271d\u00b9 : Nontrivial R\u271d\nR : Type u_4\ninst\u271d : Ring R\nf g : MulRingNorm R\nc : \u211d\nhcpos : 0 < c\nh : (fun x => f x ^ c) = \u21d1g\nx : R\n\u22a2 g x ^ (1 / c) = f x"}, {"tactic": "simpa [\u2190 congr_fun h x] using Real.rpow_rpow_inv (apply_nonneg f x) (ne_of_lt hcpos).symm", "annotated_tactic": ["simpa [\u2190 <a>congr_fun</a> h x] using <a>Real.rpow_rpow_inv</a> (<a>apply_nonneg</a> f x) (<a>ne_of_lt</a> hcpos).<a>symm</a>", [{"full_name": "congr_fun", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [78, 7], "def_end_pos": [78, 16]}, {"full_name": "Real.rpow_rpow_inv", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [506, 15], "def_end_pos": [506, 28]}, {"full_name": "NonnegHomClass.apply_nonneg", "def_path": "Mathlib/Algebra/Order/Hom/Basic.lean", "def_pos": [80, 3], "def_end_pos": [80, 15]}, {"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": [704, 9], "def_end_pos": [704, 16]}]], "state_before": "case h.right.h\nF : Type u_1\nR\u271d : Type u_2\nS : Type u_3\nx\u271d y : R\u271d\nr : \u211d\ninst\u271d\u2074 : NonAssocRing R\u271d\ninst\u271d\u00b3 : DecidableEq R\u271d\ninst\u271d\u00b2 : NoZeroDivisors R\u271d\ninst\u271d\u00b9 : Nontrivial R\u271d\nR : Type u_4\ninst\u271d : Ring R\nf g : MulRingNorm R\nc : \u211d\nhcpos : 0 < c\nh : (fun x => f x ^ c) = \u21d1g\nx : R\n\u22a2 g x ^ (1 / c) = f x", "state_after": "no goals"}, {"tactic": "simp only [one_div, inv_pos, hcpos]", "annotated_tactic": ["simp only [<a>one_div</a>, <a>inv_pos</a>, hcpos]", [{"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 16]}, {"full_name": "inv_pos", "def_path": "Mathlib/Algebra/Order/Field/Defs.lean", "def_pos": [47, 15], "def_end_pos": [47, 22]}]], "state_before": "case h.left\nF : Type u_1\nR\u271d : Type u_2\nS : Type u_3\nx y : R\u271d\nr : \u211d\ninst\u271d\u2074 : NonAssocRing R\u271d\ninst\u271d\u00b3 : DecidableEq R\u271d\ninst\u271d\u00b2 : NoZeroDivisors R\u271d\ninst\u271d\u00b9 : Nontrivial R\u271d\nR : Type u_4\ninst\u271d : Ring R\nf g : MulRingNorm R\nc : \u211d\nhcpos : 0 < c\nh : (fun x => f x ^ c) = \u21d1g\n\u22a2 0 < 1 / c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/NullMeasurable.lean", "full_name": "MeasureTheory.NullMeasurableSet.of_subsingleton", "start": [132, 1], "end": [133, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Trace.lean", "full_name": "LinearMap.trace_comp_eq_mul_of_commute_of_isNilpotent", "start": [323, 1], "end": [333, 82], "traced_tactics": [{"tactic": "set n := g - algebraMap R _ \u03bc", "annotated_tactic": ["set n := g - <a>algebraMap</a> R _ \u03bc", [{"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : Module R N\ninst\u271d\u2078 : AddCommGroup P\ninst\u271d\u2077 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : Module.Finite R M\ninst\u271d\u2074 : Module.Free R N\ninst\u271d\u00b3 : Module.Finite R N\ninst\u271d\u00b2 : Module.Free R P\ninst\u271d\u00b9 : Module.Finite R P\ninst\u271d : IsReduced R\nf g : Module.End R M\n\u03bc : R\nh_comm : Commute f g\nhg : IsNilpotent (g - (algebraMap R (Module.End R M)) \u03bc)\n\u22a2 (trace R M) (f \u2218\u2097 g) = \u03bc * (trace R M) f", "state_after": "R : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : Module R N\ninst\u271d\u2078 : AddCommGroup P\ninst\u271d\u2077 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : Module.Finite R M\ninst\u271d\u2074 : Module.Free R N\ninst\u271d\u00b3 : Module.Finite R N\ninst\u271d\u00b2 : Module.Free R P\ninst\u271d\u00b9 : Module.Finite R P\ninst\u271d : IsReduced R\nf g : Module.End R M\n\u03bc : R\nh_comm : Commute f g\nn : Module.End R M := g - (algebraMap R (Module.End R M)) \u03bc\nhg : IsNilpotent n\n\u22a2 (trace R M) (f \u2218\u2097 g) = \u03bc * (trace R M) f"}, {"tactic": "replace hg : trace R M (f \u2218\u2097 n) = 0 := by\n  rw [\u2190 isNilpotent_iff_eq_zero, \u2190 mul_eq_comp]\n  refine isNilpotent_trace_of_isNilpotent (Commute.isNilpotent_mul_right ?_ hg)\n  exact h_comm.sub_right (Algebra.commute_algebraMap_right \u03bc f)", "annotated_tactic": ["replace hg : <a>trace</a> R M (f \u2218\u2097 n) = 0 := by\n    rw [\u2190 <a>isNilpotent_iff_eq_zero</a>, \u2190 <a>mul_eq_comp</a>]\n    refine <a>isNilpotent_trace_of_isNilpotent</a> (<a>Commute.isNilpotent_mul_right</a> ?_ hg)\n    exact h_comm.sub_right (<a>Algebra.commute_algebraMap_right</a> \u03bc f)", [{"full_name": "LinearMap.trace", "def_path": "Mathlib/LinearAlgebra/Trace.lean", "def_pos": [77, 5], "def_end_pos": [77, 10]}, {"full_name": "isNilpotent_iff_eq_zero", "def_path": "Mathlib/RingTheory/Nilpotent/Defs.lean", "def_pos": [193, 9], "def_end_pos": [193, 32]}, {"full_name": "LinearMap.mul_eq_comp", "def_path": "Mathlib/Algebra/Module/LinearMap/End.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}, {"full_name": "LinearMap.isNilpotent_trace_of_isNilpotent", "def_path": "Mathlib/LinearAlgebra/Trace.lean", "def_pos": [316, 7], "def_end_pos": [316, 39]}, {"full_name": "Commute.isNilpotent_mul_right", "def_path": "Mathlib/RingTheory/Nilpotent/Defs.lean", "def_pos": [235, 9], "def_end_pos": [235, 30]}, {"full_name": "Algebra.commute_algebraMap_right", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [348, 7], "def_end_pos": [348, 31]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : Module R N\ninst\u271d\u2078 : AddCommGroup P\ninst\u271d\u2077 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : Module.Finite R M\ninst\u271d\u2074 : Module.Free R N\ninst\u271d\u00b3 : Module.Finite R N\ninst\u271d\u00b2 : Module.Free R P\ninst\u271d\u00b9 : Module.Finite R P\ninst\u271d : IsReduced R\nf g : Module.End R M\n\u03bc : R\nh_comm : Commute f g\nn : Module.End R M := g - (algebraMap R (Module.End R M)) \u03bc\nhg : IsNilpotent n\n\u22a2 (trace R M) (f \u2218\u2097 g) = \u03bc * (trace R M) f", "state_after": "R : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : Module R N\ninst\u271d\u2078 : AddCommGroup P\ninst\u271d\u2077 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : Module.Finite R M\ninst\u271d\u2074 : Module.Free R N\ninst\u271d\u00b3 : Module.Finite R N\ninst\u271d\u00b2 : Module.Free R P\ninst\u271d\u00b9 : Module.Finite R P\ninst\u271d : IsReduced R\nf g : Module.End R M\n\u03bc : R\nh_comm : Commute f g\nn : Module.End R M := g - (algebraMap R (Module.End R M)) \u03bc\nhg : (trace R M) (f \u2218\u2097 n) = 0\n\u22a2 (trace R M) (f \u2218\u2097 g) = \u03bc * (trace R M) f"}, {"tactic": "have h\u03bc : g = algebraMap R _ \u03bc + n := eq_add_of_sub_eq' rfl", "annotated_tactic": ["have h\u03bc : g = <a>algebraMap</a> R _ \u03bc + n := <a>eq_add_of_sub_eq'</a> <a>rfl</a>", [{"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "eq_add_of_sub_eq'", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1283, 3], "def_end_pos": [1283, 14]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : Module R N\ninst\u271d\u2078 : AddCommGroup P\ninst\u271d\u2077 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : Module.Finite R M\ninst\u271d\u2074 : Module.Free R N\ninst\u271d\u00b3 : Module.Finite R N\ninst\u271d\u00b2 : Module.Free R P\ninst\u271d\u00b9 : Module.Finite R P\ninst\u271d : IsReduced R\nf g : Module.End R M\n\u03bc : R\nh_comm : Commute f g\nn : Module.End R M := g - (algebraMap R (Module.End R M)) \u03bc\nhg : (trace R M) (f \u2218\u2097 n) = 0\n\u22a2 (trace R M) (f \u2218\u2097 g) = \u03bc * (trace R M) f", "state_after": "R : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : Module R N\ninst\u271d\u2078 : AddCommGroup P\ninst\u271d\u2077 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : Module.Finite R M\ninst\u271d\u2074 : Module.Free R N\ninst\u271d\u00b3 : Module.Finite R N\ninst\u271d\u00b2 : Module.Free R P\ninst\u271d\u00b9 : Module.Finite R P\ninst\u271d : IsReduced R\nf g : Module.End R M\n\u03bc : R\nh_comm : Commute f g\nn : Module.End R M := g - (algebraMap R (Module.End R M)) \u03bc\nhg : (trace R M) (f \u2218\u2097 n) = 0\nh\u03bc : g = (algebraMap R (Module.End R M)) \u03bc + n\n\u22a2 (trace R M) (f \u2218\u2097 g) = \u03bc * (trace R M) f"}, {"tactic": "have : f \u2218\u2097 algebraMap R _ \u03bc = \u03bc \u2022 f := by ext; simp", "annotated_tactic": ["have : f \u2218\u2097 <a>algebraMap</a> R _ \u03bc = \u03bc \u2022 f := by ext; simp", [{"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : Module R N\ninst\u271d\u2078 : AddCommGroup P\ninst\u271d\u2077 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : Module.Finite R M\ninst\u271d\u2074 : Module.Free R N\ninst\u271d\u00b3 : Module.Finite R N\ninst\u271d\u00b2 : Module.Free R P\ninst\u271d\u00b9 : Module.Finite R P\ninst\u271d : IsReduced R\nf g : Module.End R M\n\u03bc : R\nh_comm : Commute f g\nn : Module.End R M := g - (algebraMap R (Module.End R M)) \u03bc\nhg : (trace R M) (f \u2218\u2097 n) = 0\nh\u03bc : g = (algebraMap R (Module.End R M)) \u03bc + n\n\u22a2 (trace R M) (f \u2218\u2097 g) = \u03bc * (trace R M) f", "state_after": "R : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : Module R N\ninst\u271d\u2078 : AddCommGroup P\ninst\u271d\u2077 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : Module.Finite R M\ninst\u271d\u2074 : Module.Free R N\ninst\u271d\u00b3 : Module.Finite R N\ninst\u271d\u00b2 : Module.Free R P\ninst\u271d\u00b9 : Module.Finite R P\ninst\u271d : IsReduced R\nf g : Module.End R M\n\u03bc : R\nh_comm : Commute f g\nn : Module.End R M := g - (algebraMap R (Module.End R M)) \u03bc\nhg : (trace R M) (f \u2218\u2097 n) = 0\nh\u03bc : g = (algebraMap R (Module.End R M)) \u03bc + n\nthis : f \u2218\u2097 (algebraMap R (M \u2192\u2097[R] M)) \u03bc = \u03bc \u2022 f\n\u22a2 (trace R M) (f \u2218\u2097 g) = \u03bc * (trace R M) f"}, {"tactic": "rw [h\u03bc, comp_add, map_add, hg, add_zero, this, LinearMap.map_smul, smul_eq_mul]", "annotated_tactic": ["rw [h\u03bc, <a>comp_add</a>, <a>map_add</a>, hg, <a>add_zero</a>, this, <a>LinearMap.map_smul</a>, <a>smul_eq_mul</a>]", [{"full_name": "LinearMap.comp_add", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [919, 9], "def_end_pos": [919, 17]}, {"full_name": "map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [308, 3], "def_end_pos": [308, 14]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [482, 3], "def_end_pos": [482, 14]}, {"full_name": "LinearMap.map_smul", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [376, 19], "def_end_pos": [376, 27]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [89, 7], "def_end_pos": [89, 18]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : Module R N\ninst\u271d\u2078 : AddCommGroup P\ninst\u271d\u2077 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : Module.Finite R M\ninst\u271d\u2074 : Module.Free R N\ninst\u271d\u00b3 : Module.Finite R N\ninst\u271d\u00b2 : Module.Free R P\ninst\u271d\u00b9 : Module.Finite R P\ninst\u271d : IsReduced R\nf g : Module.End R M\n\u03bc : R\nh_comm : Commute f g\nn : Module.End R M := g - (algebraMap R (Module.End R M)) \u03bc\nhg : (trace R M) (f \u2218\u2097 n) = 0\nh\u03bc : g = (algebraMap R (Module.End R M)) \u03bc + n\nthis : f \u2218\u2097 (algebraMap R (M \u2192\u2097[R] M)) \u03bc = \u03bc \u2022 f\n\u22a2 (trace R M) (f \u2218\u2097 g) = \u03bc * (trace R M) f", "state_after": "no goals"}, {"tactic": "rw [\u2190 isNilpotent_iff_eq_zero, \u2190 mul_eq_comp]", "annotated_tactic": ["rw [\u2190 <a>isNilpotent_iff_eq_zero</a>, \u2190 <a>mul_eq_comp</a>]", [{"full_name": "isNilpotent_iff_eq_zero", "def_path": "Mathlib/RingTheory/Nilpotent/Defs.lean", "def_pos": [193, 9], "def_end_pos": [193, 32]}, {"full_name": "LinearMap.mul_eq_comp", "def_path": "Mathlib/Algebra/Module/LinearMap/End.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : Module R N\ninst\u271d\u2078 : AddCommGroup P\ninst\u271d\u2077 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : Module.Finite R M\ninst\u271d\u2074 : Module.Free R N\ninst\u271d\u00b3 : Module.Finite R N\ninst\u271d\u00b2 : Module.Free R P\ninst\u271d\u00b9 : Module.Finite R P\ninst\u271d : IsReduced R\nf g : Module.End R M\n\u03bc : R\nh_comm : Commute f g\nn : Module.End R M := g - (algebraMap R (Module.End R M)) \u03bc\nhg : IsNilpotent n\n\u22a2 (trace R M) (f \u2218\u2097 n) = 0", "state_after": "R : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : Module R N\ninst\u271d\u2078 : AddCommGroup P\ninst\u271d\u2077 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : Module.Finite R M\ninst\u271d\u2074 : Module.Free R N\ninst\u271d\u00b3 : Module.Finite R N\ninst\u271d\u00b2 : Module.Free R P\ninst\u271d\u00b9 : Module.Finite R P\ninst\u271d : IsReduced R\nf g : Module.End R M\n\u03bc : R\nh_comm : Commute f g\nn : Module.End R M := g - (algebraMap R (Module.End R M)) \u03bc\nhg : IsNilpotent n\n\u22a2 IsNilpotent ((trace R M) (f * n))"}, {"tactic": "refine isNilpotent_trace_of_isNilpotent (Commute.isNilpotent_mul_right ?_ hg)", "annotated_tactic": ["refine <a>isNilpotent_trace_of_isNilpotent</a> (<a>Commute.isNilpotent_mul_right</a> ?_ hg)", [{"full_name": "LinearMap.isNilpotent_trace_of_isNilpotent", "def_path": "Mathlib/LinearAlgebra/Trace.lean", "def_pos": [316, 7], "def_end_pos": [316, 39]}, {"full_name": "Commute.isNilpotent_mul_right", "def_path": "Mathlib/RingTheory/Nilpotent/Defs.lean", "def_pos": [235, 9], "def_end_pos": [235, 30]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : Module R N\ninst\u271d\u2078 : AddCommGroup P\ninst\u271d\u2077 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : Module.Finite R M\ninst\u271d\u2074 : Module.Free R N\ninst\u271d\u00b3 : Module.Finite R N\ninst\u271d\u00b2 : Module.Free R P\ninst\u271d\u00b9 : Module.Finite R P\ninst\u271d : IsReduced R\nf g : Module.End R M\n\u03bc : R\nh_comm : Commute f g\nn : Module.End R M := g - (algebraMap R (Module.End R M)) \u03bc\nhg : IsNilpotent n\n\u22a2 IsNilpotent ((trace R M) (f * n))", "state_after": "R : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : Module R N\ninst\u271d\u2078 : AddCommGroup P\ninst\u271d\u2077 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : Module.Finite R M\ninst\u271d\u2074 : Module.Free R N\ninst\u271d\u00b3 : Module.Finite R N\ninst\u271d\u00b2 : Module.Free R P\ninst\u271d\u00b9 : Module.Finite R P\ninst\u271d : IsReduced R\nf g : Module.End R M\n\u03bc : R\nh_comm : Commute f g\nn : Module.End R M := g - (algebraMap R (Module.End R M)) \u03bc\nhg : IsNilpotent n\n\u22a2 Commute f n"}, {"tactic": "exact h_comm.sub_right (Algebra.commute_algebraMap_right \u03bc f)", "annotated_tactic": ["exact h_comm.sub_right (<a>Algebra.commute_algebraMap_right</a> \u03bc f)", [{"full_name": "Algebra.commute_algebraMap_right", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [348, 7], "def_end_pos": [348, 31]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : Module R N\ninst\u271d\u2078 : AddCommGroup P\ninst\u271d\u2077 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : Module.Finite R M\ninst\u271d\u2074 : Module.Free R N\ninst\u271d\u00b3 : Module.Finite R N\ninst\u271d\u00b2 : Module.Free R P\ninst\u271d\u00b9 : Module.Finite R P\ninst\u271d : IsReduced R\nf g : Module.End R M\n\u03bc : R\nh_comm : Commute f g\nn : Module.End R M := g - (algebraMap R (Module.End R M)) \u03bc\nhg : IsNilpotent n\n\u22a2 Commute f n", "state_after": "no goals"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "R : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : Module R N\ninst\u271d\u2078 : AddCommGroup P\ninst\u271d\u2077 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : Module.Finite R M\ninst\u271d\u2074 : Module.Free R N\ninst\u271d\u00b3 : Module.Finite R N\ninst\u271d\u00b2 : Module.Free R P\ninst\u271d\u00b9 : Module.Finite R P\ninst\u271d : IsReduced R\nf g : Module.End R M\n\u03bc : R\nh_comm : Commute f g\nn : Module.End R M := g - (algebraMap R (Module.End R M)) \u03bc\nhg : (trace R M) (f \u2218\u2097 n) = 0\nh\u03bc : g = (algebraMap R (Module.End R M)) \u03bc + n\n\u22a2 f \u2218\u2097 (algebraMap R (M \u2192\u2097[R] M)) \u03bc = \u03bc \u2022 f", "state_after": "case h\nR : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : Module R N\ninst\u271d\u2078 : AddCommGroup P\ninst\u271d\u2077 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : Module.Finite R M\ninst\u271d\u2074 : Module.Free R N\ninst\u271d\u00b3 : Module.Finite R N\ninst\u271d\u00b2 : Module.Free R P\ninst\u271d\u00b9 : Module.Finite R P\ninst\u271d : IsReduced R\nf g : Module.End R M\n\u03bc : R\nh_comm : Commute f g\nn : Module.End R M := g - (algebraMap R (Module.End R M)) \u03bc\nhg : (trace R M) (f \u2218\u2097 n) = 0\nh\u03bc : g = (algebraMap R (Module.End R M)) \u03bc + n\nx\u271d : M\n\u22a2 (f \u2218\u2097 (algebraMap R (M \u2192\u2097[R] M)) \u03bc) x\u271d = (\u03bc \u2022 f) x\u271d"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\nR : Type u_1\ninst\u271d\u00b9\u00b3 : CommRing R\nM : Type u_2\ninst\u271d\u00b9\u00b2 : AddCommGroup M\ninst\u271d\u00b9\u00b9 : Module R M\nN : Type u_3\nP : Type u_4\ninst\u271d\u00b9\u2070 : AddCommGroup N\ninst\u271d\u2079 : Module R N\ninst\u271d\u2078 : AddCommGroup P\ninst\u271d\u2077 : Module R P\n\u03b9 : Type u_5\ninst\u271d\u2076 : Module.Free R M\ninst\u271d\u2075 : Module.Finite R M\ninst\u271d\u2074 : Module.Free R N\ninst\u271d\u00b3 : Module.Finite R N\ninst\u271d\u00b2 : Module.Free R P\ninst\u271d\u00b9 : Module.Finite R P\ninst\u271d : IsReduced R\nf g : Module.End R M\n\u03bc : R\nh_comm : Commute f g\nn : Module.End R M := g - (algebraMap R (Module.End R M)) \u03bc\nhg : (trace R M) (f \u2218\u2097 n) = 0\nh\u03bc : g = (algebraMap R (Module.End R M)) \u03bc + n\nx\u271d : M\n\u22a2 (f \u2218\u2097 (algebraMap R (M \u2192\u2097[R] M)) \u03bc) x\u271d = (\u03bc \u2022 f) x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean", "full_name": "CategoryTheory.Limits.prod.map_swap", "start": [765, 1], "end": [767, 89], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX\u271d Y\u271d A B X Y : C\nf : A \u27f6 B\ng : X \u27f6 Y\ninst\u271d : HasLimitsOfShape (Discrete WalkingPair) C\n\u22a2 map (\ud835\udfd9 X) f \u226b map g (\ud835\udfd9 B) = map g (\ud835\udfd9 A) \u226b map (\ud835\udfd9 Y) f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean", "full_name": "HasMFDerivWithinAt.mul", "start": [584, 1], "end": [587, 44], "traced_tactics": [{"tactic": "convert hp.mul' hq", "annotated_tactic": ["convert hp.mul' hq", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2076 : TopologicalSpace M\ninst\u271d\u00b9\u2075 : ChartedSpace H M\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b9 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u2070 : TopologicalSpace M'\ninst\u271d\u2079 : ChartedSpace H' M'\ninst\u271d\u2078 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I'' M''\ns : Set M\nx z : M\nF' : Type u_11\ninst\u271d\u00b9 : NormedCommRing F'\ninst\u271d : NormedAlgebra \ud835\udd5c F'\np q : M \u2192 F'\np' q' : TangentSpace I z \u2192L[\ud835\udd5c] F'\nhp : HasMFDerivWithinAt I \ud835\udcd8(\ud835\udd5c, F') p s z p'\nhq : HasMFDerivWithinAt I \ud835\udcd8(\ud835\udd5c, F') q s z q'\n\u22a2 HasMFDerivWithinAt I \ud835\udcd8(\ud835\udd5c, F') (p * q) s z (p z \u2022 q' + q z \u2022 p')", "state_after": "case h.e'_26.h.e'_6\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2076 : TopologicalSpace M\ninst\u271d\u00b9\u2075 : ChartedSpace H M\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b9 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u2070 : TopologicalSpace M'\ninst\u271d\u2079 : ChartedSpace H' M'\ninst\u271d\u2078 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I'' M''\ns : Set M\nx z : M\nF' : Type u_11\ninst\u271d\u00b9 : NormedCommRing F'\ninst\u271d : NormedAlgebra \ud835\udd5c F'\np q : M \u2192 F'\np' q' : TangentSpace I z \u2192L[\ud835\udd5c] F'\nhp : HasMFDerivWithinAt I \ud835\udcd8(\ud835\udd5c, F') p s z p'\nhq : HasMFDerivWithinAt I \ud835\udcd8(\ud835\udd5c, F') q s z q'\n\u22a2 q z \u2022 p' = p'.smulRight (q z)"}, {"tactic": "ext _", "annotated_tactic": ["ext _", []], "state_before": "case h.e'_26.h.e'_6\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2076 : TopologicalSpace M\ninst\u271d\u00b9\u2075 : ChartedSpace H M\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b9 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u2070 : TopologicalSpace M'\ninst\u271d\u2079 : ChartedSpace H' M'\ninst\u271d\u2078 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I'' M''\ns : Set M\nx z : M\nF' : Type u_11\ninst\u271d\u00b9 : NormedCommRing F'\ninst\u271d : NormedAlgebra \ud835\udd5c F'\np q : M \u2192 F'\np' q' : TangentSpace I z \u2192L[\ud835\udd5c] F'\nhp : HasMFDerivWithinAt I \ud835\udcd8(\ud835\udd5c, F') p s z p'\nhq : HasMFDerivWithinAt I \ud835\udcd8(\ud835\udd5c, F') q s z q'\n\u22a2 q z \u2022 p' = p'.smulRight (q z)", "state_after": "case h.e'_26.h.e'_6.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2076 : TopologicalSpace M\ninst\u271d\u00b9\u2075 : ChartedSpace H M\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b9 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u2070 : TopologicalSpace M'\ninst\u271d\u2079 : ChartedSpace H' M'\ninst\u271d\u2078 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I'' M''\ns : Set M\nx z : M\nF' : Type u_11\ninst\u271d\u00b9 : NormedCommRing F'\ninst\u271d : NormedAlgebra \ud835\udd5c F'\np q : M \u2192 F'\np' q' : TangentSpace I z \u2192L[\ud835\udd5c] F'\nhp : HasMFDerivWithinAt I \ud835\udcd8(\ud835\udd5c, F') p s z p'\nhq : HasMFDerivWithinAt I \ud835\udcd8(\ud835\udd5c, F') q s z q'\nx\u271d : TangentSpace I z\n\u22a2 (q z \u2022 p') x\u271d = (p'.smulRight (q z)) x\u271d"}, {"tactic": "apply mul_comm", "annotated_tactic": ["apply <a>mul_comm</a>", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "case h.e'_26.h.e'_6.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b2\u2070 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2079 : NormedAddCommGroup E\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u2077 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u2076 : TopologicalSpace M\ninst\u271d\u00b9\u2075 : ChartedSpace H M\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b9\u00b9 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b9\u2070 : TopologicalSpace M'\ninst\u271d\u2079 : ChartedSpace H' M'\ninst\u271d\u2078 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u2077 : NormedAddCommGroup E''\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u2075 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u2074 : TopologicalSpace M''\ninst\u271d\u00b3 : ChartedSpace H'' M''\ninst\u271d\u00b2 : SmoothManifoldWithCorners I'' M''\ns : Set M\nx z : M\nF' : Type u_11\ninst\u271d\u00b9 : NormedCommRing F'\ninst\u271d : NormedAlgebra \ud835\udd5c F'\np q : M \u2192 F'\np' q' : TangentSpace I z \u2192L[\ud835\udd5c] F'\nhp : HasMFDerivWithinAt I \ud835\udcd8(\ud835\udd5c, F') p s z p'\nhq : HasMFDerivWithinAt I \ud835\udcd8(\ud835\udd5c, F') q s z q'\nx\u271d : TangentSpace I z\n\u22a2 (q z \u2022 p') x\u271d = (p'.smulRight (q z)) x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "full_name": "QPF.liftpPreservation_iff_uniform", "start": [721, 1], "end": [722, 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": [707, 9], "def_end_pos": [707, 47]}, {"full_name": "QPF.suppPreservation_iff_uniform", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [699, 9], "def_end_pos": [699, 37]}]], "state_before": "F : Type u \u2192 Type u\nq : QPF F\n\u22a2 LiftpPreservation \u2194 IsUniform", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.card_singleton", "start": [793, 1], "end": [794, 76], "traced_tactics": [{"tactic": "simp only [\u2190 cons_zero, card_zero, eq_self_iff_true, zero_add, card_cons]", "annotated_tactic": ["simp only [\u2190 <a>cons_zero</a>, <a>card_zero</a>, <a>eq_self_iff_true</a>, <a>zero_add</a>, <a>card_cons</a>]", [{"full_name": "Multiset.cons_zero", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [342, 9], "def_end_pos": [342, 18]}, {"full_name": "Multiset.card_zero", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [775, 9], "def_end_pos": [775, 18]}, {"full_name": "eq_self_iff_true", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1380, 9], "def_end_pos": [1380, 25]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [476, 3], "def_end_pos": [476, 14]}, {"full_name": "Multiset.card_cons", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [788, 9], "def_end_pos": [788, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\na : \u03b1\n\u22a2 card {a} = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GroupWithZero/WithZero.lean", "full_name": "WithZero.map'_zero", "start": [127, 1], "end": [127, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Int/GCD.lean", "full_name": "Int.gcd_eq_left", "start": [320, 1], "end": [321, 90], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Defs.lean", "full_name": "Nat.pred_eq_succ_iff", "start": [204, 1], "end": [205, 48], "traced_tactics": [{"tactic": "cases n <;> constructor <;> rintro \u27e8\u27e9 <;> rfl", "annotated_tactic": ["cases n <;> constructor <;> rintro \u27e8\u27e9 <;> rfl", []], "state_before": "a b c d m n k : \u2115\np q : \u2115 \u2192 Prop\n\u22a2 n - 1 = m + 1 \u2194 n = m + 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean", "full_name": "MvPolynomial.NewtonIdentities.mem_pairs", "start": [60, 1], "end": [63, 15], "traced_tactics": [{"tactic": "simp [pairs]", "annotated_tactic": ["simp [<a>pairs</a>]", [{"full_name": "_private.Mathlib.RingTheory.MvPolynomial.NewtonIdentities.0.MvPolynomial.NewtonIdentities.pairs", "def_path": "Mathlib/RingTheory/MvPolynomial/NewtonIdentities.lean", "def_pos": [57, 13], "def_end_pos": [57, 18]}]], "state_before": "\u03c3 : Type u_1\ninst\u271d\u00b2 : Fintype \u03c3\ninst\u271d\u00b9 : DecidableEq \u03c3\nR : Type u_2\ninst\u271d : CommRing R\nk : \u2115\nt : Finset \u03c3 \u00d7 \u03c3\n\u22a2 t \u2208 MvPolynomial.NewtonIdentities.pairs \u03c3 k \u2194 t.1.card \u2264 k \u2227 (t.1.card = k \u2192 t.2 \u2208 t.1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/HNNExtension.lean", "full_name": "HNNExtension.NormalWord.prod_group_smul", "start": [490, 1], "end": [492, 47], "traced_tactics": [{"tactic": "simp [ReducedWord.prod, smul_def, mul_assoc]", "annotated_tactic": ["simp [<a>ReducedWord.prod</a>, <a>smul_def</a>, <a>mul_assoc</a>]", [{"full_name": "HNNExtension.NormalWord.ReducedWord.prod", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [209, 5], "def_end_pos": [209, 21]}, {"full_name": "Subgroup.smul_def", "def_path": "Mathlib/Algebra/Group/Subgroup/Actions.lean", "def_pos": [31, 22], "def_end_pos": [31, 30]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "G : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : \u21a5A \u2243* \u21a5B\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\ng : G\nw : NormalWord d\n\u22a2 ReducedWord.prod \u03c6 (g \u2022 w).toReducedWord = of g * ReducedWord.prod \u03c6 w.toReducedWord", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Group/Defs.lean", "full_name": "lt_or_lt_of_div_lt_div", "start": [1027, 1], "end": [1028, 50], "traced_tactics": [{"tactic": "contrapose!", "annotated_tactic": ["contrapose!", []], "state_before": "\u03b1 : Type u\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c d : \u03b1\n\u22a2 a / d < b / c \u2192 a < b \u2228 c < d", "state_after": "\u03b1 : Type u\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c d : \u03b1\n\u22a2 b \u2264 a \u2227 d \u2264 c \u2192 b / c \u2264 a / d"}, {"tactic": "exact fun h \u21a6 div_le_div'' h.1 h.2", "annotated_tactic": ["exact fun h \u21a6 <a>div_le_div''</a> h.1 h.2", [{"full_name": "div_le_div''", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [854, 9], "def_end_pos": [854, 21]}]], "state_before": "\u03b1 : Type u\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b1\ninst\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b c d : \u03b1\n\u22a2 b \u2264 a \u2227 d \u2264 c \u2192 b / c \u2264 a / d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/EqToHom.lean", "full_name": "CategoryTheory.eqToHom_iso_inv_naturality", "start": [95, 1], "end": [98, 7], "traced_tactics": [{"tactic": "simp [w]", "annotated_tactic": ["simp [w]", []], "state_before": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort u_1\nf g : \u03b2 \u2192 C\nz : (b : \u03b2) \u2192 f b \u2245 g b\nj j' : \u03b2\nw : j = j'\n\u22a2 f j = f j'", "state_after": "no goals"}, {"tactic": "simp [w]", "annotated_tactic": ["simp [w]", []], "state_before": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort u_1\nf g : \u03b2 \u2192 C\nz : (b : \u03b2) \u2192 f b \u2245 g b\nj j' : \u03b2\nw : j = j'\n\u22a2 g j = g j'", "state_after": "no goals"}, {"tactic": "cases w", "annotated_tactic": ["cases w", []], "state_before": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort u_1\nf g : \u03b2 \u2192 C\nz : (b : \u03b2) \u2192 f b \u2245 g b\nj j' : \u03b2\nw : j = j'\n\u22a2 (z j).inv \u226b eqToHom \u22ef = eqToHom \u22ef \u226b (z j').inv", "state_after": "case refl\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort u_1\nf g : \u03b2 \u2192 C\nz : (b : \u03b2) \u2192 f b \u2245 g b\nj : \u03b2\n\u22a2 (z j).inv \u226b eqToHom \u22ef = eqToHom \u22ef \u226b (z j).inv"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case refl\nC : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\n\u03b2 : Sort u_1\nf g : \u03b2 \u2192 C\nz : (b : \u03b2) \u2192 f b \u2245 g b\nj : \u03b2\n\u22a2 (z j).inv \u226b eqToHom \u22ef = eqToHom \u22ef \u226b (z j).inv", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/PowerSeries/Inverse.lean", "full_name": "PowerSeries.eq_divided_by_X_pow_order_Iff_Unit", "start": [283, 1], "end": [289, 41], "traced_tactics": [{"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "R : Type u_1\nk : Type u_2\ninst\u271d : Field k\nf : k\u27e6X\u27e7\nhf : f \u2260 0\nh : f = divided_by_X_pow_order hf\n\u22a2 IsUnit f", "state_after": "R : Type u_1\nk : Type u_2\ninst\u271d : Field k\nf : k\u27e6X\u27e7\nhf : f \u2260 0\nh : f = divided_by_X_pow_order hf\n\u22a2 IsUnit (divided_by_X_pow_order hf)"}, {"tactic": "exact isUnit_divided_by_X_pow_order hf", "annotated_tactic": ["exact <a>isUnit_divided_by_X_pow_order</a> hf", [{"full_name": "PowerSeries.isUnit_divided_by_X_pow_order", "def_path": "Mathlib/RingTheory/PowerSeries/Inverse.lean", "def_pos": [270, 9], "def_end_pos": [270, 38]}]], "state_before": "R : Type u_1\nk : Type u_2\ninst\u271d : Field k\nf : k\u27e6X\u27e7\nhf : f \u2260 0\nh : f = divided_by_X_pow_order hf\n\u22a2 IsUnit (divided_by_X_pow_order hf)", "state_after": "no goals"}, {"tactic": "have : f.order.get (order_finite_iff_ne_zero.mpr hf) = 0 := by\n  simp only [order_zero_of_unit h, PartENat.get_zero]", "annotated_tactic": ["have : f.order.get (order_finite_iff_ne_zero.mpr hf) = 0 := by\n      simp only [<a>order_zero_of_unit</a> h, <a>PartENat.get_zero</a>]", [{"full_name": "PowerSeries.order_zero_of_unit", "def_path": "Mathlib/RingTheory/PowerSeries/Order.lean", "def_pos": [324, 9], "def_end_pos": [324, 27]}, {"full_name": "PartENat.get_zero", "def_path": "Mathlib/Data/Nat/PartENat.lean", "def_pos": [203, 9], "def_end_pos": [203, 17]}]], "state_before": "R : Type u_1\nk : Type u_2\ninst\u271d : Field k\nf : k\u27e6X\u27e7\nhf : f \u2260 0\nh : IsUnit f\n\u22a2 f = divided_by_X_pow_order hf", "state_after": "R : Type u_1\nk : Type u_2\ninst\u271d : Field k\nf : k\u27e6X\u27e7\nhf : f \u2260 0\nh : IsUnit f\nthis : f.order.get \u22ef = 0\n\u22a2 f = divided_by_X_pow_order hf"}, {"tactic": "convert (self_eq_X_pow_order_mul_divided_by_X_pow_order hf).symm", "annotated_tactic": ["convert (<a>self_eq_X_pow_order_mul_divided_by_X_pow_order</a> hf).<a>symm</a>", [{"full_name": "PowerSeries.self_eq_X_pow_order_mul_divided_by_X_pow_order", "def_path": "Mathlib/RingTheory/PowerSeries/Order.lean", "def_pos": [306, 9], "def_end_pos": [306, 55]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "R : Type u_1\nk : Type u_2\ninst\u271d : Field k\nf : k\u27e6X\u27e7\nhf : f \u2260 0\nh : IsUnit f\nthis : f.order.get \u22ef = 0\n\u22a2 f = divided_by_X_pow_order hf", "state_after": "case h.e'_3\nR : Type u_1\nk : Type u_2\ninst\u271d : Field k\nf : k\u27e6X\u27e7\nhf : f \u2260 0\nh : IsUnit f\nthis : f.order.get \u22ef = 0\n\u22a2 divided_by_X_pow_order hf = X ^ f.order.get \u22ef * divided_by_X_pow_order hf"}, {"tactic": "simp only [this, pow_zero, one_mul]", "annotated_tactic": ["simp only [this, <a>pow_zero</a>, <a>one_mul</a>]", [{"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [651, 9], "def_end_pos": [651, 17]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "case h.e'_3\nR : Type u_1\nk : Type u_2\ninst\u271d : Field k\nf : k\u27e6X\u27e7\nhf : f \u2260 0\nh : IsUnit f\nthis : f.order.get \u22ef = 0\n\u22a2 divided_by_X_pow_order hf = X ^ f.order.get \u22ef * divided_by_X_pow_order hf", "state_after": "no goals"}, {"tactic": "simp only [order_zero_of_unit h, PartENat.get_zero]", "annotated_tactic": ["simp only [<a>order_zero_of_unit</a> h, <a>PartENat.get_zero</a>]", [{"full_name": "PowerSeries.order_zero_of_unit", "def_path": "Mathlib/RingTheory/PowerSeries/Order.lean", "def_pos": [324, 9], "def_end_pos": [324, 27]}, {"full_name": "PartENat.get_zero", "def_path": "Mathlib/Data/Nat/PartENat.lean", "def_pos": [203, 9], "def_end_pos": [203, 17]}]], "state_before": "R : Type u_1\nk : Type u_2\ninst\u271d : Field k\nf : k\u27e6X\u27e7\nhf : f \u2260 0\nh : IsUnit f\n\u22a2 f.order.get \u22ef = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/WithBot.lean", "full_name": "WithBot.ofDual_le_iff", "start": [1089, 1], "end": [1091, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GroupWithZero/Semiconj.lean", "full_name": "SemiconjBy.zpow_right\u2080", "start": [68, 1], "end": [70, 80], "traced_tactics": [{"tactic": "simp [h.pow_right n]", "annotated_tactic": ["simp [h.pow_right n]", []], "state_before": "\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d x\u271d y\u271d x' y' a x y : G\u2080\nh : SemiconjBy a x y\nn : \u2115\n\u22a2 SemiconjBy a (x ^ \u2191n) (y ^ \u2191n)", "state_after": "no goals"}, {"tactic": "simp only [zpow_negSucc, (h.pow_right (n + 1)).inv_right\u2080]", "annotated_tactic": ["simp only [<a>zpow_negSucc</a>, (h.pow_right (n + 1)).<a>inv_right\u2080</a>]", [{"full_name": "zpow_negSucc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1040, 9], "def_end_pos": [1040, 21]}, {"full_name": "SemiconjBy.inv_right\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Semiconj.lean", "def_pos": [45, 9], "def_end_pos": [45, 19]}]], "state_before": "\u03b1 : Type u_1\nM\u2080 : Type u_2\nG\u2080 : Type u_3\nM\u2080' : Type u_4\nG\u2080' : Type u_5\nF : Type u_6\nF' : Type u_7\ninst\u271d : GroupWithZero G\u2080\na\u271d x\u271d y\u271d x' y' a x y : G\u2080\nh : SemiconjBy a x y\nn : \u2115\n\u22a2 SemiconjBy a (x ^ Int.negSucc n) (y ^ Int.negSucc n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/BooleanRing.lean", "full_name": "BooleanRing.sup_comm", "start": [192, 1], "end": [194, 7], "traced_tactics": [{"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : BooleanRing \u03b1\ninst\u271d\u00b9 : BooleanRing \u03b2\ninst\u271d : BooleanRing \u03b3\na b : \u03b1\n\u22a2 a + b + a * b = b + a + b * a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Tropical/Basic.lean", "full_name": "Tropical.inf_eq_add", "start": [315, 1], "end": [316, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean", "full_name": "AddCircle.toCircle_apply_mk", "start": [176, 1], "end": [177, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.integral_const_of_cdf", "start": [1011, 1], "end": [1015, 64], "traced_tactics": [{"tactic": "simp only [sub_smul, \u2190 setIntegral_const]", "annotated_tactic": ["simp only [<a>sub_smul</a>, \u2190 <a>setIntegral_const</a>]", [{"full_name": "sub_smul", "def_path": "Mathlib/Algebra/Module/Defs.lean", "def_pos": [274, 9], "def_end_pos": [274, 17]}, {"full_name": "MeasureTheory.setIntegral_const", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [523, 9], "def_end_pos": [523, 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\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\na b c\u271d d : \u211d\nf g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nc : E\n\u22a2 \u222b (x : \u211d) in a..b, c \u2202\u03bc = ((\u03bc (Iic b)).toReal - (\u03bc (Iic a)).toReal) \u2022 c", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\na b c\u271d d : \u211d\nf g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nc : E\n\u22a2 \u222b (x : \u211d) in a..b, c \u2202\u03bc = \u222b (x : \u211d) in Iic b, c \u2202\u03bc - \u222b (x : \u211d) in Iic a, c \u2202\u03bc"}, {"tactic": "refine (integral_Iic_sub_Iic ?_ ?_).symm <;>\n  simp only [integrableOn_const, measure_lt_top, or_true_iff]", "annotated_tactic": ["refine (<a>integral_Iic_sub_Iic</a> ?_ ?_).<a>symm</a> <;>\n    simp only [<a>integrableOn_const</a>, <a>measure_lt_top</a>, <a>or_true_iff</a>]", [{"full_name": "intervalIntegral.integral_Iic_sub_Iic", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [989, 9], "def_end_pos": [989, 29]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}, {"full_name": "MeasureTheory.integrableOn_const", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [112, 9], "def_end_pos": [112, 27]}, {"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "def_pos": [52, 9], "def_end_pos": [52, 23]}, {"full_name": "or_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [152, 9], "def_end_pos": [152, 20]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\nE : Type u_3\nF : Type u_4\nA : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : NormedSpace \u211d E\na b c\u271d d : \u211d\nf g : \u211d \u2192 E\n\u03bc : Measure \u211d\ninst\u271d : IsFiniteMeasure \u03bc\nc : E\n\u22a2 \u222b (x : \u211d) in a..b, c \u2202\u03bc = \u222b (x : \u211d) in Iic b, c \u2202\u03bc - \u222b (x : \u211d) in Iic a, c \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/Ray.lean", "full_name": "norm_injOn_ray_left", "start": [59, 1], "end": [65, 9], "traced_tactics": [{"tactic": "rintro y hy z hz h", "annotated_tactic": ["rintro y hy z hz h", []], "state_before": "E : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : F\nhx : x \u2260 0\n\u22a2 Set.InjOn Norm.norm {y | SameRay \u211d x y}", "state_after": "E : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y\u271d : F\nhx : x \u2260 0\ny : F\nhy : y \u2208 {y | SameRay \u211d x y}\nz : F\nhz : z \u2208 {y | SameRay \u211d x y}\nh : \u2016y\u2016 = \u2016z\u2016\n\u22a2 y = z"}, {"tactic": "rcases hy.exists_nonneg_left hx with \u27e8r, hr, rfl\u27e9", "annotated_tactic": ["rcases hy.exists_nonneg_left hx with \u27e8r, hr, rfl\u27e9", []], "state_before": "E : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y\u271d : F\nhx : x \u2260 0\ny : F\nhy : y \u2208 {y | SameRay \u211d x y}\nz : F\nhz : z \u2208 {y | SameRay \u211d x y}\nh : \u2016y\u2016 = \u2016z\u2016\n\u22a2 y = z", "state_after": "case intro.intro\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : F\nhx : x \u2260 0\nz : F\nhz : z \u2208 {y | SameRay \u211d x y}\nr : \u211d\nhr : 0 \u2264 r\nhy : r \u2022 x \u2208 {y | SameRay \u211d x y}\nh : \u2016r \u2022 x\u2016 = \u2016z\u2016\n\u22a2 r \u2022 x = z"}, {"tactic": "rcases hz.exists_nonneg_left hx with \u27e8s, hs, rfl\u27e9", "annotated_tactic": ["rcases hz.exists_nonneg_left hx with \u27e8s, hs, rfl\u27e9", []], "state_before": "case intro.intro\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : F\nhx : x \u2260 0\nz : F\nhz : z \u2208 {y | SameRay \u211d x y}\nr : \u211d\nhr : 0 \u2264 r\nhy : r \u2022 x \u2208 {y | SameRay \u211d x y}\nh : \u2016r \u2022 x\u2016 = \u2016z\u2016\n\u22a2 r \u2022 x = z", "state_after": "case intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : F\nhx : x \u2260 0\nr : \u211d\nhr : 0 \u2264 r\nhy : r \u2022 x \u2208 {y | SameRay \u211d x y}\ns : \u211d\nhs : 0 \u2264 s\nhz : s \u2022 x \u2208 {y | SameRay \u211d x y}\nh : \u2016r \u2022 x\u2016 = \u2016s \u2022 x\u2016\n\u22a2 r \u2022 x = s \u2022 x"}, {"tactic": "rw [norm_smul, norm_smul, mul_left_inj' (norm_ne_zero_iff.2 hx), norm_of_nonneg hr,\n  norm_of_nonneg hs] at h", "annotated_tactic": ["rw [<a>norm_smul</a>, <a>norm_smul</a>, <a>mul_left_inj'</a> (<a>norm_ne_zero_iff</a>.2 hx), <a>norm_of_nonneg</a> hr,\n    <a>norm_of_nonneg</a> hs] at h", [{"full_name": "norm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [90, 9], "def_end_pos": [90, 18]}, {"full_name": "norm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [90, 9], "def_end_pos": [90, 18]}, {"full_name": "mul_left_inj'", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [128, 9], "def_end_pos": [128, 22]}, {"full_name": "norm_ne_zero_iff", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1530, 15], "def_end_pos": [1530, 31]}, {"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 23]}, {"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 23]}]], "state_before": "case intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : F\nhx : x \u2260 0\nr : \u211d\nhr : 0 \u2264 r\nhy : r \u2022 x \u2208 {y | SameRay \u211d x y}\ns : \u211d\nhs : 0 \u2264 s\nhz : s \u2022 x \u2208 {y | SameRay \u211d x y}\nh : \u2016r \u2022 x\u2016 = \u2016s \u2022 x\u2016\n\u22a2 r \u2022 x = s \u2022 x", "state_after": "case intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : F\nhx : x \u2260 0\nr : \u211d\nhr : 0 \u2264 r\nhy : r \u2022 x \u2208 {y | SameRay \u211d x y}\ns : \u211d\nhs : 0 \u2264 s\nhz : s \u2022 x \u2208 {y | SameRay \u211d x y}\nh : r = s\n\u22a2 r \u2022 x = s \u2022 x"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "case intro.intro.intro.intro\nE : Type u_1\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\nF : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nx y : F\nhx : x \u2260 0\nr : \u211d\nhr : 0 \u2264 r\nhy : r \u2022 x \u2208 {y | SameRay \u211d x y}\ns : \u211d\nhs : 0 \u2264 s\nhz : s \u2022 x \u2208 {y | SameRay \u211d x y}\nh : r = s\n\u22a2 r \u2022 x = s \u2022 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Function.lean", "full_name": "ConvexOn.convex_strict_epigraph", "start": [582, 1], "end": [585, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/DedekindDomain/Ideal.lean", "full_name": "Ideal.dvd_iff_le", "start": [636, 1], "end": [648, 69], "traced_tactics": [{"tactic": "by_cases hI : I = \u22a5", "annotated_tactic": ["by_cases hI : I = \u22a5", []], "state_before": "R : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDedekindDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nI J : Ideal A\nh : J \u2264 I\n\u22a2 I \u2223 J", "state_after": "case pos\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDedekindDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nI J : Ideal A\nh : J \u2264 I\nhI : I = \u22a5\n\u22a2 I \u2223 J\n\ncase neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDedekindDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nI J : Ideal A\nh : J \u2264 I\nhI : \u00acI = \u22a5\n\u22a2 I \u2223 J"}, {"tactic": "have hI' : (I : FractionalIdeal A\u2070 (FractionRing A)) \u2260 0 := coeIdeal_ne_zero.mpr hI", "annotated_tactic": ["have hI' : (I : <a>FractionalIdeal</a> A\u2070 (<a>FractionRing</a> A)) \u2260 0 := coeIdeal_ne_zero.mpr hI", [{"full_name": "FractionalIdeal", "def_path": "Mathlib/RingTheory/FractionalIdeal/Basic.lean", "def_pos": [80, 5], "def_end_pos": [80, 20]}, {"full_name": "FractionRing", "def_path": "Mathlib/RingTheory/Localization/FractionRing.lean", "def_pos": [289, 8], "def_end_pos": [289, 20]}]], "state_before": "case neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDedekindDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nI J : Ideal A\nh : J \u2264 I\nhI : \u00acI = \u22a5\n\u22a2 I \u2223 J", "state_after": "case neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDedekindDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nI J : Ideal A\nh : J \u2264 I\nhI : \u00acI = \u22a5\nhI' : \u2191I \u2260 0\n\u22a2 I \u2223 J"}, {"tactic": "have : (I : FractionalIdeal A\u2070 (FractionRing A))\u207b\u00b9 * J \u2264 1 :=\n  le_trans (mul_left_mono (\u2191I)\u207b\u00b9 ((coeIdeal_le_coeIdeal _).mpr h))\n    (le_of_eq (inv_mul_cancel hI'))", "annotated_tactic": ["have : (I : <a>FractionalIdeal</a> A\u2070 (<a>FractionRing</a> A))\u207b\u00b9 * J \u2264 1 :=\n      <a>le_trans</a> (<a>mul_left_mono</a> (\u2191I)\u207b\u00b9 ((<a>coeIdeal_le_coeIdeal</a> _).<a>mpr</a> h))\n        (<a>le_of_eq</a> (<a>inv_mul_cancel</a> hI'))", [{"full_name": "FractionalIdeal", "def_path": "Mathlib/RingTheory/FractionalIdeal/Basic.lean", "def_pos": [80, 5], "def_end_pos": [80, 20]}, {"full_name": "FractionRing", "def_path": "Mathlib/RingTheory/Localization/FractionRing.lean", "def_pos": [289, 8], "def_end_pos": [289, 20]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "FractionalIdeal.mul_left_mono", "def_path": "Mathlib/RingTheory/FractionalIdeal/Basic.lean", "def_pos": [594, 9], "def_end_pos": [594, 22]}, {"full_name": "FractionalIdeal.coeIdeal_le_coeIdeal", "def_path": "Mathlib/RingTheory/FractionalIdeal/Basic.lean", "def_pos": [289, 9], "def_end_pos": [289, 29]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}, {"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "inv_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/NeZero.lean", "def_pos": [55, 9], "def_end_pos": [55, 23]}]], "state_before": "case neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDedekindDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nI J : Ideal A\nh : J \u2264 I\nhI : \u00acI = \u22a5\nhI' : \u2191I \u2260 0\n\u22a2 I \u2223 J", "state_after": "case neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDedekindDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nI J : Ideal A\nh : J \u2264 I\nhI : \u00acI = \u22a5\nhI' : \u2191I \u2260 0\nthis : (\u2191I)\u207b\u00b9 * \u2191J \u2264 1\n\u22a2 I \u2223 J"}, {"tactic": "obtain \u27e8H, hH\u27e9 := le_one_iff_exists_coeIdeal.mp this", "annotated_tactic": ["obtain \u27e8H, hH\u27e9 := le_one_iff_exists_coeIdeal.mp this", []], "state_before": "case neg\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDedekindDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nI J : Ideal A\nh : J \u2264 I\nhI : \u00acI = \u22a5\nhI' : \u2191I \u2260 0\nthis : (\u2191I)\u207b\u00b9 * \u2191J \u2264 1\n\u22a2 I \u2223 J", "state_after": "case neg.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDedekindDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nI J : Ideal A\nh : J \u2264 I\nhI : \u00acI = \u22a5\nhI' : \u2191I \u2260 0\nthis : (\u2191I)\u207b\u00b9 * \u2191J \u2264 1\nH : Ideal A\nhH : \u2191H = (\u2191I)\u207b\u00b9 * \u2191J\n\u22a2 I \u2223 J"}, {"tactic": "use H", "annotated_tactic": ["use H", []], "state_before": "case neg.intro\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDedekindDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nI J : Ideal A\nh : J \u2264 I\nhI : \u00acI = \u22a5\nhI' : \u2191I \u2260 0\nthis : (\u2191I)\u207b\u00b9 * \u2191J \u2264 1\nH : Ideal A\nhH : \u2191H = (\u2191I)\u207b\u00b9 * \u2191J\n\u22a2 I \u2223 J", "state_after": "case h\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDedekindDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nI J : Ideal A\nh : J \u2264 I\nhI : \u00acI = \u22a5\nhI' : \u2191I \u2260 0\nthis : (\u2191I)\u207b\u00b9 * \u2191J \u2264 1\nH : Ideal A\nhH : \u2191H = (\u2191I)\u207b\u00b9 * \u2191J\n\u22a2 J = I * H"}, {"tactic": "refine coeIdeal_injective (show (J : FractionalIdeal A\u2070 (FractionRing A)) = \u2191(I * H) from ?_)", "annotated_tactic": ["refine <a>coeIdeal_injective</a> (show (J : <a>FractionalIdeal</a> A\u2070 (<a>FractionRing</a> A)) = \u2191(I * H) from ?_)", [{"full_name": "FractionalIdeal.coeIdeal_injective", "def_path": "Mathlib/RingTheory/FractionalIdeal/Operations.lean", "def_pos": [331, 9], "def_end_pos": [331, 27]}, {"full_name": "FractionalIdeal", "def_path": "Mathlib/RingTheory/FractionalIdeal/Basic.lean", "def_pos": [80, 5], "def_end_pos": [80, 20]}, {"full_name": "FractionRing", "def_path": "Mathlib/RingTheory/Localization/FractionRing.lean", "def_pos": [289, 8], "def_end_pos": [289, 20]}]], "state_before": "case h\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDedekindDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nI J : Ideal A\nh : J \u2264 I\nhI : \u00acI = \u22a5\nhI' : \u2191I \u2260 0\nthis : (\u2191I)\u207b\u00b9 * \u2191J \u2264 1\nH : Ideal A\nhH : \u2191H = (\u2191I)\u207b\u00b9 * \u2191J\n\u22a2 J = I * H", "state_after": "case h\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDedekindDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nI J : Ideal A\nh : J \u2264 I\nhI : \u00acI = \u22a5\nhI' : \u2191I \u2260 0\nthis : (\u2191I)\u207b\u00b9 * \u2191J \u2264 1\nH : Ideal A\nhH : \u2191H = (\u2191I)\u207b\u00b9 * \u2191J\n\u22a2 \u2191J = \u2191(I * H)"}, {"tactic": "rw [coeIdeal_mul, hH, \u2190 mul_assoc, mul_inv_cancel hI', one_mul]", "annotated_tactic": ["rw [<a>coeIdeal_mul</a>, hH, \u2190 <a>mul_assoc</a>, <a>mul_inv_cancel</a> hI', <a>one_mul</a>]", [{"full_name": "FractionalIdeal.coeIdeal_mul", "def_path": "Mathlib/RingTheory/FractionalIdeal/Basic.lean", "def_pos": [589, 9], "def_end_pos": [589, 21]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "mul_inv_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [223, 15], "def_end_pos": [223, 29]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "case h\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDedekindDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nI J : Ideal A\nh : J \u2264 I\nhI : \u00acI = \u22a5\nhI' : \u2191I \u2260 0\nthis : (\u2191I)\u207b\u00b9 * \u2191J \u2264 1\nH : Ideal A\nhH : \u2191H = (\u2191I)\u207b\u00b9 * \u2191J\n\u22a2 \u2191J = \u2191(I * H)", "state_after": "no goals"}, {"tactic": "have hJ : J = \u22a5 := by rwa [hI, \u2190 eq_bot_iff] at h", "annotated_tactic": ["have hJ : J = \u22a5 := by rwa [hI, \u2190 <a>eq_bot_iff</a>] at h", [{"full_name": "eq_bot_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [331, 9], "def_end_pos": [331, 19]}]], "state_before": "case pos\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDedekindDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nI J : Ideal A\nh : J \u2264 I\nhI : I = \u22a5\n\u22a2 I \u2223 J", "state_after": "case pos\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDedekindDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nI J : Ideal A\nh : J \u2264 I\nhI : I = \u22a5\nhJ : J = \u22a5\n\u22a2 I \u2223 J"}, {"tactic": "rw [hI, hJ]", "annotated_tactic": ["rw [hI, hJ]", []], "state_before": "case pos\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDedekindDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nI J : Ideal A\nh : J \u2264 I\nhI : I = \u22a5\nhJ : J = \u22a5\n\u22a2 I \u2223 J", "state_after": "no goals"}, {"tactic": "rwa [hI, \u2190 eq_bot_iff] at h", "annotated_tactic": ["rwa [hI, \u2190 <a>eq_bot_iff</a>] at h", [{"full_name": "eq_bot_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [331, 9], "def_end_pos": [331, 19]}]], "state_before": "R : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : IsDedekindDomain A\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\nI J : Ideal A\nh : J \u2264 I\nhI : I = \u22a5\n\u22a2 J = \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/Completion.lean", "full_name": "UniformSpace.Completion.map\u2082_coe_coe", "start": [667, 1], "end": [669, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.Tendsto.eventually_forall_ge_atTop", "start": [264, 1], "end": [267, 101], "traced_tactics": [{"tactic": "rw [\u2190 Filter.eventually_forall_ge_atTop] at h_evtl", "annotated_tactic": ["rw [\u2190 <a>Filter.eventually_forall_ge_atTop</a>] at h_evtl", [{"full_name": "Filter.eventually_forall_ge_atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [252, 9], "def_end_pos": [252, 35]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : Preorder \u03b2\nl : Filter \u03b1\np : \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nhf : Tendsto f l atTop\nh_evtl : \u2200\u1da0 (x : \u03b2) in atTop, p x\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2200 (y : \u03b2), f x \u2264 y \u2192 p y", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : Preorder \u03b2\nl : Filter \u03b1\np : \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nhf : Tendsto f l atTop\nh_evtl : \u2200\u1da0 (x : \u03b2) in atTop, \u2200 (y : \u03b2), x \u2264 y \u2192 p y\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2200 (y : \u03b2), f x \u2264 y \u2192 p y"}, {"tactic": "exact (h_evtl.comap f).filter_mono hf.le_comap", "annotated_tactic": ["exact (h_evtl.comap f).<a>filter_mono</a> hf.le_comap", [{"full_name": "Filter.Eventually.filter_mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1111, 9], "def_end_pos": [1111, 31]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : Preorder \u03b2\nl : Filter \u03b1\np : \u03b2 \u2192 Prop\nf : \u03b1 \u2192 \u03b2\nhf : Tendsto f l atTop\nh_evtl : \u2200\u1da0 (x : \u03b2) in atTop, \u2200 (y : \u03b2), x \u2264 y \u2192 p y\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2200 (y : \u03b2), f x \u2264 y \u2192 p y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/InitialSeg.lean", "full_name": "PrincipalSeg.ofIsEmpty_top", "start": [443, 1], "end": [445, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/List/Lemmas.lean", "full_name": "List.singleton_disjoint", "start": [870, 14], "end": [870, 87], "traced_tactics": [{"tactic": "simp [Disjoint]", "annotated_tactic": ["simp [<a>Disjoint</a>]", [{"full_name": "List.Disjoint", "def_path": ".lake/packages/batteries/Batteries/Data/List/Basic.lean", "def_pos": [773, 5], "def_end_pos": [773, 13]}]], "state_before": "\u03b1\u271d : Type u_1\na : \u03b1\u271d\nl : List \u03b1\u271d\n\u22a2 [a].Disjoint l \u2194 \u00aca \u2208 l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.RightInvOn.extendDomain", "start": [1953, 1], "end": [1955, 87], "traced_tactics": [{"tactic": "rintro _ \u27e8a, ha, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8a, ha, 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\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng g\u2081 g\u2082 : Perm \u03b1\ns t : Set \u03b1\nh : RightInvOn (\u21d1g\u2081) (\u21d1g\u2082) t\n\u22a2 RightInvOn (\u21d1(g\u2081.extendDomain f)) (\u21d1(g\u2082.extendDomain f)) (Subtype.val \u2218 \u21d1f '' t)", "state_after": "case 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\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng g\u2081 g\u2082 : Perm \u03b1\ns t : Set \u03b1\nh : RightInvOn (\u21d1g\u2081) (\u21d1g\u2082) t\na : \u03b1\nha : a \u2208 t\n\u22a2 (g\u2082.extendDomain f) ((g\u2081.extendDomain f) ((Subtype.val \u2218 \u21d1f) a)) = (Subtype.val \u2218 \u21d1f) a"}, {"tactic": "simp_rw [Function.comp_apply, extendDomain_apply_image, h ha]", "annotated_tactic": ["simp_rw [<a>Function.comp_apply</a>, <a>extendDomain_apply_image</a>, h ha]", [{"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}, {"full_name": "Equiv.Perm.extendDomain_apply_image", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1531, 9], "def_end_pos": [1531, 38]}]], "state_before": "case 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\np : \u03b2 \u2192 Prop\ninst\u271d : DecidablePred p\nf : \u03b1 \u2243 Subtype p\ng g\u2081 g\u2082 : Perm \u03b1\ns t : Set \u03b1\nh : RightInvOn (\u21d1g\u2081) (\u21d1g\u2082) t\na : \u03b1\nha : a \u2208 t\n\u22a2 (g\u2082.extendDomain f) ((g\u2081.extendDomain f) ((Subtype.val \u2218 \u21d1f) a)) = (Subtype.val \u2218 \u21d1f) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.succAbove_right_injective", "start": [1382, 1], "end": [1392, 31], "traced_tactics": [{"tactic": "rintro i j hij", "annotated_tactic": ["rintro i j hij", []], "state_before": "n m : \u2115\np : Fin (n + 1)\ni j : Fin n\n\u22a2 Injective p.succAbove", "state_after": "n m : \u2115\np : Fin (n + 1)\ni\u271d j\u271d i j : Fin n\nhij : p.succAbove i = p.succAbove j\n\u22a2 i = j"}, {"tactic": "unfold succAbove at hij", "annotated_tactic": ["unfold <a>succAbove</a> at hij", [{"full_name": "Fin.succAbove", "def_path": "Mathlib/Data/Fin/Basic.lean", "def_pos": [1298, 5], "def_end_pos": [1298, 14]}]], "state_before": "n m : \u2115\np : Fin (n + 1)\ni\u271d j\u271d i j : Fin n\nhij : p.succAbove i = p.succAbove j\n\u22a2 i = j", "state_after": "n m : \u2115\np : Fin (n + 1)\ni\u271d j\u271d i j : Fin n\nhij : (if i.castSucc < p then i.castSucc else i.succ) = if j.castSucc < p then j.castSucc else j.succ\n\u22a2 i = j"}, {"tactic": "split_ifs at hij with hi hj hj", "annotated_tactic": ["split_ifs at hij with hi hj hj", []], "state_before": "n m : \u2115\np : Fin (n + 1)\ni\u271d j\u271d i j : Fin n\nhij : (if i.castSucc < p then i.castSucc else i.succ) = if j.castSucc < p then j.castSucc else j.succ\n\u22a2 i = j", "state_after": "case pos\nn m : \u2115\np : Fin (n + 1)\ni\u271d j\u271d i j : Fin n\nhi : i.castSucc < p\nhj : j.castSucc < p\nhij : i.castSucc = j.castSucc\n\u22a2 i = j\n\ncase neg\nn m : \u2115\np : Fin (n + 1)\ni\u271d j\u271d i j : Fin n\nhi : i.castSucc < p\nhj : \u00acj.castSucc < p\nhij : i.castSucc = j.succ\n\u22a2 i = j\n\ncase pos\nn m : \u2115\np : Fin (n + 1)\ni\u271d j\u271d i j : Fin n\nhi : \u00aci.castSucc < p\nhj : j.castSucc < p\nhij : i.succ = j.castSucc\n\u22a2 i = j\n\ncase neg\nn m : \u2115\np : Fin (n + 1)\ni\u271d j\u271d i j : Fin n\nhi : \u00aci.castSucc < p\nhj : \u00acj.castSucc < p\nhij : i.succ = j.succ\n\u22a2 i = j"}, {"tactic": "exact castSucc_injective _ hij", "annotated_tactic": ["exact <a>castSucc_injective</a> _ hij", [{"full_name": "Fin.castSucc_injective", "def_path": "Mathlib/Data/Fin/Basic.lean", "def_pos": [737, 7], "def_end_pos": [737, 25]}]], "state_before": "case pos\nn m : \u2115\np : Fin (n + 1)\ni\u271d j\u271d i j : Fin n\nhi : i.castSucc < p\nhj : j.castSucc < p\nhij : i.castSucc = j.castSucc\n\u22a2 i = j", "state_after": "no goals"}, {"tactic": "rw [hij] at hi", "annotated_tactic": ["rw [hij] at hi", []], "state_before": "case neg\nn m : \u2115\np : Fin (n + 1)\ni\u271d j\u271d i j : Fin n\nhi : i.castSucc < p\nhj : \u00acj.castSucc < p\nhij : i.castSucc = j.succ\n\u22a2 i = j", "state_after": "case neg\nn m : \u2115\np : Fin (n + 1)\ni\u271d j\u271d i j : Fin n\nhi : j.succ < p\nhj : \u00acj.castSucc < p\nhij : i.castSucc = j.succ\n\u22a2 i = j"}, {"tactic": "cases hj $ Nat.lt_trans j.castSucc_lt_succ hi", "annotated_tactic": ["cases hj $ <a>Nat.lt_trans</a> j.castSucc_lt_succ hi", [{"full_name": "Nat.lt_trans", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1681, 19], "def_end_pos": [1681, 31]}]], "state_before": "case neg\nn m : \u2115\np : Fin (n + 1)\ni\u271d j\u271d i j : Fin n\nhi : j.succ < p\nhj : \u00acj.castSucc < p\nhij : i.castSucc = j.succ\n\u22a2 i = j", "state_after": "no goals"}, {"tactic": "rw [\u2190 hij] at hj", "annotated_tactic": ["rw [\u2190 hij] at hj", []], "state_before": "case pos\nn m : \u2115\np : Fin (n + 1)\ni\u271d j\u271d i j : Fin n\nhi : \u00aci.castSucc < p\nhj : j.castSucc < p\nhij : i.succ = j.castSucc\n\u22a2 i = j", "state_after": "case pos\nn m : \u2115\np : Fin (n + 1)\ni\u271d j\u271d i j : Fin n\nhi : \u00aci.castSucc < p\nhj : i.succ < p\nhij : i.succ = j.castSucc\n\u22a2 i = j"}, {"tactic": "cases hi $ Nat.lt_trans i.castSucc_lt_succ hj", "annotated_tactic": ["cases hi $ <a>Nat.lt_trans</a> i.castSucc_lt_succ hj", [{"full_name": "Nat.lt_trans", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1681, 19], "def_end_pos": [1681, 31]}]], "state_before": "case pos\nn m : \u2115\np : Fin (n + 1)\ni\u271d j\u271d i j : Fin n\nhi : \u00aci.castSucc < p\nhj : i.succ < p\nhij : i.succ = j.castSucc\n\u22a2 i = j", "state_after": "no goals"}, {"tactic": "exact succ_injective _ hij", "annotated_tactic": ["exact <a>succ_injective</a> _ hij", [{"full_name": "Fin.succ_injective", "def_path": "Mathlib/Data/Fin/Basic.lean", "def_pos": [646, 7], "def_end_pos": [646, 21]}]], "state_before": "case neg\nn m : \u2115\np : Fin (n + 1)\ni\u271d j\u271d i j : Fin n\nhi : \u00aci.castSucc < p\nhj : \u00acj.castSucc < p\nhij : i.succ = j.succ\n\u22a2 i = j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Interval/Set/Group.lean", "full_name": "Set.sub_mem_Ioc_iff_right", "start": [129, 1], "end": [130, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/FiberBundle/Basic.lean", "full_name": "FiberBundleCore.localTrivAt_def", "start": [606, 1], "end": [607, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Archimedean.lean", "full_name": "Int.subgroup_cyclic", "start": [98, 1], "end": [101, 64], "traced_tactics": [{"tactic": "simp [this]", "annotated_tactic": ["simp [this]", []], "state_before": "G : Type u_1\ninst\u271d\u00b9 : LinearOrderedAddCommGroup G\ninst\u271d : Archimedean G\nH : AddSubgroup \u2124\nthis : Ioo 0 1 = \u2205\n\u22a2 Disjoint (\u2191H) (Ioo 0 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.powerset_mono", "start": [2170, 1], "end": [2171, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Relation.lean", "full_name": "Relation.transGen_reflGen", "start": [636, 1], "end": [643, 48], "traced_tactics": [{"tactic": "ext x y", "annotated_tactic": ["ext x y", []], "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\u03b6 : Type u_6\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b c d : \u03b1\n\u22a2 TransGen (ReflGen r) = ReflTransGen r", "state_after": "case h.h.a\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\u03b6 : Type u_6\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b c d x y : \u03b1\n\u22a2 TransGen (ReflGen r) x y \u2194 ReflTransGen r x y"}, {"tactic": "refine \u27e8fun h \u21a6 ?_, fun h \u21a6 ?_\u27e9", "annotated_tactic": ["refine \u27e8fun h \u21a6 ?_, fun h \u21a6 ?_\u27e9", []], "state_before": "case h.h.a\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\u03b6 : Type u_6\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b c d x y : \u03b1\n\u22a2 TransGen (ReflGen r) x y \u2194 ReflTransGen r x y", "state_after": "case h.h.a.refine_1\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\u03b6 : Type u_6\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b c d x y : \u03b1\nh : TransGen (ReflGen r) x y\n\u22a2 ReflTransGen r x y\n\ncase h.h.a.refine_2\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\u03b6 : Type u_6\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b c d x y : \u03b1\nh : ReflTransGen r x y\n\u22a2 TransGen (ReflGen r) x y"}, {"tactic": "simpa [reflTransGen_idem]\n  using TransGen.to_reflTransGen <| TransGen.mono (fun _ _ \u21a6 ReflGen.to_reflTransGen) h", "annotated_tactic": ["simpa [<a>reflTransGen_idem</a>]\n      using <a>TransGen.to_reflTransGen</a> <| <a>TransGen.mono</a> (fun _ _ \u21a6 <a>ReflGen.to_reflTransGen</a>) h", [{"full_name": "Relation.reflTransGen_idem", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [607, 9], "def_end_pos": [607, 26]}, {"full_name": "Relation.TransGen.to_reflTransGen", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [376, 9], "def_end_pos": [376, 24]}, {"full_name": "Relation.TransGen.mono", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [534, 9], "def_end_pos": [534, 22]}, {"full_name": "Relation.ReflGen.to_reflTransGen", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [278, 9], "def_end_pos": [278, 24]}]], "state_before": "case h.h.a.refine_1\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\u03b6 : Type u_6\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b c d x y : \u03b1\nh : TransGen (ReflGen r) x y\n\u22a2 ReflTransGen r x y", "state_after": "no goals"}, {"tactic": "obtain (rfl | h) := reflTransGen_iff_eq_or_transGen.mp h", "annotated_tactic": ["obtain (rfl | h) := reflTransGen_iff_eq_or_transGen.mp h", []], "state_before": "case h.h.a.refine_2\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\u03b6 : Type u_6\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b c d x y : \u03b1\nh : ReflTransGen r x y\n\u22a2 TransGen (ReflGen r) x y", "state_after": "case h.h.a.refine_2.inl\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\u03b6 : Type u_6\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b c d y : \u03b1\nh : ReflTransGen r y y\n\u22a2 TransGen (ReflGen r) y y\n\ncase h.h.a.refine_2.inr\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\u03b6 : Type u_6\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b c d x y : \u03b1\nh\u271d : ReflTransGen r x y\nh : TransGen r x y\n\u22a2 TransGen (ReflGen r) x y"}, {"tactic": "exact .single .refl", "annotated_tactic": ["exact .single .refl", []], "state_before": "case h.h.a.refine_2.inl\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\u03b6 : Type u_6\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b c d y : \u03b1\nh : ReflTransGen r y y\n\u22a2 TransGen (ReflGen r) y y", "state_after": "no goals"}, {"tactic": "exact TransGen.mono (fun _ _ \u21a6 .single) h", "annotated_tactic": ["exact <a>TransGen.mono</a> (fun _ _ \u21a6 .single) h", [{"full_name": "Relation.TransGen.mono", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [534, 9], "def_end_pos": [534, 22]}]], "state_before": "case h.h.a.refine_2.inr\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\u03b6 : Type u_6\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b c d x y : \u03b1\nh\u271d : ReflTransGen r x y\nh : TransGen r x y\n\u22a2 TransGen (ReflGen r) x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean", "full_name": "AlgebraicTopology.DoldKan.identity_N\u2082", "start": [254, 1], "end": [260, 81], "traced_tactics": [{"tactic": "ext P : 2", "annotated_tactic": ["ext P : 2", []], "state_before": "C : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\n\u22a2 (\ud835\udfd9 N\u2082 \u25eb N\u2082\u0393\u2082.inv) \u226b (N\u2082.associator \u0393\u2082 N\u2082).inv \u226b \u0393\u2082N\u2082.natTrans \u25eb \ud835\udfd9 N\u2082 = \ud835\udfd9 N\u2082", "state_after": "case w.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\n\u22a2 ((\ud835\udfd9 N\u2082 \u25eb N\u2082\u0393\u2082.inv) \u226b (N\u2082.associator \u0393\u2082 N\u2082).inv \u226b \u0393\u2082N\u2082.natTrans \u25eb \ud835\udfd9 N\u2082).app P = (\ud835\udfd9 N\u2082).app P"}, {"tactic": "dsimp only [NatTrans.comp_app, NatTrans.hcomp_app, Functor.comp_map, Functor.associator,\n  NatTrans.id_app, Functor.comp_obj]", "annotated_tactic": ["dsimp only [<a>NatTrans.comp_app</a>, <a>NatTrans.hcomp_app</a>, <a>Functor.comp_map</a>, <a>Functor.associator</a>,\n    <a>NatTrans.id_app</a>, <a>Functor.comp_obj</a>]", [{"full_name": "CategoryTheory.NatTrans.comp_app", "def_path": "Mathlib/CategoryTheory/Functor/Category.lean", "def_pos": [76, 9], "def_end_pos": [76, 17]}, {"full_name": "CategoryTheory.NatTrans.hcomp_app", "def_path": "Mathlib/CategoryTheory/Functor/Category.lean", "def_pos": [109, 3], "def_end_pos": [109, 8]}, {"full_name": "CategoryTheory.Functor.comp_map", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 17]}, {"full_name": "CategoryTheory.Functor.associator", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [280, 5], "def_end_pos": [280, 15]}, {"full_name": "CategoryTheory.NatTrans.id_app", "def_path": "Mathlib/CategoryTheory/Functor/Category.lean", "def_pos": [72, 9], "def_end_pos": [72, 15]}, {"full_name": "CategoryTheory.Functor.comp_obj", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [110, 9], "def_end_pos": [110, 12]}]], "state_before": "case w.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\n\u22a2 ((\ud835\udfd9 N\u2082 \u25eb N\u2082\u0393\u2082.inv) \u226b (N\u2082.associator \u0393\u2082 N\u2082).inv \u226b \u0393\u2082N\u2082.natTrans \u25eb \ud835\udfd9 N\u2082).app P = (\ud835\udfd9 N\u2082).app P", "state_after": "case w.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\n\u22a2 (N\u2082\u0393\u2082.inv.app (N\u2082.obj P) \u226b N\u2082.map (\u0393\u2082.map (\ud835\udfd9 (N\u2082.obj P)))) \u226b\n      \ud835\udfd9 (N\u2082.obj (\u0393\u2082.obj (N\u2082.obj P))) \u226b \ud835\udfd9 (N\u2082.obj (\u0393\u2082.obj (N\u2082.obj P))) \u226b N\u2082.map (\u0393\u2082N\u2082.natTrans.app P) =\n    \ud835\udfd9 (N\u2082.obj P)"}, {"tactic": "rw [\u0393\u2082.map_id, N\u2082.map_id, comp_id, id_comp, id_comp, identity_N\u2082_objectwise P]", "annotated_tactic": ["rw [\u0393\u2082.map_id, N\u2082.map_id, <a>comp_id</a>, <a>id_comp</a>, <a>id_comp</a>, <a>identity_N\u2082_objectwise</a> P]", [{"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [179, 3], "def_end_pos": [179, 10]}, {"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [177, 3], "def_end_pos": [177, 10]}, {"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [177, 3], "def_end_pos": [177, 10]}, {"full_name": "AlgebraicTopology.DoldKan.identity_N\u2082_objectwise", "def_path": "Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean", "def_pos": [228, 9], "def_end_pos": [228, 31]}]], "state_before": "case w.h\nC : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\n\u22a2 (N\u2082\u0393\u2082.inv.app (N\u2082.obj P) \u226b N\u2082.map (\u0393\u2082.map (\ud835\udfd9 (N\u2082.obj P)))) \u226b\n      \ud835\udfd9 (N\u2082.obj (\u0393\u2082.obj (N\u2082.obj P))) \u226b \ud835\udfd9 (N\u2082.obj (\u0393\u2082.obj (N\u2082.obj P))) \u226b N\u2082.map (\u0393\u2082N\u2082.natTrans.app P) =\n    \ud835\udfd9 (N\u2082.obj P)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Group/Multiset.lean", "full_name": "Multiset.prod_dvd_prod_of_dvd", "start": [275, 1], "end": [280, 34], "traced_tactics": [{"tactic": "apply Multiset.induction_on' S", "annotated_tactic": ["apply <a>Multiset.induction_on'</a> S", [{"full_name": "Multiset.induction_on'", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [457, 9], "def_end_pos": [457, 22]}]], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b2\nS : Multiset \u03b1\ng1 g2 : \u03b1 \u2192 \u03b2\nh : \u2200 a \u2208 S, g1 a \u2223 g2 a\n\u22a2 (map g1 S).prod \u2223 (map g2 S).prod", "state_after": "case h\u2081\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b2\nS : Multiset \u03b1\ng1 g2 : \u03b1 \u2192 \u03b2\nh : \u2200 a \u2208 S, g1 a \u2223 g2 a\n\u22a2 (map g1 0).prod \u2223 (map g2 0).prod\n\ncase h\u2082\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b2\nS : Multiset \u03b1\ng1 g2 : \u03b1 \u2192 \u03b2\nh : \u2200 a \u2208 S, g1 a \u2223 g2 a\n\u22a2 \u2200 {a : \u03b1} {s : Multiset \u03b1},\n    a \u2208 S \u2192 s \u2286 S \u2192 (map g1 s).prod \u2223 (map g2 s).prod \u2192 (map g1 (insert a s)).prod \u2223 (map g2 (insert a s)).prod"}, {"tactic": "intro a T haS _ IH", "annotated_tactic": ["intro a T haS _ IH", []], "state_before": "case h\u2082\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b2\nS : Multiset \u03b1\ng1 g2 : \u03b1 \u2192 \u03b2\nh : \u2200 a \u2208 S, g1 a \u2223 g2 a\n\u22a2 \u2200 {a : \u03b1} {s : Multiset \u03b1},\n    a \u2208 S \u2192 s \u2286 S \u2192 (map g1 s).prod \u2223 (map g2 s).prod \u2192 (map g1 (insert a s)).prod \u2223 (map g2 (insert a s)).prod", "state_after": "case h\u2082\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b2\nS : Multiset \u03b1\ng1 g2 : \u03b1 \u2192 \u03b2\nh : \u2200 a \u2208 S, g1 a \u2223 g2 a\na : \u03b1\nT : Multiset \u03b1\nhaS : a \u2208 S\na\u271d : T \u2286 S\nIH : (map g1 T).prod \u2223 (map g2 T).prod\n\u22a2 (map g1 (insert a T)).prod \u2223 (map g2 (insert a T)).prod"}, {"tactic": "simp [mul_dvd_mul (h a haS) IH]", "annotated_tactic": ["simp [<a>mul_dvd_mul</a> (h a haS) IH]", [{"full_name": "mul_dvd_mul", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [217, 9], "def_end_pos": [217, 20]}]], "state_before": "case h\u2082\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b2\nS : Multiset \u03b1\ng1 g2 : \u03b1 \u2192 \u03b2\nh : \u2200 a \u2208 S, g1 a \u2223 g2 a\na : \u03b1\nT : Multiset \u03b1\nhaS : a \u2208 S\na\u271d : T \u2286 S\nIH : (map g1 T).prod \u2223 (map g2 T).prod\n\u22a2 (map g1 (insert a T)).prod \u2223 (map g2 (insert a T)).prod", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\u2081\nF : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : CommMonoid \u03b2\nS : Multiset \u03b1\ng1 g2 : \u03b1 \u2192 \u03b2\nh : \u2200 a \u2208 S, g1 a \u2223 g2 a\n\u22a2 (map g1 0).prod \u2223 (map g2 0).prod", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.iInter_sigma", "start": [751, 1], "end": [752, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monoidal/Bimod.lean", "full_name": "Bimod.whisker_exchange_bimod", "start": [985, 1], "end": [996, 29], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nX Y Z : Mon_ C\nM N : Bimod X Y\nP Q : Bimod Y Z\nf : M \u27f6 N\ng : P \u27f6 Q\n\u22a2 M.whiskerLeft g \u226b whiskerRight f Q = whiskerRight f P \u226b N.whiskerLeft g", "state_after": "case h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nX Y Z : Mon_ C\nM N : Bimod X Y\nP Q : Bimod Y Z\nf : M \u27f6 N\ng : P \u27f6 Q\n\u22a2 (M.whiskerLeft g \u226b whiskerRight f Q).hom = (whiskerRight f P \u226b N.whiskerLeft g).hom"}, {"tactic": "apply coequalizer.hom_ext", "annotated_tactic": ["apply <a>coequalizer.hom_ext</a>", [{"full_name": "CategoryTheory.Limits.coequalizer.hom_ext", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [1019, 9], "def_end_pos": [1019, 28]}]], "state_before": "case h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nX Y Z : Mon_ C\nM N : Bimod X Y\nP Q : Bimod Y Z\nf : M \u27f6 N\ng : P \u27f6 Q\n\u22a2 (M.whiskerLeft g \u226b whiskerRight f Q).hom = (whiskerRight f P \u226b N.whiskerLeft g).hom", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nX Y Z : Mon_ C\nM N : Bimod X Y\nP Q : Bimod Y Z\nf : M \u27f6 N\ng : P \u27f6 Q\n\u22a2 coequalizer.\u03c0 (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) \u226b (M.whiskerLeft g \u226b whiskerRight f Q).hom =\n    coequalizer.\u03c0 (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) \u226b (whiskerRight f P \u226b N.whiskerLeft g).hom"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nX Y Z : Mon_ C\nM N : Bimod X Y\nP Q : Bimod Y Z\nf : M \u27f6 N\ng : P \u27f6 Q\n\u22a2 coequalizer.\u03c0 (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) \u226b (M.whiskerLeft g \u226b whiskerRight f Q).hom =\n    coequalizer.\u03c0 (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) \u226b (whiskerRight f P \u226b N.whiskerLeft g).hom", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nX Y Z : Mon_ C\nM N : Bimod X Y\nP Q : Bimod Y Z\nf : M \u27f6 N\ng : P \u27f6 Q\n\u22a2 coequalizer.\u03c0 (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) \u226b\n      colimMap\n          (parallelPairHom (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) (M.actRight \u25b7 Q.X)\n            ((\u03b1_ M.X Y.X Q.X).hom \u226b M.X \u25c1 Q.actLeft) ((M.X \u2297 Y.X) \u25c1 g.hom) (M.X \u25c1 g.hom) \u22ef \u22ef) \u226b\n        colimMap\n          (parallelPairHom (M.actRight \u25b7 Q.X) ((\u03b1_ M.X Y.X Q.X).hom \u226b M.X \u25c1 Q.actLeft) (N.actRight \u25b7 Q.X)\n            ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft) (f.hom \u25b7 Y.X \u25b7 Q.X) (f.hom \u25b7 Q.X) \u22ef \u22ef) =\n    coequalizer.\u03c0 (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) \u226b\n      colimMap\n          (parallelPairHom (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) (N.actRight \u25b7 P.X)\n            ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef) \u226b\n        colimMap\n          (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N.actRight \u25b7 Q.X)\n            ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft) ((N.X \u2297 Y.X) \u25c1 g.hom) (N.X \u25c1 g.hom) \u22ef \u22ef)"}, {"tactic": "slice_lhs 1 2 => rw [\u03b9_colimMap, parallelPairHom_app_one]", "annotated_tactic": ["slice_lhs 1 2 => rw [<a>\u03b9_colimMap</a>, <a>parallelPairHom_app_one</a>]", [{"full_name": "CategoryTheory.Limits.\u03b9_colimMap", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [805, 9], "def_end_pos": [805, 19]}, {"full_name": "CategoryTheory.Limits.parallelPairHom_app_one", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [272, 9], "def_end_pos": [272, 32]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nX Y Z : Mon_ C\nM N : Bimod X Y\nP Q : Bimod Y Z\nf : M \u27f6 N\ng : P \u27f6 Q\n\u22a2 coequalizer.\u03c0 (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) \u226b\n      colimMap\n          (parallelPairHom (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) (M.actRight \u25b7 Q.X)\n            ((\u03b1_ M.X Y.X Q.X).hom \u226b M.X \u25c1 Q.actLeft) ((M.X \u2297 Y.X) \u25c1 g.hom) (M.X \u25c1 g.hom) \u22ef \u22ef) \u226b\n        colimMap\n          (parallelPairHom (M.actRight \u25b7 Q.X) ((\u03b1_ M.X Y.X Q.X).hom \u226b M.X \u25c1 Q.actLeft) (N.actRight \u25b7 Q.X)\n            ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft) (f.hom \u25b7 Y.X \u25b7 Q.X) (f.hom \u25b7 Q.X) \u22ef \u22ef) =\n    coequalizer.\u03c0 (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) \u226b\n      colimMap\n          (parallelPairHom (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) (N.actRight \u25b7 P.X)\n            ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef) \u226b\n        colimMap\n          (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N.actRight \u25b7 Q.X)\n            ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft) ((N.X \u2297 Y.X) \u25c1 g.hom) (N.X \u25c1 g.hom) \u22ef \u22ef)", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nX Y Z : Mon_ C\nM N : Bimod X Y\nP Q : Bimod Y Z\nf : M \u27f6 N\ng : P \u27f6 Q\n\u22a2 (M.X \u25c1 g.hom \u226b\n        colimit.\u03b9 (parallelPair (M.actRight \u25b7 Q.X) ((\u03b1_ M.X Y.X Q.X).hom \u226b M.X \u25c1 Q.actLeft)) WalkingParallelPair.one) \u226b\n      colimMap\n        (parallelPairHom (M.actRight \u25b7 Q.X) ((\u03b1_ M.X Y.X Q.X).hom \u226b M.X \u25c1 Q.actLeft) (N.actRight \u25b7 Q.X)\n          ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft) (f.hom \u25b7 Y.X \u25b7 Q.X) (f.hom \u25b7 Q.X) \u22ef \u22ef) =\n    coequalizer.\u03c0 (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) \u226b\n      colimMap\n          (parallelPairHom (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) (N.actRight \u25b7 P.X)\n            ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef) \u226b\n        colimMap\n          (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N.actRight \u25b7 Q.X)\n            ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft) ((N.X \u2297 Y.X) \u25c1 g.hom) (N.X \u25c1 g.hom) \u22ef \u22ef)"}, {"tactic": "slice_lhs 2 3 => rw [\u03b9_colimMap, parallelPairHom_app_one]", "annotated_tactic": ["slice_lhs 2 3 => rw [<a>\u03b9_colimMap</a>, <a>parallelPairHom_app_one</a>]", [{"full_name": "CategoryTheory.Limits.\u03b9_colimMap", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [805, 9], "def_end_pos": [805, 19]}, {"full_name": "CategoryTheory.Limits.parallelPairHom_app_one", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [272, 9], "def_end_pos": [272, 32]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nX Y Z : Mon_ C\nM N : Bimod X Y\nP Q : Bimod Y Z\nf : M \u27f6 N\ng : P \u27f6 Q\n\u22a2 (M.X \u25c1 g.hom \u226b\n        colimit.\u03b9 (parallelPair (M.actRight \u25b7 Q.X) ((\u03b1_ M.X Y.X Q.X).hom \u226b M.X \u25c1 Q.actLeft)) WalkingParallelPair.one) \u226b\n      colimMap\n        (parallelPairHom (M.actRight \u25b7 Q.X) ((\u03b1_ M.X Y.X Q.X).hom \u226b M.X \u25c1 Q.actLeft) (N.actRight \u25b7 Q.X)\n          ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft) (f.hom \u25b7 Y.X \u25b7 Q.X) (f.hom \u25b7 Q.X) \u22ef \u22ef) =\n    coequalizer.\u03c0 (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) \u226b\n      colimMap\n          (parallelPairHom (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) (N.actRight \u25b7 P.X)\n            ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef) \u226b\n        colimMap\n          (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N.actRight \u25b7 Q.X)\n            ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft) ((N.X \u2297 Y.X) \u25c1 g.hom) (N.X \u25c1 g.hom) \u22ef \u22ef)", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nX Y Z : Mon_ C\nM N : Bimod X Y\nP Q : Bimod Y Z\nf : M \u27f6 N\ng : P \u27f6 Q\n\u22a2 M.X \u25c1 g.hom \u226b\n      f.hom \u25b7 Q.X \u226b\n        colimit.\u03b9 (parallelPair (N.actRight \u25b7 Q.X) ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft)) WalkingParallelPair.one =\n    coequalizer.\u03c0 (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) \u226b\n      colimMap\n          (parallelPairHom (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) (N.actRight \u25b7 P.X)\n            ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef) \u226b\n        colimMap\n          (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N.actRight \u25b7 Q.X)\n            ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft) ((N.X \u2297 Y.X) \u25c1 g.hom) (N.X \u25c1 g.hom) \u22ef \u22ef)"}, {"tactic": "slice_lhs 1 2 => rw [whisker_exchange]", "annotated_tactic": ["slice_lhs 1 2 => rw [<a>whisker_exchange</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.whisker_exchange", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [278, 9], "def_end_pos": [278, 25]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nX Y Z : Mon_ C\nM N : Bimod X Y\nP Q : Bimod Y Z\nf : M \u27f6 N\ng : P \u27f6 Q\n\u22a2 M.X \u25c1 g.hom \u226b\n      f.hom \u25b7 Q.X \u226b\n        colimit.\u03b9 (parallelPair (N.actRight \u25b7 Q.X) ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft)) WalkingParallelPair.one =\n    coequalizer.\u03c0 (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) \u226b\n      colimMap\n          (parallelPairHom (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) (N.actRight \u25b7 P.X)\n            ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef) \u226b\n        colimMap\n          (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N.actRight \u25b7 Q.X)\n            ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft) ((N.X \u2297 Y.X) \u25c1 g.hom) (N.X \u25c1 g.hom) \u22ef \u22ef)", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nX Y Z : Mon_ C\nM N : Bimod X Y\nP Q : Bimod Y Z\nf : M \u27f6 N\ng : P \u27f6 Q\n\u22a2 (f.hom \u25b7 P.X \u226b N.X \u25c1 g.hom) \u226b\n      colimit.\u03b9 (parallelPair (N.actRight \u25b7 Q.X) ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft)) WalkingParallelPair.one =\n    coequalizer.\u03c0 (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) \u226b\n      colimMap\n          (parallelPairHom (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) (N.actRight \u25b7 P.X)\n            ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef) \u226b\n        colimMap\n          (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N.actRight \u25b7 Q.X)\n            ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft) ((N.X \u2297 Y.X) \u25c1 g.hom) (N.X \u25c1 g.hom) \u22ef \u22ef)"}, {"tactic": "slice_rhs 1 2 => rw [\u03b9_colimMap, parallelPairHom_app_one]", "annotated_tactic": ["slice_rhs 1 2 => rw [<a>\u03b9_colimMap</a>, <a>parallelPairHom_app_one</a>]", [{"full_name": "CategoryTheory.Limits.\u03b9_colimMap", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [805, 9], "def_end_pos": [805, 19]}, {"full_name": "CategoryTheory.Limits.parallelPairHom_app_one", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [272, 9], "def_end_pos": [272, 32]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nX Y Z : Mon_ C\nM N : Bimod X Y\nP Q : Bimod Y Z\nf : M \u27f6 N\ng : P \u27f6 Q\n\u22a2 (f.hom \u25b7 P.X \u226b N.X \u25c1 g.hom) \u226b\n      colimit.\u03b9 (parallelPair (N.actRight \u25b7 Q.X) ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft)) WalkingParallelPair.one =\n    coequalizer.\u03c0 (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) \u226b\n      colimMap\n          (parallelPairHom (M.actRight \u25b7 P.X) ((\u03b1_ M.X Y.X P.X).hom \u226b M.X \u25c1 P.actLeft) (N.actRight \u25b7 P.X)\n            ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef) \u226b\n        colimMap\n          (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N.actRight \u25b7 Q.X)\n            ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft) ((N.X \u2297 Y.X) \u25c1 g.hom) (N.X \u25c1 g.hom) \u22ef \u22ef)", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nX Y Z : Mon_ C\nM N : Bimod X Y\nP Q : Bimod Y Z\nf : M \u27f6 N\ng : P \u27f6 Q\n\u22a2 (f.hom \u25b7 P.X \u226b N.X \u25c1 g.hom) \u226b\n      colimit.\u03b9 (parallelPair (N.actRight \u25b7 Q.X) ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft)) WalkingParallelPair.one =\n    (f.hom \u25b7 P.X \u226b\n        colimit.\u03b9 (parallelPair (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft)) WalkingParallelPair.one) \u226b\n      colimMap\n        (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N.actRight \u25b7 Q.X)\n          ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft) ((N.X \u2297 Y.X) \u25c1 g.hom) (N.X \u25c1 g.hom) \u22ef \u22ef)"}, {"tactic": "slice_rhs 2 3 => rw [\u03b9_colimMap, parallelPairHom_app_one]", "annotated_tactic": ["slice_rhs 2 3 => rw [<a>\u03b9_colimMap</a>, <a>parallelPairHom_app_one</a>]", [{"full_name": "CategoryTheory.Limits.\u03b9_colimMap", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [805, 9], "def_end_pos": [805, 19]}, {"full_name": "CategoryTheory.Limits.parallelPairHom_app_one", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [272, 9], "def_end_pos": [272, 32]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nX Y Z : Mon_ C\nM N : Bimod X Y\nP Q : Bimod Y Z\nf : M \u27f6 N\ng : P \u27f6 Q\n\u22a2 (f.hom \u25b7 P.X \u226b N.X \u25c1 g.hom) \u226b\n      colimit.\u03b9 (parallelPair (N.actRight \u25b7 Q.X) ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft)) WalkingParallelPair.one =\n    (f.hom \u25b7 P.X \u226b\n        colimit.\u03b9 (parallelPair (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft)) WalkingParallelPair.one) \u226b\n      colimMap\n        (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N.actRight \u25b7 Q.X)\n          ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft) ((N.X \u2297 Y.X) \u25c1 g.hom) (N.X \u25c1 g.hom) \u22ef \u22ef)", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nX Y Z : Mon_ C\nM N : Bimod X Y\nP Q : Bimod Y Z\nf : M \u27f6 N\ng : P \u27f6 Q\n\u22a2 (f.hom \u25b7 P.X \u226b N.X \u25c1 g.hom) \u226b\n      colimit.\u03b9 (parallelPair (N.actRight \u25b7 Q.X) ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft)) WalkingParallelPair.one =\n    f.hom \u25b7 P.X \u226b\n      N.X \u25c1 g.hom \u226b\n        colimit.\u03b9 (parallelPair (N.actRight \u25b7 Q.X) ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft)) WalkingParallelPair.one"}, {"tactic": "simp only [Category.assoc]", "annotated_tactic": ["simp only [<a>Category.assoc</a>]", [{"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nX Y Z : Mon_ C\nM N : Bimod X Y\nP Q : Bimod Y Z\nf : M \u27f6 N\ng : P \u27f6 Q\n\u22a2 (f.hom \u25b7 P.X \u226b N.X \u25c1 g.hom) \u226b\n      colimit.\u03b9 (parallelPair (N.actRight \u25b7 Q.X) ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft)) WalkingParallelPair.one =\n    f.hom \u25b7 P.X \u226b\n      N.X \u25c1 g.hom \u226b\n        colimit.\u03b9 (parallelPair (N.actRight \u25b7 Q.X) ((\u03b1_ N.X Y.X Q.X).hom \u226b N.X \u25c1 Q.actLeft)) WalkingParallelPair.one", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/AlgebraicIndependent.lean", "full_name": "algebraicIndependent_comp_subtype", "start": [284, 1], "end": [291, 64], "traced_tactics": [{"tactic": "have : (aeval (x \u2218 (\u2191) : s \u2192 A) : _ \u2192\u2090[R] _) = (aeval x).comp (rename (\u2191)) := by ext; simp", "annotated_tactic": ["have : (<a>aeval</a> (x \u2218 (\u2191) : s \u2192 A) : _ \u2192\u2090[R] _) = (<a>aeval</a> x).<a>comp</a> (<a>rename</a> (\u2191)) := by ext; simp", [{"full_name": "MvPolynomial.aeval", "def_path": "Mathlib/Algebra/MvPolynomial/Basic.lean", "def_pos": [1521, 5], "def_end_pos": [1521, 10]}, {"full_name": "MvPolynomial.aeval", "def_path": "Mathlib/Algebra/MvPolynomial/Basic.lean", "def_pos": [1521, 5], "def_end_pos": [1521, 10]}, {"full_name": "AlgHom.comp", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [314, 5], "def_end_pos": [314, 9]}, {"full_name": "MvPolynomial.rename", "def_path": "Mathlib/Algebra/MvPolynomial/Rename.lean", "def_pos": [52, 5], "def_end_pos": [52, 11]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nR : Type u_3\nK : Type u_4\nA : Type u_5\nA' : Type u_6\nA'' : Type u_7\nV : Type u\nV' : Type u_8\nx : \u03b9 \u2192 A\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing A'\ninst\u271d\u00b3 : CommRing A''\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Algebra R A'\ninst\u271d : Algebra R A''\na b : R\ns : Set \u03b9\n\u22a2 AlgebraicIndependent R (x \u2218 Subtype.val) \u2194 \u2200 p \u2208 supported R s, (aeval x) p = 0 \u2192 p = 0", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nR : Type u_3\nK : Type u_4\nA : Type u_5\nA' : Type u_6\nA'' : Type u_7\nV : Type u\nV' : Type u_8\nx : \u03b9 \u2192 A\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing A'\ninst\u271d\u00b3 : CommRing A''\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Algebra R A'\ninst\u271d : Algebra R A''\na b : R\ns : Set \u03b9\nthis : aeval (x \u2218 Subtype.val) = (aeval x).comp (rename Subtype.val)\n\u22a2 AlgebraicIndependent R (x \u2218 Subtype.val) \u2194 \u2200 p \u2208 supported R s, (aeval x) p = 0 \u2192 p = 0"}, {"tactic": "have : \u2200 p : MvPolynomial s R, rename ((\u2191) : s \u2192 \u03b9) p = 0 \u2194 p = 0 :=\n  (injective_iff_map_eq_zero' (rename ((\u2191) : s \u2192 \u03b9) : MvPolynomial s R \u2192\u2090[R] _).toRingHom).1\n    (rename_injective _ Subtype.val_injective)", "annotated_tactic": ["have : \u2200 p : <a>MvPolynomial</a> s R, <a>rename</a> ((\u2191) : s \u2192 \u03b9) p = 0 \u2194 p = 0 :=\n    (<a>injective_iff_map_eq_zero'</a> (<a>rename</a> ((\u2191) : s \u2192 \u03b9) : <a>MvPolynomial</a> s R \u2192\u2090[R] _).<a>toRingHom</a>).1\n      (<a>rename_injective</a> _ <a>Subtype.val_injective</a>)", [{"full_name": "MvPolynomial", "def_path": "Mathlib/Algebra/MvPolynomial/Basic.lean", "def_pos": [84, 5], "def_end_pos": [84, 17]}, {"full_name": "MvPolynomial.rename", "def_path": "Mathlib/Algebra/MvPolynomial/Rename.lean", "def_pos": [52, 5], "def_end_pos": [52, 11]}, {"full_name": "injective_iff_map_eq_zero'", "def_path": "Mathlib/Algebra/Group/Hom/Basic.lean", "def_pos": [140, 3], "def_end_pos": [140, 14]}, {"full_name": "MvPolynomial.rename", "def_path": "Mathlib/Algebra/MvPolynomial/Rename.lean", "def_pos": [52, 5], "def_end_pos": [52, 11]}, {"full_name": "MvPolynomial", "def_path": "Mathlib/Algebra/MvPolynomial/Basic.lean", "def_pos": [84, 5], "def_end_pos": [84, 17]}, {"full_name": "AlgHom.toRingHom", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [36, 14], "def_end_pos": [36, 30]}, {"full_name": "MvPolynomial.rename_injective", "def_path": "Mathlib/Algebra/MvPolynomial/Rename.lean", "def_pos": [109, 9], "def_end_pos": [109, 25]}, {"full_name": "Subtype.val_injective", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [131, 9], "def_end_pos": [131, 22]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nR : Type u_3\nK : Type u_4\nA : Type u_5\nA' : Type u_6\nA'' : Type u_7\nV : Type u\nV' : Type u_8\nx : \u03b9 \u2192 A\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing A'\ninst\u271d\u00b3 : CommRing A''\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Algebra R A'\ninst\u271d : Algebra R A''\na b : R\ns : Set \u03b9\nthis : aeval (x \u2218 Subtype.val) = (aeval x).comp (rename Subtype.val)\n\u22a2 AlgebraicIndependent R (x \u2218 Subtype.val) \u2194 \u2200 p \u2208 supported R s, (aeval x) p = 0 \u2192 p = 0", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nR : Type u_3\nK : Type u_4\nA : Type u_5\nA' : Type u_6\nA'' : Type u_7\nV : Type u\nV' : Type u_8\nx : \u03b9 \u2192 A\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing A'\ninst\u271d\u00b3 : CommRing A''\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Algebra R A'\ninst\u271d : Algebra R A''\na b : R\ns : Set \u03b9\nthis\u271d : aeval (x \u2218 Subtype.val) = (aeval x).comp (rename Subtype.val)\nthis : \u2200 (p : MvPolynomial (\u2191s) R), (rename Subtype.val) p = 0 \u2194 p = 0\n\u22a2 AlgebraicIndependent R (x \u2218 Subtype.val) \u2194 \u2200 p \u2208 supported R s, (aeval x) p = 0 \u2192 p = 0"}, {"tactic": "simp [algebraicIndependent_iff, supported_eq_range_rename, *]", "annotated_tactic": ["simp [<a>algebraicIndependent_iff</a>, <a>supported_eq_range_rename</a>, *]", [{"full_name": "algebraicIndependent_iff", "def_path": "Mathlib/RingTheory/AlgebraicIndependent.lean", "def_pos": [73, 9], "def_end_pos": [73, 33]}, {"full_name": "MvPolynomial.supported_eq_range_rename", "def_path": "Mathlib/Algebra/MvPolynomial/Supported.lean", "def_pos": [46, 9], "def_end_pos": [46, 34]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nR : Type u_3\nK : Type u_4\nA : Type u_5\nA' : Type u_6\nA'' : Type u_7\nV : Type u\nV' : Type u_8\nx : \u03b9 \u2192 A\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing A'\ninst\u271d\u00b3 : CommRing A''\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Algebra R A'\ninst\u271d : Algebra R A''\na b : R\ns : Set \u03b9\nthis\u271d : aeval (x \u2218 Subtype.val) = (aeval x).comp (rename Subtype.val)\nthis : \u2200 (p : MvPolynomial (\u2191s) R), (rename Subtype.val) p = 0 \u2194 p = 0\n\u22a2 AlgebraicIndependent R (x \u2218 Subtype.val) \u2194 \u2200 p \u2208 supported R s, (aeval x) p = 0 \u2192 p = 0", "state_after": "no goals"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nR : Type u_3\nK : Type u_4\nA : Type u_5\nA' : Type u_6\nA'' : Type u_7\nV : Type u\nV' : Type u_8\nx : \u03b9 \u2192 A\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing A'\ninst\u271d\u00b3 : CommRing A''\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Algebra R A'\ninst\u271d : Algebra R A''\na b : R\ns : Set \u03b9\n\u22a2 aeval (x \u2218 Subtype.val) = (aeval x).comp (rename Subtype.val)", "state_after": "case hf\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nR : Type u_3\nK : Type u_4\nA : Type u_5\nA' : Type u_6\nA'' : Type u_7\nV : Type u\nV' : Type u_8\nx : \u03b9 \u2192 A\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing A'\ninst\u271d\u00b3 : CommRing A''\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Algebra R A'\ninst\u271d : Algebra R A''\na b : R\ns : Set \u03b9\ni\u271d : \u2191s\n\u22a2 (aeval (x \u2218 Subtype.val)) (X i\u271d) = ((aeval x).comp (rename Subtype.val)) (X i\u271d)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case hf\n\u03b9 : Type u_1\n\u03b9' : Type u_2\nR : Type u_3\nK : Type u_4\nA : Type u_5\nA' : Type u_6\nA'' : Type u_7\nV : Type u\nV' : Type u_8\nx : \u03b9 \u2192 A\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : CommRing A'\ninst\u271d\u00b3 : CommRing A''\ninst\u271d\u00b2 : Algebra R A\ninst\u271d\u00b9 : Algebra R A'\ninst\u271d : Algebra R A''\na b : R\ns : Set \u03b9\ni\u271d : \u2191s\n\u22a2 (aeval (x \u2218 Subtype.val)) (X i\u271d) = ((aeval x).comp (rename Subtype.val)) (X i\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Minimal.lean", "full_name": "minimals_of_symm", "start": [184, 1], "end": [185, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Complex/Isometry.lean", "full_name": "LinearIsometry.im_apply_eq_im", "start": [104, 1], "end": [116, 34], "traced_tactics": [{"tactic": "have : \u2016f z - 1\u2016 = \u2016z - 1\u2016 := by rw [\u2190 f.norm_map (z - 1), f.map_sub, h]", "annotated_tactic": ["have : \u2016f z - 1\u2016 = \u2016z - 1\u2016 := by rw [\u2190 f.norm_map (z - 1), f.map_sub, h]", []], "state_before": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\n\u22a2 z + (starRingEnd \u2102) z = f z + (starRingEnd \u2102) (f z)", "state_after": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\nthis : \u2016f z - 1\u2016 = \u2016z - 1\u2016\n\u22a2 z + (starRingEnd \u2102) z = f z + (starRingEnd \u2102) (f z)"}, {"tactic": "apply_fun fun x => x ^ 2 at this", "annotated_tactic": ["apply_fun fun x => x ^ 2 at this", []], "state_before": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\nthis : \u2016f z - 1\u2016 = \u2016z - 1\u2016\n\u22a2 z + (starRingEnd \u2102) z = f z + (starRingEnd \u2102) (f z)", "state_after": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\nthis : \u2016f z - 1\u2016 ^ 2 = \u2016z - 1\u2016 ^ 2\n\u22a2 z + (starRingEnd \u2102) z = f z + (starRingEnd \u2102) (f z)"}, {"tactic": "simp only [norm_eq_abs, \u2190 normSq_eq_abs] at this", "annotated_tactic": ["simp only [<a>norm_eq_abs</a>, \u2190 <a>normSq_eq_abs</a>] at this", [{"full_name": "Complex.norm_eq_abs", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [55, 9], "def_end_pos": [55, 20]}, {"full_name": "Complex.normSq_eq_abs", "def_path": "Mathlib/Data/Complex/Abs.lean", "def_pos": [249, 9], "def_end_pos": [249, 22]}]], "state_before": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\nthis : \u2016f z - 1\u2016 ^ 2 = \u2016z - 1\u2016 ^ 2\n\u22a2 z + (starRingEnd \u2102) z = f z + (starRingEnd \u2102) (f z)", "state_after": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\nthis : normSq (f z - 1) = normSq (z - 1)\n\u22a2 z + (starRingEnd \u2102) z = f z + (starRingEnd \u2102) (f z)"}, {"tactic": "rw [\u2190 ofReal_inj, \u2190 mul_conj, \u2190 mul_conj] at this", "annotated_tactic": ["rw [\u2190 <a>ofReal_inj</a>, \u2190 <a>mul_conj</a>, \u2190 <a>mul_conj</a>] at this", [{"full_name": "Complex.ofReal_inj", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [110, 9], "def_end_pos": [110, 19]}, {"full_name": "Complex.mul_conj", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [748, 9], "def_end_pos": [748, 17]}, {"full_name": "Complex.mul_conj", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [748, 9], "def_end_pos": [748, 17]}]], "state_before": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\nthis : normSq (f z - 1) = normSq (z - 1)\n\u22a2 z + (starRingEnd \u2102) z = f z + (starRingEnd \u2102) (f z)", "state_after": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\nthis : (f z - 1) * (starRingEnd \u2102) (f z - 1) = (z - 1) * (starRingEnd \u2102) (z - 1)\n\u22a2 z + (starRingEnd \u2102) z = f z + (starRingEnd \u2102) (f z)"}, {"tactic": "rw [RingHom.map_sub, RingHom.map_sub] at this", "annotated_tactic": ["rw [<a>RingHom.map_sub</a>, <a>RingHom.map_sub</a>] at this", [{"full_name": "RingHom.map_sub", "def_path": "Mathlib/Algebra/Ring/Hom/Defs.lean", "def_pos": [612, 19], "def_end_pos": [612, 26]}, {"full_name": "RingHom.map_sub", "def_path": "Mathlib/Algebra/Ring/Hom/Defs.lean", "def_pos": [612, 19], "def_end_pos": [612, 26]}]], "state_before": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\nthis : (f z - 1) * (starRingEnd \u2102) (f z - 1) = (z - 1) * (starRingEnd \u2102) (z - 1)\n\u22a2 z + (starRingEnd \u2102) z = f z + (starRingEnd \u2102) (f z)", "state_after": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\nthis : (f z - 1) * ((starRingEnd \u2102) (f z) - (starRingEnd \u2102) 1) = (z - 1) * ((starRingEnd \u2102) z - (starRingEnd \u2102) 1)\n\u22a2 z + (starRingEnd \u2102) z = f z + (starRingEnd \u2102) (f z)"}, {"tactic": "simp only [sub_mul, mul_sub, one_mul, mul_one] at this", "annotated_tactic": ["simp only [<a>sub_mul</a>, <a>mul_sub</a>, <a>one_mul</a>, <a>mul_one</a>] at this", [{"full_name": "sub_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [401, 7], "def_end_pos": [401, 14]}, {"full_name": "mul_sub", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [394, 7], "def_end_pos": [394, 14]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\nthis : (f z - 1) * ((starRingEnd \u2102) (f z) - (starRingEnd \u2102) 1) = (z - 1) * ((starRingEnd \u2102) z - (starRingEnd \u2102) 1)\n\u22a2 z + (starRingEnd \u2102) z = f z + (starRingEnd \u2102) (f z)", "state_after": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\nthis :\n  f z * (starRingEnd \u2102) (f z) - (starRingEnd \u2102) (f z) - (f z * (starRingEnd \u2102) 1 - (starRingEnd \u2102) 1) =\n    z * (starRingEnd \u2102) z - (starRingEnd \u2102) z - (z * (starRingEnd \u2102) 1 - (starRingEnd \u2102) 1)\n\u22a2 z + (starRingEnd \u2102) z = f z + (starRingEnd \u2102) (f z)"}, {"tactic": "rw [mul_conj, normSq_eq_abs, \u2190 norm_eq_abs, LinearIsometry.norm_map] at this", "annotated_tactic": ["rw [<a>mul_conj</a>, <a>normSq_eq_abs</a>, \u2190 <a>norm_eq_abs</a>, <a>LinearIsometry.norm_map</a>] at this", [{"full_name": "Complex.mul_conj", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [748, 9], "def_end_pos": [748, 17]}, {"full_name": "Complex.normSq_eq_abs", "def_path": "Mathlib/Data/Complex/Abs.lean", "def_pos": [249, 9], "def_end_pos": [249, 22]}, {"full_name": "Complex.norm_eq_abs", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [55, 9], "def_end_pos": [55, 20]}, {"full_name": "LinearIsometry.norm_map", "def_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "def_pos": [236, 9], "def_end_pos": [236, 17]}]], "state_before": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\nthis :\n  f z * (starRingEnd \u2102) (f z) - (starRingEnd \u2102) (f z) - (f z * (starRingEnd \u2102) 1 - (starRingEnd \u2102) 1) =\n    z * (starRingEnd \u2102) z - (starRingEnd \u2102) z - (z * (starRingEnd \u2102) 1 - (starRingEnd \u2102) 1)\n\u22a2 z + (starRingEnd \u2102) z = f z + (starRingEnd \u2102) (f z)", "state_after": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\nthis :\n  \u2191(\u2016z\u2016 ^ 2) - (starRingEnd \u2102) (f z) - (f z * (starRingEnd \u2102) 1 - (starRingEnd \u2102) 1) =\n    z * (starRingEnd \u2102) z - (starRingEnd \u2102) z - (z * (starRingEnd \u2102) 1 - (starRingEnd \u2102) 1)\n\u22a2 z + (starRingEnd \u2102) z = f z + (starRingEnd \u2102) (f z)"}, {"tactic": "rw [mul_conj, normSq_eq_abs, \u2190 norm_eq_abs] at this", "annotated_tactic": ["rw [<a>mul_conj</a>, <a>normSq_eq_abs</a>, \u2190 <a>norm_eq_abs</a>] at this", [{"full_name": "Complex.mul_conj", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [748, 9], "def_end_pos": [748, 17]}, {"full_name": "Complex.normSq_eq_abs", "def_path": "Mathlib/Data/Complex/Abs.lean", "def_pos": [249, 9], "def_end_pos": [249, 22]}, {"full_name": "Complex.norm_eq_abs", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [55, 9], "def_end_pos": [55, 20]}]], "state_before": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\nthis :\n  \u2191(\u2016z\u2016 ^ 2) - (starRingEnd \u2102) (f z) - (f z * (starRingEnd \u2102) 1 - (starRingEnd \u2102) 1) =\n    z * (starRingEnd \u2102) z - (starRingEnd \u2102) z - (z * (starRingEnd \u2102) 1 - (starRingEnd \u2102) 1)\n\u22a2 z + (starRingEnd \u2102) z = f z + (starRingEnd \u2102) (f z)", "state_after": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\nthis :\n  \u2191(\u2016z\u2016 ^ 2) - (starRingEnd \u2102) (f z) - (f z * (starRingEnd \u2102) 1 - (starRingEnd \u2102) 1) =\n    \u2191(\u2016z\u2016 ^ 2) - (starRingEnd \u2102) z - (z * (starRingEnd \u2102) 1 - (starRingEnd \u2102) 1)\n\u22a2 z + (starRingEnd \u2102) z = f z + (starRingEnd \u2102) (f z)"}, {"tactic": "simp only [sub_sub, sub_right_inj, mul_one, ofReal_pow, RingHom.map_one, norm_eq_abs] at this", "annotated_tactic": ["simp only [<a>sub_sub</a>, <a>sub_right_inj</a>, <a>mul_one</a>, <a>ofReal_pow</a>, <a>RingHom.map_one</a>, <a>norm_eq_abs</a>] at this", [{"full_name": "sub_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [770, 3], "def_end_pos": [770, 14]}, {"full_name": "sub_right_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1045, 3], "def_end_pos": [1045, 14]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "Complex.ofReal_pow", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [794, 9], "def_end_pos": [794, 19]}, {"full_name": "RingHom.map_one", "def_path": "Mathlib/Algebra/Ring/Hom/Defs.lean", "def_pos": [551, 19], "def_end_pos": [551, 26]}, {"full_name": "Complex.norm_eq_abs", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [55, 9], "def_end_pos": [55, 20]}]], "state_before": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\nthis :\n  \u2191(\u2016z\u2016 ^ 2) - (starRingEnd \u2102) (f z) - (f z * (starRingEnd \u2102) 1 - (starRingEnd \u2102) 1) =\n    \u2191(\u2016z\u2016 ^ 2) - (starRingEnd \u2102) z - (z * (starRingEnd \u2102) 1 - (starRingEnd \u2102) 1)\n\u22a2 z + (starRingEnd \u2102) z = f z + (starRingEnd \u2102) (f z)", "state_after": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\nthis : (starRingEnd \u2102) (f z) + (f z - 1) = (starRingEnd \u2102) z + (z - 1)\n\u22a2 z + (starRingEnd \u2102) z = f z + (starRingEnd \u2102) (f z)"}, {"tactic": "simp only [add_sub, sub_left_inj] at this", "annotated_tactic": ["simp only [<a>add_sub</a>, <a>sub_left_inj</a>] at this", [{"full_name": "add_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [473, 3], "def_end_pos": [473, 14]}, {"full_name": "sub_left_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1051, 3], "def_end_pos": [1051, 14]}]], "state_before": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\nthis : (starRingEnd \u2102) (f z) + (f z - 1) = (starRingEnd \u2102) z + (z - 1)\n\u22a2 z + (starRingEnd \u2102) z = f z + (starRingEnd \u2102) (f z)", "state_after": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\nthis : (starRingEnd \u2102) (f z) + f z = (starRingEnd \u2102) z + z\n\u22a2 z + (starRingEnd \u2102) z = f z + (starRingEnd \u2102) (f z)"}, {"tactic": "rw [add_comm, \u2190 this, add_comm]", "annotated_tactic": ["rw [<a>add_comm</a>, \u2190 this, <a>add_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\nthis : (starRingEnd \u2102) (f z) + f z = (starRingEnd \u2102) z + z\n\u22a2 z + (starRingEnd \u2102) z = f z + (starRingEnd \u2102) (f z)", "state_after": "no goals"}, {"tactic": "rw [\u2190 f.norm_map (z - 1), f.map_sub, h]", "annotated_tactic": ["rw [\u2190 f.norm_map (z - 1), f.map_sub, h]", []], "state_before": "f : \u2102 \u2192\u2097\u1d62[\u211d] \u2102\nh : f 1 = 1\nz : \u2102\n\u22a2 \u2016f z - 1\u2016 = \u2016z - 1\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.disjUnion_empty", "start": [1069, 1], "end": [1071, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Quaternion.lean", "full_name": "QuaternionAlgebra.zero_im", "start": [182, 9], "end": [182, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Pi.lean", "full_name": "LinearEquiv.piCongrRight_apply", "start": [365, 1], "end": [366, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/Hyperreal.lean", "full_name": "Hyperreal.Infinite.st_eq", "start": [296, 1], "end": [297, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Asymptotics/Theta.lean", "full_name": "Asymptotics.IsTheta.tendsto_norm_atTop_iff", "start": [219, 1], "end": [221, 100], "traced_tactics": [{"tactic": "simp only [Function.comp, \u2190 isLittleO_const_left_of_ne (one_ne_zero' \u211d), h.isLittleO_congr_right]", "annotated_tactic": ["simp only [<a>Function.comp</a>, \u2190 <a>isLittleO_const_left_of_ne</a> (<a>one_ne_zero'</a> \u211d), h.isLittleO_congr_right]", [{"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": "Asymptotics.isLittleO_const_left_of_ne", "def_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "def_pos": [1904, 9], "def_end_pos": [1904, 35]}, {"full_name": "one_ne_zero'", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [81, 7], "def_end_pos": [81, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nR : Type u_12\nR' : Type u_13\n\ud835\udd5c : Type u_14\n\ud835\udd5c' : Type u_15\ninst\u271d\u00b9\u00b2 : Norm E\ninst\u271d\u00b9\u00b9 : Norm F\ninst\u271d\u00b9\u2070 : Norm G\ninst\u271d\u2079 : SeminormedAddCommGroup E'\ninst\u271d\u2078 : SeminormedAddCommGroup F'\ninst\u271d\u2077 : SeminormedAddCommGroup G'\ninst\u271d\u2076 : NormedAddCommGroup E''\ninst\u271d\u2075 : NormedAddCommGroup F''\ninst\u271d\u2074 : NormedAddCommGroup G''\ninst\u271d\u00b3 : SeminormedRing R\ninst\u271d\u00b2 : SeminormedRing R'\ninst\u271d\u00b9 : NormedField \ud835\udd5c\ninst\u271d : NormedField \ud835\udd5c'\nc c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nl l' : Filter \u03b1\nh : f' =\u0398[l] g'\n\u22a2 Tendsto (norm \u2218 f') l atTop \u2194 Tendsto (norm \u2218 g') l atTop", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Group/Pointwise.lean", "full_name": "MeasurableSet.const_smul", "start": [24, 1], "end": [28, 35], "traced_tactics": [{"tactic": "rw [\u2190 preimage_smul_inv]", "annotated_tactic": ["rw [\u2190 <a>preimage_smul_inv</a>]", [{"full_name": "Set.preimage_smul_inv", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [918, 9], "def_end_pos": [918, 26]}]], "state_before": "G : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSMul G \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\na : G\n\u22a2 MeasurableSet (a \u2022 s)", "state_after": "G : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSMul G \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\na : G\n\u22a2 MeasurableSet ((fun x => a\u207b\u00b9 \u2022 x) \u207b\u00b9' s)"}, {"tactic": "exact measurable_const_smul _ hs", "annotated_tactic": ["exact <a>measurable_const_smul</a> _ hs", [{"full_name": "MeasurableSMul.measurable_const_smul", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [589, 3], "def_end_pos": [589, 24]}]], "state_before": "G : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : MulAction G \u03b1\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : MeasurableSpace \u03b1\ninst\u271d : MeasurableSMul G \u03b1\ns : Set \u03b1\nhs : MeasurableSet s\na : G\n\u22a2 MeasurableSet ((fun x => a\u207b\u00b9 \u2022 x) \u207b\u00b9' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Inductions.lean", "full_name": "Polynomial.divX_C_mul_X_pow", "start": [116, 1], "end": [117, 56], "traced_tactics": [{"tactic": "simp only [divX_C_mul, divX_X_pow, mul_ite, mul_zero]", "annotated_tactic": ["simp only [<a>divX_C_mul</a>, <a>divX_X_pow</a>, <a>mul_ite</a>, <a>mul_zero</a>]", [{"full_name": "Polynomial.divX_C_mul", "def_path": "Mathlib/Algebra/Polynomial/Inductions.lean", "def_pos": [84, 9], "def_end_pos": [84, 19]}, {"full_name": "Polynomial.divX_X_pow", "def_path": "Mathlib/Algebra/Polynomial/Inductions.lean", "def_pos": [88, 9], "def_end_pos": [88, 19]}, {"full_name": "mul_ite", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [197, 9], "def_end_pos": [197, 16]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : \u2115\ninst\u271d : Semiring R\np q : R[X]\n\u22a2 (C a * X ^ n).divX = if n = 0 then 0 else C a * X ^ (n - 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monoidal/Bimod.lean", "full_name": "Bimod.AssociatorBimod.hom_left_act_hom'", "start": [514, 1], "end": [539, 12], "traced_tactics": [{"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 ((P.tensorBimod Q).tensorBimod L).actLeft \u226b hom P Q L = R.X \u25c1 hom P Q L \u226b (P.tensorBimod (Q.tensorBimod L)).actLeft", "state_after": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 TensorBimod.actLeft (P.tensorBimod Q) L \u226b hom P Q L = R.X \u25c1 hom P Q L \u226b TensorBimod.actLeft P (Q.tensorBimod L)"}, {"tactic": "dsimp [hom, homAux]", "annotated_tactic": ["dsimp [<a>hom</a>, <a>homAux</a>]", [{"full_name": "Bimod.AssociatorBimod.hom", "def_path": "Mathlib/CategoryTheory/Monoidal/Bimod.lean", "def_pos": [493, 19], "def_end_pos": [493, 22]}, {"full_name": "Bimod.AssociatorBimod.homAux", "def_path": "Mathlib/CategoryTheory/Monoidal/Bimod.lean", "def_pos": [476, 19], "def_end_pos": [476, 25]}]], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 TensorBimod.actLeft (P.tensorBimod Q) L \u226b hom P Q L = R.X \u25c1 hom P Q L \u226b TensorBimod.actLeft P (Q.tensorBimod L)", "state_after": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 TensorBimod.actLeft (P.tensorBimod Q) L \u226b\n      coequalizer.desc\n        ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          coequalizer.desc\n            ((\u03b1_ P.X Q.X L.X).hom \u226b\n              P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                  ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n            \u22ef)\n        \u22ef =\n    R.X \u25c1\n        coequalizer.desc\n          ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                    ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef)\n          \u22ef \u226b\n      TensorBimod.actLeft P (Q.tensorBimod L)"}, {"tactic": "refine (cancel_epi ((tensorLeft _).map (coequalizer.\u03c0 _ _))).1 ?_", "annotated_tactic": ["refine (<a>cancel_epi</a> ((<a>tensorLeft</a> _).<a>map</a> (<a>coequalizer.\u03c0</a> _ _))).1 ?_", [{"full_name": "CategoryTheory.cancel_epi", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 19]}, {"full_name": "CategoryTheory.MonoidalCategory.tensorLeft", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [859, 5], "def_end_pos": [859, 15]}, {"full_name": "Prefunctor.map", "def_path": "Mathlib/Combinatorics/Quiver/Basic.lean", "def_pos": [61, 3], "def_end_pos": [61, 6]}, {"full_name": "CategoryTheory.Limits.coequalizer.\u03c0", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [954, 22], "def_end_pos": [954, 35]}]], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 TensorBimod.actLeft (P.tensorBimod Q) L \u226b\n      coequalizer.desc\n        ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          coequalizer.desc\n            ((\u03b1_ P.X Q.X L.X).hom \u226b\n              P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                  ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n            \u22ef)\n        \u22ef =\n    R.X \u25c1\n        coequalizer.desc\n          ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                    ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef)\n          \u22ef \u226b\n      TensorBimod.actLeft P (Q.tensorBimod L)", "state_after": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (tensorLeft R.X).map\n        (coequalizer.\u03c0 ((P.tensorBimod Q).actRight \u25b7 L.X)\n          ((\u03b1_ (P.tensorBimod Q).X T.X L.X).hom \u226b (P.tensorBimod Q).X \u25c1 L.actLeft)) \u226b\n      TensorBimod.actLeft (P.tensorBimod Q) L \u226b\n        coequalizer.desc\n          ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                    ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef)\n          \u22ef =\n    (tensorLeft R.X).map\n        (coequalizer.\u03c0 ((P.tensorBimod Q).actRight \u25b7 L.X)\n          ((\u03b1_ (P.tensorBimod Q).X T.X L.X).hom \u226b (P.tensorBimod Q).X \u25c1 L.actLeft)) \u226b\n      R.X \u25c1\n          coequalizer.desc\n            ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                  ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n              coequalizer.desc\n                ((\u03b1_ P.X Q.X L.X).hom \u226b\n                  P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                    coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                      ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n                \u22ef)\n            \u22ef \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)"}, {"tactic": "rw [tensorLeft_map]", "annotated_tactic": ["rw [<a>tensorLeft_map</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.tensorLeft_map", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [858, 3], "def_end_pos": [858, 9]}]], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (tensorLeft R.X).map\n        (coequalizer.\u03c0 ((P.tensorBimod Q).actRight \u25b7 L.X)\n          ((\u03b1_ (P.tensorBimod Q).X T.X L.X).hom \u226b (P.tensorBimod Q).X \u25c1 L.actLeft)) \u226b\n      TensorBimod.actLeft (P.tensorBimod Q) L \u226b\n        coequalizer.desc\n          ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                    ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef)\n          \u22ef =\n    (tensorLeft R.X).map\n        (coequalizer.\u03c0 ((P.tensorBimod Q).actRight \u25b7 L.X)\n          ((\u03b1_ (P.tensorBimod Q).X T.X L.X).hom \u226b (P.tensorBimod Q).X \u25c1 L.actLeft)) \u226b\n      R.X \u25c1\n          coequalizer.desc\n            ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                  ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n              coequalizer.desc\n                ((\u03b1_ P.X Q.X L.X).hom \u226b\n                  P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                    coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                      ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n                \u22ef)\n            \u22ef \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)", "state_after": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 R.X \u25c1\n        coequalizer.\u03c0 ((P.tensorBimod Q).actRight \u25b7 L.X)\n          ((\u03b1_ (P.tensorBimod Q).X T.X L.X).hom \u226b (P.tensorBimod Q).X \u25c1 L.actLeft) \u226b\n      TensorBimod.actLeft (P.tensorBimod Q) L \u226b\n        coequalizer.desc\n          ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                    ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef)\n          \u22ef =\n    R.X \u25c1\n        coequalizer.\u03c0 ((P.tensorBimod Q).actRight \u25b7 L.X)\n          ((\u03b1_ (P.tensorBimod Q).X T.X L.X).hom \u226b (P.tensorBimod Q).X \u25c1 L.actLeft) \u226b\n      R.X \u25c1\n          coequalizer.desc\n            ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                  ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n              coequalizer.desc\n                ((\u03b1_ P.X Q.X L.X).hom \u226b\n                  P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                    coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                      ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n                \u22ef)\n            \u22ef \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)"}, {"tactic": "slice_lhs 1 2 => rw [TensorBimod.whiskerLeft_\u03c0_actLeft]", "annotated_tactic": ["slice_lhs 1 2 => rw [<a>TensorBimod.whiskerLeft_\u03c0_actLeft</a>]", [{"full_name": "Bimod.TensorBimod.whiskerLeft_\u03c0_actLeft", "def_path": "Mathlib/CategoryTheory/Monoidal/Bimod.lean", "def_pos": [242, 9], "def_end_pos": [242, 30]}]], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 R.X \u25c1\n        coequalizer.\u03c0 ((P.tensorBimod Q).actRight \u25b7 L.X)\n          ((\u03b1_ (P.tensorBimod Q).X T.X L.X).hom \u226b (P.tensorBimod Q).X \u25c1 L.actLeft) \u226b\n      TensorBimod.actLeft (P.tensorBimod Q) L \u226b\n        coequalizer.desc\n          ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                    ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef)\n          \u22ef =\n    R.X \u25c1\n        coequalizer.\u03c0 ((P.tensorBimod Q).actRight \u25b7 L.X)\n          ((\u03b1_ (P.tensorBimod Q).X T.X L.X).hom \u226b (P.tensorBimod Q).X \u25c1 L.actLeft) \u226b\n      R.X \u25c1\n          coequalizer.desc\n            ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                  ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n              coequalizer.desc\n                ((\u03b1_ P.X Q.X L.X).hom \u226b\n                  P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                    coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                      ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n                \u22ef)\n            \u22ef \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)", "state_after": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 ((\u03b1_ R.X (P.tensorBimod Q).X L.X).inv \u226b\n        (P.tensorBimod Q).actLeft \u25b7 L.X \u226b\n          coequalizer.\u03c0 ((P.tensorBimod Q).actRight \u25b7 L.X)\n            ((\u03b1_ (P.tensorBimod Q).X T.X L.X).hom \u226b (P.tensorBimod Q).X \u25c1 L.actLeft)) \u226b\n      coequalizer.desc\n        ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          coequalizer.desc\n            ((\u03b1_ P.X Q.X L.X).hom \u226b\n              P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                  ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n            \u22ef)\n        \u22ef =\n    R.X \u25c1\n        coequalizer.\u03c0 ((P.tensorBimod Q).actRight \u25b7 L.X)\n          ((\u03b1_ (P.tensorBimod Q).X T.X L.X).hom \u226b (P.tensorBimod Q).X \u25c1 L.actLeft) \u226b\n      R.X \u25c1\n          coequalizer.desc\n            ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                  ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n              coequalizer.desc\n                ((\u03b1_ P.X Q.X L.X).hom \u226b\n                  P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                    coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                      ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n                \u22ef)\n            \u22ef \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)"}, {"tactic": "slice_lhs 3 4 => rw [coequalizer.\u03c0_desc]", "annotated_tactic": ["slice_lhs 3 4 => rw [<a>coequalizer.\u03c0_desc</a>]", [{"full_name": "CategoryTheory.Limits.coequalizer.\u03c0_desc", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [996, 9], "def_end_pos": [996, 27]}]], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 ((\u03b1_ R.X (P.tensorBimod Q).X L.X).inv \u226b\n        (P.tensorBimod Q).actLeft \u25b7 L.X \u226b\n          coequalizer.\u03c0 ((P.tensorBimod Q).actRight \u25b7 L.X)\n            ((\u03b1_ (P.tensorBimod Q).X T.X L.X).hom \u226b (P.tensorBimod Q).X \u25c1 L.actLeft)) \u226b\n      coequalizer.desc\n        ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          coequalizer.desc\n            ((\u03b1_ P.X Q.X L.X).hom \u226b\n              P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                  ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n            \u22ef)\n        \u22ef =\n    R.X \u25c1\n        coequalizer.\u03c0 ((P.tensorBimod Q).actRight \u25b7 L.X)\n          ((\u03b1_ (P.tensorBimod Q).X T.X L.X).hom \u226b (P.tensorBimod Q).X \u25c1 L.actLeft) \u226b\n      R.X \u25c1\n          coequalizer.desc\n            ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                  ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n              coequalizer.desc\n                ((\u03b1_ P.X Q.X L.X).hom \u226b\n                  P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                    coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                      ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n                \u22ef)\n            \u22ef \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)", "state_after": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (\u03b1_ R.X (P.tensorBimod Q).X L.X).inv \u226b\n      (P.tensorBimod Q).actLeft \u25b7 L.X \u226b\n        (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          coequalizer.desc\n            ((\u03b1_ P.X Q.X L.X).hom \u226b\n              P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                  ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n            \u22ef =\n    R.X \u25c1\n        coequalizer.\u03c0 ((P.tensorBimod Q).actRight \u25b7 L.X)\n          ((\u03b1_ (P.tensorBimod Q).X T.X L.X).hom \u226b (P.tensorBimod Q).X \u25c1 L.actLeft) \u226b\n      R.X \u25c1\n          coequalizer.desc\n            ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                  ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n              coequalizer.desc\n                ((\u03b1_ P.X Q.X L.X).hom \u226b\n                  P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                    coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                      ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n                \u22ef)\n            \u22ef \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)"}, {"tactic": "slice_rhs 1 2 => rw [\u2190 MonoidalCategory.whiskerLeft_comp, coequalizer.\u03c0_desc,\n  MonoidalCategory.whiskerLeft_comp]", "annotated_tactic": ["slice_rhs 1 2 => rw [\u2190 <a>MonoidalCategory.whiskerLeft_comp</a>, <a>coequalizer.\u03c0_desc</a>,\n    <a>MonoidalCategory.whiskerLeft_comp</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.whiskerLeft_comp", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [235, 9], "def_end_pos": [235, 25]}, {"full_name": "CategoryTheory.Limits.coequalizer.\u03c0_desc", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [996, 9], "def_end_pos": [996, 27]}, {"full_name": "CategoryTheory.MonoidalCategory.whiskerLeft_comp", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [235, 9], "def_end_pos": [235, 25]}]], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (\u03b1_ R.X (P.tensorBimod Q).X L.X).inv \u226b\n      (P.tensorBimod Q).actLeft \u25b7 L.X \u226b\n        (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          coequalizer.desc\n            ((\u03b1_ P.X Q.X L.X).hom \u226b\n              P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                  ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n            \u22ef =\n    R.X \u25c1\n        coequalizer.\u03c0 ((P.tensorBimod Q).actRight \u25b7 L.X)\n          ((\u03b1_ (P.tensorBimod Q).X T.X L.X).hom \u226b (P.tensorBimod Q).X \u25c1 L.actLeft) \u226b\n      R.X \u25c1\n          coequalizer.desc\n            ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                  ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n              coequalizer.desc\n                ((\u03b1_ P.X Q.X L.X).hom \u226b\n                  P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                    coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                      ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n                \u22ef)\n            \u22ef \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)", "state_after": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (\u03b1_ R.X (P.tensorBimod Q).X L.X).inv \u226b\n      (P.tensorBimod Q).actLeft \u25b7 L.X \u226b\n        (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          coequalizer.desc\n            ((\u03b1_ P.X Q.X L.X).hom \u226b\n              P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                  ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n            \u22ef =\n    (R.X \u25c1\n          (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n        R.X \u25c1\n          coequalizer.desc\n            ((\u03b1_ P.X Q.X L.X).hom \u226b\n              P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                  ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n            \u22ef) \u226b\n      TensorBimod.actLeft P (Q.tensorBimod L)"}, {"tactic": "refine (cancel_epi ((tensorRight _ \u22d9 tensorLeft _).map (coequalizer.\u03c0 _ _))).1 ?_", "annotated_tactic": ["refine (<a>cancel_epi</a> ((<a>tensorRight</a> _ \u22d9 <a>tensorLeft</a> _).<a>map</a> (<a>coequalizer.\u03c0</a> _ _))).1 ?_", [{"full_name": "CategoryTheory.cancel_epi", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 19]}, {"full_name": "CategoryTheory.MonoidalCategory.tensorRight", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [864, 5], "def_end_pos": [864, 16]}, {"full_name": "CategoryTheory.MonoidalCategory.tensorLeft", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [859, 5], "def_end_pos": [859, 15]}, {"full_name": "Prefunctor.map", "def_path": "Mathlib/Combinatorics/Quiver/Basic.lean", "def_pos": [61, 3], "def_end_pos": [61, 6]}, {"full_name": "CategoryTheory.Limits.coequalizer.\u03c0", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [954, 22], "def_end_pos": [954, 35]}]], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (\u03b1_ R.X (P.tensorBimod Q).X L.X).inv \u226b\n      (P.tensorBimod Q).actLeft \u25b7 L.X \u226b\n        (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          coequalizer.desc\n            ((\u03b1_ P.X Q.X L.X).hom \u226b\n              P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                  ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n            \u22ef =\n    (R.X \u25c1\n          (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n        R.X \u25c1\n          coequalizer.desc\n            ((\u03b1_ P.X Q.X L.X).hom \u226b\n              P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                  ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n            \u22ef) \u226b\n      TensorBimod.actLeft P (Q.tensorBimod L)", "state_after": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (tensorRight L.X \u22d9 tensorLeft R.X).map (coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)) \u226b\n      (\u03b1_ R.X (P.tensorBimod Q).X L.X).inv \u226b\n        (P.tensorBimod Q).actLeft \u25b7 L.X \u226b\n          (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                    ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef =\n    (tensorRight L.X \u22d9 tensorLeft R.X).map (coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)) \u226b\n      (R.X \u25c1\n            (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          R.X \u25c1\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                    ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef) \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (tensorRight L.X \u22d9 tensorLeft R.X).map (coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)) \u226b\n      (\u03b1_ R.X (P.tensorBimod Q).X L.X).inv \u226b\n        (P.tensorBimod Q).actLeft \u25b7 L.X \u226b\n          (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                    ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef =\n    (tensorRight L.X \u22d9 tensorLeft R.X).map (coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)) \u226b\n      (R.X \u25c1\n            (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          R.X \u25c1\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                    ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef) \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)", "state_after": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 R.X \u25c1 coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft) \u25b7 L.X \u226b\n      (\u03b1_ R.X (TensorBimod.X P Q) L.X).inv \u226b\n        TensorBimod.actLeft P Q \u25b7 L.X \u226b\n          (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                    ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef =\n    R.X \u25c1 coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft) \u25b7 L.X \u226b\n      (R.X \u25c1\n            (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          R.X \u25c1\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                    ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef) \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)"}, {"tactic": "dsimp [TensorBimod.X]", "annotated_tactic": ["dsimp [<a>TensorBimod.X</a>]", [{"full_name": "Bimod.TensorBimod.X", "def_path": "Mathlib/CategoryTheory/Monoidal/Bimod.lean", "def_pos": [211, 19], "def_end_pos": [211, 20]}]], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 R.X \u25c1 coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft) \u25b7 L.X \u226b\n      (\u03b1_ R.X (TensorBimod.X P Q) L.X).inv \u226b\n        TensorBimod.actLeft P Q \u25b7 L.X \u226b\n          (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                    ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef =\n    R.X \u25c1 coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft) \u25b7 L.X \u226b\n      (R.X \u25c1\n            (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          R.X \u25c1\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 TensorBimod.X Q L)\n                    ((\u03b1_ P.X S.X (TensorBimod.X Q L)).hom \u226b P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef) \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)", "state_after": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 R.X \u25c1 coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft) \u25b7 L.X \u226b\n      (\u03b1_ R.X (coequalizer (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)) L.X).inv \u226b\n        TensorBimod.actLeft P Q \u25b7 L.X \u226b\n          (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n                    ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n                      P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef =\n    R.X \u25c1 coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft) \u25b7 L.X \u226b\n      (R.X \u25c1\n            (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          R.X \u25c1\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n                    ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n                      P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef) \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)"}, {"tactic": "slice_lhs 1 2 => rw [associator_inv_naturality_middle]", "annotated_tactic": ["slice_lhs 1 2 => rw [<a>associator_inv_naturality_middle</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.associator_inv_naturality_middle", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [474, 9], "def_end_pos": [474, 41]}]], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 R.X \u25c1 coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft) \u25b7 L.X \u226b\n      (\u03b1_ R.X (coequalizer (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)) L.X).inv \u226b\n        TensorBimod.actLeft P Q \u25b7 L.X \u226b\n          (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n                    ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n                      P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef =\n    R.X \u25c1 coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft) \u25b7 L.X \u226b\n      (R.X \u25c1\n            (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          R.X \u25c1\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n                    ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n                      P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef) \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)", "state_after": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 ((((\u03b1_ R.X (P.X \u2297 Q.X) L.X).inv \u226b\n            (R.X \u25c1 coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)) \u25b7 L.X) \u226b\n          TensorBimod.actLeft P Q \u25b7 L.X) \u226b\n        (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv) \u226b\n      coequalizer.desc\n        ((\u03b1_ P.X Q.X L.X).hom \u226b\n          P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n            coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n              ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n                P.X \u25c1 TensorBimod.actLeft Q L))\n        \u22ef =\n    R.X \u25c1 coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft) \u25b7 L.X \u226b\n      (R.X \u25c1\n            (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          R.X \u25c1\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n                    ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n                      P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef) \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)"}, {"tactic": "slice_lhs 2 3 =>\n  rw [\u2190 comp_whiskerRight, TensorBimod.whiskerLeft_\u03c0_actLeft,\n    comp_whiskerRight, comp_whiskerRight]", "annotated_tactic": ["slice_lhs 2 3 =>\n    rw [\u2190 <a>comp_whiskerRight</a>, <a>TensorBimod.whiskerLeft_\u03c0_actLeft</a>,\n      <a>comp_whiskerRight</a>, <a>comp_whiskerRight</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.comp_whiskerRight", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [253, 9], "def_end_pos": [253, 26]}, {"full_name": "Bimod.TensorBimod.whiskerLeft_\u03c0_actLeft", "def_path": "Mathlib/CategoryTheory/Monoidal/Bimod.lean", "def_pos": [242, 9], "def_end_pos": [242, 30]}, {"full_name": "CategoryTheory.MonoidalCategory.comp_whiskerRight", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [253, 9], "def_end_pos": [253, 26]}, {"full_name": "CategoryTheory.MonoidalCategory.comp_whiskerRight", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [253, 9], "def_end_pos": [253, 26]}]], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 ((((\u03b1_ R.X (P.X \u2297 Q.X) L.X).inv \u226b\n            (R.X \u25c1 coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)) \u25b7 L.X) \u226b\n          TensorBimod.actLeft P Q \u25b7 L.X) \u226b\n        (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv) \u226b\n      coequalizer.desc\n        ((\u03b1_ P.X Q.X L.X).hom \u226b\n          P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n            coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n              ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n                P.X \u25c1 TensorBimod.actLeft Q L))\n        \u22ef =\n    R.X \u25c1 coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft) \u25b7 L.X \u226b\n      (R.X \u25c1\n            (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          R.X \u25c1\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n                    ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n                      P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef) \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)", "state_after": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (\u03b1_ R.X (P.X \u2297 Q.X) L.X).inv \u226b\n      (((\u03b1_ R.X P.X Q.X).inv \u25b7 L.X \u226b\n            P.actLeft \u25b7 Q.X \u25b7 L.X \u226b coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft) \u25b7 L.X) \u226b\n          (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n              ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv) \u226b\n        coequalizer.desc\n          ((\u03b1_ P.X Q.X L.X).hom \u226b\n            P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n              coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n                ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n                  P.X \u25c1 TensorBimod.actLeft Q L))\n          \u22ef =\n    R.X \u25c1 coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft) \u25b7 L.X \u226b\n      (R.X \u25c1\n            (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          R.X \u25c1\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n                    ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n                      P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef) \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)"}, {"tactic": "slice_lhs 4 6 => rw [\u03c0_tensor_id_preserves_coequalizer_inv_desc]", "annotated_tactic": ["slice_lhs 4 6 => rw [<a>\u03c0_tensor_id_preserves_coequalizer_inv_desc</a>]", [{"full_name": "\u03c0_tensor_id_preserves_coequalizer_inv_desc", "def_path": "Mathlib/CategoryTheory/Monoidal/Bimod.lean", "def_pos": [59, 9], "def_end_pos": [59, 51]}]], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (\u03b1_ R.X (P.X \u2297 Q.X) L.X).inv \u226b\n      (((\u03b1_ R.X P.X Q.X).inv \u25b7 L.X \u226b\n            P.actLeft \u25b7 Q.X \u25b7 L.X \u226b coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft) \u25b7 L.X) \u226b\n          (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n              ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv) \u226b\n        coequalizer.desc\n          ((\u03b1_ P.X Q.X L.X).hom \u226b\n            P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n              coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n                ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n                  P.X \u25c1 TensorBimod.actLeft Q L))\n          \u22ef =\n    R.X \u25c1 coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft) \u25b7 L.X \u226b\n      (R.X \u25c1\n            (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          R.X \u25c1\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n                    ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n                      P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef) \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)", "state_after": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (\u03b1_ R.X (P.X \u2297 Q.X) L.X).inv \u226b\n      (\u03b1_ R.X P.X Q.X).inv \u25b7 L.X \u226b\n        P.actLeft \u25b7 Q.X \u25b7 L.X \u226b\n          (\u03b1_ P.X Q.X L.X).hom \u226b\n            P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n              coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n                ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n                  P.X \u25c1 TensorBimod.actLeft Q L) =\n    R.X \u25c1 coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft) \u25b7 L.X \u226b\n      (R.X \u25c1\n            (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          R.X \u25c1\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n                    ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n                      P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef) \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)"}, {"tactic": "slice_lhs 3 4 => rw [associator_naturality_left]", "annotated_tactic": ["slice_lhs 3 4 => rw [<a>associator_naturality_left</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.associator_naturality_left", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [458, 9], "def_end_pos": [458, 35]}]], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (\u03b1_ R.X (P.X \u2297 Q.X) L.X).inv \u226b\n      (\u03b1_ R.X P.X Q.X).inv \u25b7 L.X \u226b\n        P.actLeft \u25b7 Q.X \u25b7 L.X \u226b\n          (\u03b1_ P.X Q.X L.X).hom \u226b\n            P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n              coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n                ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n                  P.X \u25c1 TensorBimod.actLeft Q L) =\n    R.X \u25c1 coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft) \u25b7 L.X \u226b\n      (R.X \u25c1\n            (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          R.X \u25c1\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n                    ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n                      P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef) \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)", "state_after": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (\u03b1_ R.X (P.X \u2297 Q.X) L.X).inv \u226b\n      (\u03b1_ R.X P.X Q.X).inv \u25b7 L.X \u226b\n        (((\u03b1_ (R.X \u2297 P.X) Q.X L.X).hom \u226b P.actLeft \u25b7 (Q.X \u2297 L.X)) \u226b\n            P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft)) \u226b\n          coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n            ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n              P.X \u25c1 TensorBimod.actLeft Q L) =\n    R.X \u25c1 coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft) \u25b7 L.X \u226b\n      (R.X \u25c1\n            (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          R.X \u25c1\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n                    ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n                      P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef) \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)"}, {"tactic": "slice_rhs 1 3 =>\n  rw [\u2190 MonoidalCategory.whiskerLeft_comp, \u2190 MonoidalCategory.whiskerLeft_comp,\n    \u03c0_tensor_id_preserves_coequalizer_inv_desc, MonoidalCategory.whiskerLeft_comp,\n    MonoidalCategory.whiskerLeft_comp]", "annotated_tactic": ["slice_rhs 1 3 =>\n    rw [\u2190 <a>MonoidalCategory.whiskerLeft_comp</a>, \u2190 <a>MonoidalCategory.whiskerLeft_comp</a>,\n      <a>\u03c0_tensor_id_preserves_coequalizer_inv_desc</a>, <a>MonoidalCategory.whiskerLeft_comp</a>,\n      <a>MonoidalCategory.whiskerLeft_comp</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.whiskerLeft_comp", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [235, 9], "def_end_pos": [235, 25]}, {"full_name": "CategoryTheory.MonoidalCategory.whiskerLeft_comp", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [235, 9], "def_end_pos": [235, 25]}, {"full_name": "\u03c0_tensor_id_preserves_coequalizer_inv_desc", "def_path": "Mathlib/CategoryTheory/Monoidal/Bimod.lean", "def_pos": [59, 9], "def_end_pos": [59, 51]}, {"full_name": "CategoryTheory.MonoidalCategory.whiskerLeft_comp", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [235, 9], "def_end_pos": [235, 25]}, {"full_name": "CategoryTheory.MonoidalCategory.whiskerLeft_comp", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [235, 9], "def_end_pos": [235, 25]}]], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (\u03b1_ R.X (P.X \u2297 Q.X) L.X).inv \u226b\n      (\u03b1_ R.X P.X Q.X).inv \u25b7 L.X \u226b\n        (((\u03b1_ (R.X \u2297 P.X) Q.X L.X).hom \u226b P.actLeft \u25b7 (Q.X \u2297 L.X)) \u226b\n            P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft)) \u226b\n          coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n            ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n              P.X \u25c1 TensorBimod.actLeft Q L) =\n    R.X \u25c1 coequalizer.\u03c0 (P.actRight \u25b7 Q.X) ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft) \u25b7 L.X \u226b\n      (R.X \u25c1\n            (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight \u25b7 Q.X)\n                ((\u03b1_ P.X S.X Q.X).hom \u226b P.X \u25c1 Q.actLeft)).inv \u226b\n          R.X \u25c1\n            coequalizer.desc\n              ((\u03b1_ P.X Q.X L.X).hom \u226b\n                P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n                  coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n                    ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n                      P.X \u25c1 TensorBimod.actLeft Q L))\n              \u22ef) \u226b\n        TensorBimod.actLeft P (Q.tensorBimod L)", "state_after": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (\u03b1_ R.X (P.X \u2297 Q.X) L.X).inv \u226b\n      (\u03b1_ R.X P.X Q.X).inv \u25b7 L.X \u226b\n        (((\u03b1_ (R.X \u2297 P.X) Q.X L.X).hom \u226b P.actLeft \u25b7 (Q.X \u2297 L.X)) \u226b\n            P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft)) \u226b\n          coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n            ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n              P.X \u25c1 TensorBimod.actLeft Q L) =\n    (R.X \u25c1 (\u03b1_ P.X Q.X L.X).hom \u226b\n        R.X \u25c1 P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n          R.X \u25c1\n            coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n              ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n                P.X \u25c1 TensorBimod.actLeft Q L)) \u226b\n      TensorBimod.actLeft P (Q.tensorBimod L)"}, {"tactic": "slice_rhs 3 4 => erw [TensorBimod.whiskerLeft_\u03c0_actLeft P (Q.tensorBimod L)]", "annotated_tactic": ["slice_rhs 3 4 => erw [<a>TensorBimod.whiskerLeft_\u03c0_actLeft</a> P (Q.tensorBimod L)]", [{"full_name": "Bimod.TensorBimod.whiskerLeft_\u03c0_actLeft", "def_path": "Mathlib/CategoryTheory/Monoidal/Bimod.lean", "def_pos": [242, 9], "def_end_pos": [242, 30]}]], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (\u03b1_ R.X (P.X \u2297 Q.X) L.X).inv \u226b\n      (\u03b1_ R.X P.X Q.X).inv \u25b7 L.X \u226b\n        (((\u03b1_ (R.X \u2297 P.X) Q.X L.X).hom \u226b P.actLeft \u25b7 (Q.X \u2297 L.X)) \u226b\n            P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft)) \u226b\n          coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n            ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n              P.X \u25c1 TensorBimod.actLeft Q L) =\n    (R.X \u25c1 (\u03b1_ P.X Q.X L.X).hom \u226b\n        R.X \u25c1 P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n          R.X \u25c1\n            coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n              ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n                P.X \u25c1 TensorBimod.actLeft Q L)) \u226b\n      TensorBimod.actLeft P (Q.tensorBimod L)", "state_after": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (\u03b1_ R.X (P.X \u2297 Q.X) L.X).inv \u226b\n      (\u03b1_ R.X P.X Q.X).inv \u25b7 L.X \u226b\n        (((\u03b1_ (R.X \u2297 P.X) Q.X L.X).hom \u226b P.actLeft \u25b7 (Q.X \u2297 L.X)) \u226b\n            P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft)) \u226b\n          coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n            ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n              P.X \u25c1 TensorBimod.actLeft Q L) =\n    R.X \u25c1 (\u03b1_ P.X Q.X L.X).hom \u226b\n      R.X \u25c1 P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n        (\u03b1_ R.X P.X (Q.tensorBimod L).X).inv \u226b\n          P.actLeft \u25b7 (Q.tensorBimod L).X \u226b\n            coequalizer.\u03c0 (P.actRight \u25b7 (Q.tensorBimod L).X)\n              ((\u03b1_ P.X S.X (Q.tensorBimod L).X).hom \u226b P.X \u25c1 (Q.tensorBimod L).actLeft)"}, {"tactic": "slice_rhs 2 3 => erw [associator_inv_naturality_right]", "annotated_tactic": ["slice_rhs 2 3 => erw [<a>associator_inv_naturality_right</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.associator_inv_naturality_right", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [486, 9], "def_end_pos": [486, 40]}]], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (\u03b1_ R.X (P.X \u2297 Q.X) L.X).inv \u226b\n      (\u03b1_ R.X P.X Q.X).inv \u25b7 L.X \u226b\n        (((\u03b1_ (R.X \u2297 P.X) Q.X L.X).hom \u226b P.actLeft \u25b7 (Q.X \u2297 L.X)) \u226b\n            P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft)) \u226b\n          coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n            ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n              P.X \u25c1 TensorBimod.actLeft Q L) =\n    R.X \u25c1 (\u03b1_ P.X Q.X L.X).hom \u226b\n      R.X \u25c1 P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft) \u226b\n        (\u03b1_ R.X P.X (Q.tensorBimod L).X).inv \u226b\n          P.actLeft \u25b7 (Q.tensorBimod L).X \u226b\n            coequalizer.\u03c0 (P.actRight \u25b7 (Q.tensorBimod L).X)\n              ((\u03b1_ P.X S.X (Q.tensorBimod L).X).hom \u226b P.X \u25c1 (Q.tensorBimod L).actLeft)", "state_after": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (\u03b1_ R.X (P.X \u2297 Q.X) L.X).inv \u226b\n      (\u03b1_ R.X P.X Q.X).inv \u25b7 L.X \u226b\n        (((\u03b1_ (R.X \u2297 P.X) Q.X L.X).hom \u226b P.actLeft \u25b7 (Q.X \u2297 L.X)) \u226b\n            P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft)) \u226b\n          coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n            ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n              P.X \u25c1 TensorBimod.actLeft Q L) =\n    R.X \u25c1 (\u03b1_ P.X Q.X L.X).hom \u226b\n      (((\u03b1_ R.X P.X (Q.X \u2297 L.X)).inv \u226b\n            (R.X \u2297 P.X) \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft)) \u226b\n          P.actLeft \u25b7 (Q.tensorBimod L).X) \u226b\n        coequalizer.\u03c0 (P.actRight \u25b7 (Q.tensorBimod L).X)\n          ((\u03b1_ P.X S.X (Q.tensorBimod L).X).hom \u226b P.X \u25c1 (Q.tensorBimod L).actLeft)"}, {"tactic": "slice_rhs 3 4 => erw [whisker_exchange]", "annotated_tactic": ["slice_rhs 3 4 => erw [<a>whisker_exchange</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.whisker_exchange", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [278, 9], "def_end_pos": [278, 25]}]], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (\u03b1_ R.X (P.X \u2297 Q.X) L.X).inv \u226b\n      (\u03b1_ R.X P.X Q.X).inv \u25b7 L.X \u226b\n        (((\u03b1_ (R.X \u2297 P.X) Q.X L.X).hom \u226b P.actLeft \u25b7 (Q.X \u2297 L.X)) \u226b\n            P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft)) \u226b\n          coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n            ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n              P.X \u25c1 TensorBimod.actLeft Q L) =\n    R.X \u25c1 (\u03b1_ P.X Q.X L.X).hom \u226b\n      (((\u03b1_ R.X P.X (Q.X \u2297 L.X)).inv \u226b\n            (R.X \u2297 P.X) \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft)) \u226b\n          P.actLeft \u25b7 (Q.tensorBimod L).X) \u226b\n        coequalizer.\u03c0 (P.actRight \u25b7 (Q.tensorBimod L).X)\n          ((\u03b1_ P.X S.X (Q.tensorBimod L).X).hom \u226b P.X \u25c1 (Q.tensorBimod L).actLeft)", "state_after": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (\u03b1_ R.X (P.X \u2297 Q.X) L.X).inv \u226b\n      (\u03b1_ R.X P.X Q.X).inv \u25b7 L.X \u226b\n        (((\u03b1_ (R.X \u2297 P.X) Q.X L.X).hom \u226b P.actLeft \u25b7 (Q.X \u2297 L.X)) \u226b\n            P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft)) \u226b\n          coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n            ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n              P.X \u25c1 TensorBimod.actLeft Q L) =\n    R.X \u25c1 (\u03b1_ P.X Q.X L.X).hom \u226b\n      (\u03b1_ R.X P.X (Q.X \u2297 L.X)).inv \u226b\n        (P.actLeft \u25b7 (Q.X \u2297 L.X) \u226b P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft)) \u226b\n          coequalizer.\u03c0 (P.actRight \u25b7 (Q.tensorBimod L).X)\n            ((\u03b1_ P.X S.X (Q.tensorBimod L).X).hom \u226b P.X \u25c1 (Q.tensorBimod L).actLeft)"}, {"tactic": "coherence", "annotated_tactic": ["coherence", []], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nR S T U : Mon_ C\nP : Bimod R S\nQ : Bimod S T\nL : Bimod T U\n\u22a2 (\u03b1_ R.X (P.X \u2297 Q.X) L.X).inv \u226b\n      (\u03b1_ R.X P.X Q.X).inv \u25b7 L.X \u226b\n        (((\u03b1_ (R.X \u2297 P.X) Q.X L.X).hom \u226b P.actLeft \u25b7 (Q.X \u2297 L.X)) \u226b\n            P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft)) \u226b\n          coequalizer.\u03c0 (P.actRight \u25b7 coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))\n            ((\u03b1_ P.X S.X (coequalizer (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft))).hom \u226b\n              P.X \u25c1 TensorBimod.actLeft Q L) =\n    R.X \u25c1 (\u03b1_ P.X Q.X L.X).hom \u226b\n      (\u03b1_ R.X P.X (Q.X \u2297 L.X)).inv \u226b\n        (P.actLeft \u25b7 (Q.X \u2297 L.X) \u226b P.X \u25c1 coequalizer.\u03c0 (Q.actRight \u25b7 L.X) ((\u03b1_ Q.X T.X L.X).hom \u226b Q.X \u25c1 L.actLeft)) \u226b\n          coequalizer.\u03c0 (P.actRight \u25b7 (Q.tensorBimod L).X)\n            ((\u03b1_ P.X S.X (Q.tensorBimod L).X).hom \u226b P.X \u25c1 (Q.tensorBimod L).actLeft)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Int/Bitwise.lean", "full_name": "Int.lor_bit", "start": [402, 1], "end": [403, 33], "traced_tactics": [{"tactic": "rw [\u2190 bitwise_or, bitwise_bit]", "annotated_tactic": ["rw [\u2190 <a>bitwise_or</a>, <a>bitwise_bit</a>]", [{"full_name": "Int.bitwise_or", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [318, 9], "def_end_pos": [318, 19]}, {"full_name": "Int.bitwise_bit", "def_path": "Mathlib/Data/Int/Bitwise.lean", "def_pos": [390, 9], "def_end_pos": [390, 20]}]], "state_before": "a : Bool\nm : \u2124\nb : Bool\nn : \u2124\n\u22a2 (bit a m).lor (bit b n) = bit (a || b) (m.lor n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/StrictConvexSpace.lean", "full_name": "combo_mem_ball_of_ne", "start": [152, 1], "end": [158, 60], "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": [212, 9], "def_end_pos": [212, 17]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nhx : x \u2208 closedBall z r\nhy : y \u2208 closedBall z r\nhne : x \u2260 y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 x + b \u2022 y \u2208 ball z r", "state_after": "case inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b : \u211d\nhne : x \u2260 y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx : x \u2208 closedBall z 0\nhy : y \u2208 closedBall z 0\n\u22a2 a \u2022 x + b \u2022 y \u2208 ball z 0\n\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nhx : x \u2208 closedBall z r\nhy : y \u2208 closedBall z r\nhne : x \u2260 y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhr : r \u2260 0\n\u22a2 a \u2022 x + b \u2022 y \u2208 ball z r"}, {"tactic": "rw [closedBall_zero, mem_singleton_iff] at hx hy", "annotated_tactic": ["rw [<a>closedBall_zero</a>, <a>mem_singleton_iff</a>] at hx hy", [{"full_name": "Metric.closedBall_zero", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [115, 17], "def_end_pos": [115, 32]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 26]}]], "state_before": "case inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b : \u211d\nhne : x \u2260 y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx : x \u2208 closedBall z 0\nhy : y \u2208 closedBall z 0\n\u22a2 a \u2022 x + b \u2022 y \u2208 ball z 0", "state_after": "case inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b : \u211d\nhne : x \u2260 y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx : x = z\nhy : y = z\n\u22a2 a \u2022 x + b \u2022 y \u2208 ball z 0"}, {"tactic": "exact (hne (hx.trans hy.symm)).elim", "annotated_tactic": ["exact (hne (hx.trans hy.symm)).<a>elim</a>", [{"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "case inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b : \u211d\nhne : x \u2260 y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx : x = z\nhy : y = z\n\u22a2 a \u2022 x + b \u2022 y \u2208 ball z 0", "state_after": "no goals"}, {"tactic": "simp only [\u2190 interior_closedBall _ hr] at hx hy \u22a2", "annotated_tactic": ["simp only [\u2190 <a>interior_closedBall</a> _ hr] at hx hy \u22a2", [{"full_name": "interior_closedBall", "def_path": "Mathlib/Analysis/NormedSpace/Real.lean", "def_pos": [81, 9], "def_end_pos": [81, 28]}]], "state_before": "case inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nhx : x \u2208 closedBall z r\nhy : y \u2208 closedBall z r\nhne : x \u2260 y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhr : r \u2260 0\n\u22a2 a \u2022 x + b \u2022 y \u2208 ball z r", "state_after": "case inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nhx : x \u2208 closedBall z r\nhy : y \u2208 closedBall z r\nhne : x \u2260 y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhr : r \u2260 0\n\u22a2 a \u2022 x + b \u2022 y \u2208 interior (closedBall z r)"}, {"tactic": "exact strictConvex_closedBall \u211d z r hx hy hne ha hb hab", "annotated_tactic": ["exact <a>strictConvex_closedBall</a> \u211d z r hx hy hne ha hb hab", [{"full_name": "strictConvex_closedBall", "def_path": "Mathlib/Analysis/Convex/StrictConvexSpace.lean", "def_pos": [76, 9], "def_end_pos": [76, 32]}]], "state_before": "case inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedLinearOrderedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : StrictConvexSpace \u211d E\nx y z : E\na b r : \u211d\nhx : x \u2208 closedBall z r\nhy : y \u2208 closedBall z r\nhne : x \u2260 y\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhr : r \u2260 0\n\u22a2 a \u2022 x + b \u2022 y \u2208 interior (closedBall z r)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.smul_monomial", "start": [288, 1], "end": [290, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SetFamily/Shadow.lean", "full_name": "Finset.mem_upShadow_iff_exists_mem_card_add_one", "start": [236, 1], "end": [244, 22], "traced_tactics": [{"tactic": "refine mem_upShadow_iff_exists_sdiff.trans <| exists_congr fun t \u21a6 and_congr_right fun _ \u21a6\n  and_congr_right fun hst \u21a6 ?_", "annotated_tactic": ["refine mem_upShadow_iff_exists_sdiff.trans <| <a>exists_congr</a> fun t \u21a6 <a>and_congr_right</a> fun _ \u21a6\n    <a>and_congr_right</a> fun hst \u21a6 ?_", [{"full_name": "exists_congr", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [210, 9], "def_end_pos": [210, 21]}, {"full_name": "and_congr_right", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [142, 9], "def_end_pos": [142, 24]}, {"full_name": "and_congr_right", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [142, 9], "def_end_pos": [142, 24]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns t : Finset \u03b1\na : \u03b1\nk r : \u2115\n\u22a2 t \u2208 \u2202\u207a \ud835\udc9c \u2194 \u2203 s \u2208 \ud835\udc9c, s \u2286 t \u2227 t.card = s.card + 1", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns t\u271d : Finset \u03b1\na : \u03b1\nk r : \u2115\nt : Finset \u03b1\nx\u271d : t \u2208 \ud835\udc9c\nhst : t \u2286 t\u271d\n\u22a2 (t\u271d \\ t).card = 1 \u2194 t\u271d.card = t.card + 1"}, {"tactic": "rw [card_sdiff hst, tsub_eq_iff_eq_add_of_le, add_comm]", "annotated_tactic": ["rw [<a>card_sdiff</a> hst, <a>tsub_eq_iff_eq_add_of_le</a>, <a>add_comm</a>]", [{"full_name": "Finset.card_sdiff", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [562, 9], "def_end_pos": [562, 19]}, {"full_name": "tsub_eq_iff_eq_add_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [210, 9], "def_end_pos": [210, 33]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns t\u271d : Finset \u03b1\na : \u03b1\nk r : \u2115\nt : Finset \u03b1\nx\u271d : t \u2208 \ud835\udc9c\nhst : t \u2286 t\u271d\n\u22a2 (t\u271d \\ t).card = 1 \u2194 t\u271d.card = t.card + 1", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns t\u271d : Finset \u03b1\na : \u03b1\nk r : \u2115\nt : Finset \u03b1\nx\u271d : t \u2208 \ud835\udc9c\nhst : t \u2286 t\u271d\n\u22a2 t.card \u2264 t\u271d.card"}, {"tactic": "exact card_mono hst", "annotated_tactic": ["exact <a>card_mono</a> hst", [{"full_name": "Finset.card_mono", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [69, 9], "def_end_pos": [69, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\ns t\u271d : Finset \u03b1\na : \u03b1\nk r : \u2115\nt : Finset \u03b1\nx\u271d : t \u2208 \ud835\udc9c\nhst : t \u2286 t\u271d\n\u22a2 t.card \u2264 t\u271d.card", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Group/List.lean", "full_name": "Commute.list_prod_right", "start": [304, 1], "end": [310, 41], "traced_tactics": [{"tactic": "induction' l with z l IH", "annotated_tactic": ["induction' l with z l IH", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na : M\nl : List M\ny : M\nh : \u2200 x \u2208 l, Commute y x\n\u22a2 Commute y l.prod", "state_after": "case nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na y : M\nh : \u2200 x \u2208 [], Commute y x\n\u22a2 Commute y [].prod\n\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na y z : M\nl : List M\nIH : (\u2200 x \u2208 l, Commute y x) \u2192 Commute y l.prod\nh : \u2200 x \u2208 z :: l, Commute y x\n\u22a2 Commute y (z :: l).prod"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na y : M\nh : \u2200 x \u2208 [], Commute y x\n\u22a2 Commute y [].prod", "state_after": "no goals"}, {"tactic": "rw [List.forall_mem_cons] at h", "annotated_tactic": ["rw [<a>List.forall_mem_cons</a>] at h", [{"full_name": "List.forall_mem_cons", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [305, 9], "def_end_pos": [305, 24]}]], "state_before": "case cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na y z : M\nl : List M\nIH : (\u2200 x \u2208 l, Commute y x) \u2192 Commute y l.prod\nh : \u2200 x \u2208 z :: l, Commute y x\n\u22a2 Commute y (z :: l).prod", "state_after": "case cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na y z : M\nl : List M\nIH : (\u2200 x \u2208 l, Commute y x) \u2192 Commute y l.prod\nh : Commute y z \u2227 \u2200 x \u2208 l, Commute y x\n\u22a2 Commute y (z :: l).prod"}, {"tactic": "rw [List.prod_cons]", "annotated_tactic": ["rw [<a>List.prod_cons</a>]", [{"full_name": "List.prod_cons", "def_path": "Mathlib/Algebra/BigOperators/Group/List.lean", "def_pos": [95, 9], "def_end_pos": [95, 18]}]], "state_before": "case cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na y z : M\nl : List M\nIH : (\u2200 x \u2208 l, Commute y x) \u2192 Commute y l.prod\nh : Commute y z \u2227 \u2200 x \u2208 l, Commute y x\n\u22a2 Commute y (z :: l).prod", "state_after": "case cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na y z : M\nl : List M\nIH : (\u2200 x \u2208 l, Commute y x) \u2192 Commute y l.prod\nh : Commute y z \u2227 \u2200 x \u2208 l, Commute y x\n\u22a2 Commute y (z * l.prod)"}, {"tactic": "exact Commute.mul_right h.1 (IH h.2)", "annotated_tactic": ["exact <a>Commute.mul_right</a> h.1 (IH h.2)", [{"full_name": "Commute.mul_right", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [107, 9], "def_end_pos": [107, 18]}]], "state_before": "case cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl\u271d l\u2081 l\u2082 : List M\na y z : M\nl : List M\nIH : (\u2200 x \u2208 l, Commute y x) \u2192 Commute y l.prod\nh : Commute y z \u2227 \u2200 x \u2208 l, Commute y x\n\u22a2 Commute y (z * l.prod)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.mapsTo_union", "start": [491, 1], "end": [495, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fin/Tuple/Reflection.lean", "full_name": "FinVec.seq_eq", "start": [44, 1], "end": [52, 12], "traced_tactics": [{"tactic": "simp_rw [seq, seq_eq]", "annotated_tactic": ["simp_rw [<a>seq</a>, seq_eq]", [{"full_name": "FinVec.seq", "def_path": "Mathlib/Data/Fin/Tuple/Reflection.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "m n\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1 \u2192 \u03b2\nv : Fin (n + 1) \u2192 \u03b1\ni : Fin (n + 1)\n\u22a2 seq f v i = f i (v i)", "state_after": "m n\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1 \u2192 \u03b2\nv : Fin (n + 1) \u2192 \u03b1\ni : Fin (n + 1)\n\u22a2 Matrix.vecCons (f 0 (v 0)) (fun i => Matrix.vecTail f i (Matrix.vecTail v i)) i = f i (v i)"}, {"tactic": "refine i.cases ?_ fun i => ?_", "annotated_tactic": ["refine i.cases ?_ fun i => ?_", []], "state_before": "m n\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1 \u2192 \u03b2\nv : Fin (n + 1) \u2192 \u03b1\ni : Fin (n + 1)\n\u22a2 Matrix.vecCons (f 0 (v 0)) (fun i => Matrix.vecTail f i (Matrix.vecTail v i)) i = f i (v i)", "state_after": "case refine_1\nm n\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1 \u2192 \u03b2\nv : Fin (n + 1) \u2192 \u03b1\ni : Fin (n + 1)\n\u22a2 Matrix.vecCons (f 0 (v 0)) (fun i => Matrix.vecTail f i (Matrix.vecTail v i)) 0 = f 0 (v 0)\n\ncase refine_2\nm n\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1 \u2192 \u03b2\nv : Fin (n + 1) \u2192 \u03b1\ni\u271d : Fin (n + 1)\ni : Fin n\n\u22a2 Matrix.vecCons (f 0 (v 0)) (fun i => Matrix.vecTail f i (Matrix.vecTail v i)) i.succ = f i.succ (v i.succ)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case refine_1\nm n\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1 \u2192 \u03b2\nv : Fin (n + 1) \u2192 \u03b1\ni : Fin (n + 1)\n\u22a2 Matrix.vecCons (f 0 (v 0)) (fun i => Matrix.vecTail f i (Matrix.vecTail v i)) 0 = f 0 (v 0)", "state_after": "no goals"}, {"tactic": "rw [Matrix.cons_val_succ]", "annotated_tactic": ["rw [<a>Matrix.cons_val_succ</a>]", [{"full_name": "Matrix.cons_val_succ", "def_path": "Mathlib/Data/Fin/VecNotation.lean", "def_pos": [141, 9], "def_end_pos": [141, 22]}]], "state_before": "case refine_2\nm n\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1 \u2192 \u03b2\nv : Fin (n + 1) \u2192 \u03b1\ni\u271d : Fin (n + 1)\ni : Fin n\n\u22a2 Matrix.vecCons (f 0 (v 0)) (fun i => Matrix.vecTail f i (Matrix.vecTail v i)) i.succ = f i.succ (v i.succ)", "state_after": "case refine_2\nm n\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1 \u2192 \u03b2\nv : Fin (n + 1) \u2192 \u03b1\ni\u271d : Fin (n + 1)\ni : Fin n\n\u22a2 Matrix.vecTail f i (Matrix.vecTail v i) = f i.succ (v i.succ)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case refine_2\nm n\u271d : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nn : \u2115\nf : Fin (n + 1) \u2192 \u03b1 \u2192 \u03b2\nv : Fin (n + 1) \u2192 \u03b1\ni\u271d : Fin (n + 1)\ni : Fin n\n\u22a2 Matrix.vecTail f i (Matrix.vecTail v i) = f i.succ (v i.succ)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Distributions/Gamma.lean", "full_name": "ProbabilityTheory.gammaPDFReal_nonneg", "start": [89, 1], "end": [93, 27], "traced_tactics": [{"tactic": "unfold gammaPDFReal", "annotated_tactic": ["unfold <a>gammaPDFReal</a>", [{"full_name": "ProbabilityTheory.gammaPDFReal", "def_path": "Mathlib/Probability/Distributions/Gamma.lean", "def_pos": [43, 5], "def_end_pos": [43, 17]}]], "state_before": "a r : \u211d\nha : 0 < a\nhr : 0 < r\nx : \u211d\n\u22a2 0 \u2264 gammaPDFReal a r x", "state_after": "a r : \u211d\nha : 0 < a\nhr : 0 < r\nx : \u211d\n\u22a2 0 \u2264 if 0 \u2264 x then r ^ a / Gamma a * x ^ (a - 1) * rexp (-(r * x)) else 0"}, {"tactic": "split_ifs <;> positivity", "annotated_tactic": ["split_ifs <;> positivity", []], "state_before": "a r : \u211d\nha : 0 < a\nhr : 0 < r\nx : \u211d\n\u22a2 0 \u2264 if 0 \u2264 x then r ^ a / Gamma a * x ^ (a - 1) * rexp (-(r * x)) else 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/ConstMulAction.lean", "full_name": "smul_closure_orbit_subset", "start": [186, 1], "end": [188, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/MeanInequalities.lean", "full_name": "NNReal.young_inequality_real", "start": [358, 1], "end": [360, 97], "traced_tactics": [{"tactic": "simpa [Real.coe_toNNReal, hpq.nonneg, hpq.symm.nonneg] using young_inequality a b hpq.toNNReal", "annotated_tactic": ["simpa [<a>Real.coe_toNNReal</a>, hpq.nonneg, hpq.symm.nonneg] using <a>young_inequality</a> a b hpq.toNNReal", [{"full_name": "Real.coe_toNNReal", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [124, 9], "def_end_pos": [124, 33]}, {"full_name": "NNReal.young_inequality", "def_path": "Mathlib/Analysis/MeanInequalities.lean", "def_pos": [352, 9], "def_end_pos": [352, 25]}]], "state_before": "\u03b9 : Type u\ns : Finset \u03b9\na b : \u211d\u22650\np q : \u211d\nhpq : p.IsConjExponent q\n\u22a2 a * b \u2264 a ^ p / p.toNNReal + b ^ q / q.toNNReal", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.Eventually.and", "start": [1121, 11], "end": [1123, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean", "full_name": "TopCat.pullbackSnd_apply", "start": [48, 1], "end": [48, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Localization/Integral.lean", "full_name": "IsFractionRing.isAlgebraic_iff", "start": [149, 1], "end": [159, 51], "traced_tactics": [{"tactic": "constructor <;> rintro \u27e8p, hp, px\u27e9", "annotated_tactic": ["constructor <;> rintro \u27e8p, hp, px\u27e9", []], "state_before": "R : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : Algebra R S\nP : Type u_3\ninst\u271d\u2079 : CommRing P\nA : Type u_4\nK : Type u_5\nC : Type u_6\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Algebra A K\ninst\u271d\u2074 : IsFractionRing A K\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra K C\ninst\u271d : IsScalarTower A K C\nx : C\n\u22a2 IsAlgebraic A x \u2194 IsAlgebraic K x", "state_after": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : Algebra R S\nP : Type u_3\ninst\u271d\u2079 : CommRing P\nA : Type u_4\nK : Type u_5\nC : Type u_6\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Algebra A K\ninst\u271d\u2074 : IsFractionRing A K\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra K C\ninst\u271d : IsScalarTower A K C\nx : C\np : A[X]\nhp : p \u2260 0\npx : (aeval x) p = 0\n\u22a2 IsAlgebraic K x\n\ncase mpr.intro.intro\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : Algebra R S\nP : Type u_3\ninst\u271d\u2079 : CommRing P\nA : Type u_4\nK : Type u_5\nC : Type u_6\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Algebra A K\ninst\u271d\u2074 : IsFractionRing A K\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra K C\ninst\u271d : IsScalarTower A K C\nx : C\np : K[X]\nhp : p \u2260 0\npx : (aeval x) p = 0\n\u22a2 IsAlgebraic A x"}, {"tactic": "refine \u27e8p.map (algebraMap A K), fun h => hp (Polynomial.ext fun i => ?_), ?_\u27e9", "annotated_tactic": ["refine \u27e8p.map (<a>algebraMap</a> A K), fun h => hp (<a>Polynomial.ext</a> fun i => ?_), ?_\u27e9", [{"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "Polynomial.ext", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [830, 9], "def_end_pos": [830, 12]}]], "state_before": "case mp.intro.intro\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : Algebra R S\nP : Type u_3\ninst\u271d\u2079 : CommRing P\nA : Type u_4\nK : Type u_5\nC : Type u_6\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Algebra A K\ninst\u271d\u2074 : IsFractionRing A K\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra K C\ninst\u271d : IsScalarTower A K C\nx : C\np : A[X]\nhp : p \u2260 0\npx : (aeval x) p = 0\n\u22a2 IsAlgebraic K x", "state_after": "case mp.intro.intro.refine_1\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : Algebra R S\nP : Type u_3\ninst\u271d\u2079 : CommRing P\nA : Type u_4\nK : Type u_5\nC : Type u_6\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Algebra A K\ninst\u271d\u2074 : IsFractionRing A K\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra K C\ninst\u271d : IsScalarTower A K C\nx : C\np : A[X]\nhp : p \u2260 0\npx : (aeval x) p = 0\nh : Polynomial.map (algebraMap A K) p = 0\ni : \u2115\n\u22a2 p.coeff i = coeff 0 i\n\ncase mp.intro.intro.refine_2\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : Algebra R S\nP : Type u_3\ninst\u271d\u2079 : CommRing P\nA : Type u_4\nK : Type u_5\nC : Type u_6\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Algebra A K\ninst\u271d\u2074 : IsFractionRing A K\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra K C\ninst\u271d : IsScalarTower A K C\nx : C\np : A[X]\nhp : p \u2260 0\npx : (aeval x) p = 0\n\u22a2 (aeval x) (Polynomial.map (algebraMap A K) p) = 0"}, {"tactic": "have : algebraMap A K (p.coeff i) = 0 :=\n  _root_.trans (Polynomial.coeff_map _ _).symm (by simp [h])", "annotated_tactic": ["have : <a>algebraMap</a> A K (p.coeff i) = 0 :=\n        <a>_root_.trans</a> (<a>Polynomial.coeff_map</a> _ _).<a>symm</a> (by simp [h])", [{"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "trans", "def_path": "Mathlib/Init/Algebra/Classes.lean", "def_pos": [299, 9], "def_end_pos": [299, 14]}, {"full_name": "Polynomial.coeff_map", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [803, 9], "def_end_pos": [803, 18]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case mp.intro.intro.refine_1\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : Algebra R S\nP : Type u_3\ninst\u271d\u2079 : CommRing P\nA : Type u_4\nK : Type u_5\nC : Type u_6\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Algebra A K\ninst\u271d\u2074 : IsFractionRing A K\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra K C\ninst\u271d : IsScalarTower A K C\nx : C\np : A[X]\nhp : p \u2260 0\npx : (aeval x) p = 0\nh : Polynomial.map (algebraMap A K) p = 0\ni : \u2115\n\u22a2 p.coeff i = coeff 0 i", "state_after": "case mp.intro.intro.refine_1\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : Algebra R S\nP : Type u_3\ninst\u271d\u2079 : CommRing P\nA : Type u_4\nK : Type u_5\nC : Type u_6\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Algebra A K\ninst\u271d\u2074 : IsFractionRing A K\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra K C\ninst\u271d : IsScalarTower A K C\nx : C\np : A[X]\nhp : p \u2260 0\npx : (aeval x) p = 0\nh : Polynomial.map (algebraMap A K) p = 0\ni : \u2115\nthis : (algebraMap A K) (p.coeff i) = 0\n\u22a2 p.coeff i = coeff 0 i"}, {"tactic": "exact to_map_eq_zero_iff.mp this", "annotated_tactic": ["exact to_map_eq_zero_iff.mp this", []], "state_before": "case mp.intro.intro.refine_1\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : Algebra R S\nP : Type u_3\ninst\u271d\u2079 : CommRing P\nA : Type u_4\nK : Type u_5\nC : Type u_6\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Algebra A K\ninst\u271d\u2074 : IsFractionRing A K\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra K C\ninst\u271d : IsScalarTower A K C\nx : C\np : A[X]\nhp : p \u2260 0\npx : (aeval x) p = 0\nh : Polynomial.map (algebraMap A K) p = 0\ni : \u2115\nthis : (algebraMap A K) (p.coeff i) = 0\n\u22a2 p.coeff i = coeff 0 i", "state_after": "no goals"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "R : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : Algebra R S\nP : Type u_3\ninst\u271d\u2079 : CommRing P\nA : Type u_4\nK : Type u_5\nC : Type u_6\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Algebra A K\ninst\u271d\u2074 : IsFractionRing A K\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra K C\ninst\u271d : IsScalarTower A K C\nx : C\np : A[X]\nhp : p \u2260 0\npx : (aeval x) p = 0\nh : Polynomial.map (algebraMap A K) p = 0\ni : \u2115\n\u22a2 (Polynomial.map (algebraMap A K) p).coeff i = 0", "state_after": "no goals"}, {"tactic": "exact (Polynomial.aeval_map_algebraMap K _ _).trans px", "annotated_tactic": ["exact (<a>Polynomial.aeval_map_algebraMap</a> K _ _).<a>trans</a> px", [{"full_name": "Polynomial.aeval_map_algebraMap", "def_path": "Mathlib/RingTheory/Polynomial/Tower.lean", "def_pos": [37, 9], "def_end_pos": [37, 29]}, {"full_name": "Eq.trans", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [335, 9], "def_end_pos": [335, 17]}]], "state_before": "case mp.intro.intro.refine_2\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : Algebra R S\nP : Type u_3\ninst\u271d\u2079 : CommRing P\nA : Type u_4\nK : Type u_5\nC : Type u_6\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Algebra A K\ninst\u271d\u2074 : IsFractionRing A K\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra K C\ninst\u271d : IsScalarTower A K C\nx : C\np : A[X]\nhp : p \u2260 0\npx : (aeval x) p = 0\n\u22a2 (aeval x) (Polynomial.map (algebraMap A K) p) = 0", "state_after": "no goals"}, {"tactic": "exact\n  \u27e8integerNormalization _ p, mt integerNormalization_eq_zero_iff.mp hp,\n    integerNormalization_aeval_eq_zero _ p px\u27e9", "annotated_tactic": ["exact\n      \u27e8<a>integerNormalization</a> _ p, <a>mt</a> integerNormalization_eq_zero_iff.mp hp,\n        <a>integerNormalization_aeval_eq_zero</a> _ p px\u27e9", [{"full_name": "IsLocalization.integerNormalization", "def_path": "Mathlib/RingTheory/Localization/Integral.lean", "def_pos": [69, 19], "def_end_pos": [69, 39]}, {"full_name": "mt", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [647, 9], "def_end_pos": [647, 11]}, {"full_name": "IsLocalization.integerNormalization_aeval_eq_zero", "def_path": "Mathlib/RingTheory/Localization/Integral.lean", "def_pos": [112, 9], "def_end_pos": [112, 43]}]], "state_before": "case mpr.intro.intro\nR : Type u_1\ninst\u271d\u00b9\u00b2 : CommRing R\nM : Submonoid R\nS : Type u_2\ninst\u271d\u00b9\u00b9 : CommRing S\ninst\u271d\u00b9\u2070 : Algebra R S\nP : Type u_3\ninst\u271d\u2079 : CommRing P\nA : Type u_4\nK : Type u_5\nC : Type u_6\ninst\u271d\u2078 : CommRing A\ninst\u271d\u2077 : IsDomain A\ninst\u271d\u2076 : Field K\ninst\u271d\u2075 : Algebra A K\ninst\u271d\u2074 : IsFractionRing A K\ninst\u271d\u00b3 : CommRing C\ninst\u271d\u00b2 : Algebra A C\ninst\u271d\u00b9 : Algebra K C\ninst\u271d : IsScalarTower A K C\nx : C\np : K[X]\nhp : p \u2260 0\npx : (aeval x) p = 0\n\u22a2 IsAlgebraic A x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Group/List.lean", "full_name": "List.Perm.prod_eq'", "start": [381, 1], "end": [393, 37], "traced_tactics": [{"tactic": "refine h.foldl_eq' ?_ _", "annotated_tactic": ["refine h.foldl_eq' ?_ _", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nh : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\n\u22a2 l\u2081.prod = l\u2082.prod", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nh : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\n\u22a2 \u2200 x \u2208 l\u2081, \u2200 y \u2208 l\u2081, \u2200 (z : M), z * x * y = z * y * x"}, {"tactic": "apply Pairwise.forall_of_forall", "annotated_tactic": ["apply <a>Pairwise.forall_of_forall</a>", [{"full_name": "List.Pairwise.forall_of_forall", "def_path": "Mathlib/Data/List/Pairwise.lean", "def_pos": [76, 9], "def_end_pos": [76, 34]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nh : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\n\u22a2 \u2200 x \u2208 l\u2081, \u2200 y \u2208 l\u2081, \u2200 (z : M), z * x * y = z * y * x", "state_after": "case H\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nh : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\n\u22a2 Symmetric fun x y => \u2200 (z : M), z * x * y = z * y * x\n\ncase H\u2081\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nh : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\n\u22a2 \u2200 x \u2208 l\u2081, \u2200 (z : M), z * x * x = z * x * x\n\ncase H\u2082\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nh : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\n\u22a2 Pairwise (fun x y => \u2200 (z : M), z * x * y = z * y * x) l\u2081"}, {"tactic": "intro x y h z", "annotated_tactic": ["intro x y h z", []], "state_before": "case H\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nh : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\n\u22a2 Symmetric fun x y => \u2200 (z : M), z * x * y = z * y * x", "state_after": "case H\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nh\u271d : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\nx y : M\nh : \u2200 (z : M), z * x * y = z * y * x\nz : M\n\u22a2 z * y * x = z * x * y"}, {"tactic": "exact (h z).symm", "annotated_tactic": ["exact (h z).<a>symm</a>", [{"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case H\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nh\u271d : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\nx y : M\nh : \u2200 (z : M), z * x * y = z * y * x\nz : M\n\u22a2 z * y * x = z * x * y", "state_after": "no goals"}, {"tactic": "intros", "annotated_tactic": ["intros", []], "state_before": "case H\u2081\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nh : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\n\u22a2 \u2200 x \u2208 l\u2081, \u2200 (z : M), z * x * x = z * x * x", "state_after": "case H\u2081\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nh : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\nx\u271d : M\na\u271d : x\u271d \u2208 l\u2081\nz\u271d : M\n\u22a2 z\u271d * x\u271d * x\u271d = z\u271d * x\u271d * x\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case H\u2081\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nh : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\nx\u271d : M\na\u271d : x\u271d \u2208 l\u2081\nz\u271d : M\n\u22a2 z\u271d * x\u271d * x\u271d = z\u271d * x\u271d * x\u271d", "state_after": "no goals"}, {"tactic": "apply hc.imp", "annotated_tactic": ["apply hc.imp", []], "state_before": "case H\u2082\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nh : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\n\u22a2 Pairwise (fun x y => \u2200 (z : M), z * x * y = z * y * x) l\u2081", "state_after": "case H\u2082\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nh : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\n\u22a2 \u2200 {a b : M}, Commute a b \u2192 \u2200 (z : M), z * a * b = z * b * a"}, {"tactic": "intro a b h z", "annotated_tactic": ["intro a b h z", []], "state_before": "case H\u2082\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na : M\nh : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\n\u22a2 \u2200 {a b : M}, Commute a b \u2192 \u2200 (z : M), z * a * b = z * b * a", "state_after": "case H\u2082\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na\u271d : M\nh\u271d : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\na b : M\nh : Commute a b\nz : M\n\u22a2 z * a * b = z * b * a"}, {"tactic": "rw [mul_assoc z, mul_assoc z, h]", "annotated_tactic": ["rw [<a>mul_assoc</a> z, <a>mul_assoc</a> z, h]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "case H\u2082\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nM : Type u_4\nN : Type u_5\nP : Type u_6\nG : Type u_7\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Monoid N\ninst\u271d : Monoid P\nl l\u2081 l\u2082 : List M\na\u271d : M\nh\u271d : l\u2081 ~ l\u2082\nhc : Pairwise Commute l\u2081\na b : M\nh : Commute a b\nz : M\n\u22a2 z * a * b = z * b * a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.mem_dite_univ_right", "start": [2206, 1], "end": [2208, 25], "traced_tactics": [{"tactic": "split_ifs <;> simp_all", "annotated_tactic": ["split_ifs <;> simp_all", []], "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 : p \u2192 Set \u03b1\nx : \u03b1\n\u22a2 (x \u2208 if h : p then t h else univ) \u2194 \u2200 (h : p), x \u2208 t h", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "full_name": "Zsqrtd.add_def", "start": [110, 1], "end": [111, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Ideal/AssociatedPrime.lean", "full_name": "IsAssociatedPrime.eq_radical", "start": [152, 1], "end": [169, 38], "traced_tactics": [{"tactic": "obtain \u27e8hJ, x, e\u27e9 := h", "annotated_tactic": ["obtain \u27e8hJ, x, e\u27e9 := h", []], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nh : IsAssociatedPrime J (R \u29f8 I)\n\u22a2 J = I.radical", "state_after": "case intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R \u29f8 I\ne : J = (Submodule.span R {x}).annihilator\n\u22a2 J = I.radical"}, {"tactic": "have : x \u2260 0 := by\n  rintro rfl\n  apply hJ.1\n  rwa [Submodule.span_singleton_eq_bot.mpr rfl, Submodule.annihilator_bot] at e", "annotated_tactic": ["have : x \u2260 0 := by\n    rintro rfl\n    apply hJ.1\n    rwa [Submodule.span_singleton_eq_bot.mpr <a>rfl</a>, <a>Submodule.annihilator_bot</a>] at e", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "Submodule.annihilator_bot", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [103, 9], "def_end_pos": [103, 24]}]], "state_before": "case intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R \u29f8 I\ne : J = (Submodule.span R {x}).annihilator\n\u22a2 J = I.radical", "state_after": "case intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R \u29f8 I\ne : J = (Submodule.span R {x}).annihilator\nthis : x \u2260 0\n\u22a2 J = I.radical"}, {"tactic": "obtain \u27e8x, rfl\u27e9 := Ideal.Quotient.mk\u2090_surjective R _ x", "annotated_tactic": ["obtain \u27e8x, rfl\u27e9 := <a>Ideal.Quotient.mk\u2090_surjective</a> R _ x", [{"full_name": "Ideal.Quotient.mk\u2090_surjective", "def_path": "Mathlib/RingTheory/Ideal/QuotientOperations.lean", "def_pos": [410, 9], "def_end_pos": [410, 32]}]], "state_before": "case intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R \u29f8 I\ne : J = (Submodule.span R {x}).annihilator\nthis : x \u2260 0\n\u22a2 J = I.radical", "state_after": "case intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R\ne : J = (Submodule.span R {(Ideal.Quotient.mk\u2090 R I) x}).annihilator\nthis : (Ideal.Quotient.mk\u2090 R I) x \u2260 0\n\u22a2 J = I.radical"}, {"tactic": "replace e : \u2200 {y}, y \u2208 J \u2194 x * y \u2208 I := by\n  intro y\n  rw [e, Submodule.mem_annihilator_span_singleton, \u2190 map_smul, smul_eq_mul, mul_comm,\n    Ideal.Quotient.mk\u2090_eq_mk, \u2190 Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero]", "annotated_tactic": ["replace e : \u2200 {y}, y \u2208 J \u2194 x * y \u2208 I := by\n    intro y\n    rw [e, <a>Submodule.mem_annihilator_span_singleton</a>, \u2190 <a>map_smul</a>, <a>smul_eq_mul</a>, <a>mul_comm</a>,\n      <a>Ideal.Quotient.mk\u2090_eq_mk</a>, \u2190 <a>Ideal.Quotient.mk_eq_mk</a>, <a>Submodule.Quotient.mk_eq_zero</a>]", [{"full_name": "Submodule.mem_annihilator_span_singleton", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [99, 9], "def_end_pos": [99, 39]}, {"full_name": "map_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Hom.lean", "def_pos": [111, 9], "def_end_pos": [111, 17]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [89, 7], "def_end_pos": [89, 18]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "Ideal.Quotient.mk\u2090_eq_mk", "def_path": "Mathlib/RingTheory/Ideal/QuotientOperations.lean", "def_pos": [388, 9], "def_end_pos": [388, 27]}, {"full_name": "Ideal.Quotient.mk_eq_mk", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [122, 9], "def_end_pos": [122, 17]}, {"full_name": "Submodule.Quotient.mk_eq_zero", "def_path": "Mathlib/LinearAlgebra/Quotient.lean", "def_pos": [100, 9], "def_end_pos": [100, 19]}]], "state_before": "case intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R\ne : J = (Submodule.span R {(Ideal.Quotient.mk\u2090 R I) x}).annihilator\nthis : (Ideal.Quotient.mk\u2090 R I) x \u2260 0\n\u22a2 J = I.radical", "state_after": "case intro.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R\nthis : (Ideal.Quotient.mk\u2090 R I) x \u2260 0\ne : \u2200 {y : R}, y \u2208 J \u2194 x * y \u2208 I\n\u22a2 J = I.radical"}, {"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.intro.intro\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R\nthis : (Ideal.Quotient.mk\u2090 R I) x \u2260 0\ne : \u2200 {y : R}, y \u2208 J \u2194 x * y \u2208 I\n\u22a2 J = I.radical", "state_after": "case intro.intro.intro.a\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R\nthis : (Ideal.Quotient.mk\u2090 R I) x \u2260 0\ne : \u2200 {y : R}, y \u2208 J \u2194 x * y \u2208 I\n\u22a2 J \u2264 I.radical\n\ncase intro.intro.intro.a\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R\nthis : (Ideal.Quotient.mk\u2090 R I) x \u2260 0\ne : \u2200 {y : R}, y \u2208 J \u2194 x * y \u2208 I\n\u22a2 I.radical \u2264 J"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R \u29f8 I\ne : J = (Submodule.span R {x}).annihilator\n\u22a2 x \u2260 0", "state_after": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\ne : J = (Submodule.span R {0}).annihilator\n\u22a2 False"}, {"tactic": "apply hJ.1", "annotated_tactic": ["apply hJ.1", []], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\ne : J = (Submodule.span R {0}).annihilator\n\u22a2 False", "state_after": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\ne : J = (Submodule.span R {0}).annihilator\n\u22a2 J = \u22a4"}, {"tactic": "rwa [Submodule.span_singleton_eq_bot.mpr rfl, Submodule.annihilator_bot] at e", "annotated_tactic": ["rwa [Submodule.span_singleton_eq_bot.mpr <a>rfl</a>, <a>Submodule.annihilator_bot</a>] at e", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "Submodule.annihilator_bot", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [103, 9], "def_end_pos": [103, 24]}]], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\ne : J = (Submodule.span R {0}).annihilator\n\u22a2 J = \u22a4", "state_after": "no goals"}, {"tactic": "intro y", "annotated_tactic": ["intro y", []], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R\ne : J = (Submodule.span R {(Ideal.Quotient.mk\u2090 R I) x}).annihilator\nthis : (Ideal.Quotient.mk\u2090 R I) x \u2260 0\n\u22a2 \u2200 {y : R}, y \u2208 J \u2194 x * y \u2208 I", "state_after": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R\ne : J = (Submodule.span R {(Ideal.Quotient.mk\u2090 R I) x}).annihilator\nthis : (Ideal.Quotient.mk\u2090 R I) x \u2260 0\ny : R\n\u22a2 y \u2208 J \u2194 x * y \u2208 I"}, {"tactic": "rw [e, Submodule.mem_annihilator_span_singleton, \u2190 map_smul, smul_eq_mul, mul_comm,\n  Ideal.Quotient.mk\u2090_eq_mk, \u2190 Ideal.Quotient.mk_eq_mk, Submodule.Quotient.mk_eq_zero]", "annotated_tactic": ["rw [e, <a>Submodule.mem_annihilator_span_singleton</a>, \u2190 <a>map_smul</a>, <a>smul_eq_mul</a>, <a>mul_comm</a>,\n      <a>Ideal.Quotient.mk\u2090_eq_mk</a>, \u2190 <a>Ideal.Quotient.mk_eq_mk</a>, <a>Submodule.Quotient.mk_eq_zero</a>]", [{"full_name": "Submodule.mem_annihilator_span_singleton", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [99, 9], "def_end_pos": [99, 39]}, {"full_name": "map_smul", "def_path": "Mathlib/GroupTheory/GroupAction/Hom.lean", "def_pos": [111, 9], "def_end_pos": [111, 17]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [89, 7], "def_end_pos": [89, 18]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "Ideal.Quotient.mk\u2090_eq_mk", "def_path": "Mathlib/RingTheory/Ideal/QuotientOperations.lean", "def_pos": [388, 9], "def_end_pos": [388, 27]}, {"full_name": "Ideal.Quotient.mk_eq_mk", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [122, 9], "def_end_pos": [122, 17]}, {"full_name": "Submodule.Quotient.mk_eq_zero", "def_path": "Mathlib/LinearAlgebra/Quotient.lean", "def_pos": [100, 9], "def_end_pos": [100, 19]}]], "state_before": "R : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R\ne : J = (Submodule.span R {(Ideal.Quotient.mk\u2090 R I) x}).annihilator\nthis : (Ideal.Quotient.mk\u2090 R I) x \u2260 0\ny : R\n\u22a2 y \u2208 J \u2194 x * y \u2208 I", "state_after": "no goals"}, {"tactic": "intro y hy", "annotated_tactic": ["intro y hy", []], "state_before": "case intro.intro.intro.a\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R\nthis : (Ideal.Quotient.mk\u2090 R I) x \u2260 0\ne : \u2200 {y : R}, y \u2208 J \u2194 x * y \u2208 I\n\u22a2 J \u2264 I.radical", "state_after": "case intro.intro.intro.a\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R\nthis : (Ideal.Quotient.mk\u2090 R I) x \u2260 0\ne : \u2200 {y : R}, y \u2208 J \u2194 x * y \u2208 I\ny : R\nhy : y \u2208 J\n\u22a2 y \u2208 I.radical"}, {"tactic": "exact (hI.2 <| e.mp hy).resolve_left ((Submodule.Quotient.mk_eq_zero I).not.mp this)", "annotated_tactic": ["exact (hI.2 <| e.mp hy).<a>resolve_left</a> ((<a>Submodule.Quotient.mk_eq_zero</a> I).not.mp this)", [{"full_name": "Or.resolve_left", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [555, 9], "def_end_pos": [555, 24]}, {"full_name": "Submodule.Quotient.mk_eq_zero", "def_path": "Mathlib/LinearAlgebra/Quotient.lean", "def_pos": [100, 9], "def_end_pos": [100, 19]}]], "state_before": "case intro.intro.intro.a\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R\nthis : (Ideal.Quotient.mk\u2090 R I) x \u2260 0\ne : \u2200 {y : R}, y \u2208 J \u2194 x * y \u2208 I\ny : R\nhy : y \u2208 J\n\u22a2 y \u2208 I.radical", "state_after": "no goals"}, {"tactic": "rw [hJ.radical_le_iff]", "annotated_tactic": ["rw [hJ.radical_le_iff]", []], "state_before": "case intro.intro.intro.a\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R\nthis : (Ideal.Quotient.mk\u2090 R I) x \u2260 0\ne : \u2200 {y : R}, y \u2208 J \u2194 x * y \u2208 I\n\u22a2 I.radical \u2264 J", "state_after": "case intro.intro.intro.a\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R\nthis : (Ideal.Quotient.mk\u2090 R I) x \u2260 0\ne : \u2200 {y : R}, y \u2208 J \u2194 x * y \u2208 I\n\u22a2 I \u2264 J"}, {"tactic": "intro y hy", "annotated_tactic": ["intro y hy", []], "state_before": "case intro.intro.intro.a\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R\nthis : (Ideal.Quotient.mk\u2090 R I) x \u2260 0\ne : \u2200 {y : R}, y \u2208 J \u2194 x * y \u2208 I\n\u22a2 I \u2264 J", "state_after": "case intro.intro.intro.a\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R\nthis : (Ideal.Quotient.mk\u2090 R I) x \u2260 0\ne : \u2200 {y : R}, y \u2208 J \u2194 x * y \u2208 I\ny : R\nhy : y \u2208 I\n\u22a2 y \u2208 J"}, {"tactic": "exact e.mpr (I.mul_mem_left x hy)", "annotated_tactic": ["exact e.mpr (I.mul_mem_left x hy)", []], "state_before": "case intro.intro.intro.a\nR : Type u_1\ninst\u271d\u2074 : CommRing R\nI J : Ideal R\nM : Type u_2\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\nM' : Type u_3\ninst\u271d\u00b9 : AddCommGroup M'\ninst\u271d : Module R M'\nf : M \u2192\u2097[R] M'\nhI : I.IsPrimary\nhJ : J.IsPrime\nx : R\nthis : (Ideal.Quotient.mk\u2090 R I) x \u2260 0\ne : \u2200 {y : R}, y \u2208 J \u2194 x * y \u2208 I\ny : R\nhy : y \u2208 I\n\u22a2 y \u2208 J", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Complex.exp_eq_exp_re_mul_sin_add_cos", "start": [780, 1], "end": [781, 34], "traced_tactics": [{"tactic": "rw [\u2190 exp_add_mul_I, re_add_im]", "annotated_tactic": ["rw [\u2190 <a>exp_add_mul_I</a>, <a>re_add_im</a>]", [{"full_name": "Complex.exp_add_mul_I", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [776, 9], "def_end_pos": [776, 22]}, {"full_name": "Complex.re_add_im", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [316, 9], "def_end_pos": [316, 18]}]], "state_before": "x y : \u2102\n\u22a2 cexp x = cexp \u2191x.re * (cos \u2191x.im + sin \u2191x.im * I)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Antidiag/Prod.lean", "full_name": "Finset.antidiagonal_subtype_ext", "start": [108, 8], "end": [109, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/BoundedVariation.lean", "full_name": "variationOnFromTo.eq_zero_iff_of_le", "start": [706, 11], "end": [711, 30], "traced_tactics": [{"tactic": "rw [variationOnFromTo.eq_of_le _ _ ab, ENNReal.toReal_eq_zero_iff, or_iff_left (hf a b ha hb),\n  eVariationOn.eq_zero_iff]", "annotated_tactic": ["rw [<a>variationOnFromTo.eq_of_le</a> _ _ ab, <a>ENNReal.toReal_eq_zero_iff</a>, <a>or_iff_left</a> (hf a b ha hb),\n    <a>eVariationOn.eq_zero_iff</a>]", [{"full_name": "variationOnFromTo.eq_of_le", "def_path": "Mathlib/Analysis/BoundedVariation.lean", "def_pos": [662, 19], "def_end_pos": [662, 27]}, {"full_name": "ENNReal.toReal_eq_zero_iff", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [297, 9], "def_end_pos": [297, 27]}, {"full_name": "or_iff_left", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [91, 9], "def_end_pos": [91, 20]}, {"full_name": "eVariationOn.eq_zero_iff", "def_path": "Mathlib/Analysis/BoundedVariation.lean", "def_pos": [164, 9], "def_end_pos": [164, 20]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : LinearOrder \u03b1\nE : Type u_2\ninst\u271d : PseudoEMetricSpace E\nf\u271d : \u03b1 \u2192 E\ns\u271d : Set \u03b1\nf : \u03b1 \u2192 E\ns : Set \u03b1\nhf : LocallyBoundedVariationOn f s\na b : \u03b1\nha : a \u2208 s\nhb : b \u2208 s\nab : a \u2264 b\n\u22a2 variationOnFromTo f s a b = 0 \u2194 \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2229 Icc a b \u2192 \u2200 \u2983y : \u03b1\u2984, y \u2208 s \u2229 Icc a b \u2192 edist (f x) (f y) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Adjunction/Limits.lean", "full_name": "CategoryTheory.Adjunction.has_limits_of_equivalence", "start": [269, 1], "end": [271, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/MonoidAlgebra/Basic.lean", "full_name": "MonoidAlgebra.ringHom_ext", "start": [786, 1], "end": [793, 96], "traced_tactics": [{"tactic": "rw [\u2190 single, \u2190 one_mul a, \u2190 mul_one b, \u2190 single_mul_single]", "annotated_tactic": ["rw [\u2190 <a>single</a>, \u2190 <a>one_mul</a> a, \u2190 <a>mul_one</a> b, \u2190 <a>single_mul_single</a>]", [{"full_name": "MonoidAlgebra.single", "def_path": "Mathlib/Algebra/MonoidAlgebra/Basic.lean", "def_pos": [110, 8], "def_end_pos": [110, 14]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "MonoidAlgebra.single_mul_single", "def_path": "Mathlib/Algebra/MonoidAlgebra/Basic.lean", "def_pos": [459, 9], "def_end_pos": [459, 26]}]], "state_before": "k : Type u\u2081\nG : Type u\u2082\nH : Type u_1\nR\u271d : Type u_2\nR : Type u_3\ninst\u271d\u00b2 : Semiring k\ninst\u271d\u00b9 : MulOneClass G\ninst\u271d : Semiring R\nf g : MonoidAlgebra k G \u2192+* R\nh\u2081 : \u2200 (b : k), f (single 1 b) = g (single 1 b)\nh_of : \u2200 (a : G), f (single a 1) = g (single a 1)\na : G\nb : k\n\u22a2 ((fun f => \u2191f) f) (Finsupp.single a b) = ((fun f => \u2191f) g) (Finsupp.single a b)", "state_after": "k : Type u\u2081\nG : Type u\u2082\nH : Type u_1\nR\u271d : Type u_2\nR : Type u_3\ninst\u271d\u00b2 : Semiring k\ninst\u271d\u00b9 : MulOneClass G\ninst\u271d : Semiring R\nf g : MonoidAlgebra k G \u2192+* R\nh\u2081 : \u2200 (b : k), f (single 1 b) = g (single 1 b)\nh_of : \u2200 (a : G), f (single a 1) = g (single a 1)\na : G\nb : k\n\u22a2 ((fun f => \u2191f) f) (single 1 b * single a 1) = ((fun f => \u2191f) g) (single 1 b * single a 1)"}, {"tactic": "erw [AddMonoidHom.coe_coe f, AddMonoidHom.coe_coe g]", "annotated_tactic": ["erw [<a>AddMonoidHom.coe_coe</a> f, <a>AddMonoidHom.coe_coe</a> g]", [{"full_name": "AddMonoidHom.coe_coe", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [410, 3], "def_end_pos": [410, 14]}, {"full_name": "AddMonoidHom.coe_coe", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [410, 3], "def_end_pos": [410, 14]}]], "state_before": "k : Type u\u2081\nG : Type u\u2082\nH : Type u_1\nR\u271d : Type u_2\nR : Type u_3\ninst\u271d\u00b2 : Semiring k\ninst\u271d\u00b9 : MulOneClass G\ninst\u271d : Semiring R\nf g : MonoidAlgebra k G \u2192+* R\nh\u2081 : \u2200 (b : k), f (single 1 b) = g (single 1 b)\nh_of : \u2200 (a : G), f (single a 1) = g (single a 1)\na : G\nb : k\n\u22a2 ((fun f => \u2191f) f) (single 1 b * single a 1) = ((fun f => \u2191f) g) (single 1 b * single a 1)", "state_after": "k : Type u\u2081\nG : Type u\u2082\nH : Type u_1\nR\u271d : Type u_2\nR : Type u_3\ninst\u271d\u00b2 : Semiring k\ninst\u271d\u00b9 : MulOneClass G\ninst\u271d : Semiring R\nf g : MonoidAlgebra k G \u2192+* R\nh\u2081 : \u2200 (b : k), f (single 1 b) = g (single 1 b)\nh_of : \u2200 (a : G), f (single a 1) = g (single a 1)\na : G\nb : k\n\u22a2 f (single 1 b * single a 1) = g (single 1 b * single a 1)"}, {"tactic": "rw [f.map_mul, g.map_mul, h\u2081, h_of]", "annotated_tactic": ["rw [f.map_mul, g.map_mul, h\u2081, h_of]", []], "state_before": "k : Type u\u2081\nG : Type u\u2082\nH : Type u_1\nR\u271d : Type u_2\nR : Type u_3\ninst\u271d\u00b2 : Semiring k\ninst\u271d\u00b9 : MulOneClass G\ninst\u271d : Semiring R\nf g : MonoidAlgebra k G \u2192+* R\nh\u2081 : \u2200 (b : k), f (single 1 b) = g (single 1 b)\nh_of : \u2200 (a : G), f (single a 1) = g (single a 1)\na : G\nb : k\n\u22a2 f (single 1 b * single a 1) = g (single 1 b * single a 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Concrete.lean", "full_name": "Equiv.Perm.nodup_toList", "start": [278, 1], "end": [308, 39], "traced_tactics": [{"tactic": "by_cases hx : p x = x", "annotated_tactic": ["by_cases hx : p x = x", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\n\u22a2 (p.toList x).Nodup", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : p x = x\n\u22a2 (p.toList x).Nodup\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : \u00acp x = x\n\u22a2 (p.toList x).Nodup"}, {"tactic": "have hc : IsCycle (cycleOf p x) := isCycle_cycleOf p hx", "annotated_tactic": ["have hc : <a>IsCycle</a> (<a>cycleOf</a> p x) := <a>isCycle_cycleOf</a> p hx", [{"full_name": "Equiv.Perm.IsCycle", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Basic.lean", "def_pos": [280, 5], "def_end_pos": [280, 12]}, {"full_name": "Equiv.Perm.cycleOf", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Factors.lean", "def_pos": [44, 5], "def_end_pos": [44, 12]}, {"full_name": "Equiv.Perm.isCycle_cycleOf", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Factors.lean", "def_pos": [171, 9], "def_end_pos": [171, 24]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : \u00acp x = x\n\u22a2 (p.toList x).Nodup", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : \u00acp x = x\nhc : (p.cycleOf x).IsCycle\n\u22a2 (p.toList x).Nodup"}, {"tactic": "rw [nodup_iff_nthLe_inj]", "annotated_tactic": ["rw [<a>nodup_iff_nthLe_inj</a>]", [{"full_name": "List.nodup_iff_nthLe_inj", "def_path": "Mathlib/Data/List/Nodup.lean", "def_pos": [111, 9], "def_end_pos": [111, 28]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : \u00acp x = x\nhc : (p.cycleOf x).IsCycle\n\u22a2 (p.toList x).Nodup", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : \u00acp x = x\nhc : (p.cycleOf x).IsCycle\n\u22a2 \u2200 (i j : \u2115) (h\u2081 : i < (p.toList x).length) (h\u2082 : j < (p.toList x).length),\n    (p.toList x).nthLe i h\u2081 = (p.toList x).nthLe j h\u2082 \u2192 i = j"}, {"tactic": "rintro n m hn hm", "annotated_tactic": ["rintro n m hn hm", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : \u00acp x = x\nhc : (p.cycleOf x).IsCycle\n\u22a2 \u2200 (i j : \u2115) (h\u2081 : i < (p.toList x).length) (h\u2082 : j < (p.toList x).length),\n    (p.toList x).nthLe i h\u2081 = (p.toList x).nthLe j h\u2082 \u2192 i = j", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : \u00acp x = x\nhc : (p.cycleOf x).IsCycle\nn m : \u2115\nhn : n < (p.toList x).length\nhm : m < (p.toList x).length\n\u22a2 (p.toList x).nthLe n hn = (p.toList x).nthLe m hm \u2192 n = m"}, {"tactic": "rw [length_toList, \u2190 hc.orderOf] at hm hn", "annotated_tactic": ["rw [<a>length_toList</a>, \u2190 hc.orderOf] at hm hn", [{"full_name": "Equiv.Perm.length_toList", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Concrete.lean", "def_pos": [229, 9], "def_end_pos": [229, 22]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : \u00acp x = x\nhc : (p.cycleOf x).IsCycle\nn m : \u2115\nhn : n < (p.toList x).length\nhm : m < (p.toList x).length\n\u22a2 (p.toList x).nthLe n hn = (p.toList x).nthLe m hm \u2192 n = m", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : \u00acp x = x\nhc : (p.cycleOf x).IsCycle\nn m : \u2115\nhn\u271d : n < (p.toList x).length\nhn : n < orderOf (p.cycleOf x)\nhm\u271d : m < (p.toList x).length\nhm : m < orderOf (p.cycleOf x)\n\u22a2 (p.toList x).nthLe n hn\u271d = (p.toList x).nthLe m hm\u271d \u2192 n = m"}, {"tactic": "rw [\u2190 cycleOf_apply_self, \u2190 Ne, \u2190 mem_support] at hx", "annotated_tactic": ["rw [\u2190 <a>cycleOf_apply_self</a>, \u2190 <a>Ne</a>, \u2190 <a>mem_support</a>] at hx", [{"full_name": "Equiv.Perm.cycleOf_apply_self", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Factors.lean", "def_pos": [121, 9], "def_end_pos": [121, 27]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "Equiv.Perm.mem_support", "def_path": "Mathlib/GroupTheory/Perm/Support.lean", "def_pos": [297, 9], "def_end_pos": [297, 20]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : \u00acp x = x\nhc : (p.cycleOf x).IsCycle\nn m : \u2115\nhn\u271d : n < (p.toList x).length\nhn : n < orderOf (p.cycleOf x)\nhm\u271d : m < (p.toList x).length\nhm : m < orderOf (p.cycleOf x)\n\u22a2 (p.toList x).nthLe n hn\u271d = (p.toList x).nthLe m hm\u271d \u2192 n = m", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn m : \u2115\nhn\u271d : n < (p.toList x).length\nhn : n < orderOf (p.cycleOf x)\nhm\u271d : m < (p.toList x).length\nhm : m < orderOf (p.cycleOf x)\n\u22a2 (p.toList x).nthLe n hn\u271d = (p.toList x).nthLe m hm\u271d \u2192 n = m"}, {"tactic": "rw [nthLe_toList, nthLe_toList, \u2190 cycleOf_pow_apply_self p x n, \u2190\n  cycleOf_pow_apply_self p x m]", "annotated_tactic": ["rw [<a>nthLe_toList</a>, <a>nthLe_toList</a>, \u2190 <a>cycleOf_pow_apply_self</a> p x n, \u2190\n    <a>cycleOf_pow_apply_self</a> p x m]", [{"full_name": "Equiv.Perm.nthLe_toList", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Concrete.lean", "def_pos": [253, 9], "def_end_pos": [253, 21]}, {"full_name": "Equiv.Perm.nthLe_toList", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Concrete.lean", "def_pos": [253, 9], "def_end_pos": [253, 21]}, {"full_name": "Equiv.Perm.cycleOf_pow_apply_self", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Factors.lean", "def_pos": [66, 9], "def_end_pos": [66, 31]}, {"full_name": "Equiv.Perm.cycleOf_pow_apply_self", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Factors.lean", "def_pos": [66, 9], "def_end_pos": [66, 31]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn m : \u2115\nhn\u271d : n < (p.toList x).length\nhn : n < orderOf (p.cycleOf x)\nhm\u271d : m < (p.toList x).length\nhm : m < orderOf (p.cycleOf x)\n\u22a2 (p.toList x).nthLe n hn\u271d = (p.toList x).nthLe m hm\u271d \u2192 n = m", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn m : \u2115\nhn\u271d : n < (p.toList x).length\nhn : n < orderOf (p.cycleOf x)\nhm\u271d : m < (p.toList x).length\nhm : m < orderOf (p.cycleOf x)\n\u22a2 (p.cycleOf x ^ n) x = (p.cycleOf x ^ m) x \u2192 n = m"}, {"tactic": "cases' n with n <;> cases' m with m", "annotated_tactic": ["cases' n with n <;> cases' m with m", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn m : \u2115\nhn\u271d : n < (p.toList x).length\nhn : n < orderOf (p.cycleOf x)\nhm\u271d : m < (p.toList x).length\nhm : m < orderOf (p.cycleOf x)\n\u22a2 (p.cycleOf x ^ n) x = (p.cycleOf x ^ m) x \u2192 n = m", "state_after": "case neg.zero.zero\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nhn\u271d : 0 < (p.toList x).length\nhn : 0 < orderOf (p.cycleOf x)\nhm\u271d : 0 < (p.toList x).length\nhm : 0 < orderOf (p.cycleOf x)\n\u22a2 (p.cycleOf x ^ 0) x = (p.cycleOf x ^ 0) x \u2192 0 = 0\n\ncase neg.zero.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nhn\u271d : 0 < (p.toList x).length\nhn : 0 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\n\u22a2 (p.cycleOf x ^ 0) x = (p.cycleOf x ^ (m + 1)) x \u2192 0 = m + 1\n\ncase neg.succ.zero\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nhm\u271d : 0 < (p.toList x).length\nhm : 0 < orderOf (p.cycleOf x)\n\u22a2 (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ 0) x \u2192 n + 1 = 0\n\ncase neg.succ.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\n\u22a2 (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x \u2192 n + 1 = m + 1"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case neg.succ.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\n\u22a2 (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x \u2192 n + 1 = m + 1", "state_after": "case neg.succ.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\n\u22a2 n + 1 = m + 1"}, {"tactic": "have hn' : \u00acorderOf (p.cycleOf x) \u2223 n.succ := Nat.not_dvd_of_pos_of_lt n.zero_lt_succ hn", "annotated_tactic": ["have hn' : \u00ac<a>orderOf</a> (p.cycleOf x) \u2223 n.succ := <a>Nat.not_dvd_of_pos_of_lt</a> n.zero_lt_succ hn", [{"full_name": "orderOf", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [155, 19], "def_end_pos": [155, 26]}, {"full_name": "Nat.not_dvd_of_pos_of_lt", "def_path": "Mathlib/Data/Nat/Defs.lean", "def_pos": [1186, 7], "def_end_pos": [1186, 27]}]], "state_before": "case neg.succ.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\n\u22a2 n + 1 = m + 1", "state_after": "case neg.succ.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : \u00acorderOf (p.cycleOf x) \u2223 n.succ\n\u22a2 n + 1 = m + 1"}, {"tactic": "have hm' : \u00acorderOf (p.cycleOf x) \u2223 m.succ := Nat.not_dvd_of_pos_of_lt m.zero_lt_succ hm", "annotated_tactic": ["have hm' : \u00ac<a>orderOf</a> (p.cycleOf x) \u2223 m.succ := <a>Nat.not_dvd_of_pos_of_lt</a> m.zero_lt_succ hm", [{"full_name": "orderOf", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [155, 19], "def_end_pos": [155, 26]}, {"full_name": "Nat.not_dvd_of_pos_of_lt", "def_path": "Mathlib/Data/Nat/Defs.lean", "def_pos": [1186, 7], "def_end_pos": [1186, 27]}]], "state_before": "case neg.succ.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : \u00acorderOf (p.cycleOf x) \u2223 n.succ\n\u22a2 n + 1 = m + 1", "state_after": "case neg.succ.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : \u00acorderOf (p.cycleOf x) \u2223 n.succ\nhm' : \u00acorderOf (p.cycleOf x) \u2223 m.succ\n\u22a2 n + 1 = m + 1"}, {"tactic": "rw [\u2190 hc.support_pow_eq_iff] at hn' hm'", "annotated_tactic": ["rw [\u2190 hc.support_pow_eq_iff] at hn' hm'", []], "state_before": "case neg.succ.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : \u00acorderOf (p.cycleOf x) \u2223 n.succ\nhm' : \u00acorderOf (p.cycleOf x) \u2223 m.succ\n\u22a2 n + 1 = m + 1", "state_after": "case neg.succ.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\n\u22a2 n + 1 = m + 1"}, {"tactic": "rw [\u2190 Nat.mod_eq_of_lt hn, \u2190 Nat.mod_eq_of_lt hm, \u2190 pow_inj_mod]", "annotated_tactic": ["rw [\u2190 <a>Nat.mod_eq_of_lt</a> hn, \u2190 <a>Nat.mod_eq_of_lt</a> hm, \u2190 <a>pow_inj_mod</a>]", [{"full_name": "Nat.mod_eq_of_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [131, 9], "def_end_pos": [131, 21]}, {"full_name": "Nat.mod_eq_of_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [131, 9], "def_end_pos": [131, 21]}, {"full_name": "pow_inj_mod", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [575, 7], "def_end_pos": [575, 18]}]], "state_before": "case neg.succ.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\n\u22a2 n + 1 = m + 1", "state_after": "case neg.succ.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\n\u22a2 p.cycleOf x ^ (n + 1) = p.cycleOf x ^ (m + 1)"}, {"tactic": "refine support_congr ?_ ?_", "annotated_tactic": ["refine <a>support_congr</a> ?_ ?_", [{"full_name": "Equiv.Perm.support_congr", "def_path": "Mathlib/GroupTheory/Perm/Support.lean", "def_pos": [324, 9], "def_end_pos": [324, 22]}]], "state_before": "case neg.succ.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\n\u22a2 p.cycleOf x ^ (n + 1) = p.cycleOf x ^ (m + 1)", "state_after": "case neg.succ.succ.refine_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\n\u22a2 (p.cycleOf x ^ (n + 1)).support \u2286 (p.cycleOf x ^ (m + 1)).support\n\ncase neg.succ.succ.refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\n\u22a2 \u2200 x_1 \u2208 (p.cycleOf x ^ (m + 1)).support, (p.cycleOf x ^ (n + 1)) x_1 = (p.cycleOf x ^ (m + 1)) x_1"}, {"tactic": "rw [\u2190 not_mem_support, \u2190 toList_eq_nil_iff] at hx", "annotated_tactic": ["rw [\u2190 <a>not_mem_support</a>, \u2190 <a>toList_eq_nil_iff</a>] at hx", [{"full_name": "Equiv.Perm.not_mem_support", "def_path": "Mathlib/GroupTheory/Perm/Support.lean", "def_pos": [301, 9], "def_end_pos": [301, 24]}, {"full_name": "Equiv.Perm.toList_eq_nil_iff", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Concrete.lean", "def_pos": [225, 9], "def_end_pos": [225, 26]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : p x = x\n\u22a2 (p.toList x).Nodup", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : p.toList x = []\n\u22a2 (p.toList x).Nodup"}, {"tactic": "simp [hx]", "annotated_tactic": ["simp [hx]", []], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : p.toList x = []\n\u22a2 (p.toList x).Nodup", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case neg.zero.zero\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nhn\u271d : 0 < (p.toList x).length\nhn : 0 < orderOf (p.cycleOf x)\nhm\u271d : 0 < (p.toList x).length\nhm : 0 < orderOf (p.cycleOf x)\n\u22a2 (p.cycleOf x ^ 0) x = (p.cycleOf x ^ 0) x \u2192 0 = 0", "state_after": "no goals"}, {"tactic": "rw [\u2190 hc.support_pow_of_pos_of_lt_orderOf m.zero_lt_succ hm, mem_support,\n  cycleOf_pow_apply_self] at hx", "annotated_tactic": ["rw [\u2190 hc.support_pow_of_pos_of_lt_orderOf m.zero_lt_succ hm, <a>mem_support</a>,\n      <a>cycleOf_pow_apply_self</a>] at hx", [{"full_name": "Equiv.Perm.mem_support", "def_path": "Mathlib/GroupTheory/Perm/Support.lean", "def_pos": [297, 9], "def_end_pos": [297, 20]}, {"full_name": "Equiv.Perm.cycleOf_pow_apply_self", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Factors.lean", "def_pos": [66, 9], "def_end_pos": [66, 31]}]], "state_before": "case neg.zero.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nhn\u271d : 0 < (p.toList x).length\nhn : 0 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\n\u22a2 (p.cycleOf x ^ 0) x = (p.cycleOf x ^ (m + 1)) x \u2192 0 = m + 1", "state_after": "case neg.zero.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhc : (p.cycleOf x).IsCycle\nhn\u271d : 0 < (p.toList x).length\nhn : 0 < orderOf (p.cycleOf x)\nm : \u2115\nhx : (p ^ m.succ) x \u2260 x\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\n\u22a2 (p.cycleOf x ^ 0) x = (p.cycleOf x ^ (m + 1)) x \u2192 0 = m + 1"}, {"tactic": "simp [hx.symm]", "annotated_tactic": ["simp [hx.symm]", []], "state_before": "case neg.zero.succ\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhc : (p.cycleOf x).IsCycle\nhn\u271d : 0 < (p.toList x).length\nhn : 0 < orderOf (p.cycleOf x)\nm : \u2115\nhx : (p ^ m.succ) x \u2260 x\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\n\u22a2 (p.cycleOf x ^ 0) x = (p.cycleOf x ^ (m + 1)) x \u2192 0 = m + 1", "state_after": "no goals"}, {"tactic": "rw [\u2190 hc.support_pow_of_pos_of_lt_orderOf n.zero_lt_succ hn, mem_support,\n  cycleOf_pow_apply_self] at hx", "annotated_tactic": ["rw [\u2190 hc.support_pow_of_pos_of_lt_orderOf n.zero_lt_succ hn, <a>mem_support</a>,\n      <a>cycleOf_pow_apply_self</a>] at hx", [{"full_name": "Equiv.Perm.mem_support", "def_path": "Mathlib/GroupTheory/Perm/Support.lean", "def_pos": [297, 9], "def_end_pos": [297, 20]}, {"full_name": "Equiv.Perm.cycleOf_pow_apply_self", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Factors.lean", "def_pos": [66, 9], "def_end_pos": [66, 31]}]], "state_before": "case neg.succ.zero\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nhm\u271d : 0 < (p.toList x).length\nhm : 0 < orderOf (p.cycleOf x)\n\u22a2 (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ 0) x \u2192 n + 1 = 0", "state_after": "case neg.succ.zero\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhx : (p ^ n.succ) x \u2260 x\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nhm\u271d : 0 < (p.toList x).length\nhm : 0 < orderOf (p.cycleOf x)\n\u22a2 (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ 0) x \u2192 n + 1 = 0"}, {"tactic": "simp [hx]", "annotated_tactic": ["simp [hx]", []], "state_before": "case neg.succ.zero\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhx : (p ^ n.succ) x \u2260 x\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nhm\u271d : 0 < (p.toList x).length\nhm : 0 < orderOf (p.cycleOf x)\n\u22a2 (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ 0) x \u2192 n + 1 = 0", "state_after": "no goals"}, {"tactic": "rw [hm', hn']", "annotated_tactic": ["rw [hm', hn']", []], "state_before": "case neg.succ.succ.refine_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\n\u22a2 (p.cycleOf x ^ (n + 1)).support \u2286 (p.cycleOf x ^ (m + 1)).support", "state_after": "no goals"}, {"tactic": "rw [hm']", "annotated_tactic": ["rw [hm']", []], "state_before": "case neg.succ.succ.refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\n\u22a2 \u2200 x_1 \u2208 (p.cycleOf x ^ (m + 1)).support, (p.cycleOf x ^ (n + 1)) x_1 = (p.cycleOf x ^ (m + 1)) x_1", "state_after": "case neg.succ.succ.refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\n\u22a2 \u2200 x_1 \u2208 (p.cycleOf x).support, (p.cycleOf x ^ (n + 1)) x_1 = (p.cycleOf x ^ (m + 1)) x_1"}, {"tactic": "intro y hy", "annotated_tactic": ["intro y hy", []], "state_before": "case neg.succ.succ.refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\n\u22a2 \u2200 x_1 \u2208 (p.cycleOf x).support, (p.cycleOf x ^ (n + 1)) x_1 = (p.cycleOf x ^ (m + 1)) x_1", "state_after": "case neg.succ.succ.refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\ny : \u03b1\nhy : y \u2208 (p.cycleOf x).support\n\u22a2 (p.cycleOf x ^ (n + 1)) y = (p.cycleOf x ^ (m + 1)) y"}, {"tactic": "obtain \u27e8k, rfl\u27e9 := hc.exists_pow_eq (mem_support.mp hx) (mem_support.mp hy)", "annotated_tactic": ["obtain \u27e8k, rfl\u27e9 := hc.exists_pow_eq (mem_support.mp hx) (mem_support.mp hy)", []], "state_before": "case neg.succ.succ.refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\ny : \u03b1\nhy : y \u2208 (p.cycleOf x).support\n\u22a2 (p.cycleOf x ^ (n + 1)) y = (p.cycleOf x ^ (m + 1)) y", "state_after": "case neg.succ.succ.refine_2.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\nk : \u2115\nhy : (p.cycleOf x ^ k) x \u2208 (p.cycleOf x).support\n\u22a2 (p.cycleOf x ^ (n + 1)) ((p.cycleOf x ^ k) x) = (p.cycleOf x ^ (m + 1)) ((p.cycleOf x ^ k) x)"}, {"tactic": "rw [\u2190 mul_apply, (Commute.pow_pow_self _ _ _).eq, mul_apply, h, \u2190 mul_apply, \u2190 mul_apply,\n  (Commute.pow_pow_self _ _ _).eq]", "annotated_tactic": ["rw [\u2190 <a>mul_apply</a>, (<a>Commute.pow_pow_self</a> _ _ _).<a>eq</a>, <a>mul_apply</a>, h, \u2190 <a>mul_apply</a>, \u2190 <a>mul_apply</a>,\n      (<a>Commute.pow_pow_self</a> _ _ _).<a>eq</a>]", [{"full_name": "Equiv.Perm.mul_apply", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 18]}, {"full_name": "Commute.pow_pow_self", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [215, 9], "def_end_pos": [215, 21]}, {"full_name": "Commute.eq", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [56, 19], "def_end_pos": [56, 21]}, {"full_name": "Equiv.Perm.mul_apply", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 18]}, {"full_name": "Equiv.Perm.mul_apply", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 18]}, {"full_name": "Equiv.Perm.mul_apply", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 18]}, {"full_name": "Commute.pow_pow_self", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [215, 9], "def_end_pos": [215, 21]}, {"full_name": "Commute.eq", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [56, 19], "def_end_pos": [56, 21]}]], "state_before": "case neg.succ.succ.refine_2.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np\u271d : Perm \u03b1\nx\u271d : \u03b1\np : Perm \u03b1\nx : \u03b1\nhx : x \u2208 (p.cycleOf x).support\nhc : (p.cycleOf x).IsCycle\nn : \u2115\nhn\u271d : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nm : \u2115\nhm\u271d : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\nh : (p.cycleOf x ^ (n + 1)) x = (p.cycleOf x ^ (m + 1)) x\nhn' : (p.cycleOf x ^ n.succ).support = (p.cycleOf x).support\nhm' : (p.cycleOf x ^ m.succ).support = (p.cycleOf x).support\nk : \u2115\nhy : (p.cycleOf x ^ k) x \u2208 (p.cycleOf x).support\n\u22a2 (p.cycleOf x ^ (n + 1)) ((p.cycleOf x ^ k) x) = (p.cycleOf x ^ (m + 1)) ((p.cycleOf x ^ k) x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean", "full_name": "exists_extension_norm_eq", "start": [73, 1], "end": [109, 93], "traced_tactics": [{"tactic": "letI : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E", "annotated_tactic": ["letI : <a>Module</a> \u211d E := <a>RestrictScalars.module</a> \u211d \ud835\udd5c E", [{"full_name": "Module", "def_path": "Mathlib/Algebra/Module/Defs.lean", "def_pos": [57, 7], "def_end_pos": [57, 13]}, {"full_name": "RestrictScalars.module", "def_path": "Mathlib/Algebra/Algebra/RestrictScalars.lean", "def_pos": [108, 10], "def_end_pos": [108, 32]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\n\u22a2 \u2203 g, (\u2200 (x : \u21a5p), g \u2191x = f x) \u2227 \u2016g\u2016 = \u2016f\u2016", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\n\u22a2 \u2203 g, (\u2200 (x : \u21a5p), g \u2191x = f x) \u2227 \u2016g\u2016 = \u2016f\u2016"}, {"tactic": "letI : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower _ _ _", "annotated_tactic": ["letI : <a>IsScalarTower</a> \u211d \ud835\udd5c E := <a>RestrictScalars.isScalarTower</a> _ _ _", [{"full_name": "IsScalarTower", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [249, 7], "def_end_pos": [249, 20]}, {"full_name": "RestrictScalars.isScalarTower", "def_path": "Mathlib/Algebra/Algebra/RestrictScalars.lean", "def_pos": [113, 10], "def_end_pos": [113, 39]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\n\u22a2 \u2203 g, (\u2200 (x : \u21a5p), g \u2191x = f x) \u2227 \u2016g\u2016 = \u2016f\u2016", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\n\u22a2 \u2203 g, (\u2200 (x : \u21a5p), g \u2191x = f x) \u2227 \u2016g\u2016 = \u2016f\u2016"}, {"tactic": "letI : NormedSpace \u211d E := NormedSpace.restrictScalars _ \ud835\udd5c _", "annotated_tactic": ["letI : <a>NormedSpace</a> \u211d E := <a>NormedSpace.restrictScalars</a> _ \ud835\udd5c _", [{"full_name": "NormedSpace", "def_path": "Mathlib/Analysis/NormedSpace/Basic.lean", "def_pos": [43, 7], "def_end_pos": [43, 18]}, {"full_name": "NormedSpace.restrictScalars", "def_path": "Mathlib/Analysis/NormedSpace/Basic.lean", "def_pos": [477, 5], "def_end_pos": [477, 32]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\n\u22a2 \u2203 g, (\u2200 (x : \u21a5p), g \u2191x = f x) \u2227 \u2016g\u2016 = \u2016f\u2016", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\n\u22a2 \u2203 g, (\u2200 (x : \u21a5p), g \u2191x = f x) \u2227 \u2016g\u2016 = \u2016f\u2016"}, {"tactic": "let fr := reCLM.comp (f.restrictScalars \u211d)", "annotated_tactic": ["let fr := reCLM.comp (f.restrictScalars \u211d)", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\n\u22a2 \u2203 g, (\u2200 (x : \u21a5p), g \u2191x = f x) \u2227 \u2016g\u2016 = \u2016f\u2016", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\n\u22a2 \u2203 g, (\u2200 (x : \u21a5p), g \u2191x = f x) \u2227 \u2016g\u2016 = \u2016f\u2016"}, {"tactic": "rcases Real.exists_extension_norm_eq (p.restrictScalars \u211d) fr with \u27e8g, \u27e8hextends, hnormeq\u27e9\u27e9", "annotated_tactic": ["rcases <a>Real.exists_extension_norm_eq</a> (p.restrictScalars \u211d) fr with \u27e8g, \u27e8hextends, hnormeq\u27e9\u27e9", [{"full_name": "Real.exists_extension_norm_eq", "def_path": "Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean", "def_pos": [44, 9], "def_end_pos": [44, 33]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\n\u22a2 \u2203 g, (\u2200 (x : \u21a5p), g \u2191x = f x) \u2227 \u2016g\u2016 = \u2016f\u2016", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\n\u22a2 \u2203 g, (\u2200 (x : \u21a5p), g \u2191x = f x) \u2227 \u2016g\u2016 = \u2016f\u2016"}, {"tactic": "refine \u27e8g.extendTo\ud835\udd5c, ?_\u27e9", "annotated_tactic": ["refine \u27e8g.extendTo\ud835\udd5c, ?_\u27e9", []], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\n\u22a2 \u2203 g, (\u2200 (x : \u21a5p), g \u2191x = f x) \u2227 \u2016g\u2016 = \u2016f\u2016", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\n\u22a2 (\u2200 (x : \u21a5p), g.extendTo\ud835\udd5c \u2191x = f x) \u2227 \u2016g.extendTo\ud835\udd5c\u2016 = \u2016f\u2016"}, {"tactic": "refine \u27e8h, le_antisymm ?_ ?_\u27e9", "annotated_tactic": ["refine \u27e8h, <a>le_antisymm</a> ?_ ?_\u27e9", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}]], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nh : \u2200 (x : \u21a5p), g.extendTo\ud835\udd5c \u2191x = f x\n\u22a2 (\u2200 (x : \u21a5p), g.extendTo\ud835\udd5c \u2191x = f x) \u2227 \u2016g.extendTo\ud835\udd5c\u2016 = \u2016f\u2016", "state_after": "case intro.intro.refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nh : \u2200 (x : \u21a5p), g.extendTo\ud835\udd5c \u2191x = f x\n\u22a2 \u2016g.extendTo\ud835\udd5c\u2016 \u2264 \u2016f\u2016\n\ncase intro.intro.refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nh : \u2200 (x : \u21a5p), g.extendTo\ud835\udd5c \u2191x = f x\n\u22a2 \u2016f\u2016 \u2264 \u2016g.extendTo\ud835\udd5c\u2016"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\n\u22a2 \u2200 (x : \u21a5p), g.extendTo\ud835\udd5c \u2191x = f x", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : \u21a5p\n\u22a2 g.extendTo\ud835\udd5c \u2191x = f x"}, {"tactic": "erw [ContinuousLinearMap.extendTo\ud835\udd5c_apply, \u2190 Submodule.coe_smul, hextends, hextends]", "annotated_tactic": ["erw [<a>ContinuousLinearMap.extendTo\ud835\udd5c_apply</a>, \u2190 <a>Submodule.coe_smul</a>, hextends, hextends]", [{"full_name": "ContinuousLinearMap.extendTo\ud835\udd5c_apply", "def_path": "Mathlib/Analysis/NormedSpace/Extend.lean", "def_pos": [163, 9], "def_end_pos": [163, 44]}, {"full_name": "Submodule.coe_smul", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [304, 9], "def_end_pos": [304, 17]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : \u21a5p\n\u22a2 g.extendTo\ud835\udd5c \u2191x = f x", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : \u21a5p\n\u22a2 \u2191(fr x) - I * \u2191(fr (I \u2022 x)) = f x"}, {"tactic": "have :\n    (fr x : \ud835\udd5c) - I * \u2191(fr ((I : \ud835\udd5c) \u2022 x)) = (re (f x) : \ud835\udd5c) - (I : \ud835\udd5c) * re (f ((I : \ud835\udd5c) \u2022 x)) := by\n  rfl", "annotated_tactic": ["have :\n        (fr x : \ud835\udd5c) - <a>I</a> * \u2191(fr ((<a>I</a> : \ud835\udd5c) \u2022 x)) = (<a>re</a> (f x) : \ud835\udd5c) - (<a>I</a> : \ud835\udd5c) * <a>re</a> (f ((<a>I</a> : \ud835\udd5c) \u2022 x)) := by\n      rfl", [{"full_name": "RCLike.I", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [58, 3], "def_end_pos": [58, 4]}, {"full_name": "RCLike.I", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [58, 3], "def_end_pos": [58, 4]}, {"full_name": "RCLike.re", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [55, 3], "def_end_pos": [55, 5]}, {"full_name": "RCLike.I", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [58, 3], "def_end_pos": [58, 4]}, {"full_name": "RCLike.re", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [55, 3], "def_end_pos": [55, 5]}, {"full_name": "RCLike.I", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [58, 3], "def_end_pos": [58, 4]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : \u21a5p\n\u22a2 \u2191(fr x) - I * \u2191(fr (I \u2022 x)) = f x", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b2 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d\u00b9 : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis\u271d : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : \u21a5p\nthis : \u2191(fr x) - I * \u2191(fr (I \u2022 x)) = \u2191(re (f x)) - I * \u2191(re (f (I \u2022 x)))\n\u22a2 \u2191(fr x) - I * \u2191(fr (I \u2022 x)) = f x"}, {"tactic": "erw [this]", "annotated_tactic": ["erw [this]", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b2 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d\u00b9 : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis\u271d : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : \u21a5p\nthis : \u2191(fr x) - I * \u2191(fr (I \u2022 x)) = \u2191(re (f x)) - I * \u2191(re (f (I \u2022 x)))\n\u22a2 \u2191(fr x) - I * \u2191(fr (I \u2022 x)) = f x", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b2 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d\u00b9 : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis\u271d : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : \u21a5p\nthis : \u2191(fr x) - I * \u2191(fr (I \u2022 x)) = \u2191(re (f x)) - I * \u2191(re (f (I \u2022 x)))\n\u22a2 \u2191(re (f x)) - I * \u2191(re (f (I \u2022 x))) = f x"}, {"tactic": "apply ext", "annotated_tactic": ["apply <a>ext</a>", [{"full_name": "RCLike.ext", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [143, 9], "def_end_pos": [143, 12]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b2 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d\u00b9 : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis\u271d : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : \u21a5p\nthis : \u2191(fr x) - I * \u2191(fr (I \u2022 x)) = \u2191(re (f x)) - I * \u2191(re (f (I \u2022 x)))\n\u22a2 \u2191(re (f x)) - I * \u2191(re (f (I \u2022 x))) = f x", "state_after": "case hre\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b2 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d\u00b9 : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis\u271d : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : \u21a5p\nthis : \u2191(fr x) - I * \u2191(fr (I \u2022 x)) = \u2191(re (f x)) - I * \u2191(re (f (I \u2022 x)))\n\u22a2 re (\u2191(re (f x)) - I * \u2191(re (f (I \u2022 x)))) = re (f x)\n\ncase him\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b2 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d\u00b9 : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis\u271d : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : \u21a5p\nthis : \u2191(fr x) - I * \u2191(fr (I \u2022 x)) = \u2191(re (f x)) - I * \u2191(re (f (I \u2022 x)))\n\u22a2 im (\u2191(re (f x)) - I * \u2191(re (f (I \u2022 x)))) = im (f x)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : \u21a5p\n\u22a2 \u2191(fr x) - I * \u2191(fr (I \u2022 x)) = \u2191(re (f x)) - I * \u2191(re (f (I \u2022 x)))", "state_after": "no goals"}, {"tactic": "simp only [add_zero, Algebra.id.smul_eq_mul, I_re, ofReal_im, AddMonoidHom.map_add, zero_sub,\n  I_im', zero_mul, ofReal_re, eq_self_iff_true, sub_zero, mul_neg, ofReal_neg,\n  mul_re, mul_zero, sub_neg_eq_add, ContinuousLinearMap.map_smul]", "annotated_tactic": ["simp only [<a>add_zero</a>, <a>Algebra.id.smul_eq_mul</a>, <a>I_re</a>, <a>ofReal_im</a>, <a>AddMonoidHom.map_add</a>, <a>zero_sub</a>,\n        <a>I_im'</a>, <a>zero_mul</a>, <a>ofReal_re</a>, <a>eq_self_iff_true</a>, <a>sub_zero</a>, <a>mul_neg</a>, <a>ofReal_neg</a>,\n        <a>mul_re</a>, <a>mul_zero</a>, <a>sub_neg_eq_add</a>, <a>ContinuousLinearMap.map_smul</a>]", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [482, 3], "def_end_pos": [482, 14]}, {"full_name": "Algebra.id.smul_eq_mul", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [449, 9], "def_end_pos": [449, 20]}, {"full_name": "RCLike.I_re", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [290, 9], "def_end_pos": [290, 13]}, {"full_name": "RCLike.ofReal_im", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [125, 9], "def_end_pos": [125, 18]}, {"full_name": "AddMonoidHom.map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [770, 3], "def_end_pos": [770, 14]}, {"full_name": "zero_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [467, 3], "def_end_pos": [467, 14]}, {"full_name": "RCLike.I_im'", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [302, 9], "def_end_pos": [302, 14]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}, {"full_name": "RCLike.ofReal_re", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [120, 9], "def_end_pos": [120, 18]}, {"full_name": "eq_self_iff_true", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1380, 9], "def_end_pos": [1380, 25]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [489, 3], "def_end_pos": [489, 14]}, {"full_name": "mul_neg", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}, {"full_name": "RCLike.ofReal_neg", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [204, 9], "def_end_pos": [204, 19]}, {"full_name": "RCLike.mul_re", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [130, 9], "def_end_pos": [130, 15]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}, {"full_name": "sub_neg_eq_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [697, 3], "def_end_pos": [697, 14]}, {"full_name": "ContinuousLinearMap.map_smul", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [509, 19], "def_end_pos": [509, 27]}]], "state_before": "case hre\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b2 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d\u00b9 : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis\u271d : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : \u21a5p\nthis : \u2191(fr x) - I * \u2191(fr (I \u2022 x)) = \u2191(re (f x)) - I * \u2191(re (f (I \u2022 x)))\n\u22a2 re (\u2191(re (f x)) - I * \u2191(re (f (I \u2022 x)))) = re (f x)", "state_after": "no goals"}, {"tactic": "simp only [Algebra.id.smul_eq_mul, I_re, ofReal_im, AddMonoidHom.map_add, zero_sub, I_im',\n  zero_mul, ofReal_re, mul_neg, mul_im, zero_add, ofReal_neg, mul_re,\n  sub_neg_eq_add, ContinuousLinearMap.map_smul]", "annotated_tactic": ["simp only [<a>Algebra.id.smul_eq_mul</a>, <a>I_re</a>, <a>ofReal_im</a>, <a>AddMonoidHom.map_add</a>, <a>zero_sub</a>, <a>I_im'</a>,\n        <a>zero_mul</a>, <a>ofReal_re</a>, <a>mul_neg</a>, <a>mul_im</a>, <a>zero_add</a>, <a>ofReal_neg</a>, <a>mul_re</a>,\n        <a>sub_neg_eq_add</a>, <a>ContinuousLinearMap.map_smul</a>]", [{"full_name": "Algebra.id.smul_eq_mul", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [449, 9], "def_end_pos": [449, 20]}, {"full_name": "RCLike.I_re", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [290, 9], "def_end_pos": [290, 13]}, {"full_name": "RCLike.ofReal_im", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [125, 9], "def_end_pos": [125, 18]}, {"full_name": "AddMonoidHom.map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [770, 3], "def_end_pos": [770, 14]}, {"full_name": "zero_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [467, 3], "def_end_pos": [467, 14]}, {"full_name": "RCLike.I_im'", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [302, 9], "def_end_pos": [302, 14]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}, {"full_name": "RCLike.ofReal_re", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [120, 9], "def_end_pos": [120, 18]}, {"full_name": "mul_neg", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}, {"full_name": "RCLike.mul_im", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [135, 9], "def_end_pos": [135, 15]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [476, 3], "def_end_pos": [476, 14]}, {"full_name": "RCLike.ofReal_neg", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [204, 9], "def_end_pos": [204, 19]}, {"full_name": "RCLike.mul_re", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [130, 9], "def_end_pos": [130, 15]}, {"full_name": "sub_neg_eq_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [697, 3], "def_end_pos": [697, 14]}, {"full_name": "ContinuousLinearMap.map_smul", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [509, 19], "def_end_pos": [509, 27]}]], "state_before": "case him\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b2 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d\u00b9 : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis\u271d : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nx : \u21a5p\nthis : \u2191(fr x) - I * \u2191(fr (I \u2022 x)) = \u2191(re (f x)) - I * \u2191(re (f (I \u2022 x)))\n\u22a2 im (\u2191(re (f x)) - I * \u2191(re (f (I \u2022 x)))) = im (f x)", "state_after": "no goals"}, {"tactic": "calc\n  \u2016g.extendTo\ud835\udd5c\u2016 = \u2016g\u2016 := g.norm_extendTo\ud835\udd5c\n  _ = \u2016fr\u2016 := hnormeq\n  _ \u2264 \u2016reCLM\u2016 * \u2016f\u2016 := ContinuousLinearMap.opNorm_comp_le _ _\n  _ = \u2016f\u2016 := by rw [reCLM_norm, one_mul]", "annotated_tactic": ["calc\n      \u2016g.extendTo\ud835\udd5c\u2016 = \u2016g\u2016 := g.norm_extendTo\ud835\udd5c\n      _ = \u2016fr\u2016 := hnormeq\n      _ \u2264 \u2016<a>reCLM</a>\u2016 * \u2016f\u2016 := <a>ContinuousLinearMap.opNorm_comp_le</a> _ _\n      _ = \u2016f\u2016 := by rw [<a>reCLM_norm</a>, <a>one_mul</a>]", [{"full_name": "RCLike.reCLM", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [1024, 19], "def_end_pos": [1024, 24]}, {"full_name": "ContinuousLinearMap.opNorm_comp_le", "def_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Basic.lean", "def_pos": [397, 9], "def_end_pos": [397, 23]}, {"full_name": "RCLike.reCLM_norm", "def_path": "Mathlib/Analysis/RCLike/Lemmas.lean", "def_pos": [71, 9], "def_end_pos": [71, 19]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "case intro.intro.refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nh : \u2200 (x : \u21a5p), g.extendTo\ud835\udd5c \u2191x = f x\n\u22a2 \u2016g.extendTo\ud835\udd5c\u2016 \u2264 \u2016f\u2016", "state_after": "no goals"}, {"tactic": "rw [reCLM_norm, one_mul]", "annotated_tactic": ["rw [<a>reCLM_norm</a>, <a>one_mul</a>]", [{"full_name": "RCLike.reCLM_norm", "def_path": "Mathlib/Analysis/RCLike/Lemmas.lean", "def_pos": [71, 9], "def_end_pos": [71, 19]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nh : \u2200 (x : \u21a5p), g.extendTo\ud835\udd5c \u2191x = f x\n\u22a2 \u2016reCLM\u2016 * \u2016f\u2016 = \u2016f\u2016", "state_after": "no goals"}, {"tactic": "exact f.opNorm_le_bound g.extendTo\ud835\udd5c.opNorm_nonneg fun x => h x \u25b8 g.extendTo\ud835\udd5c.le_opNorm x", "annotated_tactic": ["exact f.opNorm_le_bound g.extendTo\ud835\udd5c.opNorm_nonneg fun x => h x \u25b8 g.extendTo\ud835\udd5c.le_opNorm x", []], "state_before": "case intro.intro.refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_2\nF : Type u_3\ninst\u271d\u00b3 : SeminormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Subspace \ud835\udd5c E\nf : \u21a5p \u2192L[\ud835\udd5c] \ud835\udd5c\nthis\u271d\u00b9 : Module \u211d E := RestrictScalars.module \u211d \ud835\udd5c E\nthis\u271d : IsScalarTower \u211d \ud835\udd5c E := RestrictScalars.isScalarTower \u211d \ud835\udd5c E\nthis : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E\nfr : \u21a5p \u2192L[\u211d] \u211d := reCLM.comp (ContinuousLinearMap.restrictScalars \u211d f)\ng : E \u2192L[\u211d] \u211d\nhextends : \u2200 (x : \u21a5(Submodule.restrictScalars \u211d p)), g \u2191x = fr x\nhnormeq : \u2016g\u2016 = \u2016fr\u2016\nh : \u2200 (x : \u21a5p), g.extendTo\ud835\udd5c \u2191x = f x\n\u22a2 \u2016f\u2016 \u2264 \u2016g.extendTo\ud835\udd5c\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/MeanValue.lean", "full_name": "Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le", "start": [530, 1], "end": [532, 98], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.replicate_right_inj", "start": [987, 9], "end": [989, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "full_name": "AffineSubspace.comap_bot", "start": [1721, 9], "end": [1721, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/IntegralClosure.lean", "full_name": "IsIntegralClosure.mk'_add", "start": [779, 1], "end": [781, 79], "traced_tactics": [{"tactic": "simp only [algebraMap_mk', RingHom.map_add]", "annotated_tactic": ["simp only [<a>algebraMap_mk'</a>, <a>RingHom.map_add</a>]", [{"full_name": "IsIntegralClosure.algebraMap_mk'", "def_path": "Mathlib/RingTheory/IntegralClosure.lean", "def_pos": [764, 9], "def_end_pos": [764, 23]}, {"full_name": "RingHom.map_add", "def_path": "Mathlib/Algebra/Ring/Hom/Defs.lean", "def_pos": [556, 19], "def_end_pos": [556, 26]}]], "state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : CommRing B\ninst\u271d\u00b2 : Algebra R B\ninst\u271d\u00b9 : Algebra A B\ninst\u271d : IsIntegralClosure A R B\nx y : B\nhx : IsIntegral R x\nhy : IsIntegral R y\n\u22a2 (algebraMap A B) (mk' A (x + y) \u22ef) = (algebraMap A B) (mk' A x hx + mk' A y hy)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "full_name": "MeasureTheory.ProbabilityMeasure.toWeakDualBCNN_continuous", "start": [277, 1], "end": [278, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Congruence/Basic.lean", "full_name": "RingCon.ext", "start": [137, 1], "end": [138, 26], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Add R\ninst\u271d : Mul R\nc\u271d c d : RingCon R\nH : \u2200 (x y : R), c x y \u2194 d x y\n\u22a2 \u21d1c = \u21d1d", "state_after": "case h.h.a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Add R\ninst\u271d : Mul R\nc\u271d c d : RingCon R\nH : \u2200 (x y : R), c x y \u2194 d x y\nx\u271d\u00b9 x\u271d : R\n\u22a2 c x\u271d\u00b9 x\u271d \u2194 d x\u271d\u00b9 x\u271d"}, {"tactic": "apply H", "annotated_tactic": ["apply H", []], "state_before": "case h.h.a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u00b9 : Add R\ninst\u271d : Mul R\nc\u271d c d : RingCon R\nH : \u2200 (x y : R), c x y \u2194 d x y\nx\u271d\u00b9 x\u271d : R\n\u22a2 c x\u271d\u00b9 x\u271d \u2194 d x\u271d\u00b9 x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "full_name": "NNReal.rpow_left_bijective", "start": [325, 1], "end": [326, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Finiteness.lean", "full_name": "Monoid.fg_of_surjective", "start": [173, 1], "end": [179, 43], "traced_tactics": [{"tactic": "classical\n  obtain \u27e8s, hs\u27e9 := Monoid.fg_def.mp \u2039_\u203a\n  use s.image f\n  rwa [Finset.coe_image, \u2190 MonoidHom.map_mclosure, hs, \u2190 MonoidHom.mrange_eq_map,\n    MonoidHom.mrange_top_iff_surjective]", "annotated_tactic": ["classical\n    obtain \u27e8s, hs\u27e9 := Monoid.fg_def.mp \u2039_\u203a\n    use s.image f\n    rwa [<a>Finset.coe_image</a>, \u2190 <a>MonoidHom.map_mclosure</a>, hs, \u2190 <a>MonoidHom.mrange_eq_map</a>,\n      <a>MonoidHom.mrange_top_iff_surjective</a>]", [{"full_name": "Finset.coe_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [419, 9], "def_end_pos": [419, 18]}, {"full_name": "MonoidHom.map_mclosure", "def_path": "Mathlib/Algebra/Group/Submonoid/Operations.lean", "def_pos": [1020, 9], "def_end_pos": [1020, 21]}, {"full_name": "MonoidHom.mrange_eq_map", "def_path": "Mathlib/Algebra/Group/Submonoid/Operations.lean", "def_pos": [979, 9], "def_end_pos": [979, 22]}, {"full_name": "MonoidHom.mrange_top_iff_surjective", "def_path": "Mathlib/Algebra/Group/Submonoid/Operations.lean", "def_pos": [995, 9], "def_end_pos": [995, 34]}]], "state_before": "M : Type u_1\nN : Type u_2\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : AddMonoid N\nM' : Type u_3\ninst\u271d\u00b9 : Monoid M'\ninst\u271d : FG M\nf : M \u2192* M'\nhf : Function.Surjective \u21d1f\n\u22a2 FG M'", "state_after": "no goals"}, {"tactic": "obtain \u27e8s, hs\u27e9 := Monoid.fg_def.mp \u2039_\u203a", "annotated_tactic": ["obtain \u27e8s, hs\u27e9 := Monoid.fg_def.mp \u2039_\u203a", []], "state_before": "M : Type u_1\nN : Type u_2\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : AddMonoid N\nM' : Type u_3\ninst\u271d\u00b9 : Monoid M'\ninst\u271d : FG M\nf : M \u2192* M'\nhf : Function.Surjective \u21d1f\n\u22a2 FG M'", "state_after": "case intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : AddMonoid N\nM' : Type u_3\ninst\u271d\u00b9 : Monoid M'\ninst\u271d : FG M\nf : M \u2192* M'\nhf : Function.Surjective \u21d1f\ns : Finset M\nhs : Submonoid.closure \u2191s = \u22a4\n\u22a2 FG M'"}, {"tactic": "use s.image f", "annotated_tactic": ["use s.image f", []], "state_before": "case intro\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : AddMonoid N\nM' : Type u_3\ninst\u271d\u00b9 : Monoid M'\ninst\u271d : FG M\nf : M \u2192* M'\nhf : Function.Surjective \u21d1f\ns : Finset M\nhs : Submonoid.closure \u2191s = \u22a4\n\u22a2 FG M'", "state_after": "case h\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : AddMonoid N\nM' : Type u_3\ninst\u271d\u00b9 : Monoid M'\ninst\u271d : FG M\nf : M \u2192* M'\nhf : Function.Surjective \u21d1f\ns : Finset M\nhs : Submonoid.closure \u2191s = \u22a4\n\u22a2 Submonoid.closure \u2191(Finset.image (\u21d1f) s) = \u22a4"}, {"tactic": "rwa [Finset.coe_image, \u2190 MonoidHom.map_mclosure, hs, \u2190 MonoidHom.mrange_eq_map,\n  MonoidHom.mrange_top_iff_surjective]", "annotated_tactic": ["rwa [<a>Finset.coe_image</a>, \u2190 <a>MonoidHom.map_mclosure</a>, hs, \u2190 <a>MonoidHom.mrange_eq_map</a>,\n      <a>MonoidHom.mrange_top_iff_surjective</a>]", [{"full_name": "Finset.coe_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [419, 9], "def_end_pos": [419, 18]}, {"full_name": "MonoidHom.map_mclosure", "def_path": "Mathlib/Algebra/Group/Submonoid/Operations.lean", "def_pos": [1020, 9], "def_end_pos": [1020, 21]}, {"full_name": "MonoidHom.mrange_eq_map", "def_path": "Mathlib/Algebra/Group/Submonoid/Operations.lean", "def_pos": [979, 9], "def_end_pos": [979, 22]}, {"full_name": "MonoidHom.mrange_top_iff_surjective", "def_path": "Mathlib/Algebra/Group/Submonoid/Operations.lean", "def_pos": [995, 9], "def_end_pos": [995, 34]}]], "state_before": "case h\nM : Type u_1\nN : Type u_2\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : AddMonoid N\nM' : Type u_3\ninst\u271d\u00b9 : Monoid M'\ninst\u271d : FG M\nf : M \u2192* M'\nhf : Function.Surjective \u21d1f\ns : Finset M\nhs : Submonoid.closure \u2191s = \u22a4\n\u22a2 Submonoid.closure \u2191(Finset.image (\u21d1f) s) = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "full_name": "MeasureTheory.ae_restrict_uIoc_iff", "start": [608, 1], "end": [611, 43], "traced_tactics": [{"tactic": "rw [ae_restrict_uIoc_eq, eventually_sup]", "annotated_tactic": ["rw [<a>ae_restrict_uIoc_eq</a>, <a>eventually_sup</a>]", [{"full_name": "MeasureTheory.ae_restrict_uIoc_eq", "def_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "def_pos": [602, 9], "def_end_pos": [602, 28]}, {"full_name": "Filter.eventually_sup", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1239, 9], "def_end_pos": [1239, 23]}]], "state_before": "R : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b2 : MeasurableSpace \u03b2\ninst\u271d\u00b9 : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns s' t : Set \u03b1\ninst\u271d : LinearOrder \u03b1\na b : \u03b1\nP : \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc.restrict (\u0399 a b), P x) \u2194\n    (\u2200\u1d50 (x : \u03b1) \u2202\u03bc.restrict (Ioc a b), P x) \u2227 \u2200\u1d50 (x : \u03b1) \u2202\u03bc.restrict (Ioc b a), P x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Exp.lean", "full_name": "Real.tendsto_exp_neg_atTop_nhds_zero", "start": [217, 1], "end": [218, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/AddChar.lean", "full_name": "AddChar.ext", "start": [87, 1], "end": [88, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Alternating/DomCoprod.lean", "full_name": "AlternatingMap.domCoprod.summand_add_swap_smul_eq_zero", "start": [78, 1], "end": [91, 39], "traced_tactics": [{"tactic": "refine Quotient.inductionOn' \u03c3 fun \u03c3 => ?_", "annotated_tactic": ["refine <a>Quotient.inductionOn'</a> \u03c3 fun \u03c3 => ?_", [{"full_name": "Quotient.inductionOn'", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [677, 19], "def_end_pos": [677, 31]}]], "state_before": "\u03b9a : Type u_1\n\u03b9b : Type u_2\ninst\u271d\u00b9\u2070 : Fintype \u03b9a\ninst\u271d\u2079 : Fintype \u03b9b\nR' : Type u_3\nM\u1d62 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R'\ninst\u271d\u2077 : AddCommGroup N\u2081\ninst\u271d\u2076 : Module R' N\u2081\ninst\u271d\u2075 : AddCommGroup N\u2082\ninst\u271d\u2074 : Module R' N\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u1d62\ninst\u271d\u00b2 : Module R' M\u1d62\ninst\u271d\u00b9 : DecidableEq \u03b9a\ninst\u271d : DecidableEq \u03b9b\na : M\u1d62 [\u22c0^\u03b9a]\u2192\u2097[R'] N\u2081\nb : M\u1d62 [\u22c0^\u03b9b]\u2192\u2097[R'] N\u2082\n\u03c3 : Perm.ModSumCongr \u03b9a \u03b9b\nv : \u03b9a \u2295 \u03b9b \u2192 M\u1d62\ni j : \u03b9a \u2295 \u03b9b\nhv : v i = v j\nhij : i \u2260 j\n\u22a2 (summand a b \u03c3) v + (summand a b (swap i j \u2022 \u03c3)) v = 0", "state_after": "\u03b9a : Type u_1\n\u03b9b : Type u_2\ninst\u271d\u00b9\u2070 : Fintype \u03b9a\ninst\u271d\u2079 : Fintype \u03b9b\nR' : Type u_3\nM\u1d62 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R'\ninst\u271d\u2077 : AddCommGroup N\u2081\ninst\u271d\u2076 : Module R' N\u2081\ninst\u271d\u2075 : AddCommGroup N\u2082\ninst\u271d\u2074 : Module R' N\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u1d62\ninst\u271d\u00b2 : Module R' M\u1d62\ninst\u271d\u00b9 : DecidableEq \u03b9a\ninst\u271d : DecidableEq \u03b9b\na : M\u1d62 [\u22c0^\u03b9a]\u2192\u2097[R'] N\u2081\nb : M\u1d62 [\u22c0^\u03b9b]\u2192\u2097[R'] N\u2082\n\u03c3\u271d : Perm.ModSumCongr \u03b9a \u03b9b\nv : \u03b9a \u2295 \u03b9b \u2192 M\u1d62\ni j : \u03b9a \u2295 \u03b9b\nhv : v i = v j\nhij : i \u2260 j\n\u03c3 : Perm (\u03b9a \u2295 \u03b9b)\n\u22a2 (summand a b (Quotient.mk'' \u03c3)) v + (summand a b (swap i j \u2022 Quotient.mk'' \u03c3)) v = 0"}, {"tactic": "dsimp only [Quotient.liftOn'_mk'', Quotient.map'_mk'', MulAction.Quotient.smul_mk,\n  domCoprod.summand]", "annotated_tactic": ["dsimp only [<a>Quotient.liftOn'_mk''</a>, <a>Quotient.map'_mk''</a>, <a>MulAction.Quotient.smul_mk</a>,\n    <a>domCoprod.summand</a>]", [{"full_name": "Quotient.liftOn'_mk''", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [634, 19], "def_end_pos": [634, 31]}, {"full_name": "Quotient.map'_mk''", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [757, 9], "def_end_pos": [757, 18]}, {"full_name": "MulAction.Quotient.smul_mk", "def_path": "Mathlib/GroupTheory/GroupAction/Quotient.lean", "def_pos": [94, 9], "def_end_pos": [94, 25]}, {"full_name": "AlternatingMap.domCoprod.summand", "def_path": "Mathlib/LinearAlgebra/Alternating/DomCoprod.lean", "def_pos": [45, 5], "def_end_pos": [45, 22]}]], "state_before": "\u03b9a : Type u_1\n\u03b9b : Type u_2\ninst\u271d\u00b9\u2070 : Fintype \u03b9a\ninst\u271d\u2079 : Fintype \u03b9b\nR' : Type u_3\nM\u1d62 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R'\ninst\u271d\u2077 : AddCommGroup N\u2081\ninst\u271d\u2076 : Module R' N\u2081\ninst\u271d\u2075 : AddCommGroup N\u2082\ninst\u271d\u2074 : Module R' N\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u1d62\ninst\u271d\u00b2 : Module R' M\u1d62\ninst\u271d\u00b9 : DecidableEq \u03b9a\ninst\u271d : DecidableEq \u03b9b\na : M\u1d62 [\u22c0^\u03b9a]\u2192\u2097[R'] N\u2081\nb : M\u1d62 [\u22c0^\u03b9b]\u2192\u2097[R'] N\u2082\n\u03c3\u271d : Perm.ModSumCongr \u03b9a \u03b9b\nv : \u03b9a \u2295 \u03b9b \u2192 M\u1d62\ni j : \u03b9a \u2295 \u03b9b\nhv : v i = v j\nhij : i \u2260 j\n\u03c3 : Perm (\u03b9a \u2295 \u03b9b)\n\u22a2 (summand a b (Quotient.mk'' \u03c3)) v + (summand a b (swap i j \u2022 Quotient.mk'' \u03c3)) v = 0", "state_after": "\u03b9a : Type u_1\n\u03b9b : Type u_2\ninst\u271d\u00b9\u2070 : Fintype \u03b9a\ninst\u271d\u2079 : Fintype \u03b9b\nR' : Type u_3\nM\u1d62 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R'\ninst\u271d\u2077 : AddCommGroup N\u2081\ninst\u271d\u2076 : Module R' N\u2081\ninst\u271d\u2075 : AddCommGroup N\u2082\ninst\u271d\u2074 : Module R' N\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u1d62\ninst\u271d\u00b2 : Module R' M\u1d62\ninst\u271d\u00b9 : DecidableEq \u03b9a\ninst\u271d : DecidableEq \u03b9b\na : M\u1d62 [\u22c0^\u03b9a]\u2192\u2097[R'] N\u2081\nb : M\u1d62 [\u22c0^\u03b9b]\u2192\u2097[R'] N\u2082\n\u03c3\u271d : Perm.ModSumCongr \u03b9a \u03b9b\nv : \u03b9a \u2295 \u03b9b \u2192 M\u1d62\ni j : \u03b9a \u2295 \u03b9b\nhv : v i = v j\nhij : i \u2260 j\n\u03c3 : Perm (\u03b9a \u2295 \u03b9b)\n\u22a2 (Perm.sign \u03c3 \u2022 MultilinearMap.domDomCongr \u03c3 ((\u2191a).domCoprod \u2191b)) v +\n      (Perm.sign (swap i j \u2022 \u03c3) \u2022 MultilinearMap.domDomCongr (swap i j \u2022 \u03c3) ((\u2191a).domCoprod \u2191b)) v =\n    0"}, {"tactic": "rw [smul_eq_mul, Perm.sign_mul, Perm.sign_swap hij]", "annotated_tactic": ["rw [<a>smul_eq_mul</a>, <a>Perm.sign_mul</a>, <a>Perm.sign_swap</a> hij]", [{"full_name": "smul_eq_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [89, 7], "def_end_pos": [89, 18]}, {"full_name": "Equiv.Perm.sign_mul", "def_path": "Mathlib/GroupTheory/Perm/Sign.lean", "def_pos": [400, 9], "def_end_pos": [400, 17]}, {"full_name": "Equiv.Perm.sign_swap", "def_path": "Mathlib/GroupTheory/Perm/Sign.lean", "def_pos": [429, 9], "def_end_pos": [429, 18]}]], "state_before": "\u03b9a : Type u_1\n\u03b9b : Type u_2\ninst\u271d\u00b9\u2070 : Fintype \u03b9a\ninst\u271d\u2079 : Fintype \u03b9b\nR' : Type u_3\nM\u1d62 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R'\ninst\u271d\u2077 : AddCommGroup N\u2081\ninst\u271d\u2076 : Module R' N\u2081\ninst\u271d\u2075 : AddCommGroup N\u2082\ninst\u271d\u2074 : Module R' N\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u1d62\ninst\u271d\u00b2 : Module R' M\u1d62\ninst\u271d\u00b9 : DecidableEq \u03b9a\ninst\u271d : DecidableEq \u03b9b\na : M\u1d62 [\u22c0^\u03b9a]\u2192\u2097[R'] N\u2081\nb : M\u1d62 [\u22c0^\u03b9b]\u2192\u2097[R'] N\u2082\n\u03c3\u271d : Perm.ModSumCongr \u03b9a \u03b9b\nv : \u03b9a \u2295 \u03b9b \u2192 M\u1d62\ni j : \u03b9a \u2295 \u03b9b\nhv : v i = v j\nhij : i \u2260 j\n\u03c3 : Perm (\u03b9a \u2295 \u03b9b)\n\u22a2 (Perm.sign \u03c3 \u2022 MultilinearMap.domDomCongr \u03c3 ((\u2191a).domCoprod \u2191b)) v +\n      (Perm.sign (swap i j \u2022 \u03c3) \u2022 MultilinearMap.domDomCongr (swap i j \u2022 \u03c3) ((\u2191a).domCoprod \u2191b)) v =\n    0", "state_after": "\u03b9a : Type u_1\n\u03b9b : Type u_2\ninst\u271d\u00b9\u2070 : Fintype \u03b9a\ninst\u271d\u2079 : Fintype \u03b9b\nR' : Type u_3\nM\u1d62 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R'\ninst\u271d\u2077 : AddCommGroup N\u2081\ninst\u271d\u2076 : Module R' N\u2081\ninst\u271d\u2075 : AddCommGroup N\u2082\ninst\u271d\u2074 : Module R' N\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u1d62\ninst\u271d\u00b2 : Module R' M\u1d62\ninst\u271d\u00b9 : DecidableEq \u03b9a\ninst\u271d : DecidableEq \u03b9b\na : M\u1d62 [\u22c0^\u03b9a]\u2192\u2097[R'] N\u2081\nb : M\u1d62 [\u22c0^\u03b9b]\u2192\u2097[R'] N\u2082\n\u03c3\u271d : Perm.ModSumCongr \u03b9a \u03b9b\nv : \u03b9a \u2295 \u03b9b \u2192 M\u1d62\ni j : \u03b9a \u2295 \u03b9b\nhv : v i = v j\nhij : i \u2260 j\n\u03c3 : Perm (\u03b9a \u2295 \u03b9b)\n\u22a2 (Perm.sign \u03c3 \u2022 MultilinearMap.domDomCongr \u03c3 ((\u2191a).domCoprod \u2191b)) v +\n      ((-1 * Perm.sign \u03c3) \u2022 MultilinearMap.domDomCongr (swap i j * \u03c3) ((\u2191a).domCoprod \u2191b)) v =\n    0"}, {"tactic": "simp only [one_mul, neg_mul, Function.comp_apply, Units.neg_smul, Perm.coe_mul, Units.val_neg,\n  MultilinearMap.smul_apply, MultilinearMap.neg_apply, MultilinearMap.domDomCongr_apply,\n  MultilinearMap.domCoprod_apply]", "annotated_tactic": ["simp only [<a>one_mul</a>, <a>neg_mul</a>, <a>Function.comp_apply</a>, <a>Units.neg_smul</a>, <a>Perm.coe_mul</a>, <a>Units.val_neg</a>,\n    <a>MultilinearMap.smul_apply</a>, <a>MultilinearMap.neg_apply</a>, <a>MultilinearMap.domDomCongr_apply</a>,\n    <a>MultilinearMap.domCoprod_apply</a>]", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "neg_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}, {"full_name": "Units.neg_smul", "def_path": "Mathlib/Algebra/Module/Defs.lean", "def_pos": [263, 9], "def_end_pos": [263, 23]}, {"full_name": "Equiv.Perm.coe_mul", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [108, 26], "def_end_pos": [108, 33]}, {"full_name": "Units.val_neg", "def_path": "Mathlib/Algebra/Ring/Units.lean", "def_pos": [37, 19], "def_end_pos": [37, 26]}, {"full_name": "MultilinearMap.smul_apply", "def_path": "Mathlib/LinearAlgebra/Multilinear/Basic.lean", "def_pos": [221, 9], "def_end_pos": [221, 19]}, {"full_name": "MultilinearMap.neg_apply", "def_path": "Mathlib/LinearAlgebra/Multilinear/Basic.lean", "def_pos": [1297, 9], "def_end_pos": [1297, 18]}, {"full_name": "MultilinearMap.domDomCongr_apply", "def_path": "Mathlib/LinearAlgebra/Multilinear/Basic.lean", "def_pos": [706, 9], "def_end_pos": [706, 14]}, {"full_name": "MultilinearMap.domCoprod_apply", "def_path": "Mathlib/LinearAlgebra/Multilinear/TensorProduct.lean", "def_pos": [42, 9], "def_end_pos": [42, 14]}]], "state_before": "\u03b9a : Type u_1\n\u03b9b : Type u_2\ninst\u271d\u00b9\u2070 : Fintype \u03b9a\ninst\u271d\u2079 : Fintype \u03b9b\nR' : Type u_3\nM\u1d62 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R'\ninst\u271d\u2077 : AddCommGroup N\u2081\ninst\u271d\u2076 : Module R' N\u2081\ninst\u271d\u2075 : AddCommGroup N\u2082\ninst\u271d\u2074 : Module R' N\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u1d62\ninst\u271d\u00b2 : Module R' M\u1d62\ninst\u271d\u00b9 : DecidableEq \u03b9a\ninst\u271d : DecidableEq \u03b9b\na : M\u1d62 [\u22c0^\u03b9a]\u2192\u2097[R'] N\u2081\nb : M\u1d62 [\u22c0^\u03b9b]\u2192\u2097[R'] N\u2082\n\u03c3\u271d : Perm.ModSumCongr \u03b9a \u03b9b\nv : \u03b9a \u2295 \u03b9b \u2192 M\u1d62\ni j : \u03b9a \u2295 \u03b9b\nhv : v i = v j\nhij : i \u2260 j\n\u03c3 : Perm (\u03b9a \u2295 \u03b9b)\n\u22a2 (Perm.sign \u03c3 \u2022 MultilinearMap.domDomCongr \u03c3 ((\u2191a).domCoprod \u2191b)) v +\n      ((-1 * Perm.sign \u03c3) \u2022 MultilinearMap.domDomCongr (swap i j * \u03c3) ((\u2191a).domCoprod \u2191b)) v =\n    0", "state_after": "\u03b9a : Type u_1\n\u03b9b : Type u_2\ninst\u271d\u00b9\u2070 : Fintype \u03b9a\ninst\u271d\u2079 : Fintype \u03b9b\nR' : Type u_3\nM\u1d62 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R'\ninst\u271d\u2077 : AddCommGroup N\u2081\ninst\u271d\u2076 : Module R' N\u2081\ninst\u271d\u2075 : AddCommGroup N\u2082\ninst\u271d\u2074 : Module R' N\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u1d62\ninst\u271d\u00b2 : Module R' M\u1d62\ninst\u271d\u00b9 : DecidableEq \u03b9a\ninst\u271d : DecidableEq \u03b9b\na : M\u1d62 [\u22c0^\u03b9a]\u2192\u2097[R'] N\u2081\nb : M\u1d62 [\u22c0^\u03b9b]\u2192\u2097[R'] N\u2082\n\u03c3\u271d : Perm.ModSumCongr \u03b9a \u03b9b\nv : \u03b9a \u2295 \u03b9b \u2192 M\u1d62\ni j : \u03b9a \u2295 \u03b9b\nhv : v i = v j\nhij : i \u2260 j\n\u03c3 : Perm (\u03b9a \u2295 \u03b9b)\n\u22a2 (Perm.sign \u03c3 \u2022 (\u2191a fun i => v (\u03c3 (Sum.inl i))) \u2297\u209c[R'] \u2191b fun i => v (\u03c3 (Sum.inr i))) +\n      -(Perm.sign \u03c3 \u2022\n          (\u2191a fun i_1 => v ((swap i j) (\u03c3 (Sum.inl i_1)))) \u2297\u209c[R'] \u2191b fun i_1 => v ((swap i j) (\u03c3 (Sum.inr i_1)))) =\n    0"}, {"tactic": "ext k", "annotated_tactic": ["ext k", []], "state_before": "case h.e'_2.h.e'_6.h.e'_3.h.e'_6.h.e'_10.h.e'_6\n\u03b9a : Type u_1\n\u03b9b : Type u_2\ninst\u271d\u00b9\u2070 : Fintype \u03b9a\ninst\u271d\u2079 : Fintype \u03b9b\nR' : Type u_3\nM\u1d62 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R'\ninst\u271d\u2077 : AddCommGroup N\u2081\ninst\u271d\u2076 : Module R' N\u2081\ninst\u271d\u2075 : AddCommGroup N\u2082\ninst\u271d\u2074 : Module R' N\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u1d62\ninst\u271d\u00b2 : Module R' M\u1d62\ninst\u271d\u00b9 : DecidableEq \u03b9a\ninst\u271d : DecidableEq \u03b9b\na : M\u1d62 [\u22c0^\u03b9a]\u2192\u2097[R'] N\u2081\nb : M\u1d62 [\u22c0^\u03b9b]\u2192\u2097[R'] N\u2082\n\u03c3\u271d : Perm.ModSumCongr \u03b9a \u03b9b\nv : \u03b9a \u2295 \u03b9b \u2192 M\u1d62\ni j : \u03b9a \u2295 \u03b9b\nhv : v i = v j\nhij : i \u2260 j\n\u03c3 : Perm (\u03b9a \u2295 \u03b9b)\n\u22a2 (fun i_1 => v ((swap i j) (\u03c3 (Sum.inr i_1)))) = fun i => v (\u03c3 (Sum.inr i))", "state_after": "case h.e'_2.h.e'_6.h.e'_3.h.e'_6.h.e'_10.h.e'_6.h\n\u03b9a : Type u_1\n\u03b9b : Type u_2\ninst\u271d\u00b9\u2070 : Fintype \u03b9a\ninst\u271d\u2079 : Fintype \u03b9b\nR' : Type u_3\nM\u1d62 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R'\ninst\u271d\u2077 : AddCommGroup N\u2081\ninst\u271d\u2076 : Module R' N\u2081\ninst\u271d\u2075 : AddCommGroup N\u2082\ninst\u271d\u2074 : Module R' N\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u1d62\ninst\u271d\u00b2 : Module R' M\u1d62\ninst\u271d\u00b9 : DecidableEq \u03b9a\ninst\u271d : DecidableEq \u03b9b\na : M\u1d62 [\u22c0^\u03b9a]\u2192\u2097[R'] N\u2081\nb : M\u1d62 [\u22c0^\u03b9b]\u2192\u2097[R'] N\u2082\n\u03c3\u271d : Perm.ModSumCongr \u03b9a \u03b9b\nv : \u03b9a \u2295 \u03b9b \u2192 M\u1d62\ni j : \u03b9a \u2295 \u03b9b\nhv : v i = v j\nhij : i \u2260 j\n\u03c3 : Perm (\u03b9a \u2295 \u03b9b)\nk : \u03b9b\n\u22a2 v ((swap i j) (\u03c3 (Sum.inr k))) = v (\u03c3 (Sum.inr k))"}, {"tactic": "rw [Equiv.apply_swap_eq_self hv]", "annotated_tactic": ["rw [<a>Equiv.apply_swap_eq_self</a> hv]", [{"full_name": "Equiv.apply_swap_eq_self", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1713, 9], "def_end_pos": [1713, 27]}]], "state_before": "case h.e'_2.h.e'_6.h.e'_3.h.e'_6.h.e'_10.h.e'_6.h\n\u03b9a : Type u_1\n\u03b9b : Type u_2\ninst\u271d\u00b9\u2070 : Fintype \u03b9a\ninst\u271d\u2079 : Fintype \u03b9b\nR' : Type u_3\nM\u1d62 : Type u_4\nN\u2081 : Type u_5\nN\u2082 : Type u_6\ninst\u271d\u2078 : CommSemiring R'\ninst\u271d\u2077 : AddCommGroup N\u2081\ninst\u271d\u2076 : Module R' N\u2081\ninst\u271d\u2075 : AddCommGroup N\u2082\ninst\u271d\u2074 : Module R' N\u2082\ninst\u271d\u00b3 : AddCommMonoid M\u1d62\ninst\u271d\u00b2 : Module R' M\u1d62\ninst\u271d\u00b9 : DecidableEq \u03b9a\ninst\u271d : DecidableEq \u03b9b\na : M\u1d62 [\u22c0^\u03b9a]\u2192\u2097[R'] N\u2081\nb : M\u1d62 [\u22c0^\u03b9b]\u2192\u2097[R'] N\u2082\n\u03c3\u271d : Perm.ModSumCongr \u03b9a \u03b9b\nv : \u03b9a \u2295 \u03b9b \u2192 M\u1d62\ni j : \u03b9a \u2295 \u03b9b\nhv : v i = v j\nhij : i \u2260 j\n\u03c3 : Perm (\u03b9a \u2295 \u03b9b)\nk : \u03b9b\n\u22a2 v ((swap i j) (\u03c3 (Sum.inr k))) = v (\u03c3 (Sum.inr k))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Star/Order.lean", "full_name": "star_lt_star_iff", "start": [199, 1], "end": [203, 72], "traced_tactics": [{"tactic": "by_cases h : x = y", "annotated_tactic": ["by_cases h : x = y", []], "state_before": "R : Type u\ninst\u271d\u00b3 : NonUnitalSemiring R\ninst\u271d\u00b2 : PartialOrder R\ninst\u271d\u00b9 : StarRing R\ninst\u271d : StarOrderedRing R\nx y : R\n\u22a2 star x < star y \u2194 x < y", "state_after": "case pos\nR : Type u\ninst\u271d\u00b3 : NonUnitalSemiring R\ninst\u271d\u00b2 : PartialOrder R\ninst\u271d\u00b9 : StarRing R\ninst\u271d : StarOrderedRing R\nx y : R\nh : x = y\n\u22a2 star x < star y \u2194 x < y\n\ncase neg\nR : Type u\ninst\u271d\u00b3 : NonUnitalSemiring R\ninst\u271d\u00b2 : PartialOrder R\ninst\u271d\u00b9 : StarRing R\ninst\u271d : StarOrderedRing R\nx y : R\nh : \u00acx = y\n\u22a2 star x < star y \u2194 x < y"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "case pos\nR : Type u\ninst\u271d\u00b3 : NonUnitalSemiring R\ninst\u271d\u00b2 : PartialOrder R\ninst\u271d\u00b9 : StarRing R\ninst\u271d : StarOrderedRing R\nx y : R\nh : x = y\n\u22a2 star x < star y \u2194 x < y", "state_after": "no goals"}, {"tactic": "simpa [le_iff_lt_or_eq, h] using star_le_star_iff (x := x) (y := y)", "annotated_tactic": ["simpa [<a>le_iff_lt_or_eq</a>, h] using <a>star_le_star_iff</a> (x := x) (y := y)", [{"full_name": "le_iff_lt_or_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [234, 9], "def_end_pos": [234, 24]}, {"full_name": "star_le_star_iff", "def_path": "Mathlib/Algebra/Star/Order.lean", "def_pos": [188, 7], "def_end_pos": [188, 23]}]], "state_before": "case neg\nR : Type u\ninst\u271d\u00b3 : NonUnitalSemiring R\ninst\u271d\u00b2 : PartialOrder R\ninst\u271d\u00b9 : StarRing R\ninst\u271d : StarOrderedRing R\nx y : R\nh : \u00acx = y\n\u22a2 star x < star y \u2194 x < y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean", "full_name": "jacobiSym.mod_left'", "start": [246, 1], "end": [247, 31], "traced_tactics": [{"tactic": "rw [mod_left, h, \u2190 mod_left]", "annotated_tactic": ["rw [<a>mod_left</a>, h, \u2190 <a>mod_left</a>]", [{"full_name": "jacobiSym.mod_left", "def_path": "Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean", "def_pos": [231, 9], "def_end_pos": [231, 17]}, {"full_name": "jacobiSym.mod_left", "def_path": "Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean", "def_pos": [231, 9], "def_end_pos": [231, 17]}]], "state_before": "a\u2081 a\u2082 : \u2124\nb : \u2115\nh : a\u2081 % \u2191b = a\u2082 % \u2191b\n\u22a2 J(a\u2081 | b) = J(a\u2082 | b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.map_surjective_of_surjective", "start": [1403, 1], "end": [1410, 36], "traced_tactics": [{"tactic": "intro s", "annotated_tactic": ["intro s", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\n\u22a2 Surjective (map f)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\ns : Multiset \u03b2\n\u22a2 \u2203 a, map f a = s"}, {"tactic": "induction' s using Multiset.induction_on with x s ih", "annotated_tactic": ["induction' s using <a>Multiset.induction_on</a> with x s ih", [{"full_name": "Multiset.induction_on", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [174, 19], "def_end_pos": [174, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\ns : Multiset \u03b2\n\u22a2 \u2203 a, map f a = s", "state_after": "case empty\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\n\u22a2 \u2203 a, map f a = 0\n\ncase cons\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\nx : \u03b2\ns : Multiset \u03b2\nih : \u2203 a, map f a = s\n\u22a2 \u2203 a, map f a = x ::\u2098 s"}, {"tactic": "exact \u27e80, map_zero _\u27e9", "annotated_tactic": ["exact \u27e80, <a>map_zero</a> _\u27e9", [{"full_name": "Multiset.map_zero", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1220, 9], "def_end_pos": [1220, 17]}]], "state_before": "case empty\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\n\u22a2 \u2203 a, map f a = 0", "state_after": "no goals"}, {"tactic": "obtain \u27e8y, rfl\u27e9 := hf x", "annotated_tactic": ["obtain \u27e8y, rfl\u27e9 := hf x", []], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\nx : \u03b2\ns : Multiset \u03b2\nih : \u2203 a, map f a = s\n\u22a2 \u2203 a, map f a = x ::\u2098 s", "state_after": "case cons.intro\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\ns : Multiset \u03b2\nih : \u2203 a, map f a = s\ny : \u03b1\n\u22a2 \u2203 a, map f a = f y ::\u2098 s"}, {"tactic": "obtain \u27e8t, rfl\u27e9 := ih", "annotated_tactic": ["obtain \u27e8t, rfl\u27e9 := ih", []], "state_before": "case cons.intro\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\ns : Multiset \u03b2\nih : \u2203 a, map f a = s\ny : \u03b1\n\u22a2 \u2203 a, map f a = f y ::\u2098 s", "state_after": "case cons.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\ny : \u03b1\nt : Multiset \u03b1\n\u22a2 \u2203 a, map f a = f y ::\u2098 map f t"}, {"tactic": "exact \u27e8y ::\u2098 t, map_cons _ _ _\u27e9", "annotated_tactic": ["exact \u27e8y ::\u2098 t, <a>map_cons</a> _ _ _\u27e9", [{"full_name": "Multiset.map_cons", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1225, 9], "def_end_pos": [1225, 17]}]], "state_before": "case cons.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\ny : \u03b1\nt : Multiset \u03b1\n\u22a2 \u2203 a, map f a = f y ::\u2098 map f t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Nonneg/Module.lean", "full_name": "Nonneg.mk_smul", "start": [38, 1], "end": [40, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/RelIso/Basic.lean", "full_name": "RelEmbedding.isIrrefl", "start": [324, 11], "end": [325, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean", "full_name": "CategoryTheory.Abelian.epi_of_epi_of_epi_of_mono", "start": [123, 1], "end": [128, 79], "traced_tactics": [{"tactic": "simpa only [R\u2082.map'_comp 1 2 3] using hR\u2082.toIsComplex.zero 1", "annotated_tactic": ["simpa only [R\u2082.map'_comp 1 2 3] using hR\u2082.toIsComplex.zero 1", []], "state_before": "C : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\ninst\u271d : Abelian C\nR\u2081 R\u2082 : ComposableArrows C 3\n\u03c6 : R\u2081 \u27f6 R\u2082\nhR\u2081 : R\u2081.Exact\nhR\u2082 : R\u2082.Exact\nh\u2080 : Epi (app' \u03c6 0 \u22ef)\nh\u2082 : Epi (app' \u03c6 2 \u22ef)\nh\u2083 : Mono (app' \u03c6 3 \u22ef)\n\u22a2 R\u2082.map' 1 3 \u22ef \u22ef = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Subobject/Basic.lean", "full_name": "CategoryTheory.Subobject.ofMkLE_arrow", "start": [364, 1], "end": [365, 51], "traced_tactics": [{"tactic": "simp [ofMkLE]", "annotated_tactic": ["simp [<a>ofMkLE</a>]", [{"full_name": "CategoryTheory.Subobject.ofMkLE", "def_path": "Mathlib/CategoryTheory/Subobject/Basic.lean", "def_pos": [354, 5], "def_end_pos": [354, 11]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nX\u271d Y Z : C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nB A : C\nf : A \u27f6 B\ninst\u271d : Mono f\nX : Subobject B\nh : mk f \u2264 X\n\u22a2 ofMkLE f X h \u226b X.arrow = f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Shift/CommShift.lean", "full_name": "CategoryTheory.NatTrans.CommShift.of_isIso", "start": [292, 1], "end": [298, 17], "traced_tactics": [{"tactic": "haveI : NatTrans.CommShift (asIso \u03c4).hom A := by\n  dsimp\n  infer_instance", "annotated_tactic": ["haveI : <a>NatTrans.CommShift</a> (<a>asIso</a> \u03c4).<a>hom</a> A := by\n    dsimp\n    infer_instance", [{"full_name": "CategoryTheory.NatTrans.CommShift", "def_path": "Mathlib/CategoryTheory/Shift/CommShift.lean", "def_pos": [249, 7], "def_end_pos": [249, 16]}, {"full_name": "CategoryTheory.asIso", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [329, 19], "def_end_pos": [329, 24]}, {"full_name": "CategoryTheory.Iso.hom", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [53, 3], "def_end_pos": [53, 6]}]], "state_before": "C : Type u_1\nD : Type u_2\nE : Type u_3\nJ : Type u_4\ninst\u271d\u00b9\u2076 : Category.{u_7, u_1} C\ninst\u271d\u00b9\u2075 : Category.{u_6, u_2} D\ninst\u271d\u00b9\u2074 : Category.{?u.121819, u_3} E\ninst\u271d\u00b9\u00b3 : Category.{?u.121823, u_4} J\nF\u2081 F\u2082 F\u2083 : C \u2964 D\n\u03c4 : F\u2081 \u27f6 F\u2082\n\u03c4' : F\u2082 \u27f6 F\u2083\ne : F\u2081 \u2245 F\u2082\nG G' : D \u2964 E\n\u03c4'' : G \u27f6 G'\nH : E \u2964 J\nA : Type u_5\ninst\u271d\u00b9\u00b2 : AddMonoid A\ninst\u271d\u00b9\u00b9 : HasShift C A\ninst\u271d\u00b9\u2070 : HasShift D A\ninst\u271d\u2079 : HasShift E A\ninst\u271d\u2078 : HasShift J A\ninst\u271d\u2077 : F\u2081.CommShift A\ninst\u271d\u2076 : F\u2082.CommShift A\ninst\u271d\u2075 : F\u2083.CommShift A\ninst\u271d\u2074 : G.CommShift A\ninst\u271d\u00b3 : G'.CommShift A\ninst\u271d\u00b2 : H.CommShift A\ninst\u271d\u00b9 : IsIso \u03c4\ninst\u271d : CommShift \u03c4 A\n\u22a2 CommShift (inv \u03c4) A", "state_after": "C : Type u_1\nD : Type u_2\nE : Type u_3\nJ : Type u_4\ninst\u271d\u00b9\u2076 : Category.{u_7, u_1} C\ninst\u271d\u00b9\u2075 : Category.{u_6, u_2} D\ninst\u271d\u00b9\u2074 : Category.{?u.121819, u_3} E\ninst\u271d\u00b9\u00b3 : Category.{?u.121823, u_4} J\nF\u2081 F\u2082 F\u2083 : C \u2964 D\n\u03c4 : F\u2081 \u27f6 F\u2082\n\u03c4' : F\u2082 \u27f6 F\u2083\ne : F\u2081 \u2245 F\u2082\nG G' : D \u2964 E\n\u03c4'' : G \u27f6 G'\nH : E \u2964 J\nA : Type u_5\ninst\u271d\u00b9\u00b2 : AddMonoid A\ninst\u271d\u00b9\u00b9 : HasShift C A\ninst\u271d\u00b9\u2070 : HasShift D A\ninst\u271d\u2079 : HasShift E A\ninst\u271d\u2078 : HasShift J A\ninst\u271d\u2077 : F\u2081.CommShift A\ninst\u271d\u2076 : F\u2082.CommShift A\ninst\u271d\u2075 : F\u2083.CommShift A\ninst\u271d\u2074 : G.CommShift A\ninst\u271d\u00b3 : G'.CommShift A\ninst\u271d\u00b2 : H.CommShift A\ninst\u271d\u00b9 : IsIso \u03c4\ninst\u271d : CommShift \u03c4 A\nthis : CommShift (asIso \u03c4).hom A\n\u22a2 CommShift (inv \u03c4) A"}, {"tactic": "change NatTrans.CommShift (asIso \u03c4).inv A", "annotated_tactic": ["change <a>NatTrans.CommShift</a> (<a>asIso</a> \u03c4).<a>inv</a> A", [{"full_name": "CategoryTheory.NatTrans.CommShift", "def_path": "Mathlib/CategoryTheory/Shift/CommShift.lean", "def_pos": [249, 7], "def_end_pos": [249, 16]}, {"full_name": "CategoryTheory.asIso", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [329, 19], "def_end_pos": [329, 24]}, {"full_name": "CategoryTheory.Iso.inv", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [55, 3], "def_end_pos": [55, 6]}]], "state_before": "C : Type u_1\nD : Type u_2\nE : Type u_3\nJ : Type u_4\ninst\u271d\u00b9\u2076 : Category.{u_7, u_1} C\ninst\u271d\u00b9\u2075 : Category.{u_6, u_2} D\ninst\u271d\u00b9\u2074 : Category.{?u.121819, u_3} E\ninst\u271d\u00b9\u00b3 : Category.{?u.121823, u_4} J\nF\u2081 F\u2082 F\u2083 : C \u2964 D\n\u03c4 : F\u2081 \u27f6 F\u2082\n\u03c4' : F\u2082 \u27f6 F\u2083\ne : F\u2081 \u2245 F\u2082\nG G' : D \u2964 E\n\u03c4'' : G \u27f6 G'\nH : E \u2964 J\nA : Type u_5\ninst\u271d\u00b9\u00b2 : AddMonoid A\ninst\u271d\u00b9\u00b9 : HasShift C A\ninst\u271d\u00b9\u2070 : HasShift D A\ninst\u271d\u2079 : HasShift E A\ninst\u271d\u2078 : HasShift J A\ninst\u271d\u2077 : F\u2081.CommShift A\ninst\u271d\u2076 : F\u2082.CommShift A\ninst\u271d\u2075 : F\u2083.CommShift A\ninst\u271d\u2074 : G.CommShift A\ninst\u271d\u00b3 : G'.CommShift A\ninst\u271d\u00b2 : H.CommShift A\ninst\u271d\u00b9 : IsIso \u03c4\ninst\u271d : CommShift \u03c4 A\nthis : CommShift (asIso \u03c4).hom A\n\u22a2 CommShift (inv \u03c4) A", "state_after": "C : Type u_1\nD : Type u_2\nE : Type u_3\nJ : Type u_4\ninst\u271d\u00b9\u2076 : Category.{u_7, u_1} C\ninst\u271d\u00b9\u2075 : Category.{u_6, u_2} D\ninst\u271d\u00b9\u2074 : Category.{?u.121819, u_3} E\ninst\u271d\u00b9\u00b3 : Category.{?u.121823, u_4} J\nF\u2081 F\u2082 F\u2083 : C \u2964 D\n\u03c4 : F\u2081 \u27f6 F\u2082\n\u03c4' : F\u2082 \u27f6 F\u2083\ne : F\u2081 \u2245 F\u2082\nG G' : D \u2964 E\n\u03c4'' : G \u27f6 G'\nH : E \u2964 J\nA : Type u_5\ninst\u271d\u00b9\u00b2 : AddMonoid A\ninst\u271d\u00b9\u00b9 : HasShift C A\ninst\u271d\u00b9\u2070 : HasShift D A\ninst\u271d\u2079 : HasShift E A\ninst\u271d\u2078 : HasShift J A\ninst\u271d\u2077 : F\u2081.CommShift A\ninst\u271d\u2076 : F\u2082.CommShift A\ninst\u271d\u2075 : F\u2083.CommShift A\ninst\u271d\u2074 : G.CommShift A\ninst\u271d\u00b3 : G'.CommShift A\ninst\u271d\u00b2 : H.CommShift A\ninst\u271d\u00b9 : IsIso \u03c4\ninst\u271d : CommShift \u03c4 A\nthis : CommShift (asIso \u03c4).hom A\n\u22a2 CommShift (asIso \u03c4).inv A"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "C : Type u_1\nD : Type u_2\nE : Type u_3\nJ : Type u_4\ninst\u271d\u00b9\u2076 : Category.{u_7, u_1} C\ninst\u271d\u00b9\u2075 : Category.{u_6, u_2} D\ninst\u271d\u00b9\u2074 : Category.{?u.121819, u_3} E\ninst\u271d\u00b9\u00b3 : Category.{?u.121823, u_4} J\nF\u2081 F\u2082 F\u2083 : C \u2964 D\n\u03c4 : F\u2081 \u27f6 F\u2082\n\u03c4' : F\u2082 \u27f6 F\u2083\ne : F\u2081 \u2245 F\u2082\nG G' : D \u2964 E\n\u03c4'' : G \u27f6 G'\nH : E \u2964 J\nA : Type u_5\ninst\u271d\u00b9\u00b2 : AddMonoid A\ninst\u271d\u00b9\u00b9 : HasShift C A\ninst\u271d\u00b9\u2070 : HasShift D A\ninst\u271d\u2079 : HasShift E A\ninst\u271d\u2078 : HasShift J A\ninst\u271d\u2077 : F\u2081.CommShift A\ninst\u271d\u2076 : F\u2082.CommShift A\ninst\u271d\u2075 : F\u2083.CommShift A\ninst\u271d\u2074 : G.CommShift A\ninst\u271d\u00b3 : G'.CommShift A\ninst\u271d\u00b2 : H.CommShift A\ninst\u271d\u00b9 : IsIso \u03c4\ninst\u271d : CommShift \u03c4 A\nthis : CommShift (asIso \u03c4).hom A\n\u22a2 CommShift (asIso \u03c4).inv A", "state_after": "no goals"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "C : Type u_1\nD : Type u_2\nE : Type u_3\nJ : Type u_4\ninst\u271d\u00b9\u2076 : Category.{u_7, u_1} C\ninst\u271d\u00b9\u2075 : Category.{u_6, u_2} D\ninst\u271d\u00b9\u2074 : Category.{?u.121819, u_3} E\ninst\u271d\u00b9\u00b3 : Category.{?u.121823, u_4} J\nF\u2081 F\u2082 F\u2083 : C \u2964 D\n\u03c4 : F\u2081 \u27f6 F\u2082\n\u03c4' : F\u2082 \u27f6 F\u2083\ne : F\u2081 \u2245 F\u2082\nG G' : D \u2964 E\n\u03c4'' : G \u27f6 G'\nH : E \u2964 J\nA : Type u_5\ninst\u271d\u00b9\u00b2 : AddMonoid A\ninst\u271d\u00b9\u00b9 : HasShift C A\ninst\u271d\u00b9\u2070 : HasShift D A\ninst\u271d\u2079 : HasShift E A\ninst\u271d\u2078 : HasShift J A\ninst\u271d\u2077 : F\u2081.CommShift A\ninst\u271d\u2076 : F\u2082.CommShift A\ninst\u271d\u2075 : F\u2083.CommShift A\ninst\u271d\u2074 : G.CommShift A\ninst\u271d\u00b3 : G'.CommShift A\ninst\u271d\u00b2 : H.CommShift A\ninst\u271d\u00b9 : IsIso \u03c4\ninst\u271d : CommShift \u03c4 A\n\u22a2 CommShift (asIso \u03c4).hom A", "state_after": "C : Type u_1\nD : Type u_2\nE : Type u_3\nJ : Type u_4\ninst\u271d\u00b9\u2076 : Category.{u_7, u_1} C\ninst\u271d\u00b9\u2075 : Category.{u_6, u_2} D\ninst\u271d\u00b9\u2074 : Category.{?u.121819, u_3} E\ninst\u271d\u00b9\u00b3 : Category.{?u.121823, u_4} J\nF\u2081 F\u2082 F\u2083 : C \u2964 D\n\u03c4 : F\u2081 \u27f6 F\u2082\n\u03c4' : F\u2082 \u27f6 F\u2083\ne : F\u2081 \u2245 F\u2082\nG G' : D \u2964 E\n\u03c4'' : G \u27f6 G'\nH : E \u2964 J\nA : Type u_5\ninst\u271d\u00b9\u00b2 : AddMonoid A\ninst\u271d\u00b9\u00b9 : HasShift C A\ninst\u271d\u00b9\u2070 : HasShift D A\ninst\u271d\u2079 : HasShift E A\ninst\u271d\u2078 : HasShift J A\ninst\u271d\u2077 : F\u2081.CommShift A\ninst\u271d\u2076 : F\u2082.CommShift A\ninst\u271d\u2075 : F\u2083.CommShift A\ninst\u271d\u2074 : G.CommShift A\ninst\u271d\u00b3 : G'.CommShift A\ninst\u271d\u00b2 : H.CommShift A\ninst\u271d\u00b9 : IsIso \u03c4\ninst\u271d : CommShift \u03c4 A\n\u22a2 CommShift \u03c4 A"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "C : Type u_1\nD : Type u_2\nE : Type u_3\nJ : Type u_4\ninst\u271d\u00b9\u2076 : Category.{u_7, u_1} C\ninst\u271d\u00b9\u2075 : Category.{u_6, u_2} D\ninst\u271d\u00b9\u2074 : Category.{?u.121819, u_3} E\ninst\u271d\u00b9\u00b3 : Category.{?u.121823, u_4} J\nF\u2081 F\u2082 F\u2083 : C \u2964 D\n\u03c4 : F\u2081 \u27f6 F\u2082\n\u03c4' : F\u2082 \u27f6 F\u2083\ne : F\u2081 \u2245 F\u2082\nG G' : D \u2964 E\n\u03c4'' : G \u27f6 G'\nH : E \u2964 J\nA : Type u_5\ninst\u271d\u00b9\u00b2 : AddMonoid A\ninst\u271d\u00b9\u00b9 : HasShift C A\ninst\u271d\u00b9\u2070 : HasShift D A\ninst\u271d\u2079 : HasShift E A\ninst\u271d\u2078 : HasShift J A\ninst\u271d\u2077 : F\u2081.CommShift A\ninst\u271d\u2076 : F\u2082.CommShift A\ninst\u271d\u2075 : F\u2083.CommShift A\ninst\u271d\u2074 : G.CommShift A\ninst\u271d\u00b3 : G'.CommShift A\ninst\u271d\u00b2 : H.CommShift A\ninst\u271d\u00b9 : IsIso \u03c4\ninst\u271d : CommShift \u03c4 A\n\u22a2 CommShift \u03c4 A", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/List/Perm.lean", "full_name": "List.perm_comm", "start": [44, 1], "end": [44, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "full_name": "Finset.prod_range_one", "start": [1567, 1], "end": [1568, 33], "traced_tactics": [{"tactic": "rw [range_one, prod_singleton]", "annotated_tactic": ["rw [<a>range_one</a>, <a>prod_singleton</a>]", [{"full_name": "Finset.range_one", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2950, 9], "def_end_pos": [2950, 18]}, {"full_name": "Finset.prod_singleton", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [382, 9], "def_end_pos": [382, 23]}]], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\nf : \u2115 \u2192 \u03b2\n\u22a2 \u220f k \u2208 range 1, f k = f 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/NonUnitalSubring/Basic.lean", "full_name": "NonUnitalSubring.closure_univ", "start": [784, 1], "end": [785, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Concrete.lean", "full_name": "Equiv.Perm.formPerm_toList", "start": [385, 1], "end": [397, 30], "traced_tactics": [{"tactic": "by_cases hx : f x = x", "annotated_tactic": ["by_cases hx : f x = x", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx\u271d : \u03b1\nf : Perm \u03b1\nx : \u03b1\n\u22a2 (f.toList x).formPerm = f.cycleOf x", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx\u271d : \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : f x = x\n\u22a2 (f.toList x).formPerm = f.cycleOf x\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx\u271d : \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u00acf x = x\n\u22a2 (f.toList x).formPerm = f.cycleOf x"}, {"tactic": "ext y", "annotated_tactic": ["ext y", []], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx\u271d : \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u00acf x = x\n\u22a2 (f.toList x).formPerm = f.cycleOf x", "state_after": "case neg.H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx\u271d : \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u00acf x = x\ny : \u03b1\n\u22a2 (f.toList x).formPerm y = (f.cycleOf x) y"}, {"tactic": "by_cases hy : SameCycle f x y", "annotated_tactic": ["by_cases hy : <a>SameCycle</a> f x y", [{"full_name": "Equiv.Perm.SameCycle", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Basic.lean", "def_pos": [51, 5], "def_end_pos": [51, 14]}]], "state_before": "case neg.H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx\u271d : \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u00acf x = x\ny : \u03b1\n\u22a2 (f.toList x).formPerm y = (f.cycleOf x) y", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx\u271d : \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u00acf x = x\ny : \u03b1\nhy : f.SameCycle x y\n\u22a2 (f.toList x).formPerm y = (f.cycleOf x) y\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx\u271d : \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u00acf x = x\ny : \u03b1\nhy : \u00acf.SameCycle x y\n\u22a2 (f.toList x).formPerm y = (f.cycleOf x) y"}, {"tactic": "rw [(cycleOf_eq_one_iff f).mpr hx, toList_eq_nil_iff.mpr (not_mem_support.mpr hx),\n  formPerm_nil]", "annotated_tactic": ["rw [(<a>cycleOf_eq_one_iff</a> f).<a>mpr</a> hx, toList_eq_nil_iff.mpr (not_mem_support.mpr hx),\n      <a>formPerm_nil</a>]", [{"full_name": "Equiv.Perm.cycleOf_eq_one_iff", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Factors.lean", "def_pos": [134, 9], "def_end_pos": [134, 27]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}, {"full_name": "List.formPerm_nil", "def_path": "Mathlib/GroupTheory/Perm/List.lean", "def_pos": [53, 9], "def_end_pos": [53, 21]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx\u271d : \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : f x = x\n\u22a2 (f.toList x).formPerm = f.cycleOf x", "state_after": "no goals"}, {"tactic": "obtain \u27e8k, _, rfl\u27e9 := hy.exists_pow_eq_of_mem_support (mem_support.mpr hx)", "annotated_tactic": ["obtain \u27e8k, _, rfl\u27e9 := hy.exists_pow_eq_of_mem_support (mem_support.mpr hx)", []], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx\u271d : \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u00acf x = x\ny : \u03b1\nhy : f.SameCycle x y\n\u22a2 (f.toList x).formPerm y = (f.cycleOf x) y", "state_after": "case pos.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx\u271d : \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u00acf x = x\nk : \u2115\nleft\u271d : k < (f.cycleOf x).support.card\nhy : f.SameCycle x ((f ^ k) x)\n\u22a2 (f.toList x).formPerm ((f ^ k) x) = (f.cycleOf x) ((f ^ k) x)"}, {"tactic": "rw [cycleOf_apply_apply_pow_self, List.formPerm_apply_mem_eq_next (nodup_toList f x),\n  next_toList_eq_apply, pow_succ', mul_apply]", "annotated_tactic": ["rw [<a>cycleOf_apply_apply_pow_self</a>, <a>List.formPerm_apply_mem_eq_next</a> (<a>nodup_toList</a> f x),\n      <a>next_toList_eq_apply</a>, <a>pow_succ'</a>, <a>mul_apply</a>]", [{"full_name": "Equiv.Perm.cycleOf_apply_apply_pow_self", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Factors.lean", "def_pos": [109, 9], "def_end_pos": [109, 37]}, {"full_name": "List.formPerm_apply_mem_eq_next", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Concrete.lean", "def_pos": [120, 9], "def_end_pos": [120, 35]}, {"full_name": "Equiv.Perm.nodup_toList", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Concrete.lean", "def_pos": [278, 9], "def_end_pos": [278, 21]}, {"full_name": "Equiv.Perm.next_toList_eq_apply", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Concrete.lean", "def_pos": [312, 9], "def_end_pos": [312, 29]}, {"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [667, 34], "def_end_pos": [667, 43]}, {"full_name": "Equiv.Perm.mul_apply", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 18]}]], "state_before": "case pos.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx\u271d : \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u00acf x = x\nk : \u2115\nleft\u271d : k < (f.cycleOf x).support.card\nhy : f.SameCycle x ((f ^ k) x)\n\u22a2 (f.toList x).formPerm ((f ^ k) x) = (f.cycleOf x) ((f ^ k) x)", "state_after": "case pos.intro.intro.hy\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx\u271d : \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u00acf x = x\nk : \u2115\nleft\u271d : k < (f.cycleOf x).support.card\nhy : f.SameCycle x ((f ^ k) x)\n\u22a2 (f ^ k) x \u2208 f.toList x"}, {"tactic": "rw [mem_toList_iff]", "annotated_tactic": ["rw [<a>mem_toList_iff</a>]", [{"full_name": "Equiv.Perm.mem_toList_iff", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Concrete.lean", "def_pos": [265, 9], "def_end_pos": [265, 23]}]], "state_before": "case pos.intro.intro.hy\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx\u271d : \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u00acf x = x\nk : \u2115\nleft\u271d : k < (f.cycleOf x).support.card\nhy : f.SameCycle x ((f ^ k) x)\n\u22a2 (f ^ k) x \u2208 f.toList x", "state_after": "case pos.intro.intro.hy\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx\u271d : \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u00acf x = x\nk : \u2115\nleft\u271d : k < (f.cycleOf x).support.card\nhy : f.SameCycle x ((f ^ k) x)\n\u22a2 f.SameCycle x ((f ^ k) x) \u2227 x \u2208 f.support"}, {"tactic": "exact \u27e8\u27e8k, rfl\u27e9, mem_support.mpr hx\u27e9", "annotated_tactic": ["exact \u27e8\u27e8k, <a>rfl</a>\u27e9, mem_support.mpr hx\u27e9", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case pos.intro.intro.hy\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx\u271d : \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u00acf x = x\nk : \u2115\nleft\u271d : k < (f.cycleOf x).support.card\nhy : f.SameCycle x ((f ^ k) x)\n\u22a2 f.SameCycle x ((f ^ k) x) \u2227 x \u2208 f.support", "state_after": "no goals"}, {"tactic": "rw [cycleOf_apply_of_not_sameCycle hy, formPerm_apply_of_not_mem]", "annotated_tactic": ["rw [<a>cycleOf_apply_of_not_sameCycle</a> hy, <a>formPerm_apply_of_not_mem</a>]", [{"full_name": "Equiv.Perm.cycleOf_apply_of_not_sameCycle", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Factors.lean", "def_pos": [88, 9], "def_end_pos": [88, 39]}, {"full_name": "List.formPerm_apply_of_not_mem", "def_path": "Mathlib/GroupTheory/Perm/List.lean", "def_pos": [112, 9], "def_end_pos": [112, 34]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx\u271d : \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u00acf x = x\ny : \u03b1\nhy : \u00acf.SameCycle x y\n\u22a2 (f.toList x).formPerm y = (f.cycleOf x) y", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx\u271d : \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u00acf x = x\ny : \u03b1\nhy : \u00acf.SameCycle x y\n\u22a2 y \u2209 f.toList x"}, {"tactic": "simp [mem_toList_iff, hy]", "annotated_tactic": ["simp [<a>mem_toList_iff</a>, hy]", [{"full_name": "Equiv.Perm.mem_toList_iff", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Concrete.lean", "def_pos": [265, 9], "def_end_pos": [265, 23]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\np : Perm \u03b1\nx\u271d : \u03b1\nf : Perm \u03b1\nx : \u03b1\nhx : \u00acf x = x\ny : \u03b1\nhy : \u00acf.SameCycle x y\n\u22a2 y \u2209 f.toList x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "full_name": "Subgroup.Normal.of_map_subtype", "start": [3390, 1], "end": [3392, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean", "full_name": "hasDerivAt_sqrt_mul_log", "start": [115, 1], "end": [119, 40], "traced_tactics": [{"tactic": "convert (hasDerivAt_sqrt hx).mul (hasDerivAt_log hx) using 1", "annotated_tactic": ["convert (<a>hasDerivAt_sqrt</a> hx).<a>mul</a> (<a>hasDerivAt_log</a> hx) using 1", [{"full_name": "Real.hasDerivAt_sqrt", "def_path": "Mathlib/Analysis/SpecialFunctions/Sqrt.lean", "def_pos": [69, 9], "def_end_pos": [69, 24]}, {"full_name": "HasDerivAt.mul", "def_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean", "def_pos": [215, 9], "def_end_pos": [215, 23]}, {"full_name": "Real.hasDerivAt_log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean", "def_pos": [50, 9], "def_end_pos": [50, 23]}]], "state_before": "x : \u211d\nhx : x \u2260 0\n\u22a2 HasDerivAt (fun x => \u221ax * log x) ((2 + log x) / (2 * \u221ax)) x", "state_after": "case h.e'_7\nx : \u211d\nhx : x \u2260 0\n\u22a2 (2 + log x) / (2 * \u221ax) = 1 / (2 * \u221ax) * log x + \u221ax * x\u207b\u00b9"}, {"tactic": "rw [add_div, div_mul_cancel_left\u2080 two_ne_zero, \u2190 div_eq_mul_inv, sqrt_div_self', add_comm,\n  one_div, one_div, \u2190 div_eq_inv_mul]", "annotated_tactic": ["rw [<a>add_div</a>, <a>div_mul_cancel_left\u2080</a> <a>two_ne_zero</a>, \u2190 <a>div_eq_mul_inv</a>, <a>sqrt_div_self'</a>, <a>add_comm</a>,\n    <a>one_div</a>, <a>one_div</a>, \u2190 <a>div_eq_inv_mul</a>]", [{"full_name": "add_div", "def_path": "Mathlib/Algebra/Field/Basic.lean", "def_pos": [29, 9], "def_end_pos": [29, 16]}, {"full_name": "div_mul_cancel_left\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [500, 7], "def_end_pos": [500, 27]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [65, 7], "def_end_pos": [65, 18]}, {"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1059, 9], "def_end_pos": [1059, 23]}, {"full_name": "Real.sqrt_div_self'", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [410, 9], "def_end_pos": [410, 23]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 16]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 16]}, {"full_name": "div_eq_inv_mul", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [736, 9], "def_end_pos": [736, 23]}]], "state_before": "case h.e'_7\nx : \u211d\nhx : x \u2260 0\n\u22a2 (2 + log x) / (2 * \u221ax) = 1 / (2 * \u221ax) * log x + \u221ax * x\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "MeasureTheory.NullMeasurableSet.prod", "start": [539, 1], "end": [544, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/InfiniteSum/Group.lean", "full_name": "cauchySeq_finset_iff_prod_vanishing", "start": [200, 1], "end": [222, 74], "traced_tactics": [{"tactic": "rw [tendsto_atTop']", "annotated_tactic": ["rw [<a>tendsto_atTop'</a>]", [{"full_name": "Filter.tendsto_atTop'", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1374, 9], "def_end_pos": [1374, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\n\u22a2 Tendsto (fun x => (\u220f b \u2208 x.2, f b) / \u220f b \u2208 x.1, f b) atTop (\ud835\udcdd 1) \u2194\n    \u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\n\u22a2 (\u2200 s \u2208 \ud835\udcdd 1, \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 s) \u2194\n    \u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\n\u22a2 (\u2200 s \u2208 \ud835\udcdd 1, \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 s) \u2194\n    \u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\n\u22a2 (\u2200 s \u2208 \ud835\udcdd 1, \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 s) \u2192\n    \u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e\n\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\n\u22a2 (\u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e) \u2192\n    \u2200 s \u2208 \ud835\udcdd 1, \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 s"}, {"tactic": "intro h e he", "annotated_tactic": ["intro h e he", []], "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\n\u22a2 (\u2200 s \u2208 \ud835\udcdd 1, \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 s) \u2192\n    \u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : \u2200 s \u2208 \ud835\udcdd 1, \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 s\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\n\u22a2 \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e"}, {"tactic": "obtain \u27e8\u27e8s\u2081, s\u2082\u27e9, h\u27e9 := h e he", "annotated_tactic": ["obtain \u27e8\u27e8s\u2081, s\u2082\u27e9, h\u27e9 := h e he", []], "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : \u2200 s \u2208 \ud835\udcdd 1, \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 s\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\n\u22a2 \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e", "state_after": "case mp.intro.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 s \u2208 \ud835\udcdd 1, \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 s\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\ns\u2081 s\u2082 : Finset \u03b2\nh : \u2200 b \u2265 (s\u2081, s\u2082), (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 e\n\u22a2 \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e"}, {"tactic": "use s\u2081 \u222a s\u2082", "annotated_tactic": ["use s\u2081 \u222a s\u2082", []], "state_before": "case mp.intro.mk\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 s \u2208 \ud835\udcdd 1, \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 s\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\ns\u2081 s\u2082 : Finset \u03b2\nh : \u2200 b \u2265 (s\u2081, s\u2082), (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 e\n\u22a2 \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e", "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\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 s \u2208 \ud835\udcdd 1, \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 s\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\ns\u2081 s\u2082 : Finset \u03b2\nh : \u2200 b \u2265 (s\u2081, s\u2082), (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 e\n\u22a2 \u2200 (t : Finset \u03b2), Disjoint t (s\u2081 \u222a s\u2082) \u2192 \u220f b \u2208 t, f b \u2208 e"}, {"tactic": "intro t ht", "annotated_tactic": ["intro t ht", []], "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\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 s \u2208 \ud835\udcdd 1, \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 s\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\ns\u2081 s\u2082 : Finset \u03b2\nh : \u2200 b \u2265 (s\u2081, s\u2082), (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 e\n\u22a2 \u2200 (t : Finset \u03b2), Disjoint t (s\u2081 \u222a s\u2082) \u2192 \u220f b \u2208 t, f b \u2208 e", "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\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 s \u2208 \ud835\udcdd 1, \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 s\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\ns\u2081 s\u2082 : Finset \u03b2\nh : \u2200 b \u2265 (s\u2081, s\u2082), (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 e\nt : Finset \u03b2\nht : Disjoint t (s\u2081 \u222a s\u2082)\n\u22a2 \u220f b \u2208 t, f b \u2208 e"}, {"tactic": "specialize h (s\u2081 \u222a s\u2082, s\u2081 \u222a s\u2082 \u222a t) \u27e8le_sup_left, le_sup_of_le_left le_sup_right\u27e9", "annotated_tactic": ["specialize h (s\u2081 \u222a s\u2082, s\u2081 \u222a s\u2082 \u222a t) \u27e8<a>le_sup_left</a>, <a>le_sup_of_le_left</a> <a>le_sup_right</a>\u27e9", [{"full_name": "le_sup_left", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [112, 9], "def_end_pos": [112, 20]}, {"full_name": "le_sup_of_le_left", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [127, 9], "def_end_pos": [127, 26]}, {"full_name": "le_sup_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [120, 9], "def_end_pos": [120, 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\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 s \u2208 \ud835\udcdd 1, \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 s\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\ns\u2081 s\u2082 : Finset \u03b2\nh : \u2200 b \u2265 (s\u2081, s\u2082), (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 e\nt : Finset \u03b2\nht : Disjoint t (s\u2081 \u222a s\u2082)\n\u22a2 \u220f b \u2208 t, f b \u2208 e", "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\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 s \u2208 \ud835\udcdd 1, \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 s\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\ns\u2081 s\u2082 t : Finset \u03b2\nht : Disjoint t (s\u2081 \u222a s\u2082)\nh : (\u220f b \u2208 (s\u2081 \u222a s\u2082, s\u2081 \u222a s\u2082 \u222a t).2, f b) / \u220f b \u2208 (s\u2081 \u222a s\u2082, s\u2081 \u222a s\u2082 \u222a t).1, f b \u2208 e\n\u22a2 \u220f b \u2208 t, f b \u2208 e"}, {"tactic": "simpa only [Finset.prod_union ht.symm, mul_div_cancel_left] using h", "annotated_tactic": ["simpa only [<a>Finset.prod_union</a> ht.symm, <a>mul_div_cancel_left</a>] using h", [{"full_name": "Finset.prod_union", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [461, 9], "def_end_pos": [461, 19]}, {"full_name": "mul_div_cancel_left", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1326, 9], "def_end_pos": [1326, 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\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 s \u2208 \ud835\udcdd 1, \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 s\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\ns\u2081 s\u2082 t : Finset \u03b2\nht : Disjoint t (s\u2081 \u222a s\u2082)\nh : (\u220f b \u2208 (s\u2081 \u222a s\u2082, s\u2081 \u222a s\u2082 \u222a t).2, f b) / \u220f b \u2208 (s\u2081 \u222a s\u2082, s\u2081 \u222a s\u2082 \u222a t).1, f b \u2208 e\n\u22a2 \u220f b \u2208 t, f b \u2208 e", "state_after": "no goals"}, {"tactic": "rintro h e he", "annotated_tactic": ["rintro h e he", []], "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\n\u22a2 (\u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e) \u2192\n    \u2200 s \u2208 \ud835\udcdd 1, \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 s", "state_after": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : \u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\n\u22a2 \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 e"}, {"tactic": "rcases exists_nhds_split_inv he with \u27e8d, hd, hde\u27e9", "annotated_tactic": ["rcases <a>exists_nhds_split_inv</a> he with \u27e8d, hd, hde\u27e9", [{"full_name": "exists_nhds_split_inv", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [854, 9], "def_end_pos": [854, 30]}]], "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : \u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\n\u22a2 \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 e", "state_after": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : \u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\nd : Set \u03b1\nhd : d \u2208 \ud835\udcdd 1\nhde : \u2200 v \u2208 d, \u2200 w \u2208 d, v / w \u2208 e\n\u22a2 \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 e"}, {"tactic": "rcases h d hd with \u27e8s, h\u27e9", "annotated_tactic": ["rcases h d hd with \u27e8s, h\u27e9", []], "state_before": "case mpr.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh : \u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\nd : Set \u03b1\nhd : d \u2208 \ud835\udcdd 1\nhde : \u2200 v \u2208 d, \u2200 w \u2208 d, v / w \u2208 e\n\u22a2 \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 e", "state_after": "case mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\nd : Set \u03b1\nhd : d \u2208 \ud835\udcdd 1\nhde : \u2200 v \u2208 d, \u2200 w \u2208 d, v / w \u2208 e\ns : Finset \u03b2\nh : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 d\n\u22a2 \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 e"}, {"tactic": "use (s, s)", "annotated_tactic": ["use (s, s)", []], "state_before": "case mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\nd : Set \u03b1\nhd : d \u2208 \ud835\udcdd 1\nhde : \u2200 v \u2208 d, \u2200 w \u2208 d, v / w \u2208 e\ns : Finset \u03b2\nh : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 d\n\u22a2 \u2203 a, \u2200 b \u2265 a, (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 e", "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\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\nd : Set \u03b1\nhd : d \u2208 \ud835\udcdd 1\nhde : \u2200 v \u2208 d, \u2200 w \u2208 d, v / w \u2208 e\ns : Finset \u03b2\nh : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 d\n\u22a2 \u2200 b \u2265 (s, s), (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 e"}, {"tactic": "rintro \u27e8t\u2081, t\u2082\u27e9 \u27e8ht\u2081, ht\u2082\u27e9", "annotated_tactic": ["rintro \u27e8t\u2081, t\u2082\u27e9 \u27e8ht\u2081, ht\u2082\u27e9", []], "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\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\nd : Set \u03b1\nhd : d \u2208 \ud835\udcdd 1\nhde : \u2200 v \u2208 d, \u2200 w \u2208 d, v / w \u2208 e\ns : Finset \u03b2\nh : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 d\n\u22a2 \u2200 b \u2265 (s, s), (\u220f b \u2208 b.2, f b) / \u220f b \u2208 b.1, f b \u2208 e", "state_after": "case h.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\nd : Set \u03b1\nhd : d \u2208 \ud835\udcdd 1\nhde : \u2200 v \u2208 d, \u2200 w \u2208 d, v / w \u2208 e\ns : Finset \u03b2\nh : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 d\nt\u2081 t\u2082 : Finset \u03b2\nht\u2081 : (s, s).1 \u2264 (t\u2081, t\u2082).1\nht\u2082 : (s, s).2 \u2264 (t\u2081, t\u2082).2\n\u22a2 (\u220f b \u2208 (t\u2081, t\u2082).2, f b) / \u220f b \u2208 (t\u2081, t\u2082).1, f b \u2208 e"}, {"tactic": "have : ((\u220f b \u2208 t\u2082, f b) / \u220f b \u2208 t\u2081, f b) = (\u220f b \u2208 t\u2082 \\ s, f b) / \u220f b \u2208 t\u2081 \\ s, f b := by\n  rw [\u2190 Finset.prod_sdiff ht\u2081, \u2190 Finset.prod_sdiff ht\u2082, mul_div_mul_right_eq_div]", "annotated_tactic": ["have : ((\u220f b \u2208 t\u2082, f b) / \u220f b \u2208 t\u2081, f b) = (\u220f b \u2208 t\u2082 \\ s, f b) / \u220f b \u2208 t\u2081 \\ s, f b := by\n      rw [\u2190 <a>Finset.prod_sdiff</a> ht\u2081, \u2190 <a>Finset.prod_sdiff</a> ht\u2082, <a>mul_div_mul_right_eq_div</a>]", [{"full_name": "Finset.prod_sdiff", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [580, 9], "def_end_pos": [580, 19]}, {"full_name": "Finset.prod_sdiff", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [580, 9], "def_end_pos": [580, 19]}, {"full_name": "mul_div_mul_right_eq_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1020, 9], "def_end_pos": [1020, 33]}]], "state_before": "case h.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\nd : Set \u03b1\nhd : d \u2208 \ud835\udcdd 1\nhde : \u2200 v \u2208 d, \u2200 w \u2208 d, v / w \u2208 e\ns : Finset \u03b2\nh : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 d\nt\u2081 t\u2082 : Finset \u03b2\nht\u2081 : (s, s).1 \u2264 (t\u2081, t\u2082).1\nht\u2082 : (s, s).2 \u2264 (t\u2081, t\u2082).2\n\u22a2 (\u220f b \u2208 (t\u2081, t\u2082).2, f b) / \u220f b \u2208 (t\u2081, t\u2082).1, f b \u2208 e", "state_after": "case h.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\nd : Set \u03b1\nhd : d \u2208 \ud835\udcdd 1\nhde : \u2200 v \u2208 d, \u2200 w \u2208 d, v / w \u2208 e\ns : Finset \u03b2\nh : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 d\nt\u2081 t\u2082 : Finset \u03b2\nht\u2081 : (s, s).1 \u2264 (t\u2081, t\u2082).1\nht\u2082 : (s, s).2 \u2264 (t\u2081, t\u2082).2\nthis : (\u220f b \u2208 t\u2082, f b) / \u220f b \u2208 t\u2081, f b = (\u220f b \u2208 t\u2082 \\ s, f b) / \u220f b \u2208 t\u2081 \\ s, f b\n\u22a2 (\u220f b \u2208 (t\u2081, t\u2082).2, f b) / \u220f b \u2208 (t\u2081, t\u2082).1, f b \u2208 e"}, {"tactic": "simp only [this]", "annotated_tactic": ["simp only [this]", []], "state_before": "case h.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\nd : Set \u03b1\nhd : d \u2208 \ud835\udcdd 1\nhde : \u2200 v \u2208 d, \u2200 w \u2208 d, v / w \u2208 e\ns : Finset \u03b2\nh : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 d\nt\u2081 t\u2082 : Finset \u03b2\nht\u2081 : (s, s).1 \u2264 (t\u2081, t\u2082).1\nht\u2082 : (s, s).2 \u2264 (t\u2081, t\u2082).2\nthis : (\u220f b \u2208 t\u2082, f b) / \u220f b \u2208 t\u2081, f b = (\u220f b \u2208 t\u2082 \\ s, f b) / \u220f b \u2208 t\u2081 \\ s, f b\n\u22a2 (\u220f b \u2208 (t\u2081, t\u2082).2, f b) / \u220f b \u2208 (t\u2081, t\u2082).1, f b \u2208 e", "state_after": "case h.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\nd : Set \u03b1\nhd : d \u2208 \ud835\udcdd 1\nhde : \u2200 v \u2208 d, \u2200 w \u2208 d, v / w \u2208 e\ns : Finset \u03b2\nh : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 d\nt\u2081 t\u2082 : Finset \u03b2\nht\u2081 : (s, s).1 \u2264 (t\u2081, t\u2082).1\nht\u2082 : (s, s).2 \u2264 (t\u2081, t\u2082).2\nthis : (\u220f b \u2208 t\u2082, f b) / \u220f b \u2208 t\u2081, f b = (\u220f b \u2208 t\u2082 \\ s, f b) / \u220f b \u2208 t\u2081 \\ s, f b\n\u22a2 (\u220f b \u2208 t\u2082 \\ s, f b) / \u220f b \u2208 t\u2081 \\ s, f b \u2208 e"}, {"tactic": "exact hde _ (h _ Finset.sdiff_disjoint) _ (h _ Finset.sdiff_disjoint)", "annotated_tactic": ["exact hde _ (h _ <a>Finset.sdiff_disjoint</a>) _ (h _ <a>Finset.sdiff_disjoint</a>)", [{"full_name": "Finset.sdiff_disjoint", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2431, 9], "def_end_pos": [2431, 23]}, {"full_name": "Finset.sdiff_disjoint", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2431, 9], "def_end_pos": [2431, 23]}]], "state_before": "case h.mk.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\nd : Set \u03b1\nhd : d \u2208 \ud835\udcdd 1\nhde : \u2200 v \u2208 d, \u2200 w \u2208 d, v / w \u2208 e\ns : Finset \u03b2\nh : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 d\nt\u2081 t\u2082 : Finset \u03b2\nht\u2081 : (s, s).1 \u2264 (t\u2081, t\u2082).1\nht\u2082 : (s, s).2 \u2264 (t\u2081, t\u2082).2\nthis : (\u220f b \u2208 t\u2082, f b) / \u220f b \u2208 t\u2081, f b = (\u220f b \u2208 t\u2082 \\ s, f b) / \u220f b \u2208 t\u2081 \\ s, f b\n\u22a2 (\u220f b \u2208 t\u2082 \\ s, f b) / \u220f b \u2208 t\u2081 \\ s, f b \u2208 e", "state_after": "no goals"}, {"tactic": "rw [\u2190 Finset.prod_sdiff ht\u2081, \u2190 Finset.prod_sdiff ht\u2082, mul_div_mul_right_eq_div]", "annotated_tactic": ["rw [\u2190 <a>Finset.prod_sdiff</a> ht\u2081, \u2190 <a>Finset.prod_sdiff</a> ht\u2082, <a>mul_div_mul_right_eq_div</a>]", [{"full_name": "Finset.prod_sdiff", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [580, 9], "def_end_pos": [580, 19]}, {"full_name": "Finset.prod_sdiff", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [580, 9], "def_end_pos": [580, 19]}, {"full_name": "mul_div_mul_right_eq_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1020, 9], "def_end_pos": [1020, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b2 : CommGroup \u03b1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : UniformGroup \u03b1\nf g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\nh\u271d : \u2200 e \u2208 \ud835\udcdd 1, \u2203 s, \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 e\ne : Set \u03b1\nhe : e \u2208 \ud835\udcdd 1\nd : Set \u03b1\nhd : d \u2208 \ud835\udcdd 1\nhde : \u2200 v \u2208 d, \u2200 w \u2208 d, v / w \u2208 e\ns : Finset \u03b2\nh : \u2200 (t : Finset \u03b2), Disjoint t s \u2192 \u220f b \u2208 t, f b \u2208 d\nt\u2081 t\u2082 : Finset \u03b2\nht\u2081 : (s, s).1 \u2264 (t\u2081, t\u2082).1\nht\u2082 : (s, s).2 \u2264 (t\u2081, t\u2082).2\n\u22a2 (\u220f b \u2208 t\u2082, f b) / \u220f b \u2208 t\u2081, f b = (\u220f b \u2208 t\u2082 \\ s, f b) / \u220f b \u2208 t\u2081 \\ s, f b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/Reduce.lean", "full_name": "OneOneReducible.disjoin_right", "start": [299, 1], "end": [301, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Normed/Field/UnitBall.lean", "full_name": "coe_mul_unitBall", "start": [48, 1], "end": [50, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/NAry.lean", "full_name": "Filter.map_prod_eq_map\u2082'", "start": [58, 1], "end": [60, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.mem\u21121_smul_of_L1_withDensity", "start": [1051, 1], "end": [1054, 101], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Subobject/Comma.lean", "full_name": "CategoryTheory.CostructuredArrow.projectQuotient_factors", "start": [152, 1], "end": [158, 12], "traced_tactics": [{"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b3 : Category.{v, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} D\nS : C \u2964 D\nT : D\ninst\u271d\u00b9 : HasColimits C\ninst\u271d : PreservesColimits S\nA : CostructuredArrow S T\nP : (CostructuredArrow S T)\u1d52\u1d56\nf : P \u27f6 { unop := A }\nhf : Mono f\n\u22a2 S.map (projectQuotient (Subobject.mk f)).arrow.unop \u226b\n      S.map (Subobject.underlyingIso (MonoOver.mk' f).arrow.unop.left.op).unop.inv \u226b P.unop.hom =\n    A.hom", "state_after": "C : Type u\u2081\ninst\u271d\u00b3 : Category.{v, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} D\nS : C \u2964 D\nT : D\ninst\u271d\u00b9 : HasColimits C\ninst\u271d : PreservesColimits S\nA : CostructuredArrow S T\nP : (CostructuredArrow S T)\u1d52\u1d56\nf : P \u27f6 { unop := A }\nhf : Mono f\n\u22a2 S.map (Subobject.mk f.unop.left.op).arrow.unop \u226b\n      S.map (Subobject.underlyingIso f.unop.left.op).inv.unop \u226b P.unop.hom =\n    A.hom"}, {"tactic": "rw [\u2190 Category.assoc, \u2190 S.map_comp, \u2190 unop_comp]", "annotated_tactic": ["rw [\u2190 <a>Category.assoc</a>, \u2190 S.map_comp, \u2190 <a>unop_comp</a>]", [{"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.unop_comp", "def_path": "Mathlib/CategoryTheory/Opposites.lean", "def_pos": [89, 9], "def_end_pos": [89, 18]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b3 : Category.{v, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} D\nS : C \u2964 D\nT : D\ninst\u271d\u00b9 : HasColimits C\ninst\u271d : PreservesColimits S\nA : CostructuredArrow S T\nP : (CostructuredArrow S T)\u1d52\u1d56\nf : P \u27f6 { unop := A }\nhf : Mono f\n\u22a2 S.map (Subobject.mk f.unop.left.op).arrow.unop \u226b\n      S.map (Subobject.underlyingIso f.unop.left.op).inv.unop \u226b P.unop.hom =\n    A.hom", "state_after": "C : Type u\u2081\ninst\u271d\u00b3 : Category.{v, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} D\nS : C \u2964 D\nT : D\ninst\u271d\u00b9 : HasColimits C\ninst\u271d : PreservesColimits S\nA : CostructuredArrow S T\nP : (CostructuredArrow S T)\u1d52\u1d56\nf : P \u27f6 { unop := A }\nhf : Mono f\n\u22a2 S.map ((Subobject.underlyingIso f.unop.left.op).inv \u226b (Subobject.mk f.unop.left.op).arrow).unop \u226b P.unop.hom = A.hom"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b3 : Category.{v, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b2 : Category.{v, u\u2082} D\nS : C \u2964 D\nT : D\ninst\u271d\u00b9 : HasColimits C\ninst\u271d : PreservesColimits S\nA : CostructuredArrow S T\nP : (CostructuredArrow S T)\u1d52\u1d56\nf : P \u27f6 { unop := A }\nhf : Mono f\n\u22a2 S.map ((Subobject.underlyingIso f.unop.left.op).inv \u226b (Subobject.mk f.unop.left.op).arrow).unop \u226b P.unop.hom = A.hom", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/OrderClosed.lean", "full_name": "Icc_mem_nhds", "start": [771, 1], "end": [772, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Types.lean", "full_name": "CategoryTheory.ofTypeFunctor_map", "start": [311, 1], "end": [313, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Subsingleton.lean", "full_name": "Set.not_subsingleton_iff", "start": [300, 1], "end": [301, 70], "traced_tactics": [{"tactic": "simp_rw [Set.Subsingleton, Set.Nontrivial, not_forall, exists_prop]", "annotated_tactic": ["simp_rw [<a>Set.Subsingleton</a>, <a>Set.Nontrivial</a>, <a>not_forall</a>, <a>exists_prop</a>]", [{"full_name": "Set.Subsingleton", "def_path": "Mathlib/Data/Set/Subsingleton.lean", "def_pos": [31, 15], "def_end_pos": [31, 27]}, {"full_name": "Set.Nontrivial", "def_path": "Mathlib/Data/Set/Subsingleton.lean", "def_pos": [136, 15], "def_end_pos": [136, 25]}, {"full_name": "Classical.not_forall", "def_path": ".lake/packages/lean4/src/lean/Init/Classical.lean", "def_pos": [137, 21], "def_end_pos": [137, 31]}, {"full_name": "exists_prop", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [307, 17], "def_end_pos": [307, 28]}]], "state_before": "\u03b1 : Type u\na : \u03b1\ns t : Set \u03b1\n\u22a2 \u00acs.Subsingleton \u2194 s.Nontrivial", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Ring/Basic.lean", "full_name": "pow_four_le_pow_two_of_pow_two_le", "start": [392, 1], "end": [393, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/Exponential.lean", "full_name": "Commute.exp_right", "start": [172, 1], "end": [175, 67], "traced_tactics": [{"tactic": "rw [exp_eq_tsum]", "annotated_tactic": ["rw [<a>exp_eq_tsum</a>]", [{"full_name": "NormedSpace.exp_eq_tsum", "def_path": "Mathlib/Analysis/NormedSpace/Exponential.lean", "def_pos": [132, 9], "def_end_pos": [132, 20]}]], "state_before": "\ud835\udd42 : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u2075 : Field \ud835\udd42\ninst\u271d\u2074 : Ring \ud835\udd38\ninst\u271d\u00b3 : Algebra \ud835\udd42 \ud835\udd38\ninst\u271d\u00b2 : TopologicalSpace \ud835\udd38\ninst\u271d\u00b9 : TopologicalRing \ud835\udd38\ninst\u271d : T2Space \ud835\udd38\nx y : \ud835\udd38\nh : Commute x y\n\u22a2 Commute x (exp \ud835\udd42 y)", "state_after": "\ud835\udd42 : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u2075 : Field \ud835\udd42\ninst\u271d\u2074 : Ring \ud835\udd38\ninst\u271d\u00b3 : Algebra \ud835\udd42 \ud835\udd38\ninst\u271d\u00b2 : TopologicalSpace \ud835\udd38\ninst\u271d\u00b9 : TopologicalRing \ud835\udd38\ninst\u271d : T2Space \ud835\udd38\nx y : \ud835\udd38\nh : Commute x y\n\u22a2 Commute x ((fun x => \u2211' (n : \u2115), (\u2191n !)\u207b\u00b9 \u2022 x ^ n) y)"}, {"tactic": "exact Commute.tsum_right x fun n => (h.pow_right n).smul_right _", "annotated_tactic": ["exact <a>Commute.tsum_right</a> x fun n => (h.pow_right n).<a>smul_right</a> _", [{"full_name": "Commute.tsum_right", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Ring.lean", "def_pos": [62, 9], "def_end_pos": [62, 27]}, {"full_name": "Commute.smul_right", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [428, 7], "def_end_pos": [428, 25]}]], "state_before": "\ud835\udd42 : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u2075 : Field \ud835\udd42\ninst\u271d\u2074 : Ring \ud835\udd38\ninst\u271d\u00b3 : Algebra \ud835\udd42 \ud835\udd38\ninst\u271d\u00b2 : TopologicalSpace \ud835\udd38\ninst\u271d\u00b9 : TopologicalRing \ud835\udd38\ninst\u271d : T2Space \ud835\udd38\nx y : \ud835\udd38\nh : Commute x y\n\u22a2 Commute x ((fun x => \u2211' (n : \u2115), (\u2191n !)\u207b\u00b9 \u2022 x ^ n) y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "full_name": "OrthogonalFamily.inner_right_dfinsupp", "start": [2022, 1], "end": [2034, 51], "traced_tactics": [{"tactic": "simp only [DFinsupp.sum, Submodule.coe_inner, Finset.sum_ite_eq, ite_eq_left_iff,\n  DFinsupp.mem_support_toFun]", "annotated_tactic": ["simp only [<a>DFinsupp.sum</a>, <a>Submodule.coe_inner</a>, <a>Finset.sum_ite_eq</a>, <a>ite_eq_left_iff</a>,\n        <a>DFinsupp.mem_support_toFun</a>]", [{"full_name": "DFinsupp.sum", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [1693, 3], "def_end_pos": [1693, 14]}, {"full_name": "Submodule.coe_inner", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [1964, 9], "def_end_pos": [1964, 28]}, {"full_name": "Finset.sum_ite_eq", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [1319, 3], "def_end_pos": [1319, 14]}, {"full_name": "ite_eq_left_iff", "def_path": ".lake/packages/lean4/src/lean/Init/ByCases.lean", "def_pos": [48, 17], "def_end_pos": [48, 32]}, {"full_name": "DFinsupp.mem_support_toFun", "def_path": "Mathlib/Data/DFinsupp/Basic.lean", "def_pos": [1109, 9], "def_end_pos": [1109, 26]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : _root_.RCLike \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : DecidableEq \u03b9\nl : \u2a01 (i : \u03b9), G i\ni : \u03b9\nv : G i\n\u22a2 (DFinsupp.sum l fun j w => if i = j then \u27ea(V i) v, (V j) w\u27eb_\ud835\udd5c else 0) = \u27eav, l i\u27eb_\ud835\udd5c", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : _root_.RCLike \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : DecidableEq \u03b9\nl : \u2a01 (i : \u03b9), G i\ni : \u03b9\nv : G i\n\u22a2 (if l i \u2260 0 then \u27ea(V i) v, (V i) (l i)\u27eb_\ud835\udd5c else 0) = \u27eav, l i\u27eb_\ud835\udd5c"}, {"tactic": "split_ifs with h", "annotated_tactic": ["split_ifs with h", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : _root_.RCLike \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : DecidableEq \u03b9\nl : \u2a01 (i : \u03b9), G i\ni : \u03b9\nv : G i\n\u22a2 (if l i \u2260 0 then \u27ea(V i) v, (V i) (l i)\u27eb_\ud835\udd5c else 0) = \u27eav, l i\u27eb_\ud835\udd5c", "state_after": "case pos\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : _root_.RCLike \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : DecidableEq \u03b9\nl : \u2a01 (i : \u03b9), G i\ni : \u03b9\nv : G i\nh : l i \u2260 0\n\u22a2 \u27ea(V i) v, (V i) (l i)\u27eb_\ud835\udd5c = \u27eav, l i\u27eb_\ud835\udd5c\n\ncase neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : _root_.RCLike \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : DecidableEq \u03b9\nl : \u2a01 (i : \u03b9), G i\ni : \u03b9\nv : G i\nh : \u00acl i \u2260 0\n\u22a2 0 = \u27eav, l i\u27eb_\ud835\udd5c"}, {"tactic": "simp only [LinearIsometry.inner_map_map]", "annotated_tactic": ["simp only [<a>LinearIsometry.inner_map_map</a>]", [{"full_name": "LinearIsometry.inner_map_map", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [1256, 9], "def_end_pos": [1256, 37]}]], "state_before": "case pos\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : _root_.RCLike \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : DecidableEq \u03b9\nl : \u2a01 (i : \u03b9), G i\ni : \u03b9\nv : G i\nh : l i \u2260 0\n\u22a2 \u27ea(V i) v, (V i) (l i)\u27eb_\ud835\udd5c = \u27eav, l i\u27eb_\ud835\udd5c", "state_after": "no goals"}, {"tactic": "simp only [of_not_not h, inner_zero_right]", "annotated_tactic": ["simp only [<a>of_not_not</a> h, <a>inner_zero_right</a>]", [{"full_name": "of_not_not", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [255, 9], "def_end_pos": [255, 19]}, {"full_name": "inner_zero_right", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 25]}]], "state_before": "case neg\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2077 : _root_.RCLike \ud835\udd5c\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : InnerProductSpace \u211d F\n\u03b9 : Type u_4\nG : \u03b9 \u2192 Type u_5\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\ndec_V : (i : \u03b9) \u2192 (x : G i) \u2192 Decidable (x \u2260 0)\ninst\u271d : DecidableEq \u03b9\nl : \u2a01 (i : \u03b9), G i\ni : \u03b9\nv : G i\nh : \u00acl i \u2260 0\n\u22a2 0 = \u27eav, l i\u27eb_\ud835\udd5c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/Single.lean", "full_name": "HomologicalComplex.isZero_single_comp_eval", "start": [106, 1], "end": [107, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Bicategory/LocallyDiscrete.lean", "full_name": "CategoryTheory.OplaxFunctor.map\u2082_eqToHom", "start": [148, 1], "end": [151, 58], "traced_tactics": [{"tactic": "subst h", "annotated_tactic": ["subst h", []], "state_before": "B : Type u\u2081\ninst\u271d\u00b9 : Bicategory B\nC : Type u\u2082\ninst\u271d : Bicategory C\nF : OplaxFunctor B C\na b : B\nf g : a \u27f6 b\nh : f = g\n\u22a2 F.map\u2082 (eqToHom h) = eqToHom \u22ef", "state_after": "B : Type u\u2081\ninst\u271d\u00b9 : Bicategory B\nC : Type u\u2082\ninst\u271d : Bicategory C\nF : OplaxFunctor B C\na b : B\nf : a \u27f6 b\n\u22a2 F.map\u2082 (eqToHom \u22ef) = eqToHom \u22ef"}, {"tactic": "simp only [eqToHom_refl, OplaxFunctor.map\u2082_id]", "annotated_tactic": ["simp only [<a>eqToHom_refl</a>, <a>OplaxFunctor.map\u2082_id</a>]", [{"full_name": "CategoryTheory.eqToHom_refl", "def_path": "Mathlib/CategoryTheory/EqToHom.lean", "def_pos": [47, 9], "def_end_pos": [47, 21]}, {"full_name": "CategoryTheory.OplaxFunctor.map\u2082_id", "def_path": "Mathlib/CategoryTheory/Bicategory/Functor/Oplax.lean", "def_pos": [141, 3], "def_end_pos": [141, 10]}]], "state_before": "B : Type u\u2081\ninst\u271d\u00b9 : Bicategory B\nC : Type u\u2082\ninst\u271d : Bicategory C\nF : OplaxFunctor B C\na b : B\nf : a \u27f6 b\n\u22a2 F.map\u2082 (eqToHom \u22ef) = eqToHom \u22ef", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "Finset.aestronglyMeasurable_prod", "start": [1489, 1], "end": [1492, 73], "traced_tactics": [{"tactic": "simpa only [\u2190 Finset.prod_apply] using s.aestronglyMeasurable_prod' hf", "annotated_tactic": ["simpa only [\u2190 <a>Finset.prod_apply</a>] using s.aestronglyMeasurable_prod' hf", [{"full_name": "Finset.prod_apply", "def_path": "Mathlib/Algebra/BigOperators/Pi.lean", "def_pos": [39, 9], "def_end_pos": [39, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9\u271d : Type u_4\ninst\u271d\u2075 : Countable \u03b9\u271d\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : TopologicalSpace \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\nf\u271d g : \u03b1 \u2192 \u03b2\nM : Type u_5\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : TopologicalSpace M\ninst\u271d : ContinuousMul M\n\u03b9 : Type u_6\nf : \u03b9 \u2192 \u03b1 \u2192 M\ns : Finset \u03b9\nhf : \u2200 i \u2208 s, AEStronglyMeasurable (f i) \u03bc\n\u22a2 AEStronglyMeasurable (fun a => \u220f i \u2208 s, f i a) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/AddConstMap/Basic.lean", "full_name": "AddConstMap.ext", "start": [324, 18], "end": [325, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Int.lean", "full_name": "Int.ediv_two_mul_two_add_one_of_odd", "start": [245, 1], "end": [248, 22], "traced_tactics": [{"tactic": "rintro \u27e8c, rfl\u27e9", "annotated_tactic": ["rintro \u27e8c, rfl\u27e9", []], "state_before": "m n : \u2124\n\u22a2 Odd n \u2192 n / 2 * 2 + 1 = n", "state_after": "case intro\nm c : \u2124\n\u22a2 (2 * c + 1) / 2 * 2 + 1 = 2 * c + 1"}, {"tactic": "convert Int.ediv_add_emod' (2 * c + 1) 2", "annotated_tactic": ["convert <a>Int.ediv_add_emod'</a> (2 * c + 1) 2", [{"full_name": "Int.ediv_add_emod'", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean", "def_pos": [221, 9], "def_end_pos": [221, 23]}]], "state_before": "case intro\nm c : \u2124\n\u22a2 (2 * c + 1) / 2 * 2 + 1 = 2 * c + 1", "state_after": "case h.e'_2.h.e'_6\nm c : \u2124\n\u22a2 1 = (2 * c + 1) % 2"}, {"tactic": "simp [Int.add_emod]", "annotated_tactic": ["simp [<a>Int.add_emod</a>]", [{"full_name": "Int.add_emod", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean", "def_pos": [476, 9], "def_end_pos": [476, 17]}]], "state_before": "case h.e'_2.h.e'_6\nm c : \u2124\n\u22a2 1 = (2 * c + 1) % 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Sum/Interval.lean", "full_name": "Sum.Ioo_inr_inr", "start": [315, 1], "end": [316, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/Quiver/SingleObj.lean", "full_name": "Quiver.SingleObj.toPrefunctor_comp", "start": [104, 1], "end": [106, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Field/Defs.lean", "full_name": "Rat.cast_mk'", "start": [215, 1], "end": [215, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/ContDiff/Basic.lean", "full_name": "hasFTaylorSeriesUpToOn_pi", "start": [1180, 1], "end": [1197, 52], "traced_tactics": [{"tactic": "set pr := @ContinuousLinearMap.proj \ud835\udd5c _ \u03b9 F' _ _ _", "annotated_tactic": ["set pr := @<a>ContinuousLinearMap.proj</a> \ud835\udd5c _ \u03b9 F' _ _ _", [{"full_name": "ContinuousLinearMap.proj", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1329, 5], "def_end_pos": [1329, 9]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\n\u22a2 HasFTaylorSeriesUpToOn n (fun x i => \u03c6 i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s \u2194\n    \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\n\u22a2 HasFTaylorSeriesUpToOn n (fun x i => \u03c6 i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s \u2194\n    \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s"}, {"tactic": "letI : \u2200 (m : \u2115) (i : \u03b9), NormedSpace \ud835\udd5c (E[\u00d7m]\u2192L[\ud835\udd5c] F' i) := fun m i => inferInstance", "annotated_tactic": ["letI : \u2200 (m : \u2115) (i : \u03b9), <a>NormedSpace</a> \ud835\udd5c (E[\u00d7m]\u2192L[\ud835\udd5c] F' i) := fun m i => <a>inferInstance</a>", [{"full_name": "NormedSpace", "def_path": "Mathlib/Analysis/NormedSpace/Basic.lean", "def_pos": [43, 7], "def_end_pos": [43, 18]}, {"full_name": "inferInstance", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [99, 8], "def_end_pos": [99, 21]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\n\u22a2 HasFTaylorSeriesUpToOn n (fun x i => \u03c6 i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s \u2194\n    \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\n\u22a2 HasFTaylorSeriesUpToOn n (fun x i => \u03c6 i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s \u2194\n    \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s"}, {"tactic": "set L : \u2200 m : \u2115, (\u2200 i, E[\u00d7m]\u2192L[\ud835\udd5c] F' i) \u2243\u2097\u1d62[\ud835\udd5c] E[\u00d7m]\u2192L[\ud835\udd5c] \u2200 i, F' i := fun m =>\n  ContinuousMultilinearMap.pi\u2097\u1d62 _ _", "annotated_tactic": ["set L : \u2200 m : \u2115, (\u2200 i, E[\u00d7m]\u2192L[\ud835\udd5c] F' i) \u2243\u2097\u1d62[\ud835\udd5c] E[\u00d7m]\u2192L[\ud835\udd5c] \u2200 i, F' i := fun m =>\n    <a>ContinuousMultilinearMap.pi\u2097\u1d62</a> _ _", [{"full_name": "ContinuousMultilinearMap.pi\u2097\u1d62", "def_path": "Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean", "def_pos": [657, 5], "def_end_pos": [657, 9]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\n\u22a2 HasFTaylorSeriesUpToOn n (fun x i => \u03c6 i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s \u2194\n    \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : \u2115) \u2192\n  ((i : \u03b9) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) ((i : \u03b9) \u2192 F' i) :=\n  fun m => ContinuousMultilinearMap.pi\u2097\u1d62 \ud835\udd5c fun i => E\n\u22a2 HasFTaylorSeriesUpToOn n (fun x i => \u03c6 i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s \u2194\n    \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s"}, {"tactic": "refine \u27e8fun h i => ?_, fun h => \u27e8fun x hx => ?_, ?_, ?_\u27e9\u27e9", "annotated_tactic": ["refine \u27e8fun h i => ?_, fun h => \u27e8fun x hx => ?_, ?_, ?_\u27e9\u27e9", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : \u2115) \u2192\n  ((i : \u03b9) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) ((i : \u03b9) \u2192 F' i) :=\n  fun m => ContinuousMultilinearMap.pi\u2097\u1d62 \ud835\udd5c fun i => E\n\u22a2 HasFTaylorSeriesUpToOn n (fun x i => \u03c6 i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s \u2194\n    \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s", "state_after": "case refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : \u2115) \u2192\n  ((i : \u03b9) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) ((i : \u03b9) \u2192 F' i) :=\n  fun m => ContinuousMultilinearMap.pi\u2097\u1d62 \ud835\udd5c fun i => E\nh : HasFTaylorSeriesUpToOn n (fun x i => \u03c6 i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s\ni : \u03b9\n\u22a2 HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s\n\ncase refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : \u2115) \u2192\n  ((i : \u03b9) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) ((i : \u03b9) \u2192 F' i) :=\n  fun m => ContinuousMultilinearMap.pi\u2097\u1d62 \ud835\udd5c fun i => E\nh : \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s\nx : E\nhx : x \u2208 s\n\u22a2 (ContinuousMultilinearMap.pi fun i => p' i x 0).uncurry0 = fun i => \u03c6 i x\n\ncase refine_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : \u2115) \u2192\n  ((i : \u03b9) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) ((i : \u03b9) \u2192 F' i) :=\n  fun m => ContinuousMultilinearMap.pi\u2097\u1d62 \ud835\udd5c fun i => E\nh : \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s\n\u22a2 \u2200 (m : \u2115),\n    \u2191m < n \u2192\n      \u2200 x \u2208 s,\n        HasFDerivWithinAt (fun x => ContinuousMultilinearMap.pi fun i => p' i x m)\n          (ContinuousMultilinearMap.pi fun i => p' i x m.succ).curryLeft s x\n\ncase refine_4\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : \u2115) \u2192\n  ((i : \u03b9) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) ((i : \u03b9) \u2192 F' i) :=\n  fun m => ContinuousMultilinearMap.pi\u2097\u1d62 \ud835\udd5c fun i => E\nh : \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => ContinuousMultilinearMap.pi fun i => p' i x m) s"}, {"tactic": "convert h.continuousLinearMap_comp (pr i)", "annotated_tactic": ["convert h.continuousLinearMap_comp (pr i)", []], "state_before": "case refine_1\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : \u2115) \u2192\n  ((i : \u03b9) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) ((i : \u03b9) \u2192 F' i) :=\n  fun m => ContinuousMultilinearMap.pi\u2097\u1d62 \ud835\udd5c fun i => E\nh : HasFTaylorSeriesUpToOn n (fun x i => \u03c6 i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s\ni : \u03b9\n\u22a2 HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s", "state_after": "no goals"}, {"tactic": "ext1 i", "annotated_tactic": ["ext1 i", []], "state_before": "case refine_2\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : \u2115) \u2192\n  ((i : \u03b9) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) ((i : \u03b9) \u2192 F' i) :=\n  fun m => ContinuousMultilinearMap.pi\u2097\u1d62 \ud835\udd5c fun i => E\nh : \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s\nx : E\nhx : x \u2208 s\n\u22a2 (ContinuousMultilinearMap.pi fun i => p' i x 0).uncurry0 = fun i => \u03c6 i x", "state_after": "case refine_2.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : \u2115) \u2192\n  ((i : \u03b9) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) ((i : \u03b9) \u2192 F' i) :=\n  fun m => ContinuousMultilinearMap.pi\u2097\u1d62 \ud835\udd5c fun i => E\nh : \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s\nx : E\nhx : x \u2208 s\ni : \u03b9\n\u22a2 (ContinuousMultilinearMap.pi fun i => p' i x 0).uncurry0 i = \u03c6 i x"}, {"tactic": "exact (h i).zero_eq x hx", "annotated_tactic": ["exact (h i).<a>zero_eq</a> x hx", [{"full_name": "HasFTaylorSeriesUpToOn.zero_eq", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "def_pos": [190, 3], "def_end_pos": [190, 10]}]], "state_before": "case refine_2.h\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : \u2115) \u2192\n  ((i : \u03b9) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) ((i : \u03b9) \u2192 F' i) :=\n  fun m => ContinuousMultilinearMap.pi\u2097\u1d62 \ud835\udd5c fun i => E\nh : \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s\nx : E\nhx : x \u2208 s\ni : \u03b9\n\u22a2 (ContinuousMultilinearMap.pi fun i => p' i x 0).uncurry0 i = \u03c6 i x", "state_after": "no goals"}, {"tactic": "intro m hm x hx", "annotated_tactic": ["intro m hm x hx", []], "state_before": "case refine_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : \u2115) \u2192\n  ((i : \u03b9) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) ((i : \u03b9) \u2192 F' i) :=\n  fun m => ContinuousMultilinearMap.pi\u2097\u1d62 \ud835\udd5c fun i => E\nh : \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s\n\u22a2 \u2200 (m : \u2115),\n    \u2191m < n \u2192\n      \u2200 x \u2208 s,\n        HasFDerivWithinAt (fun x => ContinuousMultilinearMap.pi fun i => p' i x m)\n          (ContinuousMultilinearMap.pi fun i => p' i x m.succ).curryLeft s x", "state_after": "case refine_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : \u2115) \u2192\n  ((i : \u03b9) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) ((i : \u03b9) \u2192 F' i) :=\n  fun m => ContinuousMultilinearMap.pi\u2097\u1d62 \ud835\udd5c fun i => E\nh : \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\n\u22a2 HasFDerivWithinAt (fun x => ContinuousMultilinearMap.pi fun i => p' i x m)\n    (ContinuousMultilinearMap.pi fun i => p' i x m.succ).curryLeft s x"}, {"tactic": "have := hasFDerivWithinAt_pi.2 fun i => (h i).fderivWithin m hm x hx", "annotated_tactic": ["have := <a>hasFDerivWithinAt_pi</a>.2 fun i => (h i).<a>fderivWithin</a> m hm x hx", [{"full_name": "hasFDerivWithinAt_pi", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Prod.lean", "def_pos": [482, 9], "def_end_pos": [482, 29]}, {"full_name": "HasFTaylorSeriesUpToOn.fderivWithin", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "def_pos": [191, 13], "def_end_pos": [191, 25]}]], "state_before": "case refine_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : \u2115) \u2192\n  ((i : \u03b9) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) ((i : \u03b9) \u2192 F' i) :=\n  fun m => ContinuousMultilinearMap.pi\u2097\u1d62 \ud835\udd5c fun i => E\nh : \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\n\u22a2 HasFDerivWithinAt (fun x => ContinuousMultilinearMap.pi fun i => p' i x m)\n    (ContinuousMultilinearMap.pi fun i => p' i x m.succ).curryLeft s x", "state_after": "case refine_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis\u271d : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : \u2115) \u2192\n  ((i : \u03b9) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) ((i : \u03b9) \u2192 F' i) :=\n  fun m => ContinuousMultilinearMap.pi\u2097\u1d62 \ud835\udd5c fun i => E\nh : \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\nthis : HasFDerivWithinAt (fun x i => p' i x m) (ContinuousLinearMap.pi fun i => (p' i x m.succ).curryLeft) s x\n\u22a2 HasFDerivWithinAt (fun x => ContinuousMultilinearMap.pi fun i => p' i x m)\n    (ContinuousMultilinearMap.pi fun i => p' i x m.succ).curryLeft s x"}, {"tactic": "convert (L m).hasFDerivAt.comp_hasFDerivWithinAt x this", "annotated_tactic": ["convert (L m).hasFDerivAt.comp_hasFDerivWithinAt x this", []], "state_before": "case refine_3\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis\u271d : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : \u2115) \u2192\n  ((i : \u03b9) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) ((i : \u03b9) \u2192 F' i) :=\n  fun m => ContinuousMultilinearMap.pi\u2097\u1d62 \ud835\udd5c fun i => E\nh : \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s\nm : \u2115\nhm : \u2191m < n\nx : E\nhx : x \u2208 s\nthis : HasFDerivWithinAt (fun x i => p' i x m) (ContinuousLinearMap.pi fun i => (p' i x m.succ).curryLeft) s x\n\u22a2 HasFDerivWithinAt (fun x => ContinuousMultilinearMap.pi fun i => p' i x m)\n    (ContinuousMultilinearMap.pi fun i => p' i x m.succ).curryLeft s x", "state_after": "no goals"}, {"tactic": "intro m hm", "annotated_tactic": ["intro m hm", []], "state_before": "case refine_4\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : \u2115) \u2192\n  ((i : \u03b9) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) ((i : \u03b9) \u2192 F' i) :=\n  fun m => ContinuousMultilinearMap.pi\u2097\u1d62 \ud835\udd5c fun i => E\nh : \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => ContinuousMultilinearMap.pi fun i => p' i x m) s", "state_after": "case refine_4\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : \u2115) \u2192\n  ((i : \u03b9) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) ((i : \u03b9) \u2192 F' i) :=\n  fun m => ContinuousMultilinearMap.pi\u2097\u1d62 \ud835\udd5c fun i => E\nh : \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 ContinuousOn (fun x => ContinuousMultilinearMap.pi fun i => p' i x m) s"}, {"tactic": "have := continuousOn_pi.2 fun i => (h i).cont m hm", "annotated_tactic": ["have := <a>continuousOn_pi</a>.2 fun i => (h i).<a>cont</a> m hm", [{"full_name": "continuousOn_pi", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [625, 9], "def_end_pos": [625, 24]}, {"full_name": "HasFTaylorSeriesUpToOn.cont", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "def_pos": [193, 3], "def_end_pos": [193, 7]}]], "state_before": "case refine_4\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : \u2115) \u2192\n  ((i : \u03b9) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) ((i : \u03b9) \u2192 F' i) :=\n  fun m => ContinuousMultilinearMap.pi\u2097\u1d62 \ud835\udd5c fun i => E\nh : \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 ContinuousOn (fun x => ContinuousMultilinearMap.pi fun i => p' i x m) s", "state_after": "case refine_4\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis\u271d : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : \u2115) \u2192\n  ((i : \u03b9) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) ((i : \u03b9) \u2192 F' i) :=\n  fun m => ContinuousMultilinearMap.pi\u2097\u1d62 \ud835\udd5c fun i => E\nh : \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s\nm : \u2115\nhm : \u2191m \u2264 n\nthis : ContinuousOn (fun y i => p' i y m) s\n\u22a2 ContinuousOn (fun x => ContinuousMultilinearMap.pi fun i => p' i x m) s"}, {"tactic": "convert (L m).continuous.comp_continuousOn this", "annotated_tactic": ["convert (L m).continuous.comp_continuousOn this", []], "state_before": "case refine_4\n\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup D\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2075 : NormedAddCommGroup X\ninst\u271d\u2074 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u03b9 : Type u_3\n\u03b9' : Type u_4\ninst\u271d\u00b3 : Fintype \u03b9\ninst\u271d\u00b2 : Fintype \u03b9'\nF' : \u03b9 \u2192 Type u_5\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (F' i)\ninst\u271d : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (F' i)\n\u03c6 : (i : \u03b9) \u2192 E \u2192 F' i\np' : (i : \u03b9) \u2192 E \u2192 FormalMultilinearSeries \ud835\udd5c E (F' i)\n\u03a6 : E \u2192 (i : \u03b9) \u2192 F' i\nP' : E \u2192 FormalMultilinearSeries \ud835\udd5c E ((i : \u03b9) \u2192 F' i)\npr : (i : \u03b9) \u2192 ((i : \u03b9) \u2192 F' i) \u2192L[\ud835\udd5c] F' i := ContinuousLinearMap.proj\nthis\u271d : (m : \u2115) \u2192 (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : \u2115) \u2192\n  ((i : \u03b9) \u2192 ContinuousMultilinearMap \ud835\udd5c (fun i => E) (F' i)) \u2243\u2097\u1d62[\ud835\udd5c]\n    ContinuousMultilinearMap \ud835\udd5c (fun i => E) ((i : \u03b9) \u2192 F' i) :=\n  fun m => ContinuousMultilinearMap.pi\u2097\u1d62 \ud835\udd5c fun i => E\nh : \u2200 (i : \u03b9), HasFTaylorSeriesUpToOn n (\u03c6 i) (p' i) s\nm : \u2115\nhm : \u2191m \u2264 n\nthis : ContinuousOn (fun y i => p' i y m) s\n\u22a2 ContinuousOn (fun x => ContinuousMultilinearMap.pi fun i => p' i x m) s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.tendsto_atTop_mono", "start": [425, 1], "end": [427, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/GDelta.lean", "full_name": "IsG\u03b4.iInter", "start": [109, 11], "end": [114, 36], "traced_tactics": [{"tactic": "choose T hTo hTc hTs using hs", "annotated_tactic": ["choose T hTo hTc hTs using hs", []], "state_before": "X : Type u_1\nY : Type u_2\n\u03b9 : Type u_3\n\u03b9' : Sort u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Countable \u03b9'\ns : \u03b9' \u2192 Set X\nhs : \u2200 (i : \u03b9'), IsG\u03b4 (s i)\n\u22a2 IsG\u03b4 (\u22c2 i, s i)", "state_after": "X : Type u_1\nY : Type u_2\n\u03b9 : Type u_3\n\u03b9' : Sort u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Countable \u03b9'\ns : \u03b9' \u2192 Set X\nT : \u03b9' \u2192 Set (Set X)\nhTo : \u2200 (i : \u03b9'), \u2200 t \u2208 T i, IsOpen t\nhTc : \u2200 (i : \u03b9'), (T i).Countable\nhTs : \u2200 (i : \u03b9'), s i = \u22c2\u2080 T i\n\u22a2 IsG\u03b4 (\u22c2 i, s i)"}, {"tactic": "obtain rfl : s = fun i => \u22c2\u2080 T i := funext hTs", "annotated_tactic": ["obtain rfl : s = fun i => \u22c2\u2080 T i := <a>funext</a> hTs", [{"full_name": "funext", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1817, 9], "def_end_pos": [1817, 15]}]], "state_before": "X : Type u_1\nY : Type u_2\n\u03b9 : Type u_3\n\u03b9' : Sort u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Countable \u03b9'\ns : \u03b9' \u2192 Set X\nT : \u03b9' \u2192 Set (Set X)\nhTo : \u2200 (i : \u03b9'), \u2200 t \u2208 T i, IsOpen t\nhTc : \u2200 (i : \u03b9'), (T i).Countable\nhTs : \u2200 (i : \u03b9'), s i = \u22c2\u2080 T i\n\u22a2 IsG\u03b4 (\u22c2 i, s i)", "state_after": "X : Type u_1\nY : Type u_2\n\u03b9 : Type u_3\n\u03b9' : Sort u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Countable \u03b9'\nT : \u03b9' \u2192 Set (Set X)\nhTo : \u2200 (i : \u03b9'), \u2200 t \u2208 T i, IsOpen t\nhTc : \u2200 (i : \u03b9'), (T i).Countable\nhTs : \u2200 (i : \u03b9'), (fun i => \u22c2\u2080 T i) i = \u22c2\u2080 T i\n\u22a2 IsG\u03b4 (\u22c2 i, (fun i => \u22c2\u2080 T i) i)"}, {"tactic": "refine \u27e8\u22c3 i, T i, ?_, countable_iUnion hTc, (sInter_iUnion _).symm\u27e9", "annotated_tactic": ["refine \u27e8\u22c3 i, T i, ?_, <a>countable_iUnion</a> hTc, (<a>sInter_iUnion</a> _).<a>symm</a>\u27e9", [{"full_name": "Set.countable_iUnion", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [222, 9], "def_end_pos": [222, 25]}, {"full_name": "Set.sInter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1360, 9], "def_end_pos": [1360, 22]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "X : Type u_1\nY : Type u_2\n\u03b9 : Type u_3\n\u03b9' : Sort u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Countable \u03b9'\nT : \u03b9' \u2192 Set (Set X)\nhTo : \u2200 (i : \u03b9'), \u2200 t \u2208 T i, IsOpen t\nhTc : \u2200 (i : \u03b9'), (T i).Countable\nhTs : \u2200 (i : \u03b9'), (fun i => \u22c2\u2080 T i) i = \u22c2\u2080 T i\n\u22a2 IsG\u03b4 (\u22c2 i, (fun i => \u22c2\u2080 T i) i)", "state_after": "X : Type u_1\nY : Type u_2\n\u03b9 : Type u_3\n\u03b9' : Sort u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Countable \u03b9'\nT : \u03b9' \u2192 Set (Set X)\nhTo : \u2200 (i : \u03b9'), \u2200 t \u2208 T i, IsOpen t\nhTc : \u2200 (i : \u03b9'), (T i).Countable\nhTs : \u2200 (i : \u03b9'), (fun i => \u22c2\u2080 T i) i = \u22c2\u2080 T i\n\u22a2 \u2200 t \u2208 \u22c3 i, T i, IsOpen t"}, {"tactic": "simpa [@forall_swap \u03b9'] using hTo", "annotated_tactic": ["simpa [@<a>forall_swap</a> \u03b9'] using hTo", [{"full_name": "forall_swap", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [656, 9], "def_end_pos": [656, 20]}]], "state_before": "X : Type u_1\nY : Type u_2\n\u03b9 : Type u_3\n\u03b9' : Sort u_4\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : Countable \u03b9'\nT : \u03b9' \u2192 Set (Set X)\nhTo : \u2200 (i : \u03b9'), \u2200 t \u2208 T i, IsOpen t\nhTc : \u2200 (i : \u03b9'), (T i).Countable\nhTs : \u2200 (i : \u03b9'), (fun i => \u22c2\u2080 T i) i = \u22c2\u2080 T i\n\u22a2 \u2200 t \u2208 \u22c3 i, T i, IsOpen t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "full_name": "UniqueFactorizationMonoid.pow_eq_pow_iff", "start": [983, 1], "end": [984, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Image.lean", "full_name": "Finset.biUnion_singleton", "start": [671, 1], "end": [672, 77], "traced_tactics": [{"tactic": "simp only [mem_biUnion, mem_image, mem_singleton, eq_comm]", "annotated_tactic": ["simp only [<a>mem_biUnion</a>, <a>mem_image</a>, <a>mem_singleton</a>, <a>eq_comm</a>]", [{"full_name": "Finset.mem_biUnion", "def_path": "Mathlib/Data/Finset/Union.lean", "def_pos": [129, 15], "def_end_pos": [129, 26]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [357, 9], "def_end_pos": [357, 18]}, {"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [680, 9], "def_end_pos": [680, 22]}, {"full_name": "eq_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.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\ninst\u271d : DecidableEq \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\na : \u03b1\nb c : \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b2\n\u22a2 (x \u2208 s.biUnion fun a => {f a}) \u2194 x \u2208 image f s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Group/FundamentalDomain.lean", "full_name": "MeasureTheory.IsFundamentalDomain.quotientMeasureEqMeasurePreimage_quotientMeasure", "start": [905, 1], "end": [912, 98], "traced_tactics": [{"tactic": "rw [fund_dom_s.quotientMeasure_eq _ fund_dom_t]", "annotated_tactic": ["rw [fund_dom_s.quotientMeasure_eq _ fund_dom_t]", []], "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 : MulAction G \u03b1\ninst\u271d\u2074 : MeasurableSpace \u03b1\n\u03bd : Measure \u03b1\ninst\u271d\u00b3 : Countable G\ninst\u271d\u00b2 : MeasurableSpace G\ninst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bd\ninst\u271d : MeasurableSMul G \u03b1\ns : Set \u03b1\nfund_dom_s : IsFundamentalDomain G s \u03bd\nt : Set \u03b1\nfund_dom_t : IsFundamentalDomain G t \u03bd\n\u22a2 Measure.map (Quotient.mk \u03b1_mod_G) (\u03bd.restrict s) = Measure.map (Quotient.mk \u03b1_mod_G) (\u03bd.restrict t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Regular/SMul.lean", "full_name": "IsSMulRegular.smul", "start": [68, 1], "end": [69, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean", "full_name": "Sum.isLeft_swap", "start": [163, 9], "end": [163, 90], "traced_tactics": [{"tactic": "cases x <;> rfl", "annotated_tactic": ["cases x <;> rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nx : \u03b1 \u2295 \u03b2\n\u22a2 x.swap.isLeft = x.isRight", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Init/Data/Nat/Basic.lean", "full_name": "Nat.zero_lt_bit1", "start": [26, 11], "end": [27, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Subring/Basic.lean", "full_name": "Subring.prod_bot_sup_bot_prod", "start": [1257, 1], "end": [1262, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean", "full_name": "MeasureTheory.SignedMeasure.eq_singularPart", "start": [333, 1], "end": [341, 38], "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": [438, 5], "def_end_pos": [438, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nht\u03bc : t \u27c2\u1d65 \u03bc.toENNRealVectorMeasure\nhadd : s = t + \u03bc.withDensity\u1d65 f\n\u22a2 t = s.singularPart \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nht\u03bc : t \u27c2\u1d65 \u03bc.toENNRealVectorMeasure\nhadd : s = t + \u03bc.withDensity\u1d65 f\nhfi : Integrable f \u03bc\n\u22a2 t = s.singularPart \u03bc\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nht\u03bc : t \u27c2\u1d65 \u03bc.toENNRealVectorMeasure\nhadd : s = t + \u03bc.withDensity\u1d65 f\nhfi : \u00acIntegrable f \u03bc\n\u22a2 t = s.singularPart \u03bc"}, {"tactic": "refine eq_singularPart' t hfi.1.measurable_mk (hfi.congr hfi.1.ae_eq_mk) ht\u03bc ?_", "annotated_tactic": ["refine <a>eq_singularPart'</a> t hfi.1.<a>measurable_mk</a> (hfi.congr hfi.1.<a>ae_eq_mk</a>) ht\u03bc ?_", [{"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedLebesgue.0.MeasureTheory.SignedMeasure.eq_singularPart'", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean", "def_pos": [308, 17], "def_end_pos": [308, 33]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.measurable_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1236, 9], "def_end_pos": [1236, 22]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.ae_eq_mk", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1241, 9], "def_end_pos": [1241, 17]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nht\u03bc : t \u27c2\u1d65 \u03bc.toENNRealVectorMeasure\nhadd : s = t + \u03bc.withDensity\u1d65 f\nhfi : Integrable f \u03bc\n\u22a2 t = s.singularPart \u03bc", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nht\u03bc : t \u27c2\u1d65 \u03bc.toENNRealVectorMeasure\nhadd : s = t + \u03bc.withDensity\u1d65 f\nhfi : Integrable f \u03bc\n\u22a2 s = t + \u03bc.withDensity\u1d65 (AEStronglyMeasurable.mk f \u22ef)"}, {"tactic": "convert hadd using 2", "annotated_tactic": ["convert hadd using 2", []], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nht\u03bc : t \u27c2\u1d65 \u03bc.toENNRealVectorMeasure\nhadd : s = t + \u03bc.withDensity\u1d65 f\nhfi : Integrable f \u03bc\n\u22a2 s = t + \u03bc.withDensity\u1d65 (AEStronglyMeasurable.mk f \u22ef)", "state_after": "case h.e'_3.h.e'_6\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nht\u03bc : t \u27c2\u1d65 \u03bc.toENNRealVectorMeasure\nhadd : s = t + \u03bc.withDensity\u1d65 f\nhfi : Integrable f \u03bc\n\u22a2 \u03bc.withDensity\u1d65 (AEStronglyMeasurable.mk f \u22ef) = \u03bc.withDensity\u1d65 f"}, {"tactic": "exact WithDensity\u1d65Eq.congr_ae hfi.1.ae_eq_mk.symm", "annotated_tactic": ["exact <a>WithDensity\u1d65Eq.congr_ae</a> hfi.1.ae_eq_mk.symm", [{"full_name": "MeasureTheory.WithDensity\u1d65Eq.congr_ae", "def_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "def_pos": [165, 9], "def_end_pos": [165, 32]}]], "state_before": "case h.e'_3.h.e'_6\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nht\u03bc : t \u27c2\u1d65 \u03bc.toENNRealVectorMeasure\nhadd : s = t + \u03bc.withDensity\u1d65 f\nhfi : Integrable f \u03bc\n\u22a2 \u03bc.withDensity\u1d65 (AEStronglyMeasurable.mk f \u22ef) = \u03bc.withDensity\u1d65 f", "state_after": "no goals"}, {"tactic": "rw [withDensity\u1d65, dif_neg hfi, add_zero] at hadd", "annotated_tactic": ["rw [<a>withDensity\u1d65</a>, <a>dif_neg</a> hfi, <a>add_zero</a>] at hadd", [{"full_name": "MeasureTheory.Measure.withDensity\u1d65", "def_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "def_pos": [42, 5], "def_end_pos": [42, 25]}, {"full_name": "dif_neg", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [954, 9], "def_end_pos": [954, 16]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [482, 3], "def_end_pos": [482, 14]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nht\u03bc : t \u27c2\u1d65 \u03bc.toENNRealVectorMeasure\nhadd : s = t + \u03bc.withDensity\u1d65 f\nhfi : \u00acIntegrable f \u03bc\n\u22a2 t = s.singularPart \u03bc", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nht\u03bc : t \u27c2\u1d65 \u03bc.toENNRealVectorMeasure\nhadd : s = t\nhfi : \u00acIntegrable f \u03bc\n\u22a2 t = s.singularPart \u03bc"}, {"tactic": "refine eq_singularPart' t measurable_zero (integrable_zero _ _ \u03bc) ht\u03bc ?_", "annotated_tactic": ["refine <a>eq_singularPart'</a> t <a>measurable_zero</a> (<a>integrable_zero</a> _ _ \u03bc) ht\u03bc ?_", [{"full_name": "_private.Mathlib.MeasureTheory.Decomposition.SignedLebesgue.0.MeasureTheory.SignedMeasure.eq_singularPart'", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean", "def_pos": [308, 17], "def_end_pos": [308, 33]}, {"full_name": "measurable_zero", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [258, 3], "def_end_pos": [258, 14]}, {"full_name": "MeasureTheory.integrable_zero", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [657, 9], "def_end_pos": [657, 24]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nht\u03bc : t \u27c2\u1d65 \u03bc.toENNRealVectorMeasure\nhadd : s = t\nhfi : \u00acIntegrable f \u03bc\n\u22a2 t = s.singularPart \u03bc", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nht\u03bc : t \u27c2\u1d65 \u03bc.toENNRealVectorMeasure\nhadd : s = t\nhfi : \u00acIntegrable f \u03bc\n\u22a2 s = t + \u03bc.withDensity\u1d65 0"}, {"tactic": "rwa [withDensity\u1d65_zero, add_zero]", "annotated_tactic": ["rwa [<a>withDensity\u1d65_zero</a>, <a>add_zero</a>]", [{"full_name": "MeasureTheory.withDensity\u1d65_zero", "def_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "def_pos": [64, 9], "def_end_pos": [64, 26]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [482, 3], "def_end_pos": [482, 14]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ns t\u271d t : SignedMeasure \u03b1\nf : \u03b1 \u2192 \u211d\nht\u03bc : t \u27c2\u1d65 \u03bc.toENNRealVectorMeasure\nhadd : s = t\nhfi : \u00acIntegrable f \u03bc\n\u22a2 s = t + \u03bc.withDensity\u1d65 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/PFun.lean", "full_name": "PFun.mem_fix_iff", "start": [266, 1], "end": [302, 57], "traced_tactics": [{"tactic": "let \u27e8h\u2081, h\u2082\u27e9 := Part.mem_assert_iff.1 h", "annotated_tactic": ["let \u27e8h\u2081, h\u2082\u27e9 := <a>Part.mem_assert_iff</a>.1 h", [{"full_name": "Part.mem_assert_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [465, 9], "def_end_pos": [465, 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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 f.fix a\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'", "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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 f.fix a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a h\u2081\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'"}, {"tactic": "rw [WellFounded.fixFEq] at h\u2082", "annotated_tactic": ["rw [<a>WellFounded.fixFEq</a>] at h\u2082", [{"full_name": "WellFounded.fixFEq", "def_path": ".lake/packages/lean4/src/lean/Init/WF.lean", "def_pos": [81, 9], "def_end_pos": [81, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 f.fix a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a h\u2081\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'", "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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 f.fix a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 :\n  b \u2208\n    Part.assert (f a).Dom fun hf =>\n      match e : (f a).get hf with\n      | Sum.inl b => Part.some b\n      | Sum.inr a' =>\n        (fun y p =>\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : (f a).get hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' \u22ef)\n              y \u22ef)\n          a' \u22ef\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'"}, {"tactic": "simp only [Part.mem_assert_iff] at h\u2082", "annotated_tactic": ["simp only [<a>Part.mem_assert_iff</a>] at h\u2082", [{"full_name": "Part.mem_assert_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [465, 9], "def_end_pos": [465, 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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 f.fix a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 :\n  b \u2208\n    Part.assert (f a).Dom fun hf =>\n      match e : (f a).get hf with\n      | Sum.inl b => Part.some b\n      | Sum.inr a' =>\n        (fun y p =>\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : (f a).get hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' \u22ef)\n              y \u22ef)\n          a' \u22ef\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'", "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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 f.fix a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 :\n  \u2203 (h : (f a).Dom),\n    b \u2208\n      match e : (f a).get h with\n      | Sum.inl b => Part.some b\n      | Sum.inr a' =>\n        WellFounded.fixF\n          (fun a IH =>\n            Part.assert (f a).Dom fun hf =>\n              match e : (f a).get hf with\n              | Sum.inl b => Part.some b\n              | Sum.inr a' => IH a' \u22ef)\n          a' \u22ef\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'"}, {"tactic": "cases' h\u2082 with h\u2082 h\u2083", "annotated_tactic": ["cases' h\u2082 with h\u2082 h\u2083", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 f.fix a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 :\n  \u2203 (h : (f a).Dom),\n    b \u2208\n      match e : (f a).get h with\n      | Sum.inl b => Part.some b\n      | Sum.inr a' =>\n        WellFounded.fixF\n          (fun a IH =>\n            Part.assert (f a).Dom fun hf =>\n              match e : (f a).get hf with\n              | Sum.inl b => Part.some b\n              | Sum.inr a' => IH a' \u22ef)\n          a' \u22ef\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'", "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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 f.fix a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\nh\u2083 :\n  b \u2208\n    match e : (f a).get h\u2082 with\n    | Sum.inl b => Part.some b\n    | Sum.inr a' =>\n      WellFounded.fixF\n        (fun a IH =>\n          Part.assert (f a).Dom fun hf =>\n            match e : (f a).get hf with\n            | Sum.inl b => Part.some b\n            | Sum.inr a' => IH a' \u22ef)\n        a' \u22ef\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'"}, {"tactic": "split at h\u2083", "annotated_tactic": ["split at h\u2083", []], "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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 f.fix a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\nh\u2083 :\n  b \u2208\n    match e : (f a).get h\u2082 with\n    | Sum.inl b => Part.some b\n    | Sum.inr a' =>\n      WellFounded.fixF\n        (fun a IH =>\n          Part.assert (f a).Dom fun hf =>\n            match e : (f a).get hf with\n            | Sum.inl b => Part.some b\n            | Sum.inr a' => IH a' \u22ef)\n        a' \u22ef\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'", "state_after": "case intro.h_1\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 f.fix a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\nb\u271d : \u03b2\nheq\u271d : (f a).get h\u2082 = Sum.inl b\u271d\nh\u2083 : b \u2208 Part.some b\u271d\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'\n\ncase intro.h_2\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 f.fix a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\na'\u271d : \u03b1\nheq\u271d : (f a).get h\u2082 = Sum.inr a'\u271d\nh\u2083 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a'\u271d \u22ef\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'"}, {"tactic": "next e => simp only [Part.mem_some_iff] at h\u2083; subst b; exact Or.inl \u27e8h\u2082, e\u27e9", "annotated_tactic": ["next e => simp only [<a>Part.mem_some_iff</a>] at h\u2083; subst b; exact <a>Or.inl</a> \u27e8h\u2082, e\u27e9", [{"full_name": "Part.mem_some_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [172, 9], "def_end_pos": [172, 21]}, {"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}]], "state_before": "case intro.h_1\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 f.fix a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\nb\u271d : \u03b2\nheq\u271d : (f a).get h\u2082 = Sum.inl b\u271d\nh\u2083 : b \u2208 Part.some b\u271d\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'\n\ncase intro.h_2\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 f.fix a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\na'\u271d : \u03b1\nheq\u271d : (f a).get h\u2082 = Sum.inr a'\u271d\nh\u2083 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a'\u271d \u22ef\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'", "state_after": "case intro.h_2\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 f.fix a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\na'\u271d : \u03b1\nheq\u271d : (f a).get h\u2082 = Sum.inr a'\u271d\nh\u2083 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a'\u271d \u22ef\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'"}, {"tactic": "next e => exact Or.inr \u27e8_, \u27e8_, e\u27e9, Part.mem_assert _ h\u2083\u27e9", "annotated_tactic": ["next e => exact <a>Or.inr</a> \u27e8_, \u27e8_, e\u27e9, <a>Part.mem_assert</a> _ h\u2083\u27e9", [{"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}, {"full_name": "Part.mem_assert", "def_path": "Mathlib/Data/Part.lean", "def_pos": [460, 9], "def_end_pos": [460, 19]}]], "state_before": "case intro.h_2\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 f.fix a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\na'\u271d : \u03b1\nheq\u271d : (f a).get h\u2082 = Sum.inr a'\u271d\nh\u2083 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a'\u271d \u22ef\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'", "state_after": "no goals"}, {"tactic": "simp only [Part.mem_some_iff] at h\u2083", "annotated_tactic": ["simp only [<a>Part.mem_some_iff</a>] at h\u2083", [{"full_name": "Part.mem_some_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [172, 9], "def_end_pos": [172, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 f.fix a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\nb\u271d : \u03b2\ne : (f a).get h\u2082 = Sum.inl b\u271d\nh\u2083 : b \u2208 Part.some b\u271d\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'", "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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 f.fix a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\nb\u271d : \u03b2\ne : (f a).get h\u2082 = Sum.inl b\u271d\nh\u2083 : b = b\u271d\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'"}, {"tactic": "subst b", "annotated_tactic": ["subst b", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 f.fix a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\nb\u271d : \u03b2\ne : (f a).get h\u2082 = Sum.inl b\u271d\nh\u2083 : b = b\u271d\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'", "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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\nb\u271d : \u03b2\ne : (f a).get h\u2082 = Sum.inl b\u271d\nh : b\u271d \u2208 f.fix a\n\u22a2 Sum.inl b\u271d \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b\u271d \u2208 f.fix a'"}, {"tactic": "exact Or.inl \u27e8h\u2082, e\u27e9", "annotated_tactic": ["exact <a>Or.inl</a> \u27e8h\u2082, e\u27e9", [{"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\nb\u271d : \u03b2\ne : (f a).get h\u2082 = Sum.inl b\u271d\nh : b\u271d \u2208 f.fix a\n\u22a2 Sum.inl b\u271d \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b\u271d \u2208 f.fix a'", "state_after": "no goals"}, {"tactic": "exact Or.inr \u27e8_, \u27e8_, e\u27e9, Part.mem_assert _ h\u2083\u27e9", "annotated_tactic": ["exact <a>Or.inr</a> \u27e8_, \u27e8_, e\u27e9, <a>Part.mem_assert</a> _ h\u2083\u27e9", [{"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}, {"full_name": "Part.mem_assert", "def_path": "Mathlib/Data/Part.lean", "def_pos": [460, 9], "def_end_pos": [460, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : b \u2208 f.fix a\nh\u2081 : Acc (fun x y => Sum.inr x \u2208 f y) a\nh\u2082 : (f a).Dom\na'\u271d : \u03b1\ne : (f a).get h\u2082 = Sum.inr a'\u271d\nh\u2083 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a'\u271d \u22ef\n\u22a2 Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'", "state_after": "no goals"}, {"tactic": "simp only [fix, Part.mem_assert_iff]", "annotated_tactic": ["simp only [<a>fix</a>, <a>Part.mem_assert_iff</a>]", [{"full_name": "PFun.fix", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [250, 5], "def_end_pos": [250, 8]}, {"full_name": "Part.mem_assert_iff", "def_path": "Mathlib/Data/Part.lean", "def_pos": [465, 9], "def_end_pos": [465, 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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'\n\u22a2 b \u2208 f.fix a", "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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'\n\u22a2 \u2203 (h : Acc (fun x y => Sum.inr x \u2208 f y) a),\n    b \u2208\n      WellFounded.fixF\n        (fun a IH =>\n          Part.assert (f a).Dom fun hf =>\n            match e : (f a).get hf with\n            | Sum.inl b => Part.some b\n            | Sum.inr a' => IH a' \u22ef)\n        a h"}, {"tactic": "rcases h with (\u27e8h\u2081, h\u2082\u27e9 | \u27e8a', h, h\u2083\u27e9)", "annotated_tactic": ["rcases h with (\u27e8h\u2081, h\u2082\u27e9 | \u27e8a', h, h\u2083\u27e9)", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh : Sum.inl b \u2208 f a \u2228 \u2203 a', Sum.inr a' \u2208 f a \u2227 b \u2208 f.fix a'\n\u22a2 \u2203 (h : Acc (fun x y => Sum.inr x \u2208 f y) a),\n    b \u2208\n      WellFounded.fixF\n        (fun a IH =>\n          Part.assert (f a).Dom fun hf =>\n            match e : (f a).get hf with\n            | Sum.inl b => Part.some b\n            | Sum.inr a' => IH a' \u22ef)\n        a h", "state_after": "case inl.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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inl b\n\u22a2 \u2203 (h : Acc (fun x y => Sum.inr x \u2208 f y) a),\n    b \u2208\n      WellFounded.fixF\n        (fun a IH =>\n          Part.assert (f a).Dom fun hf =>\n            match e : (f a).get hf with\n            | Sum.inl b => Part.some b\n            | Sum.inr a' => IH a' \u22ef)\n        a h\n\ncase inr.intro.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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh : Sum.inr a' \u2208 f a\nh\u2083 : b \u2208 f.fix a'\n\u22a2 \u2203 (h : Acc (fun x y => Sum.inr x \u2208 f y) a),\n    b \u2208\n      WellFounded.fixF\n        (fun a IH =>\n          Part.assert (f a).Dom fun hf =>\n            match e : (f a).get hf with\n            | Sum.inl b => Part.some b\n            | Sum.inr a' => IH a' \u22ef)\n        a h"}, {"tactic": "refine \u27e8\u27e8_, fun y h' => ?_\u27e9, ?_\u27e9", "annotated_tactic": ["refine \u27e8\u27e8_, fun y h' => ?_\u27e9, ?_\u27e9", []], "state_before": "case inl.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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inl b\n\u22a2 \u2203 (h : Acc (fun x y => Sum.inr x \u2208 f y) a),\n    b \u2208\n      WellFounded.fixF\n        (fun a IH =>\n          Part.assert (f a).Dom fun hf =>\n            match e : (f a).get hf with\n            | Sum.inl b => Part.some b\n            | Sum.inr a' => IH a' \u22ef)\n        a h", "state_after": "case inl.intro.refine_1\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inl b\ny : \u03b1\nh' : Sum.inr y \u2208 f a\n\u22a2 Acc (fun x y => Sum.inr x \u2208 f y) y\n\ncase inl.intro.refine_2\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inl b\n\u22a2 b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a \u22ef"}, {"tactic": "injection Part.mem_unique \u27e8h\u2081, h\u2082\u27e9 h'", "annotated_tactic": ["injection <a>Part.mem_unique</a> \u27e8h\u2081, h\u2082\u27e9 h'", [{"full_name": "Part.mem_unique", "def_path": "Mathlib/Data/Part.lean", "def_pos": [146, 9], "def_end_pos": [146, 19]}]], "state_before": "case inl.intro.refine_1\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inl b\ny : \u03b1\nh' : Sum.inr y \u2208 f a\n\u22a2 Acc (fun x y => Sum.inr x \u2208 f y) y", "state_after": "no goals"}, {"tactic": "rw [WellFounded.fixFEq]", "annotated_tactic": ["rw [<a>WellFounded.fixFEq</a>]", [{"full_name": "WellFounded.fixFEq", "def_path": ".lake/packages/lean4/src/lean/Init/WF.lean", "def_pos": [81, 9], "def_end_pos": [81, 15]}]], "state_before": "case inl.intro.refine_2\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inl b\n\u22a2 b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a \u22ef", "state_after": "case inl.intro.refine_2\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inl b\n\u22a2 b \u2208\n    Part.assert (f a).Dom fun hf =>\n      match e : (f a).get hf with\n      | Sum.inl b => Part.some b\n      | Sum.inr a' =>\n        (fun y p =>\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : (f a).get hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' \u22ef)\n              y \u22ef)\n          a' \u22ef"}, {"tactic": "apply Part.mem_assert h\u2081", "annotated_tactic": ["apply <a>Part.mem_assert</a> h\u2081", [{"full_name": "Part.mem_assert", "def_path": "Mathlib/Data/Part.lean", "def_pos": [460, 9], "def_end_pos": [460, 19]}]], "state_before": "case inl.intro.refine_2\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inl b\n\u22a2 b \u2208\n    Part.assert (f a).Dom fun hf =>\n      match e : (f a).get hf with\n      | Sum.inl b => Part.some b\n      | Sum.inr a' =>\n        (fun y p =>\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : (f a).get hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' \u22ef)\n              y \u22ef)\n          a' \u22ef", "state_after": "case inl.intro.refine_2\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inl b\n\u22a2 b \u2208\n    match e : (f a).get h\u2081 with\n    | Sum.inl b => Part.some b\n    | Sum.inr a' =>\n      (fun y p =>\n          WellFounded.fixF\n            (fun a IH =>\n              Part.assert (f a).Dom fun hf =>\n                match e : (f a).get hf with\n                | Sum.inl b => Part.some b\n                | Sum.inr a' => IH a' \u22ef)\n            y \u22ef)\n        a' \u22ef"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "case inl.intro.refine_2\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inl b\n\u22a2 b \u2208\n    match e : (f a).get h\u2081 with\n    | Sum.inl b => Part.some b\n    | Sum.inr a' =>\n      (fun y p =>\n          WellFounded.fixF\n            (fun a IH =>\n              Part.assert (f a).Dom fun hf =>\n                match e : (f a).get hf with\n                | Sum.inl b => Part.some b\n                | Sum.inr a' => IH a' \u22ef)\n            y \u22ef)\n        a' \u22ef", "state_after": "case inl.intro.refine_2.h_1\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inl b\nb\u271d : \u03b2\nheq\u271d : (f a).get h\u2081 = Sum.inl b\u271d\n\u22a2 b \u2208 Part.some b\u271d\n\ncase inl.intro.refine_2.h_2\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inl b\na'\u271d : \u03b1\nheq\u271d : (f a).get h\u2081 = Sum.inr a'\u271d\n\u22a2 b \u2208\n    (fun y p =>\n        WellFounded.fixF\n          (fun a IH =>\n            Part.assert (f a).Dom fun hf =>\n              match e : (f a).get hf with\n              | Sum.inl b => Part.some b\n              | Sum.inr a' => IH a' \u22ef)\n          y \u22ef)\n      a'\u271d \u22ef"}, {"tactic": "next e =>\n  injection h\u2082.symm.trans e with h; simp [h]", "annotated_tactic": ["next e =>\n          injection h\u2082.symm.trans e with h; simp [h]", []], "state_before": "case inl.intro.refine_2.h_1\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inl b\nb\u271d : \u03b2\nheq\u271d : (f a).get h\u2081 = Sum.inl b\u271d\n\u22a2 b \u2208 Part.some b\u271d\n\ncase inl.intro.refine_2.h_2\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inl b\na'\u271d : \u03b1\nheq\u271d : (f a).get h\u2081 = Sum.inr a'\u271d\n\u22a2 b \u2208\n    (fun y p =>\n        WellFounded.fixF\n          (fun a IH =>\n            Part.assert (f a).Dom fun hf =>\n              match e : (f a).get hf with\n              | Sum.inl b => Part.some b\n              | Sum.inr a' => IH a' \u22ef)\n          y \u22ef)\n      a'\u271d \u22ef", "state_after": "case inl.intro.refine_2.h_2\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inl b\na'\u271d : \u03b1\nheq\u271d : (f a).get h\u2081 = Sum.inr a'\u271d\n\u22a2 b \u2208\n    (fun y p =>\n        WellFounded.fixF\n          (fun a IH =>\n            Part.assert (f a).Dom fun hf =>\n              match e : (f a).get hf with\n              | Sum.inl b => Part.some b\n              | Sum.inr a' => IH a' \u22ef)\n          y \u22ef)\n      a'\u271d \u22ef"}, {"tactic": "next e =>\n  injection h\u2082.symm.trans e", "annotated_tactic": ["next e =>\n          injection h\u2082.symm.trans e", []], "state_before": "case inl.intro.refine_2.h_2\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inl b\na'\u271d : \u03b1\nheq\u271d : (f a).get h\u2081 = Sum.inr a'\u271d\n\u22a2 b \u2208\n    (fun y p =>\n        WellFounded.fixF\n          (fun a IH =>\n            Part.assert (f a).Dom fun hf =>\n              match e : (f a).get hf with\n              | Sum.inl b => Part.some b\n              | Sum.inr a' => IH a' \u22ef)\n          y \u22ef)\n      a'\u271d \u22ef", "state_after": "no goals"}, {"tactic": "injection h\u2082.symm.trans e with h", "annotated_tactic": ["injection h\u2082.symm.trans e with h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inl b\nb\u271d : \u03b2\ne : (f a).get h\u2081 = Sum.inl b\u271d\n\u22a2 b \u2208 Part.some b\u271d", "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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inl b\nb\u271d : \u03b2\ne : (f a).get h\u2081 = Sum.inl b\u271d\nh : b = b\u271d\n\u22a2 b \u2208 Part.some b\u271d"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inl b\nb\u271d : \u03b2\ne : (f a).get h\u2081 = Sum.inl b\u271d\nh : b = b\u271d\n\u22a2 b \u2208 Part.some b\u271d", "state_after": "no goals"}, {"tactic": "injection h\u2082.symm.trans e", "annotated_tactic": ["injection h\u2082.symm.trans e", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inl b\na'\u271d : \u03b1\ne : (f a).get h\u2081 = Sum.inr a'\u271d\n\u22a2 b \u2208\n    (fun y p =>\n        WellFounded.fixF\n          (fun a IH =>\n            Part.assert (f a).Dom fun hf =>\n              match e : (f a).get hf with\n              | Sum.inl b => Part.some b\n              | Sum.inr a' => IH a' \u22ef)\n          y \u22ef)\n      a'\u271d \u22ef", "state_after": "no goals"}, {"tactic": "simp [fix] at h\u2083", "annotated_tactic": ["simp [<a>fix</a>] at h\u2083", [{"full_name": "PFun.fix", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [250, 5], "def_end_pos": [250, 8]}]], "state_before": "case inr.intro.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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh : Sum.inr a' \u2208 f a\nh\u2083 : b \u2208 f.fix a'\n\u22a2 \u2203 (h : Acc (fun x y => Sum.inr x \u2208 f y) a),\n    b \u2208\n      WellFounded.fixF\n        (fun a IH =>\n          Part.assert (f a).Dom fun hf =>\n            match e : (f a).get hf with\n            | Sum.inl b => Part.some b\n            | Sum.inr a' => IH a' \u22ef)\n        a h", "state_after": "case inr.intro.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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh : Sum.inr a' \u2208 f a\nh\u2083 :\n  \u2203 (h : Acc (fun x y => Sum.inr x \u2208 f y) a'),\n    b \u2208\n      WellFounded.fixF\n        (fun a IH =>\n          Part.assert (f a).Dom fun hf =>\n            match e : (f a).get hf with\n            | Sum.inl b => Part.some b\n            | Sum.inr a' => IH a' \u22ef)\n        a' h\n\u22a2 \u2203 (h : Acc (fun x y => Sum.inr x \u2208 f y) a),\n    b \u2208\n      WellFounded.fixF\n        (fun a IH =>\n          Part.assert (f a).Dom fun hf =>\n            match e : (f a).get hf with\n            | Sum.inl b => Part.some b\n            | Sum.inr a' => IH a' \u22ef)\n        a h"}, {"tactic": "cases' h\u2083 with h\u2083 h\u2084", "annotated_tactic": ["cases' h\u2083 with h\u2083 h\u2084", []], "state_before": "case inr.intro.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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh : Sum.inr a' \u2208 f a\nh\u2083 :\n  \u2203 (h : Acc (fun x y => Sum.inr x \u2208 f y) a'),\n    b \u2208\n      WellFounded.fixF\n        (fun a IH =>\n          Part.assert (f a).Dom fun hf =>\n            match e : (f a).get hf with\n            | Sum.inl b => Part.some b\n            | Sum.inr a' => IH a' \u22ef)\n        a' h\n\u22a2 \u2203 (h : Acc (fun x y => Sum.inr x \u2208 f y) a),\n    b \u2208\n      WellFounded.fixF\n        (fun a IH =>\n          Part.assert (f a).Dom fun hf =>\n            match e : (f a).get hf with\n            | Sum.inl b => Part.some b\n            | Sum.inr a' => IH a' \u22ef)\n        a h", "state_after": "case inr.intro.intro.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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh : Sum.inr a' \u2208 f a\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\n\u22a2 \u2203 (h : Acc (fun x y => Sum.inr x \u2208 f y) a),\n    b \u2208\n      WellFounded.fixF\n        (fun a IH =>\n          Part.assert (f a).Dom fun hf =>\n            match e : (f a).get hf with\n            | Sum.inl b => Part.some b\n            | Sum.inr a' => IH a' \u22ef)\n        a h"}, {"tactic": "refine \u27e8\u27e8_, fun y h' => ?_\u27e9, ?_\u27e9", "annotated_tactic": ["refine \u27e8\u27e8_, fun y h' => ?_\u27e9, ?_\u27e9", []], "state_before": "case inr.intro.intro.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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh : Sum.inr a' \u2208 f a\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\n\u22a2 \u2203 (h : Acc (fun x y => Sum.inr x \u2208 f y) a),\n    b \u2208\n      WellFounded.fixF\n        (fun a IH =>\n          Part.assert (f a).Dom fun hf =>\n            match e : (f a).get hf with\n            | Sum.inl b => Part.some b\n            | Sum.inr a' => IH a' \u22ef)\n        a h", "state_after": "case inr.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\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh : Sum.inr a' \u2208 f a\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\ny : \u03b1\nh' : Sum.inr y \u2208 f a\n\u22a2 Acc (fun x y => Sum.inr x \u2208 f y) y\n\ncase inr.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\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh : Sum.inr a' \u2208 f a\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\n\u22a2 b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a \u22ef"}, {"tactic": "injection Part.mem_unique h h' with e", "annotated_tactic": ["injection <a>Part.mem_unique</a> h h' with e", [{"full_name": "Part.mem_unique", "def_path": "Mathlib/Data/Part.lean", "def_pos": [146, 9], "def_end_pos": [146, 19]}]], "state_before": "case inr.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\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh : Sum.inr a' \u2208 f a\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\ny : \u03b1\nh' : Sum.inr y \u2208 f a\n\u22a2 Acc (fun x y => Sum.inr x \u2208 f y) y", "state_after": "case inr.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\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh : Sum.inr a' \u2208 f a\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\ny : \u03b1\nh' : Sum.inr y \u2208 f a\ne : a' = y\n\u22a2 Acc (fun x y => Sum.inr x \u2208 f y) y"}, {"tactic": "exact e \u25b8 h\u2083", "annotated_tactic": ["exact e \u25b8 h\u2083", []], "state_before": "case inr.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\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh : Sum.inr a' \u2208 f a\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\ny : \u03b1\nh' : Sum.inr y \u2208 f a\ne : a' = y\n\u22a2 Acc (fun x y => Sum.inr x \u2208 f y) y", "state_after": "no goals"}, {"tactic": "cases' h with h\u2081 h\u2082", "annotated_tactic": ["cases' h with h\u2081 h\u2082", []], "state_before": "case inr.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\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh : Sum.inr a' \u2208 f a\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\n\u22a2 b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a \u22ef", "state_after": "case inr.intro.intro.intro.refine_2.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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inr a'\n\u22a2 b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a \u22ef"}, {"tactic": "rw [WellFounded.fixFEq]", "annotated_tactic": ["rw [<a>WellFounded.fixFEq</a>]", [{"full_name": "WellFounded.fixFEq", "def_path": ".lake/packages/lean4/src/lean/Init/WF.lean", "def_pos": [81, 9], "def_end_pos": [81, 15]}]], "state_before": "case inr.intro.intro.intro.refine_2.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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inr a'\n\u22a2 b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a \u22ef", "state_after": "case inr.intro.intro.intro.refine_2.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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inr a'\n\u22a2 b \u2208\n    Part.assert (f a).Dom fun hf =>\n      match e : (f a).get hf with\n      | Sum.inl b => Part.some b\n      | Sum.inr a'_1 =>\n        (fun y p =>\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : (f a).get hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' \u22ef)\n              y \u22ef)\n          a'_1 \u22ef"}, {"tactic": "apply Part.mem_assert h\u2081", "annotated_tactic": ["apply <a>Part.mem_assert</a> h\u2081", [{"full_name": "Part.mem_assert", "def_path": "Mathlib/Data/Part.lean", "def_pos": [460, 9], "def_end_pos": [460, 19]}]], "state_before": "case inr.intro.intro.intro.refine_2.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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inr a'\n\u22a2 b \u2208\n    Part.assert (f a).Dom fun hf =>\n      match e : (f a).get hf with\n      | Sum.inl b => Part.some b\n      | Sum.inr a'_1 =>\n        (fun y p =>\n            WellFounded.fixF\n              (fun a IH =>\n                Part.assert (f a).Dom fun hf =>\n                  match e : (f a).get hf with\n                  | Sum.inl b => Part.some b\n                  | Sum.inr a' => IH a' \u22ef)\n              y \u22ef)\n          a'_1 \u22ef", "state_after": "case inr.intro.intro.intro.refine_2.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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inr a'\n\u22a2 b \u2208\n    match e : (f a).get h\u2081 with\n    | Sum.inl b => Part.some b\n    | Sum.inr a'_1 =>\n      (fun y p =>\n          WellFounded.fixF\n            (fun a IH =>\n              Part.assert (f a).Dom fun hf =>\n                match e : (f a).get hf with\n                | Sum.inl b => Part.some b\n                | Sum.inr a' => IH a' \u22ef)\n            y \u22ef)\n        a'_1 \u22ef"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "case inr.intro.intro.intro.refine_2.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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inr a'\n\u22a2 b \u2208\n    match e : (f a).get h\u2081 with\n    | Sum.inl b => Part.some b\n    | Sum.inr a'_1 =>\n      (fun y p =>\n          WellFounded.fixF\n            (fun a IH =>\n              Part.assert (f a).Dom fun hf =>\n                match e : (f a).get hf with\n                | Sum.inl b => Part.some b\n                | Sum.inr a' => IH a' \u22ef)\n            y \u22ef)\n        a'_1 \u22ef", "state_after": "case inr.intro.intro.intro.refine_2.intro.h_1\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inr a'\nb\u271d : \u03b2\nheq\u271d : (f a).get h\u2081 = Sum.inl b\u271d\n\u22a2 b \u2208 Part.some b\u271d\n\ncase inr.intro.intro.intro.refine_2.intro.h_2\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inr a'\na'\u271d : \u03b1\nheq\u271d : (f a).get h\u2081 = Sum.inr a'\u271d\n\u22a2 b \u2208\n    (fun y p =>\n        WellFounded.fixF\n          (fun a IH =>\n            Part.assert (f a).Dom fun hf =>\n              match e : (f a).get hf with\n              | Sum.inl b => Part.some b\n              | Sum.inr a' => IH a' \u22ef)\n          y \u22ef)\n      a'\u271d \u22ef"}, {"tactic": "next e =>\n  injection h\u2082.symm.trans e", "annotated_tactic": ["next e =>\n          injection h\u2082.symm.trans e", []], "state_before": "case inr.intro.intro.intro.refine_2.intro.h_1\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inr a'\nb\u271d : \u03b2\nheq\u271d : (f a).get h\u2081 = Sum.inl b\u271d\n\u22a2 b \u2208 Part.some b\u271d\n\ncase inr.intro.intro.intro.refine_2.intro.h_2\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inr a'\na'\u271d : \u03b1\nheq\u271d : (f a).get h\u2081 = Sum.inr a'\u271d\n\u22a2 b \u2208\n    (fun y p =>\n        WellFounded.fixF\n          (fun a IH =>\n            Part.assert (f a).Dom fun hf =>\n              match e : (f a).get hf with\n              | Sum.inl b => Part.some b\n              | Sum.inr a' => IH a' \u22ef)\n          y \u22ef)\n      a'\u271d \u22ef", "state_after": "case inr.intro.intro.intro.refine_2.intro.h_2\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inr a'\na'\u271d : \u03b1\nheq\u271d : (f a).get h\u2081 = Sum.inr a'\u271d\n\u22a2 b \u2208\n    (fun y p =>\n        WellFounded.fixF\n          (fun a IH =>\n            Part.assert (f a).Dom fun hf =>\n              match e : (f a).get hf with\n              | Sum.inl b => Part.some b\n              | Sum.inr a' => IH a' \u22ef)\n          y \u22ef)\n      a'\u271d \u22ef"}, {"tactic": "next e =>\n  injection h\u2082.symm.trans e; subst a'; exact h\u2084", "annotated_tactic": ["next e =>\n          injection h\u2082.symm.trans e; subst a'; exact h\u2084", []], "state_before": "case inr.intro.intro.intro.refine_2.intro.h_2\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 : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inr a'\na'\u271d : \u03b1\nheq\u271d : (f a).get h\u2081 = Sum.inr a'\u271d\n\u22a2 b \u2208\n    (fun y p =>\n        WellFounded.fixF\n          (fun a IH =>\n            Part.assert (f a).Dom fun hf =>\n              match e : (f a).get hf with\n              | Sum.inl b => Part.some b\n              | Sum.inr a' => IH a' \u22ef)\n          y \u22ef)\n      a'\u271d \u22ef", "state_after": "no goals"}, {"tactic": "injection h\u2082.symm.trans e", "annotated_tactic": ["injection h\u2082.symm.trans e", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inr a'\nb\u271d : \u03b2\ne : (f a).get h\u2081 = Sum.inl b\u271d\n\u22a2 b \u2208 Part.some b\u271d", "state_after": "no goals"}, {"tactic": "injection h\u2082.symm.trans e", "annotated_tactic": ["injection h\u2082.symm.trans e", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inr a'\na'\u271d : \u03b1\ne : (f a).get h\u2081 = Sum.inr a'\u271d\n\u22a2 b \u2208\n    (fun y p =>\n        WellFounded.fixF\n          (fun a IH =>\n            Part.assert (f a).Dom fun hf =>\n              match e : (f a).get hf with\n              | Sum.inl b => Part.some b\n              | Sum.inr a' => IH a' \u22ef)\n          y \u22ef)\n      a'\u271d \u22ef", "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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inr a'\na'\u271d : \u03b1\ne : (f a).get h\u2081 = Sum.inr a'\u271d\nval_eq\u271d : a' = a'\u271d\n\u22a2 b \u2208\n    (fun y p =>\n        WellFounded.fixF\n          (fun a IH =>\n            Part.assert (f a).Dom fun hf =>\n              match e : (f a).get hf with\n              | Sum.inl b => Part.some b\n              | Sum.inr a' => IH a' \u22ef)\n          y \u22ef)\n      a'\u271d \u22ef"}, {"tactic": "subst a'", "annotated_tactic": ["subst a'", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\na' : \u03b1\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a' h\u2083\nh\u2081 : (f a).Dom\nh\u2082 : (f a).get h\u2081 = Sum.inr a'\na'\u271d : \u03b1\ne : (f a).get h\u2081 = Sum.inr a'\u271d\nval_eq\u271d : a' = a'\u271d\n\u22a2 b \u2208\n    (fun y p =>\n        WellFounded.fixF\n          (fun a IH =>\n            Part.assert (f a).Dom fun hf =>\n              match e : (f a).get hf with\n              | Sum.inl b => Part.some b\n              | Sum.inr a' => IH a' \u22ef)\n          y \u22ef)\n      a'\u271d \u22ef", "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\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\na'\u271d : \u03b1\ne : (f a).get h\u2081 = Sum.inr a'\u271d\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\u271d\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a'\u271d h\u2083\nh\u2082 : (f a).get h\u2081 = Sum.inr a'\u271d\n\u22a2 b \u2208\n    (fun y p =>\n        WellFounded.fixF\n          (fun a IH =>\n            Part.assert (f a).Dom fun hf =>\n              match e : (f a).get hf with\n              | Sum.inl b => Part.some b\n              | Sum.inr a' => IH a' \u22ef)\n          y \u22ef)\n      a'\u271d \u22ef"}, {"tactic": "exact h\u2084", "annotated_tactic": ["exact h\u2084", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2 \u2295 \u03b1\na : \u03b1\nb : \u03b2\nh\u2081 : (f a).Dom\na'\u271d : \u03b1\ne : (f a).get h\u2081 = Sum.inr a'\u271d\nh\u2083 : Acc (fun x y => Sum.inr x \u2208 f y) a'\u271d\nh\u2084 :\n  b \u2208\n    WellFounded.fixF\n      (fun a IH =>\n        Part.assert (f a).Dom fun hf =>\n          match e : (f a).get hf with\n          | Sum.inl b => Part.some b\n          | Sum.inr a' => IH a' \u22ef)\n      a'\u271d h\u2083\nh\u2082 : (f a).get h\u2081 = Sum.inr a'\u271d\n\u22a2 b \u2208\n    (fun y p =>\n        WellFounded.fixF\n          (fun a IH =>\n            Part.assert (f a).Dom fun hf =>\n              match e : (f a).get hf with\n              | Sum.inl b => Part.some b\n              | Sum.inr a' => IH a' \u22ef)\n          y \u22ef)\n      a'\u271d \u22ef", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/Hyperreal.lean", "full_name": "Hyperreal.infinitePos_mul_infinitePos", "start": [863, 1], "end": [865, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/Support.lean", "full_name": "Equiv.Perm.apply_pow_apply_eq_iff", "start": [368, 1], "end": [370, 67], "traced_tactics": [{"tactic": "rw [\u2190 mul_apply, Commute.self_pow f, mul_apply, apply_eq_iff_eq]", "annotated_tactic": ["rw [\u2190 <a>mul_apply</a>, <a>Commute.self_pow</a> f, <a>mul_apply</a>, <a>apply_eq_iff_eq</a>]", [{"full_name": "Equiv.Perm.mul_apply", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 18]}, {"full_name": "Commute.self_pow", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [199, 9], "def_end_pos": [199, 17]}, {"full_name": "Equiv.Perm.mul_apply", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 18]}, {"full_name": "Equiv.apply_eq_iff_eq", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [316, 9], "def_end_pos": [316, 24]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nn : \u2115\nx : \u03b1\n\u22a2 f ((f ^ n) x) = (f ^ n) x \u2194 f x = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/PImage.lean", "full_name": "Finset.pimage_eq_image_filter", "start": [90, 1], "end": [97, 28], "traced_tactics": [{"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\nf g : \u03b1 \u2192. \u03b2\ninst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (f x).Dom\ninst\u271d : (x : \u03b1) \u2192 Decidable (g x).Dom\ns t : Finset \u03b1\nb : \u03b2\n\u22a2 pimage f s = image (fun x => (f \u2191x).get \u22ef) (filter (fun x => (f x).Dom) s).attach", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\nf g : \u03b1 \u2192. \u03b2\ninst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (f x).Dom\ninst\u271d : (x : \u03b1) \u2192 Decidable (g x).Dom\ns t : Finset \u03b1\nb x : \u03b2\n\u22a2 x \u2208 pimage f s \u2194 x \u2208 image (fun x => (f \u2191x).get \u22ef) (filter (fun x => (f x).Dom) s).attach"}, {"tactic": "simp [Part.mem_eq, And.exists]", "annotated_tactic": ["simp [<a>Part.mem_eq</a>, <a>And.exists</a>]", [{"full_name": "Part.mem_eq", "def_path": "Mathlib/Data/Part.lean", "def_pos": [99, 9], "def_end_pos": [99, 15]}, {"full_name": "And.exists", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [852, 9], "def_end_pos": [852, 19]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\nf g : \u03b1 \u2192. \u03b2\ninst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (f x).Dom\ninst\u271d : (x : \u03b1) \u2192 Decidable (g x).Dom\ns t : Finset \u03b1\nb x : \u03b2\n\u22a2 x \u2208 pimage f s \u2194 x \u2208 image (fun x => (f \u2191x).get \u22ef) (filter (fun x => (f x).Dom) s).attach", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\nf g : \u03b1 \u2192. \u03b2\ninst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (f x).Dom\ninst\u271d : (x : \u03b1) \u2192 Decidable (g x).Dom\ns t : Finset \u03b1\nb x : \u03b2\n\u22a2 (\u2203 a \u2208 s, \u2203 (h : (f a).Dom), (f a).get h = x) \u2194 \u2203 a, \u2203 (hp : a \u2208 s) (hq : (f a).Dom), (f a).get \u22ef = x"}, {"tactic": "simp only [\u2190 exists_prop]", "annotated_tactic": ["simp only [\u2190 <a>exists_prop</a>]", [{"full_name": "exists_prop", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [307, 17], "def_end_pos": [307, 28]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : DecidableEq \u03b2\nf g : \u03b1 \u2192. \u03b2\ninst\u271d\u00b9 : (x : \u03b1) \u2192 Decidable (f x).Dom\ninst\u271d : (x : \u03b1) \u2192 Decidable (g x).Dom\ns t : Finset \u03b1\nb x : \u03b2\n\u22a2 (\u2203 a \u2208 s, \u2203 (h : (f a).Dom), (f a).get h = x) \u2194 \u2203 a, \u2203 (hp : a \u2208 s) (hq : (f a).Dom), (f a).get \u22ef = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Chebyshev.lean", "full_name": "MonovaryOn.sum_mul_sum_le_card_mul_sum", "start": [109, 1], "end": [112, 42], "traced_tactics": [{"tactic": "rw [\u2190 nsmul_eq_mul]", "annotated_tactic": ["rw [\u2190 <a>nsmul_eq_mul</a>]", [{"full_name": "nsmul_eq_mul", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [81, 15], "def_end_pos": [81, 34]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedRing \u03b1\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf g : \u03b9 \u2192 \u03b1\nhfg : MonovaryOn f g \u2191s\n\u22a2 (\u2211 i \u2208 s, f i) * \u2211 i \u2208 s, g i \u2264 \u2191s.card * \u2211 i \u2208 s, f i * g i", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedRing \u03b1\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf g : \u03b9 \u2192 \u03b1\nhfg : MonovaryOn f g \u2191s\n\u22a2 (\u2211 i \u2208 s, f i) * \u2211 i \u2208 s, g i \u2264 s.card \u2022 \u2211 i \u2208 s, f i * g i"}, {"tactic": "exact hfg.sum_smul_sum_le_card_smul_sum", "annotated_tactic": ["exact hfg.sum_smul_sum_le_card_smul_sum", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : LinearOrderedRing \u03b1\ns : Finset \u03b9\n\u03c3 : Perm \u03b9\nf g : \u03b9 \u2192 \u03b1\nhfg : MonovaryOn f g \u2191s\n\u22a2 (\u2211 i \u2208 s, f i) * \u2211 i \u2208 s, g i \u2264 s.card \u2022 \u2211 i \u2208 s, f i * g i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "full_name": "ContinuousLinearMap.reApplyInnerSelf_smul", "start": [2318, 1], "end": [2322, 63], "traced_tactics": [{"tactic": "simp only [ContinuousLinearMap.map_smul, ContinuousLinearMap.reApplyInnerSelf_apply,\n  inner_smul_left, inner_smul_right, \u2190 mul_assoc, mul_conj, \u2190 ofReal_pow, \u2190 smul_re,\n  Algebra.smul_def (\u2016c\u2016 ^ 2) \u27eaT x, x\u27eb, algebraMap_eq_ofReal]", "annotated_tactic": ["simp only [<a>ContinuousLinearMap.map_smul</a>, <a>ContinuousLinearMap.reApplyInnerSelf_apply</a>,\n    <a>inner_smul_left</a>, <a>inner_smul_right</a>, \u2190 <a>mul_assoc</a>, <a>mul_conj</a>, \u2190 <a>ofReal_pow</a>, \u2190 <a>smul_re</a>,\n    <a>Algebra.smul_def</a> (\u2016c\u2016 ^ 2) \u27eaT x, x\u27eb, <a>algebraMap_eq_ofReal</a>]", [{"full_name": "ContinuousLinearMap.map_smul", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [509, 19], "def_end_pos": [509, 27]}, {"full_name": "ContinuousLinearMap.reApplyInnerSelf_apply", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [2308, 9], "def_end_pos": [2308, 51]}, {"full_name": "inner_smul_left", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [464, 9], "def_end_pos": [464, 24]}, {"full_name": "inner_smul_right", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [477, 9], "def_end_pos": [477, 25]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "RCLike.mul_conj", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [507, 9], "def_end_pos": [507, 17]}, {"full_name": "RCLike.ofReal_pow", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [231, 9], "def_end_pos": [231, 19]}, {"full_name": "RCLike.smul_re", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [263, 9], "def_end_pos": [263, 16]}, {"full_name": "Algebra.smul_def", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [326, 9], "def_end_pos": [326, 17]}, {"full_name": "RCLike.algebraMap_eq_ofReal", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [110, 9], "def_end_pos": [110, 29]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\nT : E \u2192L[\ud835\udd5c] E\nx : E\nc : \ud835\udd5c\n\u22a2 T.reApplyInnerSelf (c \u2022 x) = \u2016c\u2016 ^ 2 * T.reApplyInnerSelf x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.Infinite.exists_superset_ncard_eq", "start": [986, 1], "end": [991, 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": [977, 9], "def_end_pos": [977, 40]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nht : t.Infinite\nhst : s \u2286 t\nhs : s.Finite\nk : \u2115\nhsk : s.ncard \u2264 k\n\u22a2 \u2203 s', s \u2286 s' \u2227 s' \u2286 t \u2227 s'.ncard = k", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nht : t.Infinite\nhst : s \u2286 t\nhs : s.Finite\nk : \u2115\nhsk : s.ncard \u2264 k\ns\u2081 : Set \u03b1\nhs\u2081 : s\u2081 \u2286 t \\ s\nhs\u2081fin : s\u2081.Finite\nhs\u2081card : s\u2081.ncard = k - s.ncard\n\u22a2 \u2203 s', s \u2286 s' \u2227 s' \u2286 t \u2227 s'.ncard = 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": [796, 9], "def_end_pos": [796, 26]}, {"full_name": "Set.union_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [803, 9], "def_end_pos": [803, 21]}, {"full_name": "Set.diff_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1782, 9], "def_end_pos": [1782, 20]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nht : t.Infinite\nhst : s \u2286 t\nhs : s.Finite\nk : \u2115\nhsk : s.ncard \u2264 k\ns\u2081 : Set \u03b1\nhs\u2081 : s\u2081 \u2286 t \\ s\nhs\u2081fin : s\u2081.Finite\nhs\u2081card : s\u2081.ncard = k - s.ncard\n\u22a2 \u2203 s', s \u2286 s' \u2227 s' \u2286 t \u2227 s'.ncard = k", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nht : t.Infinite\nhst : s \u2286 t\nhs : s.Finite\nk : \u2115\nhsk : s.ncard \u2264 k\ns\u2081 : Set \u03b1\nhs\u2081 : s\u2081 \u2286 t \\ s\nhs\u2081fin : s\u2081.Finite\nhs\u2081card : s\u2081.ncard = k - s.ncard\n\u22a2 (s \u222a s\u2081).ncard = 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": [860, 9], "def_end_pos": [860, 23]}, {"full_name": "Set.disjoint_of_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1525, 7], "def_end_pos": [1525, 31]}, {"full_name": "Set.disjoint_sdiff_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1553, 7], "def_end_pos": [1553, 27]}, {"full_name": "add_tsub_cancel_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [25, 9], "def_end_pos": [25, 30]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b1\nht : t.Infinite\nhst : s \u2286 t\nhs : s.Finite\nk : \u2115\nhsk : s.ncard \u2264 k\ns\u2081 : Set \u03b1\nhs\u2081 : s\u2081 \u2286 t \\ s\nhs\u2081fin : s\u2081.Finite\nhs\u2081card : s\u2081.ncard = k - s.ncard\n\u22a2 (s \u222a s\u2081).ncard = k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Equiv/Defs.lean", "full_name": "Equiv.comp_surjective", "start": [393, 1], "end": [394, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean", "full_name": "Metric.isClosed_ball", "start": [75, 1], "end": [76, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Dimension/DivisionRing.lean", "full_name": "rank_fun_infinite", "start": [288, 1], "end": [300, 14], "traced_tactics": [{"tactic": "obtain \u27e8\u27e8\u03b9K, bK\u27e9\u27e9 := Module.Free.exists_basis (R := K) (M := \u03b9 \u2192 K)", "annotated_tactic": ["obtain \u27e8\u27e8\u03b9K, bK\u27e9\u27e9 := <a>Module.Free.exists_basis</a> (R := K) (M := \u03b9 \u2192 K)", [{"full_name": "Module.Free.exists_basis", "def_path": "Mathlib/LinearAlgebra/FreeModule/Basic.lean", "def_pos": [37, 3], "def_end_pos": [37, 15]}]], "state_before": "K R : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type u_1\ninst\u271d : DivisionRing K\n\u03b9 : Type v\nh\u03b9 : Infinite \u03b9\n\u22a2 Module.rank K (\u03b9 \u2192 K) = #(\u03b9 \u2192 K)", "state_after": "case intro.mk\nK R : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type u_1\ninst\u271d : DivisionRing K\n\u03b9 : Type v\nh\u03b9 : Infinite \u03b9\n\u03b9K : Type (max u v)\nbK : Basis \u03b9K K (\u03b9 \u2192 K)\n\u22a2 Module.rank K (\u03b9 \u2192 K) = #(\u03b9 \u2192 K)"}, {"tactic": "obtain \u27e8e\u27e9 := lift_mk_le'.mp ((aleph0_le_mk_iff.mpr h\u03b9).trans_eq (lift_uzero #\u03b9).symm)", "annotated_tactic": ["obtain \u27e8e\u27e9 := lift_mk_le'.mp ((aleph0_le_mk_iff.mpr h\u03b9).<a>trans_eq</a> (<a>lift_uzero</a> #\u03b9).<a>symm</a>)", [{"full_name": "LE.le.trans_eq", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [190, 7], "def_end_pos": [190, 21]}, {"full_name": "Cardinal.lift_uzero", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [231, 9], "def_end_pos": [231, 19]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case intro.mk\nK R : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type u_1\ninst\u271d : DivisionRing K\n\u03b9 : Type v\nh\u03b9 : Infinite \u03b9\n\u03b9K : Type (max u v)\nbK : Basis \u03b9K K (\u03b9 \u2192 K)\n\u22a2 Module.rank K (\u03b9 \u2192 K) = #(\u03b9 \u2192 K)", "state_after": "case intro.mk.intro\nK R : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type u_1\ninst\u271d : DivisionRing K\n\u03b9 : Type v\nh\u03b9 : Infinite \u03b9\n\u03b9K : Type (max u v)\nbK : Basis \u03b9K K (\u03b9 \u2192 K)\ne : \u2115 \u21aa \u03b9\n\u22a2 Module.rank K (\u03b9 \u2192 K) = #(\u03b9 \u2192 K)"}, {"tactic": "have := LinearMap.lift_rank_le_of_injective _ <|\n  LinearMap.funLeft_injective_of_surjective K K _ (invFun_surjective e.injective)", "annotated_tactic": ["have := <a>LinearMap.lift_rank_le_of_injective</a> _ <|\n    <a>LinearMap.funLeft_injective_of_surjective</a> K K _ (<a>invFun_surjective</a> e.injective)", [{"full_name": "LinearMap.lift_rank_le_of_injective", "def_path": "Mathlib/LinearAlgebra/Dimension/Basic.lean", "def_pos": [246, 9], "def_end_pos": [246, 44]}, {"full_name": "LinearMap.funLeft_injective_of_surjective", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [1315, 9], "def_end_pos": [1315, 40]}, {"full_name": "Function.invFun_surjective", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [470, 9], "def_end_pos": [470, 26]}]], "state_before": "case intro.mk.intro\nK R : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type u_1\ninst\u271d : DivisionRing K\n\u03b9 : Type v\nh\u03b9 : Infinite \u03b9\n\u03b9K : Type (max u v)\nbK : Basis \u03b9K K (\u03b9 \u2192 K)\ne : \u2115 \u21aa \u03b9\n\u22a2 Module.rank K (\u03b9 \u2192 K) = #(\u03b9 \u2192 K)", "state_after": "case intro.mk.intro\nK R : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type u_1\ninst\u271d : DivisionRing K\n\u03b9 : Type v\nh\u03b9 : Infinite \u03b9\n\u03b9K : Type (max u v)\nbK : Basis \u03b9K K (\u03b9 \u2192 K)\ne : \u2115 \u21aa \u03b9\nthis : lift.{max u v, u} (Module.rank K (\u2115 \u2192 K)) \u2264 lift.{u, max u v} (Module.rank K (\u03b9 \u2192 K))\n\u22a2 Module.rank K (\u03b9 \u2192 K) = #(\u03b9 \u2192 K)"}, {"tactic": "rw [lift_umax.{u,v}, lift_id'.{u,v}] at this", "annotated_tactic": ["rw [<a>lift_umax</a>.{u,v}, <a>lift_id'</a>.{u,v}] at this", [{"full_name": "Cardinal.lift_umax", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [202, 9], "def_end_pos": [202, 18]}, {"full_name": "Cardinal.lift_id'", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [218, 9], "def_end_pos": [218, 17]}]], "state_before": "case intro.mk.intro\nK R : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type u_1\ninst\u271d : DivisionRing K\n\u03b9 : Type v\nh\u03b9 : Infinite \u03b9\n\u03b9K : Type (max u v)\nbK : Basis \u03b9K K (\u03b9 \u2192 K)\ne : \u2115 \u21aa \u03b9\nthis : lift.{max u v, u} (Module.rank K (\u2115 \u2192 K)) \u2264 lift.{u, max u v} (Module.rank K (\u03b9 \u2192 K))\n\u22a2 Module.rank K (\u03b9 \u2192 K) = #(\u03b9 \u2192 K)", "state_after": "case intro.mk.intro\nK R : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type u_1\ninst\u271d : DivisionRing K\n\u03b9 : Type v\nh\u03b9 : Infinite \u03b9\n\u03b9K : Type (max u v)\nbK : Basis \u03b9K K (\u03b9 \u2192 K)\ne : \u2115 \u21aa \u03b9\nthis : lift.{v, u} (Module.rank K (\u2115 \u2192 K)) \u2264 Module.rank K (\u03b9 \u2192 K)\n\u22a2 Module.rank K (\u03b9 \u2192 K) = #(\u03b9 \u2192 K)"}, {"tactic": "have key := (lift_le.{v}.mpr <| max_aleph0_card_le_rank_fun_nat K).trans this", "annotated_tactic": ["have key := (<a>lift_le</a>.{v}.<a>mpr</a> <| <a>max_aleph0_card_le_rank_fun_nat</a> K).<a>trans</a> this", [{"full_name": "Cardinal.lift_le", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 16]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}, {"full_name": "max_aleph0_card_le_rank_fun_nat", "def_path": "Mathlib/LinearAlgebra/Dimension/DivisionRing.lean", "def_pos": [239, 9], "def_end_pos": [239, 40]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}]], "state_before": "case intro.mk.intro\nK R : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type u_1\ninst\u271d : DivisionRing K\n\u03b9 : Type v\nh\u03b9 : Infinite \u03b9\n\u03b9K : Type (max u v)\nbK : Basis \u03b9K K (\u03b9 \u2192 K)\ne : \u2115 \u21aa \u03b9\nthis : lift.{v, u} (Module.rank K (\u2115 \u2192 K)) \u2264 Module.rank K (\u03b9 \u2192 K)\n\u22a2 Module.rank K (\u03b9 \u2192 K) = #(\u03b9 \u2192 K)", "state_after": "case intro.mk.intro\nK R : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type u_1\ninst\u271d : DivisionRing K\n\u03b9 : Type v\nh\u03b9 : Infinite \u03b9\n\u03b9K : Type (max u v)\nbK : Basis \u03b9K K (\u03b9 \u2192 K)\ne : \u2115 \u21aa \u03b9\nthis : lift.{v, u} (Module.rank K (\u2115 \u2192 K)) \u2264 Module.rank K (\u03b9 \u2192 K)\nkey : lift.{v, u} (max \u2135\u2080 #K) \u2264 Module.rank K (\u03b9 \u2192 K)\n\u22a2 Module.rank K (\u03b9 \u2192 K) = #(\u03b9 \u2192 K)"}, {"tactic": "rw [lift_max, lift_aleph0, max_le_iff] at key", "annotated_tactic": ["rw [<a>lift_max</a>, <a>lift_aleph0</a>, <a>max_le_iff</a>] at key", [{"full_name": "Cardinal.lift_max", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1192, 9], "def_end_pos": [1192, 17]}, {"full_name": "Cardinal.lift_aleph0", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1274, 9], "def_end_pos": [1274, 20]}, {"full_name": "max_le_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [48, 9], "def_end_pos": [48, 19]}]], "state_before": "case intro.mk.intro\nK R : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type u_1\ninst\u271d : DivisionRing K\n\u03b9 : Type v\nh\u03b9 : Infinite \u03b9\n\u03b9K : Type (max u v)\nbK : Basis \u03b9K K (\u03b9 \u2192 K)\ne : \u2115 \u21aa \u03b9\nthis : lift.{v, u} (Module.rank K (\u2115 \u2192 K)) \u2264 Module.rank K (\u03b9 \u2192 K)\nkey : lift.{v, u} (max \u2135\u2080 #K) \u2264 Module.rank K (\u03b9 \u2192 K)\n\u22a2 Module.rank K (\u03b9 \u2192 K) = #(\u03b9 \u2192 K)", "state_after": "case intro.mk.intro\nK R : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type u_1\ninst\u271d : DivisionRing K\n\u03b9 : Type v\nh\u03b9 : Infinite \u03b9\n\u03b9K : Type (max u v)\nbK : Basis \u03b9K K (\u03b9 \u2192 K)\ne : \u2115 \u21aa \u03b9\nthis : lift.{v, u} (Module.rank K (\u2115 \u2192 K)) \u2264 Module.rank K (\u03b9 \u2192 K)\nkey : \u2135\u2080 \u2264 Module.rank K (\u03b9 \u2192 K) \u2227 lift.{v, u} #K \u2264 Module.rank K (\u03b9 \u2192 K)\n\u22a2 Module.rank K (\u03b9 \u2192 K) = #(\u03b9 \u2192 K)"}, {"tactic": "haveI : Infinite \u03b9K := by\n  rw [\u2190 aleph0_le_mk_iff, bK.mk_eq_rank'']; exact key.1", "annotated_tactic": ["haveI : <a>Infinite</a> \u03b9K := by\n    rw [\u2190 <a>aleph0_le_mk_iff</a>, bK.mk_eq_rank'']; exact key.1", [{"full_name": "Infinite", "def_path": "Mathlib/Data/Finite/Defs.lean", "def_pos": [117, 7], "def_end_pos": [117, 15]}, {"full_name": "Cardinal.aleph0_le_mk_iff", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1737, 7], "def_end_pos": [1737, 23]}]], "state_before": "case intro.mk.intro\nK R : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type u_1\ninst\u271d : DivisionRing K\n\u03b9 : Type v\nh\u03b9 : Infinite \u03b9\n\u03b9K : Type (max u v)\nbK : Basis \u03b9K K (\u03b9 \u2192 K)\ne : \u2115 \u21aa \u03b9\nthis : lift.{v, u} (Module.rank K (\u2115 \u2192 K)) \u2264 Module.rank K (\u03b9 \u2192 K)\nkey : \u2135\u2080 \u2264 Module.rank K (\u03b9 \u2192 K) \u2227 lift.{v, u} #K \u2264 Module.rank K (\u03b9 \u2192 K)\n\u22a2 Module.rank K (\u03b9 \u2192 K) = #(\u03b9 \u2192 K)", "state_after": "case intro.mk.intro\nK R : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type u_1\ninst\u271d : DivisionRing K\n\u03b9 : Type v\nh\u03b9 : Infinite \u03b9\n\u03b9K : Type (max u v)\nbK : Basis \u03b9K K (\u03b9 \u2192 K)\ne : \u2115 \u21aa \u03b9\nthis\u271d : lift.{v, u} (Module.rank K (\u2115 \u2192 K)) \u2264 Module.rank K (\u03b9 \u2192 K)\nkey : \u2135\u2080 \u2264 Module.rank K (\u03b9 \u2192 K) \u2227 lift.{v, u} #K \u2264 Module.rank K (\u03b9 \u2192 K)\nthis : Infinite \u03b9K\n\u22a2 Module.rank K (\u03b9 \u2192 K) = #(\u03b9 \u2192 K)"}, {"tactic": "rw [bK.repr.toEquiv.cardinal_eq, mk_finsupp_lift_of_infinite,\n    lift_umax.{u,v}, lift_id'.{u,v}, bK.mk_eq_rank'', eq_comm, max_eq_left]", "annotated_tactic": ["rw [bK.repr.toEquiv.cardinal_eq, <a>mk_finsupp_lift_of_infinite</a>,\n      <a>lift_umax</a>.{u,v}, <a>lift_id'</a>.{u,v}, bK.mk_eq_rank'', <a>eq_comm</a>, <a>max_eq_left</a>]", [{"full_name": "Cardinal.mk_finsupp_lift_of_infinite", "def_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "def_pos": [1206, 9], "def_end_pos": [1206, 36]}, {"full_name": "Cardinal.lift_umax", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [202, 9], "def_end_pos": [202, 18]}, {"full_name": "Cardinal.lift_id'", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [218, 9], "def_end_pos": [218, 17]}, {"full_name": "eq_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}, {"full_name": "max_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [133, 9], "def_end_pos": [133, 20]}]], "state_before": "case intro.mk.intro\nK R : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type u_1\ninst\u271d : DivisionRing K\n\u03b9 : Type v\nh\u03b9 : Infinite \u03b9\n\u03b9K : Type (max u v)\nbK : Basis \u03b9K K (\u03b9 \u2192 K)\ne : \u2115 \u21aa \u03b9\nthis\u271d : lift.{v, u} (Module.rank K (\u2115 \u2192 K)) \u2264 Module.rank K (\u03b9 \u2192 K)\nkey : \u2135\u2080 \u2264 Module.rank K (\u03b9 \u2192 K) \u2227 lift.{v, u} #K \u2264 Module.rank K (\u03b9 \u2192 K)\nthis : Infinite \u03b9K\n\u22a2 Module.rank K (\u03b9 \u2192 K) = #(\u03b9 \u2192 K)", "state_after": "case intro.mk.intro\nK R : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type u_1\ninst\u271d : DivisionRing K\n\u03b9 : Type v\nh\u03b9 : Infinite \u03b9\n\u03b9K : Type (max u v)\nbK : Basis \u03b9K K (\u03b9 \u2192 K)\ne : \u2115 \u21aa \u03b9\nthis\u271d : lift.{v, u} (Module.rank K (\u2115 \u2192 K)) \u2264 Module.rank K (\u03b9 \u2192 K)\nkey : \u2135\u2080 \u2264 Module.rank K (\u03b9 \u2192 K) \u2227 lift.{v, u} #K \u2264 Module.rank K (\u03b9 \u2192 K)\nthis : Infinite \u03b9K\n\u22a2 lift.{v, u} #K \u2264 Module.rank K (\u03b9 \u2192 K)"}, {"tactic": "exact key.2", "annotated_tactic": ["exact key.2", []], "state_before": "case intro.mk.intro\nK R : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type u_1\ninst\u271d : DivisionRing K\n\u03b9 : Type v\nh\u03b9 : Infinite \u03b9\n\u03b9K : Type (max u v)\nbK : Basis \u03b9K K (\u03b9 \u2192 K)\ne : \u2115 \u21aa \u03b9\nthis\u271d : lift.{v, u} (Module.rank K (\u2115 \u2192 K)) \u2264 Module.rank K (\u03b9 \u2192 K)\nkey : \u2135\u2080 \u2264 Module.rank K (\u03b9 \u2192 K) \u2227 lift.{v, u} #K \u2264 Module.rank K (\u03b9 \u2192 K)\nthis : Infinite \u03b9K\n\u22a2 lift.{v, u} #K \u2264 Module.rank K (\u03b9 \u2192 K)", "state_after": "no goals"}, {"tactic": "rw [\u2190 aleph0_le_mk_iff, bK.mk_eq_rank'']", "annotated_tactic": ["rw [\u2190 <a>aleph0_le_mk_iff</a>, bK.mk_eq_rank'']", [{"full_name": "Cardinal.aleph0_le_mk_iff", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1737, 7], "def_end_pos": [1737, 23]}]], "state_before": "K R : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type u_1\ninst\u271d : DivisionRing K\n\u03b9 : Type v\nh\u03b9 : Infinite \u03b9\n\u03b9K : Type (max u v)\nbK : Basis \u03b9K K (\u03b9 \u2192 K)\ne : \u2115 \u21aa \u03b9\nthis : lift.{v, u} (Module.rank K (\u2115 \u2192 K)) \u2264 Module.rank K (\u03b9 \u2192 K)\nkey : \u2135\u2080 \u2264 Module.rank K (\u03b9 \u2192 K) \u2227 lift.{v, u} #K \u2264 Module.rank K (\u03b9 \u2192 K)\n\u22a2 Infinite \u03b9K", "state_after": "K R : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type u_1\ninst\u271d : DivisionRing K\n\u03b9 : Type v\nh\u03b9 : Infinite \u03b9\n\u03b9K : Type (max u v)\nbK : Basis \u03b9K K (\u03b9 \u2192 K)\ne : \u2115 \u21aa \u03b9\nthis : lift.{v, u} (Module.rank K (\u2115 \u2192 K)) \u2264 Module.rank K (\u03b9 \u2192 K)\nkey : \u2135\u2080 \u2264 Module.rank K (\u03b9 \u2192 K) \u2227 lift.{v, u} #K \u2264 Module.rank K (\u03b9 \u2192 K)\n\u22a2 \u2135\u2080 \u2264 Module.rank K (\u03b9 \u2192 K)"}, {"tactic": "exact key.1", "annotated_tactic": ["exact key.1", []], "state_before": "K R : Type u\nV V\u2081 V\u2082 V\u2083 : Type v\nV' V'\u2081 : Type v'\nV'' : Type v''\n\u03b9\u271d : Type w\n\u03b9' : Type w'\n\u03b7 : Type u\u2081'\n\u03c6 : \u03b7 \u2192 Type u_1\ninst\u271d : DivisionRing K\n\u03b9 : Type v\nh\u03b9 : Infinite \u03b9\n\u03b9K : Type (max u v)\nbK : Basis \u03b9K K (\u03b9 \u2192 K)\ne : \u2115 \u21aa \u03b9\nthis : lift.{v, u} (Module.rank K (\u2115 \u2192 K)) \u2264 Module.rank K (\u03b9 \u2192 K)\nkey : \u2135\u2080 \u2264 Module.rank K (\u03b9 \u2192 K) \u2227 lift.{v, u} #K \u2264 Module.rank K (\u03b9 \u2192 K)\n\u22a2 \u2135\u2080 \u2264 Module.rank K (\u03b9 \u2192 K)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "full_name": "AEMeasurable.mul_const", "start": [126, 1], "end": [128, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "full_name": "Set.Finite.measurableSet_biInter", "start": [167, 1], "end": [169, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Lattice.lean", "full_name": "inf_le_iff", "start": [827, 1], "end": [828, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Category/Grp/EpiMono.lean", "full_name": "Grp.SurjectiveOfEpiAuxs.mul_smul", "start": [131, 1], "end": [136, 13], "traced_tactics": [{"tactic": "change fromCoset _ = fromCoset _", "annotated_tactic": ["change <a>fromCoset</a> _ = <a>fromCoset</a> _", [{"full_name": "Grp.SurjectiveOfEpiAuxs.XWithInfinity.fromCoset", "def_path": "Mathlib/Algebra/Category/Grp/EpiMono.lean", "def_pos": [109, 5], "def_end_pos": [109, 14]}, {"full_name": "Grp.SurjectiveOfEpiAuxs.XWithInfinity.fromCoset", "def_path": "Mathlib/Algebra/Category/Grp/EpiMono.lean", "def_pos": [109, 5], "def_end_pos": [109, 14]}]], "state_before": "A B : Grp\nf : A \u27f6 B\nb b' : \u2191B\nx : X'\ny : \u2191(Set.range fun x => x \u2022 \u2191(MonoidHom.range f))\n\u22a2 (b * b') \u2022 fromCoset y = b \u2022 b' \u2022 fromCoset y", "state_after": "A B : Grp\nf : A \u27f6 B\nb b' : \u2191B\nx : X'\ny : \u2191(Set.range fun x => x \u2022 \u2191(MonoidHom.range f))\n\u22a2 fromCoset \u27e8(b * b') \u2022 \u2191y, \u22ef\u27e9 = fromCoset \u27e8b \u2022 \u2191\u27e8b' \u2022 \u2191y, \u22ef\u27e9, \u22ef\u27e9"}, {"tactic": "simp only [leftCoset_assoc]", "annotated_tactic": ["simp only [<a>leftCoset_assoc</a>]", [{"full_name": "leftCoset_assoc", "def_path": "Mathlib/GroupTheory/Coset.lean", "def_pos": [105, 9], "def_end_pos": [105, 24]}]], "state_before": "A B : Grp\nf : A \u27f6 B\nb b' : \u2191B\nx : X'\ny : \u2191(Set.range fun x => x \u2022 \u2191(MonoidHom.range f))\n\u22a2 fromCoset \u27e8(b * b') \u2022 \u2191y, \u22ef\u27e9 = fromCoset \u27e8b \u2022 \u2191\u27e8b' \u2022 \u2191y, \u22ef\u27e9, \u22ef\u27e9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Analytic/Basic.lean", "full_name": "FormalMultilinearSeries.nnnorm_mul_pow_le_of_lt_radius", "start": [262, 1], "end": [265, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/Basic.lean", "full_name": "Set.not_mem_Ico_of_ge", "start": [757, 1], "end": [757, 96], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Algebraic.lean", "full_name": "isAlgebraic_algebraMap", "start": [109, 1], "end": [110, 77], "traced_tactics": [{"tactic": "rw [_root_.map_sub, aeval_X, aeval_C, sub_self]", "annotated_tactic": ["rw [<a>_root_.map_sub</a>, <a>aeval_X</a>, <a>aeval_C</a>, <a>sub_self</a>]", [{"full_name": "map_sub", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [461, 3], "def_end_pos": [461, 14]}, {"full_name": "Polynomial.aeval_X", "def_path": "Mathlib/Algebra/Polynomial/AlgebraMap.lean", "def_pos": [209, 9], "def_end_pos": [209, 16]}, {"full_name": "Polynomial.aeval_C", "def_path": "Mathlib/Algebra/Polynomial/AlgebraMap.lean", "def_pos": [215, 9], "def_end_pos": [215, 16]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1003, 30], "def_end_pos": [1003, 38]}]], "state_before": "R : Type u\nS : Type u_1\nA : Type v\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : CommRing S\ninst\u271d\u2075 : Ring A\ninst\u271d\u2074 : Algebra R A\ninst\u271d\u00b3 : Algebra R S\ninst\u271d\u00b2 : Algebra S A\ninst\u271d\u00b9 : IsScalarTower R S A\ninst\u271d : Nontrivial R\nx : R\n\u22a2 (aeval ((algebraMap R A) x)) (X - C x) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "List.toFinset_nonempty_iff", "start": [3326, 1], "end": [3327, 38], "traced_tactics": [{"tactic": "simp [Finset.nonempty_iff_ne_empty]", "annotated_tactic": ["simp [<a>Finset.nonempty_iff_ne_empty</a>]", [{"full_name": "Finset.nonempty_iff_ne_empty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [606, 9], "def_end_pos": [606, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\nl\u271d l' : List \u03b1\na : \u03b1\nl : List \u03b1\n\u22a2 l.toFinset.Nonempty \u2194 l \u2260 []", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.cauchySeq_Lp_iff_cauchySeq_\u2112p", "start": [1471, 1], "end": [1477, 23], "traced_tactics": [{"tactic": "simp_rw [cauchySeq_iff_tendsto_dist_atTop_0, dist_def]", "annotated_tactic": ["simp_rw [<a>cauchySeq_iff_tendsto_dist_atTop_0</a>, <a>dist_def</a>]", [{"full_name": "cauchySeq_iff_tendsto_dist_atTop_0", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [1393, 9], "def_end_pos": [1393, 43]}, {"full_name": "MeasureTheory.Lp.dist_def", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [296, 9], "def_end_pos": [296, 17]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : SemilatticeSup \u03b9\nhp : Fact (1 \u2264 p)\nf : \u03b9 \u2192 \u21a5(Lp E p \u03bc)\n\u22a2 CauchySeq f \u2194 Tendsto (fun n => snorm (\u2191\u2191(f n.1) - \u2191\u2191(f n.2)) p \u03bc) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : SemilatticeSup \u03b9\nhp : Fact (1 \u2264 p)\nf : \u03b9 \u2192 \u21a5(Lp E p \u03bc)\n\u22a2 Tendsto (fun n => (snorm (\u2191\u2191(f n.1) - \u2191\u2191(f n.2)) p \u03bc).toReal) atTop (\ud835\udcdd 0) \u2194\n    Tendsto (fun n => snorm (\u2191\u2191(f n.1) - \u2191\u2191(f n.2)) p \u03bc) atTop (\ud835\udcdd 0)"}, {"tactic": "rw [\u2190 ENNReal.zero_toReal, ENNReal.tendsto_toReal_iff (fun n => ?_) ENNReal.zero_ne_top]", "annotated_tactic": ["rw [\u2190 <a>ENNReal.zero_toReal</a>, <a>ENNReal.tendsto_toReal_iff</a> (fun n => ?_) <a>ENNReal.zero_ne_top</a>]", [{"full_name": "ENNReal.zero_toReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [268, 17], "def_end_pos": [268, 28]}, {"full_name": "ENNReal.tendsto_toReal_iff", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1100, 9], "def_end_pos": [1100, 27]}, {"full_name": "ENNReal.zero_ne_top", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [357, 17], "def_end_pos": [357, 28]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : SemilatticeSup \u03b9\nhp : Fact (1 \u2264 p)\nf : \u03b9 \u2192 \u21a5(Lp E p \u03bc)\n\u22a2 Tendsto (fun n => (snorm (\u2191\u2191(f n.1) - \u2191\u2191(f n.2)) p \u03bc).toReal) atTop (\ud835\udcdd 0) \u2194\n    Tendsto (fun n => snorm (\u2191\u2191(f n.1) - \u2191\u2191(f n.2)) p \u03bc) atTop (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : SemilatticeSup \u03b9\nhp : Fact (1 \u2264 p)\nf : \u03b9 \u2192 \u21a5(Lp E p \u03bc)\nn : \u03b9 \u00d7 \u03b9\n\u22a2 snorm (\u2191\u2191(f n.1) - \u2191\u2191(f n.2)) p \u03bc \u2260 \u22a4"}, {"tactic": "rw [snorm_congr_ae (Lp.coeFn_sub _ _).symm]", "annotated_tactic": ["rw [<a>snorm_congr_ae</a> (<a>Lp.coeFn_sub</a> _ _).<a>symm</a>]", [{"full_name": "MeasureTheory.snorm_congr_ae", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [553, 9], "def_end_pos": [553, 23]}, {"full_name": "MeasureTheory.Lp.coeFn_sub", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [242, 9], "def_end_pos": [242, 18]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1530, 9], "def_end_pos": [1530, 26]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : SemilatticeSup \u03b9\nhp : Fact (1 \u2264 p)\nf : \u03b9 \u2192 \u21a5(Lp E p \u03bc)\nn : \u03b9 \u00d7 \u03b9\n\u22a2 snorm (\u2191\u2191(f n.1) - \u2191\u2191(f n.2)) p \u03bc \u2260 \u22a4", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : SemilatticeSup \u03b9\nhp : Fact (1 \u2264 p)\nf : \u03b9 \u2192 \u21a5(Lp E p \u03bc)\nn : \u03b9 \u00d7 \u03b9\n\u22a2 snorm (\u2191\u2191(f n.1 - f n.2)) p \u03bc \u2260 \u22a4"}, {"tactic": "exact snorm_ne_top _", "annotated_tactic": ["exact <a>snorm_ne_top</a> _", [{"full_name": "MeasureTheory.Lp.snorm_ne_top", "def_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "def_pos": [208, 9], "def_end_pos": [208, 21]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm m0 : MeasurableSpace \u03b1\np : \u211d\u22650\u221e\nq : \u211d\n\u03bc \u03bd : Measure \u03b1\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedAddCommGroup G\n\u03b9 : Type u_5\ninst\u271d\u00b9 : Nonempty \u03b9\ninst\u271d : SemilatticeSup \u03b9\nhp : Fact (1 \u2264 p)\nf : \u03b9 \u2192 \u21a5(Lp E p \u03bc)\nn : \u03b9 \u00d7 \u03b9\n\u22a2 snorm (\u2191\u2191(f n.1 - f n.2)) p \u03bc \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Quasiconvex.lean", "full_name": "MonotoneOn.quasiconvexOn", "start": [182, 1], "end": [183, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean", "full_name": "PiTensorProduct.injectiveSeminorm_le_projectiveSeminorm", "start": [204, 1], "end": [219, 65], "traced_tactics": [{"tactic": "rw [injectiveSeminorm]", "annotated_tactic": ["rw [<a>injectiveSeminorm</a>]", [{"full_name": "PiTensorProduct.injectiveSeminorm", "def_path": "Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean", "def_pos": [126, 31], "def_end_pos": [126, 48]}]], "state_before": "\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\n\u22a2 injectiveSeminorm \u2264 projectiveSeminorm", "state_after": "\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\n\u22a2 sSup\n      {p |\n        \u2203 G x x_1,\n          p = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)} \u2264\n    projectiveSeminorm"}, {"tactic": "refine csSup_le ?_ ?_", "annotated_tactic": ["refine <a>csSup_le</a> ?_ ?_", [{"full_name": "csSup_le", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [463, 9], "def_end_pos": [463, 17]}]], "state_before": "\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\n\u22a2 sSup\n      {p |\n        \u2203 G x x_1,\n          p = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)} \u2264\n    projectiveSeminorm", "state_after": "case refine_1\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\n\u22a2 {p |\n      \u2203 G x x_1,\n        p = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)}.Nonempty\n\ncase refine_2\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\n\u22a2 \u2200\n    b \u2208\n      {p |\n        \u2203 G x x_1,\n          p = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)},\n    b \u2264 projectiveSeminorm"}, {"tactic": "existsi 0", "annotated_tactic": ["existsi 0", []], "state_before": "case refine_1\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\n\u22a2 {p |\n      \u2203 G x x_1,\n        p = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)}.Nonempty", "state_after": "case refine_1\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\n\u22a2 0 \u2208\n    {p |\n      \u2203 G x x_1, p = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)}"}, {"tactic": "simp only [Set.mem_setOf_eq]", "annotated_tactic": ["simp only [<a>Set.mem_setOf_eq</a>]", [{"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [79, 29], "def_end_pos": [79, 41]}]], "state_before": "case refine_1\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\n\u22a2 0 \u2208\n    {p |\n      \u2203 G x x_1, p = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)}", "state_after": "case refine_1\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\n\u22a2 \u2203 G x x_1, 0 = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)"}, {"tactic": "existsi PUnit, inferInstance, inferInstance", "annotated_tactic": ["existsi <a>PUnit</a>, <a>inferInstance</a>, <a>inferInstance</a>", [{"full_name": "PUnit", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [122, 11], "def_end_pos": [122, 16]}, {"full_name": "inferInstance", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [99, 8], "def_end_pos": [99, 21]}, {"full_name": "inferInstance", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [99, 8], "def_end_pos": [99, 21]}]], "state_before": "case refine_1\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\n\u22a2 \u2203 G x x_1, 0 = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)", "state_after": "case refine_1\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\n\u22a2 0 =\n    (normSeminorm \ud835\udd5c\n          (ContinuousMultilinearMap \ud835\udd5c E PUnit.{max (max (uE + 1) (u\u03b9 + 1)) (u\ud835\udd5c + 1)} \u2192L[\ud835\udd5c]\n            PUnit.{max (max (uE + 1) (u\u03b9 + 1)) (u\ud835\udd5c + 1)})).comp\n      (toDualContinuousMultilinearMap PUnit.{max (max (uE + 1) (u\u03b9 + 1)) (u\ud835\udd5c + 1)})"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case refine_1\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\n\u22a2 0 =\n    (normSeminorm \ud835\udd5c\n          (ContinuousMultilinearMap \ud835\udd5c E PUnit.{max (max (uE + 1) (u\u03b9 + 1)) (u\ud835\udd5c + 1)} \u2192L[\ud835\udd5c]\n            PUnit.{max (max (uE + 1) (u\u03b9 + 1)) (u\ud835\udd5c + 1)})).comp\n      (toDualContinuousMultilinearMap PUnit.{max (max (uE + 1) (u\u03b9 + 1)) (u\ud835\udd5c + 1)})", "state_after": "case refine_1.h\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nx : \u2a02[\ud835\udd5c] (i : \u03b9), E i\n\u22a2 0 x =\n    ((normSeminorm \ud835\udd5c\n            (ContinuousMultilinearMap \ud835\udd5c E PUnit.{max (max (uE + 1) (u\u03b9 + 1)) (u\ud835\udd5c + 1)} \u2192L[\ud835\udd5c]\n              PUnit.{max (max (uE + 1) (u\u03b9 + 1)) (u\ud835\udd5c + 1)})).comp\n        (toDualContinuousMultilinearMap PUnit.{max (max (uE + 1) (u\u03b9 + 1)) (u\ud835\udd5c + 1)}))\n      x"}, {"tactic": "simp only [Seminorm.zero_apply, Seminorm.comp_apply, coe_normSeminorm]", "annotated_tactic": ["simp only [<a>Seminorm.zero_apply</a>, <a>Seminorm.comp_apply</a>, <a>coe_normSeminorm</a>]", [{"full_name": "Seminorm.zero_apply", "def_path": "Mathlib/Analysis/Seminorm.lean", "def_pos": [145, 9], "def_end_pos": [145, 19]}, {"full_name": "Seminorm.comp_apply", "def_path": "Mathlib/Analysis/Seminorm.lean", "def_pos": [312, 9], "def_end_pos": [312, 19]}, {"full_name": "coe_normSeminorm", "def_path": "Mathlib/Analysis/Seminorm.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 25]}]], "state_before": "case refine_1.h\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nx : \u2a02[\ud835\udd5c] (i : \u03b9), E i\n\u22a2 0 x =\n    ((normSeminorm \ud835\udd5c\n            (ContinuousMultilinearMap \ud835\udd5c E PUnit.{max (max (uE + 1) (u\u03b9 + 1)) (u\ud835\udd5c + 1)} \u2192L[\ud835\udd5c]\n              PUnit.{max (max (uE + 1) (u\u03b9 + 1)) (u\ud835\udd5c + 1)})).comp\n        (toDualContinuousMultilinearMap PUnit.{max (max (uE + 1) (u\u03b9 + 1)) (u\ud835\udd5c + 1)}))\n      x", "state_after": "case refine_1.h\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nx : \u2a02[\ud835\udd5c] (i : \u03b9), E i\n\u22a2 0 = \u2016(toDualContinuousMultilinearMap PUnit.{max (max (uE + 1) (u\u03b9 + 1)) (u\ud835\udd5c + 1)}) x\u2016"}, {"tactic": "have heq : toDualContinuousMultilinearMap PUnit x = 0 := by ext _", "annotated_tactic": ["have heq : <a>toDualContinuousMultilinearMap</a> <a>PUnit</a> x = 0 := by ext _", [{"full_name": "PiTensorProduct.toDualContinuousMultilinearMap", "def_path": "Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean", "def_pos": [94, 19], "def_end_pos": [94, 49]}, {"full_name": "PUnit", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [122, 11], "def_end_pos": [122, 16]}]], "state_before": "case refine_1.h\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nx : \u2a02[\ud835\udd5c] (i : \u03b9), E i\n\u22a2 0 = \u2016(toDualContinuousMultilinearMap PUnit.{max (max (uE + 1) (u\u03b9 + 1)) (u\ud835\udd5c + 1)}) x\u2016", "state_after": "case refine_1.h\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nx : \u2a02[\ud835\udd5c] (i : \u03b9), E i\nheq : (toDualContinuousMultilinearMap PUnit.{?u.263108 + 1}) x = 0\n\u22a2 0 = \u2016(toDualContinuousMultilinearMap PUnit.{max (max (uE + 1) (u\u03b9 + 1)) (u\ud835\udd5c + 1)}) x\u2016"}, {"tactic": "rw [heq, norm_zero]", "annotated_tactic": ["rw [heq, <a>norm_zero</a>]", [{"full_name": "norm_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [507, 30], "def_end_pos": [507, 39]}]], "state_before": "case refine_1.h\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nx : \u2a02[\ud835\udd5c] (i : \u03b9), E i\nheq : (toDualContinuousMultilinearMap PUnit.{?u.263108 + 1}) x = 0\n\u22a2 0 = \u2016(toDualContinuousMultilinearMap PUnit.{max (max (uE + 1) (u\u03b9 + 1)) (u\ud835\udd5c + 1)}) x\u2016", "state_after": "no goals"}, {"tactic": "ext _", "annotated_tactic": ["ext _", []], "state_before": "\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\nx : \u2a02[\ud835\udd5c] (i : \u03b9), E i\n\u22a2 (toDualContinuousMultilinearMap PUnit.{?u.263108 + 1}) x = 0", "state_after": "no goals"}, {"tactic": "intro p hp", "annotated_tactic": ["intro p hp", []], "state_before": "case refine_2\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\n\u22a2 \u2200\n    b \u2208\n      {p |\n        \u2203 G x x_1,\n          p = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)},\n    b \u2264 projectiveSeminorm", "state_after": "case refine_2\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Seminorm \ud835\udd5c (\u2a02[\ud835\udd5c] (i : \u03b9), E i)\nhp :\n  p \u2208\n    {p |\n      \u2203 G x x_1, p = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)}\n\u22a2 p \u2264 projectiveSeminorm"}, {"tactic": "simp only [Set.mem_setOf_eq] at hp", "annotated_tactic": ["simp only [<a>Set.mem_setOf_eq</a>] at hp", [{"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [79, 29], "def_end_pos": [79, 41]}]], "state_before": "case refine_2\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Seminorm \ud835\udd5c (\u2a02[\ud835\udd5c] (i : \u03b9), E i)\nhp :\n  p \u2208\n    {p |\n      \u2203 G x x_1, p = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)}\n\u22a2 p \u2264 projectiveSeminorm", "state_after": "case refine_2\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Seminorm \ud835\udd5c (\u2a02[\ud835\udd5c] (i : \u03b9), E i)\nhp : \u2203 G x x_1, p = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)\n\u22a2 p \u2264 projectiveSeminorm"}, {"tactic": "obtain \u27e8G, _, _, h\u27e9 := hp", "annotated_tactic": ["obtain \u27e8G, _, _, h\u27e9 := hp", []], "state_before": "case refine_2\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Seminorm \ud835\udd5c (\u2a02[\ud835\udd5c] (i : \u03b9), E i)\nhp : \u2203 G x x_1, p = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)\n\u22a2 p \u2264 projectiveSeminorm", "state_after": "case refine_2.intro.intro.intro\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Seminorm \ud835\udd5c (\u2a02[\ud835\udd5c] (i : \u03b9), E i)\nG : Type (max u\u03b9 u\ud835\udd5c uE)\nw\u271d\u00b9 : SeminormedAddCommGroup G\nw\u271d : NormedSpace \ud835\udd5c G\nh : p = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)\n\u22a2 p \u2264 projectiveSeminorm"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "case refine_2.intro.intro.intro\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Seminorm \ud835\udd5c (\u2a02[\ud835\udd5c] (i : \u03b9), E i)\nG : Type (max u\u03b9 u\ud835\udd5c uE)\nw\u271d\u00b9 : SeminormedAddCommGroup G\nw\u271d : NormedSpace \ud835\udd5c G\nh : p = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)\n\u22a2 p \u2264 projectiveSeminorm", "state_after": "case refine_2.intro.intro.intro\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Seminorm \ud835\udd5c (\u2a02[\ud835\udd5c] (i : \u03b9), E i)\nG : Type (max u\u03b9 u\ud835\udd5c uE)\nw\u271d\u00b9 : SeminormedAddCommGroup G\nw\u271d : NormedSpace \ud835\udd5c G\nh : p = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)\n\u22a2 (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G) \u2264 projectiveSeminorm"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "case refine_2.intro.intro.intro\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Seminorm \ud835\udd5c (\u2a02[\ud835\udd5c] (i : \u03b9), E i)\nG : Type (max u\u03b9 u\ud835\udd5c uE)\nw\u271d\u00b9 : SeminormedAddCommGroup G\nw\u271d : NormedSpace \ud835\udd5c G\nh : p = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)\n\u22a2 (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G) \u2264 projectiveSeminorm", "state_after": "case refine_2.intro.intro.intro\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Seminorm \ud835\udd5c (\u2a02[\ud835\udd5c] (i : \u03b9), E i)\nG : Type (max u\u03b9 u\ud835\udd5c uE)\nw\u271d\u00b9 : SeminormedAddCommGroup G\nw\u271d : NormedSpace \ud835\udd5c G\nh : p = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)\nx : \u2a02[\ud835\udd5c] (i : \u03b9), E i\n\u22a2 (fun f => \u21d1f) ((normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)) x \u2264\n    (fun f => \u21d1f) projectiveSeminorm x"}, {"tactic": "simp only [Seminorm.comp_apply, coe_normSeminorm]", "annotated_tactic": ["simp only [<a>Seminorm.comp_apply</a>, <a>coe_normSeminorm</a>]", [{"full_name": "Seminorm.comp_apply", "def_path": "Mathlib/Analysis/Seminorm.lean", "def_pos": [312, 9], "def_end_pos": [312, 19]}, {"full_name": "coe_normSeminorm", "def_path": "Mathlib/Analysis/Seminorm.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 25]}]], "state_before": "case refine_2.intro.intro.intro\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Seminorm \ud835\udd5c (\u2a02[\ud835\udd5c] (i : \u03b9), E i)\nG : Type (max u\u03b9 u\ud835\udd5c uE)\nw\u271d\u00b9 : SeminormedAddCommGroup G\nw\u271d : NormedSpace \ud835\udd5c G\nh : p = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)\nx : \u2a02[\ud835\udd5c] (i : \u03b9), E i\n\u22a2 (fun f => \u21d1f) ((normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)) x \u2264\n    (fun f => \u21d1f) projectiveSeminorm x", "state_after": "case refine_2.intro.intro.intro\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Seminorm \ud835\udd5c (\u2a02[\ud835\udd5c] (i : \u03b9), E i)\nG : Type (max u\u03b9 u\ud835\udd5c uE)\nw\u271d\u00b9 : SeminormedAddCommGroup G\nw\u271d : NormedSpace \ud835\udd5c G\nh : p = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)\nx : \u2a02[\ud835\udd5c] (i : \u03b9), E i\n\u22a2 \u2016(toDualContinuousMultilinearMap G) x\u2016 \u2264 projectiveSeminorm x"}, {"tactic": "exact toDualContinuousMultilinearMap_le_projectiveSeminorm _", "annotated_tactic": ["exact <a>toDualContinuousMultilinearMap_le_projectiveSeminorm</a> _", [{"full_name": "PiTensorProduct.toDualContinuousMultilinearMap_le_projectiveSeminorm", "def_path": "Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean", "def_pos": [116, 9], "def_end_pos": [116, 61]}]], "state_before": "case refine_2.intro.intro.intro\n\u03b9 : Type u\u03b9\ninst\u271d\u2075 : Fintype \u03b9\n\ud835\udd5c : Type u\ud835\udd5c\ninst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c\nE : \u03b9 \u2192 Type uE\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (E i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i)\nF : Type uF\ninst\u271d\u00b9 : SeminormedAddCommGroup F\ninst\u271d : NormedSpace \ud835\udd5c F\np : Seminorm \ud835\udd5c (\u2a02[\ud835\udd5c] (i : \u03b9), E i)\nG : Type (max u\u03b9 u\ud835\udd5c uE)\nw\u271d\u00b9 : SeminormedAddCommGroup G\nw\u271d : NormedSpace \ud835\udd5c G\nh : p = (normSeminorm \ud835\udd5c (ContinuousMultilinearMap \ud835\udd5c E G \u2192L[\ud835\udd5c] G)).comp (toDualContinuousMultilinearMap G)\nx : \u2a02[\ud835\udd5c] (i : \u03b9), E i\n\u22a2 \u2016(toDualContinuousMultilinearMap G) x\u2016 \u2264 projectiveSeminorm x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/BilinearForm/Properties.lean", "full_name": "LinearMap.BilinForm.comp_symmCompOfNondegenerate_apply", "start": [521, 1], "end": [526, 52], "traced_tactics": [{"tactic": "erw [symmCompOfNondegenerate]", "annotated_tactic": ["erw [<a>symmCompOfNondegenerate</a>]", [{"full_name": "LinearMap.BilinForm.symmCompOfNondegenerate", "def_path": "Mathlib/LinearAlgebra/BilinearForm/Properties.lean", "def_pos": [516, 19], "def_end_pos": [516, 42]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2075 : CommSemiring R\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b2 : CommRing R\u2081\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module K V\nM'\u271d : Type u_7\nM'' : Type u_8\ninst\u271d\u2076 : AddCommMonoid M'\u271d\ninst\u271d\u2075 : AddCommMonoid M''\ninst\u271d\u2074 : Module R M'\u271d\ninst\u271d\u00b3 : Module R M''\nB : BilinForm R M\nB\u2081\u271d : BilinForm R\u2081 M\u2081\nM' : Type u_9\ninst\u271d\u00b2 : AddCommMonoid M'\ninst\u271d\u00b9 : Module R M'\ninst\u271d : FiniteDimensional K V\nB\u2081 B\u2082 : BilinForm K V\nb\u2082 : B\u2082.Nondegenerate\nv : V\n\u22a2 B\u2082 ((B\u2081.symmCompOfNondegenerate B\u2082 b\u2082) v) = B\u2081 v", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2075 : CommSemiring R\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b2 : CommRing R\u2081\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module K V\nM'\u271d : Type u_7\nM'' : Type u_8\ninst\u271d\u2076 : AddCommMonoid M'\u271d\ninst\u271d\u2075 : AddCommMonoid M''\ninst\u271d\u2074 : Module R M'\u271d\ninst\u271d\u00b3 : Module R M''\nB : BilinForm R M\nB\u2081\u271d : BilinForm R\u2081 M\u2081\nM' : Type u_9\ninst\u271d\u00b2 : AddCommMonoid M'\ninst\u271d\u00b9 : Module R M'\ninst\u271d : FiniteDimensional K V\nB\u2081 B\u2082 : BilinForm K V\nb\u2082 : B\u2082.Nondegenerate\nv : V\n\u22a2 B\u2082 ((\u2191(B\u2082.toDual b\u2082).symm \u2218\u2097 B\u2081) v) = B\u2081 v"}, {"tactic": "simp only [coe_comp, LinearEquiv.coe_coe, Function.comp_apply, DFunLike.coe_fn_eq]", "annotated_tactic": ["simp only [<a>coe_comp</a>, <a>LinearEquiv.coe_coe</a>, <a>Function.comp_apply</a>, <a>DFunLike.coe_fn_eq</a>]", [{"full_name": "LinearMap.coe_comp", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [557, 9], "def_end_pos": [557, 17]}, {"full_name": "LinearEquiv.coe_coe", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [222, 9], "def_end_pos": [222, 16]}, {"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}, {"full_name": "DFunLike.coe_fn_eq", "def_path": "Mathlib/Data/FunLike/Basic.lean", "def_pos": [189, 9], "def_end_pos": [189, 18]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2075 : CommSemiring R\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b2 : CommRing R\u2081\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module K V\nM'\u271d : Type u_7\nM'' : Type u_8\ninst\u271d\u2076 : AddCommMonoid M'\u271d\ninst\u271d\u2075 : AddCommMonoid M''\ninst\u271d\u2074 : Module R M'\u271d\ninst\u271d\u00b3 : Module R M''\nB : BilinForm R M\nB\u2081\u271d : BilinForm R\u2081 M\u2081\nM' : Type u_9\ninst\u271d\u00b2 : AddCommMonoid M'\ninst\u271d\u00b9 : Module R M'\ninst\u271d : FiniteDimensional K V\nB\u2081 B\u2082 : BilinForm K V\nb\u2082 : B\u2082.Nondegenerate\nv : V\n\u22a2 B\u2082 ((\u2191(B\u2082.toDual b\u2082).symm \u2218\u2097 B\u2081) v) = B\u2081 v", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2075 : CommSemiring R\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b2 : CommRing R\u2081\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module K V\nM'\u271d : Type u_7\nM'' : Type u_8\ninst\u271d\u2076 : AddCommMonoid M'\u271d\ninst\u271d\u2075 : AddCommMonoid M''\ninst\u271d\u2074 : Module R M'\u271d\ninst\u271d\u00b3 : Module R M''\nB : BilinForm R M\nB\u2081\u271d : BilinForm R\u2081 M\u2081\nM' : Type u_9\ninst\u271d\u00b2 : AddCommMonoid M'\ninst\u271d\u00b9 : Module R M'\ninst\u271d : FiniteDimensional K V\nB\u2081 B\u2082 : BilinForm K V\nb\u2082 : B\u2082.Nondegenerate\nv : V\n\u22a2 B\u2082 ((B\u2082.toDual b\u2082).symm (B\u2081 v)) = B\u2081 v"}, {"tactic": "erw [LinearEquiv.apply_symm_apply (B\u2082.toDual b\u2082)]", "annotated_tactic": ["erw [<a>LinearEquiv.apply_symm_apply</a> (B\u2082.toDual b\u2082)]", [{"full_name": "LinearEquiv.apply_symm_apply", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [376, 9], "def_end_pos": [376, 25]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9\u2075 : CommSemiring R\ninst\u271d\u00b9\u2074 : AddCommMonoid M\ninst\u271d\u00b9\u00b3 : Module R M\nR\u2081 : Type u_3\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b2 : CommRing R\u2081\ninst\u271d\u00b9\u00b9 : AddCommGroup M\u2081\ninst\u271d\u00b9\u2070 : Module R\u2081 M\u2081\nV : Type u_5\nK : Type u_6\ninst\u271d\u2079 : Field K\ninst\u271d\u2078 : AddCommGroup V\ninst\u271d\u2077 : Module K V\nM'\u271d : Type u_7\nM'' : Type u_8\ninst\u271d\u2076 : AddCommMonoid M'\u271d\ninst\u271d\u2075 : AddCommMonoid M''\ninst\u271d\u2074 : Module R M'\u271d\ninst\u271d\u00b3 : Module R M''\nB : BilinForm R M\nB\u2081\u271d : BilinForm R\u2081 M\u2081\nM' : Type u_9\ninst\u271d\u00b2 : AddCommMonoid M'\ninst\u271d\u00b9 : Module R M'\ninst\u271d : FiniteDimensional K V\nB\u2081 B\u2082 : BilinForm K V\nb\u2082 : B\u2082.Nondegenerate\nv : V\n\u22a2 B\u2082 ((B\u2082.toDual b\u2082).symm (B\u2081 v)) = B\u2081 v", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Finiteness.lean", "full_name": "GroupFG.iff_add_fg", "start": [330, 1], "end": [331, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean", "full_name": "fderiv_csinh", "start": [547, 1], "end": [549, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Equiv.lean", "full_name": "LinearEquiv.funCongrLeft_symm", "start": [1357, 1], "end": [1358, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/Disjoint.lean", "full_name": "IsGLB.biUnion_Ici_eq_Ioi", "start": [220, 1], "end": [226, 27], "traced_tactics": [{"tactic": "refine (iUnion\u2082_subset fun x hx => ?_).antisymm fun x hx => ?_", "annotated_tactic": ["refine (<a>iUnion\u2082_subset</a> fun x hx => ?_).<a>antisymm</a> fun x hx => ?_", [{"full_name": "Set.iUnion\u2082_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [242, 9], "def_end_pos": [242, 23]}, {"full_name": "HasSubset.Subset.antisymm", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [658, 7], "def_end_pos": [658, 32]}]], "state_before": "\u03b9 : Sort u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\na : \u03b1\nf : \u03b9 \u2192 \u03b1\na_glb : IsGLB s a\na_not_mem : a \u2209 s\n\u22a2 \u22c3 x \u2208 s, Ici x = Ioi a", "state_after": "case refine_1\n\u03b9 : Sort u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\na : \u03b1\nf : \u03b9 \u2192 \u03b1\na_glb : IsGLB s a\na_not_mem : a \u2209 s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 Ici x \u2286 Ioi a\n\ncase refine_2\n\u03b9 : Sort u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\na : \u03b1\nf : \u03b9 \u2192 \u03b1\na_glb : IsGLB s a\na_not_mem : a \u2209 s\nx : \u03b1\nhx : x \u2208 Ioi a\n\u22a2 x \u2208 \u22c3 i \u2208 s, Ici i"}, {"tactic": "exact Ici_subset_Ioi.mpr (lt_of_le_of_ne (a_glb.1 hx) fun h => (h \u25b8 a_not_mem) hx)", "annotated_tactic": ["exact Ici_subset_Ioi.mpr (<a>lt_of_le_of_ne</a> (a_glb.1 hx) fun h => (h \u25b8 a_not_mem) hx)", [{"full_name": "lt_of_le_of_ne", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [198, 9], "def_end_pos": [198, 23]}]], "state_before": "case refine_1\n\u03b9 : Sort u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\na : \u03b1\nf : \u03b9 \u2192 \u03b1\na_glb : IsGLB s a\na_not_mem : a \u2209 s\nx : \u03b1\nhx : x \u2208 s\n\u22a2 Ici x \u2286 Ioi a", "state_after": "no goals"}, {"tactic": "rcases a_glb.exists_between hx with \u27e8y, hys, _, hyx\u27e9", "annotated_tactic": ["rcases a_glb.exists_between hx with \u27e8y, hys, _, hyx\u27e9", []], "state_before": "case refine_2\n\u03b9 : Sort u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\na : \u03b1\nf : \u03b9 \u2192 \u03b1\na_glb : IsGLB s a\na_not_mem : a \u2209 s\nx : \u03b1\nhx : x \u2208 Ioi a\n\u22a2 x \u2208 \u22c3 i \u2208 s, Ici i", "state_after": "case refine_2.intro.intro.intro\n\u03b9 : Sort u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\na : \u03b1\nf : \u03b9 \u2192 \u03b1\na_glb : IsGLB s a\na_not_mem : a \u2209 s\nx : \u03b1\nhx : x \u2208 Ioi a\ny : \u03b1\nhys : y \u2208 s\nleft\u271d : a \u2264 y\nhyx : y < x\n\u22a2 x \u2208 \u22c3 i \u2208 s, Ici i"}, {"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": [67, 9], "def_end_pos": [67, 20]}]], "state_before": "case refine_2.intro.intro.intro\n\u03b9 : Sort u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\na : \u03b1\nf : \u03b9 \u2192 \u03b1\na_glb : IsGLB s a\na_not_mem : a \u2209 s\nx : \u03b1\nhx : x \u2208 Ioi a\ny : \u03b1\nhys : y \u2208 s\nleft\u271d : a \u2264 y\nhyx : y < x\n\u22a2 x \u2208 \u22c3 i \u2208 s, Ici i", "state_after": "case refine_2.intro.intro.intro\n\u03b9 : Sort u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\na : \u03b1\nf : \u03b9 \u2192 \u03b1\na_glb : IsGLB s a\na_not_mem : a \u2209 s\nx : \u03b1\nhx : x \u2208 Ioi a\ny : \u03b1\nhys : y \u2208 s\nleft\u271d : a \u2264 y\nhyx : y < x\n\u22a2 \u2203 i, \u2203 (_ : i \u2208 s), x \u2208 Ici i"}, {"tactic": "exact \u27e8y, hys, hyx.le\u27e9", "annotated_tactic": ["exact \u27e8y, hys, hyx.le\u27e9", []], "state_before": "case refine_2.intro.intro.intro\n\u03b9 : Sort u\n\u03b1 : Type v\n\u03b2 : Type w\ninst\u271d : LinearOrder \u03b1\ns : Set \u03b1\na : \u03b1\nf : \u03b9 \u2192 \u03b1\na_glb : IsGLB s a\na_not_mem : a \u2209 s\nx : \u03b1\nhx : x \u2208 Ioi a\ny : \u03b1\nhys : y \u2208 s\nleft\u271d : a \u2264 y\nhyx : y < x\n\u22a2 \u2203 i, \u2203 (_ : i \u2208 s), x \u2208 Ici i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean", "full_name": "one_add_mul_self_le_rpow_one_add", "start": [127, 1], "end": [133, 56], "traced_tactics": [{"tactic": "rcases eq_or_lt_of_le hp with (rfl | hp)", "annotated_tactic": ["rcases <a>eq_or_lt_of_le</a> hp with (rfl | hp)", [{"full_name": "eq_or_lt_of_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [381, 9], "def_end_pos": [381, 23]}]], "state_before": "s : \u211d\nhs : -1 \u2264 s\np : \u211d\nhp : 1 \u2264 p\n\u22a2 1 + p * s \u2264 (1 + s) ^ p", "state_after": "case inl\ns : \u211d\nhs : -1 \u2264 s\nhp : 1 \u2264 1\n\u22a2 1 + 1 * s \u2264 (1 + s) ^ 1\n\ncase inr\ns : \u211d\nhs : -1 \u2264 s\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\n\u22a2 1 + p * s \u2264 (1 + s) ^ p"}, {"tactic": "by_cases hs' : s = 0", "annotated_tactic": ["by_cases hs' : s = 0", []], "state_before": "case inr\ns : \u211d\nhs : -1 \u2264 s\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\n\u22a2 1 + p * s \u2264 (1 + s) ^ p", "state_after": "case pos\ns : \u211d\nhs : -1 \u2264 s\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nhs' : s = 0\n\u22a2 1 + p * s \u2264 (1 + s) ^ p\n\ncase neg\ns : \u211d\nhs : -1 \u2264 s\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nhs' : \u00acs = 0\n\u22a2 1 + p * s \u2264 (1 + s) ^ p"}, {"tactic": "exact (one_add_mul_self_lt_rpow_one_add hs hs' hp).le", "annotated_tactic": ["exact (<a>one_add_mul_self_lt_rpow_one_add</a> hs hs' hp).<a>le</a>", [{"full_name": "one_add_mul_self_lt_rpow_one_add", "def_path": "Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean", "def_pos": [99, 9], "def_end_pos": [99, 41]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}]], "state_before": "case neg\ns : \u211d\nhs : -1 \u2264 s\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nhs' : \u00acs = 0\n\u22a2 1 + p * s \u2264 (1 + s) ^ p", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case inl\ns : \u211d\nhs : -1 \u2264 s\nhp : 1 \u2264 1\n\u22a2 1 + 1 * s \u2264 (1 + s) ^ 1", "state_after": "no goals"}, {"tactic": "simp [hs']", "annotated_tactic": ["simp [hs']", []], "state_before": "case pos\ns : \u211d\nhs : -1 \u2264 s\np : \u211d\nhp\u271d : 1 \u2264 p\nhp : 1 < p\nhs' : s = 0\n\u22a2 1 + p * s \u2264 (1 + s) ^ p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/Implicit.lean", "full_name": "ImplicitFunctionData.map_nhds_eq", "start": [196, 1], "end": [198, 75], "traced_tactics": [{"tactic": "rw [\u2190 map_map, \u03c6.hasStrictFDerivAt.map_nhds_eq_of_equiv, map_fst_nhds]", "annotated_tactic": ["rw [\u2190 <a>map_map</a>, \u03c6.hasStrictFDerivAt.map_nhds_eq_of_equiv, <a>map_fst_nhds</a>]", [{"full_name": "Filter.map_map", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1976, 9], "def_end_pos": [1976, 16]}, {"full_name": "map_fst_nhds", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [773, 9], "def_end_pos": [773, 21]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \ud835\udd5c E\ninst\u271d\u2076 : CompleteSpace E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b3 : CompleteSpace F\nG : Type u_4\ninst\u271d\u00b2 : NormedAddCommGroup G\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c G\ninst\u271d : CompleteSpace G\n\u03c6 : ImplicitFunctionData \ud835\udd5c E F G\n\u22a2 map (Prod.fst \u2218 \u03c6.prodFun) (\ud835\udcdd \u03c6.pt) = \ud835\udcdd (\u03c6.prodFun \u03c6.pt).1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "full_name": "Polynomial.natDegree_X_sub_C_le", "start": [1461, 1], "end": [1462, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean", "full_name": "CircleDeg1Lift.tendsto_translationNumber", "start": [792, 1], "end": [796, 41], "traced_tactics": [{"tactic": "rw [\u2190 translationNumber_conj_eq' (translate <| Multiplicative.ofAdd x)]", "annotated_tactic": ["rw [\u2190 <a>translationNumber_conj_eq'</a> (<a>translate</a> <| <a>Multiplicative.ofAdd</a> x)]", [{"full_name": "CircleDeg1Lift.translationNumber_conj_eq'", "def_path": "Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean", "def_pos": [763, 9], "def_end_pos": [763, 35]}, {"full_name": "CircleDeg1Lift.translate", "def_path": "Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean", "def_pos": [288, 5], "def_end_pos": [288, 14]}, {"full_name": "Multiplicative.ofAdd", "def_path": "Mathlib/Algebra/Group/TypeTags.lean", "def_pos": [84, 5], "def_end_pos": [84, 10]}]], "state_before": "f g : CircleDeg1Lift\nx : \u211d\n\u22a2 Tendsto (fun n => ((f ^ n) x - x) / \u2191n) atTop (\ud835\udcdd (\u03c4 f))", "state_after": "f g : CircleDeg1Lift\nx : \u211d\n\u22a2 Tendsto (fun n => ((f ^ n) x - x) / \u2191n) atTop\n    (\ud835\udcdd (\u03c4 (\u2191(translate (Multiplicative.ofAdd x))\u207b\u00b9 * f * \u2191(translate (Multiplicative.ofAdd x)))))"}, {"tactic": "refine (tendsto_translation_number\u2080 _).congr fun n \u21a6 ?_", "annotated_tactic": ["refine (<a>tendsto_translation_number\u2080</a> _).<a>congr</a> fun n \u21a6 ?_", [{"full_name": "CircleDeg1Lift.tendsto_translation_number\u2080", "def_path": "Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean", "def_pos": [786, 9], "def_end_pos": [786, 36]}, {"full_name": "Filter.Tendsto.congr", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3085, 9], "def_end_pos": [3085, 22]}]], "state_before": "f g : CircleDeg1Lift\nx : \u211d\n\u22a2 Tendsto (fun n => ((f ^ n) x - x) / \u2191n) atTop\n    (\ud835\udcdd (\u03c4 (\u2191(translate (Multiplicative.ofAdd x))\u207b\u00b9 * f * \u2191(translate (Multiplicative.ofAdd x)))))", "state_after": "f g : CircleDeg1Lift\nx : \u211d\nn : \u2115\n\u22a2 ((\u2191(translate (Multiplicative.ofAdd x))\u207b\u00b9 * f * \u2191(translate (Multiplicative.ofAdd x))) ^ n) 0 / \u2191n =\n    ((f ^ n) x - x) / \u2191n"}, {"tactic": "simp [sub_eq_neg_add, Units.conj_pow']", "annotated_tactic": ["simp [<a>sub_eq_neg_add</a>, <a>Units.conj_pow'</a>]", [{"full_name": "sub_eq_neg_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [735, 3], "def_end_pos": [735, 14]}, {"full_name": "Units.conj_pow'", "def_path": "Mathlib/Algebra/Group/Semiconj/Units.lean", "def_pos": [121, 7], "def_end_pos": [121, 16]}]], "state_before": "f g : CircleDeg1Lift\nx : \u211d\nn : \u2115\n\u22a2 ((\u2191(translate (Multiplicative.ofAdd x))\u207b\u00b9 * f * \u2191(translate (Multiplicative.ofAdd x))) ^ n) 0 / \u2191n =\n    ((f ^ n) x - x) / \u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "full_name": "MeasureTheory.addHaar_image_le_mul_of_det_lt", "start": [282, 1], "end": [385, 26], "traced_tactics": [{"tactic": "apply nhdsWithin_le_nhds", "annotated_tactic": ["apply <a>nhdsWithin_le_nhds</a>", [{"full_name": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [196, 9], "def_end_pos": [196, 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 : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\n\u22a2 \u2200\u1da0 (\u03b4 : \u211d\u22650) in \ud835\udcdd[>] 0, \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u03bc (f '' s) \u2264 \u2191m * \u03bc s", "state_after": "case a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u03bc (f '' s) \u2264 \u2191m * \u03bc s) x} \u2208 \ud835\udcdd 0"}, {"tactic": "let d := ENNReal.ofReal |A.det|", "annotated_tactic": ["let d := <a>ENNReal.ofReal</a> |A.det|", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [198, 29], "def_end_pos": [198, 35]}]], "state_before": "case a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u03bc (f '' s) \u2264 \u2191m * \u03bc s) x} \u2208 \ud835\udcdd 0", "state_after": "case a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u03bc (f '' s) \u2264 \u2191m * \u03bc s) x} \u2208 \ud835\udcdd 0"}, {"tactic": "obtain \u27e8\u03b5, h\u03b5, \u03b5pos\u27e9 :\n  \u2203 \u03b5 : \u211d, \u03bc (closedBall 0 \u03b5 + A '' closedBall 0 1) < m * \u03bc (closedBall 0 1) \u2227 0 < \u03b5 := by\n  have HC : IsCompact (A '' closedBall 0 1) :=\n    (ProperSpace.isCompact_closedBall _ _).image A.continuous\n  have L0 :\n    Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (A '' closedBall 0 1))) (\ud835\udcdd[>] 0)\n      (\ud835\udcdd (\u03bc (A '' closedBall 0 1))) := by\n    apply Tendsto.mono_left _ nhdsWithin_le_nhds\n    exact tendsto_measure_cthickening_of_isCompact HC\n  have L1 :\n    Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + A '' closedBall 0 1)) (\ud835\udcdd[>] 0)\n      (\ud835\udcdd (\u03bc (A '' closedBall 0 1))) := by\n    apply L0.congr' _\n    filter_upwards [self_mem_nhdsWithin] with r hr\n    rw [\u2190 HC.add_closedBall_zero (le_of_lt hr), add_comm]\n  have L2 :\n    Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + A '' closedBall 0 1)) (\ud835\udcdd[>] 0)\n      (\ud835\udcdd (d * \u03bc (closedBall 0 1))) := by\n    convert L1\n    exact (addHaar_image_continuousLinearMap _ _ _).symm\n  have I : d * \u03bc (closedBall 0 1) < m * \u03bc (closedBall 0 1) :=\n    (ENNReal.mul_lt_mul_right (measure_closedBall_pos \u03bc _ zero_lt_one).ne'\n          measure_closedBall_lt_top.ne).2\n      hm\n  have H :\n    \u2200\u1da0 b : \u211d in \ud835\udcdd[>] 0, \u03bc (closedBall 0 b + A '' closedBall 0 1) < m * \u03bc (closedBall 0 1) :=\n    (tendsto_order.1 L2).2 _ I\n  exact (H.and self_mem_nhdsWithin).exists", "annotated_tactic": ["obtain \u27e8\u03b5, h\u03b5, \u03b5pos\u27e9 :\n    \u2203 \u03b5 : \u211d, \u03bc (<a>closedBall</a> 0 \u03b5 + A '' <a>closedBall</a> 0 1) < m * \u03bc (<a>closedBall</a> 0 1) \u2227 0 < \u03b5 := by\n    have HC : <a>IsCompact</a> (A '' <a>closedBall</a> 0 1) :=\n      (<a>ProperSpace.isCompact_closedBall</a> _ _).<a>image</a> A.continuous\n    have L0 :\n      <a>Tendsto</a> (fun \u03b5 => \u03bc (<a>cthickening</a> \u03b5 (A '' <a>closedBall</a> 0 1))) (\ud835\udcdd[>] 0)\n        (\ud835\udcdd (\u03bc (A '' <a>closedBall</a> 0 1))) := by\n      apply <a>Tendsto.mono_left</a> _ <a>nhdsWithin_le_nhds</a>\n      exact <a>tendsto_measure_cthickening_of_isCompact</a> HC\n    have L1 :\n      <a>Tendsto</a> (fun \u03b5 => \u03bc (<a>closedBall</a> 0 \u03b5 + A '' <a>closedBall</a> 0 1)) (\ud835\udcdd[>] 0)\n        (\ud835\udcdd (\u03bc (A '' <a>closedBall</a> 0 1))) := by\n      apply L0.congr' _\n      filter_upwards [<a>self_mem_nhdsWithin</a>] with r hr\n      rw [\u2190 HC.add_closedBall_zero (<a>le_of_lt</a> hr), <a>add_comm</a>]\n    have L2 :\n      <a>Tendsto</a> (fun \u03b5 => \u03bc (<a>closedBall</a> 0 \u03b5 + A '' <a>closedBall</a> 0 1)) (\ud835\udcdd[>] 0)\n        (\ud835\udcdd (d * \u03bc (<a>closedBall</a> 0 1))) := by\n      convert L1\n      exact (<a>addHaar_image_continuousLinearMap</a> _ _ _).<a>symm</a>\n    have I : d * \u03bc (<a>closedBall</a> 0 1) < m * \u03bc (<a>closedBall</a> 0 1) :=\n      (<a>ENNReal.mul_lt_mul_right</a> (<a>measure_closedBall_pos</a> \u03bc _ <a>zero_lt_one</a>).<a>ne'</a>\n            measure_closedBall_lt_top.ne).2\n        hm\n    have H :\n      \u2200\u1da0 b : \u211d in \ud835\udcdd[>] 0, \u03bc (<a>closedBall</a> 0 b + A '' <a>closedBall</a> 0 1) < m * \u03bc (<a>closedBall</a> 0 1) :=\n      (<a>tendsto_order</a>.1 L2).2 _ I\n    exact (H.and <a>self_mem_nhdsWithin</a>).<a>exists</a>", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "IsCompact", "def_path": "Mathlib/Topology/Defs/Filter.lean", "def_pos": [273, 5], "def_end_pos": [273, 14]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "ProperSpace.isCompact_closedBall", "def_path": "Mathlib/Topology/MetricSpace/ProperSpace.lean", "def_pos": [34, 3], "def_end_pos": [34, 23]}, {"full_name": "IsCompact.image", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [119, 9], "def_end_pos": [119, 24]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2991, 5], "def_end_pos": [2991, 12]}, {"full_name": "Metric.cthickening", "def_path": "Mathlib/Topology/MetricSpace/Thickening.lean", "def_pos": [195, 5], "def_end_pos": [195, 16]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Filter.Tendsto.mono_left", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3107, 9], "def_end_pos": [3107, 26]}, {"full_name": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [196, 9], "def_end_pos": [196, 27]}, {"full_name": "tendsto_measure_cthickening_of_isCompact", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Metric.lean", "def_pos": [207, 9], "def_end_pos": [207, 49]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2991, 5], "def_end_pos": [2991, 12]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [147, 9], "def_end_pos": [147, 28]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2991, 5], "def_end_pos": [2991, 12]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "MeasureTheory.Measure.addHaar_image_continuousLinearMap", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [329, 9], "def_end_pos": [329, 42]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "ENNReal.mul_lt_mul_right", "def_path": "Mathlib/Data/ENNReal/Operations.lean", "def_pos": [114, 9], "def_end_pos": [114, 25]}, {"full_name": "Metric.measure_closedBall_pos", "def_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "def_pos": [229, 9], "def_end_pos": [229, 31]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 22]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [147, 9], "def_end_pos": [147, 28]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1328, 9], "def_end_pos": [1328, 26]}]], "state_before": "case a\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u03bc (f '' s) \u2264 \u2191m * \u03bc s) x} \u2208 \ud835\udcdd 0", "state_after": "case a.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u03bc (f '' s) \u2264 \u2191m * \u03bc s) x} \u2208 \ud835\udcdd 0"}, {"tactic": "have : Iio (\u27e8\u03b5, \u03b5pos.le\u27e9 : \u211d\u22650) \u2208 \ud835\udcdd (0 : \u211d\u22650) := by apply Iio_mem_nhds; exact \u03b5pos", "annotated_tactic": ["have : <a>Iio</a> (\u27e8\u03b5, \u03b5pos.le\u27e9 : \u211d\u22650) \u2208 \ud835\udcdd (0 : \u211d\u22650) := by apply <a>Iio_mem_nhds</a>; exact \u03b5pos", [{"full_name": "Set.Iio", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [54, 5], "def_end_pos": [54, 8]}, {"full_name": "Iio_mem_nhds", "def_path": "Mathlib/Topology/Order/OrderClosed.lean", "def_pos": [455, 9], "def_end_pos": [455, 21]}]], "state_before": "case a.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u03bc (f '' s) \u2264 \u2191m * \u03bc s) x} \u2208 \ud835\udcdd 0", "state_after": "case a.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u03bc (f '' s) \u2264 \u2191m * \u03bc s) x} \u2208 \ud835\udcdd 0"}, {"tactic": "filter_upwards [this]", "annotated_tactic": ["filter_upwards [this]", []], "state_before": "case a.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u22a2 {x | (fun \u03b4 => \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s \u03b4 \u2192 \u03bc (f '' s) \u2264 \u2191m * \u03bc s) x} \u2208 \ud835\udcdd 0", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u22a2 \u2200 a \u2208 Iio \u27e8\u03b5, \u22ef\u27e9, \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s a \u2192 \u03bc (f '' s) \u2264 \u2191m * \u03bc s"}, {"tactic": "intro \u03b4 h\u03b4 s f hf", "annotated_tactic": ["intro \u03b4 h\u03b4 s f hf", []], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u22a2 \u2200 a \u2208 Iio \u27e8\u03b5, \u22ef\u27e9, \u2200 (s : Set E) (f : E \u2192 E), ApproximatesLinearOn f A s a \u2192 \u03bc (f '' s) \u2264 \u2191m * \u03bc s", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio \u27e8\u03b5, \u22ef\u27e9\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\n\u22a2 \u03bc (f '' s) \u2264 \u2191m * \u03bc s"}, {"tactic": "simp only [mem_Iio, \u2190 NNReal.coe_lt_coe, NNReal.coe_mk] at h\u03b4", "annotated_tactic": ["simp only [<a>mem_Iio</a>, \u2190 <a>NNReal.coe_lt_coe</a>, <a>NNReal.coe_mk</a>] at h\u03b4", [{"full_name": "Set.mem_Iio", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 16]}, {"full_name": "NNReal.coe_lt_coe", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [375, 26], "def_end_pos": [375, 36]}, {"full_name": "NNReal.coe_mk", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [139, 28], "def_end_pos": [139, 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\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 : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\nh\u03b4 : \u03b4 \u2208 Iio \u27e8\u03b5, \u22ef\u27e9\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\n\u22a2 \u03bc (f '' s) \u2264 \u2191m * \u03bc s", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\n\u22a2 \u03bc (f '' s) \u2264 \u2191m * \u03bc s"}, {"tactic": "have J : \u2200\u1da0 a in \ud835\udcdd[>] (0 : \u211d\u22650\u221e), \u03bc (f '' s) \u2264 m * (\u03bc s + a) := by\n  filter_upwards [self_mem_nhdsWithin] with a ha\n  rw [mem_Ioi] at ha\n  obtain \u27e8t, r, t_count, ts, rpos, st, \u03bct\u27e9 :\n    \u2203 (t : Set E) (r : E \u2192 \u211d),\n      t.Countable \u2227\n        t \u2286 s \u2227\n          (\u2200 x : E, x \u2208 t \u2192 0 < r x) \u2227\n            (s \u2286 \u22c3 x \u2208 t, closedBall x (r x)) \u2227\n              (\u2211' x : \u21a5t, \u03bc (closedBall (\u2191x) (r \u2191x))) \u2264 \u03bc s + a :=\n    Besicovitch.exists_closedBall_covering_tsum_measure_le \u03bc ha.ne' (fun _ => Ioi 0) s\n      fun x _ \u03b4 \u03b4pos => \u27e8\u03b4 / 2, by simp [half_pos \u03b4pos, \u03b4pos]\u27e9\n  haveI : Encodable t := t_count.toEncodable\n  calc\n    \u03bc (f '' s) \u2264 \u03bc (\u22c3 x : t, f '' (s \u2229 closedBall x (r x))) := by\n      rw [biUnion_eq_iUnion] at st\n      apply measure_mono\n      rw [\u2190 image_iUnion, \u2190 inter_iUnion]\n      exact image_subset _ (subset_inter (Subset.refl _) st)\n    _ \u2264 \u2211' x : t, \u03bc (f '' (s \u2229 closedBall x (r x))) := measure_iUnion_le _\n    _ \u2264 \u2211' x : t, m * \u03bc (closedBall x (r x)) :=\n      (ENNReal.tsum_le_tsum fun x => I x (r x) (ts x.2) (rpos x x.2).le)\n    _ \u2264 m * (\u03bc s + a) := by rw [ENNReal.tsum_mul_left]; gcongr", "annotated_tactic": ["have J : \u2200\u1da0 a in \ud835\udcdd[>] (0 : \u211d\u22650\u221e), \u03bc (f '' s) \u2264 m * (\u03bc s + a) := by\n    filter_upwards [<a>self_mem_nhdsWithin</a>] with a ha\n    rw [<a>mem_Ioi</a>] at ha\n    obtain \u27e8t, r, t_count, ts, rpos, st, \u03bct\u27e9 :\n      \u2203 (t : <a>Set</a> E) (r : E \u2192 \u211d),\n        t.Countable \u2227\n          t \u2286 s \u2227\n            (\u2200 x : E, x \u2208 t \u2192 0 < r x) \u2227\n              (s \u2286 \u22c3 x \u2208 t, <a>closedBall</a> x (r x)) \u2227\n                (\u2211' x : \u21a5t, \u03bc (<a>closedBall</a> (\u2191x) (r \u2191x))) \u2264 \u03bc s + a :=\n      <a>Besicovitch.exists_closedBall_covering_tsum_measure_le</a> \u03bc ha.ne' (fun _ => <a>Ioi</a> 0) s\n        fun x _ \u03b4 \u03b4pos => \u27e8\u03b4 / 2, by simp [<a>half_pos</a> \u03b4pos, \u03b4pos]\u27e9\n    haveI : <a>Encodable</a> t := t_count.toEncodable\n    calc\n      \u03bc (f '' s) \u2264 \u03bc (\u22c3 x : t, f '' (s \u2229 <a>closedBall</a> x (r x))) := by\n        rw [<a>biUnion_eq_iUnion</a>] at st\n        apply <a>measure_mono</a>\n        rw [\u2190 <a>image_iUnion</a>, \u2190 <a>inter_iUnion</a>]\n        exact <a>image_subset</a> _ (<a>subset_inter</a> (<a>Subset.refl</a> _) st)\n      _ \u2264 \u2211' x : t, \u03bc (f '' (s \u2229 <a>closedBall</a> x (r x))) := <a>measure_iUnion_le</a> _\n      _ \u2264 \u2211' x : t, m * \u03bc (<a>closedBall</a> x (r x)) :=\n        (<a>ENNReal.tsum_le_tsum</a> fun x => I x (r x) (ts x.2) (rpos x x.2).<a>le</a>)\n      _ \u2264 m * (\u03bc s + a) := by rw [<a>ENNReal.tsum_mul_left</a>]; gcongr", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [147, 9], "def_end_pos": [147, 28]}, {"full_name": "Set.mem_Ioi", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 16]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Besicovitch.exists_closedBall_covering_tsum_measure_le", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [889, 9], "def_end_pos": [889, 51]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 17]}, {"full_name": "Encodable", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [44, 7], "def_end_pos": [44, 16]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Set.biUnion_eq_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [862, 9], "def_end_pos": [862, 26]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 21]}, {"full_name": "Set.image_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1650, 9], "def_end_pos": [1650, 21]}, {"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [482, 9], "def_end_pos": [482, 21]}, {"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [291, 9], "def_end_pos": [291, 21]}, {"full_name": "Set.subset_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [937, 9], "def_end_pos": [937, 21]}, {"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [331, 9], "def_end_pos": [331, 20]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "MeasureTheory.measure_iUnion_le", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [63, 9], "def_end_pos": [63, 26]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "ENNReal.tsum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [854, 19], "def_end_pos": [854, 31]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}, {"full_name": "ENNReal.tsum_mul_left", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [928, 19], "def_end_pos": [928, 32]}]], "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 : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\n\u22a2 \u03bc (f '' s) \u2264 \u2191m * \u03bc s", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[>] 0, \u03bc (f '' s) \u2264 \u2191m * (\u03bc s + a)\n\u22a2 \u03bc (f '' s) \u2264 \u2191m * \u03bc s"}, {"tactic": "have L : Tendsto (fun a => (m : \u211d\u22650\u221e) * (\u03bc s + a)) (\ud835\udcdd[>] 0) (\ud835\udcdd (m * (\u03bc s + 0))) := by\n  apply Tendsto.mono_left _ nhdsWithin_le_nhds\n  apply ENNReal.Tendsto.const_mul (tendsto_const_nhds.add tendsto_id)\n  simp only [ENNReal.coe_ne_top, Ne, or_true_iff, not_false_iff]", "annotated_tactic": ["have L : <a>Tendsto</a> (fun a => (m : \u211d\u22650\u221e) * (\u03bc s + a)) (\ud835\udcdd[>] 0) (\ud835\udcdd (m * (\u03bc s + 0))) := by\n    apply <a>Tendsto.mono_left</a> _ <a>nhdsWithin_le_nhds</a>\n    apply <a>ENNReal.Tendsto.const_mul</a> (tendsto_const_nhds.add <a>tendsto_id</a>)\n    simp only [<a>ENNReal.coe_ne_top</a>, <a>Ne</a>, <a>or_true_iff</a>, <a>not_false_iff</a>]", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2991, 5], "def_end_pos": [2991, 12]}, {"full_name": "Filter.Tendsto.mono_left", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3107, 9], "def_end_pos": [3107, 26]}, {"full_name": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [196, 9], "def_end_pos": [196, 27]}, {"full_name": "ENNReal.Tendsto.const_mul", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [387, 19], "def_end_pos": [387, 36]}, {"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3094, 9], "def_end_pos": [3094, 19]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [325, 17], "def_end_pos": [325, 27]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "or_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [152, 9], "def_end_pos": [152, 20]}, {"full_name": "not_false_iff", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1371, 9], "def_end_pos": [1371, 22]}]], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[>] 0, \u03bc (f '' s) \u2264 \u2191m * (\u03bc s + a)\n\u22a2 \u03bc (f '' s) \u2264 \u2191m * \u03bc s", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[>] 0, \u03bc (f '' s) \u2264 \u2191m * (\u03bc s + a)\nL : Tendsto (fun a => \u2191m * (\u03bc s + a)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u2191m * (\u03bc s + 0)))\n\u22a2 \u03bc (f '' s) \u2264 \u2191m * \u03bc s"}, {"tactic": "rw [add_zero] at L", "annotated_tactic": ["rw [<a>add_zero</a>] at L", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [482, 3], "def_end_pos": [482, 14]}]], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[>] 0, \u03bc (f '' s) \u2264 \u2191m * (\u03bc s + a)\nL : Tendsto (fun a => \u2191m * (\u03bc s + a)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u2191m * (\u03bc s + 0)))\n\u22a2 \u03bc (f '' s) \u2264 \u2191m * \u03bc s", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[>] 0, \u03bc (f '' s) \u2264 \u2191m * (\u03bc s + a)\nL : Tendsto (fun a => \u2191m * (\u03bc s + a)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u2191m * \u03bc s))\n\u22a2 \u03bc (f '' s) \u2264 \u2191m * \u03bc s"}, {"tactic": "exact ge_of_tendsto L J", "annotated_tactic": ["exact <a>ge_of_tendsto</a> L J", [{"full_name": "ge_of_tendsto", "def_path": "Mathlib/Topology/Order/OrderClosed.lean", "def_pos": [394, 9], "def_end_pos": [394, 22]}]], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[>] 0, \u03bc (f '' s) \u2264 \u2191m * (\u03bc s + a)\nL : Tendsto (fun a => \u2191m * (\u03bc s + a)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u2191m * \u03bc s))\n\u22a2 \u03bc (f '' s) \u2264 \u2191m * \u03bc s", "state_after": "no goals"}, {"tactic": "have HC : IsCompact (A '' closedBall 0 1) :=\n  (ProperSpace.isCompact_closedBall _ _).image A.continuous", "annotated_tactic": ["have HC : <a>IsCompact</a> (A '' <a>closedBall</a> 0 1) :=\n      (<a>ProperSpace.isCompact_closedBall</a> _ _).<a>image</a> A.continuous", [{"full_name": "IsCompact", "def_path": "Mathlib/Topology/Defs/Filter.lean", "def_pos": [273, 5], "def_end_pos": [273, 14]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "ProperSpace.isCompact_closedBall", "def_path": "Mathlib/Topology/MetricSpace/ProperSpace.lean", "def_pos": [34, 3], "def_end_pos": [34, 23]}, {"full_name": "IsCompact.image", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [119, 9], "def_end_pos": [119, 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 : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u22a2 \u2203 \u03b5, \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1) \u2227 0 < \u03b5", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\n\u22a2 \u2203 \u03b5, \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1) \u2227 0 < \u03b5"}, {"tactic": "have L0 :\n  Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (A '' closedBall 0 1))) (\ud835\udcdd[>] 0)\n    (\ud835\udcdd (\u03bc (A '' closedBall 0 1))) := by\n  apply Tendsto.mono_left _ nhdsWithin_le_nhds\n  exact tendsto_measure_cthickening_of_isCompact HC", "annotated_tactic": ["have L0 :\n      <a>Tendsto</a> (fun \u03b5 => \u03bc (<a>cthickening</a> \u03b5 (A '' <a>closedBall</a> 0 1))) (\ud835\udcdd[>] 0)\n        (\ud835\udcdd (\u03bc (A '' <a>closedBall</a> 0 1))) := by\n      apply <a>Tendsto.mono_left</a> _ <a>nhdsWithin_le_nhds</a>\n      exact <a>tendsto_measure_cthickening_of_isCompact</a> HC", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2991, 5], "def_end_pos": [2991, 12]}, {"full_name": "Metric.cthickening", "def_path": "Mathlib/Topology/MetricSpace/Thickening.lean", "def_pos": [195, 5], "def_end_pos": [195, 16]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Filter.Tendsto.mono_left", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3107, 9], "def_end_pos": [3107, 26]}, {"full_name": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [196, 9], "def_end_pos": [196, 27]}, {"full_name": "tendsto_measure_cthickening_of_isCompact", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Metric.lean", "def_pos": [207, 9], "def_end_pos": [207, 49]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\n\u22a2 \u2203 \u03b5, \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1) \u2227 0 < \u03b5", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\n\u22a2 \u2203 \u03b5, \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1) \u2227 0 < \u03b5"}, {"tactic": "have L1 :\n  Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + A '' closedBall 0 1)) (\ud835\udcdd[>] 0)\n    (\ud835\udcdd (\u03bc (A '' closedBall 0 1))) := by\n  apply L0.congr' _\n  filter_upwards [self_mem_nhdsWithin] with r hr\n  rw [\u2190 HC.add_closedBall_zero (le_of_lt hr), add_comm]", "annotated_tactic": ["have L1 :\n      <a>Tendsto</a> (fun \u03b5 => \u03bc (<a>closedBall</a> 0 \u03b5 + A '' <a>closedBall</a> 0 1)) (\ud835\udcdd[>] 0)\n        (\ud835\udcdd (\u03bc (A '' <a>closedBall</a> 0 1))) := by\n      apply L0.congr' _\n      filter_upwards [<a>self_mem_nhdsWithin</a>] with r hr\n      rw [\u2190 HC.add_closedBall_zero (<a>le_of_lt</a> hr), <a>add_comm</a>]", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2991, 5], "def_end_pos": [2991, 12]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [147, 9], "def_end_pos": [147, 28]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\n\u22a2 \u2203 \u03b5, \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1) \u2227 0 < \u03b5", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\n\u22a2 \u2203 \u03b5, \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1) \u2227 0 < \u03b5"}, {"tactic": "have L2 :\n  Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + A '' closedBall 0 1)) (\ud835\udcdd[>] 0)\n    (\ud835\udcdd (d * \u03bc (closedBall 0 1))) := by\n  convert L1\n  exact (addHaar_image_continuousLinearMap _ _ _).symm", "annotated_tactic": ["have L2 :\n      <a>Tendsto</a> (fun \u03b5 => \u03bc (<a>closedBall</a> 0 \u03b5 + A '' <a>closedBall</a> 0 1)) (\ud835\udcdd[>] 0)\n        (\ud835\udcdd (d * \u03bc (<a>closedBall</a> 0 1))) := by\n      convert L1\n      exact (<a>addHaar_image_continuousLinearMap</a> _ _ _).<a>symm</a>", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2991, 5], "def_end_pos": [2991, 12]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "MeasureTheory.Measure.addHaar_image_continuousLinearMap", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [329, 9], "def_end_pos": [329, 42]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\n\u22a2 \u2203 \u03b5, \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1) \u2227 0 < \u03b5", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\nL2 : Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (d * \u03bc (closedBall 0 1)))\n\u22a2 \u2203 \u03b5, \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1) \u2227 0 < \u03b5"}, {"tactic": "have I : d * \u03bc (closedBall 0 1) < m * \u03bc (closedBall 0 1) :=\n  (ENNReal.mul_lt_mul_right (measure_closedBall_pos \u03bc _ zero_lt_one).ne'\n        measure_closedBall_lt_top.ne).2\n    hm", "annotated_tactic": ["have I : d * \u03bc (<a>closedBall</a> 0 1) < m * \u03bc (<a>closedBall</a> 0 1) :=\n      (<a>ENNReal.mul_lt_mul_right</a> (<a>measure_closedBall_pos</a> \u03bc _ <a>zero_lt_one</a>).<a>ne'</a>\n            measure_closedBall_lt_top.ne).2\n        hm", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "ENNReal.mul_lt_mul_right", "def_path": "Mathlib/Data/ENNReal/Operations.lean", "def_pos": [114, 9], "def_end_pos": [114, 25]}, {"full_name": "Metric.measure_closedBall_pos", "def_path": "Mathlib/MeasureTheory/Measure/OpenPos.lean", "def_pos": [229, 9], "def_end_pos": [229, 31]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 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 : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\nL2 : Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (d * \u03bc (closedBall 0 1)))\n\u22a2 \u2203 \u03b5, \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1) \u2227 0 < \u03b5", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\nL2 : Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (d * \u03bc (closedBall 0 1)))\nI : d * \u03bc (closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u22a2 \u2203 \u03b5, \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1) \u2227 0 < \u03b5"}, {"tactic": "have H :\n  \u2200\u1da0 b : \u211d in \ud835\udcdd[>] 0, \u03bc (closedBall 0 b + A '' closedBall 0 1) < m * \u03bc (closedBall 0 1) :=\n  (tendsto_order.1 L2).2 _ I", "annotated_tactic": ["have H :\n      \u2200\u1da0 b : \u211d in \ud835\udcdd[>] 0, \u03bc (<a>closedBall</a> 0 b + A '' <a>closedBall</a> 0 1) < m * \u03bc (<a>closedBall</a> 0 1) :=\n      (<a>tendsto_order</a>.1 L2).2 _ I", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [126, 9], "def_end_pos": [126, 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 : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\nL2 : Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (d * \u03bc (closedBall 0 1)))\nI : d * \u03bc (closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u22a2 \u2203 \u03b5, \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1) \u2227 0 < \u03b5", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\nL2 : Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (d * \u03bc (closedBall 0 1)))\nI : d * \u03bc (closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\nH : \u2200\u1da0 (b : \u211d) in \ud835\udcdd[>] 0, \u03bc (closedBall 0 b + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u22a2 \u2203 \u03b5, \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1) \u2227 0 < \u03b5"}, {"tactic": "exact (H.and self_mem_nhdsWithin).exists", "annotated_tactic": ["exact (H.and <a>self_mem_nhdsWithin</a>).<a>exists</a>", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [147, 9], "def_end_pos": [147, 28]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1328, 9], "def_end_pos": [1328, 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 : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\nL2 : Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (d * \u03bc (closedBall 0 1)))\nI : d * \u03bc (closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\nH : \u2200\u1da0 (b : \u211d) in \ud835\udcdd[>] 0, \u03bc (closedBall 0 b + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u22a2 \u2203 \u03b5, \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1) \u2227 0 < \u03b5", "state_after": "no goals"}, {"tactic": "apply Tendsto.mono_left _ nhdsWithin_le_nhds", "annotated_tactic": ["apply <a>Tendsto.mono_left</a> _ <a>nhdsWithin_le_nhds</a>", [{"full_name": "Filter.Tendsto.mono_left", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3107, 9], "def_end_pos": [3107, 26]}, {"full_name": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [196, 9], "def_end_pos": [196, 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 : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\n\u22a2 Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\n\u22a2 Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) (\ud835\udcdd 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))"}, {"tactic": "exact tendsto_measure_cthickening_of_isCompact HC", "annotated_tactic": ["exact <a>tendsto_measure_cthickening_of_isCompact</a> HC", [{"full_name": "tendsto_measure_cthickening_of_isCompact", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Metric.lean", "def_pos": [207, 9], "def_end_pos": [207, 49]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\n\u22a2 Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) (\ud835\udcdd 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))", "state_after": "no goals"}, {"tactic": "apply L0.congr' _", "annotated_tactic": ["apply L0.congr' _", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\n\u22a2 Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\n\u22a2 (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) =\u1da0[\ud835\udcdd[>] 0] fun \u03b5 => \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1)"}, {"tactic": "filter_upwards [self_mem_nhdsWithin] with r hr", "annotated_tactic": ["filter_upwards [<a>self_mem_nhdsWithin</a>] with r hr", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [147, 9], "def_end_pos": [147, 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 : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\n\u22a2 (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) =\u1da0[\ud835\udcdd[>] 0] fun \u03b5 => \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1)", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\nr : \u211d\nhr : r \u2208 Ioi 0\n\u22a2 \u03bc (cthickening r (\u21d1A '' closedBall 0 1)) = \u03bc (closedBall 0 r + \u21d1A '' closedBall 0 1)"}, {"tactic": "rw [\u2190 HC.add_closedBall_zero (le_of_lt hr), add_comm]", "annotated_tactic": ["rw [\u2190 HC.add_closedBall_zero (<a>le_of_lt</a> hr), <a>add_comm</a>]", [{"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\nr : \u211d\nhr : r \u2208 Ioi 0\n\u22a2 \u03bc (cthickening r (\u21d1A '' closedBall 0 1)) = \u03bc (closedBall 0 r + \u21d1A '' closedBall 0 1)", "state_after": "no goals"}, {"tactic": "convert L1", "annotated_tactic": ["convert L1", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\n\u22a2 Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (d * \u03bc (closedBall 0 1)))", "state_after": "case h.e'_5.h.e'_3\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\n\u22a2 d * \u03bc (closedBall 0 1) = \u03bc (\u21d1A '' closedBall 0 1)"}, {"tactic": "exact (addHaar_image_continuousLinearMap _ _ _).symm", "annotated_tactic": ["exact (<a>addHaar_image_continuousLinearMap</a> _ _ _).<a>symm</a>", [{"full_name": "MeasureTheory.Measure.addHaar_image_continuousLinearMap", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [329, 9], "def_end_pos": [329, 42]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case h.e'_5.h.e'_3\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\nHC : IsCompact (\u21d1A '' closedBall 0 1)\nL0 : Tendsto (fun \u03b5 => \u03bc (cthickening \u03b5 (\u21d1A '' closedBall 0 1))) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\nL1 : Tendsto (fun \u03b5 => \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u03bc (\u21d1A '' closedBall 0 1)))\n\u22a2 d * \u03bc (closedBall 0 1) = \u03bc (\u21d1A '' closedBall 0 1)", "state_after": "no goals"}, {"tactic": "apply Iio_mem_nhds", "annotated_tactic": ["apply <a>Iio_mem_nhds</a>", [{"full_name": "Iio_mem_nhds", "def_path": "Mathlib/Topology/Order/OrderClosed.lean", "def_pos": [455, 9], "def_end_pos": [455, 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 : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\n\u22a2 Iio \u27e8\u03b5, \u22ef\u27e9 \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 : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\n\u22a2 0 < \u27e8\u03b5, \u22ef\u27e9"}, {"tactic": "exact \u03b5pos", "annotated_tactic": ["exact \u03b5pos", []], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns : Set E\nf : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\n\u22a2 0 < \u27e8\u03b5, \u22ef\u27e9", "state_after": "no goals"}, {"tactic": "intro x r xs r0", "annotated_tactic": ["intro x r xs r0", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\n\u22a2 \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\n\u22a2 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)"}, {"tactic": "have :\n  A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) =\n    {f x} + r \u2022 (A '' closedBall 0 1 + closedBall 0 \u03b5) := by\n  rw [smul_add, \u2190 add_assoc, add_comm {f x}, add_assoc, smul_closedBall _ _ \u03b5pos.le, smul_zero,\n    singleton_add_closedBall_zero, \u2190 image_smul_set \u211d E E A, smul_closedBall _ _ zero_le_one,\n    smul_zero, Real.norm_eq_abs, abs_of_nonneg r0, mul_one, mul_comm]", "annotated_tactic": ["have :\n      A '' <a>closedBall</a> 0 r + <a>closedBall</a> (f x) (\u03b5 * r) =\n        {f x} + r \u2022 (A '' <a>closedBall</a> 0 1 + <a>closedBall</a> 0 \u03b5) := by\n      rw [<a>smul_add</a>, \u2190 <a>add_assoc</a>, <a>add_comm</a> {f x}, <a>add_assoc</a>, <a>smul_closedBall</a> _ _ \u03b5pos.le, <a>smul_zero</a>,\n        <a>singleton_add_closedBall_zero</a>, \u2190 <a>image_smul_set</a> \u211d E E A, <a>smul_closedBall</a> _ _ <a>zero_le_one</a>,\n        <a>smul_zero</a>, <a>Real.norm_eq_abs</a>, <a>abs_of_nonneg</a> r0, <a>mul_one</a>, <a>mul_comm</a>]", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "smul_add", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [145, 9], "def_end_pos": [145, 17]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [288, 3], "def_end_pos": [288, 14]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [288, 3], "def_end_pos": [288, 14]}, {"full_name": "smul_closedBall", "def_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "def_pos": [400, 9], "def_end_pos": [400, 24]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [66, 9], "def_end_pos": [66, 18]}, {"full_name": "singleton_add_closedBall_zero", "def_path": "Mathlib/Analysis/Normed/Group/Pointwise.lean", "def_pos": [195, 3], "def_end_pos": [195, 14]}, {"full_name": "image_smul_set", "def_path": "Mathlib/GroupTheory/GroupAction/Pointwise.lean", "def_pos": [129, 9], "def_end_pos": [129, 23]}, {"full_name": "smul_closedBall", "def_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "def_pos": [400, 9], "def_end_pos": [400, 24]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [66, 9], "def_end_pos": [66, 18]}, {"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1434, 9], "def_end_pos": [1434, 20]}, {"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 \u21d1A '' closedBall 0 r + closedBall (f x) (\u03b5 * r)\n\u22a2 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 \u21d1A '' closedBall 0 r + closedBall (f x) (\u03b5 * r)\nthis : \u21d1A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)"}, {"tactic": "rw [this] at K", "annotated_tactic": ["rw [this] at K", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 \u21d1A '' closedBall 0 r + closedBall (f x) (\u03b5 * r)\nthis : \u21d1A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u21d1A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)"}, {"tactic": "calc\n  \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u03bc ({f x} + r \u2022 (A '' closedBall 0 1 + closedBall 0 \u03b5)) :=\n    measure_mono K\n  _ = ENNReal.ofReal (r ^ finrank \u211d E) * \u03bc (A '' closedBall 0 1 + closedBall 0 \u03b5) := by\n    simp only [abs_of_nonneg r0, addHaar_smul, image_add_left, abs_pow, singleton_add,\n      measure_preimage_add]\n  _ \u2264 ENNReal.ofReal (r ^ finrank \u211d E) * (m * \u03bc (closedBall 0 1)) := by\n    rw [add_comm]; gcongr\n  _ = m * \u03bc (closedBall x r) := by simp only [addHaar_closedBall' \u03bc _ r0]; ring", "annotated_tactic": ["calc\n      \u03bc (f '' (s \u2229 <a>closedBall</a> x r)) \u2264 \u03bc ({f x} + r \u2022 (A '' <a>closedBall</a> 0 1 + <a>closedBall</a> 0 \u03b5)) :=\n        <a>measure_mono</a> K\n      _ = <a>ENNReal.ofReal</a> (r ^ <a>finrank</a> \u211d E) * \u03bc (A '' <a>closedBall</a> 0 1 + <a>closedBall</a> 0 \u03b5) := by\n        simp only [<a>abs_of_nonneg</a> r0, <a>addHaar_smul</a>, <a>image_add_left</a>, <a>abs_pow</a>, <a>singleton_add</a>,\n          <a>measure_preimage_add</a>]\n      _ \u2264 <a>ENNReal.ofReal</a> (r ^ <a>finrank</a> \u211d E) * (m * \u03bc (<a>closedBall</a> 0 1)) := by\n        rw [<a>add_comm</a>]; gcongr\n      _ = m * \u03bc (<a>closedBall</a> x r) := by simp only [<a>addHaar_closedBall'</a> \u03bc _ r0]; ring", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 21]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [198, 29], "def_end_pos": [198, 35]}, {"full_name": "FiniteDimensional.finrank", "def_path": "Mathlib/LinearAlgebra/Dimension/Finrank.lean", "def_pos": [54, 19], "def_end_pos": [54, 26]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"full_name": "MeasureTheory.Measure.addHaar_smul", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [386, 9], "def_end_pos": [386, 21]}, {"full_name": "Set.image_add_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1203, 3], "def_end_pos": [1203, 14]}, {"full_name": "abs_pow", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [64, 7], "def_end_pos": [64, 14]}, {"full_name": "Set.singleton_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [405, 3], "def_end_pos": [405, 14]}, {"full_name": "MeasureTheory.measure_preimage_add", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [313, 3], "def_end_pos": [313, 14]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [198, 29], "def_end_pos": [198, 35]}, {"full_name": "FiniteDimensional.finrank", "def_path": "Mathlib/LinearAlgebra/Dimension/Finrank.lean", "def_pos": [54, 19], "def_end_pos": [54, 26]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "MeasureTheory.Measure.addHaar_closedBall'", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [496, 9], "def_end_pos": [496, 28]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u21d1A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)", "state_after": "no goals"}, {"tactic": "rintro y \u27e8z, \u27e8zs, zr\u27e9, rfl\u27e9", "annotated_tactic": ["rintro y \u27e8z, \u27e8zs, zr\u27e9, rfl\u27e9", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\n\u22a2 f '' (s \u2229 closedBall x r) \u2286 \u21d1A '' closedBall 0 r + closedBall (f x) (\u03b5 * r)", "state_after": "case intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 f z \u2208 \u21d1A '' closedBall 0 r + closedBall (f x) (\u03b5 * r)"}, {"tactic": "rw [mem_closedBall_iff_norm] at zr", "annotated_tactic": ["rw [<a>mem_closedBall_iff_norm</a>] at zr", [{"full_name": "mem_closedBall_iff_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [625, 15], "def_end_pos": [625, 38]}]], "state_before": "case intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : z \u2208 closedBall x r\n\u22a2 f z \u2208 \u21d1A '' closedBall 0 r + closedBall (f x) (\u03b5 * r)", "state_after": "case intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : \u2016z - x\u2016 \u2264 r\n\u22a2 f z \u2208 \u21d1A '' closedBall 0 r + closedBall (f x) (\u03b5 * r)"}, {"tactic": "apply Set.mem_add.2 \u27e8A (z - x), _, f z - f x - A (z - x) + f x, _, _\u27e9", "annotated_tactic": ["apply <a>Set.mem_add</a>.2 \u27e8A (z - x), _, f z - f x - A (z - x) + f x, _, _\u27e9", [{"full_name": "Set.mem_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [339, 3], "def_end_pos": [339, 14]}]], "state_before": "case intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : \u2016z - x\u2016 \u2264 r\n\u22a2 f z \u2208 \u21d1A '' closedBall 0 r + closedBall (f x) (\u03b5 * r)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : \u2016z - x\u2016 \u2264 r\n\u22a2 A (z - x) \u2208 \u21d1A '' closedBall 0 r\n\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : \u2016z - x\u2016 \u2264 r\n\u22a2 f z - f x - A (z - x) + f x \u2208 closedBall (f x) (\u03b5 * r)\n\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : \u2016z - x\u2016 \u2264 r\n\u22a2 A (z - x) + (f z - f x - A (z - x) + f x) = f z"}, {"tactic": "apply mem_image_of_mem", "annotated_tactic": ["apply <a>mem_image_of_mem</a>", [{"full_name": "Set.mem_image_of_mem", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [132, 9], "def_end_pos": [132, 25]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : \u2016z - x\u2016 \u2264 r\n\u22a2 A (z - x) \u2208 \u21d1A '' closedBall 0 r", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : \u2016z - x\u2016 \u2264 r\n\u22a2 z - x \u2208 closedBall 0 r"}, {"tactic": "simpa only [dist_eq_norm, mem_closedBall, mem_closedBall_zero_iff, sub_zero] using zr", "annotated_tactic": ["simpa only [<a>dist_eq_norm</a>, <a>mem_closedBall</a>, <a>mem_closedBall_zero_iff</a>, <a>sub_zero</a>] using zr", [{"full_name": "dist_eq_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [401, 7], "def_end_pos": [401, 19]}, {"full_name": "Metric.mem_closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [468, 17], "def_end_pos": [468, 31]}, {"full_name": "mem_closedBall_zero_iff", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [631, 3], "def_end_pos": [631, 14]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [489, 3], "def_end_pos": [489, 14]}]], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : \u2016z - x\u2016 \u2264 r\n\u22a2 z - x \u2208 closedBall 0 r", "state_after": "no goals"}, {"tactic": "rw [mem_closedBall_iff_norm, add_sub_cancel_right]", "annotated_tactic": ["rw [<a>mem_closedBall_iff_norm</a>, <a>add_sub_cancel_right</a>]", [{"full_name": "mem_closedBall_iff_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [625, 15], "def_end_pos": [625, 38]}, {"full_name": "add_sub_cancel_right", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1008, 3], "def_end_pos": [1008, 14]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : \u2016z - x\u2016 \u2264 r\n\u22a2 f z - f x - A (z - x) + f x \u2208 closedBall (f x) (\u03b5 * r)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : \u2016z - x\u2016 \u2264 r\n\u22a2 \u2016f z - f x - A (z - x)\u2016 \u2264 \u03b5 * r"}, {"tactic": "calc\n  \u2016f z - f x - A (z - x)\u2016 \u2264 \u03b4 * \u2016z - x\u2016 := hf _ zs _ xs\n  _ \u2264 \u03b5 * r := by gcongr", "annotated_tactic": ["calc\n          \u2016f z - f x - A (z - x)\u2016 \u2264 \u03b4 * \u2016z - x\u2016 := hf _ zs _ xs\n          _ \u2264 \u03b5 * r := by gcongr", []], "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 : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : \u2016z - x\u2016 \u2264 r\n\u22a2 \u2016f z - f x - A (z - x)\u2016 \u2264 \u03b5 * r", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "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 : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : \u2016z - x\u2016 \u2264 r\n\u22a2 \u2191\u03b4 * \u2016z - x\u2016 \u2264 \u03b5 * r", "state_after": "no goals"}, {"tactic": "simp only [map_sub, Pi.sub_apply]", "annotated_tactic": ["simp only [<a>map_sub</a>, <a>Pi.sub_apply</a>]", [{"full_name": "map_sub", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [461, 3], "def_end_pos": [461, 14]}, {"full_name": "Pi.sub_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [195, 3], "def_end_pos": [195, 14]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : \u2016z - x\u2016 \u2264 r\n\u22a2 A (z - x) + (f z - f x - A (z - x) + f x) = f z", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : \u2016z - x\u2016 \u2264 r\n\u22a2 A z - A x + (f z - f x - (A z - A x) + f x) = f z"}, {"tactic": "abel", "annotated_tactic": ["abel", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nz : E\nzs : z \u2208 s\nzr : \u2016z - x\u2016 \u2264 r\n\u22a2 A z - A x + (f z - f x - (A z - A x) + f x) = f z", "state_after": "no goals"}, {"tactic": "rw [smul_add, \u2190 add_assoc, add_comm {f x}, add_assoc, smul_closedBall _ _ \u03b5pos.le, smul_zero,\n  singleton_add_closedBall_zero, \u2190 image_smul_set \u211d E E A, smul_closedBall _ _ zero_le_one,\n  smul_zero, Real.norm_eq_abs, abs_of_nonneg r0, mul_one, mul_comm]", "annotated_tactic": ["rw [<a>smul_add</a>, \u2190 <a>add_assoc</a>, <a>add_comm</a> {f x}, <a>add_assoc</a>, <a>smul_closedBall</a> _ _ \u03b5pos.le, <a>smul_zero</a>,\n        <a>singleton_add_closedBall_zero</a>, \u2190 <a>image_smul_set</a> \u211d E E A, <a>smul_closedBall</a> _ _ <a>zero_le_one</a>,\n        <a>smul_zero</a>, <a>Real.norm_eq_abs</a>, <a>abs_of_nonneg</a> r0, <a>mul_one</a>, <a>mul_comm</a>]", [{"full_name": "smul_add", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [145, 9], "def_end_pos": [145, 17]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [288, 3], "def_end_pos": [288, 14]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [288, 3], "def_end_pos": [288, 14]}, {"full_name": "smul_closedBall", "def_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "def_pos": [400, 9], "def_end_pos": [400, 24]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [66, 9], "def_end_pos": [66, 18]}, {"full_name": "singleton_add_closedBall_zero", "def_path": "Mathlib/Analysis/Normed/Group/Pointwise.lean", "def_pos": [195, 3], "def_end_pos": [195, 14]}, {"full_name": "image_smul_set", "def_path": "Mathlib/GroupTheory/GroupAction/Pointwise.lean", "def_pos": [129, 9], "def_end_pos": [129, 23]}, {"full_name": "smul_closedBall", "def_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "def_pos": [400, 9], "def_end_pos": [400, 24]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [66, 9], "def_end_pos": [66, 18]}, {"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1434, 9], "def_end_pos": [1434, 20]}, {"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 \u21d1A '' closedBall 0 r + closedBall (f x) (\u03b5 * r)\n\u22a2 \u21d1A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)", "state_after": "no goals"}, {"tactic": "simp only [abs_of_nonneg r0, addHaar_smul, image_add_left, abs_pow, singleton_add,\n  measure_preimage_add]", "annotated_tactic": ["simp only [<a>abs_of_nonneg</a> r0, <a>addHaar_smul</a>, <a>image_add_left</a>, <a>abs_pow</a>, <a>singleton_add</a>,\n          <a>measure_preimage_add</a>]", [{"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"full_name": "MeasureTheory.Measure.addHaar_smul", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [386, 9], "def_end_pos": [386, 21]}, {"full_name": "Set.image_add_left", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [1203, 3], "def_end_pos": [1203, 14]}, {"full_name": "abs_pow", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [64, 7], "def_end_pos": [64, 14]}, {"full_name": "Set.singleton_add", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [405, 3], "def_end_pos": [405, 14]}, {"full_name": "MeasureTheory.measure_preimage_add", "def_path": "Mathlib/MeasureTheory/Group/Measure.lean", "def_pos": [313, 3], "def_end_pos": [313, 14]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u21d1A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 \u03bc ({f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)) =\n    ENNReal.ofReal (r ^ finrank \u211d E) * \u03bc (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)", "state_after": "no goals"}, {"tactic": "rw [add_comm]", "annotated_tactic": ["rw [<a>add_comm</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u21d1A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 ENNReal.ofReal (r ^ finrank \u211d E) * \u03bc (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5) \u2264\n    ENNReal.ofReal (r ^ finrank \u211d E) * (\u2191m * \u03bc (closedBall 0 1))", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u21d1A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 ENNReal.ofReal (r ^ finrank \u211d E) * \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) \u2264\n    ENNReal.ofReal (r ^ finrank \u211d E) * (\u2191m * \u03bc (closedBall 0 1))"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "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 : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u21d1A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 ENNReal.ofReal (r ^ finrank \u211d E) * \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) \u2264\n    ENNReal.ofReal (r ^ finrank \u211d E) * (\u2191m * \u03bc (closedBall 0 1))", "state_after": "no goals"}, {"tactic": "simp only [addHaar_closedBall' \u03bc _ r0]", "annotated_tactic": ["simp only [<a>addHaar_closedBall'</a> \u03bc _ r0]", [{"full_name": "MeasureTheory.Measure.addHaar_closedBall'", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [496, 9], "def_end_pos": [496, 28]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u21d1A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 ENNReal.ofReal (r ^ finrank \u211d E) * (\u2191m * \u03bc (closedBall 0 1)) = \u2191m * \u03bc (closedBall x r)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u21d1A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 ENNReal.ofReal (r ^ finrank \u211d E) * (\u2191m * \u03bc (closedBall 0 1)) =\n    \u2191m * (ENNReal.ofReal (r ^ finrank \u211d E) * \u03bc (closedBall 0 1))"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nx : E\nr : \u211d\nxs : x \u2208 s\nr0 : 0 \u2264 r\nK : f '' (s \u2229 closedBall x r) \u2286 {f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)\nthis : \u21d1A '' closedBall 0 r + closedBall (f x) (\u03b5 * r) = {f x} + r \u2022 (\u21d1A '' closedBall 0 1 + closedBall 0 \u03b5)\n\u22a2 ENNReal.ofReal (r ^ finrank \u211d E) * (\u2191m * \u03bc (closedBall 0 1)) =\n    \u2191m * (ENNReal.ofReal (r ^ finrank \u211d E) * \u03bc (closedBall 0 1))", "state_after": "no goals"}, {"tactic": "filter_upwards [self_mem_nhdsWithin] with a ha", "annotated_tactic": ["filter_upwards [<a>self_mem_nhdsWithin</a>] with a ha", [{"full_name": "self_mem_nhdsWithin", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [147, 9], "def_end_pos": [147, 28]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\n\u22a2 \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[>] 0, \u03bc (f '' s) \u2264 \u2191m * (\u03bc s + a)", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : a \u2208 Ioi 0\n\u22a2 \u03bc (f '' s) \u2264 \u2191m * (\u03bc s + a)"}, {"tactic": "rw [mem_Ioi] at ha", "annotated_tactic": ["rw [<a>mem_Ioi</a>] at ha", [{"full_name": "Set.mem_Ioi", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 16]}]], "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 : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : a \u2208 Ioi 0\n\u22a2 \u03bc (f '' s) \u2264 \u2191m * (\u03bc s + a)", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\n\u22a2 \u03bc (f '' s) \u2264 \u2191m * (\u03bc s + a)"}, {"tactic": "obtain \u27e8t, r, t_count, ts, rpos, st, \u03bct\u27e9 :\n  \u2203 (t : Set E) (r : E \u2192 \u211d),\n    t.Countable \u2227\n      t \u2286 s \u2227\n        (\u2200 x : E, x \u2208 t \u2192 0 < r x) \u2227\n          (s \u2286 \u22c3 x \u2208 t, closedBall x (r x)) \u2227\n            (\u2211' x : \u21a5t, \u03bc (closedBall (\u2191x) (r \u2191x))) \u2264 \u03bc s + a :=\n  Besicovitch.exists_closedBall_covering_tsum_measure_le \u03bc ha.ne' (fun _ => Ioi 0) s\n    fun x _ \u03b4 \u03b4pos => \u27e8\u03b4 / 2, by simp [half_pos \u03b4pos, \u03b4pos]\u27e9", "annotated_tactic": ["obtain \u27e8t, r, t_count, ts, rpos, st, \u03bct\u27e9 :\n      \u2203 (t : <a>Set</a> E) (r : E \u2192 \u211d),\n        t.Countable \u2227\n          t \u2286 s \u2227\n            (\u2200 x : E, x \u2208 t \u2192 0 < r x) \u2227\n              (s \u2286 \u22c3 x \u2208 t, <a>closedBall</a> x (r x)) \u2227\n                (\u2211' x : \u21a5t, \u03bc (<a>closedBall</a> (\u2191x) (r \u2191x))) \u2264 \u03bc s + a :=\n      <a>Besicovitch.exists_closedBall_covering_tsum_measure_le</a> \u03bc ha.ne' (fun _ => <a>Ioi</a> 0) s\n        fun x _ \u03b4 \u03b4pos => \u27e8\u03b4 / 2, by simp [<a>half_pos</a> \u03b4pos, \u03b4pos]\u27e9", [{"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Besicovitch.exists_closedBall_covering_tsum_measure_le", "def_path": "Mathlib/MeasureTheory/Covering/Besicovitch.lean", "def_pos": [889, 9], "def_end_pos": [889, 51]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 17]}]], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\n\u22a2 \u03bc (f '' s) \u2264 \u2191m * (\u03bc s + a)", "state_after": "case h.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : t.Countable\nts : t \u2286 s\nrpos : \u2200 x \u2208 t, 0 < r x\nst : s \u2286 \u22c3 x \u2208 t, closedBall x (r x)\n\u03bct : \u2211' (x : \u2191t), \u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03bc s + a\n\u22a2 \u03bc (f '' s) \u2264 \u2191m * (\u03bc s + a)"}, {"tactic": "haveI : Encodable t := t_count.toEncodable", "annotated_tactic": ["haveI : <a>Encodable</a> t := t_count.toEncodable", [{"full_name": "Encodable", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [44, 7], "def_end_pos": [44, 16]}]], "state_before": "case h.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : t.Countable\nts : t \u2286 s\nrpos : \u2200 x \u2208 t, 0 < r x\nst : s \u2286 \u22c3 x \u2208 t, closedBall x (r x)\n\u03bct : \u2211' (x : \u2191t), \u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03bc s + a\n\u22a2 \u03bc (f '' s) \u2264 \u2191m * (\u03bc s + a)", "state_after": "case h.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : t.Countable\nts : t \u2286 s\nrpos : \u2200 x \u2208 t, 0 < r x\nst : s \u2286 \u22c3 x \u2208 t, closedBall x (r x)\n\u03bct : \u2211' (x : \u2191t), \u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03bc s + a\nthis : Encodable \u2191t\n\u22a2 \u03bc (f '' s) \u2264 \u2191m * (\u03bc s + a)"}, {"tactic": "calc\n  \u03bc (f '' s) \u2264 \u03bc (\u22c3 x : t, f '' (s \u2229 closedBall x (r x))) := by\n    rw [biUnion_eq_iUnion] at st\n    apply measure_mono\n    rw [\u2190 image_iUnion, \u2190 inter_iUnion]\n    exact image_subset _ (subset_inter (Subset.refl _) st)\n  _ \u2264 \u2211' x : t, \u03bc (f '' (s \u2229 closedBall x (r x))) := measure_iUnion_le _\n  _ \u2264 \u2211' x : t, m * \u03bc (closedBall x (r x)) :=\n    (ENNReal.tsum_le_tsum fun x => I x (r x) (ts x.2) (rpos x x.2).le)\n  _ \u2264 m * (\u03bc s + a) := by rw [ENNReal.tsum_mul_left]; gcongr", "annotated_tactic": ["calc\n      \u03bc (f '' s) \u2264 \u03bc (\u22c3 x : t, f '' (s \u2229 <a>closedBall</a> x (r x))) := by\n        rw [<a>biUnion_eq_iUnion</a>] at st\n        apply <a>measure_mono</a>\n        rw [\u2190 <a>image_iUnion</a>, \u2190 <a>inter_iUnion</a>]\n        exact <a>image_subset</a> _ (<a>subset_inter</a> (<a>Subset.refl</a> _) st)\n      _ \u2264 \u2211' x : t, \u03bc (f '' (s \u2229 <a>closedBall</a> x (r x))) := <a>measure_iUnion_le</a> _\n      _ \u2264 \u2211' x : t, m * \u03bc (<a>closedBall</a> x (r x)) :=\n        (<a>ENNReal.tsum_le_tsum</a> fun x => I x (r x) (ts x.2) (rpos x x.2).<a>le</a>)\n      _ \u2264 m * (\u03bc s + a) := by rw [<a>ENNReal.tsum_mul_left</a>]; gcongr", [{"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "Set.biUnion_eq_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [862, 9], "def_end_pos": [862, 26]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 21]}, {"full_name": "Set.image_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1650, 9], "def_end_pos": [1650, 21]}, {"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [482, 9], "def_end_pos": [482, 21]}, {"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [291, 9], "def_end_pos": [291, 21]}, {"full_name": "Set.subset_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [937, 9], "def_end_pos": [937, 21]}, {"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [331, 9], "def_end_pos": [331, 20]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "MeasureTheory.measure_iUnion_le", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [63, 9], "def_end_pos": [63, 26]}, {"full_name": "Metric.closedBall", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [464, 5], "def_end_pos": [464, 15]}, {"full_name": "ENNReal.tsum_le_tsum", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [854, 19], "def_end_pos": [854, 31]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}, {"full_name": "ENNReal.tsum_mul_left", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [928, 19], "def_end_pos": [928, 32]}]], "state_before": "case h.intro.intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : t.Countable\nts : t \u2286 s\nrpos : \u2200 x \u2208 t, 0 < r x\nst : s \u2286 \u22c3 x \u2208 t, closedBall x (r x)\n\u03bct : \u2211' (x : \u2191t), \u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03bc s + a\nthis : Encodable \u2191t\n\u22a2 \u03bc (f '' s) \u2264 \u2191m * (\u03bc s + a)", "state_after": "no goals"}, {"tactic": "simp [half_pos \u03b4pos, \u03b4pos]", "annotated_tactic": ["simp [<a>half_pos</a> \u03b4pos, \u03b4pos]", [{"full_name": "half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 17]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4\u271d : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\u271d\nh\u03b4 : \u2191\u03b4\u271d < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nx : E\nx\u271d : x \u2208 s\n\u03b4 : \u211d\n\u03b4pos : \u03b4 > 0\n\u22a2 \u03b4 / 2 \u2208 (fun x => Ioi 0) x \u2229 Ioo 0 \u03b4", "state_after": "no goals"}, {"tactic": "rw [biUnion_eq_iUnion] at st", "annotated_tactic": ["rw [<a>biUnion_eq_iUnion</a>] at st", [{"full_name": "Set.biUnion_eq_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [862, 9], "def_end_pos": [862, 26]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : t.Countable\nts : t \u2286 s\nrpos : \u2200 x \u2208 t, 0 < r x\nst : s \u2286 \u22c3 x \u2208 t, closedBall x (r x)\n\u03bct : \u2211' (x : \u2191t), \u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03bc s + a\nthis : Encodable \u2191t\n\u22a2 \u03bc (f '' s) \u2264 \u03bc (\u22c3 x, f '' (s \u2229 closedBall (\u2191x) (r \u2191x)))", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : t.Countable\nts : t \u2286 s\nrpos : \u2200 x \u2208 t, 0 < r x\nst : s \u2286 \u22c3 x, closedBall (\u2191x) (r \u2191x)\n\u03bct : \u2211' (x : \u2191t), \u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03bc s + a\nthis : Encodable \u2191t\n\u22a2 \u03bc (f '' s) \u2264 \u03bc (\u22c3 x, f '' (s \u2229 closedBall (\u2191x) (r \u2191x)))"}, {"tactic": "apply measure_mono", "annotated_tactic": ["apply <a>measure_mono</a>", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 21]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : t.Countable\nts : t \u2286 s\nrpos : \u2200 x \u2208 t, 0 < r x\nst : s \u2286 \u22c3 x, closedBall (\u2191x) (r \u2191x)\n\u03bct : \u2211' (x : \u2191t), \u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03bc s + a\nthis : Encodable \u2191t\n\u22a2 \u03bc (f '' s) \u2264 \u03bc (\u22c3 x, f '' (s \u2229 closedBall (\u2191x) (r \u2191x)))", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : t.Countable\nts : t \u2286 s\nrpos : \u2200 x \u2208 t, 0 < r x\nst : s \u2286 \u22c3 x, closedBall (\u2191x) (r \u2191x)\n\u03bct : \u2211' (x : \u2191t), \u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03bc s + a\nthis : Encodable \u2191t\n\u22a2 f '' s \u2286 \u22c3 x, f '' (s \u2229 closedBall (\u2191x) (r \u2191x))"}, {"tactic": "rw [\u2190 image_iUnion, \u2190 inter_iUnion]", "annotated_tactic": ["rw [\u2190 <a>image_iUnion</a>, \u2190 <a>inter_iUnion</a>]", [{"full_name": "Set.image_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [1650, 9], "def_end_pos": [1650, 21]}, {"full_name": "Set.inter_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [482, 9], "def_end_pos": [482, 21]}]], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : t.Countable\nts : t \u2286 s\nrpos : \u2200 x \u2208 t, 0 < r x\nst : s \u2286 \u22c3 x, closedBall (\u2191x) (r \u2191x)\n\u03bct : \u2211' (x : \u2191t), \u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03bc s + a\nthis : Encodable \u2191t\n\u22a2 f '' s \u2286 \u22c3 x, f '' (s \u2229 closedBall (\u2191x) (r \u2191x))", "state_after": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : t.Countable\nts : t \u2286 s\nrpos : \u2200 x \u2208 t, 0 < r x\nst : s \u2286 \u22c3 x, closedBall (\u2191x) (r \u2191x)\n\u03bct : \u2211' (x : \u2191t), \u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03bc s + a\nthis : Encodable \u2191t\n\u22a2 f '' s \u2286 f '' (s \u2229 \u22c3 i, closedBall (\u2191i) (r \u2191i))"}, {"tactic": "exact image_subset _ (subset_inter (Subset.refl _) st)", "annotated_tactic": ["exact <a>image_subset</a> _ (<a>subset_inter</a> (<a>Subset.refl</a> _) st)", [{"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [291, 9], "def_end_pos": [291, 21]}, {"full_name": "Set.subset_inter", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [937, 9], "def_end_pos": [937, 21]}, {"full_name": "Set.Subset.refl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [331, 9], "def_end_pos": [331, 20]}]], "state_before": "case h\nE : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : t.Countable\nts : t \u2286 s\nrpos : \u2200 x \u2208 t, 0 < r x\nst : s \u2286 \u22c3 x, closedBall (\u2191x) (r \u2191x)\n\u03bct : \u2211' (x : \u2191t), \u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03bc s + a\nthis : Encodable \u2191t\n\u22a2 f '' s \u2286 f '' (s \u2229 \u22c3 i, closedBall (\u2191i) (r \u2191i))", "state_after": "no goals"}, {"tactic": "rw [ENNReal.tsum_mul_left]", "annotated_tactic": ["rw [<a>ENNReal.tsum_mul_left</a>]", [{"full_name": "ENNReal.tsum_mul_left", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [928, 19], "def_end_pos": [928, 32]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : t.Countable\nts : t \u2286 s\nrpos : \u2200 x \u2208 t, 0 < r x\nst : s \u2286 \u22c3 x \u2208 t, closedBall x (r x)\n\u03bct : \u2211' (x : \u2191t), \u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03bc s + a\nthis : Encodable \u2191t\n\u22a2 \u2211' (x : \u2191t), \u2191m * \u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u2191m * (\u03bc s + a)", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : t.Countable\nts : t \u2286 s\nrpos : \u2200 x \u2208 t, 0 < r x\nst : s \u2286 \u22c3 x \u2208 t, closedBall x (r x)\n\u03bct : \u2211' (x : \u2191t), \u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03bc s + a\nthis : Encodable \u2191t\n\u22a2 \u2191m * \u2211' (i : \u2191t), \u03bc (closedBall (\u2191i) (r \u2191i)) \u2264 \u2191m * (\u03bc s + a)"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "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 : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis\u271d : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\na : \u211d\u22650\u221e\nha : 0 < a\nt : Set E\nr : E \u2192 \u211d\nt_count : t.Countable\nts : t \u2286 s\nrpos : \u2200 x \u2208 t, 0 < r x\nst : s \u2286 \u22c3 x \u2208 t, closedBall x (r x)\n\u03bct : \u2211' (x : \u2191t), \u03bc (closedBall (\u2191x) (r \u2191x)) \u2264 \u03bc s + a\nthis : Encodable \u2191t\n\u22a2 \u2191m * \u2211' (i : \u2191t), \u03bc (closedBall (\u2191i) (r \u2191i)) \u2264 \u2191m * (\u03bc s + a)", "state_after": "no goals"}, {"tactic": "apply Tendsto.mono_left _ nhdsWithin_le_nhds", "annotated_tactic": ["apply <a>Tendsto.mono_left</a> _ <a>nhdsWithin_le_nhds</a>", [{"full_name": "Filter.Tendsto.mono_left", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3107, 9], "def_end_pos": [3107, 26]}, {"full_name": "nhdsWithin_le_nhds", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [196, 9], "def_end_pos": [196, 27]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[>] 0, \u03bc (f '' s) \u2264 \u2191m * (\u03bc s + a)\n\u22a2 Tendsto (fun a => \u2191m * (\u03bc s + a)) (\ud835\udcdd[>] 0) (\ud835\udcdd (\u2191m * (\u03bc s + 0)))", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[>] 0, \u03bc (f '' s) \u2264 \u2191m * (\u03bc s + a)\n\u22a2 Tendsto (fun a => \u2191m * (\u03bc s + a)) (\ud835\udcdd 0) (\ud835\udcdd (\u2191m * (\u03bc s + 0)))"}, {"tactic": "apply ENNReal.Tendsto.const_mul (tendsto_const_nhds.add tendsto_id)", "annotated_tactic": ["apply <a>ENNReal.Tendsto.const_mul</a> (tendsto_const_nhds.add <a>tendsto_id</a>)", [{"full_name": "ENNReal.Tendsto.const_mul", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [387, 19], "def_end_pos": [387, 36]}, {"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3094, 9], "def_end_pos": [3094, 19]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[>] 0, \u03bc (f '' s) \u2264 \u2191m * (\u03bc s + a)\n\u22a2 Tendsto (fun a => \u2191m * (\u03bc s + a)) (\ud835\udcdd 0) (\ud835\udcdd (\u2191m * (\u03bc s + 0)))", "state_after": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[>] 0, \u03bc (f '' s) \u2264 \u2191m * (\u03bc s + a)\n\u22a2 \u03bc s + 0 \u2260 0 \u2228 \u2191m \u2260 \u22a4"}, {"tactic": "simp only [ENNReal.coe_ne_top, Ne, or_true_iff, not_false_iff]", "annotated_tactic": ["simp only [<a>ENNReal.coe_ne_top</a>, <a>Ne</a>, <a>or_true_iff</a>, <a>not_false_iff</a>]", [{"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [325, 17], "def_end_pos": [325, 27]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "or_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [152, 9], "def_end_pos": [152, 20]}, {"full_name": "not_false_iff", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1371, 9], "def_end_pos": [1371, 22]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \u211d E\ninst\u271d\u2075 : FiniteDimensional \u211d E\ninst\u271d\u2074 : NormedAddCommGroup F\ninst\u271d\u00b3 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf' : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b2 : MeasurableSpace E\ninst\u271d\u00b9 : BorelSpace E\n\u03bc : Measure E\ninst\u271d : \u03bc.IsAddHaarMeasure\nA : E \u2192L[\u211d] E\nm : \u211d\u22650\nhm : ENNReal.ofReal |A.det| < \u2191m\nd : \u211d\u22650\u221e := ENNReal.ofReal |A.det|\n\u03b5 : \u211d\nh\u03b5 : \u03bc (closedBall 0 \u03b5 + \u21d1A '' closedBall 0 1) < \u2191m * \u03bc (closedBall 0 1)\n\u03b5pos : 0 < \u03b5\nthis : Iio \u27e8\u03b5, \u22ef\u27e9 \u2208 \ud835\udcdd 0\n\u03b4 : \u211d\u22650\ns : Set E\nf : E \u2192 E\nhf : ApproximatesLinearOn f A s \u03b4\nh\u03b4 : \u2191\u03b4 < \u03b5\nI : \u2200 (x : E) (r : \u211d), x \u2208 s \u2192 0 \u2264 r \u2192 \u03bc (f '' (s \u2229 closedBall x r)) \u2264 \u2191m * \u03bc (closedBall x r)\nJ : \u2200\u1da0 (a : \u211d\u22650\u221e) in \ud835\udcdd[>] 0, \u03bc (f '' s) \u2264 \u2191m * (\u03bc s + a)\n\u22a2 \u03bc s + 0 \u2260 0 \u2228 \u2191m \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.succAbove_of_succ_le", "start": [1307, 1], "end": [1309, 63], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean", "full_name": "CategoryTheory.LaxBraidedFunctor.comp_toNatTrans", "start": [427, 1], "end": [429, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Concept.lean", "full_name": "extentClosure_union", "start": [99, 1], "end": [101, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Subring/Basic.lean", "full_name": "SubringClass.coe_natCast", "start": [131, 1], "end": [132, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/IsLimit.lean", "full_name": "CategoryTheory.Limits.IsLimit.lift_comp_conePointUniqueUpToIso_inv", "start": [171, 1], "end": [173, 23], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "J : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\nC : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} C\nF : J \u2964 C\nr s t : Cone F\nP : IsLimit s\nQ : IsLimit t\n\u22a2 \u2200 (j : J), (Q.lift r \u226b (P.conePointUniqueUpToIso Q).inv) \u226b s.\u03c0.app j = r.\u03c0.app j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/NatAntidiagonal.lean", "full_name": "List.Nat.map_swap_antidiagonal", "start": [95, 1], "end": [100, 78], "traced_tactics": [{"tactic": "rw [antidiagonal, map_map, \u2190 List.map_reverse, range_eq_range', reverse_range', \u2190\n  range_eq_range', map_map]", "annotated_tactic": ["rw [<a>antidiagonal</a>, <a>map_map</a>, \u2190 <a>List.map_reverse</a>, <a>range_eq_range'</a>, <a>reverse_range'</a>, \u2190\n    <a>range_eq_range'</a>, <a>map_map</a>]", [{"full_name": "List.Nat.antidiagonal", "def_path": "Mathlib/Data/List/NatAntidiagonal.lean", "def_pos": [32, 5], "def_end_pos": [32, 17]}, {"full_name": "List.map_map", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [842, 17], "def_end_pos": [842, 24]}, {"full_name": "List.map_reverse", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [1480, 17], "def_end_pos": [1480, 28]}, {"full_name": "List.range_eq_range'", "def_path": ".lake/packages/batteries/Batteries/Data/List/Lemmas.lean", "def_pos": [1396, 9], "def_end_pos": [1396, 24]}, {"full_name": "List.reverse_range'", "def_path": ".lake/packages/batteries/Batteries/Data/List/Lemmas.lean", "def_pos": [1459, 9], "def_end_pos": [1459, 23]}, {"full_name": "List.range_eq_range'", "def_path": ".lake/packages/batteries/Batteries/Data/List/Lemmas.lean", "def_pos": [1396, 9], "def_end_pos": [1396, 24]}, {"full_name": "List.map_map", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [842, 17], "def_end_pos": [842, 24]}]], "state_before": "n : \u2115\n\u22a2 map Prod.swap (antidiagonal n) = (antidiagonal n).reverse", "state_after": "n : \u2115\n\u22a2 map (Prod.swap \u2218 fun i => (i, n - i)) (range (n + 1)) =\n    map ((fun i => (i, n - i)) \u2218 fun x => 0 + (n + 1) - 1 - x) (range (n + 1))"}, {"tactic": "apply map_congr_left", "annotated_tactic": ["apply <a>map_congr_left</a>", [{"full_name": "List.map_congr_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [761, 9], "def_end_pos": [761, 23]}]], "state_before": "n : \u2115\n\u22a2 map (Prod.swap \u2218 fun i => (i, n - i)) (range (n + 1)) =\n    map ((fun i => (i, n - i)) \u2218 fun x => 0 + (n + 1) - 1 - x) (range (n + 1))", "state_after": "case h\nn : \u2115\n\u22a2 \u2200 a \u2208 range (n + 1), (Prod.swap \u2218 fun i => (i, n - i)) a = ((fun i => (i, n - i)) \u2218 fun x => 0 + (n + 1) - 1 - x) a"}, {"tactic": "simp (config := { contextual := true }) [Nat.sub_sub_self, Nat.lt_succ_iff]", "annotated_tactic": ["simp (config := { contextual := <a>true</a> }) [<a>Nat.sub_sub_self</a>, <a>Nat.lt_succ_iff</a>]", [{"full_name": "Bool.true", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [571, 5], "def_end_pos": [571, 9]}, {"full_name": "Nat.sub_sub_self", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean", "def_pos": [122, 19], "def_end_pos": [122, 31]}, {"full_name": "Nat.lt_succ_iff", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [573, 19], "def_end_pos": [573, 30]}]], "state_before": "case h\nn : \u2115\n\u22a2 \u2200 a \u2208 range (n + 1), (Prod.swap \u2218 fun i => (i, n - i)) a = ((fun i => (i, n - i)) \u2218 fun x => 0 + (n + 1) - 1 - x) a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean", "full_name": "Matrix.det_fromBlocks_zero\u2081\u2082", "start": [726, 1], "end": [729, 19], "traced_tactics": [{"tactic": "rw [\u2190 det_transpose, fromBlocks_transpose, transpose_zero, det_fromBlocks_zero\u2082\u2081, det_transpose,\n  det_transpose]", "annotated_tactic": ["rw [\u2190 <a>det_transpose</a>, <a>fromBlocks_transpose</a>, <a>transpose_zero</a>, <a>det_fromBlocks_zero\u2082\u2081</a>, <a>det_transpose</a>,\n    <a>det_transpose</a>]", [{"full_name": "Matrix.det_transpose", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean", "def_pos": [222, 9], "def_end_pos": [222, 22]}, {"full_name": "Matrix.fromBlocks_transpose", "def_path": "Mathlib/Data/Matrix/Block.lean", "def_pos": [149, 9], "def_end_pos": [149, 29]}, {"full_name": "Matrix.transpose_zero", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [2080, 9], "def_end_pos": [2080, 23]}, {"full_name": "Matrix.det_fromBlocks_zero\u2082\u2081", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean", "def_pos": [671, 9], "def_end_pos": [671, 30]}, {"full_name": "Matrix.det_transpose", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean", "def_pos": [222, 9], "def_end_pos": [222, 22]}, {"full_name": "Matrix.det_transpose", "def_path": 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"https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Subsemiring/Pointwise.lean", "full_name": "Subsemiring.smul_mem_pointwise_smul", "start": [67, 1], "end": [68, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/MonoidLocalization.lean", "full_name": "Submonoid.LocalizationMap.mk'_eq_iff_eq'", "start": [799, 1], "end": [801, 40], "traced_tactics": [{"tactic": "simp only [f.mk'_eq_iff_eq, mul_comm]", "annotated_tactic": ["simp only [f.mk'_eq_iff_eq, <a>mul_comm</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "M : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf : 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v\ninst\u271d : Add \u03b1\na\u271d b c d : WithTop \u03b1\nx y : \u03b1\na : WithTop \u03b1\n\u22a2 a + \u22a4 = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean", "full_name": "CategoryTheory.Limits.prodZeroIso_hom", "start": [84, 1], "end": [85, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/List/Lemmas.lean", "full_name": "List.IsSuffix.filter", "start": [1237, 1], "end": [1240, 42], "traced_tactics": [{"tactic": "obtain \u27e8xs, rfl\u27e9 := h", "annotated_tactic": ["obtain \u27e8xs, rfl\u27e9 := h", []], "state_before": "\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u2081 l\u2082 : List \u03b1\nh : l\u2081 <:+ l\u2082\n\u22a2 List.filter p l\u2081 <:+ List.filter p l\u2082", "state_after": "case intro\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u2081 xs : List \u03b1\n\u22a2 List.filter p l\u2081 <:+ List.filter p (xs ++ l\u2081)"}, {"tactic": "rw [filter_append]", "annotated_tactic": ["rw [<a>filter_append</a>]", [{"full_name": "List.filter_append", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [920, 17], "def_end_pos": [920, 30]}]], "state_before": "case intro\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u2081 xs : List \u03b1\n\u22a2 List.filter p l\u2081 <:+ List.filter p (xs ++ l\u2081)", "state_after": "case intro\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u2081 xs : List \u03b1\n\u22a2 List.filter p l\u2081 <:+ List.filter p xs ++ List.filter p l\u2081"}, {"tactic": "apply suffix_append", "annotated_tactic": ["apply <a>suffix_append</a>", [{"full_name": "List.suffix_append", "def_path": ".lake/packages/batteries/Batteries/Data/List/Lemmas.lean", "def_pos": [1056, 17], "def_end_pos": [1056, 30]}]], "state_before": "case intro\n\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl\u2081 xs : List \u03b1\n\u22a2 List.filter p l\u2081 <:+ List.filter p xs ++ List.filter p l\u2081", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "BoundedContinuousFunction.coeFn_toLp", "start": [1843, 1], "end": [1845, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean", "full_name": "ContinuousMultilinearMap.pi_apply", "start": [264, 1], "end": [267, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean", "full_name": "ContinuousLinearMap.hasMFDerivWithinAt", "start": [42, 11], "end": [43, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Coprime/Basic.lean", "full_name": "IsCoprime.neg_neg_iff", "start": [415, 1], "end": [416, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Module/Defs.lean", "full_name": "smul_lt_of_lt_one_left", "start": [690, 1], "end": [691, 61], "traced_tactics": [{"tactic": "simpa only [one_smul] using smul_lt_smul_of_pos_right h hb", "annotated_tactic": ["simpa only [<a>one_smul</a>] using <a>smul_lt_smul_of_pos_right</a> h hb", [{"full_name": "one_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [453, 7], "def_end_pos": [453, 15]}, {"full_name": "smul_lt_smul_of_pos_right", "def_path": "Mathlib/Algebra/Order/Module/Defs.lean", "def_pos": [317, 17], "def_end_pos": [317, 42]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u2076 : Monoid \u03b1\ninst\u271d\u2075 : Zero \u03b1\ninst\u271d\u2074 : Zero \u03b2\ninst\u271d\u00b3 : MulAction \u03b1 \u03b2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : SMulPosStrictMono \u03b1 \u03b2\nhb : 0 < b\nh : a < 1\n\u22a2 a \u2022 b < b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Bounded.lean", "full_name": "Bornology.IsBounded.subset_closedBall", "start": [76, 1], "end": [78, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/SplittingField/IsSplittingField.lean", "full_name": "Polynomial.IsSplittingField.of_algEquiv", "start": [136, 1], "end": [142, 64], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "F : Type u\nK : Type v\nL : Type w\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\np : K[X]\nf : F \u2243\u2090[K] L\ninst\u271d : IsSplittingField K F p\n\u22a2 IsSplittingField K L p", "state_after": "case splits'\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\np : K[X]\nf : F \u2243\u2090[K] L\ninst\u271d : IsSplittingField K F p\n\u22a2 Splits (algebraMap K L) p\n\ncase adjoin_rootSet'\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\np : K[X]\nf : F \u2243\u2090[K] L\ninst\u271d : IsSplittingField K F p\n\u22a2 Algebra.adjoin K (p.rootSet L) = \u22a4"}, {"tactic": "rw [\u2190 f.toAlgHom.comp_algebraMap]", "annotated_tactic": ["rw [\u2190 f.toAlgHom.comp_algebraMap]", []], "state_before": "case splits'\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\np : K[X]\nf : F \u2243\u2090[K] L\ninst\u271d : IsSplittingField K F p\n\u22a2 Splits (algebraMap K L) p", "state_after": "case splits'\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\np : K[X]\nf : F \u2243\u2090[K] L\ninst\u271d : IsSplittingField K F p\n\u22a2 Splits ((\u2191\u2191f).comp (algebraMap K F)) p"}, {"tactic": "exact splits_comp_of_splits _ _ (splits F p)", "annotated_tactic": ["exact <a>splits_comp_of_splits</a> _ _ (<a>splits</a> F p)", [{"full_name": "Polynomial.splits_comp_of_splits", "def_path": "Mathlib/Algebra/Polynomial/Splits.lean", "def_pos": [440, 9], "def_end_pos": [440, 30]}, {"full_name": "Polynomial.IsSplittingField.splits", "def_path": "Mathlib/FieldTheory/SplittingField/IsSplittingField.lean", "def_pos": [57, 9], "def_end_pos": [57, 15]}]], "state_before": "case splits'\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\np : K[X]\nf : F \u2243\u2090[K] L\ninst\u271d : IsSplittingField K F p\n\u22a2 Splits ((\u2191\u2191f).comp (algebraMap K F)) p", "state_after": "no goals"}, {"tactic": "rw [\u2190 (Algebra.range_top_iff_surjective f.toAlgHom).mpr f.surjective,\n  adjoin_rootSet_eq_range (splits F p), adjoin_rootSet F p]", "annotated_tactic": ["rw [\u2190 (<a>Algebra.range_top_iff_surjective</a> f.toAlgHom).<a>mpr</a> f.surjective,\n      <a>adjoin_rootSet_eq_range</a> (<a>splits</a> F p), <a>adjoin_rootSet</a> F p]", [{"full_name": "Algebra.range_top_iff_surjective", "def_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "def_pos": [899, 9], "def_end_pos": [899, 33]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}, {"full_name": "Polynomial.adjoin_rootSet_eq_range", "def_path": "Mathlib/Algebra/Polynomial/Splits.lean", "def_pos": [338, 9], "def_end_pos": [338, 32]}, {"full_name": "Polynomial.IsSplittingField.splits", "def_path": "Mathlib/FieldTheory/SplittingField/IsSplittingField.lean", "def_pos": [57, 9], "def_end_pos": [57, 15]}, {"full_name": "Polynomial.IsSplittingField.adjoin_rootSet", "def_path": "Mathlib/FieldTheory/SplittingField/IsSplittingField.lean", "def_pos": [63, 9], "def_end_pos": [63, 23]}]], "state_before": "case adjoin_rootSet'\nF : Type u\nK : Type v\nL : Type w\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Field L\ninst\u271d\u00b3 : Field F\ninst\u271d\u00b2 : Algebra K L\ninst\u271d\u00b9 : Algebra K F\np : K[X]\nf : F \u2243\u2090[K] L\ninst\u271d : IsSplittingField K F p\n\u22a2 Algebra.adjoin K (p.rootSet L) = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/RCLike/Basic.lean", "full_name": "RCLike.ofReal_zero", "start": [148, 1], "end": [149, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/AddConstMap/Basic.lean", "full_name": "AddConstMap.one_def", "start": [388, 1], "end": [388, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Connected/PathConnected.lean", "full_name": "IsPathConnected.mem_pathComponent", "start": [1021, 1], "end": [1023, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/RCLike/Basic.lean", "full_name": "RCLike.conj_inv", "start": [562, 1], "end": [563, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Normed/Field/InfiniteSum.lean", "full_name": "Summable.mul_of_nonneg", "start": [32, 1], "end": [35, 71], "traced_tactics": [{"tactic": "simpa only [hg.tsum_mul_left _] using hf.mul_right (\u2211' x, g x)", "annotated_tactic": ["simpa only [hg.tsum_mul_left _] using hf.mul_right (\u2211' x, g x)", []], "state_before": "R : Type u_1\n\u03b9 : Type u_2\n\u03b9' : Type u_3\ninst\u271d : NormedRing R\nf : \u03b9 \u2192 \u211d\ng : \u03b9' \u2192 \u211d\nhf : Summable f\nhg : Summable g\nhf' : 0 \u2264 f\nhg' : 0 \u2264 g\n\u22a2 Summable fun x => \u2211' (y : \u03b9'), f (x, y).1 * g (x, y).2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/InfiniteSum/Module.lean", "full_name": "HasSum.smul_eq", "start": [96, 1], "end": [101, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/ModEq.lean", "full_name": "Nat.ModEq.mul_right_cancel'", "start": [199, 11], "end": [201, 86], "traced_tactics": [{"tactic": "simp [modEq_iff_dvd, \u2190 sub_mul, mul_dvd_mul_iff_right (by simp [hc] : (c : \u2124) \u2260 0)]", "annotated_tactic": ["simp [<a>modEq_iff_dvd</a>, \u2190 <a>sub_mul</a>, <a>mul_dvd_mul_iff_right</a> (by simp [hc] : (c : \u2124) \u2260 0)]", [{"full_name": "Nat.modEq_iff_dvd", "def_path": "Mathlib/Data/Nat/ModEq.lean", "def_pos": [89, 9], "def_end_pos": [89, 22]}, {"full_name": "sub_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [401, 7], "def_end_pos": [401, 14]}, {"full_name": "mul_dvd_mul_iff_right", "def_path": "Mathlib/Algebra/GroupWithZero/Divisibility.lean", "def_pos": [56, 9], "def_end_pos": [56, 30]}]], "state_before": "m\u271d n a\u271d b\u271d c\u271d d a b c m : \u2115\nhc : c \u2260 0\n\u22a2 a * c \u2261 b * c [MOD m * c] \u2192 a \u2261 b [MOD m]", "state_after": "no goals"}, {"tactic": "simp [hc]", "annotated_tactic": ["simp [hc]", []], "state_before": "m\u271d n a\u271d b\u271d c\u271d d a b c m : \u2115\nhc : c \u2260 0\n\u22a2 \u2191c \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/RCLike/Basic.lean", "full_name": "RCLike.re_ofReal_mul", "start": [253, 1], "end": [254, 63], "traced_tactics": [{"tactic": "simp only [mul_re, ofReal_im, zero_mul, ofReal_re, sub_zero]", "annotated_tactic": ["simp only [<a>mul_re</a>, <a>ofReal_im</a>, <a>zero_mul</a>, <a>ofReal_re</a>, <a>sub_zero</a>]", [{"full_name": "RCLike.mul_re", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [130, 9], "def_end_pos": [130, 15]}, {"full_name": "RCLike.ofReal_im", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [125, 9], "def_end_pos": [125, 18]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}, {"full_name": "RCLike.ofReal_re", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [120, 9], "def_end_pos": [120, 18]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [489, 3], "def_end_pos": [489, 14]}]], "state_before": "K : Type u_1\nE : Type u_2\ninst\u271d : RCLike K\nr : \u211d\nz : K\n\u22a2 re (\u2191r * z) = r * re z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/Colex.lean", "full_name": "Finset.Colex.IsInitSeg.exists_initSeg", "start": [365, 1], "end": [378, 39], "traced_tactics": [{"tactic": "have hs := sup'_mem (ofColex \u207b\u00b9' \ud835\udc9c) (LinearOrder.supClosed _) \ud835\udc9c h\ud835\udc9c\u2080 toColex\n  (fun a ha \u21a6 by simpa using ha)", "annotated_tactic": ["have hs := <a>sup'_mem</a> (<a>ofColex</a> \u207b\u00b9' \ud835\udc9c) (<a>LinearOrder.supClosed</a> _) \ud835\udc9c h\ud835\udc9c\u2080 <a>toColex</a>\n    (fun a ha \u21a6 by simpa using ha)", [{"full_name": "Finset.sup'_mem", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [927, 9], "def_end_pos": [927, 17]}, {"full_name": "Finset.Colex.ofColex", "def_path": "Mathlib/Combinatorics/Colex.lean", "def_pos": [75, 4], "def_end_pos": [75, 11]}, {"full_name": "LinearOrder.supClosed", "def_path": "Mathlib/Order/SupClosed.lean", "def_pos": [242, 25], "def_end_pos": [242, 46]}, {"full_name": "Finset.Colex.toColex", "def_path": "Mathlib/Combinatorics/Colex.lean", "def_pos": [73, 3], "def_end_pos": [73, 10]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t u : Finset \u03b1\na b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\n\u22a2 \u2203 s, s.card = r \u2227 \ud835\udc9c = initSeg s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t u : Finset \u03b1\na b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\nhs : \ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex \u2208 ofColex \u207b\u00b9' \u2191\ud835\udc9c\n\u22a2 \u2203 s, s.card = r \u2227 \ud835\udc9c = initSeg s"}, {"tactic": "refine \u27e8_, h\ud835\udc9c.1 hs, ?_\u27e9", "annotated_tactic": ["refine \u27e8_, h\ud835\udc9c.1 hs, ?_\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t u : Finset \u03b1\na b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\nhs : \ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex \u2208 ofColex \u207b\u00b9' \u2191\ud835\udc9c\n\u22a2 \u2203 s, s.card = r \u2227 \ud835\udc9c = initSeg s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t u : Finset \u03b1\na b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\nhs : \ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex \u2208 ofColex \u207b\u00b9' \u2191\ud835\udc9c\n\u22a2 \ud835\udc9c = initSeg (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex"}, {"tactic": "ext t", "annotated_tactic": ["ext t", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t u : Finset \u03b1\na b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\nhs : \ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex \u2208 ofColex \u207b\u00b9' \u2191\ud835\udc9c\n\u22a2 \ud835\udc9c = initSeg (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t\u271d u : Finset \u03b1\na b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\nhs : \ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex \u2208 ofColex \u207b\u00b9' \u2191\ud835\udc9c\nt : Finset \u03b1\n\u22a2 t \u2208 \ud835\udc9c \u2194 t \u2208 initSeg (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex"}, {"tactic": "rw [mem_initSeg]", "annotated_tactic": ["rw [<a>mem_initSeg</a>]", [{"full_name": "Finset.Colex.mem_initSeg", "def_path": "Mathlib/Combinatorics/Colex.lean", "def_pos": [355, 7], "def_end_pos": [355, 18]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t\u271d u : Finset \u03b1\na b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\nhs : \ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex \u2208 ofColex \u207b\u00b9' \u2191\ud835\udc9c\nt : Finset \u03b1\n\u22a2 t \u2208 \ud835\udc9c \u2194 t \u2208 initSeg (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t\u271d u : Finset \u03b1\na b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\nhs : \ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex \u2208 ofColex \u207b\u00b9' \u2191\ud835\udc9c\nt : Finset \u03b1\n\u22a2 t \u2208 \ud835\udc9c \u2194 (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex.card = t.card \u2227 { ofColex := t } \u2264 { ofColex := (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex }"}, {"tactic": "refine \u27e8fun p \u21a6 ?_, ?_\u27e9", "annotated_tactic": ["refine \u27e8fun p \u21a6 ?_, ?_\u27e9", []], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t\u271d u : Finset \u03b1\na b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\nhs : \ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex \u2208 ofColex \u207b\u00b9' \u2191\ud835\udc9c\nt : Finset \u03b1\n\u22a2 t \u2208 \ud835\udc9c \u2194 (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex.card = t.card \u2227 { ofColex := t } \u2264 { ofColex := (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex }", "state_after": "case a.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t\u271d u : Finset \u03b1\na b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\nhs : \ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex \u2208 ofColex \u207b\u00b9' \u2191\ud835\udc9c\nt : Finset \u03b1\np : t \u2208 \ud835\udc9c\n\u22a2 (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex.card = t.card \u2227 { ofColex := t } \u2264 { ofColex := (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex }\n\ncase a.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t\u271d u : Finset \u03b1\na b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\nhs : \ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex \u2208 ofColex \u207b\u00b9' \u2191\ud835\udc9c\nt : Finset \u03b1\n\u22a2 (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex.card = t.card \u2227 { ofColex := t } \u2264 { ofColex := (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex } \u2192 t \u2208 \ud835\udc9c"}, {"tactic": "rintro \u27e8cards, le\u27e9", "annotated_tactic": ["rintro \u27e8cards, le\u27e9", []], "state_before": "case a.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t\u271d u : Finset \u03b1\na b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\nhs : \ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex \u2208 ofColex \u207b\u00b9' \u2191\ud835\udc9c\nt : Finset \u03b1\n\u22a2 (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex.card = t.card \u2227 { ofColex := t } \u2264 { ofColex := (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex } \u2192 t \u2208 \ud835\udc9c", "state_after": "case a.refine_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t\u271d u : Finset \u03b1\na b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\nhs : \ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex \u2208 ofColex \u207b\u00b9' \u2191\ud835\udc9c\nt : Finset \u03b1\ncards : (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex.card = t.card\nle : { ofColex := t } \u2264 { ofColex := (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex }\n\u22a2 t \u2208 \ud835\udc9c"}, {"tactic": "obtain p | p := le.eq_or_lt", "annotated_tactic": ["obtain p | p := le.eq_or_lt", []], "state_before": "case a.refine_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t\u271d u : Finset \u03b1\na b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\nhs : \ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex \u2208 ofColex \u207b\u00b9' \u2191\ud835\udc9c\nt : Finset \u03b1\ncards : (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex.card = t.card\nle : { ofColex := t } \u2264 { ofColex := (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex }\n\u22a2 t \u2208 \ud835\udc9c", "state_after": "case a.refine_2.intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t\u271d u : Finset \u03b1\na b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\nhs : \ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex \u2208 ofColex \u207b\u00b9' \u2191\ud835\udc9c\nt : Finset \u03b1\ncards : (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex.card = t.card\nle : { ofColex := t } \u2264 { ofColex := (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex }\np : { ofColex := t } = { ofColex := (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex }\n\u22a2 t \u2208 \ud835\udc9c\n\ncase a.refine_2.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t\u271d u : Finset \u03b1\na b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\nhs : \ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex \u2208 ofColex \u207b\u00b9' \u2191\ud835\udc9c\nt : Finset \u03b1\ncards : (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex.card = t.card\nle : { ofColex := t } \u2264 { ofColex := (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex }\np : { ofColex := t } < { ofColex := (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex }\n\u22a2 t \u2208 \ud835\udc9c"}, {"tactic": "simpa using ha", "annotated_tactic": ["simpa using ha", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t u : Finset \u03b1\na\u271d b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\na : Finset \u03b1\nha : a \u2208 \ud835\udc9c\n\u22a2 { ofColex := a } \u2208 ofColex \u207b\u00b9' \u2191\ud835\udc9c", "state_after": "no goals"}, {"tactic": "rw [h\ud835\udc9c.1 p, h\ud835\udc9c.1 hs]", "annotated_tactic": ["rw [h\ud835\udc9c.1 p, h\ud835\udc9c.1 hs]", []], "state_before": "case a.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t\u271d u : Finset \u03b1\na b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\nhs : \ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex \u2208 ofColex \u207b\u00b9' \u2191\ud835\udc9c\nt : Finset \u03b1\np : t \u2208 \ud835\udc9c\n\u22a2 (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex.card = t.card \u2227 { ofColex := t } \u2264 { ofColex := (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex }", "state_after": "case a.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t\u271d u : Finset \u03b1\na b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\nhs : \ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex \u2208 ofColex \u207b\u00b9' \u2191\ud835\udc9c\nt : Finset \u03b1\np : t \u2208 \ud835\udc9c\n\u22a2 r = r \u2227 { ofColex := t } \u2264 { ofColex := (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex }"}, {"tactic": "exact \u27e8rfl, le_sup' _ p\u27e9", "annotated_tactic": ["exact \u27e8<a>rfl</a>, <a>le_sup'</a> _ p\u27e9", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "Finset.le_sup'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [852, 9], "def_end_pos": [852, 16]}]], "state_before": "case a.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t\u271d u : Finset \u03b1\na b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\nhs : \ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex \u2208 ofColex \u207b\u00b9' \u2191\ud835\udc9c\nt : Finset \u03b1\np : t \u2208 \ud835\udc9c\n\u22a2 r = r \u2227 { ofColex := t } \u2264 { ofColex := (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex }", "state_after": "no goals"}, {"tactic": "rwa [toColex_inj.1 p]", "annotated_tactic": ["rwa [<a>toColex_inj</a>.1 p]", [{"full_name": "Finset.toColex_inj", "def_path": "Mathlib/Combinatorics/Colex.lean", "def_pos": [86, 7], "def_end_pos": [86, 18]}]], "state_before": "case a.refine_2.intro.inl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t\u271d u : Finset \u03b1\na b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\nhs : \ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex \u2208 ofColex \u207b\u00b9' \u2191\ud835\udc9c\nt : Finset \u03b1\ncards : (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex.card = t.card\nle : { ofColex := t } \u2264 { ofColex := (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex }\np : { ofColex := t } = { ofColex := (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex }\n\u22a2 t \u2208 \ud835\udc9c", "state_after": "no goals"}, {"tactic": "exact h\ud835\udc9c.2 hs \u27e8p, cards \u25b8 h\ud835\udc9c.1 hs\u27e9", "annotated_tactic": ["exact h\ud835\udc9c.2 hs \u27e8p, cards \u25b8 h\ud835\udc9c.1 hs\u27e9", []], "state_before": "case a.refine_2.intro.inr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : LinearOrder \u03b2\nf : \u03b1 \u2192 \u03b2\n\ud835\udc9c \ud835\udc9c\u2081 \ud835\udc9c\u2082 : Finset (Finset \u03b1)\ns t\u271d u : Finset \u03b1\na b : \u03b1\nr : \u2115\ninst\u271d : Fintype \u03b1\nh\ud835\udc9c : IsInitSeg \ud835\udc9c r\nh\ud835\udc9c\u2080 : \ud835\udc9c.Nonempty\nhs : \ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex \u2208 ofColex \u207b\u00b9' \u2191\ud835\udc9c\nt : Finset \u03b1\ncards : (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex.card = t.card\nle : { ofColex := t } \u2264 { ofColex := (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex }\np : { ofColex := t } < { ofColex := (\ud835\udc9c.sup' h\ud835\udc9c\u2080 toColex).ofColex }\n\u22a2 t \u2208 \ud835\udc9c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Star/NonUnitalSubalgebra.lean", "full_name": "NonUnitalStarSubalgebra.toNonUnitalSubring_injective", "start": [190, 1], "end": [193, 90], "traced_tactics": [{"tactic": "rw [\u2190 mem_toNonUnitalSubring, \u2190 mem_toNonUnitalSubring, h]", "annotated_tactic": ["rw [\u2190 <a>mem_toNonUnitalSubring</a>, \u2190 <a>mem_toNonUnitalSubring</a>, h]", [{"full_name": "NonUnitalStarSubalgebra.mem_toNonUnitalSubring", "def_path": "Mathlib/Algebra/Star/NonUnitalSubalgebra.lean", "def_pos": [181, 9], "def_end_pos": [181, 31]}, {"full_name": "NonUnitalStarSubalgebra.mem_toNonUnitalSubring", "def_path": "Mathlib/Algebra/Star/NonUnitalSubalgebra.lean", "def_pos": [181, 9], "def_end_pos": [181, 31]}]], "state_before": "F : Type v'\nR' : Type u'\nR\u271d : Type u\nA\u271d : Type v\nB : Type w\nC : Type w'\ninst\u271d\u00b9\u2076 : CommSemiring R\u271d\ninst\u271d\u00b9\u2075 : NonUnitalNonAssocSemiring A\u271d\ninst\u271d\u00b9\u2074 : Module R\u271d A\u271d\ninst\u271d\u00b9\u00b3 : Star A\u271d\ninst\u271d\u00b9\u00b2 : NonUnitalNonAssocSemiring B\ninst\u271d\u00b9\u00b9 : Module R\u271d B\ninst\u271d\u00b9\u2070 : Star B\ninst\u271d\u2079 : NonUnitalNonAssocSemiring C\ninst\u271d\u2078 : Module R\u271d C\ninst\u271d\u2077 : Star C\ninst\u271d\u2076 : FunLike F A\u271d B\ninst\u271d\u2075 : NonUnitalAlgHomClass F R\u271d A\u271d B\ninst\u271d\u2074 : NonUnitalStarAlgHomClass F R\u271d A\u271d B\nS\u271d : NonUnitalStarSubalgebra R\u271d A\u271d\nR : Type u\nA : Type v\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : NonUnitalRing A\ninst\u271d\u00b9 : Module R A\ninst\u271d : Star A\nS T : NonUnitalStarSubalgebra R A\nh : S.toNonUnitalSubring = T.toNonUnitalSubring\nx : A\n\u22a2 x \u2208 S \u2194 x \u2208 T", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/Int.lean", "full_name": "Int.nnnorm_coe_units", "start": [24, 1], "end": [26, 73], "traced_tactics": [{"tactic": "obtain rfl | rfl := units_eq_one_or e <;>\n  simp only [Units.coe_neg_one, Units.val_one, nnnorm_neg, nnnorm_one]", "annotated_tactic": ["obtain rfl | rfl := <a>units_eq_one_or</a> e <;>\n    simp only [<a>Units.coe_neg_one</a>, <a>Units.val_one</a>, <a>nnnorm_neg</a>, <a>nnnorm_one</a>]", [{"full_name": "Int.units_eq_one_or", "def_path": "Mathlib/Algebra/Ring/Int.lean", "def_pos": [89, 7], "def_end_pos": [89, 22]}, {"full_name": "Units.coe_neg_one", "def_path": "Mathlib/Algebra/Ring/Units.lean", "def_pos": [42, 19], "def_end_pos": [42, 30]}, {"full_name": "Units.val_one", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [232, 9], "def_end_pos": [232, 16]}, {"full_name": "nnnorm_neg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [811, 30], "def_end_pos": [811, 40]}, {"full_name": "nnnorm_one", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [178, 9], "def_end_pos": [178, 19]}]], "state_before": "e : \u2124\u02e3\n\u22a2 \u2016\u2191e\u2016\u208a = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "full_name": "HasCompactSupport.rpow_const", "start": [266, 11], "end": [268, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/RegularExpressions.lean", "full_name": "RegularExpression.one_def", "start": [93, 1], "end": [94, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean", "full_name": "TopCat.Sheaf.interUnionPullbackCone_snd", "start": [335, 1], "end": [337, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.one_le_lift_iff", "start": [1370, 1], "end": [1372, 39], "traced_tactics": [{"tactic": "simpa using nat_le_lift_iff (n := 1)", "annotated_tactic": ["simpa using <a>nat_le_lift_iff</a> (n := 1)", [{"full_name": "Cardinal.nat_le_lift_iff", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1365, 9], "def_end_pos": [1365, 24]}]], "state_before": "\u03b1 \u03b2 : Type u\na : Cardinal.{u}\n\u22a2 1 \u2264 lift.{v, u} a \u2194 1 \u2264 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GradedMonoid.lean", "full_name": "GradedMonoid.fst_pow", "start": [257, 9], "end": [258, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.tr_push", "start": [1091, 1], "end": [1092, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Normed/Group/Quotient.lean", "full_name": "Ideal.Quotient.norm_mk_lt", "start": [491, 8], "end": [493, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Artinian.lean", "full_name": "isArtinian_of_submodule_of_artinian", "start": [377, 1], "end": [378, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Function/Basic.lean", "full_name": "Function.FactorsThrough.extend_comp", "start": [774, 1], "end": [776, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Regular/IsSMulRegular.lean", "full_name": "mem_of_isSMulRegular_on_quot_of_smul_mem", "start": [108, 1], "end": [110, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean", "full_name": "HurwitzZeta.hurwitzZeta_neg_two_mul_nat", "start": [182, 1], "end": [188, 90], "traced_tactics": [{"tactic": "suffices hurwitzZetaEven x (-(2 * k)) = 0 by\n  rw [hurwitzZeta, this, zero_add, hurwitzZetaOdd_neg_two_mul_nat hk hx]", "annotated_tactic": ["suffices <a>hurwitzZetaEven</a> x (-(2 * k)) = 0 by\n    rw [<a>hurwitzZeta</a>, this, <a>zero_add</a>, <a>hurwitzZetaOdd_neg_two_mul_nat</a> hk hx]", [{"full_name": "HurwitzZeta.hurwitzZetaEven", "def_path": "Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean", "def_pos": [601, 19], "def_end_pos": [601, 34]}, {"full_name": "HurwitzZeta.hurwitzZeta", "def_path": "Mathlib/NumberTheory/LSeries/HurwitzZeta.lean", "def_pos": [49, 19], "def_end_pos": [49, 30]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [476, 3], "def_end_pos": [476, 14]}, {"full_name": "HurwitzZeta.hurwitzZetaOdd_neg_two_mul_nat", "def_path": "Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean", "def_pos": [148, 9], "def_end_pos": [148, 39]}]], "state_before": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\n\u22a2 hurwitzZeta (\u2191x) (-(2 * \u2191k)) =\n    -1 / (2 * \u2191k + 1) * Polynomial.eval (\u2191x) (Polynomial.map (algebraMap \u211a \u2102) (Polynomial.bernoulli (2 * k + 1)))", "state_after": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\n\u22a2 hurwitzZetaEven (\u2191x) (-(2 * \u2191k)) = 0"}, {"tactic": "obtain \u27e8k, rfl\u27e9 := Nat.exists_eq_succ_of_ne_zero hk", "annotated_tactic": ["obtain \u27e8k, rfl\u27e9 := <a>Nat.exists_eq_succ_of_ne_zero</a> hk", [{"full_name": "Nat.exists_eq_succ_of_ne_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [691, 9], "def_end_pos": [691, 34]}]], "state_before": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\n\u22a2 hurwitzZetaEven (\u2191x) (-(2 * \u2191k)) = 0", "state_after": "case intro\nx : \u211d\nhx : x \u2208 Icc 0 1\nk : \u2115\nhk : k.succ \u2260 0\n\u22a2 hurwitzZetaEven (\u2191x) (-(2 * \u2191k.succ)) = 0"}, {"tactic": "simpa only [Nat.cast_succ, \u2190 neg_mul] using hurwitzZetaEven_neg_two_mul_nat_add_one x k", "annotated_tactic": ["simpa only [<a>Nat.cast_succ</a>, \u2190 <a>neg_mul</a>] using <a>hurwitzZetaEven_neg_two_mul_nat_add_one</a> x k", [{"full_name": "Nat.cast_succ", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [135, 9], "def_end_pos": [135, 18]}, {"full_name": "neg_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "HurwitzZeta.hurwitzZetaEven_neg_two_mul_nat_add_one", "def_path": "Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean", "def_pos": [623, 9], "def_end_pos": [623, 48]}]], "state_before": "case intro\nx : \u211d\nhx : x \u2208 Icc 0 1\nk : \u2115\nhk : k.succ \u2260 0\n\u22a2 hurwitzZetaEven (\u2191x) (-(2 * \u2191k.succ)) = 0", "state_after": "no goals"}, {"tactic": "rw [hurwitzZeta, this, zero_add, hurwitzZetaOdd_neg_two_mul_nat hk hx]", "annotated_tactic": ["rw [<a>hurwitzZeta</a>, this, <a>zero_add</a>, <a>hurwitzZetaOdd_neg_two_mul_nat</a> hk hx]", [{"full_name": "HurwitzZeta.hurwitzZeta", "def_path": "Mathlib/NumberTheory/LSeries/HurwitzZeta.lean", "def_pos": [49, 19], "def_end_pos": [49, 30]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [476, 3], "def_end_pos": [476, 14]}, {"full_name": "HurwitzZeta.hurwitzZetaOdd_neg_two_mul_nat", "def_path": "Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean", "def_pos": [148, 9], "def_end_pos": [148, 39]}]], "state_before": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\nthis : hurwitzZetaEven (\u2191x) (-(2 * \u2191k)) = 0\n\u22a2 hurwitzZeta (\u2191x) (-(2 * \u2191k)) =\n    -1 / (2 * \u2191k + 1) * Polynomial.eval (\u2191x) (Polynomial.map (algebraMap \u211a \u2102) (Polynomial.bernoulli (2 * k + 1)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/Irrational.lean", "full_name": "irrational_sqrt_two", "start": [143, 1], "end": [144, 44], "traced_tactics": [{"tactic": "simpa using Nat.prime_two.irrational_sqrt", "annotated_tactic": ["simpa using Nat.prime_two.irrational_sqrt", []], "state_before": "\u22a2 Irrational \u221a2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/ContinuousFunction/ContinuousMapZero.lean", "full_name": "ContinuousMapZero.comp_apply", "start": [71, 1], "end": [72, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "full_name": "equicontinuous_restrict_iff", "start": [191, 1], "end": [193, 73], "traced_tactics": [{"tactic": "simp [Equicontinuous, EquicontinuousOn, equicontinuousAt_restrict_iff]", "annotated_tactic": ["simp [<a>Equicontinuous</a>, <a>EquicontinuousOn</a>, <a>equicontinuousAt_restrict_iff</a>]", [{"full_name": "Equicontinuous", "def_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "def_pos": [117, 5], "def_end_pos": [117, 19]}, {"full_name": "EquicontinuousOn", "def_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "def_pos": [129, 5], "def_end_pos": [129, 21]}, {"full_name": "equicontinuousAt_restrict_iff", "def_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "def_pos": [174, 7], "def_end_pos": [174, 36]}]], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\nX : Type u_3\nX' : Type u_4\nY : Type u_5\nZ : Type u_6\n\u03b1 : Type u_7\n\u03b1' : Type u_8\n\u03b2 : Type u_9\n\u03b2' : Type u_10\n\u03b3 : Type u_11\n\ud835\udcd5 : Type u_12\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\ntZ : TopologicalSpace Z\nu\u03b1 : UniformSpace \u03b1\nu\u03b2 : UniformSpace \u03b2\nu\u03b3 : UniformSpace \u03b3\nF : \u03b9 \u2192 X \u2192 \u03b1\nS : Set X\n\u22a2 Equicontinuous (S.restrict \u2218 F) \u2194 EquicontinuousOn F S", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ENNReal/Real.lean", "full_name": "ENNReal.trichotomy\u2082", "start": [471, 11], "end": [484, 94], "traced_tactics": [{"tactic": "rcases eq_or_lt_of_le (bot_le : 0 \u2264 p) with ((rfl : 0 = p) | (hp : 0 < p))", "annotated_tactic": ["rcases <a>eq_or_lt_of_le</a> (<a>bot_le</a> : 0 \u2264 p) with ((rfl : 0 = p) | (hp : 0 < p))", [{"full_name": "eq_or_lt_of_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [381, 9], "def_end_pos": [381, 23]}, {"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [224, 9], "def_end_pos": [224, 15]}]], "state_before": "a b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\n\u22a2 p = 0 \u2227 q = 0 \u2228\n    p = 0 \u2227 q = \u22a4 \u2228\n      p = 0 \u2227 0 < q.toReal \u2228 p = \u22a4 \u2227 q = \u22a4 \u2228 0 < p.toReal \u2227 q = \u22a4 \u2228 0 < p.toReal \u2227 0 < q.toReal \u2227 p.toReal \u2264 q.toReal", "state_after": "case inl\na b c d : \u211d\u22650\u221e\nr p q\u271d : \u211d\u22650\nq : \u211d\u22650\u221e\nhpq : 0 \u2264 q\n\u22a2 0 = 0 \u2227 q = 0 \u2228\n    0 = 0 \u2227 q = \u22a4 \u2228\n      0 = 0 \u2227 0 < q.toReal \u2228\n        0 = \u22a4 \u2227 q = \u22a4 \u2228 0 < ENNReal.toReal 0 \u2227 q = \u22a4 \u2228 0 < ENNReal.toReal 0 \u2227 0 < q.toReal \u2227 ENNReal.toReal 0 \u2264 q.toReal\n\ncase inr\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\n\u22a2 p = 0 \u2227 q = 0 \u2228\n    p = 0 \u2227 q = \u22a4 \u2228\n      p = 0 \u2227 0 < q.toReal \u2228 p = \u22a4 \u2227 q = \u22a4 \u2228 0 < p.toReal \u2227 q = \u22a4 \u2228 0 < p.toReal \u2227 0 < q.toReal \u2227 p.toReal \u2264 q.toReal"}, {"tactic": "rcases eq_or_lt_of_le (le_top : q \u2264 \u221e) with (rfl | hq)", "annotated_tactic": ["rcases <a>eq_or_lt_of_le</a> (<a>le_top</a> : q \u2264 \u221e) with (rfl | hq)", [{"full_name": "eq_or_lt_of_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [381, 9], "def_end_pos": [381, 23]}, {"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [66, 9], "def_end_pos": [66, 15]}]], "state_before": "case inr\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\n\u22a2 p = 0 \u2227 q = 0 \u2228\n    p = 0 \u2227 q = \u22a4 \u2228\n      p = 0 \u2227 0 < q.toReal \u2228 p = \u22a4 \u2227 q = \u22a4 \u2228 0 < p.toReal \u2227 q = \u22a4 \u2228 0 < p.toReal \u2227 0 < q.toReal \u2227 p.toReal \u2264 q.toReal", "state_after": "case inr.inl\na b c d : \u211d\u22650\u221e\nr p\u271d q : \u211d\u22650\np : \u211d\u22650\u221e\nhp : 0 < p\nhpq : p \u2264 \u22a4\n\u22a2 p = 0 \u2227 \u22a4 = 0 \u2228\n    p = 0 \u2227 \u22a4 = \u22a4 \u2228\n      p = 0 \u2227 0 < \u22a4.toReal \u2228 p = \u22a4 \u2227 \u22a4 = \u22a4 \u2228 0 < p.toReal \u2227 \u22a4 = \u22a4 \u2228 0 < p.toReal \u2227 0 < \u22a4.toReal \u2227 p.toReal \u2264 \u22a4.toReal\n\ncase inr.inr\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\n\u22a2 p = 0 \u2227 q = 0 \u2228\n    p = 0 \u2227 q = \u22a4 \u2228\n      p = 0 \u2227 0 < q.toReal \u2228 p = \u22a4 \u2227 q = \u22a4 \u2228 0 < p.toReal \u2227 q = \u22a4 \u2228 0 < p.toReal \u2227 0 < q.toReal \u2227 p.toReal \u2264 q.toReal"}, {"tactic": "repeat' right", "annotated_tactic": ["repeat' right", []], "state_before": "case inr.inr\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\n\u22a2 p = 0 \u2227 q = 0 \u2228\n    p = 0 \u2227 q = \u22a4 \u2228\n      p = 0 \u2227 0 < q.toReal \u2228 p = \u22a4 \u2227 q = \u22a4 \u2228 0 < p.toReal \u2227 q = \u22a4 \u2228 0 < p.toReal \u2227 0 < q.toReal \u2227 p.toReal \u2264 q.toReal", "state_after": "case inr.inr.h.h.h.h.h\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\n\u22a2 0 < p.toReal \u2227 0 < q.toReal \u2227 p.toReal \u2264 q.toReal"}, {"tactic": "have hq' : 0 < q := lt_of_lt_of_le hp hpq", "annotated_tactic": ["have hq' : 0 < q := <a>lt_of_lt_of_le</a> hp 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": "case inr.inr.h.h.h.h.h\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\n\u22a2 0 < p.toReal \u2227 0 < q.toReal \u2227 p.toReal \u2264 q.toReal", "state_after": "case inr.inr.h.h.h.h.h\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\nhq' : 0 < q\n\u22a2 0 < p.toReal \u2227 0 < q.toReal \u2227 p.toReal \u2264 q.toReal"}, {"tactic": "have hp' : p < \u221e := lt_of_le_of_lt hpq hq", "annotated_tactic": ["have hp' : p < \u221e := <a>lt_of_le_of_lt</a> hpq hq", [{"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 inr.inr.h.h.h.h.h\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\nhq' : 0 < q\n\u22a2 0 < p.toReal \u2227 0 < q.toReal \u2227 p.toReal \u2264 q.toReal", "state_after": "case inr.inr.h.h.h.h.h\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\nhq' : 0 < q\nhp' : p < \u22a4\n\u22a2 0 < p.toReal \u2227 0 < q.toReal \u2227 p.toReal \u2264 q.toReal"}, {"tactic": "simp [ENNReal.toReal_le_toReal hp'.ne hq.ne, ENNReal.toReal_pos_iff, hpq, hp, hp', hq', hq]", "annotated_tactic": ["simp [<a>ENNReal.toReal_le_toReal</a> hp'.ne hq.ne, <a>ENNReal.toReal_pos_iff</a>, hpq, hp, hp', hq', hq]", [{"full_name": "ENNReal.toReal_le_toReal", "def_path": "Mathlib/Data/ENNReal/Real.lean", "def_pos": [76, 9], "def_end_pos": [76, 25]}, {"full_name": "ENNReal.toReal_pos_iff", "def_path": "Mathlib/Data/ENNReal/Real.lean", "def_pos": [167, 9], "def_end_pos": [167, 23]}]], "state_before": "case inr.inr.h.h.h.h.h\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\nhq' : 0 < q\nhp' : p < \u22a4\n\u22a2 0 < p.toReal \u2227 0 < q.toReal \u2227 p.toReal \u2264 q.toReal", "state_after": "no goals"}, {"tactic": "simpa using q.trichotomy", "annotated_tactic": ["simpa using q.trichotomy", []], "state_before": "case inl\na b c d : \u211d\u22650\u221e\nr p q\u271d : \u211d\u22650\nq : \u211d\u22650\u221e\nhpq : 0 \u2264 q\n\u22a2 0 = 0 \u2227 q = 0 \u2228\n    0 = 0 \u2227 q = \u22a4 \u2228\n      0 = 0 \u2227 0 < q.toReal \u2228\n        0 = \u22a4 \u2227 q = \u22a4 \u2228 0 < ENNReal.toReal 0 \u2227 q = \u22a4 \u2228 0 < ENNReal.toReal 0 \u2227 0 < q.toReal \u2227 ENNReal.toReal 0 \u2264 q.toReal", "state_after": "no goals"}, {"tactic": "simpa using p.trichotomy", "annotated_tactic": ["simpa using p.trichotomy", []], "state_before": "case inr.inl\na b c d : \u211d\u22650\u221e\nr p\u271d q : \u211d\u22650\np : \u211d\u22650\u221e\nhp : 0 < p\nhpq : p \u2264 \u22a4\n\u22a2 p = 0 \u2227 \u22a4 = 0 \u2228\n    p = 0 \u2227 \u22a4 = \u22a4 \u2228\n      p = 0 \u2227 0 < \u22a4.toReal \u2228 p = \u22a4 \u2227 \u22a4 = \u22a4 \u2228 0 < p.toReal \u2227 \u22a4 = \u22a4 \u2228 0 < p.toReal \u2227 0 < \u22a4.toReal \u2227 p.toReal \u2264 \u22a4.toReal", "state_after": "no goals"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "case inr.inr.h.h.h.h\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\n\u22a2 0 < p.toReal \u2227 q = \u22a4 \u2228 0 < p.toReal \u2227 0 < q.toReal \u2227 p.toReal \u2264 q.toReal", "state_after": "case inr.inr.h.h.h.h.h\na b c d : \u211d\u22650\u221e\nr p\u271d q\u271d : \u211d\u22650\np q : \u211d\u22650\u221e\nhpq : p \u2264 q\nhp : 0 < p\nhq : q < \u22a4\n\u22a2 0 < p.toReal \u2227 0 < q.toReal \u2227 p.toReal \u2264 q.toReal"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Order/Field.lean", "full_name": "Filter.Tendsto.mul_atTop", "start": [72, 1], "end": [74, 49], "traced_tactics": [{"tactic": "simpa only [mul_comm] using hg.atTop_mul hC hf", "annotated_tactic": ["simpa only [<a>mul_comm</a>] using hg.atTop_mul hC hf", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "\ud835\udd5c : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \ud835\udd5c\ninst\u271d\u00b9 : TopologicalSpace \ud835\udd5c\ninst\u271d : OrderTopology \ud835\udd5c\nl : Filter \u03b1\nf g : \u03b1 \u2192 \ud835\udd5c\nC : \ud835\udd5c\nhC : 0 < C\nhf : Tendsto f l (\ud835\udcdd C)\nhg : Tendsto g l atTop\n\u22a2 Tendsto (fun x => f x * g x) l atTop", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "full_name": "Set.Nonempty.of_mul_right", "start": [394, 1], "end": [395, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Analytic/Constructions.lean", "full_name": "HasFPowerSeriesOnBall.prod", "start": [68, 1], "end": [81, 72], "traced_tactics": [{"tactic": "rw [p.radius_prod_eq_min]", "annotated_tactic": ["rw [p.radius_prod_eq_min]", []], "state_before": "\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_3\nF : Type u_4\nG : Type u_5\nH : Type u_6\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\ninst\u271d\u2075 : NormedAddCommGroup H\ninst\u271d\u2074 : NormedSpace \ud835\udd5c H\n\ud835\udd5d : Type u_7\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5d\nA : Type u_8\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\ne : E\nf : E \u2192 F\ng : E \u2192 G\nr s : \u211d\u22650\u221e\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c E G\nhf : HasFPowerSeriesOnBall f p e r\nhg : HasFPowerSeriesOnBall g q e s\n\u22a2 min r s \u2264 (p.prod q).radius", "state_after": "\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_3\nF : Type u_4\nG : Type u_5\nH : Type u_6\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\ninst\u271d\u2075 : NormedAddCommGroup H\ninst\u271d\u2074 : NormedSpace \ud835\udd5c H\n\ud835\udd5d : Type u_7\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5d\nA : Type u_8\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\ne : E\nf : E \u2192 F\ng : E \u2192 G\nr s : \u211d\u22650\u221e\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c E G\nhf : HasFPowerSeriesOnBall f p e r\nhg : HasFPowerSeriesOnBall g q e s\n\u22a2 min r s \u2264 min p.radius q.radius"}, {"tactic": "exact min_le_min hf.r_le hg.r_le", "annotated_tactic": ["exact <a>min_le_min</a> hf.r_le hg.r_le", [{"full_name": "min_le_min", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [82, 9], "def_end_pos": [82, 19]}]], "state_before": "\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_3\nF : Type u_4\nG : Type u_5\nH : Type u_6\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\ninst\u271d\u2075 : NormedAddCommGroup H\ninst\u271d\u2074 : NormedSpace \ud835\udd5c H\n\ud835\udd5d : Type u_7\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5d\nA : Type u_8\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\ne : E\nf : E \u2192 F\ng : E \u2192 G\nr s : \u211d\u22650\u221e\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c E G\nhf : HasFPowerSeriesOnBall f p e r\nhg : HasFPowerSeriesOnBall g q e s\n\u22a2 min r s \u2264 min p.radius q.radius", "state_after": "no goals"}, {"tactic": "intro y hy", "annotated_tactic": ["intro y hy", []], "state_before": "\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_3\nF : Type u_4\nG : Type u_5\nH : Type u_6\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\ninst\u271d\u2075 : NormedAddCommGroup H\ninst\u271d\u2074 : NormedSpace \ud835\udd5c H\n\ud835\udd5d : Type u_7\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5d\nA : Type u_8\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\ne : E\nf : E \u2192 F\ng : E \u2192 G\nr s : \u211d\u22650\u221e\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c E G\nhf : HasFPowerSeriesOnBall f p e r\nhg : HasFPowerSeriesOnBall g q e s\n\u22a2 \u2200 {y : E}, y \u2208 EMetric.ball 0 (min r s) \u2192 HasSum (fun n => (p.prod q n) fun x => y) (f (e + y), g (e + y))", "state_after": "\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_3\nF : Type u_4\nG : Type u_5\nH : Type u_6\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\ninst\u271d\u2075 : NormedAddCommGroup H\ninst\u271d\u2074 : NormedSpace \ud835\udd5c H\n\ud835\udd5d : Type u_7\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5d\nA : Type u_8\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\ne : E\nf : E \u2192 F\ng : E \u2192 G\nr s : \u211d\u22650\u221e\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c E G\nhf : HasFPowerSeriesOnBall f p e r\nhg : HasFPowerSeriesOnBall g q e s\ny : E\nhy : y \u2208 EMetric.ball 0 (min r s)\n\u22a2 HasSum (fun n => (p.prod q n) fun x => y) (f (e + y), g (e + y))"}, {"tactic": "simp_rw [FormalMultilinearSeries.prod, ContinuousMultilinearMap.prod_apply]", "annotated_tactic": ["simp_rw [<a>FormalMultilinearSeries.prod</a>, <a>ContinuousMultilinearMap.prod_apply</a>]", [{"full_name": "FormalMultilinearSeries.prod", "def_path": "Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean", "def_pos": [95, 5], "def_end_pos": [95, 9]}, {"full_name": "ContinuousMultilinearMap.prod_apply", "def_path": "Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean", "def_pos": [243, 9], "def_end_pos": [243, 19]}]], "state_before": "\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_3\nF : Type u_4\nG : Type u_5\nH : Type u_6\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\ninst\u271d\u2075 : NormedAddCommGroup H\ninst\u271d\u2074 : NormedSpace \ud835\udd5c H\n\ud835\udd5d : Type u_7\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5d\nA : Type u_8\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\ne : E\nf : E \u2192 F\ng : E \u2192 G\nr s : \u211d\u22650\u221e\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c E G\nhf : HasFPowerSeriesOnBall f p e r\nhg : HasFPowerSeriesOnBall g q e s\ny : E\nhy : y \u2208 EMetric.ball 0 (min r s)\n\u22a2 HasSum (fun n => (p.prod q n) fun x => y) (f (e + y), g (e + y))", "state_after": "\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_3\nF : Type u_4\nG : Type u_5\nH : Type u_6\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\ninst\u271d\u2075 : NormedAddCommGroup H\ninst\u271d\u2074 : NormedSpace \ud835\udd5c H\n\ud835\udd5d : Type u_7\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5d\nA : Type u_8\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\ne : E\nf : E \u2192 F\ng : E \u2192 G\nr s : \u211d\u22650\u221e\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c E G\nhf : HasFPowerSeriesOnBall f p e r\nhg : HasFPowerSeriesOnBall g q e s\ny : E\nhy : y \u2208 EMetric.ball 0 (min r s)\n\u22a2 HasSum (fun n => ((p n) fun x => y, (q n) fun x => y)) (f (e + y), g (e + y))"}, {"tactic": "refine (hf.hasSum ?_).prod_mk (hg.hasSum ?_)", "annotated_tactic": ["refine (hf.hasSum ?_).<a>prod_mk</a> (hg.hasSum ?_)", [{"full_name": "HasSum.prod_mk", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean", "def_pos": [67, 15], "def_end_pos": [67, 29]}]], "state_before": "\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_3\nF : Type u_4\nG : Type u_5\nH : Type u_6\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\ninst\u271d\u2075 : NormedAddCommGroup H\ninst\u271d\u2074 : NormedSpace \ud835\udd5c H\n\ud835\udd5d : Type u_7\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5d\nA : Type u_8\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\ne : E\nf : E \u2192 F\ng : E \u2192 G\nr s : \u211d\u22650\u221e\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c E G\nhf : HasFPowerSeriesOnBall f p e r\nhg : HasFPowerSeriesOnBall g q e s\ny : E\nhy : y \u2208 EMetric.ball 0 (min r s)\n\u22a2 HasSum (fun n => ((p n) fun x => y, (q n) fun x => y)) (f (e + y), g (e + y))", "state_after": "case refine_1\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_3\nF : Type u_4\nG : Type u_5\nH : Type u_6\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\ninst\u271d\u2075 : NormedAddCommGroup H\ninst\u271d\u2074 : NormedSpace \ud835\udd5c H\n\ud835\udd5d : Type u_7\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5d\nA : Type u_8\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\ne : E\nf : E \u2192 F\ng : E \u2192 G\nr s : \u211d\u22650\u221e\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c E G\nhf : HasFPowerSeriesOnBall f p e r\nhg : HasFPowerSeriesOnBall g q e s\ny : E\nhy : y \u2208 EMetric.ball 0 (min r s)\n\u22a2 y \u2208 EMetric.ball 0 r\n\ncase refine_2\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_3\nF : Type u_4\nG : Type u_5\nH : Type u_6\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\ninst\u271d\u2075 : NormedAddCommGroup H\ninst\u271d\u2074 : NormedSpace \ud835\udd5c H\n\ud835\udd5d : Type u_7\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5d\nA : Type u_8\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\ne : E\nf : E \u2192 F\ng : E \u2192 G\nr s : \u211d\u22650\u221e\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c E G\nhf : HasFPowerSeriesOnBall f p e r\nhg : HasFPowerSeriesOnBall g q e s\ny : E\nhy : y \u2208 EMetric.ball 0 (min r s)\n\u22a2 y \u2208 EMetric.ball 0 s"}, {"tactic": "exact EMetric.mem_ball.mpr (lt_of_lt_of_le hy (min_le_left _ _))", "annotated_tactic": ["exact EMetric.mem_ball.mpr (<a>lt_of_lt_of_le</a> hy (<a>min_le_left</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": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 20]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_3\nF : Type u_4\nG : Type u_5\nH : Type u_6\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\ninst\u271d\u2075 : NormedAddCommGroup H\ninst\u271d\u2074 : NormedSpace \ud835\udd5c H\n\ud835\udd5d : Type u_7\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5d\nA : Type u_8\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\ne : E\nf : E \u2192 F\ng : E \u2192 G\nr s : \u211d\u22650\u221e\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c E G\nhf : HasFPowerSeriesOnBall f p e r\nhg : HasFPowerSeriesOnBall g q e s\ny : E\nhy : y \u2208 EMetric.ball 0 (min r s)\n\u22a2 y \u2208 EMetric.ball 0 r", "state_after": "no goals"}, {"tactic": "exact EMetric.mem_ball.mpr (lt_of_lt_of_le hy (min_le_right _ _))", "annotated_tactic": ["exact EMetric.mem_ball.mpr (<a>lt_of_lt_of_le</a> hy (<a>min_le_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": "min_le_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [40, 9], "def_end_pos": [40, 21]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c\nE : Type u_3\nF : Type u_4\nG : Type u_5\nH : Type u_6\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E\ninst\u271d\u2079 : NormedAddCommGroup F\ninst\u271d\u2078 : NormedSpace \ud835\udd5c F\ninst\u271d\u2077 : NormedAddCommGroup G\ninst\u271d\u2076 : NormedSpace \ud835\udd5c G\ninst\u271d\u2075 : NormedAddCommGroup H\ninst\u271d\u2074 : NormedSpace \ud835\udd5c H\n\ud835\udd5d : Type u_7\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5d\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5d\nA : Type u_8\ninst\u271d\u00b9 : NormedRing A\ninst\u271d : NormedAlgebra \ud835\udd5c A\ne : E\nf : E \u2192 F\ng : E \u2192 G\nr s : \u211d\u22650\u221e\np : FormalMultilinearSeries \ud835\udd5c E F\nq : FormalMultilinearSeries \ud835\udd5c E G\nhf : HasFPowerSeriesOnBall f p e r\nhg : HasFPowerSeriesOnBall g q e s\ny : E\nhy : y \u2208 EMetric.ball 0 (min r s)\n\u22a2 y \u2208 EMetric.ball 0 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/StarSubalgebra.lean", "full_name": "elementalStarAlgebra.starAlgHomClass_ext", "start": [270, 1], "end": [281, 94], "traced_tactics": [{"tactic": "have : StarAlgHomClass F R (\u21a5(topologicalClosure (adjoin R {a}))) B :=\n  inferInstanceAs (StarAlgHomClass F R (elementalStarAlgebra R a) B)", "annotated_tactic": ["have : <a>StarAlgHomClass</a> F R (\u21a5(<a>topologicalClosure</a> (<a>adjoin</a> R {a}))) B :=\n    <a>inferInstanceAs</a> (<a>StarAlgHomClass</a> F R (<a>elementalStarAlgebra</a> R a) B)", [{"full_name": "StarAlgHomClass", "def_path": "Mathlib/Algebra/Star/StarAlgHom.lean", "def_pos": [336, 7], "def_end_pos": [336, 22]}, {"full_name": "StarSubalgebra.topologicalClosure", "def_path": "Mathlib/Topology/Algebra/StarSubalgebra.lean", "def_pos": [65, 5], "def_end_pos": [65, 23]}, {"full_name": "StarAlgebra.adjoin", "def_path": "Mathlib/Algebra/Star/Subalgebra.lean", "def_pos": [431, 5], "def_end_pos": [431, 11]}, {"full_name": "inferInstanceAs", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [113, 8], "def_end_pos": [113, 23]}, {"full_name": "StarAlgHomClass", "def_path": "Mathlib/Algebra/Star/StarAlgHom.lean", "def_pos": [336, 7], "def_end_pos": [336, 22]}, {"full_name": "elementalStarAlgebra", "def_path": "Mathlib/Topology/Algebra/StarSubalgebra.lean", "def_pos": [194, 5], "def_end_pos": [194, 25]}]], "state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : StarRing R\ninst\u271d\u00b9\u2074 : TopologicalSpace A\ninst\u271d\u00b9\u00b3 : Semiring A\ninst\u271d\u00b9\u00b2 : StarRing A\ninst\u271d\u00b9\u00b9 : TopologicalSemiring A\ninst\u271d\u00b9\u2070 : ContinuousStar A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : StarRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : T2Space B\nF : Type u_4\na : A\ninst\u271d\u00b2 : FunLike F (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d\u00b9 : AlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d : StarAlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u21d1\u03c6\nh\u03c8 : Continuous \u21d1\u03c8\nh : \u03c6 \u27e8a, \u22ef\u27e9 = \u03c8 \u27e8a, \u22ef\u27e9\n\u22a2 \u03c6 = \u03c8", "state_after": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : StarRing R\ninst\u271d\u00b9\u2074 : TopologicalSpace A\ninst\u271d\u00b9\u00b3 : Semiring A\ninst\u271d\u00b9\u00b2 : StarRing A\ninst\u271d\u00b9\u00b9 : TopologicalSemiring A\ninst\u271d\u00b9\u2070 : ContinuousStar A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : StarRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : T2Space B\nF : Type u_4\na : A\ninst\u271d\u00b2 : FunLike F (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d\u00b9 : AlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d : StarAlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u21d1\u03c6\nh\u03c8 : Continuous \u21d1\u03c8\nh : \u03c6 \u27e8a, \u22ef\u27e9 = \u03c8 \u27e8a, \u22ef\u27e9\nthis : StarAlgHomClass F R (\u21a5(adjoin R {a}).topologicalClosure) B\n\u22a2 \u03c6 = \u03c8"}, {"tactic": "refine StarAlgHomClass.ext_topologicalClosure h\u03c6 h\u03c8 fun x => ?_", "annotated_tactic": ["refine <a>StarAlgHomClass.ext_topologicalClosure</a> h\u03c6 h\u03c8 fun x => ?_", [{"full_name": "StarAlgHomClass.ext_topologicalClosure", "def_path": "Mathlib/Topology/Algebra/StarSubalgebra.lean", "def_pos": [166, 9], "def_end_pos": [166, 54]}]], "state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : StarRing R\ninst\u271d\u00b9\u2074 : TopologicalSpace A\ninst\u271d\u00b9\u00b3 : Semiring A\ninst\u271d\u00b9\u00b2 : StarRing A\ninst\u271d\u00b9\u00b9 : TopologicalSemiring A\ninst\u271d\u00b9\u2070 : ContinuousStar A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : StarRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : T2Space B\nF : Type u_4\na : A\ninst\u271d\u00b2 : FunLike F (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d\u00b9 : AlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d : StarAlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u21d1\u03c6\nh\u03c8 : Continuous \u21d1\u03c8\nh : \u03c6 \u27e8a, \u22ef\u27e9 = \u03c8 \u27e8a, \u22ef\u27e9\nthis : StarAlgHomClass F R (\u21a5(adjoin R {a}).topologicalClosure) B\n\u22a2 \u03c6 = \u03c8", "state_after": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : StarRing R\ninst\u271d\u00b9\u2074 : TopologicalSpace A\ninst\u271d\u00b9\u00b3 : Semiring A\ninst\u271d\u00b9\u00b2 : StarRing A\ninst\u271d\u00b9\u00b9 : TopologicalSemiring A\ninst\u271d\u00b9\u2070 : ContinuousStar A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : StarRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : T2Space B\nF : Type u_4\na : A\ninst\u271d\u00b2 : FunLike F (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d\u00b9 : AlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d : StarAlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u21d1\u03c6\nh\u03c8 : Continuous \u21d1\u03c8\nh : \u03c6 \u27e8a, \u22ef\u27e9 = \u03c8 \u27e8a, \u22ef\u27e9\nthis : StarAlgHomClass F R (\u21a5(adjoin R {a}).topologicalClosure) B\nx : \u21a5(adjoin R {a})\n\u22a2 \u03c6 ((StarSubalgebra.inclusion \u22ef) x) = \u03c8 ((StarSubalgebra.inclusion \u22ef) x)"}, {"tactic": "refine adjoin_induction' x ?_ ?_ ?_ ?_ ?_", "annotated_tactic": ["refine <a>adjoin_induction'</a> x ?_ ?_ ?_ ?_ ?_", [{"full_name": "StarAlgebra.adjoin_induction'", "def_path": "Mathlib/Algebra/Star/Subalgebra.lean", "def_pos": [548, 9], "def_end_pos": [548, 26]}]], "state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : StarRing R\ninst\u271d\u00b9\u2074 : TopologicalSpace A\ninst\u271d\u00b9\u00b3 : Semiring A\ninst\u271d\u00b9\u00b2 : StarRing A\ninst\u271d\u00b9\u00b9 : TopologicalSemiring A\ninst\u271d\u00b9\u2070 : ContinuousStar A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : StarRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : T2Space B\nF : Type u_4\na : A\ninst\u271d\u00b2 : FunLike F (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d\u00b9 : AlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d : StarAlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u21d1\u03c6\nh\u03c8 : Continuous \u21d1\u03c8\nh : \u03c6 \u27e8a, \u22ef\u27e9 = \u03c8 \u27e8a, \u22ef\u27e9\nthis : StarAlgHomClass F R (\u21a5(adjoin R {a}).topologicalClosure) B\nx : \u21a5(adjoin R {a})\n\u22a2 \u03c6 ((StarSubalgebra.inclusion \u22ef) x) = \u03c8 ((StarSubalgebra.inclusion \u22ef) x)", "state_after": "case refine_1\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : StarRing R\ninst\u271d\u00b9\u2074 : TopologicalSpace A\ninst\u271d\u00b9\u00b3 : Semiring A\ninst\u271d\u00b9\u00b2 : StarRing A\ninst\u271d\u00b9\u00b9 : TopologicalSemiring A\ninst\u271d\u00b9\u2070 : ContinuousStar A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : StarRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : T2Space B\nF : Type u_4\na : A\ninst\u271d\u00b2 : FunLike F (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d\u00b9 : AlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d : StarAlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u21d1\u03c6\nh\u03c8 : Continuous \u21d1\u03c8\nh : \u03c6 \u27e8a, \u22ef\u27e9 = \u03c8 \u27e8a, \u22ef\u27e9\nthis : StarAlgHomClass F R (\u21a5(adjoin R {a}).topologicalClosure) B\nx : \u21a5(adjoin R {a})\n\u22a2 \u2200 (x : A) (h : x \u2208 {a}), \u03c6 ((StarSubalgebra.inclusion \u22ef) \u27e8x, \u22ef\u27e9) = \u03c8 ((StarSubalgebra.inclusion \u22ef) \u27e8x, \u22ef\u27e9)\n\ncase refine_2\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : StarRing R\ninst\u271d\u00b9\u2074 : TopologicalSpace A\ninst\u271d\u00b9\u00b3 : Semiring A\ninst\u271d\u00b9\u00b2 : StarRing A\ninst\u271d\u00b9\u00b9 : TopologicalSemiring A\ninst\u271d\u00b9\u2070 : ContinuousStar A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : StarRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : T2Space B\nF : Type u_4\na : A\ninst\u271d\u00b2 : FunLike F (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d\u00b9 : AlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d : StarAlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u21d1\u03c6\nh\u03c8 : Continuous \u21d1\u03c8\nh : \u03c6 \u27e8a, \u22ef\u27e9 = \u03c8 \u27e8a, \u22ef\u27e9\nthis : StarAlgHomClass F R (\u21a5(adjoin R {a}).topologicalClosure) B\nx : \u21a5(adjoin R {a})\n\u22a2 \u2200 (r : R),\n    \u03c6 ((StarSubalgebra.inclusion \u22ef) ((algebraMap R \u21a5(adjoin R {a})) r)) =\n      \u03c8 ((StarSubalgebra.inclusion \u22ef) ((algebraMap R \u21a5(adjoin R {a})) r))\n\ncase refine_3\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : StarRing R\ninst\u271d\u00b9\u2074 : TopologicalSpace A\ninst\u271d\u00b9\u00b3 : Semiring A\ninst\u271d\u00b9\u00b2 : StarRing A\ninst\u271d\u00b9\u00b9 : TopologicalSemiring A\ninst\u271d\u00b9\u2070 : ContinuousStar A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : StarRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : T2Space B\nF : Type u_4\na : A\ninst\u271d\u00b2 : FunLike F (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d\u00b9 : AlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d : StarAlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u21d1\u03c6\nh\u03c8 : Continuous \u21d1\u03c8\nh : \u03c6 \u27e8a, \u22ef\u27e9 = \u03c8 \u27e8a, \u22ef\u27e9\nthis : StarAlgHomClass F R (\u21a5(adjoin R {a}).topologicalClosure) B\nx : \u21a5(adjoin R {a})\n\u22a2 \u2200 (x y : \u21a5(adjoin R {a})),\n    \u03c6 ((StarSubalgebra.inclusion \u22ef) x) = \u03c8 ((StarSubalgebra.inclusion \u22ef) x) \u2192\n      \u03c6 ((StarSubalgebra.inclusion \u22ef) y) = \u03c8 ((StarSubalgebra.inclusion \u22ef) y) \u2192\n        \u03c6 ((StarSubalgebra.inclusion \u22ef) (x + y)) = \u03c8 ((StarSubalgebra.inclusion \u22ef) (x + y))\n\ncase refine_4\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : StarRing R\ninst\u271d\u00b9\u2074 : TopologicalSpace A\ninst\u271d\u00b9\u00b3 : Semiring A\ninst\u271d\u00b9\u00b2 : StarRing A\ninst\u271d\u00b9\u00b9 : TopologicalSemiring A\ninst\u271d\u00b9\u2070 : ContinuousStar A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : StarRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : T2Space B\nF : Type u_4\na : A\ninst\u271d\u00b2 : FunLike F (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d\u00b9 : AlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d : StarAlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u21d1\u03c6\nh\u03c8 : Continuous \u21d1\u03c8\nh : \u03c6 \u27e8a, \u22ef\u27e9 = \u03c8 \u27e8a, \u22ef\u27e9\nthis : StarAlgHomClass F R (\u21a5(adjoin R {a}).topologicalClosure) B\nx : \u21a5(adjoin R {a})\n\u22a2 \u2200 (x y : \u21a5(adjoin R {a})),\n    \u03c6 ((StarSubalgebra.inclusion \u22ef) x) = \u03c8 ((StarSubalgebra.inclusion \u22ef) x) \u2192\n      \u03c6 ((StarSubalgebra.inclusion \u22ef) y) = \u03c8 ((StarSubalgebra.inclusion \u22ef) y) \u2192\n        \u03c6 ((StarSubalgebra.inclusion \u22ef) (x * y)) = \u03c8 ((StarSubalgebra.inclusion \u22ef) (x * y))\n\ncase refine_5\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : StarRing R\ninst\u271d\u00b9\u2074 : TopologicalSpace A\ninst\u271d\u00b9\u00b3 : Semiring A\ninst\u271d\u00b9\u00b2 : StarRing A\ninst\u271d\u00b9\u00b9 : TopologicalSemiring A\ninst\u271d\u00b9\u2070 : ContinuousStar A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : StarRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : T2Space B\nF : Type u_4\na : A\ninst\u271d\u00b2 : FunLike F (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d\u00b9 : AlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d : StarAlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u21d1\u03c6\nh\u03c8 : Continuous \u21d1\u03c8\nh : \u03c6 \u27e8a, \u22ef\u27e9 = \u03c8 \u27e8a, \u22ef\u27e9\nthis : StarAlgHomClass F R (\u21a5(adjoin R {a}).topologicalClosure) B\nx : \u21a5(adjoin R {a})\n\u22a2 \u2200 (x : \u21a5(adjoin R {a})),\n    \u03c6 ((StarSubalgebra.inclusion \u22ef) x) = \u03c8 ((StarSubalgebra.inclusion \u22ef) x) \u2192\n      \u03c6 ((StarSubalgebra.inclusion \u22ef) (star x)) = \u03c8 ((StarSubalgebra.inclusion \u22ef) (star x))"}, {"tactic": "exacts [fun y hy => by simpa only [Set.mem_singleton_iff.mp hy] using h, fun r => by\n  simp only [AlgHomClass.commutes], fun x y hx hy => by simp only [map_add, hx, hy],\n  fun x y hx hy => by simp only [map_mul, hx, hy], fun x hx => by simp only [map_star, hx]]", "annotated_tactic": ["exacts [fun y hy => by simpa only [Set.mem_singleton_iff.mp hy] using h, fun r => by\n    simp only [<a>AlgHomClass.commutes</a>], fun x y hx hy => by simp only [<a>map_add</a>, hx, hy],\n    fun x y hx hy => by simp only [<a>map_mul</a>, hx, hy], fun x hx => by simp only [<a>map_star</a>, hx]]", [{"full_name": "AlgHomClass.commutes", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [49, 3], "def_end_pos": [49, 11]}, {"full_name": "map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [308, 3], "def_end_pos": [308, 14]}, {"full_name": "map_mul", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [309, 9], "def_end_pos": [309, 16]}, {"full_name": "StarHomClass.map_star", "def_path": "Mathlib/Algebra/Star/Basic.lean", "def_pos": [503, 3], "def_end_pos": [503, 11]}]], "state_before": "case refine_1\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : StarRing R\ninst\u271d\u00b9\u2074 : TopologicalSpace A\ninst\u271d\u00b9\u00b3 : Semiring A\ninst\u271d\u00b9\u00b2 : StarRing A\ninst\u271d\u00b9\u00b9 : TopologicalSemiring A\ninst\u271d\u00b9\u2070 : ContinuousStar A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : StarRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : T2Space B\nF : Type u_4\na : A\ninst\u271d\u00b2 : FunLike F (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d\u00b9 : AlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d : StarAlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u21d1\u03c6\nh\u03c8 : Continuous \u21d1\u03c8\nh : \u03c6 \u27e8a, \u22ef\u27e9 = \u03c8 \u27e8a, \u22ef\u27e9\nthis : StarAlgHomClass F R (\u21a5(adjoin R {a}).topologicalClosure) B\nx : \u21a5(adjoin R {a})\n\u22a2 \u2200 (x : A) (h : x \u2208 {a}), \u03c6 ((StarSubalgebra.inclusion \u22ef) \u27e8x, \u22ef\u27e9) = \u03c8 ((StarSubalgebra.inclusion \u22ef) \u27e8x, \u22ef\u27e9)\n\ncase refine_2\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : StarRing R\ninst\u271d\u00b9\u2074 : TopologicalSpace A\ninst\u271d\u00b9\u00b3 : Semiring A\ninst\u271d\u00b9\u00b2 : StarRing A\ninst\u271d\u00b9\u00b9 : TopologicalSemiring A\ninst\u271d\u00b9\u2070 : ContinuousStar A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : StarRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : T2Space B\nF : Type u_4\na : A\ninst\u271d\u00b2 : FunLike F (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d\u00b9 : AlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d : StarAlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u21d1\u03c6\nh\u03c8 : Continuous \u21d1\u03c8\nh : \u03c6 \u27e8a, \u22ef\u27e9 = \u03c8 \u27e8a, \u22ef\u27e9\nthis : StarAlgHomClass F R (\u21a5(adjoin R {a}).topologicalClosure) B\nx : \u21a5(adjoin R {a})\n\u22a2 \u2200 (r : R),\n    \u03c6 ((StarSubalgebra.inclusion \u22ef) ((algebraMap R \u21a5(adjoin R {a})) r)) =\n      \u03c8 ((StarSubalgebra.inclusion \u22ef) ((algebraMap R \u21a5(adjoin R {a})) r))\n\ncase refine_3\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : StarRing R\ninst\u271d\u00b9\u2074 : TopologicalSpace A\ninst\u271d\u00b9\u00b3 : Semiring A\ninst\u271d\u00b9\u00b2 : StarRing A\ninst\u271d\u00b9\u00b9 : TopologicalSemiring A\ninst\u271d\u00b9\u2070 : ContinuousStar A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : StarRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : T2Space B\nF : Type u_4\na : A\ninst\u271d\u00b2 : FunLike F (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d\u00b9 : AlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d : StarAlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u21d1\u03c6\nh\u03c8 : Continuous \u21d1\u03c8\nh : \u03c6 \u27e8a, \u22ef\u27e9 = \u03c8 \u27e8a, \u22ef\u27e9\nthis : StarAlgHomClass F R (\u21a5(adjoin R {a}).topologicalClosure) B\nx : \u21a5(adjoin R {a})\n\u22a2 \u2200 (x y : \u21a5(adjoin R {a})),\n    \u03c6 ((StarSubalgebra.inclusion \u22ef) x) = \u03c8 ((StarSubalgebra.inclusion \u22ef) x) \u2192\n      \u03c6 ((StarSubalgebra.inclusion \u22ef) y) = \u03c8 ((StarSubalgebra.inclusion \u22ef) y) \u2192\n        \u03c6 ((StarSubalgebra.inclusion \u22ef) (x + y)) = \u03c8 ((StarSubalgebra.inclusion \u22ef) (x + y))\n\ncase refine_4\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : StarRing R\ninst\u271d\u00b9\u2074 : TopologicalSpace A\ninst\u271d\u00b9\u00b3 : Semiring A\ninst\u271d\u00b9\u00b2 : StarRing A\ninst\u271d\u00b9\u00b9 : TopologicalSemiring A\ninst\u271d\u00b9\u2070 : ContinuousStar A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : StarRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : T2Space B\nF : Type u_4\na : A\ninst\u271d\u00b2 : FunLike F (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d\u00b9 : AlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d : StarAlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u21d1\u03c6\nh\u03c8 : Continuous \u21d1\u03c8\nh : \u03c6 \u27e8a, \u22ef\u27e9 = \u03c8 \u27e8a, \u22ef\u27e9\nthis : StarAlgHomClass F R (\u21a5(adjoin R {a}).topologicalClosure) B\nx : \u21a5(adjoin R {a})\n\u22a2 \u2200 (x y : \u21a5(adjoin R {a})),\n    \u03c6 ((StarSubalgebra.inclusion \u22ef) x) = \u03c8 ((StarSubalgebra.inclusion \u22ef) x) \u2192\n      \u03c6 ((StarSubalgebra.inclusion \u22ef) y) = \u03c8 ((StarSubalgebra.inclusion \u22ef) y) \u2192\n        \u03c6 ((StarSubalgebra.inclusion \u22ef) (x * y)) = \u03c8 ((StarSubalgebra.inclusion \u22ef) (x * y))\n\ncase refine_5\nR : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : StarRing R\ninst\u271d\u00b9\u2074 : TopologicalSpace A\ninst\u271d\u00b9\u00b3 : Semiring A\ninst\u271d\u00b9\u00b2 : StarRing A\ninst\u271d\u00b9\u00b9 : TopologicalSemiring A\ninst\u271d\u00b9\u2070 : ContinuousStar A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : StarRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : T2Space B\nF : Type u_4\na : A\ninst\u271d\u00b2 : FunLike F (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d\u00b9 : AlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d : StarAlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u21d1\u03c6\nh\u03c8 : Continuous \u21d1\u03c8\nh : \u03c6 \u27e8a, \u22ef\u27e9 = \u03c8 \u27e8a, \u22ef\u27e9\nthis : StarAlgHomClass F R (\u21a5(adjoin R {a}).topologicalClosure) B\nx : \u21a5(adjoin R {a})\n\u22a2 \u2200 (x : \u21a5(adjoin R {a})),\n    \u03c6 ((StarSubalgebra.inclusion \u22ef) x) = \u03c8 ((StarSubalgebra.inclusion \u22ef) x) \u2192\n      \u03c6 ((StarSubalgebra.inclusion \u22ef) (star x)) = \u03c8 ((StarSubalgebra.inclusion \u22ef) (star x))", "state_after": "no goals"}, {"tactic": "simpa only [Set.mem_singleton_iff.mp hy] using h", "annotated_tactic": ["simpa only [Set.mem_singleton_iff.mp hy] using h", []], "state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : StarRing R\ninst\u271d\u00b9\u2074 : TopologicalSpace A\ninst\u271d\u00b9\u00b3 : Semiring A\ninst\u271d\u00b9\u00b2 : StarRing A\ninst\u271d\u00b9\u00b9 : TopologicalSemiring A\ninst\u271d\u00b9\u2070 : ContinuousStar A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : StarRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : T2Space B\nF : Type u_4\na : A\ninst\u271d\u00b2 : FunLike F (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d\u00b9 : AlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d : StarAlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u21d1\u03c6\nh\u03c8 : Continuous \u21d1\u03c8\nh : \u03c6 \u27e8a, \u22ef\u27e9 = \u03c8 \u27e8a, \u22ef\u27e9\nthis : StarAlgHomClass F R (\u21a5(adjoin R {a}).topologicalClosure) B\nx : \u21a5(adjoin R {a})\ny : A\nhy : y \u2208 {a}\n\u22a2 \u03c6 ((StarSubalgebra.inclusion \u22ef) \u27e8y, \u22ef\u27e9) = \u03c8 ((StarSubalgebra.inclusion \u22ef) \u27e8y, \u22ef\u27e9)", "state_after": "no goals"}, {"tactic": "simp only [AlgHomClass.commutes]", "annotated_tactic": ["simp only [<a>AlgHomClass.commutes</a>]", [{"full_name": "AlgHomClass.commutes", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [49, 3], "def_end_pos": [49, 11]}]], "state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : StarRing R\ninst\u271d\u00b9\u2074 : TopologicalSpace A\ninst\u271d\u00b9\u00b3 : Semiring A\ninst\u271d\u00b9\u00b2 : StarRing A\ninst\u271d\u00b9\u00b9 : TopologicalSemiring A\ninst\u271d\u00b9\u2070 : ContinuousStar A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : StarRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : T2Space B\nF : Type u_4\na : A\ninst\u271d\u00b2 : FunLike F (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d\u00b9 : AlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d : StarAlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u21d1\u03c6\nh\u03c8 : Continuous \u21d1\u03c8\nh : \u03c6 \u27e8a, \u22ef\u27e9 = \u03c8 \u27e8a, \u22ef\u27e9\nthis : StarAlgHomClass F R (\u21a5(adjoin R {a}).topologicalClosure) B\nx : \u21a5(adjoin R {a})\nr : R\n\u22a2 \u03c6 ((StarSubalgebra.inclusion \u22ef) ((algebraMap R \u21a5(adjoin R {a})) r)) =\n    \u03c8 ((StarSubalgebra.inclusion \u22ef) ((algebraMap R \u21a5(adjoin R {a})) r))", "state_after": "no goals"}, {"tactic": "simp only [map_add, hx, hy]", "annotated_tactic": ["simp only [<a>map_add</a>, hx, hy]", [{"full_name": "map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [308, 3], "def_end_pos": [308, 14]}]], "state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : StarRing R\ninst\u271d\u00b9\u2074 : TopologicalSpace A\ninst\u271d\u00b9\u00b3 : Semiring A\ninst\u271d\u00b9\u00b2 : StarRing A\ninst\u271d\u00b9\u00b9 : TopologicalSemiring A\ninst\u271d\u00b9\u2070 : ContinuousStar A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : StarRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : T2Space B\nF : Type u_4\na : A\ninst\u271d\u00b2 : FunLike F (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d\u00b9 : AlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d : StarAlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u21d1\u03c6\nh\u03c8 : Continuous \u21d1\u03c8\nh : \u03c6 \u27e8a, \u22ef\u27e9 = \u03c8 \u27e8a, \u22ef\u27e9\nthis : StarAlgHomClass F R (\u21a5(adjoin R {a}).topologicalClosure) B\nx\u271d x y : \u21a5(adjoin R {a})\nhx : \u03c6 ((StarSubalgebra.inclusion \u22ef) x) = \u03c8 ((StarSubalgebra.inclusion \u22ef) x)\nhy : \u03c6 ((StarSubalgebra.inclusion \u22ef) y) = \u03c8 ((StarSubalgebra.inclusion \u22ef) y)\n\u22a2 \u03c6 ((StarSubalgebra.inclusion \u22ef) (x + y)) = \u03c8 ((StarSubalgebra.inclusion \u22ef) (x + y))", "state_after": "no goals"}, {"tactic": "simp only [map_mul, hx, hy]", "annotated_tactic": ["simp only [<a>map_mul</a>, hx, hy]", [{"full_name": "map_mul", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [309, 9], "def_end_pos": [309, 16]}]], "state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : StarRing R\ninst\u271d\u00b9\u2074 : TopologicalSpace A\ninst\u271d\u00b9\u00b3 : Semiring A\ninst\u271d\u00b9\u00b2 : StarRing A\ninst\u271d\u00b9\u00b9 : TopologicalSemiring A\ninst\u271d\u00b9\u2070 : ContinuousStar A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : StarRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : T2Space B\nF : Type u_4\na : A\ninst\u271d\u00b2 : FunLike F (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d\u00b9 : AlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d : StarAlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u21d1\u03c6\nh\u03c8 : Continuous \u21d1\u03c8\nh : \u03c6 \u27e8a, \u22ef\u27e9 = \u03c8 \u27e8a, \u22ef\u27e9\nthis : StarAlgHomClass F R (\u21a5(adjoin R {a}).topologicalClosure) B\nx\u271d x y : \u21a5(adjoin R {a})\nhx : \u03c6 ((StarSubalgebra.inclusion \u22ef) x) = \u03c8 ((StarSubalgebra.inclusion \u22ef) x)\nhy : \u03c6 ((StarSubalgebra.inclusion \u22ef) y) = \u03c8 ((StarSubalgebra.inclusion \u22ef) y)\n\u22a2 \u03c6 ((StarSubalgebra.inclusion \u22ef) (x * y)) = \u03c8 ((StarSubalgebra.inclusion \u22ef) (x * y))", "state_after": "no goals"}, {"tactic": "simp only [map_star, hx]", "annotated_tactic": ["simp only [<a>map_star</a>, hx]", [{"full_name": "StarHomClass.map_star", "def_path": "Mathlib/Algebra/Star/Basic.lean", "def_pos": [503, 3], "def_end_pos": [503, 11]}]], "state_before": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : StarRing R\ninst\u271d\u00b9\u2074 : TopologicalSpace A\ninst\u271d\u00b9\u00b3 : Semiring A\ninst\u271d\u00b9\u00b2 : StarRing A\ninst\u271d\u00b9\u00b9 : TopologicalSemiring A\ninst\u271d\u00b9\u2070 : ContinuousStar A\ninst\u271d\u2079 : Algebra R A\ninst\u271d\u2078 : StarModule R A\ninst\u271d\u2077 : TopologicalSpace B\ninst\u271d\u2076 : Semiring B\ninst\u271d\u2075 : StarRing B\ninst\u271d\u2074 : Algebra R B\ninst\u271d\u00b3 : T2Space B\nF : Type u_4\na : A\ninst\u271d\u00b2 : FunLike F (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d\u00b9 : AlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\ninst\u271d : StarAlgHomClass F R (\u21a5(elementalStarAlgebra R a)) B\n\u03c6 \u03c8 : F\nh\u03c6 : Continuous \u21d1\u03c6\nh\u03c8 : Continuous \u21d1\u03c8\nh : \u03c6 \u27e8a, \u22ef\u27e9 = \u03c8 \u27e8a, \u22ef\u27e9\nthis : StarAlgHomClass F R (\u21a5(adjoin R {a}).topologicalClosure) B\nx\u271d x : \u21a5(adjoin R {a})\nhx : \u03c6 ((StarSubalgebra.inclusion \u22ef) x) = \u03c8 ((StarSubalgebra.inclusion \u22ef) x)\n\u22a2 \u03c6 ((StarSubalgebra.inclusion \u22ef) (star x)) = \u03c8 ((StarSubalgebra.inclusion \u22ef) (star x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/UpperLower/Basic.lean", "full_name": "lowerClosure_empty", "start": [1534, 1], "end": [1535, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean", "full_name": "ContinuousAffineMap.norm_def", "start": [164, 1], "end": [165, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/PiLp.lean", "full_name": "PiLp.nnnorm_equiv_symm_single", "start": [845, 1], "end": [865, 95], "traced_tactics": [{"tactic": "haveI : Nonempty \u03b9 := \u27e8i\u27e9", "annotated_tactic": ["haveI : <a>Nonempty</a> \u03b9 := \u27e8i\u27e9", [{"full_name": "Nonempty", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [709, 17], "def_end_pos": [709, 25]}]], "state_before": "p : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\nhp : Fact (1 \u2264 p)\ni : \u03b9\nb : \u03b2 i\n\u22a2 \u2016(WithLp.equiv p ((i : \u03b9) \u2192 \u03b2 i)).symm (Pi.single i b)\u2016\u208a = \u2016b\u2016\u208a", "state_after": "p : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\nhp : Fact (1 \u2264 p)\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\n\u22a2 \u2016(WithLp.equiv p ((i : \u03b9) \u2192 \u03b2 i)).symm (Pi.single i b)\u2016\u208a = \u2016b\u2016\u208a"}, {"tactic": "simp_rw [nnnorm_eq_ciSup, WithLp.equiv_symm_pi_apply]", "annotated_tactic": ["simp_rw [<a>nnnorm_eq_ciSup</a>, <a>WithLp.equiv_symm_pi_apply</a>]", [{"full_name": "PiLp.nnnorm_eq_ciSup", "def_path": "Mathlib/Analysis/NormedSpace/PiLp.lean", "def_pos": [594, 9], "def_end_pos": [594, 24]}, {"full_name": "WithLp.equiv_symm_pi_apply", "def_path": "Mathlib/Analysis/NormedSpace/PiLp.lean", "def_pos": [144, 10], "def_end_pos": [144, 43]}]], "state_before": "case top\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\nhp : Fact (1 \u2264 \u22a4)\n\u22a2 \u2016(WithLp.equiv \u22a4 ((i : \u03b9) \u2192 \u03b2 i)).symm (Pi.single i b)\u2016\u208a = \u2016b\u2016\u208a", "state_after": "case top\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\nhp : Fact (1 \u2264 \u22a4)\n\u22a2 \u2a06 i_1, \u2016Pi.single i b i_1\u2016\u208a = \u2016b\u2016\u208a"}, {"tactic": "refine\n  ciSup_eq_of_forall_le_of_forall_lt_exists_gt (fun j => ?_) fun n hn => \u27e8i, hn.trans_eq ?_\u27e9", "annotated_tactic": ["refine\n      <a>ciSup_eq_of_forall_le_of_forall_lt_exists_gt</a> (fun j => ?_) fun n hn => \u27e8i, hn.trans_eq ?_\u27e9", [{"full_name": "ciSup_eq_of_forall_le_of_forall_lt_exists_gt", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [920, 9], "def_end_pos": [920, 53]}]], "state_before": "case top\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\nhp : Fact (1 \u2264 \u22a4)\n\u22a2 \u2a06 i_1, \u2016Pi.single i b i_1\u2016\u208a = \u2016b\u2016\u208a", "state_after": "case top.refine_1\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\nhp : Fact (1 \u2264 \u22a4)\nj : \u03b9\n\u22a2 \u2016Pi.single i b j\u2016\u208a \u2264 \u2016b\u2016\u208a\n\ncase top.refine_2\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\nhp : Fact (1 \u2264 \u22a4)\nn : \u211d\u22650\nhn : n < \u2016b\u2016\u208a\n\u22a2 \u2016b\u2016\u208a = \u2016Pi.single i b i\u2016\u208a"}, {"tactic": "obtain rfl | hij := Decidable.eq_or_ne i j", "annotated_tactic": ["obtain rfl | hij := <a>Decidable.eq_or_ne</a> i j", [{"full_name": "Decidable.eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [204, 9], "def_end_pos": [204, 27]}]], "state_before": "case top.refine_1\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\nhp : Fact (1 \u2264 \u22a4)\nj : \u03b9\n\u22a2 \u2016Pi.single i b j\u2016\u208a \u2264 \u2016b\u2016\u208a", "state_after": "case top.refine_1.inl\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\nhp : Fact (1 \u2264 \u22a4)\n\u22a2 \u2016Pi.single i b i\u2016\u208a \u2264 \u2016b\u2016\u208a\n\ncase top.refine_1.inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\nhp : Fact (1 \u2264 \u22a4)\nj : \u03b9\nhij : i \u2260 j\n\u22a2 \u2016Pi.single i b j\u2016\u208a \u2264 \u2016b\u2016\u208a"}, {"tactic": "rw [Pi.single_eq_same]", "annotated_tactic": ["rw [<a>Pi.single_eq_same</a>]", [{"full_name": "Pi.single_eq_same", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [354, 3], "def_end_pos": [354, 14]}]], "state_before": "case top.refine_1.inl\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\nhp : Fact (1 \u2264 \u22a4)\n\u22a2 \u2016Pi.single i b i\u2016\u208a \u2264 \u2016b\u2016\u208a", "state_after": "no goals"}, {"tactic": "rw [Pi.single_eq_of_ne' hij, nnnorm_zero]", "annotated_tactic": ["rw [<a>Pi.single_eq_of_ne'</a> hij, <a>nnnorm_zero</a>]", [{"full_name": "Pi.single_eq_of_ne'", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [367, 3], "def_end_pos": [367, 14]}, {"full_name": "nnnorm_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [791, 30], "def_end_pos": [791, 41]}]], "state_before": "case top.refine_1.inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\nhp : Fact (1 \u2264 \u22a4)\nj : \u03b9\nhij : i \u2260 j\n\u22a2 \u2016Pi.single i b j\u2016\u208a \u2264 \u2016b\u2016\u208a", "state_after": "case top.refine_1.inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\nhp : Fact (1 \u2264 \u22a4)\nj : \u03b9\nhij : i \u2260 j\n\u22a2 0 \u2264 \u2016b\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": [204, 30], "def_end_pos": [204, 37]}]], "state_before": "case top.refine_1.inr\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\nhp : Fact (1 \u2264 \u22a4)\nj : \u03b9\nhij : i \u2260 j\n\u22a2 0 \u2264 \u2016b\u2016\u208a", "state_after": "no goals"}, {"tactic": "rw [Pi.single_eq_same]", "annotated_tactic": ["rw [<a>Pi.single_eq_same</a>]", [{"full_name": "Pi.single_eq_same", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [354, 3], "def_end_pos": [354, 14]}]], "state_before": "case top.refine_2\np : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\nhp : Fact (1 \u2264 \u22a4)\nn : \u211d\u22650\nhn : n < \u2016b\u2016\u208a\n\u22a2 \u2016b\u2016\u208a = \u2016Pi.single i b i\u2016\u208a", "state_after": "no goals"}, {"tactic": "have hp0 : (p : \u211d) \u2260 0 :=\n  mod_cast (zero_lt_one.trans_le <| Fact.out (p := 1 \u2264 (p : \u211d\u22650\u221e))).ne'", "annotated_tactic": ["have hp0 : (p : \u211d) \u2260 0 :=\n      mod_cast (zero_lt_one.trans_le <| <a>Fact.out</a> (p := 1 \u2264 (p : \u211d\u22650\u221e))).<a>ne'</a>", [{"full_name": "Fact.out", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [117, 3], "def_end_pos": [117, 6]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}]], "state_before": "case coe\np\u271d : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p\u271d)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\np : \u211d\u22650\nhp : Fact (1 \u2264 \u2191p)\n\u22a2 \u2016(WithLp.equiv (\u2191p) ((i : \u03b9) \u2192 \u03b2 i)).symm (Pi.single i b)\u2016\u208a = \u2016b\u2016\u208a", "state_after": "case coe\np\u271d : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p\u271d)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\np : \u211d\u22650\nhp : Fact (1 \u2264 \u2191p)\nhp0 : \u2191p \u2260 0\n\u22a2 \u2016(WithLp.equiv (\u2191p) ((i : \u03b9) \u2192 \u03b2 i)).symm (Pi.single i b)\u2016\u208a = \u2016b\u2016\u208a"}, {"tactic": "rw [nnnorm_eq_sum ENNReal.coe_ne_top, ENNReal.coe_toReal, Fintype.sum_eq_single i,\n  WithLp.equiv_symm_pi_apply, Pi.single_eq_same, \u2190 NNReal.rpow_mul, one_div, mul_inv_cancel hp0,\n  NNReal.rpow_one]", "annotated_tactic": ["rw [<a>nnnorm_eq_sum</a> <a>ENNReal.coe_ne_top</a>, <a>ENNReal.coe_toReal</a>, <a>Fintype.sum_eq_single</a> i,\n      <a>WithLp.equiv_symm_pi_apply</a>, <a>Pi.single_eq_same</a>, \u2190 <a>NNReal.rpow_mul</a>, <a>one_div</a>, <a>mul_inv_cancel</a> hp0,\n      <a>NNReal.rpow_one</a>]", [{"full_name": "PiLp.nnnorm_eq_sum", "def_path": "Mathlib/Analysis/NormedSpace/PiLp.lean", "def_pos": [583, 9], "def_end_pos": [583, 22]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [325, 17], "def_end_pos": [325, 27]}, {"full_name": "ENNReal.coe_toReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [262, 17], "def_end_pos": [262, 27]}, {"full_name": "Fintype.sum_eq_single", "def_path": "Mathlib/Data/Fintype/BigOperators.lean", "def_pos": [76, 3], "def_end_pos": [76, 14]}, {"full_name": "WithLp.equiv_symm_pi_apply", "def_path": "Mathlib/Analysis/NormedSpace/PiLp.lean", "def_pos": [144, 10], "def_end_pos": [144, 43]}, {"full_name": "Pi.single_eq_same", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [354, 3], "def_end_pos": [354, 14]}, {"full_name": "NNReal.rpow_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [93, 9], "def_end_pos": [93, 17]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 16]}, {"full_name": "mul_inv_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [223, 15], "def_end_pos": [223, 29]}, {"full_name": "NNReal.rpow_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [68, 9], "def_end_pos": [68, 17]}]], "state_before": "case coe\np\u271d : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p\u271d)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\np : \u211d\u22650\nhp : Fact (1 \u2264 \u2191p)\nhp0 : \u2191p \u2260 0\n\u22a2 \u2016(WithLp.equiv (\u2191p) ((i : \u03b9) \u2192 \u03b2 i)).symm (Pi.single i b)\u2016\u208a = \u2016b\u2016\u208a", "state_after": "case coe\np\u271d : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p\u271d)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\np : \u211d\u22650\nhp : Fact (1 \u2264 \u2191p)\nhp0 : \u2191p \u2260 0\n\u22a2 \u2200 (x : \u03b9), x \u2260 i \u2192 \u2016(WithLp.equiv (\u2191p) ((i : \u03b9) \u2192 \u03b2 i)).symm (Pi.single i b) x\u2016\u208a ^ \u2191p = 0"}, {"tactic": "intro j hij", "annotated_tactic": ["intro j hij", []], "state_before": "case coe\np\u271d : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p\u271d)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\np : \u211d\u22650\nhp : Fact (1 \u2264 \u2191p)\nhp0 : \u2191p \u2260 0\n\u22a2 \u2200 (x : \u03b9), x \u2260 i \u2192 \u2016(WithLp.equiv (\u2191p) ((i : \u03b9) \u2192 \u03b2 i)).symm (Pi.single i b) x\u2016\u208a ^ \u2191p = 0", "state_after": "case coe\np\u271d : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p\u271d)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\np : \u211d\u22650\nhp : Fact (1 \u2264 \u2191p)\nhp0 : \u2191p \u2260 0\nj : \u03b9\nhij : j \u2260 i\n\u22a2 \u2016(WithLp.equiv (\u2191p) ((i : \u03b9) \u2192 \u03b2 i)).symm (Pi.single i b) j\u2016\u208a ^ \u2191p = 0"}, {"tactic": "rw [WithLp.equiv_symm_pi_apply, Pi.single_eq_of_ne hij, nnnorm_zero, NNReal.zero_rpow hp0]", "annotated_tactic": ["rw [<a>WithLp.equiv_symm_pi_apply</a>, <a>Pi.single_eq_of_ne</a> hij, <a>nnnorm_zero</a>, <a>NNReal.zero_rpow</a> hp0]", [{"full_name": "WithLp.equiv_symm_pi_apply", "def_path": "Mathlib/Analysis/NormedSpace/PiLp.lean", "def_pos": [144, 10], "def_end_pos": [144, 43]}, {"full_name": "Pi.single_eq_of_ne", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [360, 3], "def_end_pos": [360, 14]}, {"full_name": "nnnorm_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [791, 30], "def_end_pos": [791, 41]}, {"full_name": "NNReal.zero_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [63, 9], "def_end_pos": [63, 18]}]], "state_before": "case coe\np\u271d : \u211d\u22650\u221e\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ninst\u271d\u2077 : Fact (1 \u2264 p\u271d)\ninst\u271d\u2076 : Fintype \u03b9\ninst\u271d\u2075 : Semiring \ud835\udd5c\ninst\u271d\u2074 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 SeminormedAddCommGroup (\u03b2 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b1 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module \ud835\udd5c (\u03b2 i)\nc : \ud835\udd5c\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nb : \u03b2 i\nthis : Nonempty \u03b9\np : \u211d\u22650\nhp : Fact (1 \u2264 \u2191p)\nhp0 : \u2191p \u2260 0\nj : \u03b9\nhij : j \u2260 i\n\u22a2 \u2016(WithLp.equiv (\u2191p) ((i : \u03b9) \u2192 \u03b2 i)).symm (Pi.single i b) j\u2016\u208a ^ \u2191p = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean", "full_name": "GaussianInt.natAbs_norm_eq", "start": [195, 1], "end": [197, 47], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "x : \u2124[i]\n\u22a2 Int.ofNat (norm x).natAbs = Int.ofNat (x.re.natAbs * x.re.natAbs + x.im.natAbs * x.im.natAbs)", "state_after": "x : \u2124[i]\n\u22a2 norm x = x.re * x.re + x.im * x.im"}, {"tactic": "simp [Zsqrtd.norm]", "annotated_tactic": ["simp [<a>Zsqrtd.norm</a>]", [{"full_name": "Zsqrtd.norm", "def_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "def_pos": [500, 5], "def_end_pos": [500, 9]}]], "state_before": "x : \u2124[i]\n\u22a2 norm x = x.re * x.re + x.im * x.im", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/Deriv/Mul.lean", "full_name": "deriv_mul_const", "start": [269, 1], "end": [271, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Dynamics/PeriodicPts.lean", "full_name": "Function.IsPeriodicPt.gcd", "start": [171, 11], "end": [177, 28], "traced_tactics": [{"tactic": "revert hm hn", "annotated_tactic": ["revert hm hn", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\nhm : IsPeriodicPt f m x\nhn : IsPeriodicPt f n x\n\u22a2 IsPeriodicPt f (m.gcd n) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\n\u22a2 IsPeriodicPt f m x \u2192 IsPeriodicPt f n x \u2192 IsPeriodicPt f (m.gcd n) x"}, {"tactic": "refine Nat.gcd.induction m n (fun n _ hn => ?_) fun m n _ ih hm hn => ?_", "annotated_tactic": ["refine <a>Nat.gcd.induction</a> m n (fun n _ hn => ?_) fun m n _ ih hm hn => ?_", [{"full_name": "Nat.gcd.induction", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [73, 25], "def_end_pos": [73, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n : \u2115\n\u22a2 IsPeriodicPt f m x \u2192 IsPeriodicPt f n x \u2192 IsPeriodicPt f (m.gcd n) x", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n\u271d n : \u2115\nx\u271d : IsPeriodicPt f 0 x\nhn : IsPeriodicPt f n x\n\u22a2 IsPeriodicPt f (Nat.gcd 0 n) x\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm\u271d n\u271d m n : \u2115\nx\u271d : 0 < m\nih : IsPeriodicPt f (n % m) x \u2192 IsPeriodicPt f m x \u2192 IsPeriodicPt f ((n % m).gcd m) x\nhm : IsPeriodicPt f m x\nhn : IsPeriodicPt f n x\n\u22a2 IsPeriodicPt f (m.gcd n) x"}, {"tactic": "rwa [Nat.gcd_zero_left]", "annotated_tactic": ["rwa [<a>Nat.gcd_zero_left</a>]", [{"full_name": "Nat.gcd_zero_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [40, 17], "def_end_pos": [40, 30]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm n\u271d n : \u2115\nx\u271d : IsPeriodicPt f 0 x\nhn : IsPeriodicPt f n x\n\u22a2 IsPeriodicPt f (Nat.gcd 0 n) x", "state_after": "no goals"}, {"tactic": "rw [Nat.gcd_rec]", "annotated_tactic": ["rw [<a>Nat.gcd_rec</a>]", [{"full_name": "Nat.gcd_rec", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [68, 9], "def_end_pos": [68, 16]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm\u271d n\u271d m n : \u2115\nx\u271d : 0 < m\nih : IsPeriodicPt f (n % m) x \u2192 IsPeriodicPt f m x \u2192 IsPeriodicPt f ((n % m).gcd m) x\nhm : IsPeriodicPt f m x\nhn : IsPeriodicPt f n x\n\u22a2 IsPeriodicPt f (m.gcd n) x", "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm\u271d n\u271d m n : \u2115\nx\u271d : 0 < m\nih : IsPeriodicPt f (n % m) x \u2192 IsPeriodicPt f m x \u2192 IsPeriodicPt f ((n % m).gcd m) x\nhm : IsPeriodicPt f m x\nhn : IsPeriodicPt f n x\n\u22a2 IsPeriodicPt f ((n % m).gcd m) x"}, {"tactic": "exact ih (hn.mod hm) hm", "annotated_tactic": ["exact ih (hn.mod hm) hm", []], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf fa : \u03b1 \u2192 \u03b1\nfb : \u03b2 \u2192 \u03b2\nx y : \u03b1\nm\u271d n\u271d m n : \u2115\nx\u271d : 0 < m\nih : IsPeriodicPt f (n % m) x \u2192 IsPeriodicPt f m x \u2192 IsPeriodicPt f ((n % m).gcd m) x\nhm : IsPeriodicPt f m x\nhn : IsPeriodicPt f n x\n\u22a2 IsPeriodicPt f ((n % m).gcd m) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/Enumerative/Composition.lean", "full_name": "List.length_splitWrtComposition", "start": [662, 1], "end": [664, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/LeftRight.lean", "full_name": "frequently_gt_nhds", "start": [37, 1], "end": [38, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Moments.lean", "full_name": "ProbabilityTheory.measure_le_le_exp_mul_mgf", "start": [349, 1], "end": [357, 16], "traced_tactics": [{"tactic": "rw [\u2190 neg_neg t, \u2190 mgf_neg, neg_neg, \u2190 neg_mul_neg (-t)]", "annotated_tactic": ["rw [\u2190 <a>neg_neg</a> t, \u2190 <a>mgf_neg</a>, <a>neg_neg</a>, \u2190 <a>neg_mul_neg</a> (-t)]", [{"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [869, 3], "def_end_pos": [869, 14]}, {"full_name": "ProbabilityTheory.mgf_neg", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [210, 9], "def_end_pos": [210, 16]}, {"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [869, 3], "def_end_pos": [869, 14]}, {"full_name": "neg_mul_neg", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [327, 9], "def_end_pos": [327, 20]}]], "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)) \u03bc\n\u22a2 (\u03bc {\u03c9 | X \u03c9 \u2264 \u03b5}).toReal \u2264 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 : t \u2264 0\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\n\u22a2 (\u03bc {\u03c9 | X \u03c9 \u2264 \u03b5}).toReal \u2264 rexp (- -t * -\u03b5) * mgf (-X) \u03bc (-t)"}, {"tactic": "refine Eq.trans_le ?_ (measure_ge_le_exp_mul_mgf (-\u03b5) (neg_nonneg.mpr ht) ?_)", "annotated_tactic": ["refine <a>Eq.trans_le</a> ?_ (<a>measure_ge_le_exp_mul_mgf</a> (-\u03b5) (neg_nonneg.mpr ht) ?_)", [{"full_name": "Eq.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [198, 7], "def_end_pos": [198, 18]}, {"full_name": "ProbabilityTheory.measure_ge_le_exp_mul_mgf", "def_path": "Mathlib/Probability/Moments.lean", "def_pos": [326, 9], "def_end_pos": [326, 34]}]], "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)) \u03bc\n\u22a2 (\u03bc {\u03c9 | X \u03c9 \u2264 \u03b5}).toReal \u2264 rexp (- -t * -\u03b5) * mgf (-X) \u03bc (-t)", "state_after": "case refine_1\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 : t \u2264 0\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\n\u22a2 (\u03bc {\u03c9 | X \u03c9 \u2264 \u03b5}).toReal = (\u03bc {\u03c9 | -\u03b5 \u2264 (-X) \u03c9}).toReal\n\ncase refine_2\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 : t \u2264 0\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\n\u22a2 Integrable (fun \u03c9 => rexp (-t * (-X) \u03c9)) \u03bc"}, {"tactic": "congr with \u03c9", "annotated_tactic": ["congr with \u03c9", []], "state_before": "case refine_1\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 : t \u2264 0\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\n\u22a2 (\u03bc {\u03c9 | X \u03c9 \u2264 \u03b5}).toReal = (\u03bc {\u03c9 | -\u03b5 \u2264 (-X) \u03c9}).toReal", "state_after": "case refine_1.e_a.h.e_6.h.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 : t \u2264 0\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\n\u03c9 : \u03a9\n\u22a2 \u03c9 \u2208 {\u03c9 | X \u03c9 \u2264 \u03b5} \u2194 \u03c9 \u2208 {\u03c9 | -\u03b5 \u2264 (-X) \u03c9}"}, {"tactic": "simp only [Pi.neg_apply, neg_le_neg_iff]", "annotated_tactic": ["simp only [<a>Pi.neg_apply</a>, <a>neg_le_neg_iff</a>]", [{"full_name": "Pi.neg_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [166, 3], "def_end_pos": [166, 14]}, {"full_name": "neg_le_neg_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [342, 3], "def_end_pos": [342, 14]}]], "state_before": "case refine_1.e_a.h.e_6.h.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 : t \u2264 0\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\n\u03c9 : \u03a9\n\u22a2 \u03c9 \u2208 {\u03c9 | X \u03c9 \u2264 \u03b5} \u2194 \u03c9 \u2208 {\u03c9 | -\u03b5 \u2264 (-X) \u03c9}", "state_after": "no goals"}, {"tactic": "simp_rw [Pi.neg_apply, neg_mul_neg]", "annotated_tactic": ["simp_rw [<a>Pi.neg_apply</a>, <a>neg_mul_neg</a>]", [{"full_name": "Pi.neg_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [166, 3], "def_end_pos": [166, 14]}, {"full_name": "neg_mul_neg", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [327, 9], "def_end_pos": [327, 20]}]], "state_before": "case refine_2\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 : t \u2264 0\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\n\u22a2 Integrable (fun \u03c9 => rexp (-t * (-X) \u03c9)) \u03bc", "state_after": "case refine_2\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 : t \u2264 0\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\n\u22a2 Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc"}, {"tactic": "exact h_int", "annotated_tactic": ["exact h_int", []], "state_before": "case refine_2\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 : t \u2264 0\nh_int : Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc\n\u22a2 Integrable (fun \u03c9 => rexp (t * X \u03c9)) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "not_surjective_of_ordinal", "start": [2133, 1], "end": [2134, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Injective.lean", "full_name": "Module.injective_module_of_injective_object", "start": [70, 1], "end": [76, 27], "traced_tactics": [{"tactic": "have : CategoryTheory.Mono (ModuleCat.ofHom f) := (ModuleCat.mono_iff_injective _).mpr hf", "annotated_tactic": ["have : <a>CategoryTheory.Mono</a> (<a>ModuleCat.ofHom</a> f) := (<a>ModuleCat.mono_iff_injective</a> _).<a>mpr</a> hf", [{"full_name": "CategoryTheory.Mono", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [295, 7], "def_end_pos": [295, 11]}, {"full_name": "ModuleCat.ofHom", "def_path": "Mathlib/Algebra/Category/ModuleCat/Basic.lean", "def_pos": [165, 5], "def_end_pos": [165, 10]}, {"full_name": "ModuleCat.mono_iff_injective", "def_path": "Mathlib/Algebra/Category/ModuleCat/EpiMono.lean", "def_pos": [45, 9], "def_end_pos": [45, 27]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}]], "state_before": "R : Type u\ninst\u271d\u00b2 : Ring R\nQ : Type v\ninst\u271d\u00b9 : AddCommGroup Q\ninst\u271d : Module R Q\ninj : CategoryTheory.Injective (ModuleCat.of R Q)\nX Y : Type v\nx\u271d\u00b3 : AddCommGroup X\nx\u271d\u00b2 : AddCommGroup Y\nx\u271d\u00b9 : Module R X\nx\u271d : Module R Y\nf : X \u2192\u2097[R] Y\nhf : Function.Injective \u21d1f\ng : X \u2192\u2097[R] Q\n\u22a2 \u2203 h, \u2200 (x : X), h (f x) = g x", "state_after": "R : Type u\ninst\u271d\u00b2 : Ring R\nQ : Type v\ninst\u271d\u00b9 : AddCommGroup Q\ninst\u271d : Module R Q\ninj : CategoryTheory.Injective (ModuleCat.of R Q)\nX Y : Type v\nx\u271d\u00b3 : AddCommGroup X\nx\u271d\u00b2 : AddCommGroup Y\nx\u271d\u00b9 : Module R X\nx\u271d : Module R Y\nf : X \u2192\u2097[R] Y\nhf : Function.Injective \u21d1f\ng : X \u2192\u2097[R] Q\nthis : CategoryTheory.Mono (ModuleCat.ofHom f)\n\u22a2 \u2203 h, \u2200 (x : X), h (f x) = g x"}, {"tactic": "obtain \u27e8l, rfl\u27e9 := inj.factors (ModuleCat.ofHom g) (ModuleCat.ofHom f)", "annotated_tactic": ["obtain \u27e8l, rfl\u27e9 := inj.factors (<a>ModuleCat.ofHom</a> g) (<a>ModuleCat.ofHom</a> f)", [{"full_name": "ModuleCat.ofHom", "def_path": "Mathlib/Algebra/Category/ModuleCat/Basic.lean", "def_pos": [165, 5], "def_end_pos": [165, 10]}, {"full_name": "ModuleCat.ofHom", "def_path": "Mathlib/Algebra/Category/ModuleCat/Basic.lean", "def_pos": [165, 5], "def_end_pos": [165, 10]}]], "state_before": "R : Type u\ninst\u271d\u00b2 : Ring R\nQ : Type v\ninst\u271d\u00b9 : AddCommGroup Q\ninst\u271d : Module R Q\ninj : CategoryTheory.Injective (ModuleCat.of R Q)\nX Y : Type v\nx\u271d\u00b3 : AddCommGroup X\nx\u271d\u00b2 : AddCommGroup Y\nx\u271d\u00b9 : Module R X\nx\u271d : Module R Y\nf : X \u2192\u2097[R] Y\nhf : Function.Injective \u21d1f\ng : X \u2192\u2097[R] Q\nthis : CategoryTheory.Mono (ModuleCat.ofHom f)\n\u22a2 \u2203 h, \u2200 (x : X), h (f x) = g x", "state_after": "case intro\nR : Type u\ninst\u271d\u00b2 : Ring R\nQ : Type v\ninst\u271d\u00b9 : AddCommGroup Q\ninst\u271d : Module R Q\ninj : CategoryTheory.Injective (ModuleCat.of R Q)\nX Y : Type v\nx\u271d\u00b3 : AddCommGroup X\nx\u271d\u00b2 : AddCommGroup Y\nx\u271d\u00b9 : Module R X\nx\u271d : Module R Y\nf : X \u2192\u2097[R] Y\nhf : Function.Injective \u21d1f\nthis : CategoryTheory.Mono (ModuleCat.ofHom f)\nl : ModuleCat.of R Y \u27f6 ModuleCat.of R Q\n\u22a2 \u2203 h, \u2200 (x : X), h (f x) = (CategoryTheory.CategoryStruct.comp (ModuleCat.ofHom f) l) x"}, {"tactic": "exact \u27e8l, fun _ \u21a6 rfl\u27e9", "annotated_tactic": ["exact \u27e8l, fun _ \u21a6 <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case intro\nR : Type u\ninst\u271d\u00b2 : Ring R\nQ : Type v\ninst\u271d\u00b9 : AddCommGroup Q\ninst\u271d : Module R Q\ninj : CategoryTheory.Injective (ModuleCat.of R Q)\nX Y : Type v\nx\u271d\u00b3 : AddCommGroup X\nx\u271d\u00b2 : AddCommGroup Y\nx\u271d\u00b9 : Module R X\nx\u271d : Module R Y\nf : X \u2192\u2097[R] Y\nhf : Function.Injective \u21d1f\nthis : CategoryTheory.Mono (ModuleCat.ofHom f)\nl : ModuleCat.of R Y \u27f6 ModuleCat.of R Q\n\u22a2 \u2203 h, \u2200 (x : X), h (f x) = (CategoryTheory.CategoryStruct.comp (ModuleCat.ofHom f) l) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Compactness/Compact.lean", "full_name": "isCompact_iff_ultrafilter_le_nhds'", "start": [145, 1], "end": [147, 87], "traced_tactics": [{"tactic": "simp only [isCompact_iff_ultrafilter_le_nhds, le_principal_iff, Ultrafilter.mem_coe]", "annotated_tactic": ["simp only [<a>isCompact_iff_ultrafilter_le_nhds</a>, <a>le_principal_iff</a>, <a>Ultrafilter.mem_coe</a>]", [{"full_name": "isCompact_iff_ultrafilter_le_nhds", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [134, 9], "def_end_pos": [134, 42]}, {"full_name": "Filter.le_principal_iff", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [668, 9], "def_end_pos": [668, 25]}, {"full_name": "Ultrafilter.mem_coe", "def_path": "Mathlib/Order/Filter/Ultrafilter.lean", "def_pos": [76, 9], "def_end_pos": [76, 16]}]], "state_before": "X : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t : Set X\n\u22a2 IsCompact s \u2194 \u2200 (f : Ultrafilter X), s \u2208 f \u2192 \u2203 x \u2208 s, \u2191f \u2264 \ud835\udcdd x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/ExactFunctor.lean", "full_name": "CategoryTheory.ExactFunctor.forget_obj", "start": [176, 1], "end": [177, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_max", "start": [1435, 1], "end": [1443, 74], "traced_tactics": [{"tactic": "have hm : MeasurableSet { x | f x \u2264 g x } := measurableSet_le hf hg", "annotated_tactic": ["have hm : <a>MeasurableSet</a> { x | f x \u2264 g x } := <a>measurableSet_le</a> hf hg", [{"full_name": "MeasurableSet", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [64, 5], "def_end_pos": [64, 18]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean", "def_pos": [167, 9], "def_end_pos": [167, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\n\u22a2 \u222b\u207b (x : \u03b1), max (f x) (g x) \u2202\u03bc = \u222b\u207b (x : \u03b1) in {x | f x \u2264 g x}, g x \u2202\u03bc + \u222b\u207b (x : \u03b1) in {x | g x < f x}, f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nhm : MeasurableSet {x | f x \u2264 g x}\n\u22a2 \u222b\u207b (x : \u03b1), max (f x) (g x) \u2202\u03bc = \u222b\u207b (x : \u03b1) in {x | f x \u2264 g x}, g x \u2202\u03bc + \u222b\u207b (x : \u03b1) in {x | g x < f x}, f x \u2202\u03bc"}, {"tactic": "rw [\u2190 lintegral_add_compl (fun x => max (f x) (g x)) hm]", "annotated_tactic": ["rw [\u2190 <a>lintegral_add_compl</a> (fun x => <a>max</a> (f x) (g x)) hm]", [{"full_name": "MeasureTheory.lintegral_add_compl", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [1430, 9], "def_end_pos": [1430, 28]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nhm : MeasurableSet {x | f x \u2264 g x}\n\u22a2 \u222b\u207b (x : \u03b1), max (f x) (g x) \u2202\u03bc = \u222b\u207b (x : \u03b1) in {x | f x \u2264 g x}, g x \u2202\u03bc + \u222b\u207b (x : \u03b1) in {x | g x < f x}, f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nhm : MeasurableSet {x | f x \u2264 g x}\n\u22a2 \u222b\u207b (x : \u03b1) in {x | f x \u2264 g x}, max (f x) (g x) \u2202\u03bc + \u222b\u207b (x : \u03b1) in {x | f x \u2264 g x}\u1d9c, max (f x) (g x) \u2202\u03bc =\n    \u222b\u207b (x : \u03b1) in {x | f x \u2264 g x}, g x \u2202\u03bc + \u222b\u207b (x : \u03b1) in {x | g x < f x}, f x \u2202\u03bc"}, {"tactic": "simp only [\u2190 compl_setOf, \u2190 not_le]", "annotated_tactic": ["simp only [\u2190 <a>compl_setOf</a>, \u2190 <a>not_le</a>]", [{"full_name": "Set.compl_setOf", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1605, 9], "def_end_pos": [1605, 20]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [375, 9], "def_end_pos": [375, 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\nhf : Measurable f\nhg : Measurable g\nhm : MeasurableSet {x | f x \u2264 g x}\n\u22a2 \u222b\u207b (x : \u03b1) in {x | f x \u2264 g x}, max (f x) (g x) \u2202\u03bc + \u222b\u207b (x : \u03b1) in {x | f x \u2264 g x}\u1d9c, max (f x) (g x) \u2202\u03bc =\n    \u222b\u207b (x : \u03b1) in {x | f x \u2264 g x}, g x \u2202\u03bc + \u222b\u207b (x : \u03b1) in {x | g x < f x}, f x \u2202\u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nhm : MeasurableSet {x | f x \u2264 g x}\n\u22a2 \u222b\u207b (x : \u03b1) in {x | f x \u2264 g x}, max (f x) (g x) \u2202\u03bc + \u222b\u207b (x : \u03b1) in {x | f x \u2264 g x}\u1d9c, max (f x) (g x) \u2202\u03bc =\n    \u222b\u207b (x : \u03b1) in {x | f x \u2264 g x}, g x \u2202\u03bc + \u222b\u207b (x : \u03b1) in {a | f a \u2264 g a}\u1d9c, f x \u2202\u03bc"}, {"tactic": "exacts [ae_of_all _ fun x => max_eq_right (a := f x) (b := g x),\n  ae_of_all _ fun x (hx : \u00ac f x \u2264 g x) => max_eq_left (not_le.1 hx).le]", "annotated_tactic": ["exacts [<a>ae_of_all</a> _ fun x => <a>max_eq_right</a> (a := f x) (b := g x),\n    <a>ae_of_all</a> _ fun x (hx : \u00ac f x \u2264 g x) => <a>max_eq_left</a> (<a>not_le</a>.1 hx).<a>le</a>]", [{"full_name": "MeasureTheory.ae_of_all", "def_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean", "def_pos": [94, 9], "def_end_pos": [94, 18]}, {"full_name": "max_eq_right", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [137, 9], "def_end_pos": [137, 21]}, {"full_name": "MeasureTheory.ae_of_all", "def_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean", "def_pos": [94, 9], "def_end_pos": [94, 18]}, {"full_name": "max_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [133, 9], "def_end_pos": [133, 20]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [375, 9], "def_end_pos": [375, 15]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}]], "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 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nhm : MeasurableSet {x | f x \u2264 g x}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 {x | f x \u2264 g x} \u2192 max (f x) (g x) = g x\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 \u211d\u22650\u221e\nhf : Measurable f\nhg : Measurable g\nhm : MeasurableSet {x | f x \u2264 g x}\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 {x | f x \u2264 g x}\u1d9c \u2192 max (f x) (g x) = f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Group/Multiset.lean", "full_name": "Multiset.prod_add", "start": [111, 1], "end": [112, 49], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "F : Type u_1\n\u03b9 : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : CommMonoid \u03b2\ns\u271d t\u271d : Multiset \u03b1\na : \u03b1\nm : Multiset \u03b9\nf g : \u03b9 \u2192 \u03b1\ns t : Multiset \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 (\u27e6l\u2081\u27e7 + \u27e6l\u2082\u27e7).prod = prod \u27e6l\u2081\u27e7 * prod \u27e6l\u2082\u27e7", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/BumpFunction/Convolution.lean", "full_name": "ContDiffBump.convolution_tendsto_right", "start": [88, 8], "end": [95, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Dist.lean", "full_name": "Nat.dist_eq_sub_of_le", "start": [45, 1], "end": [46, 49], "traced_tactics": [{"tactic": "rw [dist, tsub_eq_zero_iff_le.mpr h, zero_add]", "annotated_tactic": ["rw [<a>dist</a>, tsub_eq_zero_iff_le.mpr h, <a>zero_add</a>]", [{"full_name": "Nat.dist", "def_path": "Mathlib/Data/Nat/Dist.lean", "def_pos": [20, 5], "def_end_pos": [20, 9]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [476, 3], "def_end_pos": [476, 14]}]], "state_before": "n m : \u2115\nh : n \u2264 m\n\u22a2 n.dist m = m - n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Equiv/Fin.lean", "full_name": "finRotate_zero", "start": [388, 1], "end": [388, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Equiv/Defs.lean", "full_name": "Equiv.symm_bijective", "start": [353, 1], "end": [354, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Sober.lean", "full_name": "IsGenericPoint.def", "start": [42, 1], "end": [44, 4], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Submodule/Lattice.lean", "full_name": "Submodule.sum_mem_biSup", "start": [304, 1], "end": [306, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/ContDiff/Basic.lean", "full_name": "ContDiffWithinAt.prod_map'", "start": [1690, 1], "end": [1694, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/LocallyConvex/WithSeminorms.lean", "full_name": "SeminormFamily.withSeminorms_iff_nhds_eq_iInf", "start": [430, 1], "end": [435, 43], "traced_tactics": [{"tactic": "rw [\u2190 p.filter_eq_iInf]", "annotated_tactic": ["rw [\u2190 p.filter_eq_iInf]", []], "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5d : Type u_3\n\ud835\udd5d\u2082 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\n\u03b9 : Type u_8\n\u03b9' : Type u_9\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Nonempty \u03b9\nt : TopologicalSpace E\ninst\u271d : TopologicalAddGroup E\np : SeminormFamily \ud835\udd5c E \u03b9\n\u22a2 WithSeminorms p \u2194 \ud835\udcdd 0 = \u2a05 i, comap (\u21d1(p i)) (\ud835\udcdd 0)", "state_after": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5d : Type u_3\n\ud835\udd5d\u2082 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\n\u03b9 : Type u_8\n\u03b9' : Type u_9\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Nonempty \u03b9\nt : TopologicalSpace E\ninst\u271d : TopologicalAddGroup E\np : SeminormFamily \ud835\udd5c E \u03b9\n\u22a2 WithSeminorms p \u2194 \ud835\udcdd 0 = AddGroupFilterBasis.toFilterBasis.filter"}, {"tactic": "refine \u27e8fun h => ?_, p.withSeminorms_of_nhds\u27e9", "annotated_tactic": ["refine \u27e8fun h => ?_, p.withSeminorms_of_nhds\u27e9", []], "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5d : Type u_3\n\ud835\udd5d\u2082 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\n\u03b9 : Type u_8\n\u03b9' : Type u_9\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Nonempty \u03b9\nt : TopologicalSpace E\ninst\u271d : TopologicalAddGroup E\np : SeminormFamily \ud835\udd5c E \u03b9\n\u22a2 WithSeminorms p \u2194 \ud835\udcdd 0 = AddGroupFilterBasis.toFilterBasis.filter", "state_after": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5d : Type u_3\n\ud835\udd5d\u2082 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\n\u03b9 : Type u_8\n\u03b9' : Type u_9\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Nonempty \u03b9\nt : TopologicalSpace E\ninst\u271d : TopologicalAddGroup E\np : SeminormFamily \ud835\udd5c E \u03b9\nh : WithSeminorms p\n\u22a2 \ud835\udcdd 0 = AddGroupFilterBasis.toFilterBasis.filter"}, {"tactic": "exact AddGroupFilterBasis.nhds_zero_eq _", "annotated_tactic": ["exact <a>AddGroupFilterBasis.nhds_zero_eq</a> _", [{"full_name": "AddGroupFilterBasis.nhds_zero_eq", "def_path": "Mathlib/Topology/Algebra/FilterBasis.lean", "def_pos": [187, 3], "def_end_pos": [187, 14]}]], "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c\u2082 : Type u_2\n\ud835\udd5d : Type u_3\n\ud835\udd5d\u2082 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\n\u03b9 : Type u_8\n\u03b9' : Type u_9\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : Nonempty \u03b9\nt : TopologicalSpace E\ninst\u271d : TopologicalAddGroup E\np : SeminormFamily \ud835\udd5c E \u03b9\nh : WithSeminorms p\n\u22a2 \ud835\udcdd 0 = AddGroupFilterBasis.toFilterBasis.filter", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/KrullTopology.lean", "full_name": "IntermediateField.map_id", "start": [60, 1], "end": [62, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.setToL1_mono", "start": [1188, 1], "end": [1193, 40], "traced_tactics": [{"tactic": "rw [\u2190 sub_nonneg] at hfg \u22a2", "annotated_tactic": ["rw [\u2190 <a>sub_nonneg</a>] at hfg \u22a2", [{"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\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 \u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 (T s) x\nf g : \u21a5(Lp G' 1 \u03bc)\nhfg : f \u2264 g\n\u22a2 (setToL1 hT) f \u2264 (setToL1 hT) g", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\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 \u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 (T s) x\nf g : \u21a5(Lp G' 1 \u03bc)\nhfg : 0 \u2264 g - f\n\u22a2 0 \u2264 (setToL1 hT) g - (setToL1 hT) f"}, {"tactic": "rw [\u2190 (setToL1 hT).map_sub]", "annotated_tactic": ["rw [\u2190 (<a>setToL1</a> hT).<a>map_sub</a>]", [{"full_name": "MeasureTheory.L1.setToL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1019, 5], "def_end_pos": [1019, 12]}, {"full_name": "ContinuousLinearMap.map_sub", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [1401, 19], "def_end_pos": [1401, 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\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 \u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 (T s) x\nf g : \u21a5(Lp G' 1 \u03bc)\nhfg : 0 \u2264 g - f\n\u22a2 0 \u2264 (setToL1 hT) g - (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 \u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 (T s) x\nf g : \u21a5(Lp G' 1 \u03bc)\nhfg : 0 \u2264 g - f\n\u22a2 0 \u2264 (setToL1 hT) (g - f)"}, {"tactic": "exact setToL1_nonneg hT hT_nonneg hfg", "annotated_tactic": ["exact <a>setToL1_nonneg</a> hT hT_nonneg hfg", [{"full_name": "MeasureTheory.L1.setToL1_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1172, 9], "def_end_pos": [1172, 23]}]], "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 \u03bc s < \u22a4 \u2192 \u2200 (x : G'), 0 \u2264 x \u2192 0 \u2264 (T s) x\nf g : \u21a5(Lp G' 1 \u03bc)\nhfg : 0 \u2264 g - f\n\u22a2 0 \u2264 (setToL1 hT) (g - f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Sites/Sheaf.lean", "full_name": "CategoryTheory.Presheaf.isSheaf_iff_isSheaf_forget", "start": [755, 1], "end": [759, 33], "traced_tactics": [{"tactic": "have : HasLimitsOfSize.{v\u2081, max v\u2081 u\u2081} A' := hasLimitsOfSizeShrink.{_, _, u\u2081, 0} A'", "annotated_tactic": ["have : <a>HasLimitsOfSize</a>.{v\u2081, max v\u2081 u\u2081} A' := <a>hasLimitsOfSizeShrink</a>.{_, _, u\u2081, 0} A'", [{"full_name": "CategoryTheory.Limits.HasLimitsOfSize", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [112, 7], "def_end_pos": [112, 22]}, {"full_name": "CategoryTheory.Limits.hasLimitsOfSizeShrink", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [648, 9], "def_end_pos": [648, 30]}]], "state_before": "C : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nB : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ns : A' \u2964 Type (max v\u2081 u\u2081)\ninst\u271d\u00b2 : HasLimits A'\ninst\u271d\u00b9 : PreservesLimits s\ninst\u271d : s.ReflectsIsomorphisms\n\u22a2 IsSheaf J P' \u2194 IsSheaf J (P' \u22d9 s)", "state_after": "C : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nB : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ns : A' \u2964 Type (max v\u2081 u\u2081)\ninst\u271d\u00b2 : HasLimits A'\ninst\u271d\u00b9 : PreservesLimits s\ninst\u271d : s.ReflectsIsomorphisms\nthis : HasLimitsOfSize.{v\u2081, max v\u2081 u\u2081, max u\u2081 v\u2081, u\u2082} A'\n\u22a2 IsSheaf J P' \u2194 IsSheaf J (P' \u22d9 s)"}, {"tactic": "have : PreservesLimitsOfSize.{v\u2081, max v\u2081 u\u2081} s := preservesLimitsOfSizeShrink.{_, 0, _, u\u2081} s", "annotated_tactic": ["have : <a>PreservesLimitsOfSize</a>.{v\u2081, max v\u2081 u\u2081} s := <a>preservesLimitsOfSizeShrink</a>.{_, 0, _, u\u2081} s", [{"full_name": "CategoryTheory.Limits.PreservesLimitsOfSize", "def_path": "Mathlib/CategoryTheory/Limits/Preserves/Basic.lean", "def_pos": [83, 7], "def_end_pos": [83, 28]}, {"full_name": "CategoryTheory.Limits.preservesLimitsOfSizeShrink", "def_path": "Mathlib/CategoryTheory/Limits/Preserves/Basic.lean", "def_pos": [288, 5], "def_end_pos": [288, 32]}]], "state_before": "C : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nB : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ns : A' \u2964 Type (max v\u2081 u\u2081)\ninst\u271d\u00b2 : HasLimits A'\ninst\u271d\u00b9 : PreservesLimits s\ninst\u271d : s.ReflectsIsomorphisms\nthis : HasLimitsOfSize.{v\u2081, max v\u2081 u\u2081, max u\u2081 v\u2081, u\u2082} A'\n\u22a2 IsSheaf J P' \u2194 IsSheaf J (P' \u22d9 s)", "state_after": "C : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nB : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ns : A' \u2964 Type (max v\u2081 u\u2081)\ninst\u271d\u00b2 : HasLimits A'\ninst\u271d\u00b9 : PreservesLimits s\ninst\u271d : s.ReflectsIsomorphisms\nthis\u271d : HasLimitsOfSize.{v\u2081, max v\u2081 u\u2081, max u\u2081 v\u2081, u\u2082} A'\nthis : PreservesLimitsOfSize.{v\u2081, max v\u2081 u\u2081, max u\u2081 v\u2081, max u\u2081 v\u2081, u\u2082, max (u\u2081 + 1) (v\u2081 + 1)} s\n\u22a2 IsSheaf J P' \u2194 IsSheaf J (P' \u22d9 s)"}, {"tactic": "apply isSheaf_iff_isSheaf_comp", "annotated_tactic": ["apply <a>isSheaf_iff_isSheaf_comp</a>", [{"full_name": "CategoryTheory.Presheaf.isSheaf_iff_isSheaf_comp", "def_path": "Mathlib/CategoryTheory/Sites/Sheaf.lean", "def_pos": [740, 9], "def_end_pos": [740, 33]}]], "state_before": "C : Type u\u2081\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\nA : Type u\u2082\ninst\u271d\u2075 : Category.{v\u2082, u\u2082} A\nA' : Type u\u2082\ninst\u271d\u2074 : Category.{max v\u2081 u\u2081, u\u2082} A'\nB : Type u\u2083\ninst\u271d\u00b3 : Category.{v\u2083, u\u2083} B\nJ : GrothendieckTopology C\nU : C\nR : Presieve U\nP : C\u1d52\u1d56 \u2964 A\nP' : C\u1d52\u1d56 \u2964 A'\ns : A' \u2964 Type (max v\u2081 u\u2081)\ninst\u271d\u00b2 : HasLimits A'\ninst\u271d\u00b9 : PreservesLimits s\ninst\u271d : s.ReflectsIsomorphisms\nthis\u271d : HasLimitsOfSize.{v\u2081, max v\u2081 u\u2081, max u\u2081 v\u2081, u\u2082} A'\nthis : PreservesLimitsOfSize.{v\u2081, max v\u2081 u\u2081, max u\u2081 v\u2081, max u\u2081 v\u2081, u\u2082, max (u\u2081 + 1) (v\u2081 + 1)} s\n\u22a2 IsSheaf J P' \u2194 IsSheaf J (P' \u22d9 s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/ADEInequality.lean", "full_name": "ADEInequality.admissible_E6", "start": [148, 1], "end": [149, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Eigenspace/Matrix.lean", "full_name": "hasEigenvector_toLin'_diagonal", "start": [37, 1], "end": [40, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Field/Defs.lean", "full_name": "div_nonneg", "start": [83, 1], "end": [84, 61], "traced_tactics": [{"tactic": "rw [div_eq_mul_inv]", "annotated_tactic": ["rw [<a>div_eq_mul_inv</a>]", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1059, 9], "def_end_pos": [1059, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\na b : \u03b1\nha : 0 \u2264 a\nhb : 0 \u2264 b\n\u22a2 0 \u2264 a / b", "state_after": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\na b : \u03b1\nha : 0 \u2264 a\nhb : 0 \u2264 b\n\u22a2 0 \u2264 a * b\u207b\u00b9"}, {"tactic": "exact mul_nonneg ha (inv_nonneg.2 hb)", "annotated_tactic": ["exact <a>mul_nonneg</a> ha (<a>inv_nonneg</a>.2 hb)", [{"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [437, 7], "def_end_pos": [437, 17]}, {"full_name": "inv_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Defs.lean", "def_pos": [55, 15], "def_end_pos": [55, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : LinearOrderedSemifield \u03b1\na b : \u03b1\nha : 0 \u2264 a\nhb : 0 \u2264 b\n\u22a2 0 \u2264 a * b\u207b\u00b9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ZMod/Basic.lean", "full_name": "ZMod.cast_sub_one", "start": [651, 1], "end": [662, 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 (k - 1).cast = (if k = 0 then \u2191n else k.cast) - 1", "state_after": "case pos\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\nk : ZMod n\nhk : k = 0\n\u22a2 (k - 1).cast = \u2191n - 1\n\ncase neg\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\nk : ZMod n\nhk : \u00ack = 0\n\u22a2 (k - 1).cast = k.cast - 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": [467, 3], "def_end_pos": [467, 14]}, {"full_name": "ZMod.cast_neg_one", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [645, 9], "def_end_pos": [645, 21]}]], "state_before": "case pos\nR : Type u_1\ninst\u271d : Ring R\nn : \u2115\nk : ZMod n\nhk : k = 0\n\u22a2 (k - 1).cast = \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 (k - 1).cast = k.cast - 1", "state_after": "case neg.zero\nR : Type u_1\ninst\u271d : Ring R\nk : ZMod 0\nhk : \u00ack = 0\n\u22a2 (k - 1).cast = k.cast - 1\n\ncase neg.succ\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nk : ZMod (n\u271d + 1)\nhk : \u00ack = 0\n\u22a2 (k - 1).cast = k.cast - 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": [95, 5], "def_end_pos": [95, 9]}, {"full_name": "ZMod.cast", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [169, 5], "def_end_pos": [169, 9]}]], "state_before": "case neg.zero\nR : Type u_1\ninst\u271d : Ring R\nk : ZMod 0\nhk : \u00ack = 0\n\u22a2 (k - 1).cast = k.cast - 1", "state_after": "case neg.zero\nR : Type u_1\ninst\u271d : Ring R\nk : ZMod 0\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": [123, 9], "def_end_pos": [123, 17]}, {"full_name": "Int.cast_one", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [79, 9], "def_end_pos": [79, 17]}]], "state_before": "case neg.zero\nR : Type u_1\ninst\u271d : Ring R\nk : ZMod 0\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": [95, 5], "def_end_pos": [95, 9]}, {"full_name": "ZMod.cast", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [169, 5], "def_end_pos": [169, 9]}, {"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 (n\u271d + 1)\nhk : \u00ack = 0\n\u22a2 (k - 1).cast = k.cast - 1", "state_after": "case neg.succ\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nk : ZMod (n\u271d + 1)\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/Algebra/Group/Fin.lean", "def_pos": [90, 7], "def_end_pos": [90, 18]}, {"full_name": "if_neg", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [937, 9], "def_end_pos": [937, 15]}]], "state_before": "case neg.succ\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nk : ZMod (n\u271d + 1)\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 (n\u271d + 1)\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 (n\u271d + 1)\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": [154, 9], "def_end_pos": [154, 17]}]], "state_before": "case neg.succ\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nk : ZMod (n\u271d + 1)\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 (n\u271d + 1)\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/lean4/src/lean/Init/Data/Fin/Lemmas.lean", "def_pos": [42, 9], "def_end_pos": [42, 16]}, {"full_name": "Fin.val_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean", "def_pos": [126, 17], "def_end_pos": [126, 25]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "Nat.one_le_iff_ne_zero", "def_path": "Mathlib/Data/Nat/Defs.lean", "def_pos": [166, 7], "def_end_pos": [166, 25]}]], "state_before": "case neg.succ\nR : Type u_1\ninst\u271d : Ring R\nn\u271d : \u2115\nk : ZMod (n\u271d + 1)\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 (n\u271d + 1)\nhk : \u00ack = 0\n\u22a2 \u00ack = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finite/Defs.lean", "full_name": "not_infinite_iff_finite", "start": [128, 1], "end": [129, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/EffectiveEpi/RegularEpi.lean", "full_name": "CategoryTheory.effectiveEpiOfKernelPair", "start": [34, 1], "end": [37, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Iso.lean", "full_name": "CategoryTheory.comp_hom_eq_id", "start": [503, 1], "end": [504, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.coe_neg", "start": [1856, 11], "end": [1857, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "full_name": "EMetric.closedBall_subset_closedBall", "start": [595, 1], "end": [596, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Regular/SMul.lean", "full_name": "Units.isSMulRegular", "start": [252, 1], "end": [253, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/Irrational.lean", "full_name": "Irrational.int_mul", "start": [403, 1], "end": [404, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean", "full_name": "UniformOnFun.isClosed_setOf_continuous_of_le", "start": [1101, 1], "end": [1104, 98], "traced_tactics": [{"tactic": "simpa only [isOpen_iSup_iff, isOpen_coinduced]", "annotated_tactic": ["simpa only [<a>isOpen_iSup_iff</a>, <a>isOpen_coinduced</a>]", [{"full_name": "isOpen_iSup_iff", "def_path": "Mathlib/Topology/Order.lean", "def_pos": [987, 9], "def_end_pos": [987, 24]}, {"full_name": "isOpen_coinduced", "def_path": "Mathlib/Topology/Order.lean", "def_pos": [392, 9], "def_end_pos": [392, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\ng : \u03b9 \u2192 \u03b1\ninst\u271d : UniformSpace \u03b2\n\ud835\udd16 : Set (Set \u03b1)\nt : TopologicalSpace \u03b1\nh : t \u2264 \u2a06 s \u2208 \ud835\udd16, TopologicalSpace.coinduced Subtype.val inferInstance\nu : Set \u03b1\nhu : \u2200 s \u2208 \ud835\udd16, IsOpen (Subtype.val \u207b\u00b9' u)\n\u22a2 IsOpen u", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.sup_singleton", "start": [79, 1], "end": [80, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monoidal/Mon_.lean", "full_name": "Mon_.tensorUnit_mul", "start": [486, 1], "end": [486, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "full_name": "Algebra.coe_sInf", 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"def_pos": [57, 9], "def_end_pos": [57, 27]}]], "state_before": "C : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} C\nX Y : C\u1d52\u1d56\nf : X \u27f6 Y\n\u22a2 IsIso f.unop \u2194 IsIso f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Function.lean", "full_name": "ConvexOn.translate_right", "start": [281, 1], "end": [288, 8], "traced_tactics": [{"tactic": "rw [smul_add, smul_add, add_add_add_comm, Convex.combo_self hab]", "annotated_tactic": ["rw [<a>smul_add</a>, <a>smul_add</a>, <a>add_add_add_comm</a>, <a>Convex.combo_self</a> hab]", [{"full_name": "smul_add", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [145, 9], "def_end_pos": [145, 17]}, {"full_name": "smul_add", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [145, 9], "def_end_pos": [145, 17]}, {"full_name": "add_add_add_comm", "def_path": 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"file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.le_measure_diff", "start": [243, 1], "end": [245, 37], "traced_tactics": [{"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u22a2 \u03bc (s\u2081 \u2229 s\u2082) + \u03bc (s\u2081 \\ s\u2082) \u2264 \u03bc s\u2082 + \u03bc (s\u2081 \\ s\u2082)", "state_after": "case bc.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\nm : MeasurableSpace \u03b1\n\u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1\ns s\u2081 s\u2082 t : Set \u03b1\n\u22a2 s\u2081 \u2229 s\u2082 \u2286 s\u2082"}, {"tactic": "apply inter_subset_right", "annotated_tactic": ["apply 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u_4\n\u03b9 : Sort u_5\ns\u271d : Set \u03b1\nt : Set \u03b2\nf : Filter \u03b1\ng : Filter \u03b2\ns : Set (\u03b1 \u00d7 \u03b2)\nx\u271d : Set \u03b1\n\u22a2 (\u2203 t\u2082 \u2208 g, \u2200 x \u2208 x\u271d, \u2200 y \u2208 t\u2082, (x, y) \u2208 s) \u2194 \u2203 t \u2208 g, t \u2286 {x | (fun y => \u2200 x \u2208 x\u271d, (x, y) \u2208 s) x}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/Enumerative/Composition.lean", "full_name": "Composition.monotone_sizeUpTo", "start": [239, 1], "end": [240, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Ideal/Norm.lean", "full_name": "Ideal.absNorm_apply", "start": [265, 1], "end": [265, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean", "full_name": "MeasureTheory.exists_upperSemicontinuous_le_integral_le", "start": [421, 1], "end": [447, 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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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)) \u03bc \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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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)) \u03bc \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": [129, 9], "def_end_pos": [129, 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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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)) \u03bc \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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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)) \u03bc \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": [386, 9], "def_end_pos": [386, 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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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)) \u03bc \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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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)) \u03bc \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": [100, 9], "def_end_pos": [100, 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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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)) \u03bc \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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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)) \u03bc \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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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)) \u03bc \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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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)) \u03bc\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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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": [100, 9], "def_end_pos": [100, 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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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": [460, 9], "def_end_pos": [460, 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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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)) \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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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": [1129, 9], "def_end_pos": [1129, 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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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": [1143, 9], "def_end_pos": [1143, 43]}, {"full_name": "MeasureTheory.integral_eq_lintegral_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1143, 9], "def_end_pos": [1143, 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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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 (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc).toReal - \u2191\u03b5 \u2264 (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc).toReal\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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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 (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc).toReal - \u2191\u03b5 \u2264 (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc).toReal", "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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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 (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc).toReal \u2264 (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc).toReal + \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/ENNReal/Real.lean", "def_pos": [83, 9], "def_end_pos": [83, 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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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 (a : \u03b1), ENNReal.ofReal \u2191(f a) \u2202\u03bc).toReal \u2264 (\u222b\u207b (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc).toReal + \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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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 (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc).toReal + \u2191\u03b5 = (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5).toReal\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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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/ENNReal/Real.lean", "def_pos": [37, 9], "def_end_pos": [37, 19]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [325, 17], "def_end_pos": [325, 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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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 (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc).toReal + \u2191\u03b5 = (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc + \u2191\u03b5).toReal", "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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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 (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc).toReal + \u2191\u03b5 = (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc).toReal + (\u2191\u03b5).toReal"}, {"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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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 (a : \u03b1), ENNReal.ofReal \u2191(g a) \u2202\u03bc).toReal + \u2191\u03b5 = (\u222b\u207b (x : \u03b1), \u2191(g x) \u2202\u03bc).toReal + (\u2191\u03b5).toReal", "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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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": [1129, 9], "def_end_pos": [1129, 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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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), 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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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), 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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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": [1129, 9], "def_end_pos": [1129, 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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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), 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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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), 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 : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 \u211d\u22650\nfint : Integrable (fun x => \u2191(f x)) \u03bc\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": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/ContinuousMapDense.lean", "full_name": "MeasureTheory.Integrable.exists_boundedContinuous_lintegral_sub_le", "start": [308, 1], "end": [312, 71], "traced_tactics": [{"tactic": "simp only [\u2190 mem\u2112p_one_iff_integrable, \u2190 snorm_one_eq_lintegral_nnnorm] at hf \u22a2", "annotated_tactic": ["simp only [\u2190 <a>mem\u2112p_one_iff_integrable</a>, \u2190 <a>snorm_one_eq_lintegral_nnnorm</a>] at hf \u22a2", [{"full_name": "MeasureTheory.mem\u2112p_one_iff_integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 9], "def_end_pos": [442, 33]}, {"full_name": "MeasureTheory.snorm_one_eq_lintegral_nnnorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [102, 9], "def_end_pos": [102, 38]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 E\nhf : Integrable f \u03bc\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\n\u22a2 \u2203 g, \u222b\u207b (x : \u03b1), \u2191\u2016f x - g x\u2016\u208a \u2202\u03bc \u2264 \u03b5 \u2227 Integrable (\u21d1g) \u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 E\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhf : Mem\u2112p f 1 \u03bc\n\u22a2 \u2203 g, snorm (fun x => f x - g x) 1 \u03bc \u2264 \u03b5 \u2227 Mem\u2112p (\u21d1g) 1 \u03bc"}, {"tactic": "exact hf.exists_boundedContinuous_snorm_sub_le ENNReal.one_ne_top h\u03b5", "annotated_tactic": ["exact hf.exists_boundedContinuous_snorm_sub_le <a>ENNReal.one_ne_top</a> h\u03b5", [{"full_name": "ENNReal.one_ne_top", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [363, 17], "def_end_pos": [363, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b1\ninst\u271d\u2074 : T4Space \u03b1\ninst\u271d\u00b3 : BorelSpace \u03b1\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03bc : Measure \u03b1\np : \u211d\u22650\u221e\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : \u03bc.WeaklyRegular\nf : \u03b1 \u2192 E\n\u03b5 : \u211d\u22650\u221e\nh\u03b5 : \u03b5 \u2260 0\nhf : Mem\u2112p f 1 \u03bc\n\u22a2 \u2203 g, snorm (fun x => f x - g x) 1 \u03bc \u2264 \u03b5 \u2227 Mem\u2112p (\u21d1g) 1 \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean", "full_name": "MeasureTheory.SignedMeasure.rnDeriv_neg", "start": [430, 1], "end": [436, 45], "traced_tactics": [{"tactic": "refine\n  Integrable.ae_eq_of_withDensity\u1d65_eq (integrable_rnDeriv _ _) (integrable_rnDeriv _ _).neg ?_", "annotated_tactic": ["refine\n    <a>Integrable.ae_eq_of_withDensity\u1d65_eq</a> (<a>integrable_rnDeriv</a> _ _) (<a>integrable_rnDeriv</a> _ _).<a>neg</a> ?_", [{"full_name": "MeasureTheory.Integrable.ae_eq_of_withDensity\u1d65_eq", "def_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "def_pos": [159, 9], "def_end_pos": [159, 44]}, {"full_name": "MeasureTheory.SignedMeasure.integrable_rnDeriv", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean", "def_pos": [200, 9], "def_end_pos": [200, 27]}, {"full_name": "MeasureTheory.SignedMeasure.integrable_rnDeriv", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean", "def_pos": [200, 9], "def_end_pos": [200, 27]}, {"full_name": "MeasureTheory.Integrable.neg", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [691, 9], "def_end_pos": [691, 23]}]], "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\ninst\u271d : s.HaveLebesgueDecomposition \u03bc\n\u22a2 (-s).rnDeriv \u03bc =\u1da0[ae \u03bc] -s.rnDeriv \u03bc", "state_after": "\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\ninst\u271d : s.HaveLebesgueDecomposition \u03bc\n\u22a2 \u03bc.withDensity\u1d65 ((-s).rnDeriv \u03bc) = \u03bc.withDensity\u1d65 (-s.rnDeriv \u03bc)"}, {"tactic": "rw [withDensity\u1d65_neg, \u2190 add_right_inj ((-s).singularPart \u03bc),\n  singularPart_add_withDensity_rnDeriv_eq, singularPart_neg, \u2190 neg_add,\n  singularPart_add_withDensity_rnDeriv_eq]", "annotated_tactic": ["rw [<a>withDensity\u1d65_neg</a>, \u2190 <a>add_right_inj</a> ((-s).<a>singularPart</a> \u03bc),\n    <a>singularPart_add_withDensity_rnDeriv_eq</a>, <a>singularPart_neg</a>, \u2190 <a>neg_add</a>,\n    <a>singularPart_add_withDensity_rnDeriv_eq</a>]", [{"full_name": "MeasureTheory.withDensity\u1d65_neg", "def_path": "Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean", "def_pos": [69, 9], "def_end_pos": [69, 25]}, {"full_name": "add_right_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}, {"full_name": "MeasureTheory.SignedMeasure.singularPart", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean", "def_pos": [124, 5], "def_end_pos": [124, 17]}, {"full_name": "MeasureTheory.SignedMeasure.singularPart_add_withDensity_rnDeriv_eq", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean", "def_pos": [217, 9], "def_end_pos": [217, 48]}, {"full_name": "MeasureTheory.SignedMeasure.singularPart_neg", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean", "def_pos": [349, 9], "def_end_pos": [349, 25]}, {"full_name": "neg_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [725, 15], "def_end_pos": [725, 22]}, {"full_name": "MeasureTheory.SignedMeasure.singularPart_add_withDensity_rnDeriv_eq", "def_path": "Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean", "def_pos": [217, 9], "def_end_pos": [217, 48]}]], "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\ninst\u271d : s.HaveLebesgueDecomposition \u03bc\n\u22a2 \u03bc.withDensity\u1d65 ((-s).rnDeriv \u03bc) = \u03bc.withDensity\u1d65 (-s.rnDeriv \u03bc)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "full_name": "Polynomial.cyclotomic_dvd_geom_sum_of_dvd", "start": [420, 1], "end": [429, 25], "traced_tactics": [{"tactic": "suffices cyclotomic d \u2124 \u2223 \u2211 i \u2208 Finset.range n, X ^ i by\n  simpa only [map_cyclotomic_int, Polynomial.map_sum, Polynomial.map_pow, Polynomial.map_X] using\n    map_dvd (Int.castRingHom R) this", "annotated_tactic": ["suffices <a>cyclotomic</a> d \u2124 \u2223 \u2211 i \u2208 <a>Finset.range</a> n, <a>X</a> ^ i by\n    simpa only [<a>map_cyclotomic_int</a>, <a>Polynomial.map_sum</a>, <a>Polynomial.map_pow</a>, <a>Polynomial.map_X</a>] using\n      <a>map_dvd</a> (<a>Int.castRingHom</a> R) this", [{"full_name": "Polynomial.cyclotomic", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [253, 5], "def_end_pos": [253, 15]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2925, 5], "def_end_pos": [2925, 10]}, {"full_name": "Polynomial.X", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [564, 5], "def_end_pos": [564, 6]}, {"full_name": "Polynomial.map_cyclotomic_int", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [266, 9], "def_end_pos": [266, 27]}, {"full_name": "Polynomial.map_sum", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [968, 19], "def_end_pos": [968, 26]}, {"full_name": "Polynomial.map_pow", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [936, 19], "def_end_pos": [936, 26]}, {"full_name": "Polynomial.map_X", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [723, 9], "def_end_pos": [723, 14]}, {"full_name": "Polynomial.map_dvd", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [798, 9], "def_end_pos": [798, 16]}, {"full_name": "Int.castRingHom", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [98, 5], "def_end_pos": [98, 16]}]], "state_before": "R : Type u_1\ninst\u271d : Ring R\nd n : \u2115\nhdn : d \u2223 n\nhd : d \u2260 1\n\u22a2 cyclotomic d R \u2223 \u2211 i \u2208 range n, X ^ i", "state_after": "R : Type u_1\ninst\u271d : Ring R\nd n : \u2115\nhdn : d \u2223 n\nhd : d \u2260 1\n\u22a2 cyclotomic d \u2124 \u2223 \u2211 i \u2208 range n, X ^ i"}, {"tactic": "rcases n.eq_zero_or_pos with (rfl | hn)", "annotated_tactic": ["rcases n.eq_zero_or_pos with (rfl | hn)", []], "state_before": "R : Type u_1\ninst\u271d : Ring R\nd n : \u2115\nhdn : d \u2223 n\nhd : d \u2260 1\n\u22a2 cyclotomic d \u2124 \u2223 \u2211 i \u2208 range n, X ^ i", "state_after": "case inl\nR : Type u_1\ninst\u271d : Ring R\nd : \u2115\nhd : d \u2260 1\nhdn : d \u2223 0\n\u22a2 cyclotomic d \u2124 \u2223 \u2211 i \u2208 range 0, X ^ i\n\ncase inr\nR : Type u_1\ninst\u271d : Ring R\nd n : \u2115\nhdn : d \u2223 n\nhd : d \u2260 1\nhn : n > 0\n\u22a2 cyclotomic d \u2124 \u2223 \u2211 i \u2208 range n, X ^ i"}, {"tactic": "rw [\u2190 prod_cyclotomic_eq_geom_sum hn]", "annotated_tactic": ["rw [\u2190 <a>prod_cyclotomic_eq_geom_sum</a> hn]", [{"full_name": "Polynomial.prod_cyclotomic_eq_geom_sum", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [387, 9], "def_end_pos": [387, 36]}]], "state_before": "case inr\nR : Type u_1\ninst\u271d : Ring R\nd n : \u2115\nhdn : d \u2223 n\nhd : d \u2260 1\nhn : n > 0\n\u22a2 cyclotomic d \u2124 \u2223 \u2211 i \u2208 range n, X ^ i", "state_after": "case inr\nR : Type u_1\ninst\u271d : Ring R\nd n : \u2115\nhdn : d \u2223 n\nhd : d \u2260 1\nhn : n > 0\n\u22a2 cyclotomic d \u2124 \u2223 \u220f i \u2208 n.divisors.erase 1, cyclotomic i \u2124"}, {"tactic": "apply Finset.dvd_prod_of_mem", "annotated_tactic": ["apply <a>Finset.dvd_prod_of_mem</a>", [{"full_name": "Finset.dvd_prod_of_mem", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [1886, 9], "def_end_pos": [1886, 24]}]], "state_before": "case inr\nR : Type u_1\ninst\u271d : Ring R\nd n : \u2115\nhdn : d \u2223 n\nhd : d \u2260 1\nhn : n > 0\n\u22a2 cyclotomic d \u2124 \u2223 \u220f i \u2208 n.divisors.erase 1, cyclotomic i \u2124", "state_after": "case inr.ha\nR : Type u_1\ninst\u271d : Ring R\nd n : \u2115\nhdn : d \u2223 n\nhd : d \u2260 1\nhn : n > 0\n\u22a2 d \u2208 n.divisors.erase 1"}, {"tactic": "simp [hd, hdn, hn.ne']", "annotated_tactic": ["simp [hd, hdn, hn.ne']", []], "state_before": "case inr.ha\nR : Type u_1\ninst\u271d : Ring R\nd n : \u2115\nhdn : d \u2223 n\nhd : d \u2260 1\nhn : n > 0\n\u22a2 d \u2208 n.divisors.erase 1", "state_after": "no goals"}, {"tactic": "simpa only [map_cyclotomic_int, Polynomial.map_sum, Polynomial.map_pow, Polynomial.map_X] using\n  map_dvd (Int.castRingHom R) this", 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"CategoryTheory.ConcreteCategory.mono_of_injective", "def_path": "Mathlib/CategoryTheory/ConcreteCategory/EpiMono.lean", "def_pos": [38, 9], "def_end_pos": [38, 26]}]], "state_before": "A B : NonemptyFinLinOrd\nf : A \u27f6 B\n\u22a2 Mono f \u2194 Function.Injective \u21d1f", "state_after": "A B : NonemptyFinLinOrd\nf : A \u27f6 B\n\u22a2 Mono f \u2192 Function.Injective \u21d1f"}, {"tactic": "intro", "annotated_tactic": ["intro", []], "state_before": "A B : NonemptyFinLinOrd\nf : A \u27f6 B\n\u22a2 Mono f \u2192 Function.Injective \u21d1f", "state_after": "A B : NonemptyFinLinOrd\nf : A \u27f6 B\na\u271d : Mono f\n\u22a2 Function.Injective \u21d1f"}, {"tactic": "intro a\u2081 a\u2082 h", "annotated_tactic": ["intro a\u2081 a\u2082 h", []], "state_before": "A B : NonemptyFinLinOrd\nf : A \u27f6 B\na\u271d : Mono f\n\u22a2 Function.Injective \u21d1f", "state_after": "A B : NonemptyFinLinOrd\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : f a\u2081 = f a\u2082\n\u22a2 a\u2081 = a\u2082"}, {"tactic": "let X := NonemptyFinLinOrd.of (ULift (Fin 1))", "annotated_tactic": ["let X := <a>NonemptyFinLinOrd.of</a> (<a>ULift</a> (<a>Fin</a> 1))", [{"full_name": "NonemptyFinLinOrd.of", "def_path": "Mathlib/Order/Category/NonemptyFinLinOrd.lean", "def_pos": [79, 5], "def_end_pos": [79, 7]}, {"full_name": "ULift", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [809, 11], "def_end_pos": [809, 16]}, {"full_name": "Fin", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1831, 11], "def_end_pos": [1831, 14]}]], "state_before": "A B : NonemptyFinLinOrd\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : f a\u2081 = f a\u2082\n\u22a2 a\u2081 = a\u2082", "state_after": "A B : NonemptyFinLinOrd\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : f a\u2081 = f a\u2082\nX : NonemptyFinLinOrd := of (ULift.{?u.17762, 0} (Fin 1))\n\u22a2 a\u2081 = a\u2082"}, {"tactic": "let g\u2081 : X \u27f6 A := \u27e8fun _ => a\u2081, fun _ _ _ => by rfl\u27e9", "annotated_tactic": ["let g\u2081 : X \u27f6 A := \u27e8fun _ => a\u2081, fun _ _ _ => by rfl\u27e9", []], "state_before": "A B : NonemptyFinLinOrd\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : f a\u2081 = f a\u2082\nX : NonemptyFinLinOrd := of (ULift.{?u.17762, 0} (Fin 1))\n\u22a2 a\u2081 = a\u2082", "state_after": "A B : NonemptyFinLinOrd\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : f a\u2081 = f a\u2082\nX : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 1))\ng\u2081 : X \u27f6 A := { toFun := fun x => a\u2081, monotone' := \u22ef }\n\u22a2 a\u2081 = a\u2082"}, {"tactic": "let g\u2082 : X \u27f6 A := \u27e8fun _ => a\u2082, fun _ _ _ => by rfl\u27e9", "annotated_tactic": ["let g\u2082 : X \u27f6 A := \u27e8fun _ => a\u2082, fun _ _ _ => by rfl\u27e9", []], "state_before": "A B : NonemptyFinLinOrd\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : f a\u2081 = f a\u2082\nX : 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g\u2082 { down := 0 }"}, {"tactic": "have eq : g\u2081 \u226b f = g\u2082 \u226b f := by\n  ext\n  exact h", "annotated_tactic": ["have eq : g\u2081 \u226b f = g\u2082 \u226b f := by\n    ext\n    exact h", []], "state_before": "A B : NonemptyFinLinOrd\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : f a\u2081 = f a\u2082\nX : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 1))\ng\u2081 : X \u27f6 A := { toFun := fun x => a\u2081, monotone' := \u22ef }\ng\u2082 : X \u27f6 A := { toFun := fun x => a\u2082, monotone' := \u22ef }\n\u22a2 g\u2081 { down := 0 } = g\u2082 { down := 0 }", "state_after": "A B : NonemptyFinLinOrd\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : f a\u2081 = f a\u2082\nX : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 1))\ng\u2081 : X \u27f6 A := { toFun := fun x => a\u2081, monotone' := \u22ef }\ng\u2082 : X \u27f6 A := { toFun := fun x => a\u2082, monotone' := \u22ef }\neq : g\u2081 \u226b f = g\u2082 \u226b f\n\u22a2 g\u2081 { down := 0 } = g\u2082 { down := 0 }"}, {"tactic": "rw [cancel_mono] at eq", "annotated_tactic": ["rw [<a>cancel_mono</a>] at eq", [{"full_name": "CategoryTheory.cancel_mono", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [310, 9], "def_end_pos": [310, 20]}]], "state_before": "A B : NonemptyFinLinOrd\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : f a\u2081 = f a\u2082\nX : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 1))\ng\u2081 : X \u27f6 A := { toFun := fun x => a\u2081, monotone' := \u22ef }\ng\u2082 : X \u27f6 A := { toFun := fun x => a\u2082, monotone' := \u22ef }\neq : g\u2081 \u226b f = g\u2082 \u226b f\n\u22a2 g\u2081 { down := 0 } = g\u2082 { down := 0 }", "state_after": "A B : NonemptyFinLinOrd\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : f a\u2081 = f a\u2082\nX : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 1))\ng\u2081 : X \u27f6 A := { toFun := fun x => a\u2081, monotone' := \u22ef }\ng\u2082 : X \u27f6 A := { toFun := fun x => a\u2082, monotone' := \u22ef }\neq : g\u2081 = g\u2082\n\u22a2 g\u2081 { down := 0 } = g\u2082 { down := 0 }"}, {"tactic": "rw [eq]", "annotated_tactic": ["rw [eq]", []], "state_before": "A B : NonemptyFinLinOrd\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : f a\u2081 = f a\u2082\nX : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 1))\ng\u2081 : X \u27f6 A := { toFun := fun x => a\u2081, monotone' := \u22ef }\ng\u2082 : X \u27f6 A := { toFun := fun x => a\u2082, monotone' := \u22ef }\neq : g\u2081 = g\u2082\n\u22a2 g\u2081 { down := 0 } = g\u2082 { down := 0 }", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "A B : NonemptyFinLinOrd\nf : A \u27f6 B\na\u271d : Mono f\na\u2081 a\u2082 : \u2191A\nh : f a\u2081 = f a\u2082\nX : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 1))\nx\u271d\u00b2 x\u271d\u00b9 : \u2191X\nx\u271d : x\u271d\u00b2 \u2264 x\u271d\u00b9\n\u22a2 (fun x => a\u2081) x\u271d\u00b2 \u2264 (fun x => 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"29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Hom/Lattice.lean", "full_name": "SupHom.coe_mk", "start": [376, 1], "end": [376, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/GaloisConnection.lean", "full_name": "OrderIso.bddBelow_preimage", "start": [451, 1], "end": [452, 44], "traced_tactics": [{"tactic": "rw [\u2190 e.bddBelow_image, e.image_preimage]", "annotated_tactic": ["rw [\u2190 e.bddBelow_image, e.image_preimage]", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\n\u03ba : \u03b9 \u2192 Sort u_1\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u2243o \u03b2\ns : Set \u03b2\n\u22a2 BddBelow (\u21d1e \u207b\u00b9' s) \u2194 BddBelow s", "state_after": "no goals"}]}, {"url": 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x.fst = r \u2022> x.snd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "full_name": "csSup_eq_of_forall_le_of_forall_lt_exists_gt", "start": [626, 1], "end": [630, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Equiv.lean", "full_name": "LinearEquiv.toLinearMap_injective", "start": [171, 1], "end": [172, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Padics/PadicVal.lean", "full_name": "padicValNat.zero", "start": [87, 11], "end": [87, 70], "traced_tactics": [{"tactic": "simp [padicValNat]", "annotated_tactic": ["simp [<a>padicValNat</a>]", [{"full_name": 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\u27ea-o.rightAngleRotation x, y\u27eb_\u211d"}, {"tactic": "rw [inner_rightAngleRotation_left]", "annotated_tactic": ["rw [<a>inner_rightAngleRotation_left</a>]", [{"full_name": "Orientation.inner_rightAngleRotation_left", "def_path": "Mathlib/Analysis/InnerProductSpace/TwoDim.lean", "def_pos": [263, 9], "def_end_pos": [263, 38]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 \u27ea(-o).rightAngleRotation x, y\u27eb_\u211d = \u27ea-o.rightAngleRotation x, y\u27eb_\u211d", "state_after": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 ((-o).areaForm x) y = \u27ea-o.rightAngleRotation x, y\u27eb_\u211d"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : InnerProductSpace \u211d E\ninst\u271d : Fact (finrank \u211d E = 2)\no : Orientation \u211d E (Fin 2)\nx y : E\n\u22a2 ((-o).areaForm x) y = \u27ea-o.rightAngleRotation x, y\u27eb_\u211d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Galois/Prorepresentability.lean", "full_name": "CategoryTheory.PreGaloisCategory.AutGalois.ext", "start": [224, 1], "end": [230, 12], "traced_tactics": [{"tactic": "ext A", "annotated_tactic": ["ext A", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{u\u2082, u\u2081} C\ninst\u271d\u00b9 : GaloisCategory C\nF : C \u2964 FintypeCat\ninst\u271d : FiberFunctor F\nf g : AutGalois F\nh : \u2200 (A : PointedGaloisObject F), (\u03c0 F A) f = (\u03c0 F A) g\n\u22a2 f = g", "state_after": "case a.h\nC : Type u\u2081\ninst\u271d\u00b2 : 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s (m.toMeasure h)\n\u22a2 (m.toMeasure h) s = m 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\nms : MeasurableSpace \u03b1\ns\u271d t : Set \u03b1\nm : OuterMeasure \u03b1\nh : ms \u2264 m.caratheodory\ns : Set \u03b1\nhs : NullMeasurableSet s (m.toMeasure h)\n\u22a2 (m.toMeasure h) s \u2264 m s"}, {"tactic": "rcases hs.exists_measurable_subset_ae_eq with \u27e8t, hts, htm, heq\u27e9", "annotated_tactic": ["rcases hs.exists_measurable_subset_ae_eq with \u27e8t, hts, htm, heq\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\nms : MeasurableSpace \u03b1\ns\u271d t : Set \u03b1\nm : OuterMeasure \u03b1\nh : ms \u2264 m.caratheodory\ns : Set \u03b1\nhs : NullMeasurableSet s (m.toMeasure h)\n\u22a2 (m.toMeasure h) s \u2264 m s", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Type u_5\nR : Type u_6\nR' : Type u_7\nms : MeasurableSpace \u03b1\ns\u271d t\u271d : Set \u03b1\nm : OuterMeasure \u03b1\nh : ms \u2264 m.caratheodory\ns : Set \u03b1\nhs : NullMeasurableSet s (m.toMeasure h)\nt : Set \u03b1\nhts : t \u2286 s\nhtm : MeasurableSet t\nheq : t =\u1da0[ae (m.toMeasure h)] s\n\u22a2 (m.toMeasure h) s \u2264 m s"}, {"tactic": "calc\n  m.toMeasure h s = m.toMeasure h t := measure_congr heq.symm\n  _ = m t := toMeasure_apply m h htm\n  _ \u2264 m s := m.mono hts", "annotated_tactic": ["calc\n    m.toMeasure h s = m.toMeasure h t := <a>measure_congr</a> heq.symm\n    _ = m t := <a>toMeasure_apply</a> m h htm\n    _ \u2264 m s := m.mono hts", [{"full_name": "MeasureTheory.measure_congr", "def_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean", "def_pos": [269, 9], "def_end_pos": [269, 22]}, {"full_name": "MeasureTheory.toMeasure_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", 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"start": [298, 1], "end": [298, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/RootsOfUnity/Basic.lean", "full_name": "IsPrimitiveRoot.pow_eq_one_iff_dvd", "start": [354, 1], "end": [356, 79], "traced_tactics": [{"tactic": "rintro \u27e8i, rfl\u27e9", "annotated_tactic": ["rintro \u27e8i, rfl\u27e9", []], "state_before": "M : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l\u271d : \u2115\n\u03b6 : M\nf : F\nh : IsPrimitiveRoot \u03b6 k\nl : \u2115\n\u22a2 k \u2223 l \u2192 \u03b6 ^ l = 1", "state_after": "case intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh : IsPrimitiveRoot \u03b6 k\ni : \u2115\n\u22a2 \u03b6 ^ (k * i) = 1"}, {"tactic": "simp only [pow_mul, h.pow_eq_one, one_pow, PNat.mul_coe]", "annotated_tactic": ["simp only [<a>pow_mul</a>, h.pow_eq_one, <a>one_pow</a>, <a>PNat.mul_coe</a>]", [{"full_name": "pow_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [713, 32], "def_end_pos": [713, 39]}, {"full_name": "one_pow", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [696, 39], "def_end_pos": [696, 46]}, {"full_name": "PNat.mul_coe", "def_path": "Mathlib/Data/PNat/Basic.lean", "def_pos": [236, 9], "def_end_pos": [236, 16]}]], "state_before": "case intro\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst\u271d\u00b2 : CommMonoid M\ninst\u271d\u00b9 : CommMonoid N\ninst\u271d : DivisionCommMonoid G\nk l : \u2115\n\u03b6 : M\nf : F\nh : IsPrimitiveRoot \u03b6 k\ni : \u2115\n\u22a2 \u03b6 ^ (k * i) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.comp_bfamilyOfFamily", "start": [1220, 1], "end": [1222, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/ContinuousFunction/Bounded.lean", "full_name": "BoundedContinuousFunction.one_compContinuous", "start": [637, 1], "end": [638, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "full_name": "Matrix.toLin_toMatrix", "start": [582, 1], "end": [584, 57], "traced_tactics": [{"tactic": "rw [\u2190 Matrix.toLin_symm, LinearEquiv.apply_symm_apply]", "annotated_tactic": ["rw [\u2190 <a>Matrix.toLin_symm</a>, 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"29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/MeanValue.lean", "full_name": "image_le_of_deriv_right_lt_deriv_boundary'", "start": [187, 1], "end": [193, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "full_name": "AddMonoid.End.coe_pow", "start": [1107, 1], "end": [1107, 99], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Kernel/MeasureCompProd.lean", "full_name": "MeasureTheory.Measure.ae_compProd_iff", "start": [77, 1], "end": [80, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Quaternion.lean", "full_name": "QuaternionAlgebra.coe_starAe", "start": [787, 1], "end": [788, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Submonoid/Operations.lean", "full_name": "Submonoid.eq_bot_of_subsingleton", "start": [1306, 1], "end": [1309, 82], "traced_tactics": [{"tactic": "rw [eq_bot_iff_forall]", "annotated_tactic": ["rw [<a>eq_bot_iff_forall</a>]", [{"full_name": "Submonoid.eq_bot_iff_forall", "def_path": "Mathlib/Algebra/Group/Submonoid/Operations.lean", "def_pos": [1300, 9], "def_end_pos": [1300, 26]}]], "state_before": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u2074 : MulOneClass M\ninst\u271d\u00b3 : MulOneClass N\ninst\u271d\u00b2 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d\u00b9 : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ninst\u271d : Subsingleton \u21a5S\n\u22a2 S = \u22a5", "state_after": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u2074 : MulOneClass M\ninst\u271d\u00b3 : MulOneClass N\ninst\u271d\u00b2 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d\u00b9 : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ninst\u271d : Subsingleton \u21a5S\n\u22a2 \u2200 x \u2208 S, x = 1"}, {"tactic": "intro y hy", "annotated_tactic": ["intro y hy", []], "state_before": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u2074 : MulOneClass M\ninst\u271d\u00b3 : MulOneClass N\ninst\u271d\u00b2 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d\u00b9 : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ninst\u271d : Subsingleton \u21a5S\n\u22a2 \u2200 x \u2208 S, x = 1", "state_after": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u2074 : MulOneClass M\ninst\u271d\u00b3 : MulOneClass N\ninst\u271d\u00b2 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d\u00b9 : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ninst\u271d : Subsingleton \u21a5S\ny : M\nhy : y \u2208 S\n\u22a2 y = 1"}, {"tactic": "simpa using _root_.congr_arg ((\u2191) : S \u2192 M) <| Subsingleton.elim (\u27e8y, hy\u27e9 : S) 1", "annotated_tactic": ["simpa using <a>_root_.congr_arg</a> ((\u2191) : S \u2192 M) <| <a>Subsingleton.elim</a> (\u27e8y, hy\u27e9 : S) 1", [{"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "Subsingleton.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1015, 19], "def_end_pos": [1015, 36]}]], "state_before": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u2074 : MulOneClass M\ninst\u271d\u00b3 : MulOneClass N\ninst\u271d\u00b2 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d\u00b9 : SetLike A M\nhA : SubmonoidClass A M\nS' : A\ninst\u271d : Subsingleton \u21a5S\ny : M\nhy : y \u2208 S\n\u22a2 y = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "full_name": "Real.nnnorm_of_nonneg", "start": [1476, 1], "end": [1477, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Star/Subalgebra.lean", "full_name": "StarAlgebra.adjoin_eq_starClosure_adjoin", "start": [438, 1], "end": [441, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "full_name": "LinearIsometryEquiv.self_comp_symm", "start": [855, 1], "end": [856, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.le_iff_subset", "start": [120, 1], "end": [121, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.integral_pos_iff_support_of_nonneg_ae", "start": [1270, 1], "end": [1274, 22], "traced_tactics": [{"tactic": "simp_rw [(integral_nonneg_of_ae hf).lt_iff_ne, pos_iff_ne_zero, Ne, @eq_comm \u211d 0,\n  integral_eq_zero_iff_of_nonneg_ae hf hfi, Filter.EventuallyEq, ae_iff, Pi.zero_apply,\n  Function.support]", "annotated_tactic": ["simp_rw [(<a>integral_nonneg_of_ae</a> hf).<a>lt_iff_ne</a>, <a>pos_iff_ne_zero</a>, <a>Ne</a>, @<a>eq_comm</a> \u211d 0,\n    <a>integral_eq_zero_iff_of_nonneg_ae</a> hf hfi, <a>Filter.EventuallyEq</a>, <a>ae_iff</a>, <a>Pi.zero_apply</a>,\n    <a>Function.support</a>]", [{"full_name": "MeasureTheory.integral_nonneg_of_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1186, 9], "def_end_pos": [1186, 30]}, {"full_name": "LE.le.lt_iff_ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [254, 9], "def_end_pos": [254, 18]}, {"full_name": "pos_iff_ne_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [230, 3], "def_end_pos": [230, 14]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "eq_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}, {"full_name": "MeasureTheory.integral_eq_zero_iff_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1246, 9], "def_end_pos": [1246, 42]}, {"full_name": "Filter.EventuallyEq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1470, 5], "def_end_pos": [1470, 17]}, {"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean", "def_pos": [75, 9], "def_end_pos": [75, 15]}, {"full_name": "Pi.zero_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [62, 3], "def_end_pos": [62, 14]}, {"full_name": "Function.support", "def_path": "Mathlib/Algebra/Group/Support.lean", "def_pos": [30, 3], "def_end_pos": [30, 14]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\n\ud835\udd5c : Type u_4\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup E\ninst\u271d\u00b9\u2070 : NormedSpace \u211d E\nhE : CompleteSpace E\ninst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedSpace \u211d F\ninst\u271d\u2074 : CompleteSpace F\nG : Type u_5\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \u211d G\nf\u271d g : \u03b1 \u2192 E\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nX : Type u_6\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : FirstCountableTopology X\nf : \u03b1 \u2192 \u211d\nhf : 0 \u2264\u1da0[ae \u03bc] f\nhfi : Integrable f \u03bc\n\u22a2 0 < \u222b (x : \u03b1), f x \u2202\u03bc \u2194 0 < \u03bc (Function.support f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/VectorBundle/SmoothSection.lean", "full_name": "ContMDiffSection.coe_smul", "start": [175, 1], "end": [176, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/ClassGroup.lean", "full_name": "mem_principal_ideals_iff", "start": [72, 1], "end": [79, 32], "traced_tactics": [{"tactic": "simp only [MonoidHom.mem_range, toPrincipalIdeal_eq_iff]", "annotated_tactic": ["simp only [<a>MonoidHom.mem_range</a>, <a>toPrincipalIdeal_eq_iff</a>]", [{"full_name": "MonoidHom.mem_range", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [2503, 9], "def_end_pos": [2503, 18]}, {"full_name": "toPrincipalIdeal_eq_iff", "def_path": "Mathlib/RingTheory/ClassGroup.lean", "def_pos": [67, 9], "def_end_pos": [67, 32]}]], "state_before": "R : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : DecidableEq L\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsFractionRing R K\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : FiniteDimensional K L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nI : (FractionalIdeal R\u2070 K)\u02e3\n\u22a2 I \u2208 (toPrincipalIdeal R K).range \u2194 \u2203 x, spanSingleton R\u2070 x = \u2191I", "state_after": "R : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : DecidableEq L\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsFractionRing R K\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : FiniteDimensional K L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nI : (FractionalIdeal R\u2070 K)\u02e3\n\u22a2 (\u2203 x, spanSingleton R\u2070 \u2191x = \u2191I) \u2194 \u2203 x, spanSingleton R\u2070 x = \u2191I"}, {"tactic": "constructor <;> rintro \u27e8x, hx\u27e9", "annotated_tactic": ["constructor <;> rintro \u27e8x, hx\u27e9", []], "state_before": "R : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : DecidableEq L\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsFractionRing R K\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : FiniteDimensional K L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nI : (FractionalIdeal R\u2070 K)\u02e3\n\u22a2 (\u2203 x, spanSingleton R\u2070 \u2191x = \u2191I) \u2194 \u2203 x, spanSingleton R\u2070 x = \u2191I", "state_after": "case mp.intro\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : DecidableEq L\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsFractionRing R K\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : FiniteDimensional K L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nI : (FractionalIdeal R\u2070 K)\u02e3\nx : K\u02e3\nhx : spanSingleton R\u2070 \u2191x = \u2191I\n\u22a2 \u2203 x, spanSingleton R\u2070 x = \u2191I\n\ncase mpr.intro\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : DecidableEq L\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsFractionRing R K\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : FiniteDimensional K L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nI : (FractionalIdeal R\u2070 K)\u02e3\nx : K\nhx : spanSingleton R\u2070 x = \u2191I\n\u22a2 \u2203 x, spanSingleton R\u2070 \u2191x = \u2191I"}, {"tactic": "exact \u27e8x, hx\u27e9", "annotated_tactic": ["exact \u27e8x, hx\u27e9", []], "state_before": "case mp.intro\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : DecidableEq L\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsFractionRing R K\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : FiniteDimensional K L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nI : (FractionalIdeal R\u2070 K)\u02e3\nx : K\u02e3\nhx : spanSingleton R\u2070 \u2191x = \u2191I\n\u22a2 \u2203 x, spanSingleton R\u2070 x = \u2191I", "state_after": "no goals"}, {"tactic": "refine \u27e8Units.mk0 x ?_, hx\u27e9", "annotated_tactic": ["refine \u27e8<a>Units.mk0</a> x ?_, hx\u27e9", [{"full_name": "Units.mk0", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [186, 5], "def_end_pos": [186, 8]}]], "state_before": "case mpr.intro\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : DecidableEq L\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsFractionRing R K\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : FiniteDimensional K L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nI : (FractionalIdeal R\u2070 K)\u02e3\nx : K\nhx : spanSingleton R\u2070 x = \u2191I\n\u22a2 \u2203 x, spanSingleton R\u2070 \u2191x = \u2191I", "state_after": "case mpr.intro\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : DecidableEq L\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsFractionRing R K\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : FiniteDimensional K L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nI : (FractionalIdeal R\u2070 K)\u02e3\nx : K\nhx : spanSingleton R\u2070 x = \u2191I\n\u22a2 x \u2260 0"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "case mpr.intro\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : DecidableEq L\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsFractionRing R K\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : FiniteDimensional K L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nI : (FractionalIdeal R\u2070 K)\u02e3\nx : K\nhx : spanSingleton R\u2070 x = \u2191I\n\u22a2 x \u2260 0", "state_after": "case mpr.intro\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : DecidableEq L\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsFractionRing R K\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : FiniteDimensional K L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nI : (FractionalIdeal R\u2070 K)\u02e3\nhx : spanSingleton R\u2070 0 = \u2191I\n\u22a2 False"}, {"tactic": "simp [I.ne_zero.symm] at hx", "annotated_tactic": ["simp [I.ne_zero.symm] at hx", []], "state_before": "case mpr.intro\nR : Type u_1\nK : Type u_2\nL : Type u_3\ninst\u271d\u2079 : CommRing R\ninst\u271d\u2078 : Field K\ninst\u271d\u2077 : Field L\ninst\u271d\u2076 : DecidableEq L\ninst\u271d\u2075 : Algebra R K\ninst\u271d\u2074 : IsFractionRing R K\ninst\u271d\u00b3 : Algebra K L\ninst\u271d\u00b2 : FiniteDimensional K L\ninst\u271d\u00b9 : Algebra R L\ninst\u271d : IsScalarTower R K L\nI : (FractionalIdeal R\u2070 K)\u02e3\nhx : spanSingleton R\u2070 0 = \u2191I\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Cast/Synonym.lean", "full_name": "ofDual_natCast", "start": [55, 1], "end": [56, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/AffineSpace/Independent.lean", "full_name": "AffineIndependent.indicator_eq_of_affineCombination_eq", "start": [265, 1], "end": [269, 94], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Field/Subfield.lean", "full_name": "Subfield.closure_eq_of_le", "start": [692, 1], "end": [694, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "IsClosed.inv", "start": [387, 1], "end": [388, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Preadditive/Basic.lean", "full_name": "CategoryTheory.Preadditive.neg_iso_inv", "start": [489, 1], "end": [490, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "full_name": "Pi.apply_mulSingle", "start": [402, 1], "end": [404, 83], "traced_tactics": [{"tactic": "simpa only [Pi.one_apply, hf', mulSingle] using Function.apply_update f' 1 i x j", "annotated_tactic": ["simpa only [<a>Pi.one_apply</a>, hf', <a>mulSingle</a>] using <a>Function.apply_update</a> f' 1 i x j", [{"full_name": "Pi.one_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [63, 9], "def_end_pos": [63, 18]}, {"full_name": "Pi.mulSingle", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [349, 5], "def_end_pos": [349, 14]}, {"full_name": "Function.apply_update", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [651, 9], "def_end_pos": [651, 21]}]], "state_before": "I : Type u\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : I \u2192 Type v\u2081\ng : I \u2192 Type v\u2082\nh : I \u2192 Type v\u2083\nx\u271d y : (i : I) \u2192 f i\ni\u271d : I\ninst\u271d\u00b3 : DecidableEq I\ninst\u271d\u00b2 : (i : I) \u2192 One (f i)\ninst\u271d\u00b9 : (i : I) \u2192 One (g i)\ninst\u271d : (i : I) \u2192 One (h i)\nf' : (i : I) \u2192 f i \u2192 g i\nhf' : \u2200 (i : I), f' i 1 = 1\ni : I\nx : f i\nj : I\n\u22a2 f' j (mulSingle i x j) = mulSingle i (f' i x) j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Game/Basic.lean", "full_name": "SetTheory.PGame.quot_mul_zero", "start": [469, 1], "end": [470, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/QuotientGroup.lean", "full_name": "QuotientGroup.homQuotientZPowOfHom_comp_of_rightInverse", "start": [566, 1], "end": [568, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Star/NonUnitalSubalgebra.lean", "full_name": "NonUnitalStarAlgebra.iInf_toNonUnitalSubalgebra", "start": [768, 1], "end": [770, 35], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : StarRing R\ninst\u271d\u00b9\u2074 : NonUnitalSemiring A\ninst\u271d\u00b9\u00b3 : StarRing A\ninst\u271d\u00b9\u00b2 : Module R A\ninst\u271d\u00b9\u00b9 : IsScalarTower R A A\ninst\u271d\u00b9\u2070 : SMulCommClass R A A\ninst\u271d\u2079 : StarModule R A\ninst\u271d\u2078 : NonUnitalSemiring B\ninst\u271d\u2077 : StarRing B\ninst\u271d\u2076 : Module R B\ninst\u271d\u2075 : IsScalarTower R B B\ninst\u271d\u2074 : SMulCommClass R B B\ninst\u271d\u00b3 : StarModule R B\ninst\u271d\u00b2 : FunLike F A B\ninst\u271d\u00b9 : NonUnitalAlgHomClass F R A B\ninst\u271d : NonUnitalStarAlgHomClass F R A B\n\u03b9 : Sort u_1\nS : \u03b9 \u2192 NonUnitalStarSubalgebra R A\n\u22a2 \u2191(\u2a05 i, S i).toNonUnitalSubalgebra = \u2191(\u2a05 i, (S i).toNonUnitalSubalgebra)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.biUnion_eq_iUnion", "start": [862, 1], "end": [864, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/LocalInvariantProperties.lean", "full_name": "StructureGroupoid.LocalInvariantProp.liftPropWithinAt_mono_of_mem", "start": [448, 1], "end": [454, 56], "traced_tactics": [{"tactic": "simp only [liftPropWithinAt_iff'] at h \u22a2", "annotated_tactic": ["simp only [<a>liftPropWithinAt_iff'</a>] at h \u22a2", [{"full_name": "ChartedSpace.liftPropWithinAt_iff'", "def_path": "Mathlib/Geometry/Manifold/LocalInvariantProperties.lean", "def_pos": [164, 10], "def_end_pos": [164, 31]}]], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\nmono_of_mem : \u2200 \u2983s : Set H\u2984 \u2983x : H\u2984 \u2983t : Set H\u2984 \u2983f : H \u2192 H'\u2984, s \u2208 \ud835\udcdd[t] x \u2192 P f s x \u2192 P f t x\nh : LiftPropWithinAt P g s x\nhst : s \u2208 \ud835\udcdd[t] x\n\u22a2 LiftPropWithinAt P g t x", "state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\nmono_of_mem : \u2200 \u2983s : Set H\u2984 \u2983x : H\u2984 \u2983t : Set H\u2984 \u2983f : H \u2192 H'\u2984, s \u2208 \ud835\udcdd[t] x \u2192 P f s x \u2192 P f t x\nhst : s \u2208 \ud835\udcdd[t] x\nh :\n  ContinuousWithinAt g s x \u2227\n    P (\u2191(chartAt H' (g x)) \u2218 g \u2218 \u2191(chartAt H x).symm) (\u2191(chartAt H x).symm \u207b\u00b9' s) (\u2191(chartAt H x) x)\n\u22a2 ContinuousWithinAt g t x \u2227\n    P (\u2191(chartAt H' (g x)) \u2218 g \u2218 \u2191(chartAt H x).symm) (\u2191(chartAt H x).symm \u207b\u00b9' t) (\u2191(chartAt H x) x)"}, {"tactic": "refine \u27e8h.1.mono_of_mem hst, mono_of_mem ?_ h.2\u27e9", "annotated_tactic": ["refine \u27e8h.1.<a>mono_of_mem</a> hst, mono_of_mem ?_ h.2\u27e9", [{"full_name": "ContinuousWithinAt.mono_of_mem", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [752, 9], "def_end_pos": [752, 39]}]], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\nmono_of_mem : \u2200 \u2983s : Set H\u2984 \u2983x : H\u2984 \u2983t : Set H\u2984 \u2983f : H \u2192 H'\u2984, s \u2208 \ud835\udcdd[t] x \u2192 P f s x \u2192 P f t x\nhst : s \u2208 \ud835\udcdd[t] x\nh :\n  ContinuousWithinAt g s x \u2227\n    P (\u2191(chartAt H' (g x)) \u2218 g \u2218 \u2191(chartAt H x).symm) (\u2191(chartAt H x).symm \u207b\u00b9' s) (\u2191(chartAt H x) x)\n\u22a2 ContinuousWithinAt g t x \u2227\n    P (\u2191(chartAt H' (g x)) \u2218 g \u2218 \u2191(chartAt H x).symm) (\u2191(chartAt H x).symm \u207b\u00b9' t) (\u2191(chartAt H x) x)", "state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\nmono_of_mem : \u2200 \u2983s : Set H\u2984 \u2983x : H\u2984 \u2983t : Set H\u2984 \u2983f : H \u2192 H'\u2984, s \u2208 \ud835\udcdd[t] x \u2192 P f s x \u2192 P f t x\nhst : s \u2208 \ud835\udcdd[t] x\nh :\n  ContinuousWithinAt g s x \u2227\n    P (\u2191(chartAt H' (g x)) \u2218 g \u2218 \u2191(chartAt H x).symm) (\u2191(chartAt H x).symm \u207b\u00b9' s) (\u2191(chartAt H x) x)\n\u22a2 \u2191(chartAt H x).symm \u207b\u00b9' s \u2208 \ud835\udcdd[\u2191(chartAt H x).symm \u207b\u00b9' t] \u2191(chartAt H x) x"}, {"tactic": "simp_rw [\u2190 mem_map, (chartAt H x).symm.map_nhdsWithin_preimage_eq (mem_chart_target H x),\n  (chartAt H x).left_inv (mem_chart_source H x), hst]", "annotated_tactic": ["simp_rw [\u2190 <a>mem_map</a>, (<a>chartAt</a> H x).symm.map_nhdsWithin_preimage_eq (<a>mem_chart_target</a> H x),\n    (<a>chartAt</a> H x).<a>left_inv</a> (<a>mem_chart_source</a> H x), hst]", [{"full_name": "Filter.mem_map", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1930, 9], "def_end_pos": [1930, 16]}, {"full_name": "chartAt", "def_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "def_pos": [589, 8], "def_end_pos": [589, 15]}, {"full_name": "mem_chart_target", "def_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "def_pos": [631, 9], "def_end_pos": [631, 25]}, {"full_name": "chartAt", "def_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "def_pos": [589, 8], "def_end_pos": [589, 15]}, {"full_name": "PartialHomeomorph.left_inv", "def_path": "Mathlib/Topology/PartialHomeomorph.lean", "def_pos": [161, 9], "def_end_pos": [161, 17]}, {"full_name": "mem_chart_source", "def_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "def_pos": [594, 7], "def_end_pos": [594, 23]}]], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\nmono_of_mem : \u2200 \u2983s : Set H\u2984 \u2983x : H\u2984 \u2983t : Set H\u2984 \u2983f : H \u2192 H'\u2984, s \u2208 \ud835\udcdd[t] x \u2192 P f s x \u2192 P f t x\nhst : s \u2208 \ud835\udcdd[t] x\nh :\n  ContinuousWithinAt g s x \u2227\n    P (\u2191(chartAt H' (g x)) \u2218 g \u2218 \u2191(chartAt H x).symm) (\u2191(chartAt H x).symm \u207b\u00b9' s) (\u2191(chartAt H x) x)\n\u22a2 \u2191(chartAt H x).symm \u207b\u00b9' s \u2208 \ud835\udcdd[\u2191(chartAt H x).symm \u207b\u00b9' t] \u2191(chartAt H x) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Bitwise.lean", "full_name": "Nat.xor_zero", "start": [355, 1], "end": [355, 76], "traced_tactics": [{"tactic": "simp [HXor.hXor, Xor.xor, xor]", "annotated_tactic": ["simp [<a>HXor.hXor</a>, <a>Xor.xor</a>, <a>xor</a>]", [{"full_name": "HXor.hXor", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1282, 3], "def_end_pos": [1282, 7]}, {"full_name": "Xor.xor", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1416, 3], "def_end_pos": [1416, 6]}, {"full_name": "Nat.xor", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Bitwise/Basic.lean", "def_pos": [38, 5], "def_end_pos": [38, 8]}]], "state_before": "n : \u2115\n\u22a2 n ^^^ 0 = n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Bilinear.lean", "full_name": "IsBoundedBilinearMap.hasFDerivWithinAt", "start": [84, 1], "end": [86, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Arsinh.lean", "full_name": "DifferentiableAt.arsinh", "start": [257, 1], "end": [259, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subgroup/Pointwise.lean", "full_name": "Subgroup.inf_mul_assoc", "start": [254, 1], "end": [265, 37], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\n\u22a2 \u2191(A \u2293 B) * \u2191C = \u2191A \u2229 (\u2191B * \u2191C)", "state_after": "case h\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\nx\u271d : G\n\u22a2 x\u271d \u2208 \u2191(A \u2293 B) * \u2191C \u2194 x\u271d \u2208 \u2191A \u2229 (\u2191B * \u2191C)"}, {"tactic": "simp only [coe_inf, Set.mem_mul, Set.mem_inter_iff]", "annotated_tactic": ["simp only [<a>coe_inf</a>, <a>Set.mem_mul</a>, <a>Set.mem_inter_iff</a>]", [{"full_name": "Subgroup.coe_inf", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [950, 9], "def_end_pos": [950, 16]}, {"full_name": "Set.mem_mul", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 16]}, {"full_name": "Set.mem_inter_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [874, 9], "def_end_pos": [874, 22]}]], "state_before": "case h\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\nx\u271d : G\n\u22a2 x\u271d \u2208 \u2191(A \u2293 B) * \u2191C \u2194 x\u271d \u2208 \u2191A \u2229 (\u2191B * \u2191C)", "state_after": "case h\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\nx\u271d : G\n\u22a2 (\u2203 x, (x \u2208 \u2191A \u2227 x \u2208 \u2191B) \u2227 \u2203 y \u2208 \u2191C, x * y = x\u271d) \u2194 x\u271d \u2208 \u2191A \u2227 \u2203 x \u2208 \u2191B, \u2203 y \u2208 \u2191C, x * y = x\u271d"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\nx\u271d : G\n\u22a2 (\u2203 x, (x \u2208 \u2191A \u2227 x \u2208 \u2191B) \u2227 \u2203 y \u2208 \u2191C, x * y = x\u271d) \u2194 x\u271d \u2208 \u2191A \u2227 \u2203 x \u2208 \u2191B, \u2203 y \u2208 \u2191C, x * y = x\u271d", "state_after": "case h.mp\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\nx\u271d : G\n\u22a2 (\u2203 x, (x \u2208 \u2191A \u2227 x \u2208 \u2191B) \u2227 \u2203 y \u2208 \u2191C, x * y = x\u271d) \u2192 x\u271d \u2208 \u2191A \u2227 \u2203 x \u2208 \u2191B, \u2203 y \u2208 \u2191C, x * y = x\u271d\n\ncase h.mpr\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\nx\u271d : G\n\u22a2 (x\u271d \u2208 \u2191A \u2227 \u2203 x \u2208 \u2191B, \u2203 y \u2208 \u2191C, x * y = x\u271d) \u2192 \u2203 x, (x \u2208 \u2191A \u2227 x \u2208 \u2191B) \u2227 \u2203 y \u2208 \u2191C, x * y = x\u271d"}, {"tactic": "rintro \u27e8hyz, y, hy, z, hz, rfl\u27e9", "annotated_tactic": ["rintro \u27e8hyz, y, hy, z, hz, rfl\u27e9", []], "state_before": "case h.mpr\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\nx\u271d : G\n\u22a2 (x\u271d \u2208 \u2191A \u2227 \u2203 x \u2208 \u2191B, \u2203 y \u2208 \u2191C, x * y = x\u271d) \u2192 \u2203 x, (x \u2208 \u2191A \u2227 x \u2208 \u2191B) \u2227 \u2203 y \u2208 \u2191C, x * y = x\u271d", "state_after": "case h.mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\ny : G\nhy : y \u2208 \u2191B\nz : G\nhz : z \u2208 \u2191C\nhyz : y * z \u2208 \u2191A\n\u22a2 \u2203 x, (x \u2208 \u2191A \u2227 x \u2208 \u2191B) \u2227 \u2203 y_1 \u2208 \u2191C, x * y_1 = y * z"}, {"tactic": "refine \u27e8y, \u27e8?_, hy\u27e9, z, hz, rfl\u27e9", "annotated_tactic": ["refine \u27e8y, \u27e8?_, hy\u27e9, z, hz, <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case h.mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\ny : G\nhy : y \u2208 \u2191B\nz : G\nhz : z \u2208 \u2191C\nhyz : y * z \u2208 \u2191A\n\u22a2 \u2203 x, (x \u2208 \u2191A \u2227 x \u2208 \u2191B) \u2227 \u2203 y_1 \u2208 \u2191C, x * y_1 = y * z", "state_after": "case h.mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\ny : G\nhy : y \u2208 \u2191B\nz : G\nhz : z \u2208 \u2191C\nhyz : y * z \u2208 \u2191A\n\u22a2 y \u2208 \u2191A"}, {"tactic": "suffices y * z * z\u207b\u00b9 \u2208 A by simpa", "annotated_tactic": ["suffices y * z * z\u207b\u00b9 \u2208 A by simpa", []], "state_before": "case h.mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\ny : G\nhy : y \u2208 \u2191B\nz : G\nhz : z \u2208 \u2191C\nhyz : y * z \u2208 \u2191A\n\u22a2 y \u2208 \u2191A", "state_after": "case h.mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\ny : G\nhy : y \u2208 \u2191B\nz : G\nhz : z \u2208 \u2191C\nhyz : y * z \u2208 \u2191A\n\u22a2 y * z * z\u207b\u00b9 \u2208 A"}, {"tactic": "exact mul_mem hyz (inv_mem (h hz))", "annotated_tactic": ["exact <a>mul_mem</a> hyz (<a>inv_mem</a> (h hz))", [{"full_name": "MulMemClass.mul_mem", "def_path": "Mathlib/Algebra/Group/Subsemigroup/Basic.lean", "def_pos": [64, 3], "def_end_pos": [64, 10]}, {"full_name": "InvMemClass.inv_mem", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [100, 3], "def_end_pos": [100, 10]}]], "state_before": "case h.mpr.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\ny : G\nhy : y \u2208 \u2191B\nz : G\nhz : z \u2208 \u2191C\nhyz : y * z \u2208 \u2191A\n\u22a2 y * z * z\u207b\u00b9 \u2208 A", "state_after": "no goals"}, {"tactic": "rintro \u27e8y, \u27e8hyA, hyB\u27e9, z, hz, rfl\u27e9", "annotated_tactic": ["rintro \u27e8y, \u27e8hyA, hyB\u27e9, z, hz, rfl\u27e9", []], "state_before": "case h.mp\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\nx\u271d : G\n\u22a2 (\u2203 x, (x \u2208 \u2191A \u2227 x \u2208 \u2191B) \u2227 \u2203 y \u2208 \u2191C, x * y = x\u271d) \u2192 x\u271d \u2208 \u2191A \u2227 \u2203 x \u2208 \u2191B, \u2203 y \u2208 \u2191C, x * y = x\u271d", "state_after": "case h.mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\ny : G\nhyA : y \u2208 \u2191A\nhyB : y \u2208 \u2191B\nz : G\nhz : z \u2208 \u2191C\n\u22a2 y * z \u2208 \u2191A \u2227 \u2203 x \u2208 \u2191B, \u2203 y_1 \u2208 \u2191C, x * y_1 = y * z"}, {"tactic": "refine \u27e8A.mul_mem hyA (h hz), ?_\u27e9", "annotated_tactic": ["refine \u27e8A.mul_mem hyA (h hz), ?_\u27e9", []], "state_before": "case h.mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\ny : G\nhyA : y \u2208 \u2191A\nhyB : y \u2208 \u2191B\nz : G\nhz : z \u2208 \u2191C\n\u22a2 y * z \u2208 \u2191A \u2227 \u2203 x \u2208 \u2191B, \u2203 y_1 \u2208 \u2191C, x * y_1 = y * z", "state_after": "case h.mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\ny : G\nhyA : y \u2208 \u2191A\nhyB : y \u2208 \u2191B\nz : G\nhz : z \u2208 \u2191C\n\u22a2 \u2203 x \u2208 \u2191B, \u2203 y_1 \u2208 \u2191C, x * y_1 = y * z"}, {"tactic": "exact \u27e8y, hyB, z, hz, rfl\u27e9", "annotated_tactic": ["exact \u27e8y, hyB, z, hz, <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case h.mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\ny : G\nhyA : y \u2208 \u2191A\nhyB : y \u2208 \u2191B\nz : G\nhz : z \u2208 \u2191C\n\u22a2 \u2203 x \u2208 \u2191B, \u2203 y_1 \u2208 \u2191C, x * y_1 = y * z", "state_after": "no goals"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "\u03b1 : Type u_1\nG : Type u_2\nA\u271d : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\u271d\ns : Set G\nA B C : Subgroup G\nh : C \u2264 A\ny : G\nhy : y \u2208 \u2191B\nz : G\nhz : z \u2208 \u2191C\nhyz : y * z \u2208 \u2191A\nthis : y * z * z\u207b\u00b9 \u2208 A\n\u22a2 y \u2208 \u2191A", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.Infinite.encard_eq", "start": [85, 1], "end": [88, 70], "traced_tactics": [{"tactic": "have := h.to_subtype", "annotated_tactic": ["have := h.to_subtype", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t s : Set \u03b1\nh : s.Infinite\n\u22a2 s.encard = \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t s : Set \u03b1\nh : s.Infinite\nthis : Infinite \u2191s\n\u22a2 s.encard = \u22a4"}, {"tactic": "rw [encard, \u2190 PartENat.withTopEquiv.symm.injective.eq_iff, Equiv.symm_apply_apply,\n  PartENat.withTopEquiv_symm_top, PartENat.card_eq_top_of_infinite]", "annotated_tactic": ["rw [<a>encard</a>, \u2190 PartENat.withTopEquiv.symm.injective.eq_iff, <a>Equiv.symm_apply_apply</a>,\n    <a>PartENat.withTopEquiv_symm_top</a>, <a>PartENat.card_eq_top_of_infinite</a>]", [{"full_name": "Set.encard", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [64, 19], "def_end_pos": [64, 25]}, {"full_name": "Equiv.symm_apply_apply", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [282, 17], "def_end_pos": [282, 33]}, {"full_name": "PartENat.withTopEquiv_symm_top", "def_path": "Mathlib/Data/Nat/PartENat.lean", "def_pos": [776, 9], "def_end_pos": [776, 30]}, {"full_name": "PartENat.card_eq_top_of_infinite", "def_path": "Mathlib/SetTheory/Cardinal/Finite.lean", "def_pos": [229, 9], "def_end_pos": [229, 32]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t s : Set \u03b1\nh : s.Infinite\nthis : Infinite \u2191s\n\u22a2 s.encard = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.coe_div", "start": [641, 1], "end": [642, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/KernelPair.lean", "full_name": "CategoryTheory.IsKernelPair.comp_of_mono", "start": [139, 1], "end": [150, 58], "traced_tactics": [{"tactic": "rw [small_k.w_assoc]", "annotated_tactic": ["rw [small_k.w_assoc]", []], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\n\u22a2 a \u226b f\u2081 \u226b f\u2082 = b \u226b f\u2081 \u226b f\u2082", "state_after": "no goals"}, {"tactic": "refine PullbackCone.isLimitAux _\n  (fun s => small_k.lift s.fst s.snd (by rw [\u2190 cancel_mono f\u2082, assoc, s.condition, assoc]))\n  (by simp) (by simp) ?_", "annotated_tactic": ["refine <a>PullbackCone.isLimitAux</a> _\n        (fun s => small_k.lift s.fst s.snd (by rw [\u2190 <a>cancel_mono</a> f\u2082, <a>assoc</a>, s.condition, <a>assoc</a>]))\n        (by simp) (by simp) ?_", [{"full_name": "CategoryTheory.Limits.PullbackCone.isLimitAux", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [545, 5], "def_end_pos": [545, 15]}, {"full_name": "CategoryTheory.cancel_mono", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [310, 9], "def_end_pos": [310, 20]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}]], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\n\u22a2 IsLimit (PullbackCone.mk a b \u22ef)", "state_after": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\n\u22a2 \u2200 (s : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)) (m : s.pt \u27f6 (PullbackCone.mk a b \u22ef).pt),\n    (\u2200 (j : WalkingCospan), m \u226b (PullbackCone.mk a b \u22ef).\u03c0.app j = s.\u03c0.app j) \u2192\n      m = (fun s => small_k.lift s.fst s.snd \u22ef) s"}, {"tactic": "intro s m hm", "annotated_tactic": ["intro s m hm", []], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\n\u22a2 \u2200 (s : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)) (m : s.pt \u27f6 (PullbackCone.mk a b \u22ef).pt),\n    (\u2200 (j : WalkingCospan), m \u226b (PullbackCone.mk a b \u22ef).\u03c0.app j = s.\u03c0.app j) \u2192\n      m = (fun s => small_k.lift s.fst s.snd \u22ef) s", "state_after": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\ns : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)\nm : s.pt \u27f6 (PullbackCone.mk a b \u22ef).pt\nhm : \u2200 (j : WalkingCospan), m \u226b (PullbackCone.mk a b \u22ef).\u03c0.app j = s.\u03c0.app j\n\u22a2 m = (fun s => small_k.lift s.fst s.snd \u22ef) s"}, {"tactic": "apply small_k.isLimit.hom_ext", "annotated_tactic": ["apply small_k.isLimit.hom_ext", []], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\ns : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)\nm : s.pt \u27f6 (PullbackCone.mk a b \u22ef).pt\nhm : \u2200 (j : WalkingCospan), m \u226b (PullbackCone.mk a b \u22ef).\u03c0.app j = s.\u03c0.app j\n\u22a2 m = (fun s => small_k.lift s.fst s.snd \u22ef) s", "state_after": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\ns : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)\nm : s.pt \u27f6 (PullbackCone.mk a b \u22ef).pt\nhm : \u2200 (j : WalkingCospan), m \u226b (PullbackCone.mk a b \u22ef).\u03c0.app j = s.\u03c0.app j\n\u22a2 \u2200 (j : WalkingCospan),\n    m \u226b (IsPullback.cone small_k).\u03c0.app j = (fun s => small_k.lift s.fst s.snd \u22ef) s \u226b (IsPullback.cone small_k).\u03c0.app j"}, {"tactic": "apply PullbackCone.equalizer_ext small_k.cone _ _", "annotated_tactic": ["apply <a>PullbackCone.equalizer_ext</a> small_k.cone _ _", [{"full_name": "CategoryTheory.Limits.PullbackCone.equalizer_ext", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [615, 9], "def_end_pos": [615, 22]}]], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\ns : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)\nm : s.pt \u27f6 (PullbackCone.mk a b \u22ef).pt\nhm : \u2200 (j : WalkingCospan), m \u226b (PullbackCone.mk a b \u22ef).\u03c0.app j = s.\u03c0.app j\n\u22a2 \u2200 (j : WalkingCospan),\n    m \u226b (IsPullback.cone small_k).\u03c0.app j = (fun s => small_k.lift s.fst s.snd \u22ef) s \u226b (IsPullback.cone small_k).\u03c0.app j", "state_after": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\ns : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)\nm : s.pt \u27f6 (PullbackCone.mk a b \u22ef).pt\nhm : \u2200 (j : WalkingCospan), m \u226b (PullbackCone.mk a b \u22ef).\u03c0.app j = s.\u03c0.app j\n\u22a2 m \u226b (IsPullback.cone small_k).fst = (fun s => small_k.lift s.fst s.snd \u22ef) s \u226b (IsPullback.cone small_k).fst\n\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\ns : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)\nm : s.pt \u27f6 (PullbackCone.mk a b \u22ef).pt\nhm : \u2200 (j : WalkingCospan), m \u226b (PullbackCone.mk a b \u22ef).\u03c0.app j = s.\u03c0.app j\n\u22a2 m \u226b (IsPullback.cone small_k).snd = (fun s => small_k.lift s.fst s.snd \u22ef) s \u226b (IsPullback.cone small_k).snd"}, {"tactic": "rw [\u2190 cancel_mono f\u2082, assoc, s.condition, assoc]", "annotated_tactic": ["rw [\u2190 <a>cancel_mono</a> f\u2082, <a>assoc</a>, s.condition, <a>assoc</a>]", [{"full_name": "CategoryTheory.cancel_mono", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [310, 9], "def_end_pos": [310, 20]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}]], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\ns : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)\n\u22a2 s.fst \u226b f\u2081 = s.snd \u226b f\u2081", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\n\u22a2 \u2200 (s : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)),\n    (fun s => small_k.lift s.fst s.snd \u22ef) s \u226b (PullbackCone.mk a b \u22ef).fst = s.fst", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\n\u22a2 \u2200 (s : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)),\n    (fun s => small_k.lift s.fst s.snd \u22ef) s \u226b (PullbackCone.mk a b \u22ef).snd = s.snd", "state_after": "no goals"}, {"tactic": "exact (hm WalkingCospan.left).trans (by simp)", "annotated_tactic": ["exact (hm <a>WalkingCospan.left</a>).<a>trans</a> (by simp)", [{"full_name": "CategoryTheory.Limits.WalkingCospan.left", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [44, 8], "def_end_pos": [44, 26]}, {"full_name": "Eq.trans", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [335, 9], "def_end_pos": [335, 17]}]], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\ns : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)\nm : s.pt \u27f6 (PullbackCone.mk a b \u22ef).pt\nhm : \u2200 (j : WalkingCospan), m \u226b (PullbackCone.mk a b \u22ef).\u03c0.app j = s.\u03c0.app j\n\u22a2 m \u226b (IsPullback.cone small_k).fst = (fun s => small_k.lift s.fst s.snd \u22ef) s \u226b (IsPullback.cone small_k).fst", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\ns : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)\nm : s.pt \u27f6 (PullbackCone.mk a b \u22ef).pt\nhm : \u2200 (j : WalkingCospan), m \u226b (PullbackCone.mk a b \u22ef).\u03c0.app j = s.\u03c0.app j\n\u22a2 s.\u03c0.app WalkingCospan.left = (fun s => small_k.lift s.fst s.snd \u22ef) s \u226b (IsPullback.cone small_k).fst", "state_after": "no goals"}, {"tactic": "exact (hm WalkingCospan.right).trans (by simp)", "annotated_tactic": ["exact (hm <a>WalkingCospan.right</a>).<a>trans</a> (by simp)", [{"full_name": "CategoryTheory.Limits.WalkingCospan.right", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [50, 8], "def_end_pos": [50, 27]}, {"full_name": "Eq.trans", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [335, 9], "def_end_pos": [335, 17]}]], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\ns : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)\nm : s.pt \u27f6 (PullbackCone.mk a b \u22ef).pt\nhm : \u2200 (j : WalkingCospan), m \u226b (PullbackCone.mk a b \u22ef).\u03c0.app j = s.\u03c0.app j\n\u22a2 m \u226b (IsPullback.cone small_k).snd = (fun s => small_k.lift s.fst s.snd \u22ef) s \u226b (IsPullback.cone small_k).snd", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nR X Y Z : C\nf : X \u27f6 Y\na b : R \u27f6 X\nf\u2081 : X \u27f6 Y\nf\u2082 : Y \u27f6 Z\ninst\u271d : Mono f\u2082\nsmall_k : IsKernelPair f\u2081 a b\ns : PullbackCone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082)\nm : s.pt \u27f6 (PullbackCone.mk a b \u22ef).pt\nhm : \u2200 (j : WalkingCospan), m \u226b (PullbackCone.mk a b \u22ef).\u03c0.app j = s.\u03c0.app j\n\u22a2 s.\u03c0.app WalkingCospan.right = (fun s => small_k.lift s.fst s.snd \u22ef) s \u226b (IsPullback.cone small_k).snd", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/ContinuousAffineMap.lean", "full_name": "ContinuousAffineMap.toContinuousMap_coe", "start": [97, 1], "end": [97, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "full_name": "AlgHom.mem_range", "start": [597, 1], "end": [598, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/ToLin.lean", "full_name": "LinearMap.toMatrixAlgEquiv'_id", "start": [516, 1], "end": [518, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Symmetrized.lean", "full_name": "SymAlg.unsym_bijective", "start": [95, 1], "end": [96, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/HahnSeries/Multiplication.lean", "full_name": "HahnModule.smul_coeff", "start": [139, 1], "end": [143, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Regular/Basic.lean", "full_name": "Units.isRegular", "start": [346, 1], "end": [347, 82], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean", "full_name": "Complex.cos_eq_tsum", "start": [92, 1], "end": [94, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/NAry.lean", "full_name": "Filter.map\u2082_inf_subset_right", "start": [132, 1], "end": [133, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "full_name": "Asymptotics.isLittleO_of_subsingleton", "start": [308, 1], "end": [309, 88], "traced_tactics": [{"tactic": "simp [Subsingleton.elim (f' _) 0, mul_nonneg hc.le]", "annotated_tactic": ["simp [<a>Subsingleton.elim</a> (f' _) 0, <a>mul_nonneg</a> hc.le]", [{"full_name": "Subsingleton.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1015, 19], "def_end_pos": [1015, 36]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [437, 7], "def_end_pos": [437, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nE' : Type u_6\nF' : Type u_7\nG' : Type u_8\nE'' : Type u_9\nF'' : Type u_10\nG'' : Type u_11\nE''' : Type u_12\nR : Type u_13\nR' : Type u_14\n\ud835\udd5c : Type u_15\n\ud835\udd5c' : Type u_16\ninst\u271d\u00b9\u2074 : Norm E\ninst\u271d\u00b9\u00b3 : Norm F\ninst\u271d\u00b9\u00b2 : Norm G\ninst\u271d\u00b9\u00b9 : SeminormedAddCommGroup E'\ninst\u271d\u00b9\u2070 : SeminormedAddCommGroup F'\ninst\u271d\u2079 : SeminormedAddCommGroup G'\ninst\u271d\u2078 : NormedAddCommGroup E''\ninst\u271d\u2077 : NormedAddCommGroup F''\ninst\u271d\u2076 : NormedAddCommGroup G''\ninst\u271d\u2075 : SeminormedRing R\ninst\u271d\u2074 : SeminormedAddGroup E'''\ninst\u271d\u00b3 : SeminormedRing R'\ninst\u271d\u00b2 : NormedDivisionRing \ud835\udd5c\ninst\u271d\u00b9 : NormedDivisionRing \ud835\udd5c'\nc\u271d c' c\u2081 c\u2082 : \u211d\nf : \u03b1 \u2192 E\ng : \u03b1 \u2192 F\nk : \u03b1 \u2192 G\nf' : \u03b1 \u2192 E'\ng' : \u03b1 \u2192 F'\nk' : \u03b1 \u2192 G'\nf'' : \u03b1 \u2192 E''\ng'' : \u03b1 \u2192 F''\nk'' : \u03b1 \u2192 G''\nl l' : Filter \u03b1\ninst\u271d : Subsingleton E'\nc : \u211d\nhc : 0 < c\n\u22a2 \u2200\u1da0 (x : \u03b1) in l, \u2016f' x\u2016 \u2264 c * \u2016g' x\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "full_name": "Ultrafilter.cauchy_of_totallyBounded", "start": [588, 1], "end": [598, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean", "full_name": "CategoryTheory.Pretriangulated.complete_distinguished_triangle_morphism\u2081", "start": [192, 1], "end": [206, 48], "traced_tactics": [{"tactic": "obtain \u27e8a, \u27e8ha\u2081, ha\u2082\u27e9\u27e9 := complete_distinguished_triangle_morphism _ _\n  (rot_of_distTriang _ hT\u2081) (rot_of_distTriang _ hT\u2082) b c comm", "annotated_tactic": ["obtain \u27e8a, \u27e8ha\u2081, ha\u2082\u27e9\u27e9 := <a>complete_distinguished_triangle_morphism</a> _ _\n    (<a>rot_of_distTriang</a> _ hT\u2081) (<a>rot_of_distTriang</a> _ hT\u2082) b c comm", [{"full_name": "CategoryTheory.Pretriangulated.complete_distinguished_triangle_morphism", "def_path": "Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean", "def_pos": [80, 3], "def_end_pos": [80, 43]}, {"full_name": "CategoryTheory.Pretriangulated.rot_of_distTriang", "def_path": "Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean", "def_pos": [105, 9], "def_end_pos": [105, 26]}, {"full_name": "CategoryTheory.Pretriangulated.rot_of_distTriang", "def_path": "Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean", "def_pos": [105, 9], "def_end_pos": [105, 26]}]], "state_before": "C : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroObject C\ninst\u271d\u00b2 : HasShift C \u2124\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : \u2200 (n : \u2124), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nT\u2081 T\u2082 : Triangle C\nhT\u2081 : T\u2081 \u2208 distinguishedTriangles\nhT\u2082 : T\u2082 \u2208 distinguishedTriangles\nb : T\u2081.obj\u2082 \u27f6 T\u2082.obj\u2082\nc : T\u2081.obj\u2083 \u27f6 T\u2082.obj\u2083\ncomm : T\u2081.mor\u2082 \u226b c = b \u226b T\u2082.mor\u2082\n\u22a2 \u2203 a, T\u2081.mor\u2081 \u226b b = a \u226b T\u2082.mor\u2081 \u2227 T\u2081.mor\u2083 \u226b (shiftFunctor C 1).map a = c \u226b T\u2082.mor\u2083", "state_after": "case intro.intro\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroObject C\ninst\u271d\u00b2 : HasShift C \u2124\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : \u2200 (n : \u2124), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nT\u2081 T\u2082 : Triangle C\nhT\u2081 : T\u2081 \u2208 distinguishedTriangles\nhT\u2082 : T\u2082 \u2208 distinguishedTriangles\nb : T\u2081.obj\u2082 \u27f6 T\u2082.obj\u2082\nc : T\u2081.obj\u2083 \u27f6 T\u2082.obj\u2083\ncomm : T\u2081.mor\u2082 \u226b c = b \u226b T\u2082.mor\u2082\na : T\u2081.rotate.obj\u2083 \u27f6 T\u2082.rotate.obj\u2083\nha\u2081 : T\u2081.rotate.mor\u2082 \u226b a = c \u226b T\u2082.rotate.mor\u2082\nha\u2082 : T\u2081.rotate.mor\u2083 \u226b (shiftFunctor C 1).map b = a \u226b T\u2082.rotate.mor\u2083\n\u22a2 \u2203 a, T\u2081.mor\u2081 \u226b b = a \u226b T\u2082.mor\u2081 \u2227 T\u2081.mor\u2083 \u226b (shiftFunctor C 1).map a = c \u226b T\u2082.mor\u2083"}, {"tactic": "refine \u27e8(shiftFunctor C (1 : \u2124)).preimage a, \u27e8?_, ?_\u27e9\u27e9", "annotated_tactic": ["refine \u27e8(<a>shiftFunctor</a> C (1 : \u2124)).<a>preimage</a> a, \u27e8?_, ?_\u27e9\u27e9", [{"full_name": "CategoryTheory.shiftFunctor", "def_path": "Mathlib/CategoryTheory/Shift/Basic.lean", "def_pos": [168, 5], "def_end_pos": [168, 17]}, {"full_name": "CategoryTheory.Functor.preimage", "def_path": "Mathlib/CategoryTheory/Functor/FullyFaithful.lean", "def_pos": [82, 19], "def_end_pos": [82, 27]}]], "state_before": "case intro.intro\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroObject C\ninst\u271d\u00b2 : HasShift C \u2124\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : \u2200 (n : \u2124), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nT\u2081 T\u2082 : Triangle C\nhT\u2081 : T\u2081 \u2208 distinguishedTriangles\nhT\u2082 : T\u2082 \u2208 distinguishedTriangles\nb : T\u2081.obj\u2082 \u27f6 T\u2082.obj\u2082\nc : T\u2081.obj\u2083 \u27f6 T\u2082.obj\u2083\ncomm : T\u2081.mor\u2082 \u226b c = b \u226b T\u2082.mor\u2082\na : T\u2081.rotate.obj\u2083 \u27f6 T\u2082.rotate.obj\u2083\nha\u2081 : T\u2081.rotate.mor\u2082 \u226b a = c \u226b T\u2082.rotate.mor\u2082\nha\u2082 : T\u2081.rotate.mor\u2083 \u226b (shiftFunctor C 1).map b = a \u226b T\u2082.rotate.mor\u2083\n\u22a2 \u2203 a, T\u2081.mor\u2081 \u226b b = a \u226b T\u2082.mor\u2081 \u2227 T\u2081.mor\u2083 \u226b (shiftFunctor C 1).map a = c \u226b T\u2082.mor\u2083", "state_after": "case intro.intro.refine_1\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroObject C\ninst\u271d\u00b2 : HasShift C \u2124\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : \u2200 (n : \u2124), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nT\u2081 T\u2082 : Triangle C\nhT\u2081 : T\u2081 \u2208 distinguishedTriangles\nhT\u2082 : T\u2082 \u2208 distinguishedTriangles\nb : T\u2081.obj\u2082 \u27f6 T\u2082.obj\u2082\nc : T\u2081.obj\u2083 \u27f6 T\u2082.obj\u2083\ncomm : T\u2081.mor\u2082 \u226b c = b \u226b T\u2082.mor\u2082\na : T\u2081.rotate.obj\u2083 \u27f6 T\u2082.rotate.obj\u2083\nha\u2081 : T\u2081.rotate.mor\u2082 \u226b a = c \u226b T\u2082.rotate.mor\u2082\nha\u2082 : T\u2081.rotate.mor\u2083 \u226b (shiftFunctor C 1).map b = a \u226b T\u2082.rotate.mor\u2083\n\u22a2 T\u2081.mor\u2081 \u226b b = (shiftFunctor C 1).preimage a \u226b T\u2082.mor\u2081\n\ncase intro.intro.refine_2\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroObject C\ninst\u271d\u00b2 : HasShift C \u2124\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : \u2200 (n : \u2124), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nT\u2081 T\u2082 : Triangle C\nhT\u2081 : T\u2081 \u2208 distinguishedTriangles\nhT\u2082 : T\u2082 \u2208 distinguishedTriangles\nb : T\u2081.obj\u2082 \u27f6 T\u2082.obj\u2082\nc : T\u2081.obj\u2083 \u27f6 T\u2082.obj\u2083\ncomm : T\u2081.mor\u2082 \u226b c = b \u226b T\u2082.mor\u2082\na : T\u2081.rotate.obj\u2083 \u27f6 T\u2082.rotate.obj\u2083\nha\u2081 : T\u2081.rotate.mor\u2082 \u226b a = c \u226b T\u2082.rotate.mor\u2082\nha\u2082 : T\u2081.rotate.mor\u2083 \u226b (shiftFunctor C 1).map b = a \u226b T\u2082.rotate.mor\u2083\n\u22a2 T\u2081.mor\u2083 \u226b (shiftFunctor C 1).map ((shiftFunctor C 1).preimage a) = c \u226b T\u2082.mor\u2083"}, {"tactic": "apply (shiftFunctor C (1 : \u2124)).map_injective", "annotated_tactic": ["apply (<a>shiftFunctor</a> C (1 : \u2124)).<a>map_injective</a>", [{"full_name": "CategoryTheory.shiftFunctor", "def_path": "Mathlib/CategoryTheory/Shift/Basic.lean", "def_pos": [168, 5], "def_end_pos": [168, 17]}, {"full_name": "CategoryTheory.Functor.map_injective", "def_path": "Mathlib/CategoryTheory/Functor/FullyFaithful.lean", "def_pos": [62, 9], "def_end_pos": [62, 22]}]], "state_before": "case intro.intro.refine_1\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroObject C\ninst\u271d\u00b2 : HasShift C \u2124\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : \u2200 (n : \u2124), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nT\u2081 T\u2082 : Triangle C\nhT\u2081 : T\u2081 \u2208 distinguishedTriangles\nhT\u2082 : T\u2082 \u2208 distinguishedTriangles\nb : T\u2081.obj\u2082 \u27f6 T\u2082.obj\u2082\nc : T\u2081.obj\u2083 \u27f6 T\u2082.obj\u2083\ncomm : T\u2081.mor\u2082 \u226b c = b \u226b T\u2082.mor\u2082\na : T\u2081.rotate.obj\u2083 \u27f6 T\u2082.rotate.obj\u2083\nha\u2081 : T\u2081.rotate.mor\u2082 \u226b a = c \u226b T\u2082.rotate.mor\u2082\nha\u2082 : T\u2081.rotate.mor\u2083 \u226b (shiftFunctor C 1).map b = a \u226b T\u2082.rotate.mor\u2083\n\u22a2 T\u2081.mor\u2081 \u226b b = (shiftFunctor C 1).preimage a \u226b T\u2082.mor\u2081", "state_after": "case intro.intro.refine_1.a\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroObject C\ninst\u271d\u00b2 : HasShift C \u2124\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : \u2200 (n : \u2124), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nT\u2081 T\u2082 : Triangle C\nhT\u2081 : T\u2081 \u2208 distinguishedTriangles\nhT\u2082 : T\u2082 \u2208 distinguishedTriangles\nb : T\u2081.obj\u2082 \u27f6 T\u2082.obj\u2082\nc : T\u2081.obj\u2083 \u27f6 T\u2082.obj\u2083\ncomm : T\u2081.mor\u2082 \u226b c = b \u226b T\u2082.mor\u2082\na : T\u2081.rotate.obj\u2083 \u27f6 T\u2082.rotate.obj\u2083\nha\u2081 : T\u2081.rotate.mor\u2082 \u226b a = c \u226b T\u2082.rotate.mor\u2082\nha\u2082 : T\u2081.rotate.mor\u2083 \u226b (shiftFunctor C 1).map b = a \u226b T\u2082.rotate.mor\u2083\n\u22a2 (shiftFunctor C 1).map (T\u2081.mor\u2081 \u226b b) = (shiftFunctor C 1).map ((shiftFunctor C 1).preimage a \u226b T\u2082.mor\u2081)"}, {"tactic": "dsimp at ha\u2082", "annotated_tactic": ["dsimp at ha\u2082", []], "state_before": "case intro.intro.refine_1.a\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroObject C\ninst\u271d\u00b2 : HasShift C \u2124\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : \u2200 (n : \u2124), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nT\u2081 T\u2082 : Triangle C\nhT\u2081 : T\u2081 \u2208 distinguishedTriangles\nhT\u2082 : T\u2082 \u2208 distinguishedTriangles\nb : T\u2081.obj\u2082 \u27f6 T\u2082.obj\u2082\nc : T\u2081.obj\u2083 \u27f6 T\u2082.obj\u2083\ncomm : T\u2081.mor\u2082 \u226b c = b \u226b T\u2082.mor\u2082\na : T\u2081.rotate.obj\u2083 \u27f6 T\u2082.rotate.obj\u2083\nha\u2081 : T\u2081.rotate.mor\u2082 \u226b a = c \u226b T\u2082.rotate.mor\u2082\nha\u2082 : T\u2081.rotate.mor\u2083 \u226b (shiftFunctor C 1).map b = a \u226b T\u2082.rotate.mor\u2083\n\u22a2 (shiftFunctor C 1).map (T\u2081.mor\u2081 \u226b b) = (shiftFunctor C 1).map ((shiftFunctor C 1).preimage a \u226b T\u2082.mor\u2081)", "state_after": "case intro.intro.refine_1.a\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroObject C\ninst\u271d\u00b2 : HasShift C \u2124\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : \u2200 (n : \u2124), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nT\u2081 T\u2082 : Triangle C\nhT\u2081 : T\u2081 \u2208 distinguishedTriangles\nhT\u2082 : T\u2082 \u2208 distinguishedTriangles\nb : T\u2081.obj\u2082 \u27f6 T\u2082.obj\u2082\nc : T\u2081.obj\u2083 \u27f6 T\u2082.obj\u2083\ncomm : T\u2081.mor\u2082 \u226b c = b \u226b T\u2082.mor\u2082\na : T\u2081.rotate.obj\u2083 \u27f6 T\u2082.rotate.obj\u2083\nha\u2081 : T\u2081.rotate.mor\u2082 \u226b a = c \u226b T\u2082.rotate.mor\u2082\nha\u2082 : (-(shiftFunctor C 1).map T\u2081.mor\u2081) \u226b (shiftFunctor C 1).map b = a \u226b (-(shiftFunctor C 1).map T\u2082.mor\u2081)\n\u22a2 (shiftFunctor C 1).map (T\u2081.mor\u2081 \u226b b) = (shiftFunctor C 1).map ((shiftFunctor C 1).preimage a \u226b T\u2082.mor\u2081)"}, {"tactic": "rw [neg_comp, comp_neg, neg_inj] at ha\u2082", "annotated_tactic": ["rw [<a>neg_comp</a>, <a>comp_neg</a>, <a>neg_inj</a>] at ha\u2082", [{"full_name": "CategoryTheory.Preadditive.neg_comp", "def_path": "Mathlib/CategoryTheory/Preadditive/Basic.lean", "def_pos": [150, 9], "def_end_pos": [150, 17]}, {"full_name": "CategoryTheory.Preadditive.comp_neg", "def_path": "Mathlib/CategoryTheory/Preadditive/Basic.lean", "def_pos": [156, 9], "def_end_pos": [156, 17]}, {"full_name": "neg_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [406, 3], "def_end_pos": [406, 14]}]], "state_before": "case intro.intro.refine_1.a\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroObject C\ninst\u271d\u00b2 : HasShift C \u2124\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : \u2200 (n : \u2124), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nT\u2081 T\u2082 : Triangle C\nhT\u2081 : T\u2081 \u2208 distinguishedTriangles\nhT\u2082 : T\u2082 \u2208 distinguishedTriangles\nb : T\u2081.obj\u2082 \u27f6 T\u2082.obj\u2082\nc : T\u2081.obj\u2083 \u27f6 T\u2082.obj\u2083\ncomm : T\u2081.mor\u2082 \u226b c = b \u226b T\u2082.mor\u2082\na : T\u2081.rotate.obj\u2083 \u27f6 T\u2082.rotate.obj\u2083\nha\u2081 : T\u2081.rotate.mor\u2082 \u226b a = c \u226b T\u2082.rotate.mor\u2082\nha\u2082 : (-(shiftFunctor C 1).map T\u2081.mor\u2081) \u226b (shiftFunctor C 1).map b = a \u226b (-(shiftFunctor C 1).map T\u2082.mor\u2081)\n\u22a2 (shiftFunctor C 1).map (T\u2081.mor\u2081 \u226b b) = (shiftFunctor C 1).map ((shiftFunctor C 1).preimage a \u226b T\u2082.mor\u2081)", "state_after": "case intro.intro.refine_1.a\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroObject C\ninst\u271d\u00b2 : HasShift C \u2124\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : \u2200 (n : \u2124), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nT\u2081 T\u2082 : Triangle C\nhT\u2081 : T\u2081 \u2208 distinguishedTriangles\nhT\u2082 : T\u2082 \u2208 distinguishedTriangles\nb : T\u2081.obj\u2082 \u27f6 T\u2082.obj\u2082\nc : T\u2081.obj\u2083 \u27f6 T\u2082.obj\u2083\ncomm : T\u2081.mor\u2082 \u226b c = b \u226b T\u2082.mor\u2082\na : T\u2081.rotate.obj\u2083 \u27f6 T\u2082.rotate.obj\u2083\nha\u2081 : T\u2081.rotate.mor\u2082 \u226b a = c \u226b T\u2082.rotate.mor\u2082\nha\u2082 : (shiftFunctor C 1).map T\u2081.mor\u2081 \u226b (shiftFunctor C 1).map b = a \u226b (shiftFunctor C 1).map T\u2082.mor\u2081\n\u22a2 (shiftFunctor C 1).map (T\u2081.mor\u2081 \u226b b) = (shiftFunctor C 1).map ((shiftFunctor C 1).preimage a \u226b T\u2082.mor\u2081)"}, {"tactic": "simpa only [Functor.map_comp, Functor.map_preimage] using ha\u2082", "annotated_tactic": ["simpa only [<a>Functor.map_comp</a>, <a>Functor.map_preimage</a>] using ha\u2082", [{"full_name": "CategoryTheory.Functor.map_comp", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [43, 3], "def_end_pos": [43, 11]}, {"full_name": "CategoryTheory.Functor.map_preimage", "def_path": "Mathlib/CategoryTheory/Functor/FullyFaithful.lean", "def_pos": [87, 9], "def_end_pos": [87, 21]}]], "state_before": "case intro.intro.refine_1.a\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroObject C\ninst\u271d\u00b2 : HasShift C \u2124\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : \u2200 (n : \u2124), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nT\u2081 T\u2082 : Triangle C\nhT\u2081 : T\u2081 \u2208 distinguishedTriangles\nhT\u2082 : T\u2082 \u2208 distinguishedTriangles\nb : T\u2081.obj\u2082 \u27f6 T\u2082.obj\u2082\nc : T\u2081.obj\u2083 \u27f6 T\u2082.obj\u2083\ncomm : T\u2081.mor\u2082 \u226b c = b \u226b T\u2082.mor\u2082\na : T\u2081.rotate.obj\u2083 \u27f6 T\u2082.rotate.obj\u2083\nha\u2081 : T\u2081.rotate.mor\u2082 \u226b a = c \u226b T\u2082.rotate.mor\u2082\nha\u2082 : (shiftFunctor C 1).map T\u2081.mor\u2081 \u226b (shiftFunctor C 1).map b = a \u226b (shiftFunctor C 1).map T\u2082.mor\u2081\n\u22a2 (shiftFunctor C 1).map (T\u2081.mor\u2081 \u226b b) = (shiftFunctor C 1).map ((shiftFunctor C 1).preimage a \u226b T\u2082.mor\u2081)", "state_after": "no goals"}, {"tactic": "simpa only [Functor.map_preimage] using ha\u2081", "annotated_tactic": ["simpa only [<a>Functor.map_preimage</a>] using ha\u2081", [{"full_name": "CategoryTheory.Functor.map_preimage", "def_path": "Mathlib/CategoryTheory/Functor/FullyFaithful.lean", "def_pos": [87, 9], "def_end_pos": [87, 21]}]], "state_before": "case intro.intro.refine_2\nC : Type u\ninst\u271d\u2074 : Category.{v, u} C\ninst\u271d\u00b3 : HasZeroObject C\ninst\u271d\u00b2 : HasShift C \u2124\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : \u2200 (n : \u2124), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nT\u2081 T\u2082 : Triangle C\nhT\u2081 : T\u2081 \u2208 distinguishedTriangles\nhT\u2082 : T\u2082 \u2208 distinguishedTriangles\nb : T\u2081.obj\u2082 \u27f6 T\u2082.obj\u2082\nc : T\u2081.obj\u2083 \u27f6 T\u2082.obj\u2083\ncomm : T\u2081.mor\u2082 \u226b c = b \u226b T\u2082.mor\u2082\na : T\u2081.rotate.obj\u2083 \u27f6 T\u2082.rotate.obj\u2083\nha\u2081 : T\u2081.rotate.mor\u2082 \u226b a = c \u226b T\u2082.rotate.mor\u2082\nha\u2082 : T\u2081.rotate.mor\u2083 \u226b (shiftFunctor C 1).map b = a \u226b T\u2082.rotate.mor\u2083\n\u22a2 T\u2081.mor\u2083 \u226b (shiftFunctor C 1).map ((shiftFunctor C 1).preimage a) = c \u226b T\u2082.mor\u2083", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Equiv.lean", "full_name": "AlgEquiv.bijective", "start": [291, 11], "end": [292, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean", "full_name": "InnerProductGeometry.angle_smul_smul", "start": [57, 1], "end": [60, 90], "traced_tactics": [{"tactic": "have : c * c \u2260 0 := mul_ne_zero hc hc", "annotated_tactic": ["have : c * c \u2260 0 := <a>mul_ne_zero</a> hc hc", [{"full_name": "mul_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 20]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d : V\nc : \u211d\nhc : c \u2260 0\nx y : V\n\u22a2 angle (c \u2022 x) (c \u2022 y) = angle x y", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d : V\nc : \u211d\nhc : c \u2260 0\nx y : V\nthis : c * c \u2260 0\n\u22a2 angle (c \u2022 x) (c \u2022 y) = angle x y"}, {"tactic": "rw [angle, angle, real_inner_smul_left, inner_smul_right, norm_smul, norm_smul, Real.norm_eq_abs,\n  mul_mul_mul_comm _ \u2016x\u2016, abs_mul_abs_self, \u2190 mul_assoc c c, mul_div_mul_left _ _ this]", "annotated_tactic": ["rw [<a>angle</a>, <a>angle</a>, <a>real_inner_smul_left</a>, <a>inner_smul_right</a>, <a>norm_smul</a>, <a>norm_smul</a>, <a>Real.norm_eq_abs</a>,\n    <a>mul_mul_mul_comm</a> _ \u2016x\u2016, <a>abs_mul_abs_self</a>, \u2190 <a>mul_assoc</a> c c, <a>mul_div_mul_left</a> _ _ this]", [{"full_name": "InnerProductGeometry.angle", "def_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean", "def_pos": [45, 5], "def_end_pos": [45, 10]}, {"full_name": "InnerProductGeometry.angle", "def_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean", "def_pos": [45, 5], "def_end_pos": [45, 10]}, {"full_name": "real_inner_smul_left", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 29]}, {"full_name": "inner_smul_right", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [477, 9], "def_end_pos": [477, 25]}, {"full_name": "norm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [90, 9], "def_end_pos": [90, 18]}, {"full_name": "norm_smul", "def_path": "Mathlib/Analysis/Normed/MulAction.lean", "def_pos": [90, 9], "def_end_pos": [90, 18]}, {"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1434, 9], "def_end_pos": [1434, 20]}, {"full_name": "mul_mul_mul_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [196, 9], "def_end_pos": [196, 25]}, {"full_name": "abs_mul_abs_self", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [82, 15], "def_end_pos": [82, 31]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "mul_div_mul_left", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [520, 7], "def_end_pos": [520, 23]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx\u271d y\u271d : V\nc : \u211d\nhc : c \u2260 0\nx y : V\nthis : c * c \u2260 0\n\u22a2 angle (c \u2022 x) (c \u2022 y) = angle x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "full_name": "AffineSubspace.mem_direction_iff_eq_vsub_left", "start": [316, 1], "end": [319, 91], "traced_tactics": [{"tactic": "rw [\u2190 SetLike.mem_coe, coe_direction_eq_vsub_set_left hp]", "annotated_tactic": ["rw [\u2190 <a>SetLike.mem_coe</a>, <a>coe_direction_eq_vsub_set_left</a> hp]", [{"full_name": "SetLike.mem_coe", "def_path": "Mathlib/Data/SetLike/Basic.lean", "def_pos": [180, 9], "def_end_pos": [180, 16]}, {"full_name": "AffineSubspace.coe_direction_eq_vsub_set_left", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [295, 9], "def_end_pos": [295, 39]}]], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\nv : V\n\u22a2 v \u2208 s.direction \u2194 \u2203 p2 \u2208 s, v = p -\u1d65 p2", "state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\nv : V\n\u22a2 v \u2208 (fun x => p -\u1d65 x) '' \u2191s \u2194 \u2203 p2 \u2208 s, v = p -\u1d65 p2"}, {"tactic": "exact \u27e8fun \u27e8p2, hp2, hv\u27e9 => \u27e8p2, hp2, hv.symm\u27e9, fun \u27e8p2, hp2, hv\u27e9 => \u27e8p2, hp2, hv.symm\u27e9\u27e9", "annotated_tactic": ["exact \u27e8fun \u27e8p2, hp2, hv\u27e9 => \u27e8p2, hp2, hv.symm\u27e9, fun \u27e8p2, hp2, hv\u27e9 => \u27e8p2, hp2, hv.symm\u27e9\u27e9", []], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p \u2208 s\nv : V\n\u22a2 v \u2208 (fun x => p -\u1d65 x) '' \u2191s \u2194 \u2203 p2 \u2208 s, v = p -\u1d65 p2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Determinant.lean", "full_name": "LinearMap.associated_det_of_eq_comp", "start": [481, 1], "end": [488, 76], "traced_tactics": [{"tactic": "suffices Associated (LinearMap.det (f' \u2218\u2097 \u2191e)) (LinearMap.det f') by\n  convert this using 2\n  ext x\n  exact h x", "annotated_tactic": ["suffices <a>Associated</a> (<a>LinearMap.det</a> (f' \u2218\u2097 \u2191e)) (<a>LinearMap.det</a> f') by\n    convert this using 2\n    ext x\n    exact h x", [{"full_name": "Associated", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [397, 5], "def_end_pos": [397, 15]}, {"full_name": "LinearMap.det", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [175, 27], "def_end_pos": [175, 30]}, {"full_name": "LinearMap.det", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [175, 27], "def_end_pos": [175, 30]}]], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\ne : M \u2243\u2097[R] M\nf f' : M \u2192\u2097[R] M\nh : \u2200 (x : M), f x = f' (e x)\n\u22a2 Associated (LinearMap.det f) (LinearMap.det f')", "state_after": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\ne : M \u2243\u2097[R] M\nf f' : M \u2192\u2097[R] M\nh : \u2200 (x : M), f x = f' (e x)\n\u22a2 Associated (LinearMap.det (f' \u2218\u2097 \u2191e)) (LinearMap.det f')"}, {"tactic": "rw [\u2190 mul_one (LinearMap.det f'), LinearMap.det_comp]", "annotated_tactic": ["rw [\u2190 <a>mul_one</a> (<a>LinearMap.det</a> f'), <a>LinearMap.det_comp</a>]", [{"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "LinearMap.det", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [175, 27], "def_end_pos": [175, 30]}, {"full_name": "LinearMap.det_comp", "def_path": "Mathlib/LinearAlgebra/Determinant.lean", "def_pos": [244, 9], "def_end_pos": [244, 17]}]], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\ne : M \u2243\u2097[R] M\nf f' : M \u2192\u2097[R] M\nh : \u2200 (x : M), f x = f' (e x)\n\u22a2 Associated (LinearMap.det (f' \u2218\u2097 \u2191e)) (LinearMap.det f')", "state_after": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\ne : M \u2243\u2097[R] M\nf f' : M \u2192\u2097[R] M\nh : \u2200 (x : M), f x = f' (e x)\n\u22a2 Associated (LinearMap.det f' * LinearMap.det \u2191e) (LinearMap.det f' * 1)"}, {"tactic": "exact Associated.mul_left _ (associated_one_iff_isUnit.mpr e.isUnit_det')", "annotated_tactic": ["exact <a>Associated.mul_left</a> _ (associated_one_iff_isUnit.mpr e.isUnit_det')", [{"full_name": "Associated.mul_left", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [551, 9], "def_end_pos": [551, 28]}]], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\ne : M \u2243\u2097[R] M\nf f' : M \u2192\u2097[R] M\nh : \u2200 (x : M), f x = f' (e x)\n\u22a2 Associated (LinearMap.det f' * LinearMap.det \u2191e) (LinearMap.det f' * 1)", "state_after": "no goals"}, {"tactic": "convert this using 2", "annotated_tactic": ["convert this using 2", []], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\ne : M \u2243\u2097[R] M\nf f' : M \u2192\u2097[R] M\nh : \u2200 (x : M), f x = f' (e x)\nthis : Associated (LinearMap.det (f' \u2218\u2097 \u2191e)) (LinearMap.det f')\n\u22a2 Associated (LinearMap.det f) (LinearMap.det f')", "state_after": "case h.e'_3.h.e'_6\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\ne : M \u2243\u2097[R] M\nf f' : M \u2192\u2097[R] M\nh : \u2200 (x : M), f x = f' (e x)\nthis : Associated (LinearMap.det (f' \u2218\u2097 \u2191e)) (LinearMap.det f')\n\u22a2 f = f' \u2218\u2097 \u2191e"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case h.e'_3.h.e'_6\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\ne : M \u2243\u2097[R] M\nf f' : M \u2192\u2097[R] M\nh : \u2200 (x : M), f x = f' (e x)\nthis : Associated (LinearMap.det (f' \u2218\u2097 \u2191e)) (LinearMap.det f')\n\u22a2 f = f' \u2218\u2097 \u2191e", "state_after": "case h.e'_3.h.e'_6.h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\ne : M \u2243\u2097[R] M\nf f' : M \u2192\u2097[R] M\nh : \u2200 (x : M), f x = f' (e x)\nthis : Associated (LinearMap.det (f' \u2218\u2097 \u2191e)) (LinearMap.det f')\nx : M\n\u22a2 f x = (f' \u2218\u2097 \u2191e) x"}, {"tactic": "exact h x", "annotated_tactic": ["exact h x", []], "state_before": "case h.e'_3.h.e'_6.h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nM : Type u_2\ninst\u271d\u2075 : AddCommGroup M\ninst\u271d\u2074 : Module R M\nM' : Type u_3\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\n\u03b9 : Type u_4\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : Fintype \u03b9\ne\u271d : Basis \u03b9 R M\ne : M \u2243\u2097[R] M\nf f' : M \u2192\u2097[R] M\nh : \u2200 (x : M), f x = f' (e x)\nthis : Associated (LinearMap.det (f' \u2218\u2097 \u2191e)) (LinearMap.det f')\nx : M\n\u22a2 f x = (f' \u2218\u2097 \u2191e) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Sylow.lean", "full_name": "not_dvd_card_sylow", "start": [342, 1], "end": [347, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Tactic/CategoryTheory/Coherence.lean", "full_name": "Mathlib.Tactic.Coherence.insert_id_lhs", "start": [196, 1], "end": [198, 16], "traced_tactics": [{"tactic": "simpa using w", "annotated_tactic": ["simpa using w", []], "state_before": "C\u271d : Type u\ninst\u271d\u00b2 : Category.{v, u} C\u271d\ninst\u271d\u00b9 : MonoidalCategory C\u271d\nC : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nX Y : C\nf g : X \u27f6 Y\nw : f \u226b \ud835\udfd9 Y = g\n\u22a2 f = g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.filter_eq", "start": [123, 1], "end": [124, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subsemigroup/Operations.lean", "full_name": "Subsemigroup.monotone_map", "start": [306, 1], "end": [307, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "full_name": "Ordinal.lift_lift", "start": [706, 1], "end": [710, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Squarefree.lean", "full_name": "Nat.squarefree_and_prime_pow_iff_prime", "start": [106, 1], "end": [111, 21], "traced_tactics": [{"tactic": "refine \u27e8?_, fun hn => \u27e8hn.squarefree, hn.isPrimePow\u27e9\u27e9", "annotated_tactic": ["refine \u27e8?_, fun hn => \u27e8hn.squarefree, hn.isPrimePow\u27e9\u27e9", []], "state_before": "s : Finset \u2115\nm n\u271d p n : \u2115\n\u22a2 Squarefree n \u2227 IsPrimePow n \u2194 Prime n", "state_after": "s : Finset \u2115\nm n\u271d p n : \u2115\n\u22a2 Squarefree n \u2227 IsPrimePow n \u2192 Prime n"}, {"tactic": "rw [isPrimePow_nat_iff]", "annotated_tactic": ["rw [<a>isPrimePow_nat_iff</a>]", [{"full_name": "isPrimePow_nat_iff", "def_path": "Mathlib/Algebra/IsPrimePow.lean", "def_pos": [76, 9], "def_end_pos": [76, 27]}]], "state_before": "s : Finset \u2115\nm n\u271d p n : \u2115\n\u22a2 Squarefree n \u2227 IsPrimePow n \u2192 Prime n", "state_after": "s : Finset \u2115\nm n\u271d p n : \u2115\n\u22a2 (Squarefree n \u2227 \u2203 p k, Prime p \u2227 0 < k \u2227 p ^ k = n) \u2192 Prime n"}, {"tactic": "rintro \u27e8h, p, k, hp, hk, rfl\u27e9", "annotated_tactic": ["rintro \u27e8h, p, k, hp, hk, rfl\u27e9", []], "state_before": "s : Finset \u2115\nm n\u271d p n : \u2115\n\u22a2 (Squarefree n \u2227 \u2203 p k, Prime p \u2227 0 < k \u2227 p ^ k = n) \u2192 Prime n", "state_after": "case intro.intro.intro.intro.intro\ns : Finset \u2115\nm n p\u271d p k : \u2115\nhp : Prime p\nhk : 0 < k\nh : Squarefree (p ^ k)\n\u22a2 Prime (p ^ k)"}, {"tactic": "rw [squarefree_pow_iff hp.ne_one hk.ne'] at h", "annotated_tactic": ["rw [<a>squarefree_pow_iff</a> hp.ne_one hk.ne'] at h", [{"full_name": "Nat.squarefree_pow_iff", "def_path": "Mathlib/Data/Nat/Squarefree.lean", "def_pos": [94, 9], "def_end_pos": [94, 27]}]], "state_before": "case intro.intro.intro.intro.intro\ns : Finset \u2115\nm n p\u271d p k : \u2115\nhp : Prime p\nhk : 0 < k\nh : Squarefree (p ^ k)\n\u22a2 Prime (p ^ k)", "state_after": "case intro.intro.intro.intro.intro\ns : Finset \u2115\nm n p\u271d p k : \u2115\nhp : Prime p\nhk : 0 < k\nh : Squarefree p \u2227 k = 1\n\u22a2 Prime (p ^ k)"}, {"tactic": "rwa [h.2, pow_one]", "annotated_tactic": ["rwa [h.2, <a>pow_one</a>]", [{"full_name": "pow_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [663, 7], "def_end_pos": [663, 14]}]], "state_before": "case intro.intro.intro.intro.intro\ns : Finset \u2115\nm n p\u271d p k : \u2115\nhp : Prime p\nhk : 0 < k\nh : Squarefree p \u2227 k = 1\n\u22a2 Prime (p ^ k)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Variance.lean", "full_name": "ProbabilityTheory.evariance_eq_zero_iff", "start": [143, 1], "end": [151, 47], "traced_tactics": [{"tactic": "rw [evariance, lintegral_eq_zero_iff']", "annotated_tactic": ["rw [<a>evariance</a>, <a>lintegral_eq_zero_iff'</a>]", [{"full_name": "ProbabilityTheory.evariance", "def_path": "Mathlib/Probability/Variance.lean", "def_pos": [50, 5], "def_end_pos": [50, 14]}, {"full_name": "MeasureTheory.lintegral_eq_zero_iff'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [986, 9], "def_end_pos": [986, 31]}]], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\nhX : AEMeasurable X \u03bc\n\u22a2 evariance X \u03bc = 0 \u2194 X =\u1da0[ae \u03bc] fun x => \u222b (x : \u03a9), X x \u2202\u03bc", "state_after": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\nhX : AEMeasurable X \u03bc\n\u22a2 (fun \u03c9 => \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2) =\u1da0[ae \u03bc] 0 \u2194 X =\u1da0[ae \u03bc] fun x => \u222b (x : \u03a9), X x \u2202\u03bc\n\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\nhX : AEMeasurable X \u03bc\n\u22a2 AEMeasurable (fun \u03c9 => \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2) \u03bc"}, {"tactic": "constructor <;> intro hX <;> filter_upwards [hX] with \u03c9 h\u03c9", "annotated_tactic": ["constructor <;> intro hX <;> filter_upwards [hX] with \u03c9 h\u03c9", []], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\nhX : AEMeasurable X \u03bc\n\u22a2 (fun \u03c9 => \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2) =\u1da0[ae \u03bc] 0 \u2194 X =\u1da0[ae \u03bc] fun x => \u222b (x : \u03a9), X x \u2202\u03bc\n\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\nhX : AEMeasurable X \u03bc\n\u22a2 AEMeasurable (fun \u03c9 => \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2) \u03bc", "state_after": "case h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\nhX\u271d : AEMeasurable X \u03bc\nhX : (fun \u03c9 => \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2) =\u1da0[ae \u03bc] 0\n\u03c9 : \u03a9\nh\u03c9 : \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2 = 0 \u03c9\n\u22a2 X \u03c9 = \u222b (x : \u03a9), X x \u2202\u03bc\n\ncase h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\nhX\u271d : AEMeasurable X \u03bc\nhX : X =\u1da0[ae \u03bc] fun x => \u222b (x : \u03a9), X x \u2202\u03bc\n\u03c9 : \u03a9\nh\u03c9 : X \u03c9 = \u222b (x : \u03a9), X x \u2202\u03bc\n\u22a2 \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2 = 0 \u03c9\n\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\nhX : AEMeasurable X \u03bc\n\u22a2 AEMeasurable (fun \u03c9 => \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2) \u03bc"}, {"tactic": "simpa only [Pi.zero_apply, sq_eq_zero_iff, ENNReal.coe_eq_zero, nnnorm_eq_zero, sub_eq_zero]\n  using h\u03c9", "annotated_tactic": ["simpa only [<a>Pi.zero_apply</a>, <a>sq_eq_zero_iff</a>, <a>ENNReal.coe_eq_zero</a>, <a>nnnorm_eq_zero</a>, <a>sub_eq_zero</a>]\n      using h\u03c9", [{"full_name": "Pi.zero_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [62, 3], "def_end_pos": [62, 14]}, {"full_name": "sq_eq_zero_iff", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [206, 7], "def_end_pos": [206, 21]}, {"full_name": "ENNReal.coe_eq_zero", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [390, 28], "def_end_pos": [390, 39]}, {"full_name": "nnnorm_eq_zero", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1571, 30], "def_end_pos": [1571, 44]}, {"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1070, 3], "def_end_pos": [1070, 14]}]], "state_before": "case h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\nhX\u271d : AEMeasurable X \u03bc\nhX : (fun \u03c9 => \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2) =\u1da0[ae \u03bc] 0\n\u03c9 : \u03a9\nh\u03c9 : \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2 = 0 \u03c9\n\u22a2 X \u03c9 = \u222b (x : \u03a9), X x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [h\u03c9]", "annotated_tactic": ["rw [h\u03c9]", []], "state_before": "case h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\nhX\u271d : AEMeasurable X \u03bc\nhX : X =\u1da0[ae \u03bc] fun x => \u222b (x : \u03a9), X x \u2202\u03bc\n\u03c9 : \u03a9\nh\u03c9 : X \u03c9 = \u222b (x : \u03a9), X x \u2202\u03bc\n\u22a2 \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2 = 0 \u03c9", "state_after": "case h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\nhX\u271d : AEMeasurable X \u03bc\nhX : X =\u1da0[ae \u03bc] fun x => \u222b (x : \u03a9), X x \u2202\u03bc\n\u03c9 : \u03a9\nh\u03c9 : X \u03c9 = \u222b (x : \u03a9), X x \u2202\u03bc\n\u22a2 \u2191\u2016\u222b (x : \u03a9), X x \u2202\u03bc - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2 = 0 \u03c9"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\n\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\nhX\u271d : AEMeasurable X \u03bc\nhX : X =\u1da0[ae \u03bc] fun x => \u222b (x : \u03a9), X x \u2202\u03bc\n\u03c9 : \u03a9\nh\u03c9 : X \u03c9 = \u222b (x : \u03a9), X x \u2202\u03bc\n\u22a2 \u2191\u2016\u222b (x : \u03a9), X x \u2202\u03bc - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2 = 0 \u03c9", "state_after": "no goals"}, {"tactic": "exact (hX.sub_const _).ennnorm.pow_const _", "annotated_tactic": ["exact (hX.sub_const _).ennnorm.pow_const _", []], "state_before": "\u03a9 : Type u_1\nm : MeasurableSpace \u03a9\nX : \u03a9 \u2192 \u211d\n\u03bc : Measure \u03a9\nhX : AEMeasurable X \u03bc\n\u22a2 AEMeasurable (fun \u03c9 => \u2191\u2016X \u03c9 - \u222b (x : \u03a9), X x \u2202\u03bc\u2016\u208a ^ 2) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "ContinuousMap.toLp_inj", "start": [1934, 1], "end": [1936, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/GoldenRatio.lean", "full_name": "Real.coe_fib_eq", "start": [233, 1], "end": [234, 47], "traced_tactics": [{"tactic": "rw [\u2190 Function.funext_iff, Real.coe_fib_eq']", "annotated_tactic": ["rw [\u2190 <a>Function.funext_iff</a>, <a>Real.coe_fib_eq'</a>]", [{"full_name": "Function.funext_iff", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [63, 9], "def_end_pos": [63, 28]}, {"full_name": "Real.coe_fib_eq'", "def_path": "Mathlib/Data/Real/GoldenRatio.lean", "def_pos": [210, 9], "def_end_pos": [210, 25]}]], "state_before": "\u22a2 \u2200 (n : \u2115), \u2191(Nat.fib n) = (\u03c6 ^ n - \u03c8 ^ n) / \u221a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Nat.ceil_int", "start": [1722, 1], "end": [1723, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Prime.lean", "full_name": "Nat.minFac_eq_two_iff", "start": [442, 1], "end": [454, 60], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "n\u271d n : \u2115\n\u22a2 n.minFac = 2 \u2194 2 \u2223 n", "state_after": "case mp\nn\u271d n : \u2115\n\u22a2 n.minFac = 2 \u2192 2 \u2223 n\n\ncase mpr\nn\u271d n : \u2115\n\u22a2 2 \u2223 n \u2192 n.minFac = 2"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mp\nn\u271d n : \u2115\n\u22a2 n.minFac = 2 \u2192 2 \u2223 n", "state_after": "case mp\nn\u271d n : \u2115\nh : n.minFac = 2\n\u22a2 2 \u2223 n"}, {"tactic": "rw [\u2190 h]", "annotated_tactic": ["rw [\u2190 h]", []], "state_before": "case mp\nn\u271d n : \u2115\nh : n.minFac = 2\n\u22a2 2 \u2223 n", "state_after": "case mp\nn\u271d n : \u2115\nh : n.minFac = 2\n\u22a2 n.minFac \u2223 n"}, {"tactic": "exact minFac_dvd n", "annotated_tactic": ["exact <a>minFac_dvd</a> n", [{"full_name": "Nat.minFac_dvd", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [338, 9], "def_end_pos": [338, 19]}]], "state_before": "case mp\nn\u271d n : \u2115\nh : n.minFac = 2\n\u22a2 n.minFac \u2223 n", "state_after": "no goals"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mpr\nn\u271d n : \u2115\n\u22a2 2 \u2223 n \u2192 n.minFac = 2", "state_after": "case mpr\nn\u271d n : \u2115\nh : 2 \u2223 n\n\u22a2 n.minFac = 2"}, {"tactic": "have ub := minFac_le_of_dvd (le_refl 2) h", "annotated_tactic": ["have ub := <a>minFac_le_of_dvd</a> (<a>le_refl</a> 2) h", [{"full_name": "Nat.minFac_le_of_dvd", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [347, 9], "def_end_pos": [347, 25]}, {"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [45, 9], "def_end_pos": [45, 16]}]], "state_before": "case mpr\nn\u271d n : \u2115\nh : 2 \u2223 n\n\u22a2 n.minFac = 2", "state_after": "case mpr\nn\u271d n : \u2115\nh : 2 \u2223 n\nub : n.minFac \u2264 2\n\u22a2 n.minFac = 2"}, {"tactic": "have lb := minFac_pos n", "annotated_tactic": ["have lb := <a>minFac_pos</a> n", [{"full_name": "Nat.minFac_pos", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [352, 9], "def_end_pos": [352, 19]}]], "state_before": "case mpr\nn\u271d n : \u2115\nh : 2 \u2223 n\nub : n.minFac \u2264 2\n\u22a2 n.minFac = 2", "state_after": "case mpr\nn\u271d n : \u2115\nh : 2 \u2223 n\nub : n.minFac \u2264 2\nlb : 0 < n.minFac\n\u22a2 n.minFac = 2"}, {"tactic": "refine ub.eq_or_lt.resolve_right fun h' => ?_", "annotated_tactic": ["refine ub.eq_or_lt.resolve_right fun h' => ?_", []], "state_before": "case mpr\nn\u271d n : \u2115\nh : 2 \u2223 n\nub : n.minFac \u2264 2\nlb : 0 < n.minFac\n\u22a2 n.minFac = 2", "state_after": "case mpr\nn\u271d n : \u2115\nh : 2 \u2223 n\nub : n.minFac \u2264 2\nlb : 0 < n.minFac\nh' : n.minFac < 2\n\u22a2 False"}, {"tactic": "have := le_antisymm (Nat.succ_le_of_lt lb) (Nat.lt_succ_iff.mp h')", "annotated_tactic": ["have := <a>le_antisymm</a> (<a>Nat.succ_le_of_lt</a> lb) (Nat.lt_succ_iff.mp h')", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "Nat.succ_le_of_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [348, 9], "def_end_pos": [348, 22]}]], "state_before": "case mpr\nn\u271d n : \u2115\nh : 2 \u2223 n\nub : n.minFac \u2264 2\nlb : 0 < n.minFac\nh' : n.minFac < 2\n\u22a2 False", "state_after": "case mpr\nn\u271d n : \u2115\nh : 2 \u2223 n\nub : n.minFac \u2264 2\nlb : 0 < n.minFac\nh' : n.minFac < 2\nthis : succ 0 = n.minFac\n\u22a2 False"}, {"tactic": "rw [eq_comm, Nat.minFac_eq_one_iff] at this", "annotated_tactic": ["rw [<a>eq_comm</a>, <a>Nat.minFac_eq_one_iff</a>] at this", [{"full_name": "eq_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}, {"full_name": "Nat.minFac_eq_one_iff", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [430, 9], "def_end_pos": [430, 26]}]], "state_before": "case mpr\nn\u271d n : \u2115\nh : 2 \u2223 n\nub : n.minFac \u2264 2\nlb : 0 < n.minFac\nh' : n.minFac < 2\nthis : succ 0 = n.minFac\n\u22a2 False", "state_after": "case mpr\nn\u271d n : \u2115\nh : 2 \u2223 n\nub : n.minFac \u2264 2\nlb : 0 < n.minFac\nh' : n.minFac < 2\nthis : n = 1\n\u22a2 False"}, {"tactic": "subst this", "annotated_tactic": ["subst this", []], "state_before": "case mpr\nn\u271d n : \u2115\nh : 2 \u2223 n\nub : n.minFac \u2264 2\nlb : 0 < n.minFac\nh' : n.minFac < 2\nthis : n = 1\n\u22a2 False", "state_after": "case mpr\nn : \u2115\nh : 2 \u2223 1\nub : minFac 1 \u2264 2\nlb : 0 < minFac 1\nh' : minFac 1 < 2\n\u22a2 False"}, {"tactic": "exact not_lt_of_le (le_of_dvd zero_lt_one h) one_lt_two", "annotated_tactic": ["exact <a>not_lt_of_le</a> (<a>le_of_dvd</a> <a>zero_lt_one</a> h) <a>one_lt_two</a>", [{"full_name": "not_lt_of_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [343, 9], "def_end_pos": [343, 21]}, {"full_name": "Nat.le_of_dvd", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Dvd.lean", "def_pos": [46, 9], "def_end_pos": [46, 18]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}, {"full_name": "one_lt_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [106, 7], "def_end_pos": [106, 17]}]], "state_before": "case mpr\nn : \u2115\nh : 2 \u2223 1\nub : minFac 1 \u2264 2\nlb : 0 < minFac 1\nh' : minFac 1 < 2\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/EReal.lean", "full_name": "EReal.coe_toReal", "start": [411, 1], "end": [413, 6], "traced_tactics": [{"tactic": "lift x to \u211d using \u27e8hx, h'x\u27e9", "annotated_tactic": ["lift x to \u211d using \u27e8hx, h'x\u27e9", []], "state_before": "x : EReal\nhx : x \u2260 \u22a4\nh'x : x \u2260 \u22a5\n\u22a2 \u2191x.toReal = x", "state_after": "case intro\nx : \u211d\nhx : \u2191x \u2260 \u22a4\nh'x : \u2191x \u2260 \u22a5\n\u22a2 \u2191(\u2191x).toReal = \u2191x"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case intro\nx : \u211d\nhx : \u2191x \u2260 \u22a4\nh'x : \u2191x \u2260 \u22a5\n\u22a2 \u2191(\u2191x).toReal = \u2191x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean", "full_name": "CategoryTheory.ShortComplex.opcyclesMap_sub", "start": [275, 1], "end": [277, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Homotopy/HomotopyGroup.lean", "full_name": "GenLoop.continuous_toLoop", "start": [209, 1], "end": [217, 8], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Holor.lean", "full_name": "Holor.holor_index_cons_decomp", "start": [195, 1], "end": [199, 36], "traced_tactics": [{"tactic": "rw [\u2190 h]", "annotated_tactic": ["rw [\u2190 h]", []], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\np : HolorIndex (d :: ds) \u2192 Prop\nt : HolorIndex (d :: ds)\ni : \u2115\nis : List \u2115\nh : \u2191t = i :: is\n\u22a2 Forall\u2082 (fun x x_1 => x < x_1) (i :: is) (d :: ds)", "state_after": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\np : HolorIndex (d :: ds) \u2192 Prop\nt : HolorIndex (d :: ds)\ni : \u2115\nis : List \u2115\nh : \u2191t = i :: is\n\u22a2 Forall\u2082 (fun x x_1 => x < x_1) (\u2191t) (d :: ds)"}, {"tactic": "exact t.2", "annotated_tactic": ["exact t.2", []], "state_before": "\u03b1 : Type\nd : \u2115\nds ds\u2081 ds\u2082 ds\u2083 : List \u2115\np : HolorIndex (d :: ds) \u2192 Prop\nt : HolorIndex (d :: ds)\ni : \u2115\nis : List \u2115\nh : \u2191t = i :: is\n\u22a2 Forall\u2082 (fun x x_1 => x < x_1) (\u2191t) (d :: ds)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/ConditionalProbability.lean", "full_name": "ProbabilityTheory.cond_inter_self", "start": [150, 1], "end": [151, 79], "traced_tactics": [{"tactic": "rw [cond_apply _ hms, \u2190 Set.inter_assoc, Set.inter_self, \u2190 cond_apply _ hms]", "annotated_tactic": ["rw [<a>cond_apply</a> _ hms, \u2190 <a>Set.inter_assoc</a>, <a>Set.inter_self</a>, \u2190 <a>cond_apply</a> _ hms]", [{"full_name": "ProbabilityTheory.cond_apply", "def_path": "Mathlib/Probability/ConditionalProbability.lean", "def_pos": [143, 9], "def_end_pos": [143, 19]}, {"full_name": "Set.inter_assoc", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 20]}, {"full_name": "Set.inter_self", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [891, 9], "def_end_pos": [891, 19]}, {"full_name": "ProbabilityTheory.cond_apply", "def_path": "Mathlib/Probability/ConditionalProbability.lean", "def_pos": [143, 9], "def_end_pos": [143, 19]}]], "state_before": "\u03a9 : Type u_1\n\u03a9' : Type u_2\n\u03b1 : Type u_3\nm : MeasurableSpace \u03a9\nm' : MeasurableSpace \u03a9'\n\u03bc : Measure \u03a9\ns t\u271d : Set \u03a9\nhms : MeasurableSet s\nt : Set \u03a9\n\u22a2 \u03bc[|s] (s \u2229 t) = \u03bc[|s] t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/IsAdjoinRoot.lean", "full_name": "AdjoinRoot.isAdjoinRootMonic_root_eq_root", "start": [345, 1], "end": [347, 89], "traced_tactics": [{"tactic": "simp only [IsAdjoinRoot.root, AdjoinRoot.root, AdjoinRoot.isAdjoinRootMonic_map_eq_mk]", "annotated_tactic": ["simp only [<a>IsAdjoinRoot.root</a>, <a>AdjoinRoot.root</a>, <a>AdjoinRoot.isAdjoinRootMonic_map_eq_mk</a>]", [{"full_name": "IsAdjoinRoot.root", "def_path": "Mathlib/RingTheory/IsAdjoinRoot.lean", "def_pos": [119, 5], "def_end_pos": [119, 9]}, {"full_name": "AdjoinRoot.root", "def_path": "Mathlib/RingTheory/AdjoinRoot.lean", "def_pos": [169, 5], "def_end_pos": [169, 9]}, {"full_name": "AdjoinRoot.isAdjoinRootMonic_map_eq_mk", "def_path": "Mathlib/RingTheory/IsAdjoinRoot.lean", "def_pos": [334, 9], "def_end_pos": [334, 36]}]], "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : Ring S\nf : R[X]\ninst\u271d : Algebra R S\nhf : f.Monic\n\u22a2 (AdjoinRoot.isAdjoinRootMonic f hf).root = root f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/ContMDiff/Product.lean", "full_name": "contMDiff_prod_iff", "start": [301, 1], "end": [304, 65], "traced_tactics": [{"tactic": "convert h.1.prod_mk h.2", "annotated_tactic": ["convert h.1.<a>prod_mk</a> h.2", [{"full_name": "ContMDiff.prod_mk", "def_path": "Mathlib/Geometry/Manifold/ContMDiff/Product.lean", "def_pos": [94, 16], "def_end_pos": [94, 33]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b3\u2076 : NormedAddCommGroup E\ninst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b3\u2074 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b3\u00b3 : TopologicalSpace M\ninst\u271d\u00b3\u00b2 : ChartedSpace H M\ninst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M\nE' : Type u_5\ninst\u271d\u00b3\u2070 : NormedAddCommGroup E'\ninst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u00b2\u2078 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u00b2\u2077 : TopologicalSpace M'\ninst\u271d\u00b2\u2076 : ChartedSpace H' M'\ninst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M'\nE'' : Type u_8\ninst\u271d\u00b2\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b2\u00b9 : TopologicalSpace M''\ninst\u271d\u00b2\u2070 : ChartedSpace H'' M''\nF : Type u_11\ninst\u271d\u00b9\u2079 : NormedAddCommGroup F\ninst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F\nG : Type u_12\ninst\u271d\u00b9\u2077 : TopologicalSpace G\nJ : ModelWithCorners \ud835\udd5c F G\nN : Type u_13\ninst\u271d\u00b9\u2076 : TopologicalSpace N\ninst\u271d\u00b9\u2075 : ChartedSpace G N\ninst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N\nF' : Type u_14\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup F'\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F'\nG' : Type u_15\ninst\u271d\u00b9\u00b9 : TopologicalSpace G'\nJ' : ModelWithCorners \ud835\udd5c F' G'\nN' : Type u_16\ninst\u271d\u00b9\u2070 : TopologicalSpace N'\ninst\u271d\u2079 : ChartedSpace G' N'\ninst\u271d\u2078 : SmoothManifoldWithCorners J' N'\nF\u2081 : Type u_17\ninst\u271d\u2077 : NormedAddCommGroup F\u2081\ninst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081\nF\u2082 : Type u_18\ninst\u271d\u2075 : NormedAddCommGroup F\u2082\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082\nF\u2083 : Type u_19\ninst\u271d\u00b3 : NormedAddCommGroup F\u2083\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083\nF\u2084 : Type u_20\ninst\u271d\u00b9 : NormedAddCommGroup F\u2084\ninst\u271d : NormedSpace \ud835\udd5c F\u2084\ne : PartialHomeomorph M H\ne' : PartialHomeomorph M' H'\nf\u271d f\u2081 : M \u2192 M'\ns s\u2081 t : Set M\nx : M\nm n : \u2115\u221e\nf : M \u2192 M' \u00d7 N'\nh : ContMDiff I I' n (Prod.fst \u2218 f) \u2227 ContMDiff I J' n (Prod.snd \u2218 f)\n\u22a2 ContMDiff I (I'.prod J') n f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/HomologicalComplex.lean", "full_name": "CochainComplex.mk_X_0", "start": [1027, 1], "end": [1028, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean", "full_name": "Real.GammaIntegral_convergent", "start": [71, 1], "end": [82, 63], "traced_tactics": [{"tactic": "rw [\u2190 Ioc_union_Ioi_eq_Ioi (@zero_le_one \u211d _ _ _ _), integrableOn_union]", "annotated_tactic": ["rw [\u2190 <a>Ioc_union_Ioi_eq_Ioi</a> (@<a>zero_le_one</a> \u211d _ _ _ _), <a>integrableOn_union</a>]", [{"full_name": "Set.Ioc_union_Ioi_eq_Ioi", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [1338, 9], "def_end_pos": [1338, 29]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}, {"full_name": "MeasureTheory.integrableOn_union", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [191, 9], "def_end_pos": [191, 27]}]], "state_before": "s : \u211d\nh : 0 < s\n\u22a2 IntegrableOn (fun x => rexp (-x) * x ^ (s - 1)) (Ioi 0) volume", "state_after": "s : \u211d\nh : 0 < s\n\u22a2 IntegrableOn (fun x => rexp (-x) * x ^ (s - 1)) (Ioc 0 1) volume \u2227\n    IntegrableOn (fun x => rexp (-x) * x ^ (s - 1)) (Ioi 1) volume"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "s : \u211d\nh : 0 < s\n\u22a2 IntegrableOn (fun x => rexp (-x) * x ^ (s - 1)) (Ioc 0 1) volume \u2227\n    IntegrableOn (fun x => rexp (-x) * x ^ (s - 1)) (Ioi 1) volume", "state_after": "case left\ns : \u211d\nh : 0 < s\n\u22a2 IntegrableOn (fun x => rexp (-x) * x ^ (s - 1)) (Ioc 0 1) volume\n\ncase right\ns : \u211d\nh : 0 < s\n\u22a2 IntegrableOn (fun x => rexp (-x) * x ^ (s - 1)) (Ioi 1) volume"}, {"tactic": "rw [\u2190 integrableOn_Icc_iff_integrableOn_Ioc]", "annotated_tactic": ["rw [\u2190 <a>integrableOn_Icc_iff_integrableOn_Ioc</a>]", [{"full_name": "integrableOn_Icc_iff_integrableOn_Ioc", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [757, 9], "def_end_pos": [757, 46]}]], "state_before": "case left\ns : \u211d\nh : 0 < s\n\u22a2 IntegrableOn (fun x => rexp (-x) * x ^ (s - 1)) (Ioc 0 1) volume", "state_after": "case left\ns : \u211d\nh : 0 < s\n\u22a2 IntegrableOn (fun x => rexp (-x) * x ^ (s - 1)) (Icc 0 1) volume"}, {"tactic": "refine IntegrableOn.continuousOn_mul continuousOn_id.neg.rexp ?_ isCompact_Icc", "annotated_tactic": ["refine <a>IntegrableOn.continuousOn_mul</a> continuousOn_id.neg.rexp ?_ <a>isCompact_Icc</a>", [{"full_name": "MeasureTheory.IntegrableOn.continuousOn_mul", "def_path": "Mathlib/MeasureTheory/Function/LocallyIntegrable.lean", "def_pos": [631, 9], "def_end_pos": [631, 38]}, {"full_name": "CompactIccSpace.isCompact_Icc", "def_path": "Mathlib/Topology/Algebra/Order/Compact.lean", "def_pos": [52, 3], "def_end_pos": [52, 16]}]], "state_before": "case left\ns : \u211d\nh : 0 < s\n\u22a2 IntegrableOn (fun x => rexp (-x) * x ^ (s - 1)) (Icc 0 1) volume", "state_after": "case left\ns : \u211d\nh : 0 < s\n\u22a2 IntegrableOn (fun x => x ^ (s - 1)) (Icc 0 1) volume"}, {"tactic": "refine (intervalIntegrable_iff_integrableOn_Icc_of_le zero_le_one).mp ?_", "annotated_tactic": ["refine (<a>intervalIntegrable_iff_integrableOn_Icc_of_le</a> <a>zero_le_one</a>).<a>mp</a> ?_", [{"full_name": "intervalIntegrable_iff_integrableOn_Icc_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [103, 9], "def_end_pos": [103, 54]}, {"full_name": "zero_le_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [26, 15], "def_end_pos": [26, 26]}, {"full_name": "Iff.mp", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [118, 3], "def_end_pos": [118, 5]}]], "state_before": "case left\ns : \u211d\nh : 0 < s\n\u22a2 IntegrableOn (fun x => x ^ (s - 1)) (Icc 0 1) volume", "state_after": "case left\ns : \u211d\nh : 0 < s\n\u22a2 IntervalIntegrable (fun x => x ^ (s - 1)) volume 0 1"}, {"tactic": "exact intervalIntegrable_rpow' (by linarith)", "annotated_tactic": ["exact <a>intervalIntegrable_rpow'</a> (by linarith)", [{"full_name": "intervalIntegral.intervalIntegrable_rpow'", "def_path": "Mathlib/Analysis/SpecialFunctions/Integrals.lean", "def_pos": [73, 9], "def_end_pos": [73, 33]}]], "state_before": "case left\ns : \u211d\nh : 0 < s\n\u22a2 IntervalIntegrable (fun x => x ^ (s - 1)) volume 0 1", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "s : \u211d\nh : 0 < s\n\u22a2 -1 < s - 1", "state_after": "no goals"}, {"tactic": "refine integrable_of_isBigO_exp_neg one_half_pos ?_ (Gamma_integrand_isLittleO _).isBigO", "annotated_tactic": ["refine <a>integrable_of_isBigO_exp_neg</a> <a>one_half_pos</a> ?_ (<a>Gamma_integrand_isLittleO</a> _).<a>isBigO</a>", [{"full_name": "integrable_of_isBigO_exp_neg", "def_path": "Mathlib/MeasureTheory/Integral/ExpDecay.lean", "def_pos": [41, 9], "def_end_pos": [41, 37]}, {"full_name": "one_half_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [449, 9], "def_end_pos": [449, 21]}, {"full_name": "Real.Gamma_integrand_isLittleO", "def_path": "Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean", "def_pos": [56, 9], "def_end_pos": [56, 34]}, {"full_name": "Asymptotics.IsLittleO.isBigO", "def_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "def_pos": [217, 9], "def_end_pos": [217, 25]}]], "state_before": "case right\ns : \u211d\nh : 0 < s\n\u22a2 IntegrableOn (fun x => rexp (-x) * x ^ (s - 1)) (Ioi 1) volume", "state_after": "case right\ns : \u211d\nh : 0 < s\n\u22a2 ContinuousOn (fun x => rexp (-x) * x ^ (s - 1)) (Ici 1)"}, {"tactic": "refine continuousOn_id.neg.rexp.mul (continuousOn_id.rpow_const ?_)", "annotated_tactic": ["refine continuousOn_id.neg.rexp.mul (continuousOn_id.rpow_const ?_)", []], "state_before": "case right\ns : \u211d\nh : 0 < s\n\u22a2 ContinuousOn (fun x => rexp (-x) * x ^ (s - 1)) (Ici 1)", "state_after": "case right\ns : \u211d\nh : 0 < s\n\u22a2 \u2200 x \u2208 Ici 1, id x \u2260 0 \u2228 0 \u2264 s - 1"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case right\ns : \u211d\nh : 0 < s\n\u22a2 \u2200 x \u2208 Ici 1, id x \u2260 0 \u2228 0 \u2264 s - 1", "state_after": "case right\ns : \u211d\nh : 0 < s\nx : \u211d\nhx : x \u2208 Ici 1\n\u22a2 id x \u2260 0 \u2228 0 \u2264 s - 1"}, {"tactic": "exact Or.inl ((zero_lt_one : (0 : \u211d) < 1).trans_le hx).ne'", "annotated_tactic": ["exact <a>Or.inl</a> ((<a>zero_lt_one</a> : (0 : \u211d) < 1).<a>trans_le</a> hx).<a>ne'</a>", [{"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [143, 7], "def_end_pos": [143, 21]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}]], "state_before": "case right\ns : \u211d\nh : 0 < s\nx : \u211d\nhx : x \u2208 Ici 1\n\u22a2 id x \u2260 0 \u2228 0 \u2264 s - 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Degree/Lemmas.lean", "full_name": "Polynomial.natDegree_iterate_comp", "start": [406, 1], "end": [410, 80], "traced_tactics": [{"tactic": "induction' k with k IH", "annotated_tactic": ["induction' k with k IH", []], "state_before": "R : Type u\nS : Type v\n\u03b9 : Type w\na\u271d b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\na : R\nk : \u2115\n\u22a2 (p.comp^[k] q).natDegree = p.natDegree ^ k * q.natDegree", "state_after": "case zero\nR : Type u\nS : Type v\n\u03b9 : Type w\na\u271d b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\na : R\n\u22a2 (p.comp^[0] q).natDegree = p.natDegree ^ 0 * q.natDegree\n\ncase succ\nR : Type u\nS : Type v\n\u03b9 : Type w\na\u271d b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\na : R\nk : \u2115\nIH : (p.comp^[k] q).natDegree = p.natDegree ^ k * q.natDegree\n\u22a2 (p.comp^[k + 1] q).natDegree = p.natDegree ^ (k + 1) * q.natDegree"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case zero\nR : Type u\nS : Type v\n\u03b9 : Type w\na\u271d b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\na : R\n\u22a2 (p.comp^[0] q).natDegree = p.natDegree ^ 0 * q.natDegree", "state_after": "no goals"}, {"tactic": "rw [Function.iterate_succ_apply', natDegree_comp, IH, pow_succ', mul_assoc]", "annotated_tactic": ["rw [<a>Function.iterate_succ_apply'</a>, <a>natDegree_comp</a>, IH, <a>pow_succ'</a>, <a>mul_assoc</a>]", [{"full_name": "Function.iterate_succ_apply'", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [191, 9], "def_end_pos": [191, 28]}, {"full_name": "Polynomial.natDegree_comp", "def_path": "Mathlib/Algebra/Polynomial/Degree/Lemmas.lean", "def_pos": [394, 9], "def_end_pos": [394, 23]}, {"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [667, 34], "def_end_pos": [667, 43]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "case succ\nR : Type u\nS : Type v\n\u03b9 : Type w\na\u271d b : R\nm n : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\na : R\nk : \u2115\nIH : (p.comp^[k] q).natDegree = p.natDegree ^ k * q.natDegree\n\u22a2 (p.comp^[k + 1] q).natDegree = p.natDegree ^ (k + 1) * q.natDegree", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Weights/Cartan.lean", "full_name": "LieAlgebra.toEnd_pow_apply_mem", "start": [72, 1], "end": [81, 51], "traced_tactics": [{"tactic": "induction n with\n| zero => simpa using hm\n| succ n IH =>\n  simp only [pow_succ', LinearMap.mul_apply, toEnd_apply_apply,\n    Nat.cast_add, Nat.cast_one, rootSpace]\n  convert lie_mem_weightSpace_of_mem_weightSpace hx IH using 2\n  rw [succ_nsmul, \u2190 add_assoc, add_comm (n \u2022 _)]", "annotated_tactic": ["induction n with\n  | zero => simpa using hm\n  | succ n IH =>\n    simp only [<a>pow_succ'</a>, <a>LinearMap.mul_apply</a>, <a>toEnd_apply_apply</a>,\n      <a>Nat.cast_add</a>, <a>Nat.cast_one</a>, <a>rootSpace</a>]\n    convert <a>lie_mem_weightSpace_of_mem_weightSpace</a> hx IH using 2\n    rw [<a>succ_nsmul</a>, \u2190 <a>add_assoc</a>, <a>add_comm</a> (n \u2022 _)]", [{"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [667, 34], "def_end_pos": [667, 43]}, {"full_name": "LinearMap.mul_apply", "def_path": "Mathlib/Algebra/Module/LinearMap/End.lean", "def_pos": [62, 9], "def_end_pos": [62, 18]}, {"full_name": "LieModule.toEnd_apply_apply", "def_path": "Mathlib/Algebra/Lie/OfAssociative.lean", "def_pos": [195, 3], "def_end_pos": [195, 8]}, {"full_name": "Nat.cast_add", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 17]}, {"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [154, 9], "def_end_pos": [154, 17]}, {"full_name": "LieAlgebra.rootSpace", "def_path": "Mathlib/Algebra/Lie/Weights/Cartan.lean", "def_pos": [44, 8], "def_end_pos": [44, 17]}, {"full_name": "LieAlgebra.lie_mem_weightSpace_of_mem_weightSpace", "def_path": "Mathlib/Algebra/Lie/Weights/Cartan.lean", "def_pos": [61, 9], "def_end_pos": [61, 47]}, {"full_name": "succ_nsmul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [656, 15], "def_end_pos": [656, 25]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [288, 3], "def_end_pos": [288, 14]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "R : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 : \u21a5H \u2192 R\nx : L\nm : M\nhx : x \u2208 rootSpace H \u03c7\u2081\nhm : m \u2208 weightSpace M \u03c7\u2082\nn : \u2115\n\u22a2 ((toEnd R L M) x ^ n) m \u2208 weightSpace M (n \u2022 \u03c7\u2081 + \u03c7\u2082)", "state_after": "no goals"}, {"tactic": "simpa using hm", "annotated_tactic": ["simpa using hm", []], "state_before": "case zero\nR : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 : \u21a5H \u2192 R\nx : L\nm : M\nhx : x \u2208 rootSpace H \u03c7\u2081\nhm : m \u2208 weightSpace M \u03c7\u2082\n\u22a2 ((toEnd R L M) x ^ 0) m \u2208 weightSpace M (0 \u2022 \u03c7\u2081 + \u03c7\u2082)", "state_after": "no goals"}, {"tactic": "simp only [pow_succ', LinearMap.mul_apply, toEnd_apply_apply,\n  Nat.cast_add, Nat.cast_one, rootSpace]", "annotated_tactic": ["simp only [<a>pow_succ'</a>, <a>LinearMap.mul_apply</a>, <a>toEnd_apply_apply</a>,\n      <a>Nat.cast_add</a>, <a>Nat.cast_one</a>, <a>rootSpace</a>]", [{"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [667, 34], "def_end_pos": [667, 43]}, {"full_name": "LinearMap.mul_apply", "def_path": "Mathlib/Algebra/Module/LinearMap/End.lean", "def_pos": [62, 9], "def_end_pos": [62, 18]}, {"full_name": "LieModule.toEnd_apply_apply", "def_path": "Mathlib/Algebra/Lie/OfAssociative.lean", "def_pos": [195, 3], "def_end_pos": [195, 8]}, {"full_name": "Nat.cast_add", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 17]}, {"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [154, 9], "def_end_pos": [154, 17]}, {"full_name": "LieAlgebra.rootSpace", "def_path": "Mathlib/Algebra/Lie/Weights/Cartan.lean", "def_pos": [44, 8], "def_end_pos": [44, 17]}]], "state_before": "case succ\nR : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 : \u21a5H \u2192 R\nx : L\nm : M\nhx : x \u2208 rootSpace H \u03c7\u2081\nhm : m \u2208 weightSpace M \u03c7\u2082\nn : \u2115\nIH : ((toEnd R L M) x ^ n) m \u2208 weightSpace M (n \u2022 \u03c7\u2081 + \u03c7\u2082)\n\u22a2 ((toEnd R L M) x ^ (n + 1)) m \u2208 weightSpace M ((n + 1) \u2022 \u03c7\u2081 + \u03c7\u2082)", "state_after": "case succ\nR : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 : \u21a5H \u2192 R\nx : L\nm : M\nhx : x \u2208 rootSpace H \u03c7\u2081\nhm : m \u2208 weightSpace M \u03c7\u2082\nn : \u2115\nIH : ((toEnd R L M) x ^ n) m \u2208 weightSpace M (n \u2022 \u03c7\u2081 + \u03c7\u2082)\n\u22a2 \u2045x, ((toEnd R L M) x ^ n) m\u2046 \u2208 weightSpace M ((n + 1) \u2022 \u03c7\u2081 + \u03c7\u2082)"}, {"tactic": "convert lie_mem_weightSpace_of_mem_weightSpace hx IH using 2", "annotated_tactic": ["convert <a>lie_mem_weightSpace_of_mem_weightSpace</a> hx IH using 2", [{"full_name": "LieAlgebra.lie_mem_weightSpace_of_mem_weightSpace", "def_path": "Mathlib/Algebra/Lie/Weights/Cartan.lean", "def_pos": [61, 9], "def_end_pos": [61, 47]}]], "state_before": "case succ\nR : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 : \u21a5H \u2192 R\nx : L\nm : M\nhx : x \u2208 rootSpace H \u03c7\u2081\nhm : m \u2208 weightSpace M \u03c7\u2082\nn : \u2115\nIH : ((toEnd R L M) x ^ n) m \u2208 weightSpace M (n \u2022 \u03c7\u2081 + \u03c7\u2082)\n\u22a2 \u2045x, ((toEnd R L M) x ^ n) m\u2046 \u2208 weightSpace M ((n + 1) \u2022 \u03c7\u2081 + \u03c7\u2082)", "state_after": "case h.e'_5.h.e'_12\nR : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 : \u21a5H \u2192 R\nx : L\nm : M\nhx : x \u2208 rootSpace H \u03c7\u2081\nhm : m \u2208 weightSpace M \u03c7\u2082\nn : \u2115\nIH : ((toEnd R L M) x ^ n) m \u2208 weightSpace M (n \u2022 \u03c7\u2081 + \u03c7\u2082)\n\u22a2 (n + 1) \u2022 \u03c7\u2081 + \u03c7\u2082 = \u03c7\u2081 + (n \u2022 \u03c7\u2081 + \u03c7\u2082)"}, {"tactic": "rw [succ_nsmul, \u2190 add_assoc, add_comm (n \u2022 _)]", "annotated_tactic": ["rw [<a>succ_nsmul</a>, \u2190 <a>add_assoc</a>, <a>add_comm</a> (n \u2022 _)]", [{"full_name": "succ_nsmul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [656, 15], "def_end_pos": [656, 25]}, {"full_name": "add_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [288, 3], "def_end_pos": [288, 14]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "case h.e'_5.h.e'_12\nR : Type u_1\nL : Type u_2\ninst\u271d\u2077 : CommRing R\ninst\u271d\u2076 : LieRing L\ninst\u271d\u2075 : LieAlgebra R L\nH : LieSubalgebra R L\ninst\u271d\u2074 : IsNilpotent R \u21a5H\nM : Type u_3\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : LieRingModule L M\ninst\u271d : LieModule R L M\n\u03c7\u2081 \u03c7\u2082 : \u21a5H \u2192 R\nx : L\nm : M\nhx : x \u2208 rootSpace H \u03c7\u2081\nhm : m \u2208 weightSpace M \u03c7\u2082\nn : \u2115\nIH : ((toEnd R L M) x ^ n) m \u2208 weightSpace M (n \u2022 \u03c7\u2081 + \u03c7\u2082)\n\u22a2 (n + 1) \u2022 \u03c7\u2081 + \u03c7\u2082 = \u03c7\u2081 + (n \u2022 \u03c7\u2081 + \u03c7\u2082)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Skeletal.lean", "full_name": "CategoryTheory.skeleton_isSkeleton", "start": [114, 1], "end": [117, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/BumpFunction.lean", "full_name": "SmoothBumpFunction.smoothAt", "start": [324, 11], "end": [325, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "continuous_of_const", "start": [1684, 1], "end": [1686, 76], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Exponential.lean", "full_name": "hasStrictFDerivAt_exp_zero_of_radius_pos", "start": [67, 1], "end": [72, 43], "traced_tactics": [{"tactic": "convert (hasFPowerSeriesAt_exp_zero_of_radius_pos h).hasStrictFDerivAt", "annotated_tactic": ["convert (<a>hasFPowerSeriesAt_exp_zero_of_radius_pos</a> h).<a>hasStrictFDerivAt</a>", [{"full_name": "NormedSpace.hasFPowerSeriesAt_exp_zero_of_radius_pos", "def_path": "Mathlib/Analysis/NormedSpace/Exponential.lean", "def_pos": [268, 9], "def_end_pos": [268, 49]}, {"full_name": "HasFPowerSeriesAt.hasStrictFDerivAt", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Analytic.lean", "def_pos": [39, 9], "def_end_pos": [39, 44]}]], "state_before": "\ud835\udd42 : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd42\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nh : 0 < (expSeries \ud835\udd42 \ud835\udd38).radius\n\u22a2 HasStrictFDerivAt (exp \ud835\udd42) 1 0", "state_after": "case h.e'_10\n\ud835\udd42 : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd42\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nh : 0 < (expSeries \ud835\udd42 \ud835\udd38).radius\n\u22a2 1 = (continuousMultilinearCurryFin1 \ud835\udd42 \ud835\udd38 \ud835\udd38) (expSeries \ud835\udd42 \ud835\udd38 1)"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case h.e'_10\n\ud835\udd42 : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd42\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nh : 0 < (expSeries \ud835\udd42 \ud835\udd38).radius\n\u22a2 1 = (continuousMultilinearCurryFin1 \ud835\udd42 \ud835\udd38 \ud835\udd38) (expSeries \ud835\udd42 \ud835\udd38 1)", "state_after": "case h.e'_10.h\n\ud835\udd42 : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd42\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nh : 0 < (expSeries \ud835\udd42 \ud835\udd38).radius\nx : \ud835\udd38\n\u22a2 1 x = ((continuousMultilinearCurryFin1 \ud835\udd42 \ud835\udd38 \ud835\udd38) (expSeries \ud835\udd42 \ud835\udd38 1)) x"}, {"tactic": "change x = expSeries \ud835\udd42 \ud835\udd38 1 fun _ => x", "annotated_tactic": ["change x = <a>expSeries</a> \ud835\udd42 \ud835\udd38 1 fun _ => x", [{"full_name": "NormedSpace.expSeries", "def_path": "Mathlib/Analysis/NormedSpace/Exponential.lean", "def_pos": [101, 5], "def_end_pos": [101, 14]}]], "state_before": "case h.e'_10.h\n\ud835\udd42 : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd42\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nh : 0 < (expSeries \ud835\udd42 \ud835\udd38).radius\nx : \ud835\udd38\n\u22a2 1 x = ((continuousMultilinearCurryFin1 \ud835\udd42 \ud835\udd38 \ud835\udd38) (expSeries \ud835\udd42 \ud835\udd38 1)) x", "state_after": "case h.e'_10.h\n\ud835\udd42 : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd42\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nh : 0 < (expSeries \ud835\udd42 \ud835\udd38).radius\nx : \ud835\udd38\n\u22a2 x = (expSeries \ud835\udd42 \ud835\udd38 1) fun x_1 => x"}, {"tactic": "simp [expSeries_apply_eq, Nat.factorial]", "annotated_tactic": ["simp [<a>expSeries_apply_eq</a>, <a>Nat.factorial</a>]", [{"full_name": "NormedSpace.expSeries_apply_eq", "def_path": "Mathlib/Analysis/NormedSpace/Exponential.lean", "def_pos": [119, 9], "def_end_pos": [119, 27]}, {"full_name": "Nat.factorial", "def_path": "Mathlib/Data/Nat/Factorial/Basic.lean", "def_pos": [31, 5], "def_end_pos": [31, 14]}]], "state_before": "case h.e'_10.h\n\ud835\udd42 : Type u_1\n\ud835\udd38 : Type u_2\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd42\ninst\u271d\u00b2 : NormedRing \ud835\udd38\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd38\ninst\u271d : CompleteSpace \ud835\udd38\nh : 0 < (expSeries \ud835\udd42 \ud835\udd38).radius\nx : \ud835\udd38\n\u22a2 x = (expSeries \ud835\udd42 \ud835\udd38 1) fun x_1 => x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "closure_inter_subset_inter_closure", "start": [455, 1], "end": [457, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Abelian.lean", "full_name": "LieAlgebra.isLieAbelian_iff_center_eq_top", "start": [279, 1], "end": [280, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Sigma.lean", "full_name": "Set.image_sigmaMk_subset_sigma_left", "start": [251, 1], "end": [253, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean", "full_name": "HomologicalComplex.pOpcycles_opcyclesIsoSc'_inv", "start": [759, 1], "end": [765, 6], "traced_tactics": [{"tactic": "dsimp [opcyclesIsoSc']", "annotated_tactic": ["dsimp [<a>opcyclesIsoSc'</a>]", [{"full_name": "HomologicalComplex.opcyclesIsoSc'", "def_path": "Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean", "def_pos": [756, 19], "def_end_pos": [756, 33]}]], "state_before": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_3, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\n\u03b9 : Type u_2\nc : ComplexShape \u03b9\nK : HomologicalComplex C c\ni j k : \u03b9\nhi : c.prev j = i\nhk : c.next j = k\ninst\u271d\u00b9 : K.HasHomology j\ninst\u271d : (K.sc' i j k).HasHomology\n\u22a2 (K.sc' i j k).pOpcycles \u226b (K.opcyclesIsoSc' i j k hi hk).inv = K.pOpcycles j", "state_after": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_3, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\n\u03b9 : Type u_2\nc : ComplexShape \u03b9\nK : HomologicalComplex C c\ni j k : \u03b9\nhi : c.prev j = i\nhk : c.next j = k\ninst\u271d\u00b9 : K.HasHomology j\ninst\u271d : (K.sc' i j k).HasHomology\n\u22a2 (K.sc' i j k).pOpcycles \u226b ShortComplex.opcyclesMap ((natIsoSc' C c i j k hi hk).inv.app K) = K.pOpcycles j"}, {"tactic": "simp only [ShortComplex.p_opcyclesMap, shortComplexFunctor'_obj_X\u2082, shortComplexFunctor_obj_X\u2082,\n  natIsoSc'_inv_app_\u03c4\u2082, id_comp]", "annotated_tactic": ["simp only [<a>ShortComplex.p_opcyclesMap</a>, <a>shortComplexFunctor'_obj_X\u2082</a>, <a>shortComplexFunctor_obj_X\u2082</a>,\n    <a>natIsoSc'_inv_app_\u03c4\u2082</a>, <a>id_comp</a>]", [{"full_name": "CategoryTheory.ShortComplex.p_opcyclesMap", "def_path": "Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean", "def_pos": [618, 7], "def_end_pos": [618, 20]}, {"full_name": "HomologicalComplex.shortComplexFunctor'_obj_X\u2082", "def_path": "Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean", "def_pos": [33, 3], "def_end_pos": [33, 8]}, {"full_name": "HomologicalComplex.shortComplexFunctor_obj_X\u2082", "def_path": "Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean", "def_pos": [43, 3], "def_end_pos": [43, 9]}, {"full_name": "HomologicalComplex.natIsoSc'_inv_app_\u03c4\u2082", "def_path": "Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean", "def_pos": [49, 3], "def_end_pos": [49, 9]}, {"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [177, 3], "def_end_pos": [177, 10]}]], "state_before": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_3, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\n\u03b9 : Type u_2\nc : ComplexShape \u03b9\nK : HomologicalComplex C c\ni j k : \u03b9\nhi : c.prev j = i\nhk : c.next j = k\ninst\u271d\u00b9 : K.HasHomology j\ninst\u271d : (K.sc' i j k).HasHomology\n\u22a2 (K.sc' i j k).pOpcycles \u226b ShortComplex.opcyclesMap ((natIsoSc' C c i j k hi hk).inv.app K) = K.pOpcycles j", "state_after": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_3, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\n\u03b9 : Type u_2\nc : ComplexShape \u03b9\nK : HomologicalComplex C c\ni j k : \u03b9\nhi : c.prev j = i\nhk : c.next j = k\ninst\u271d\u00b9 : K.HasHomology j\ninst\u271d : (K.sc' i j k).HasHomology\n\u22a2 (K.sc j).pOpcycles = K.pOpcycles j"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_3, u_1} C\ninst\u271d\u00b2 : HasZeroMorphisms C\n\u03b9 : Type u_2\nc : ComplexShape \u03b9\nK : HomologicalComplex C c\ni j k : \u03b9\nhi : c.prev j = i\nhk : c.next j = k\ninst\u271d\u00b9 : K.HasHomology j\ninst\u271d : (K.sc' i j k).HasHomology\n\u22a2 (K.sc j).pOpcycles = K.pOpcycles j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Comp.lean", "full_name": "HasStrictFDerivAt.iterate", "start": [243, 11], "end": [250, 24], "traced_tactics": [{"tactic": "induction' n with n ihn", "annotated_tactic": ["induction' n with n ihn", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasStrictFDerivAt f f' x\nhx : f x = x\nn : \u2115\n\u22a2 HasStrictFDerivAt f^[n] (f' ^ n) x", "state_after": "case zero\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasStrictFDerivAt f f' x\nhx : f x = x\n\u22a2 HasStrictFDerivAt f^[0] (f' ^ 0) x\n\ncase succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasStrictFDerivAt f f' x\nhx : f x = x\nn : \u2115\nihn : HasStrictFDerivAt f^[n] (f' ^ n) x\n\u22a2 HasStrictFDerivAt f^[n + 1] (f' ^ (n + 1)) x"}, {"tactic": "exact hasStrictFDerivAt_id x", "annotated_tactic": ["exact <a>hasStrictFDerivAt_id</a> x", [{"full_name": "hasStrictFDerivAt_id", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [1077, 9], "def_end_pos": [1077, 29]}]], "state_before": "case zero\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasStrictFDerivAt f f' x\nhx : f x = x\n\u22a2 HasStrictFDerivAt f^[0] (f' ^ 0) x", "state_after": "no goals"}, {"tactic": "rw [Function.iterate_succ, pow_succ]", "annotated_tactic": ["rw [<a>Function.iterate_succ</a>, <a>pow_succ</a>]", [{"full_name": "Function.iterate_succ", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [62, 9], "def_end_pos": [62, 21]}, {"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [657, 9], "def_end_pos": [657, 17]}]], "state_before": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasStrictFDerivAt f f' x\nhx : f x = x\nn : \u2115\nihn : HasStrictFDerivAt f^[n] (f' ^ n) x\n\u22a2 HasStrictFDerivAt f^[n + 1] (f' ^ (n + 1)) x", "state_after": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasStrictFDerivAt f f' x\nhx : f x = x\nn : \u2115\nihn : HasStrictFDerivAt f^[n] (f' ^ n) x\n\u22a2 HasStrictFDerivAt (f^[n] \u2218 f) (f' ^ n * f') x"}, {"tactic": "rw [\u2190 hx] at ihn", "annotated_tactic": ["rw [\u2190 hx] at ihn", []], "state_before": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasStrictFDerivAt f f' x\nhx : f x = x\nn : \u2115\nihn : HasStrictFDerivAt f^[n] (f' ^ n) x\n\u22a2 HasStrictFDerivAt (f^[n] \u2218 f) (f' ^ n * f') x", "state_after": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasStrictFDerivAt f f' x\nhx : f x = x\nn : \u2115\nihn : HasStrictFDerivAt f^[n] (f' ^ n) (f x)\n\u22a2 HasStrictFDerivAt (f^[n] \u2218 f) (f' ^ n * f') x"}, {"tactic": "exact ihn.comp x hf", "annotated_tactic": ["exact ihn.comp x hf", []], "state_before": "case succ\n\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf'\u271d f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\nf : E \u2192 E\nf' : E \u2192L[\ud835\udd5c] E\nhf : HasStrictFDerivAt f f' x\nhx : f x = x\nn : \u2115\nihn : HasStrictFDerivAt f^[n] (f' ^ n) (f x)\n\u22a2 HasStrictFDerivAt (f^[n] \u2218 f) (f' ^ n * f') x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "full_name": "ContinuousLinearMap.coe_zero", "start": [638, 1], "end": [639, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Extr.lean", "full_name": "IsMinFilter.congr", "start": [660, 1], "end": [662, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/PythagoreanTriples.lean", "full_name": "PythagoreanTriple.gcd_dvd", "start": [164, 1], "end": [182, 26], "traced_tactics": [{"tactic": "by_cases h0 : Int.gcd x y = 0", "annotated_tactic": ["by_cases h0 : <a>Int.gcd</a> x y = 0", [{"full_name": "Int.gcd", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Gcd.lean", "def_pos": [20, 5], "def_end_pos": [20, 8]}]], "state_before": "x y z : \u2124\nh : PythagoreanTriple x y z\n\u22a2 \u2191(x.gcd y) \u2223 z", "state_after": "case pos\nx y z : \u2124\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\n\u22a2 \u2191(x.gcd y) \u2223 z\n\ncase neg\nx y z : \u2124\nh : PythagoreanTriple x y z\nh0 : \u00acx.gcd y = 0\n\u22a2 \u2191(x.gcd y) \u2223 z"}, {"tactic": "obtain \u27e8k, x0, y0, _, h2, rfl, rfl\u27e9 :\n  \u2203 (k : \u2115) (x0 y0 : _), 0 < k \u2227 Int.gcd x0 y0 = 1 \u2227 x = x0 * k \u2227 y = y0 * k :=\n  Int.exists_gcd_one' (Nat.pos_of_ne_zero h0)", "annotated_tactic": ["obtain \u27e8k, x0, y0, _, h2, rfl, rfl\u27e9 :\n    \u2203 (k : \u2115) (x0 y0 : _), 0 < k \u2227 <a>Int.gcd</a> x0 y0 = 1 \u2227 x = x0 * k \u2227 y = y0 * k :=\n    <a>Int.exists_gcd_one'</a> (<a>Nat.pos_of_ne_zero</a> h0)", [{"full_name": "Int.gcd", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Gcd.lean", "def_pos": [20, 5], "def_end_pos": [20, 8]}, {"full_name": "Int.exists_gcd_one'", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [338, 9], "def_end_pos": [338, 24]}, {"full_name": "Nat.pos_of_ne_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [354, 19], "def_end_pos": [354, 33]}]], "state_before": "case neg\nx y z : \u2124\nh : PythagoreanTriple x y z\nh0 : \u00acx.gcd y = 0\n\u22a2 \u2191(x.gcd y) \u2223 z", "state_after": "case neg.intro.intro.intro.intro.intro.intro\nz : \u2124\nk : \u2115\nx0 y0 : \u2124\nleft\u271d : 0 < k\nh2 : x0.gcd y0 = 1\nh : PythagoreanTriple (x0 * \u2191k) (y0 * \u2191k) z\nh0 : \u00ac(x0 * \u2191k).gcd (y0 * \u2191k) = 0\n\u22a2 \u2191((x0 * \u2191k).gcd (y0 * \u2191k)) \u2223 z"}, {"tactic": "rw [Int.gcd_mul_right, h2, Int.natAbs_ofNat, one_mul]", "annotated_tactic": ["rw [<a>Int.gcd_mul_right</a>, h2, <a>Int.natAbs_ofNat</a>, <a>one_mul</a>]", [{"full_name": "Int.gcd_mul_right", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [254, 9], "def_end_pos": [254, 22]}, {"full_name": "Int.natAbs_ofNat", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Order.lean", "def_pos": [413, 17], "def_end_pos": [413, 29]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "case neg.intro.intro.intro.intro.intro.intro\nz : \u2124\nk : \u2115\nx0 y0 : \u2124\nleft\u271d : 0 < k\nh2 : x0.gcd y0 = 1\nh : PythagoreanTriple (x0 * \u2191k) (y0 * \u2191k) z\nh0 : \u00ac(x0 * \u2191k).gcd (y0 * \u2191k) = 0\n\u22a2 \u2191((x0 * \u2191k).gcd (y0 * \u2191k)) \u2223 z", "state_after": "case neg.intro.intro.intro.intro.intro.intro\nz : \u2124\nk : \u2115\nx0 y0 : \u2124\nleft\u271d : 0 < k\nh2 : x0.gcd y0 = 1\nh : PythagoreanTriple (x0 * \u2191k) (y0 * \u2191k) z\nh0 : \u00ac(x0 * \u2191k).gcd (y0 * \u2191k) = 0\n\u22a2 \u2191k \u2223 z"}, {"tactic": "rw [\u2190 Int.pow_dvd_pow_iff two_ne_zero, sq z, \u2190 h.eq]", "annotated_tactic": ["rw [\u2190 <a>Int.pow_dvd_pow_iff</a> <a>two_ne_zero</a>, <a>sq</a> z, \u2190 h.eq]", [{"full_name": "Int.pow_dvd_pow_iff", "def_path": "Mathlib/Data/Int/GCD.lean", "def_pos": [344, 9], "def_end_pos": [344, 24]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [65, 7], "def_end_pos": [65, 18]}, {"full_name": "sq", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [684, 41], "def_end_pos": [684, 43]}]], "state_before": "case neg.intro.intro.intro.intro.intro.intro\nz : \u2124\nk : \u2115\nx0 y0 : \u2124\nleft\u271d : 0 < k\nh2 : x0.gcd y0 = 1\nh : PythagoreanTriple (x0 * \u2191k) (y0 * \u2191k) z\nh0 : \u00ac(x0 * \u2191k).gcd (y0 * \u2191k) = 0\n\u22a2 \u2191k \u2223 z", "state_after": "case neg.intro.intro.intro.intro.intro.intro\nz : \u2124\nk : \u2115\nx0 y0 : \u2124\nleft\u271d : 0 < k\nh2 : x0.gcd y0 = 1\nh : PythagoreanTriple (x0 * \u2191k) (y0 * \u2191k) z\nh0 : \u00ac(x0 * \u2191k).gcd (y0 * \u2191k) = 0\n\u22a2 \u2191k ^ 2 \u2223 x0 * \u2191k * (x0 * \u2191k) + y0 * \u2191k * (y0 * \u2191k)"}, {"tactic": "rw [(by ring : x0 * k * (x0 * k) + y0 * k * (y0 * k) = (k : \u2124) ^ 2 * (x0 * x0 + y0 * y0))]", "annotated_tactic": ["rw [(by ring : x0 * k * (x0 * k) + y0 * k * (y0 * k) = (k : \u2124) ^ 2 * (x0 * x0 + y0 * y0))]", []], "state_before": "case neg.intro.intro.intro.intro.intro.intro\nz : \u2124\nk : \u2115\nx0 y0 : \u2124\nleft\u271d : 0 < k\nh2 : x0.gcd y0 = 1\nh : PythagoreanTriple (x0 * \u2191k) (y0 * \u2191k) z\nh0 : \u00ac(x0 * \u2191k).gcd (y0 * \u2191k) = 0\n\u22a2 \u2191k ^ 2 \u2223 x0 * \u2191k * (x0 * \u2191k) + y0 * \u2191k * (y0 * \u2191k)", "state_after": "case neg.intro.intro.intro.intro.intro.intro\nz : \u2124\nk : \u2115\nx0 y0 : \u2124\nleft\u271d : 0 < k\nh2 : x0.gcd y0 = 1\nh : PythagoreanTriple (x0 * \u2191k) (y0 * \u2191k) z\nh0 : \u00ac(x0 * \u2191k).gcd (y0 * \u2191k) = 0\n\u22a2 \u2191k ^ 2 \u2223 \u2191k ^ 2 * (x0 * x0 + y0 * y0)"}, {"tactic": "exact dvd_mul_right _ _", "annotated_tactic": ["exact <a>dvd_mul_right</a> _ _", [{"full_name": "dvd_mul_right", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 22]}]], "state_before": "case neg.intro.intro.intro.intro.intro.intro\nz : \u2124\nk : \u2115\nx0 y0 : \u2124\nleft\u271d : 0 < k\nh2 : x0.gcd y0 = 1\nh : PythagoreanTriple (x0 * \u2191k) (y0 * \u2191k) z\nh0 : \u00ac(x0 * \u2191k).gcd (y0 * \u2191k) = 0\n\u22a2 \u2191k ^ 2 \u2223 \u2191k ^ 2 * (x0 * x0 + y0 * y0)", "state_after": "no goals"}, {"tactic": "have hx : x = 0 := by\n  apply Int.natAbs_eq_zero.mp\n  apply Nat.eq_zero_of_gcd_eq_zero_left h0", "annotated_tactic": ["have hx : x = 0 := by\n      apply Int.natAbs_eq_zero.mp\n      apply <a>Nat.eq_zero_of_gcd_eq_zero_left</a> h0", [{"full_name": "Nat.eq_zero_of_gcd_eq_zero_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [149, 9], "def_end_pos": [149, 36]}]], "state_before": "case pos\nx y z : \u2124\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\n\u22a2 \u2191(x.gcd y) \u2223 z", "state_after": "case pos\nx y z : \u2124\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\n\u22a2 \u2191(x.gcd y) \u2223 z"}, {"tactic": "have hy : y = 0 := by\n  apply Int.natAbs_eq_zero.mp\n  apply Nat.eq_zero_of_gcd_eq_zero_right h0", "annotated_tactic": ["have hy : y = 0 := by\n      apply Int.natAbs_eq_zero.mp\n      apply <a>Nat.eq_zero_of_gcd_eq_zero_right</a> h0", [{"full_name": "Nat.eq_zero_of_gcd_eq_zero_right", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [154, 9], "def_end_pos": [154, 37]}]], "state_before": "case pos\nx y z : \u2124\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\n\u22a2 \u2191(x.gcd y) \u2223 z", "state_after": "case pos\nx y z : \u2124\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\nhy : y = 0\n\u22a2 \u2191(x.gcd y) \u2223 z"}, {"tactic": "have hz : z = 0 := by\n  simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero,\n    or_self_iff] using h", "annotated_tactic": ["have hz : z = 0 := by\n      simpa only [<a>PythagoreanTriple</a>, hx, hy, <a>add_zero</a>, <a>zero_eq_mul</a>, <a>mul_zero</a>,\n        <a>or_self_iff</a>] using h", [{"full_name": "PythagoreanTriple", "def_path": "Mathlib/NumberTheory/PythagoreanTriples.lean", "def_pos": [48, 5], "def_end_pos": [48, 22]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [482, 3], "def_end_pos": [482, 14]}, {"full_name": "zero_eq_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [290, 9], "def_end_pos": [290, 20]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}, {"full_name": "or_self_iff", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [74, 9], "def_end_pos": [74, 20]}]], "state_before": "case pos\nx y z : \u2124\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\nhy : y = 0\n\u22a2 \u2191(x.gcd y) \u2223 z", "state_after": "case pos\nx y z : \u2124\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\nhy : y = 0\nhz : z = 0\n\u22a2 \u2191(x.gcd y) \u2223 z"}, {"tactic": "simp only [hz, dvd_zero]", "annotated_tactic": ["simp only [hz, <a>dvd_zero</a>]", [{"full_name": "dvd_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Divisibility.lean", "def_pos": [41, 9], "def_end_pos": [41, 17]}]], "state_before": "case pos\nx y z : \u2124\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\nhy : y = 0\nhz : z = 0\n\u22a2 \u2191(x.gcd y) \u2223 z", "state_after": "no goals"}, {"tactic": "apply Int.natAbs_eq_zero.mp", "annotated_tactic": ["apply Int.natAbs_eq_zero.mp", []], "state_before": "x y z : \u2124\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\n\u22a2 x = 0", "state_after": "x y z : \u2124\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\n\u22a2 x.natAbs = 0"}, {"tactic": "apply Nat.eq_zero_of_gcd_eq_zero_left h0", "annotated_tactic": ["apply <a>Nat.eq_zero_of_gcd_eq_zero_left</a> h0", [{"full_name": "Nat.eq_zero_of_gcd_eq_zero_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [149, 9], "def_end_pos": [149, 36]}]], "state_before": "x y z : \u2124\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\n\u22a2 x.natAbs = 0", "state_after": "no goals"}, {"tactic": "apply Int.natAbs_eq_zero.mp", "annotated_tactic": ["apply Int.natAbs_eq_zero.mp", []], "state_before": "x y z : \u2124\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\n\u22a2 y = 0", "state_after": "x y z : \u2124\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\n\u22a2 y.natAbs = 0"}, {"tactic": "apply Nat.eq_zero_of_gcd_eq_zero_right h0", "annotated_tactic": ["apply <a>Nat.eq_zero_of_gcd_eq_zero_right</a> h0", [{"full_name": "Nat.eq_zero_of_gcd_eq_zero_right", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [154, 9], "def_end_pos": [154, 37]}]], "state_before": "x y z : \u2124\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\n\u22a2 y.natAbs = 0", "state_after": "no goals"}, {"tactic": "simpa only [PythagoreanTriple, hx, hy, add_zero, zero_eq_mul, mul_zero,\n  or_self_iff] using h", "annotated_tactic": ["simpa only [<a>PythagoreanTriple</a>, hx, hy, <a>add_zero</a>, <a>zero_eq_mul</a>, <a>mul_zero</a>,\n        <a>or_self_iff</a>] using h", [{"full_name": "PythagoreanTriple", "def_path": "Mathlib/NumberTheory/PythagoreanTriples.lean", "def_pos": [48, 5], "def_end_pos": [48, 22]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [482, 3], "def_end_pos": [482, 14]}, {"full_name": "zero_eq_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [290, 9], "def_end_pos": [290, 20]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}, {"full_name": "or_self_iff", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [74, 9], "def_end_pos": [74, 20]}]], "state_before": "x y z : \u2124\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\nhx : x = 0\nhy : y = 0\n\u22a2 z = 0", "state_after": "no goals"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "z : \u2124\nk : \u2115\nx0 y0 : \u2124\nleft\u271d : 0 < k\nh2 : x0.gcd y0 = 1\nh : PythagoreanTriple (x0 * \u2191k) (y0 * \u2191k) z\nh0 : \u00ac(x0 * \u2191k).gcd (y0 * \u2191k) = 0\n\u22a2 x0 * \u2191k * (x0 * \u2191k) + y0 * \u2191k * (y0 * \u2191k) = \u2191k ^ 2 * (x0 * x0 + y0 * y0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/SesquilinearForm.lean", "full_name": "LinearMap.isSymm_zero", "start": [242, 1], "end": [242, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/Unitization.lean", "full_name": "Unitization.nnnorm_def", "start": [135, 1], "end": [136, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_eq_top_of_measure_eq_top_ne_zero", "start": [935, 1], "end": [940, 57], "traced_tactics": [{"tactic": "simp [mul_eq_top, h\u03bcf]", "annotated_tactic": ["simp [<a>mul_eq_top</a>, h\u03bcf]", [{"full_name": "ENNReal.mul_eq_top", "def_path": "Mathlib/Data/ENNReal/Operations.lean", "def_pos": [225, 9], "def_end_pos": [225, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f \u03bc\nh\u03bcf : \u03bc {x | f x = \u22a4} \u2260 0\n\u22a2 \u22a4 = \u22a4 * \u03bc {x | \u22a4 \u2264 f x}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/ContinuousFunction/Ideals.lean", "full_name": "ContinuousMap.idealOfSet_closed", "start": [94, 1], "end": [98, 59], "traced_tactics": [{"tactic": "simp only [idealOfSet, Submodule.coe_set_mk, Set.setOf_forall]", "annotated_tactic": ["simp only [<a>idealOfSet</a>, <a>Submodule.coe_set_mk</a>, <a>Set.setOf_forall</a>]", [{"full_name": "ContinuousMap.idealOfSet", "def_path": "Mathlib/Topology/ContinuousFunction/Ideals.lean", "def_pos": [87, 5], "def_end_pos": [87, 15]}, {"full_name": "Submodule.coe_set_mk", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [80, 9], "def_end_pos": [80, 19]}, {"full_name": "Set.setOf_forall", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [233, 9], "def_end_pos": [233, 21]}]], "state_before": "X : Type u_1\nR : Type u_2\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : TopologicalSemiring R\ninst\u271d : T2Space R\ns : Set X\n\u22a2 IsClosed \u2191(idealOfSet R s)", "state_after": "X : Type u_1\nR : Type u_2\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : TopologicalSemiring R\ninst\u271d : T2Space R\ns : Set X\n\u22a2 IsClosed \u2191{ carrier := \u22c2 i \u2208 s\u1d9c, {x | x i = 0}, add_mem' := \u22ef, zero_mem' := \u22ef }"}, {"tactic": "exact isClosed_iInter fun x => isClosed_iInter fun _ =>\n  isClosed_eq (continuous_eval_const x) continuous_const", "annotated_tactic": ["exact <a>isClosed_iInter</a> fun x => <a>isClosed_iInter</a> fun _ =>\n    <a>isClosed_eq</a> (<a>continuous_eval_const</a> x) <a>continuous_const</a>", [{"full_name": "isClosed_iInter", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [187, 9], "def_end_pos": [187, 24]}, {"full_name": "isClosed_iInter", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [187, 9], "def_end_pos": [187, 24]}, {"full_name": "isClosed_eq", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [1822, 9], "def_end_pos": [1822, 20]}, {"full_name": "ContinuousMap.continuous_eval_const", "def_path": "Mathlib/Topology/CompactOpen.lean", "def_pos": [193, 9], "def_end_pos": [193, 30]}, {"full_name": "continuous_const", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1675, 9], "def_end_pos": [1675, 25]}]], "state_before": "X : Type u_1\nR : Type u_2\ninst\u271d\u2074 : TopologicalSpace X\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : TopologicalSpace R\ninst\u271d\u00b9 : TopologicalSemiring R\ninst\u271d : T2Space R\ns : Set X\n\u22a2 IsClosed \u2191{ carrier := \u22c2 i \u2208 s\u1d9c, {x | x i = 0}, add_mem' := \u22ef, zero_mem' := \u22ef }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Adjunction/Limits.lean", "full_name": "CategoryTheory.Adjunction.hasColimit_comp_equivalence", "start": [134, 1], "end": [138, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.EqOn.trans", "start": [216, 1], "end": [217, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean", "full_name": "AffineEquiv.one_def", "start": [389, 1], "end": [390, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.injOn_of_ncard_image_eq", "start": [660, 1], "end": [663, 38], "traced_tactics": [{"tactic": "rw [\u2190 Nat.cast_inj (R := \u2115\u221e), hs.cast_ncard_eq, (hs.image _).cast_ncard_eq] at h", "annotated_tactic": ["rw [\u2190 <a>Nat.cast_inj</a> (R := \u2115\u221e), hs.cast_ncard_eq, (hs.image _).<a>cast_ncard_eq</a>] at h", [{"full_name": "Nat.cast_inj", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [74, 9], "def_end_pos": [74, 17]}, {"full_name": "Set.Finite.cast_ncard_eq", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [479, 9], "def_end_pos": [479, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nh : (f '' s).ncard = s.ncard\nhs : autoParam s.Finite _auto\u271d\n\u22a2 InjOn f s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nf : \u03b1 \u2192 \u03b2\nh : (f '' s).encard = s.encard\nhs : autoParam s.Finite _auto\u271d\n\u22a2 InjOn f s"}, {"tactic": "exact 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"29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "full_name": "gcd_zero_left'", "start": [389, 1], "end": [390, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Analytic/CPolynomial.lean", "full_name": "ContinuousLinearMap.comp_cPolynomialOn", "start": [248, 1], "end": [252, 87], "traced_tactics": [{"tactic": "rintro x hx", "annotated_tactic": ["rintro x hx", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g\u271d : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx : E\nr r' : \u211d\u22650\u221e\nn m : \u2115\ns : Set E\ng : F \u2192L[\ud835\udd5c] G\nh : CPolynomialOn \ud835\udd5c f s\n\u22a2 CPolynomialOn \ud835\udd5c (\u21d1g \u2218 f) s", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g\u271d : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx\u271d : E\nr r' : \u211d\u22650\u221e\nn m : \u2115\ns : Set E\ng : F \u2192L[\ud835\udd5c] G\nh : CPolynomialOn \ud835\udd5c f s\nx : E\nhx : x \u2208 s\n\u22a2 CPolynomialAt \ud835\udd5c (\u21d1g \u2218 f) x"}, {"tactic": "rcases h x hx with \u27e8p, n, r, hp\u27e9", "annotated_tactic": ["rcases h x hx with \u27e8p, n, r, hp\u27e9", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g\u271d : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx\u271d : E\nr r' : \u211d\u22650\u221e\nn m : \u2115\ns : Set E\ng : F \u2192L[\ud835\udd5c] G\nh : CPolynomialOn \ud835\udd5c f s\nx : E\nhx : x \u2208 s\n\u22a2 CPolynomialAt \ud835\udd5c (\u21d1g \u2218 f) x", "state_after": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g\u271d : E \u2192 F\np\u271d pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx\u271d : E\nr\u271d r' : \u211d\u22650\u221e\nn\u271d m : \u2115\ns : Set E\ng : F \u2192L[\ud835\udd5c] G\nh : CPolynomialOn \ud835\udd5c f s\nx : E\nhx : x \u2208 s\np : FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nr : \u211d\u22650\u221e\nhp : HasFiniteFPowerSeriesOnBall f p x n r\n\u22a2 CPolynomialAt \ud835\udd5c (\u21d1g \u2218 f) x"}, {"tactic": "exact \u27e8g.compFormalMultilinearSeries p, n, r, g.comp_hasFiniteFPowerSeriesOnBall hp\u27e9", "annotated_tactic": ["exact \u27e8g.compFormalMultilinearSeries p, n, r, g.comp_hasFiniteFPowerSeriesOnBall hp\u27e9", []], "state_before": "case intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g\u271d : E \u2192 F\np\u271d pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx\u271d : E\nr\u271d r' : \u211d\u22650\u221e\nn\u271d m : \u2115\ns : Set E\ng : F \u2192L[\ud835\udd5c] G\nh : CPolynomialOn \ud835\udd5c f s\nx : E\nhx : x \u2208 s\np : FormalMultilinearSeries \ud835\udd5c E F\nn : \u2115\nr : \u211d\u22650\u221e\nhp : HasFiniteFPowerSeriesOnBall f p x n r\n\u22a2 CPolynomialAt \ud835\udd5c (\u21d1g \u2218 f) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Types.lean", "full_name": "CategoryTheory.Limits.Types.Colimit.\u03b9_desc_apply", "start": [549, 1], "end": [551, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "full_name": "CategoryTheory.Sieve.sieveOfSubfunctor_functorInclusion", "start": [860, 1], "end": [867, 25], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R : Sieve X\n\u22a2 sieveOfSubfunctor S.functorInclusion = S", "state_after": "case h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R : Sieve X\nY\u271d : C\nf\u271d : Y\u271d \u27f6 X\n\u22a2 (sieveOfSubfunctor S.functorInclusion).arrows f\u271d \u2194 S.arrows f\u271d"}, {"tactic": "simp only [functorInclusion_app, sieveOfSubfunctor_apply]", "annotated_tactic": ["simp only [<a>functorInclusion_app</a>, <a>sieveOfSubfunctor_apply</a>]", [{"full_name": "CategoryTheory.Sieve.functorInclusion_app", "def_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "def_pos": [830, 3], "def_end_pos": [830, 8]}, {"full_name": "CategoryTheory.Sieve.sieveOfSubfunctor_apply", "def_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "def_pos": [850, 3], "def_end_pos": [850, 8]}]], "state_before": "case h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R : Sieve X\nY\u271d : C\nf\u271d : Y\u271d \u27f6 X\n\u22a2 (sieveOfSubfunctor S.functorInclusion).arrows f\u271d \u2194 S.arrows f\u271d", "state_after": "case h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R : Sieve X\nY\u271d : C\nf\u271d : Y\u271d \u27f6 X\n\u22a2 (\u2203 t, \u2191t = f\u271d) \u2194 S.arrows f\u271d"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R : Sieve X\nY\u271d : C\nf\u271d : Y\u271d \u27f6 X\n\u22a2 (\u2203 t, \u2191t = f\u271d) \u2194 S.arrows f\u271d", "state_after": "case h.mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R : Sieve X\nY\u271d : C\nf\u271d : Y\u271d \u27f6 X\n\u22a2 (\u2203 t, \u2191t = f\u271d) \u2192 S.arrows f\u271d\n\ncase h.mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R : Sieve X\nY\u271d : C\nf\u271d : Y\u271d \u27f6 X\n\u22a2 S.arrows f\u271d \u2192 \u2203 t, \u2191t = f\u271d"}, {"tactic": "rintro \u27e8\u27e8f, hf\u27e9, rfl\u27e9", "annotated_tactic": ["rintro \u27e8\u27e8f, hf\u27e9, rfl\u27e9", []], "state_before": "case h.mp\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R : Sieve X\nY\u271d : C\nf\u271d : Y\u271d \u27f6 X\n\u22a2 (\u2203 t, \u2191t = f\u271d) \u2192 S.arrows f\u271d", "state_after": "case h.mp.intro.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R : Sieve X\nY\u271d : C\nf : { unop := Y\u271d }.unop \u27f6 X\nhf : S.arrows f\n\u22a2 S.arrows \u2191\u27e8f, hf\u27e9"}, {"tactic": "exact hf", "annotated_tactic": ["exact hf", []], "state_before": "case h.mp.intro.mk\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf\u271d : Y \u27f6 X\nS R : Sieve X\nY\u271d : C\nf : { unop := Y\u271d }.unop \u27f6 X\nhf : S.arrows f\n\u22a2 S.arrows \u2191\u27e8f, hf\u27e9", "state_after": "no goals"}, {"tactic": "intro hf", "annotated_tactic": ["intro hf", []], "state_before": "case h.mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R : Sieve X\nY\u271d : C\nf\u271d : Y\u271d \u27f6 X\n\u22a2 S.arrows f\u271d \u2192 \u2203 t, \u2191t = f\u271d", "state_after": "case h.mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R : Sieve X\nY\u271d : C\nf\u271d : Y\u271d \u27f6 X\nhf : S.arrows f\u271d\n\u22a2 \u2203 t, \u2191t = f\u271d"}, {"tactic": "exact \u27e8\u27e8_, hf\u27e9, rfl\u27e9", "annotated_tactic": ["exact \u27e8\u27e8_, hf\u27e9, <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case h.mpr\nC : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX Y Z : C\nf : Y \u27f6 X\nS R : Sieve X\nY\u271d : C\nf\u271d : Y\u271d \u27f6 X\nhf : S.arrows f\u271d\n\u22a2 \u2203 t, \u2191t = f\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Choose/Sum.lean", "full_name": "Commute.add_pow'", "start": [72, 1], "end": [75, 47], "traced_tactics": [{"tactic": "simp_rw [Finset.Nat.sum_antidiagonal_eq_sum_range_succ fun m p \u21a6 choose n m \u2022 (x ^ m * y ^ p),\n  _root_.nsmul_eq_mul, cast_comm, h.add_pow]", "annotated_tactic": ["simp_rw [<a>Finset.Nat.sum_antidiagonal_eq_sum_range_succ</a> fun m p \u21a6 <a>choose</a> n m \u2022 (x ^ m * y ^ p),\n    <a>_root_.nsmul_eq_mul</a>, <a>cast_comm</a>, h.add_pow]", [{"full_name": "Finset.Nat.sum_antidiagonal_eq_sum_range_succ", "def_path": "Mathlib/Algebra/BigOperators/NatAntidiagonal.lean", "def_pos": [69, 3], "def_end_pos": [69, 14]}, {"full_name": "Nat.choose", "def_path": "Mathlib/Data/Nat/Choose/Basic.lean", "def_pos": [47, 5], "def_end_pos": [47, 11]}, {"full_name": "nsmul_eq_mul", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [81, 15], "def_end_pos": [81, 34]}, {"full_name": "Nat.cast_comm", "def_path": "Mathlib/Data/Nat/Cast/Commute.lean", "def_pos": [33, 9], "def_end_pos": [33, 18]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\nx y : R\nh : Commute x y\nn : \u2115\n\u22a2 (x + y) ^ n = \u2211 m \u2208 antidiagonal n, n.choose m.1 \u2022 (x ^ m.1 * y ^ m.2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Countable/Defs.lean", "full_name": "Function.Surjective.countable", "start": [58, 11], "end": [60, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Padics/PadicIntegers.lean", "full_name": "PadicInt.norm_lt_one_iff_dvd", "start": [576, 1], "end": [580, 80], "traced_tactics": [{"tactic": "have := norm_le_pow_iff_mem_span_pow x 1", "annotated_tactic": ["have := <a>norm_le_pow_iff_mem_span_pow</a> x 1", [{"full_name": "PadicInt.norm_le_pow_iff_mem_span_pow", "def_path": "Mathlib/NumberTheory/Padics/PadicIntegers.lean", "def_pos": [557, 9], "def_end_pos": [557, 37]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\n\u22a2 \u2016x\u2016 < 1 \u2194 \u2191p \u2223 x", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nthis : \u2016x\u2016 \u2264 \u2191p ^ (-\u21911) \u2194 x \u2208 Ideal.span {\u2191p ^ 1}\n\u22a2 \u2016x\u2016 < 1 \u2194 \u2191p \u2223 x"}, {"tactic": "rw [Ideal.mem_span_singleton, pow_one] at this", "annotated_tactic": ["rw [<a>Ideal.mem_span_singleton</a>, <a>pow_one</a>] at this", [{"full_name": "Ideal.mem_span_singleton", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [503, 9], "def_end_pos": [503, 27]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [663, 7], "def_end_pos": [663, 14]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nthis : \u2016x\u2016 \u2264 \u2191p ^ (-\u21911) \u2194 x \u2208 Ideal.span {\u2191p ^ 1}\n\u22a2 \u2016x\u2016 < 1 \u2194 \u2191p \u2223 x", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nthis : \u2016x\u2016 \u2264 \u2191p ^ (-\u21911) \u2194 \u2191p \u2223 x\n\u22a2 \u2016x\u2016 < 1 \u2194 \u2191p \u2223 x"}, {"tactic": "rw [\u2190 this, norm_le_pow_iff_norm_lt_pow_add_one]", "annotated_tactic": ["rw [\u2190 this, <a>norm_le_pow_iff_norm_lt_pow_add_one</a>]", [{"full_name": "PadicInt.norm_le_pow_iff_norm_lt_pow_add_one", "def_path": "Mathlib/NumberTheory/Padics/PadicIntegers.lean", "def_pos": [566, 9], "def_end_pos": [566, 44]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nthis : \u2016x\u2016 \u2264 \u2191p ^ (-\u21911) \u2194 \u2191p \u2223 x\n\u22a2 \u2016x\u2016 < 1 \u2194 \u2191p \u2223 x", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nthis : \u2016x\u2016 \u2264 \u2191p ^ (-\u21911) \u2194 \u2191p \u2223 x\n\u22a2 \u2016x\u2016 < 1 \u2194 \u2016x\u2016 < \u2191p ^ (-\u21911 + 1)"}, {"tactic": "simp only [zpow_zero, Int.ofNat_zero, Int.ofNat_succ, add_left_neg, zero_add]", "annotated_tactic": ["simp only [<a>zpow_zero</a>, <a>Int.ofNat_zero</a>, <a>Int.ofNat_succ</a>, <a>add_left_neg</a>, <a>zero_add</a>]", [{"full_name": "zpow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1015, 50], "def_end_pos": [1015, 59]}, {"full_name": "Int.ofNat_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [73, 17], "def_end_pos": [73, 27]}, {"full_name": "Int.ofNat_succ", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean", "def_pos": [27, 9], "def_end_pos": [27, 19]}, {"full_name": "add_left_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1221, 3], "def_end_pos": [1221, 14]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [476, 3], "def_end_pos": [476, 14]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nx : \u2124_[p]\nthis : \u2016x\u2016 \u2264 \u2191p ^ (-\u21911) \u2194 \u2191p \u2223 x\n\u22a2 \u2016x\u2016 < 1 \u2194 \u2016x\u2016 < \u2191p ^ (-\u21911 + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "full_name": "SimpleGraph.Walk.fst_mem_support_of_mem_edges", "start": [815, 1], "end": [821, 49], "traced_tactics": [{"tactic": "obtain \u27e8d, hd, he\u27e9 := List.mem_map.mp he", "annotated_tactic": ["obtain \u27e8d, hd, he\u27e9 := List.mem_map.mp he", []], "state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nt u v w : V\np : G.Walk v w\nhe : s(t, u) \u2208 p.edges\n\u22a2 t \u2208 p.support", "state_after": "case intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nt u v w : V\np : G.Walk v w\nhe\u271d : s(t, u) \u2208 p.edges\nd : G.Dart\nhd : d \u2208 p.darts\nhe : d.edge = s(t, u)\n\u22a2 t \u2208 p.support"}, {"tactic": "rw [dart_edge_eq_mk'_iff'] at he", "annotated_tactic": ["rw [<a>dart_edge_eq_mk'_iff'</a>] at he", [{"full_name": "SimpleGraph.dart_edge_eq_mk'_iff'", "def_path": "Mathlib/Combinatorics/SimpleGraph/Dart.lean", "def_pos": [118, 9], "def_end_pos": [118, 30]}]], "state_before": "case intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nt u v w : V\np : G.Walk v w\nhe\u271d : s(t, u) \u2208 p.edges\nd : G.Dart\nhd : d \u2208 p.darts\nhe : d.edge = s(t, u)\n\u22a2 t \u2208 p.support", "state_after": "case intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nt u v w : V\np : G.Walk v w\nhe\u271d : s(t, u) \u2208 p.edges\nd : G.Dart\nhd : d \u2208 p.darts\nhe : d.toProd.1 = t \u2227 d.toProd.2 = u \u2228 d.toProd.1 = u \u2227 d.toProd.2 = t\n\u22a2 t \u2208 p.support"}, {"tactic": "rcases he with (\u27e8rfl, rfl\u27e9 | \u27e8rfl, rfl\u27e9)", "annotated_tactic": ["rcases he with (\u27e8rfl, rfl\u27e9 | \u27e8rfl, rfl\u27e9)", []], "state_before": "case intro.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nt u v w : V\np : G.Walk v w\nhe\u271d : s(t, u) \u2208 p.edges\nd : G.Dart\nhd : d \u2208 p.darts\nhe : d.toProd.1 = t \u2227 d.toProd.2 = u \u2228 d.toProd.1 = u \u2227 d.toProd.2 = t\n\u22a2 t \u2208 p.support", "state_after": "case intro.intro.inl.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\np : G.Walk v w\nd : G.Dart\nhd : d \u2208 p.darts\nhe : s(d.toProd.1, d.toProd.2) \u2208 p.edges\n\u22a2 d.toProd.1 \u2208 p.support\n\ncase intro.intro.inr.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\np : G.Walk v w\nd : G.Dart\nhd : d \u2208 p.darts\nhe : s(d.toProd.2, d.toProd.1) \u2208 p.edges\n\u22a2 d.toProd.2 \u2208 p.support"}, {"tactic": "exact dart_fst_mem_support_of_mem_darts _ hd", "annotated_tactic": ["exact <a>dart_fst_mem_support_of_mem_darts</a> _ hd", [{"full_name": "SimpleGraph.Walk.dart_fst_mem_support_of_mem_darts", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [801, 9], "def_end_pos": [801, 42]}]], "state_before": "case intro.intro.inl.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\np : G.Walk v w\nd : G.Dart\nhd : d \u2208 p.darts\nhe : s(d.toProd.1, d.toProd.2) \u2208 p.edges\n\u22a2 d.toProd.1 \u2208 p.support", "state_after": "no goals"}, {"tactic": "exact dart_snd_mem_support_of_mem_darts _ hd", "annotated_tactic": ["exact <a>dart_snd_mem_support_of_mem_darts</a> _ hd", [{"full_name": "SimpleGraph.Walk.dart_snd_mem_support_of_mem_darts", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [810, 9], "def_end_pos": [810, 42]}]], "state_before": "case intro.intro.inr.intro\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nv w : V\np : G.Walk v w\nd : G.Dart\nhd : d \u2208 p.darts\nhe : s(d.toProd.2, d.toProd.1) \u2208 p.edges\n\u22a2 d.toProd.2 \u2208 p.support", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Ker.lean", "full_name": "Filter.ker_iInf", "start": [44, 1], "end": [45, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Surreal/Dyadic.lean", "full_name": "SetTheory.PGame.powHalf_succ_le_powHalf", "start": [102, 1], "end": [103, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "full_name": "LinearEquiv.rTensor_refl_apply", "start": [1469, 1], "end": [1469, 92], "traced_tactics": [{"tactic": "rw [rTensor_refl, refl_apply]", "annotated_tactic": ["rw [<a>rTensor_refl</a>, <a>refl_apply</a>]", [{"full_name": "LinearEquiv.rTensor_refl", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [1467, 17], "def_end_pos": [1467, 29]}, {"full_name": "LinearEquiv.refl_apply", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [276, 9], "def_end_pos": [276, 19]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9\u2076 : CommSemiring R\nR' : Type u_2\ninst\u271d\u00b9\u2075 : Monoid R'\nR'' : Type u_3\ninst\u271d\u00b9\u2074 : Semiring R''\nM : Type u_4\nN : Type u_5\nP : Type u_6\nQ : Type u_7\nS : Type u_8\nT : Type u_9\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\ninst\u271d\u00b9\u00b2 : AddCommMonoid N\ninst\u271d\u00b9\u00b9 : AddCommMonoid P\ninst\u271d\u00b9\u2070 : AddCommMonoid Q\ninst\u271d\u2079 : AddCommMonoid S\ninst\u271d\u2078 : AddCommMonoid T\ninst\u271d\u2077 : Module R M\ninst\u271d\u2076 : Module R N\ninst\u271d\u2075 : Module R P\ninst\u271d\u2074 : Module R Q\ninst\u271d\u00b3 : Module R S\ninst\u271d\u00b2 : Module R T\ninst\u271d\u00b9 : DistribMulAction R' M\ninst\u271d : Module R'' M\ng : P \u2243\u2097[R] Q\nf : N \u2243\u2097[R] P\nm : M\nn : N\np : P\nx : M \u2297[R] N\ny : N \u2297[R] M\n\u22a2 (rTensor M (refl R N)) y = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/ContDiff/RCLike.lean", "full_name": "ContDiff.hasStrictFDerivAt", "start": [74, 1], "end": [76, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Image.lean", "full_name": "Finset.coe_image_subset_range", "start": [504, 1], "end": [507, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "full_name": "MeasureTheory.SignedMeasure.mutuallySingular_iff", "start": [550, 1], "end": [569, 80], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\n\u22a2 s \u27c2\u1d65 t \u2194 s.totalVariation \u27c2\u2098 t.totalVariation", "state_after": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\n\u22a2 s \u27c2\u1d65 t \u2192 s.totalVariation \u27c2\u2098 t.totalVariation\n\ncase mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\n\u22a2 s.totalVariation \u27c2\u2098 t.totalVariation \u2192 s \u27c2\u1d65 t"}, {"tactic": "rintro \u27e8u, hmeas, hu\u2081, hu\u2082\u27e9", "annotated_tactic": ["rintro \u27e8u, hmeas, hu\u2081, hu\u2082\u27e9", []], "state_before": "case mp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\n\u22a2 s \u27c2\u1d65 t \u2192 s.totalVariation \u27c2\u2098 t.totalVariation", "state_after": "case mp.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 t \u2286 u, \u2191s t = 0\nhu\u2082 : \u2200 t_1 \u2286 u\u1d9c, \u2191t t_1 = 0\n\u22a2 s.totalVariation \u27c2\u2098 t.totalVariation"}, {"tactic": "obtain \u27e8i, hi\u2081, hi\u2082, hi\u2083, hipos, hineg\u27e9 := s.toJordanDecomposition_spec", "annotated_tactic": ["obtain \u27e8i, hi\u2081, hi\u2082, hi\u2083, hipos, hineg\u27e9 := s.toJordanDecomposition_spec", []], "state_before": "case mp.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 t \u2286 u, \u2191s t = 0\nhu\u2082 : \u2200 t_1 \u2286 u\u1d9c, \u2191t t_1 = 0\n\u22a2 s.totalVariation \u27c2\u2098 t.totalVariation", "state_after": "case mp.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 t \u2286 u, \u2191s t = 0\nhu\u2082 : \u2200 t_1 \u2286 u\u1d9c, \u2191t t_1 = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhipos : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi\u2081 hi\u2082\nhineg : s.toJordanDecomposition.negPart = s.toMeasureOfLEZero i\u1d9c \u22ef hi\u2083\n\u22a2 s.totalVariation \u27c2\u2098 t.totalVariation"}, {"tactic": "obtain \u27e8j, hj\u2081, hj\u2082, hj\u2083, hjpos, hjneg\u27e9 := t.toJordanDecomposition_spec", "annotated_tactic": ["obtain \u27e8j, hj\u2081, hj\u2082, hj\u2083, hjpos, hjneg\u27e9 := t.toJordanDecomposition_spec", []], "state_before": "case mp.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 t \u2286 u, \u2191s t = 0\nhu\u2082 : \u2200 t_1 \u2286 u\u1d9c, \u2191t t_1 = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhipos : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi\u2081 hi\u2082\nhineg : s.toJordanDecomposition.negPart = s.toMeasureOfLEZero i\u1d9c \u22ef hi\u2083\n\u22a2 s.totalVariation \u27c2\u2098 t.totalVariation", "state_after": "case mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 t \u2286 u, \u2191s t = 0\nhu\u2082 : \u2200 t_1 \u2286 u\u1d9c, \u2191t t_1 = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhipos : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi\u2081 hi\u2082\nhineg : s.toJordanDecomposition.negPart = s.toMeasureOfLEZero i\u1d9c \u22ef hi\u2083\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : VectorMeasure.restrict 0 j \u2264 VectorMeasure.restrict t j\nhj\u2083 : VectorMeasure.restrict t j\u1d9c \u2264 VectorMeasure.restrict 0 j\u1d9c\nhjpos : t.toJordanDecomposition.posPart = t.toMeasureOfZeroLE j hj\u2081 hj\u2082\nhjneg : t.toJordanDecomposition.negPart = t.toMeasureOfLEZero j\u1d9c \u22ef hj\u2083\n\u22a2 s.totalVariation \u27c2\u2098 t.totalVariation"}, {"tactic": "refine \u27e8u, hmeas, ?_, ?_\u27e9", "annotated_tactic": ["refine \u27e8u, hmeas, ?_, ?_\u27e9", []], "state_before": "case mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 t \u2286 u, \u2191s t = 0\nhu\u2082 : \u2200 t_1 \u2286 u\u1d9c, \u2191t t_1 = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhipos : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi\u2081 hi\u2082\nhineg : s.toJordanDecomposition.negPart = s.toMeasureOfLEZero i\u1d9c \u22ef hi\u2083\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : VectorMeasure.restrict 0 j \u2264 VectorMeasure.restrict t j\nhj\u2083 : VectorMeasure.restrict t j\u1d9c \u2264 VectorMeasure.restrict 0 j\u1d9c\nhjpos : t.toJordanDecomposition.posPart = t.toMeasureOfZeroLE j hj\u2081 hj\u2082\nhjneg : t.toJordanDecomposition.negPart = t.toMeasureOfLEZero j\u1d9c \u22ef hj\u2083\n\u22a2 s.totalVariation \u27c2\u2098 t.totalVariation", "state_after": "case mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 t \u2286 u, \u2191s t = 0\nhu\u2082 : \u2200 t_1 \u2286 u\u1d9c, \u2191t t_1 = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhipos : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi\u2081 hi\u2082\nhineg : s.toJordanDecomposition.negPart = s.toMeasureOfLEZero i\u1d9c \u22ef hi\u2083\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : VectorMeasure.restrict 0 j \u2264 VectorMeasure.restrict t j\nhj\u2083 : VectorMeasure.restrict t j\u1d9c \u2264 VectorMeasure.restrict 0 j\u1d9c\nhjpos : t.toJordanDecomposition.posPart = t.toMeasureOfZeroLE j hj\u2081 hj\u2082\nhjneg : t.toJordanDecomposition.negPart = t.toMeasureOfLEZero j\u1d9c \u22ef hj\u2083\n\u22a2 s.totalVariation u = 0\n\ncase mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 t \u2286 u, \u2191s t = 0\nhu\u2082 : \u2200 t_1 \u2286 u\u1d9c, \u2191t t_1 = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhipos : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi\u2081 hi\u2082\nhineg : s.toJordanDecomposition.negPart = s.toMeasureOfLEZero i\u1d9c \u22ef hi\u2083\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : VectorMeasure.restrict 0 j \u2264 VectorMeasure.restrict t j\nhj\u2083 : VectorMeasure.restrict t j\u1d9c \u2264 VectorMeasure.restrict 0 j\u1d9c\nhjpos : t.toJordanDecomposition.posPart = t.toMeasureOfZeroLE j hj\u2081 hj\u2082\nhjneg : t.toJordanDecomposition.negPart = t.toMeasureOfLEZero j\u1d9c \u22ef hj\u2083\n\u22a2 t.totalVariation u\u1d9c = 0"}, {"tactic": "rw [totalVariation, Measure.add_apply, hipos, hineg, toMeasureOfZeroLE_apply _ _ _ hmeas,\n  toMeasureOfLEZero_apply _ _ _ hmeas]", "annotated_tactic": ["rw [<a>totalVariation</a>, <a>Measure.add_apply</a>, hipos, hineg, <a>toMeasureOfZeroLE_apply</a> _ _ _ hmeas,\n        <a>toMeasureOfLEZero_apply</a> _ _ _ hmeas]", [{"full_name": "MeasureTheory.SignedMeasure.totalVariation", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [497, 5], "def_end_pos": [497, 19]}, {"full_name": "MeasureTheory.Measure.add_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [862, 9], "def_end_pos": [862, 18]}, {"full_name": "MeasureTheory.SignedMeasure.toMeasureOfZeroLE_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [1324, 9], "def_end_pos": [1324, 32]}, {"full_name": "MeasureTheory.SignedMeasure.toMeasureOfLEZero_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [1338, 9], "def_end_pos": [1338, 32]}]], "state_before": "case mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 t \u2286 u, \u2191s t = 0\nhu\u2082 : \u2200 t_1 \u2286 u\u1d9c, \u2191t t_1 = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhipos : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi\u2081 hi\u2082\nhineg : s.toJordanDecomposition.negPart = s.toMeasureOfLEZero i\u1d9c \u22ef hi\u2083\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : VectorMeasure.restrict 0 j \u2264 VectorMeasure.restrict t j\nhj\u2083 : VectorMeasure.restrict t j\u1d9c \u2264 VectorMeasure.restrict 0 j\u1d9c\nhjpos : t.toJordanDecomposition.posPart = t.toMeasureOfZeroLE j hj\u2081 hj\u2082\nhjneg : t.toJordanDecomposition.negPart = t.toMeasureOfLEZero j\u1d9c \u22ef hj\u2083\n\u22a2 s.totalVariation u = 0", "state_after": "case mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 t \u2286 u, \u2191s t = 0\nhu\u2082 : \u2200 t_1 \u2286 u\u1d9c, \u2191t t_1 = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhipos : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi\u2081 hi\u2082\nhineg : s.toJordanDecomposition.negPart = s.toMeasureOfLEZero i\u1d9c \u22ef hi\u2083\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : VectorMeasure.restrict 0 j \u2264 VectorMeasure.restrict t j\nhj\u2083 : VectorMeasure.restrict t j\u1d9c \u2264 VectorMeasure.restrict 0 j\u1d9c\nhjpos : t.toJordanDecomposition.posPart = t.toMeasureOfZeroLE j hj\u2081 hj\u2082\nhjneg : t.toJordanDecomposition.negPart = t.toMeasureOfLEZero j\u1d9c \u22ef hj\u2083\n\u22a2 \u2191\u27e8\u2191s (i \u2229 u), \u22ef\u27e9 + \u2191\u27e8-\u2191s (i\u1d9c \u2229 u), \u22ef\u27e9 = 0"}, {"tactic": "simp [hu\u2081 _ Set.inter_subset_right, \u2190 NNReal.eq_iff]", "annotated_tactic": ["simp [hu\u2081 _ <a>Set.inter_subset_right</a>, \u2190 <a>NNReal.eq_iff</a>]", [{"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [934, 9], "def_end_pos": [934, 27]}, {"full_name": "NNReal.eq_iff", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [101, 19], "def_end_pos": [101, 25]}]], "state_before": "case mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 t \u2286 u, \u2191s t = 0\nhu\u2082 : \u2200 t_1 \u2286 u\u1d9c, \u2191t t_1 = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhipos : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi\u2081 hi\u2082\nhineg : s.toJordanDecomposition.negPart = s.toMeasureOfLEZero i\u1d9c \u22ef hi\u2083\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : VectorMeasure.restrict 0 j \u2264 VectorMeasure.restrict t j\nhj\u2083 : VectorMeasure.restrict t j\u1d9c \u2264 VectorMeasure.restrict 0 j\u1d9c\nhjpos : t.toJordanDecomposition.posPart = t.toMeasureOfZeroLE j hj\u2081 hj\u2082\nhjneg : t.toJordanDecomposition.negPart = t.toMeasureOfLEZero j\u1d9c \u22ef hj\u2083\n\u22a2 \u2191\u27e8\u2191s (i \u2229 u), \u22ef\u27e9 + \u2191\u27e8-\u2191s (i\u1d9c \u2229 u), \u22ef\u27e9 = 0", "state_after": "no goals"}, {"tactic": "rw [totalVariation, Measure.add_apply, hjpos, hjneg,\n  toMeasureOfZeroLE_apply _ _ _ hmeas.compl,\n  toMeasureOfLEZero_apply _ _ _ hmeas.compl]", "annotated_tactic": ["rw [<a>totalVariation</a>, <a>Measure.add_apply</a>, hjpos, hjneg,\n        <a>toMeasureOfZeroLE_apply</a> _ _ _ hmeas.compl,\n        <a>toMeasureOfLEZero_apply</a> _ _ _ hmeas.compl]", [{"full_name": "MeasureTheory.SignedMeasure.totalVariation", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [497, 5], "def_end_pos": [497, 19]}, {"full_name": "MeasureTheory.Measure.add_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [862, 9], "def_end_pos": [862, 18]}, {"full_name": "MeasureTheory.SignedMeasure.toMeasureOfZeroLE_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [1324, 9], "def_end_pos": [1324, 32]}, {"full_name": "MeasureTheory.SignedMeasure.toMeasureOfLEZero_apply", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [1338, 9], "def_end_pos": [1338, 32]}]], "state_before": "case mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 t \u2286 u, \u2191s t = 0\nhu\u2082 : \u2200 t_1 \u2286 u\u1d9c, \u2191t t_1 = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhipos : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi\u2081 hi\u2082\nhineg : s.toJordanDecomposition.negPart = s.toMeasureOfLEZero i\u1d9c \u22ef hi\u2083\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : VectorMeasure.restrict 0 j \u2264 VectorMeasure.restrict t j\nhj\u2083 : VectorMeasure.restrict t j\u1d9c \u2264 VectorMeasure.restrict 0 j\u1d9c\nhjpos : t.toJordanDecomposition.posPart = t.toMeasureOfZeroLE j hj\u2081 hj\u2082\nhjneg : t.toJordanDecomposition.negPart = t.toMeasureOfLEZero j\u1d9c \u22ef hj\u2083\n\u22a2 t.totalVariation u\u1d9c = 0", "state_after": "case mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 t \u2286 u, \u2191s t = 0\nhu\u2082 : \u2200 t_1 \u2286 u\u1d9c, \u2191t t_1 = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhipos : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi\u2081 hi\u2082\nhineg : s.toJordanDecomposition.negPart = s.toMeasureOfLEZero i\u1d9c \u22ef hi\u2083\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : VectorMeasure.restrict 0 j \u2264 VectorMeasure.restrict t j\nhj\u2083 : VectorMeasure.restrict t j\u1d9c \u2264 VectorMeasure.restrict 0 j\u1d9c\nhjpos : t.toJordanDecomposition.posPart = t.toMeasureOfZeroLE j hj\u2081 hj\u2082\nhjneg : t.toJordanDecomposition.negPart = t.toMeasureOfLEZero j\u1d9c \u22ef hj\u2083\n\u22a2 \u2191\u27e8\u2191t (j \u2229 u\u1d9c), \u22ef\u27e9 + \u2191\u27e8-\u2191t (j\u1d9c \u2229 u\u1d9c), \u22ef\u27e9 = 0"}, {"tactic": "simp [hu\u2082 _ Set.inter_subset_right, \u2190 NNReal.eq_iff]", "annotated_tactic": ["simp [hu\u2082 _ <a>Set.inter_subset_right</a>, \u2190 <a>NNReal.eq_iff</a>]", [{"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [934, 9], "def_end_pos": [934, 27]}, {"full_name": "NNReal.eq_iff", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [101, 19], "def_end_pos": [101, 25]}]], "state_before": "case mp.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : \u2200 t \u2286 u, \u2191s t = 0\nhu\u2082 : \u2200 t_1 \u2286 u\u1d9c, \u2191t t_1 = 0\ni : Set \u03b1\nhi\u2081 : MeasurableSet i\nhi\u2082 : VectorMeasure.restrict 0 i \u2264 VectorMeasure.restrict s i\nhi\u2083 : VectorMeasure.restrict s i\u1d9c \u2264 VectorMeasure.restrict 0 i\u1d9c\nhipos : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi\u2081 hi\u2082\nhineg : s.toJordanDecomposition.negPart = s.toMeasureOfLEZero i\u1d9c \u22ef hi\u2083\nj : Set \u03b1\nhj\u2081 : MeasurableSet j\nhj\u2082 : VectorMeasure.restrict 0 j \u2264 VectorMeasure.restrict t j\nhj\u2083 : VectorMeasure.restrict t j\u1d9c \u2264 VectorMeasure.restrict 0 j\u1d9c\nhjpos : t.toJordanDecomposition.posPart = t.toMeasureOfZeroLE j hj\u2081 hj\u2082\nhjneg : t.toJordanDecomposition.negPart = t.toMeasureOfLEZero j\u1d9c \u22ef hj\u2083\n\u22a2 \u2191\u27e8\u2191t (j \u2229 u\u1d9c), \u22ef\u27e9 + \u2191\u27e8-\u2191t (j\u1d9c \u2229 u\u1d9c), \u22ef\u27e9 = 0", "state_after": "no goals"}, {"tactic": "rintro \u27e8u, hmeas, hu\u2081, hu\u2082\u27e9", "annotated_tactic": ["rintro \u27e8u, hmeas, hu\u2081, hu\u2082\u27e9", []], "state_before": "case mpr\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\n\u22a2 s.totalVariation \u27c2\u2098 t.totalVariation \u2192 s \u27c2\u1d65 t", "state_after": "case mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : s.totalVariation u = 0\nhu\u2082 : t.totalVariation u\u1d9c = 0\n\u22a2 s \u27c2\u1d65 t"}, {"tactic": "exact\n  \u27e8u, hmeas, fun t htu => null_of_totalVariation_zero _ (measure_mono_null htu hu\u2081),\n    fun t htv => null_of_totalVariation_zero _ (measure_mono_null htv hu\u2082)\u27e9", "annotated_tactic": ["exact\n      \u27e8u, hmeas, fun t htu => <a>null_of_totalVariation_zero</a> _ (<a>measure_mono_null</a> htu hu\u2081),\n        fun t htv => <a>null_of_totalVariation_zero</a> _ (<a>measure_mono_null</a> htv hu\u2082)\u27e9", [{"full_name": "MeasureTheory.SignedMeasure.null_of_totalVariation_zero", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [509, 9], "def_end_pos": [509, 36]}, {"full_name": "MeasureTheory.measure_mono_null", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [56, 9], "def_end_pos": [56, 26]}, {"full_name": "MeasureTheory.SignedMeasure.null_of_totalVariation_zero", "def_path": "Mathlib/MeasureTheory/Decomposition/Jordan.lean", "def_pos": [509, 9], "def_end_pos": [509, 36]}, {"full_name": "MeasureTheory.measure_mono_null", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [56, 9], "def_end_pos": [56, 26]}]], "state_before": "case mpr.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : MeasurableSpace \u03b1\ns t : SignedMeasure \u03b1\nu : Set \u03b1\nhmeas : MeasurableSet u\nhu\u2081 : s.totalVariation u = 0\nhu\u2082 : t.totalVariation u\u1d9c = 0\n\u22a2 s \u27c2\u1d65 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean", "full_name": "Complex.natCast_cpow_natCast_mul", "start": [230, 1], "end": [233, 49], "traced_tactics": [{"tactic": "refine cpow_nat_mul' (x := n) (n := m) ?_ ?_ z", "annotated_tactic": ["refine <a>cpow_nat_mul'</a> (x := n) (n := m) ?_ ?_ z", [{"full_name": "Complex.cpow_nat_mul'", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean", "def_pos": [188, 7], "def_end_pos": [188, 20]}]], "state_before": "n m : \u2115\nz : \u2102\n\u22a2 \u2191n ^ (\u2191m * z) = (\u2191n ^ m) ^ z", "state_after": "case refine_1\nn m : \u2115\nz : \u2102\n\u22a2 -\u03c0 < \u2191m * (\u2191n).arg\n\ncase refine_2\nn m : \u2115\nz : \u2102\n\u22a2 \u2191m * (\u2191n).arg \u2264 \u03c0"}, {"tactic": "simp only [natCast_arg, mul_zero, Left.neg_neg_iff, pi_pos]", "annotated_tactic": ["simp only [<a>natCast_arg</a>, <a>mul_zero</a>, <a>Left.neg_neg_iff</a>, <a>pi_pos</a>]", [{"full_name": "Complex.natCast_arg", "def_path": "Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean", "def_pos": [237, 7], "def_end_pos": [237, 18]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}, {"full_name": "Left.neg_neg_iff", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [164, 3], "def_end_pos": [164, 14]}, {"full_name": "Real.pi_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "def_pos": [169, 9], "def_end_pos": [169, 15]}]], "state_before": "case refine_1\nn m : \u2115\nz : \u2102\n\u22a2 -\u03c0 < \u2191m * (\u2191n).arg", "state_after": "no goals"}, {"tactic": "simp only [natCast_arg, mul_zero, pi_pos.le]", "annotated_tactic": ["simp only [<a>natCast_arg</a>, <a>mul_zero</a>, pi_pos.le]", [{"full_name": "Complex.natCast_arg", "def_path": "Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean", "def_pos": [237, 7], "def_end_pos": [237, 18]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}]], "state_before": "case refine_2\nn m : \u2115\nz : \u2102\n\u22a2 \u2191m * (\u2191n).arg \u2264 \u03c0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "full_name": "CategoryTheory.Limits.pullbackConeOfRightIso_x", "start": [1722, 1], "end": [1722, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/MonoCoprod.lean", "full_name": "CategoryTheory.Limits.MonoCoprod.binaryCofan_inr", "start": [63, 1], "end": [69, 30], "traced_tactics": [{"tactic": "haveI hc' : IsColimit (BinaryCofan.mk c.inr c.inl) :=\n  BinaryCofan.IsColimit.mk _ (fun f\u2081 f\u2082 => hc.desc (BinaryCofan.mk f\u2082 f\u2081))\n    (by aesop_cat) (by aesop_cat)\n    (fun f\u2081 f\u2082 m h\u2081 h\u2082 => BinaryCofan.IsColimit.hom_ext hc (by aesop_cat) (by aesop_cat))", "annotated_tactic": ["haveI hc' : <a>IsColimit</a> (<a>BinaryCofan.mk</a> c.inr c.inl) :=\n    <a>BinaryCofan.IsColimit.mk</a> _ (fun f\u2081 f\u2082 => hc.desc (<a>BinaryCofan.mk</a> f\u2082 f\u2081))\n      (by aesop_cat) (by aesop_cat)\n      (fun f\u2081 f\u2082 m h\u2081 h\u2082 => <a>BinaryCofan.IsColimit.hom_ext</a> hc (by aesop_cat) (by aesop_cat))", [{"full_name": "CategoryTheory.Limits.IsColimit", "def_path": "Mathlib/CategoryTheory/Limits/IsLimit.lean", "def_pos": [566, 11], "def_end_pos": [566, 20]}, {"full_name": "CategoryTheory.Limits.BinaryCofan.mk", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean", "def_pos": [306, 5], "def_end_pos": [306, 19]}, {"full_name": "CategoryTheory.Limits.BinaryCofan.IsColimit.mk", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean", "def_pos": [266, 5], "def_end_pos": [266, 29]}, {"full_name": "CategoryTheory.Limits.BinaryCofan.mk", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean", "def_pos": [306, 5], "def_end_pos": [306, 19]}, {"full_name": "CategoryTheory.Limits.BinaryCofan.IsColimit.hom_ext", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean", "def_pos": [282, 9], "def_end_pos": [282, 38]}]], "state_before": "C : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\nA B : C\ninst\u271d : MonoCoprod C\nc : BinaryCofan A B\nhc : IsColimit c\n\u22a2 Mono c.inr", "state_after": "C : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\nA B : C\ninst\u271d : MonoCoprod C\nc : BinaryCofan A B\nhc : IsColimit c\nhc' : IsColimit (BinaryCofan.mk c.inr c.inl)\n\u22a2 Mono c.inr"}, {"tactic": "exact binaryCofan_inl _ hc'", "annotated_tactic": ["exact <a>binaryCofan_inl</a> _ hc'", [{"full_name": "CategoryTheory.Limits.MonoCoprod.binaryCofan_inl", "def_path": "Mathlib/CategoryTheory/Limits/MonoCoprod.lean", "def_pos": [49, 3], "def_end_pos": [49, 18]}]], "state_before": "C : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\nA B : C\ninst\u271d : MonoCoprod C\nc : BinaryCofan A B\nhc : IsColimit c\nhc' : IsColimit (BinaryCofan.mk c.inr c.inl)\n\u22a2 Mono c.inr", "state_after": "no goals"}, {"tactic": "aesop_cat", "annotated_tactic": ["aesop_cat", []], "state_before": "C : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\nA B : C\ninst\u271d : MonoCoprod C\nc : BinaryCofan A B\nhc : IsColimit c\n\u22a2 \u2200 {T : C} (f : (pair A B).obj { as := WalkingPair.right } \u27f6 T) (g : (pair A B).obj { as := WalkingPair.left } \u27f6 T),\n    (BinaryCofan.mk c.inr c.inl).inl \u226b (fun {T} f\u2081 f\u2082 => hc.desc (BinaryCofan.mk f\u2082 f\u2081)) f g = f", "state_after": "no goals"}, {"tactic": "aesop_cat", "annotated_tactic": ["aesop_cat", []], "state_before": "C : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\nA B : C\ninst\u271d : MonoCoprod C\nc : BinaryCofan A B\nhc : IsColimit c\n\u22a2 \u2200 {T : C} (f : (pair A B).obj { as := WalkingPair.right } \u27f6 T) (g : (pair A B).obj { as := WalkingPair.left } \u27f6 T),\n    (BinaryCofan.mk c.inr c.inl).inr \u226b (fun {T} f\u2081 f\u2082 => hc.desc (BinaryCofan.mk f\u2082 f\u2081)) f g = g", "state_after": "no goals"}, {"tactic": "aesop_cat", "annotated_tactic": ["aesop_cat", []], "state_before": "C : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\nA B : C\ninst\u271d : MonoCoprod C\nc : BinaryCofan A B\nhc : IsColimit c\nT\u271d : C\nf\u2081 : (pair A B).obj { as := WalkingPair.right } \u27f6 T\u271d\nf\u2082 : (pair A B).obj { as := WalkingPair.left } \u27f6 T\u271d\nm : (BinaryCofan.mk c.inr c.inl).pt \u27f6 T\u271d\nh\u2081 : (BinaryCofan.mk c.inr c.inl).inl \u226b m = f\u2081\nh\u2082 : (BinaryCofan.mk c.inr c.inl).inr \u226b m = f\u2082\n\u22a2 c.inl \u226b m = c.inl \u226b (fun {T} f\u2081 f\u2082 => hc.desc (BinaryCofan.mk f\u2082 f\u2081)) f\u2081 f\u2082", "state_after": "no goals"}, {"tactic": "aesop_cat", "annotated_tactic": ["aesop_cat", []], "state_before": "C : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} C\nA B : C\ninst\u271d : MonoCoprod C\nc : BinaryCofan A B\nhc : IsColimit c\nT\u271d : C\nf\u2081 : (pair A B).obj { as := WalkingPair.right } \u27f6 T\u271d\nf\u2082 : (pair A B).obj { as := WalkingPair.left } \u27f6 T\u271d\nm : (BinaryCofan.mk c.inr c.inl).pt \u27f6 T\u271d\nh\u2081 : (BinaryCofan.mk c.inr c.inl).inl \u226b m = f\u2081\nh\u2082 : (BinaryCofan.mk c.inr c.inl).inr \u226b m = f\u2082\n\u22a2 c.inr \u226b m = c.inr \u226b (fun {T} f\u2081 f\u2082 => hc.desc (BinaryCofan.mk f\u2082 f\u2081)) f\u2081 f\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.tendsto_Iic_atBot", "start": [1748, 1], "end": [1750, 58], "traced_tactics": [{"tactic": "rw [atBot_Iic_eq, tendsto_comap_iff, Function.comp_def]", "annotated_tactic": ["rw [<a>atBot_Iic_eq</a>, <a>tendsto_comap_iff</a>, <a>Function.comp_def</a>]", [{"full_name": "Filter.atBot_Iic_eq", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1729, 9], "def_end_pos": [1729, 21]}, {"full_name": "Filter.tendsto_comap_iff", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3140, 9], "def_end_pos": [3140, 26]}, {"full_name": "Function.comp_def", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [37, 9], "def_end_pos": [37, 26]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ninst\u271d : SemilatticeInf \u03b1\na : \u03b1\nf : \u03b2 \u2192 \u2191(Iic a)\nl : Filter \u03b2\n\u22a2 Tendsto f l atBot \u2194 Tendsto (fun x => \u2191(f x)) l atBot", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean", "full_name": "ContinuousMap.adjoin_id_eq_span_one_add", "start": [500, 1], "end": [507, 21], "traced_tactics": [{"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "X : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\n\ud835\udd5c : Type u_2\ninst\u271d : RCLike \ud835\udd5c\ns : Set \ud835\udd5c\n\u22a2 \u2191(StarAlgebra.adjoin \ud835\udd5c {restrict s (ContinuousMap.id \ud835\udd5c)}) =\n    \u2191(span \ud835\udd5c {1}) + \u2191(adjoin \ud835\udd5c {restrict s (ContinuousMap.id \ud835\udd5c)})", "state_after": "case h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\n\ud835\udd5c : Type u_2\ninst\u271d : RCLike \ud835\udd5c\ns : Set \ud835\udd5c\nx : C(\u2191s, \ud835\udd5c)\n\u22a2 x \u2208 \u2191(StarAlgebra.adjoin \ud835\udd5c {restrict s (ContinuousMap.id \ud835\udd5c)}) \u2194\n    x \u2208 \u2191(span \ud835\udd5c {1}) + \u2191(adjoin \ud835\udd5c {restrict s (ContinuousMap.id \ud835\udd5c)})"}, {"tactic": "rw [SetLike.mem_coe, \u2190 StarAlgebra.adjoin_nonUnitalStarSubalgebra,\n  \u2190 StarSubalgebra.mem_toSubalgebra, \u2190 Subalgebra.mem_toSubmodule,\n  StarAlgebra.adjoin_nonUnitalStarSubalgebra_eq_span, mem_sup]", "annotated_tactic": ["rw [<a>SetLike.mem_coe</a>, \u2190 <a>StarAlgebra.adjoin_nonUnitalStarSubalgebra</a>,\n    \u2190 <a>StarSubalgebra.mem_toSubalgebra</a>, \u2190 <a>Subalgebra.mem_toSubmodule</a>,\n    <a>StarAlgebra.adjoin_nonUnitalStarSubalgebra_eq_span</a>, <a>mem_sup</a>]", [{"full_name": "SetLike.mem_coe", "def_path": "Mathlib/Data/SetLike/Basic.lean", "def_pos": [180, 9], "def_end_pos": [180, 16]}, {"full_name": "StarAlgebra.adjoin_nonUnitalStarSubalgebra", "def_path": "Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean", "def_pos": [342, 7], "def_end_pos": [342, 49]}, {"full_name": "StarSubalgebra.mem_toSubalgebra", "def_path": "Mathlib/Algebra/Star/Subalgebra.lean", "def_pos": [93, 9], "def_end_pos": [93, 25]}, {"full_name": "Subalgebra.mem_toSubmodule", "def_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "def_pos": [294, 9], "def_end_pos": [294, 24]}, {"full_name": "StarAlgebra.adjoin_nonUnitalStarSubalgebra_eq_span", "def_path": "Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean", "def_pos": [331, 7], "def_end_pos": [331, 57]}, {"full_name": "Submodule.mem_sup", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [425, 9], "def_end_pos": [425, 16]}]], "state_before": "case h\nX : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\n\ud835\udd5c : Type u_2\ninst\u271d : RCLike \ud835\udd5c\ns : Set \ud835\udd5c\nx : C(\u2191s, \ud835\udd5c)\n\u22a2 x \u2208 \u2191(StarAlgebra.adjoin \ud835\udd5c {restrict s (ContinuousMap.id \ud835\udd5c)}) \u2194\n    x \u2208 \u2191(span \ud835\udd5c {1}) + \u2191(adjoin \ud835\udd5c {restrict s 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"PrimeSpectrum.closedEmbedding_comap_of_surjective", "start": [763, 1], "end": [766, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/List/Lemmas.lean", "full_name": "List.getElem_range'", "start": [1372, 9], "end": [1374, 70], "traced_tactics": [{"tactic": "simpa using H", "annotated_tactic": ["simpa using H", []], "state_before": "n m step i : Nat\nH : i < (range' n m step).length\n\u22a2 i < m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Strict.lean", "full_name": "StrictConvex.smul_mem_of_zero_mem", "start": [353, 1], "end": [355, 71], "traced_tactics": [{"tactic": "simpa using hs.add_smul_mem zero_mem (by simpa using hx) hx\u2080 ht\u2080 ht\u2081", "annotated_tactic": ["simpa using hs.add_smul_mem zero_mem (by simpa using hx) hx\u2080 ht\u2080 ht\u2081", []], "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5d : Type u_2\nE : Type u_3\nF : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2076 : OrderedRing \ud835\udd5c\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : TopologicalSpace F\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns t\u271d : Set E\nx y : E\nhs : StrictConvex \ud835\udd5c s\nzero_mem : 0 \u2208 s\nhx : x \u2208 s\nhx\u2080 : x \u2260 0\nt : \ud835\udd5c\nht\u2080 : 0 < t\nht\u2081 : t < 1\n\u22a2 t \u2022 x \u2208 interior s", "state_after": "no goals"}, {"tactic": "simpa using hx", "annotated_tactic": ["simpa using hx", []], "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5d : Type u_2\nE : Type u_3\nF : Type u_4\n\u03b2 : Type u_5\ninst\u271d\u2076 : OrderedRing \ud835\udd5c\ninst\u271d\u2075 : TopologicalSpace E\ninst\u271d\u2074 : TopologicalSpace F\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : AddCommGroup F\ninst\u271d\u00b9 : Module \ud835\udd5c E\ninst\u271d : Module \ud835\udd5c F\ns t\u271d : Set E\nx y : E\nhs : StrictConvex \ud835\udd5c s\nzero_mem : 0 \u2208 s\nhx : x \u2208 s\nhx\u2080 : x \u2260 0\nt : \ud835\udd5c\nht\u2080 : 0 < t\nht\u2081 : t < 1\n\u22a2 0 + x \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Sites/Subsheaf.lean", "full_name": "CategoryTheory.GrothendieckTopology.Subpresheaf.eq_sheafify_iff", "start": [258, 1], "end": [260, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "full_name": "Orthonormal.inner_finsupp_eq_zero", "start": [893, 1], "end": [897, 55], "traced_tactics": [{"tactic": "rw [Finsupp.mem_supported'] at hl", "annotated_tactic": ["rw [<a>Finsupp.mem_supported'</a>] at hl", [{"full_name": "Finsupp.mem_supported'", "def_path": "Mathlib/LinearAlgebra/Finsupp.lean", "def_pos": [314, 9], "def_end_pos": [314, 23]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\n\u03b9 : Type u_4\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\ns : Set \u03b9\ni : \u03b9\nhi : i \u2209 s\nl : \u03b9 \u2192\u2080 \ud835\udd5c\nhl : l \u2208 Finsupp.supported \ud835\udd5c \ud835\udd5c s\n\u22a2 \u27ea(Finsupp.total \u03b9 E \ud835\udd5c v) l, v i\u27eb_\ud835\udd5c = 0", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\n\u03b9 : Type u_4\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\ns : Set \u03b9\ni : \u03b9\nhi : i \u2209 s\nl : \u03b9 \u2192\u2080 \ud835\udd5c\nhl : \u2200 x \u2209 s, l x = 0\n\u22a2 \u27ea(Finsupp.total \u03b9 E \ud835\udd5c v) l, v i\u27eb_\ud835\udd5c = 0"}, {"tactic": "simp only [hv.inner_left_finsupp, hl i hi, map_zero]", "annotated_tactic": ["simp only [hv.inner_left_finsupp, hl i hi, <a>map_zero</a>]", [{"full_name": "map_zero", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [203, 3], "def_end_pos": [203, 14]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\n\u03b9 : Type u_4\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\ns : Set \u03b9\ni : \u03b9\nhi : i \u2209 s\nl : \u03b9 \u2192\u2080 \ud835\udd5c\nhl : \u2200 x \u2209 s, l x = 0\n\u22a2 \u27ea(Finsupp.total \u03b9 E \ud835\udd5c v) l, v i\u27eb_\ud835\udd5c = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/MulAction.lean", "full_name": "AddTorsor.connectedSpace", "start": [293, 11], "end": [299, 34], "traced_tactics": [{"tactic": "convert\n  isPreconnected_univ.image (Equiv.vaddConst (Classical.arbitrary P) : G \u2192 P)\n    (continuous_id.vadd continuous_const).continuousOn", "annotated_tactic": ["convert\n        isPreconnected_univ.image (<a>Equiv.vaddConst</a> (<a>Classical.arbitrary</a> P) : G \u2192 P)\n          (continuous_id.vadd <a>continuous_const</a>).<a>continuousOn</a>", [{"full_name": "Equiv.vaddConst", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [355, 5], "def_end_pos": [355, 14]}, {"full_name": "Classical.arbitrary", "def_path": "Mathlib/Logic/Nonempty.lean", "def_pos": [99, 32], "def_end_pos": [99, 51]}, {"full_name": "continuous_const", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1675, 9], "def_end_pos": [1675, 25]}, {"full_name": "Continuous.continuousOn", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [968, 9], "def_end_pos": [968, 32]}]], "state_before": "G : Type u_1\nP : Type u_2\ninst\u271d\u2075 : AddGroup G\ninst\u271d\u2074 : AddTorsor G P\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : PreconnectedSpace G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : ContinuousVAdd G P\n\u22a2 IsPreconnected Set.univ", "state_after": "case h.e'_3\nG : Type u_1\nP : Type u_2\ninst\u271d\u2075 : AddGroup G\ninst\u271d\u2074 : AddTorsor G P\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : PreconnectedSpace G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : ContinuousVAdd G P\n\u22a2 Set.univ = \u21d1(Equiv.vaddConst (Classical.arbitrary P)) '' Set.univ"}, {"tactic": "rw [Set.image_univ, Equiv.range_eq_univ]", "annotated_tactic": ["rw [<a>Set.image_univ</a>, <a>Equiv.range_eq_univ</a>]", [{"full_name": "Set.image_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [701, 9], "def_end_pos": [701, 19]}, {"full_name": "Equiv.range_eq_univ", "def_path": "Mathlib/Logic/Equiv/Set.lean", "def_pos": [37, 9], "def_end_pos": [37, 22]}]], "state_before": "case h.e'_3\nG : Type u_1\nP : Type u_2\ninst\u271d\u2075 : AddGroup G\ninst\u271d\u2074 : AddTorsor G P\ninst\u271d\u00b3 : TopologicalSpace G\ninst\u271d\u00b2 : PreconnectedSpace G\ninst\u271d\u00b9 : TopologicalSpace P\ninst\u271d : ContinuousVAdd G P\n\u22a2 Set.univ = \u21d1(Equiv.vaddConst (Classical.arbitrary P)) '' Set.univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convolution.lean", "full_name": "MeasureTheory.ConvolutionExistsAt.integrable", "start": [180, 1], "end": [182, 4], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/BigOperators.lean", "full_name": "Polynomial.coeff_prod_of_natDegree_le", "start": [225, 1], "end": [231, 18], "traced_tactics": [{"tactic": "cases' s with l hl", "annotated_tactic": ["cases' s with l hl", []], "state_before": "R : Type u\n\u03b9 : Type w\ns : Finset \u03b9\ninst\u271d : CommSemiring R\nf\u271d : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nf : \u03b9 \u2192 R[X]\nn : \u2115\nh : \u2200 p \u2208 s, (f p).natDegree \u2264 n\n\u22a2 (\u220f i \u2208 s, f i).coeff (s.card * n) = \u220f i \u2208 s, (f i).coeff n", "state_after": "case mk\nR : Type u\n\u03b9 : Type w\ninst\u271d : CommSemiring R\nf\u271d : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nf : \u03b9 \u2192 R[X]\nn : \u2115\nl : Multiset \u03b9\nhl : l.Nodup\nh : \u2200 p \u2208 { val := l, nodup := hl }, (f p).natDegree \u2264 n\n\u22a2 (\u220f i \u2208 { val := l, nodup := hl }, f i).coeff ({ val := l, nodup := hl }.card * n) =\n    \u220f i \u2208 { val := l, nodup := hl }, (f i).coeff n"}, {"tactic": "convert coeff_multiset_prod_of_natDegree_le (l.map f) n ?_", "annotated_tactic": ["convert <a>coeff_multiset_prod_of_natDegree_le</a> (l.map f) n ?_", [{"full_name": "Polynomial.coeff_multiset_prod_of_natDegree_le", "def_path": "Mathlib/Algebra/Polynomial/BigOperators.lean", "def_pos": [219, 9], "def_end_pos": [219, 44]}]], "state_before": "case mk\nR : Type u\n\u03b9 : Type w\ninst\u271d : CommSemiring R\nf\u271d : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nf : \u03b9 \u2192 R[X]\nn : \u2115\nl : Multiset \u03b9\nhl : l.Nodup\nh : \u2200 p \u2208 { val := l, nodup := hl }, (f p).natDegree \u2264 n\n\u22a2 (\u220f i \u2208 { val := l, nodup := hl }, f i).coeff ({ val := l, nodup := hl }.card * n) =\n    \u220f i \u2208 { val := l, nodup := hl }, (f i).coeff n", "state_after": "case h.e'_2.h.e'_4.h.e'_5\nR : Type u\n\u03b9 : Type w\ninst\u271d : CommSemiring R\nf\u271d : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nf : \u03b9 \u2192 R[X]\nn : \u2115\nl : Multiset \u03b9\nhl : l.Nodup\nh : \u2200 p \u2208 { val := l, nodup := hl }, (f p).natDegree \u2264 n\n\u22a2 { val := l, nodup := hl }.card = Multiset.card (Multiset.map f l)\n\ncase h.e'_3\nR : Type u\n\u03b9 : Type w\ninst\u271d : CommSemiring R\nf\u271d : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nf : \u03b9 \u2192 R[X]\nn : \u2115\nl : Multiset \u03b9\nhl : l.Nodup\nh : \u2200 p \u2208 { val := l, nodup := hl }, (f p).natDegree \u2264 n\n\u22a2 \u220f i \u2208 { val := l, nodup := hl }, (f i).coeff n = (Multiset.map (fun p => p.coeff n) (Multiset.map f l)).prod\n\ncase mk\nR : Type u\n\u03b9 : Type w\ninst\u271d : CommSemiring R\nf\u271d : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nf : \u03b9 \u2192 R[X]\nn : \u2115\nl : Multiset \u03b9\nhl : l.Nodup\nh : \u2200 p \u2208 { val := l, nodup := hl }, (f p).natDegree \u2264 n\n\u22a2 \u2200 p \u2208 Multiset.map f l, p.natDegree \u2264 n"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_2.h.e'_4.h.e'_5\nR : Type u\n\u03b9 : Type w\ninst\u271d : CommSemiring R\nf\u271d : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nf : \u03b9 \u2192 R[X]\nn : \u2115\nl : Multiset \u03b9\nhl : l.Nodup\nh : \u2200 p \u2208 { val := l, nodup := hl }, (f p).natDegree \u2264 n\n\u22a2 { val := l, nodup := hl }.card = Multiset.card (Multiset.map f l)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_3\nR : Type u\n\u03b9 : Type w\ninst\u271d : CommSemiring R\nf\u271d : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nf : \u03b9 \u2192 R[X]\nn : \u2115\nl : Multiset \u03b9\nhl : l.Nodup\nh : \u2200 p \u2208 { val := l, nodup := hl }, (f p).natDegree \u2264 n\n\u22a2 \u220f i \u2208 { val := l, nodup := hl }, (f i).coeff n = (Multiset.map (fun p => p.coeff n) (Multiset.map f l)).prod", "state_after": "no goals"}, {"tactic": "simpa using h", "annotated_tactic": ["simpa using h", []], "state_before": "case mk\nR : Type u\n\u03b9 : Type w\ninst\u271d : CommSemiring R\nf\u271d : \u03b9 \u2192 R[X]\nt : Multiset R[X]\nf : \u03b9 \u2192 R[X]\nn : \u2115\nl : Multiset \u03b9\nhl : l.Nodup\nh : \u2200 p \u2208 { val := l, nodup := hl }, (f p).natDegree \u2264 n\n\u22a2 \u2200 p \u2208 Multiset.map f l, p.natDegree \u2264 n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/TMToPartrec.lean", "full_name": "Turing.PartrecToTM2.copy_ok", "start": [1420, 1], "end": [1430, 17], "traced_tactics": [{"tactic": "induction' b with x b IH generalizing a d s", "annotated_tactic": ["induction' b with x b IH generalizing a d s", []], "state_before": "q : \u039b'\ns : Option \u0393'\na b c d : List \u0393'\n\u22a2 Reaches\u2081 (TM2.step tr) { l := some q.copy, var := s, stk := elim a b c d }\n    { l := some q, var := none, stk := elim (b.reverseAux a) [] c (b.reverseAux d) }", "state_after": "case nil\nq : \u039b'\nc : List \u0393'\ns : Option \u0393'\na d : List \u0393'\n\u22a2 Reaches\u2081 (TM2.step tr) { l := some q.copy, var := s, stk := elim a [] c d }\n    { l := some q, var := none, stk := elim ([].reverseAux a) [] c ([].reverseAux d) }\n\ncase cons\nq : \u039b'\nc : List \u0393'\nx : \u0393'\nb : List \u0393'\nIH :\n  \u2200 (s : Option \u0393') (a d : List \u0393'),\n    Reaches\u2081 (TM2.step tr) { l := some q.copy, var := s, stk := elim a b c d }\n      { l := some q, var := none, stk := elim (b.reverseAux a) [] c (b.reverseAux d) }\ns : Option \u0393'\na d : List \u0393'\n\u22a2 Reaches\u2081 (TM2.step tr) { l := some q.copy, var := s, stk := elim a (x :: b) c d }\n    { l := some q, var := none, stk := elim ((x :: b).reverseAux a) [] c ((x :: b).reverseAux d) }"}, {"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": [410, 9], "def_end_pos": [410, 13]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case cons\nq : \u039b'\nc : List \u0393'\nx : \u0393'\nb : List \u0393'\nIH :\n  \u2200 (s : Option \u0393') (a d : List \u0393'),\n    Reaches\u2081 (TM2.step tr) { l := some q.copy, var := s, stk := elim a b c d }\n      { l := some q, var := none, stk := elim (b.reverseAux a) [] c (b.reverseAux d) }\ns : Option \u0393'\na d : List \u0393'\n\u22a2 Reaches\u2081 (TM2.step tr) { l := some q.copy, var := s, stk := elim a (x :: b) c d }\n    { l := some q, var := none, stk := elim ((x :: b).reverseAux a) [] c ((x :: b).reverseAux d) }", "state_after": "case cons\nq : \u039b'\nc : List \u0393'\nx : \u0393'\nb : List \u0393'\nIH :\n  \u2200 (s : Option \u0393') (a d : List \u0393'),\n    Reaches\u2081 (TM2.step tr) { l := some q.copy, var := s, stk := elim a b c d }\n      { l := some q, var := none, stk := elim (b.reverseAux a) [] c (b.reverseAux d) }\ns : Option \u0393'\na d : List \u0393'\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a) (TM2.stepAux (tr q.copy) s (elim a (x :: b) c d))\n    { l := some q, var := none, stk := elim ((x :: b).reverseAux a) [] c ((x :: b).reverseAux d) }"}, {"tactic": "simp only [TM2.step, Option.mem_def, TM2.stepAux, elim_rev, List.head?_cons, Option.isSome_some,\n  List.tail_cons, elim_update_rev, ne_eq, Function.update_noteq, elim_main, elim_update_main,\n  elim_stack, elim_update_stack, cond_true, List.reverseAux_cons]", "annotated_tactic": ["simp only [<a>TM2.step</a>, <a>Option.mem_def</a>, <a>TM2.stepAux</a>, <a>elim_rev</a>, <a>List.head?_cons</a>, <a>Option.isSome_some</a>,\n    <a>List.tail_cons</a>, <a>elim_update_rev</a>, <a>ne_eq</a>, <a>Function.update_noteq</a>, <a>elim_main</a>, <a>elim_update_main</a>,\n    <a>elim_stack</a>, <a>elim_update_stack</a>, <a>cond_true</a>, <a>List.reverseAux_cons</a>]", [{"full_name": "Turing.TM2.step", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2161, 5], "def_end_pos": [2161, 9]}, {"full_name": "Option.mem_def", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Option/Instances.lean", "def_pos": [24, 17], "def_end_pos": [24, 24]}, {"full_name": "Turing.TM2.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [2149, 5], "def_end_pos": [2149, 12]}, {"full_name": "Turing.PartrecToTM2.K'.elim_rev", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 20]}, {"full_name": "List.head?_cons", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [301, 17], "def_end_pos": [301, 27]}, {"full_name": "Option.isSome_some", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Option/Lemmas.lean", "def_pos": [57, 17], "def_end_pos": [57, 28]}, {"full_name": "List.tail_cons", "def_path": ".lake/packages/batteries/Batteries/Data/List/Basic.lean", "def_pos": [47, 17], "def_end_pos": [47, 26]}, {"full_name": "Turing.PartrecToTM2.K'.elim_update_rev", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1265, 9], "def_end_pos": [1265, 27]}, {"full_name": "ne_eq", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [89, 17], "def_end_pos": [89, 22]}, {"full_name": "Function.update_noteq", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [563, 9], "def_end_pos": [563, 21]}, {"full_name": "Turing.PartrecToTM2.K'.elim_main", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1249, 9], "def_end_pos": [1249, 21]}, {"full_name": "Turing.PartrecToTM2.K'.elim_update_main", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1260, 9], "def_end_pos": [1260, 28]}, {"full_name": "Turing.PartrecToTM2.K'.elim_stack", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1255, 9], "def_end_pos": [1255, 22]}, {"full_name": "Turing.PartrecToTM2.K'.elim_update_stack", "def_path": "Mathlib/Computability/TMToPartrec.lean", "def_pos": [1275, 9], "def_end_pos": [1275, 29]}, {"full_name": "cond_true", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [253, 17], "def_end_pos": [253, 26]}, {"full_name": "List.reverseAux_cons", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [431, 17], "def_end_pos": [431, 32]}]], "state_before": "case cons\nq : \u039b'\nc : List \u0393'\nx : \u0393'\nb : List \u0393'\nIH :\n  \u2200 (s : Option \u0393') (a d : List \u0393'),\n    Reaches\u2081 (TM2.step tr) { l := some q.copy, var := s, stk := elim a b c d }\n      { l := some q, var := none, stk := elim (b.reverseAux a) [] c (b.reverseAux d) }\ns : Option \u0393'\na d : List \u0393'\n\u22a2 TransGen (fun a b => b \u2208 TM2.step tr a) (TM2.stepAux (tr q.copy) s (elim a (x :: b) c d))\n    { l := some q, var := none, stk := elim ((x :: b).reverseAux a) [] c ((x :: b).reverseAux d) }", "state_after": "case cons\nq : \u039b'\nc : List \u0393'\nx : \u0393'\nb : List \u0393'\nIH :\n  \u2200 (s : Option \u0393') (a d : List \u0393'),\n    Reaches\u2081 (TM2.step tr) { l := some q.copy, var := s, stk := elim a b c d }\n      { l := some q, var := none, stk := elim (b.reverseAux a) [] c (b.reverseAux d) }\ns : Option \u0393'\na d : 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 q.copy, var := some x, stk := elim ((some x).iget :: a) b c ((some x).iget :: d) }\n    { l := some q, var := none, stk := elim (b.reverseAux (x :: a)) [] c (b.reverseAux (x :: d)) }"}, {"tactic": "exact IH _ _ _", "annotated_tactic": ["exact IH _ _ _", []], "state_before": "case cons\nq : \u039b'\nc : List \u0393'\nx : \u0393'\nb : List \u0393'\nIH :\n  \u2200 (s : Option \u0393') (a d : List \u0393'),\n    Reaches\u2081 (TM2.step tr) { l := some q.copy, var := s, stk := elim a b c d }\n      { l := some q, var := none, stk := elim (b.reverseAux a) [] c (b.reverseAux d) }\ns : Option \u0393'\na d : 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 q.copy, var := some x, stk := elim ((some x).iget :: a) b c ((some x).iget :: d) }\n    { l := some q, var := none, stk := elim (b.reverseAux (x :: a)) [] c (b.reverseAux (x :: d)) }", "state_after": "no goals"}, {"tactic": "refine TransGen.single ?_", "annotated_tactic": ["refine <a>TransGen.single</a> ?_", [{"full_name": "Relation.TransGen.single", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [268, 5], "def_end_pos": [268, 11]}]], "state_before": "case nil\nq : \u039b'\nc : List \u0393'\ns : Option \u0393'\na d : List \u0393'\n\u22a2 Reaches\u2081 (TM2.step tr) { l := some q.copy, var := s, stk := elim a [] c d }\n    { l := some q, var := none, stk := elim ([].reverseAux a) [] c ([].reverseAux d) }", "state_after": "case nil\nq : \u039b'\nc : List \u0393'\ns : Option \u0393'\na d : List \u0393'\n\u22a2 { l := some q, var := none, stk := elim ([].reverseAux a) [] c ([].reverseAux d) } \u2208\n    TM2.step tr { l := some q.copy, var := s, stk := elim a [] c d }"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case nil\nq : \u039b'\nc : List \u0393'\ns : Option \u0393'\na d : List \u0393'\n\u22a2 { l := some q, var := none, stk := elim ([].reverseAux a) [] c ([].reverseAux d) } \u2208\n    TM2.step tr { l := some q.copy, var := s, stk := elim a [] c d }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Subsemiring/Basic.lean", "full_name": "Subsemiring.coe_sSup_of_directedOn", "start": [1070, 1], "end": [1072, 59], "traced_tactics": [{"tactic": "simp [mem_sSup_of_directedOn Sne hS]", "annotated_tactic": ["simp [<a>mem_sSup_of_directedOn</a> Sne hS]", [{"full_name": "Subsemiring.mem_sSup_of_directedOn", "def_path": "Mathlib/Algebra/Ring/Subsemiring/Basic.lean", "def_pos": [1063, 9], "def_end_pos": [1063, 31]}]], "state_before": "R : Type u\nS\u271d : Type v\nT : Type w\ninst\u271d\u00b2 : NonAssocSemiring R\nM : Submonoid R\ninst\u271d\u00b9 : NonAssocSemiring S\u271d\ninst\u271d : NonAssocSemiring T\nS : Set (Subsemiring R)\nSne : S.Nonempty\nhS : DirectedOn (fun x x_1 => x \u2264 x_1) S\nx : R\n\u22a2 x \u2208 \u2191(sSup S) \u2194 x \u2208 \u22c3 s \u2208 S, \u2191s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "Dense.open_subset_closure_inter", "start": [1409, 1], "end": [1413, 44], "traced_tactics": [{"tactic": "rw [hs.closure_eq, inter_univ]", "annotated_tactic": ["rw [hs.closure_eq, <a>inter_univ</a>]", [{"full_name": "Set.inter_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [981, 9], "def_end_pos": [981, 19]}]], "state_before": "X : Type u\nY : Type v\n\u03b9 : Sort w\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nx : X\ns s\u2081 s\u2082 t : Set X\np p\u2081 p\u2082 : X \u2192 Prop\ninst\u271d : TopologicalSpace X\nhs : Dense s\nht : IsOpen t\n\u22a2 t = t \u2229 closure s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Zsqrtd/Basic.lean", "full_name": "Zsqrtd.intCast_val", "start": [287, 1], "end": [287, 68], "traced_tactics": [{"tactic": "ext <;> simp", "annotated_tactic": ["ext <;> simp", []], "state_before": "d n : \u2124\n\u22a2 \u2191n = { re := n, im := 0 }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Equiv/Basic.lean", "full_name": "MulEquiv.refl_symm", "start": [358, 1], "end": [358, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/LiminfLimsup.lean", "full_name": "OrderIso.liminf_apply", "start": [1525, 1], "end": [1532, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/AffineSpace/ContinuousAffineEquiv.lean", "full_name": "ContinuousAffineEquiv.bijective", "start": [201, 11], "end": [202, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean", "full_name": "qrSign.eq_iff_eq", "start": [427, 1], "end": [434, 61], "traced_tactics": [{"tactic": "refine\n    \u27e8fun h' =>\n      let h := h'.symm\n      ?_,\n      fun h => ?_\u27e9 <;>\n  rw [h, \u2190 mul_assoc, \u2190 pow_two, sq_eq_one hm hn, one_mul]", "annotated_tactic": ["refine\n      \u27e8fun h' =>\n        let h := h'.symm\n        ?_,\n        fun h => ?_\u27e9 <;>\n    rw [h, \u2190 <a>mul_assoc</a>, \u2190 <a>pow_two</a>, <a>sq_eq_one</a> hm hn, <a>one_mul</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "pow_two", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [679, 32], "def_end_pos": [679, 39]}, {"full_name": "qrSign.sq_eq_one", "def_path": "Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean", "def_pos": [406, 9], "def_end_pos": [406, 18]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "m n : \u2115\nhm : Odd m\nhn : Odd n\nx y : \u2124\n\u22a2 qrSign m n * x = y \u2194 x = qrSign m n * y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fintype/Basic.lean", "full_name": "Finset.subset_univ", "start": [128, 1], "end": [128, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/RamificationInertia.lean", "full_name": "Ideal.ramificationIdx_ne_zero", "start": [116, 1], "end": [118, 38], "traced_tactics": [{"tactic": "rwa [ramificationIdx_spec hle hnle]", "annotated_tactic": ["rwa [<a>ramificationIdx_spec</a> hle hnle]", [{"full_name": "Ideal.ramificationIdx_spec", "def_path": "Mathlib/NumberTheory/RamificationInertia.lean", "def_pos": [82, 9], "def_end_pos": [82, 29]}]], "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\nS : Type v\ninst\u271d : CommRing S\nf : R \u2192+* S\np : Ideal R\nP : Ideal S\ne : \u2115\nhe : e \u2260 0\nhle : map f p \u2264 P ^ e\nhnle : \u00acmap f p \u2264 P ^ (e + 1)\n\u22a2 ramificationIdx f p P \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Control/ULift.lean", "full_name": "ULift.bind_up", "start": [123, 1], "end": [124, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": 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".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\na b : \u03b1\nm : Multiset \u03b1\n\u22a2 (a = b \u2227 0 = m \u2228 a \u2260 b \u2227 \u2203 cs, 0 = b ::\u2098 cs \u2227 m = a ::\u2098 cs) \u2194 a = b \u2227 m = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Bounded.lean", "full_name": "Metric.tendsto_dist_left_atTop_iff", "start": [142, 1], "end": [144, 56], "traced_tactics": [{"tactic": "simp only [dist_comm c, tendsto_dist_right_atTop_iff]", "annotated_tactic": ["simp only [<a>dist_comm</a> c, <a>tendsto_dist_right_atTop_iff</a>]", [{"full_name": "dist_comm", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 18]}, {"full_name": "Metric.tendsto_dist_right_atTop_iff", "def_path": "Mathlib/Topology/MetricSpace/Bounded.lean", "def_pos": [137, 9], "def_end_pos": [137, 37]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\n\u03b9 : Type u_2\ninst\u271d : PseudoMetricSpace \u03b1\nx : \u03b1\ns t : Set \u03b1\nr : \u211d\nc : \u03b1\nf : \u03b2 \u2192 \u03b1\nl : Filter \u03b2\n\u22a2 Tendsto (fun x => dist c (f x)) l atTop \u2194 Tendsto f l (cobounded \u03b1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/IsLimit.lean", "full_name": "CategoryTheory.Limits.IsColimit.OfNatIso.cocone_fac", "start": [1048, 1], "end": [1051, 35], "traced_tactics": [{"tactic": "rw [\u2190 coconeOfHom_homOfCocone h s]", "annotated_tactic": ["rw [\u2190 <a>coconeOfHom_homOfCocone</a> h s]", [{"full_name": "CategoryTheory.Limits.IsColimit.OfNatIso.coconeOfHom_homOfCocone", "def_path": "Mathlib/CategoryTheory/Limits/IsLimit.lean", "def_pos": [1016, 9], "def_end_pos": [1016, 32]}]], "state_before": "J : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\nC : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} C\nF : J \u2964 C\nt : Cocone F\nX : C\nh : coyoneda.obj { unop := X } \u22d9 uliftFunctor.{u\u2081, v\u2083} \u2245 F.cocones\ns : Cocone F\n\u22a2 (colimitCocone h).extend (homOfCocone h s) = s", "state_after": "J : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\nC : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} C\nF : J \u2964 C\nt : Cocone F\nX : C\nh : coyoneda.obj { unop := X } \u22d9 uliftFunctor.{u\u2081, v\u2083} \u2245 F.cocones\ns : Cocone F\n\u22a2 (colimitCocone h).extend (homOfCocone h (coconeOfHom h (homOfCocone h s))) = coconeOfHom h (homOfCocone h s)"}, {"tactic": "conv_lhs => simp only [homOfCocone_cooneOfHom]", "annotated_tactic": ["conv_lhs => simp only [<a>homOfCocone_cooneOfHom</a>]", [{"full_name": "CategoryTheory.Limits.IsColimit.OfNatIso.homOfCocone_cooneOfHom", "def_path": "Mathlib/CategoryTheory/Limits/IsLimit.lean", "def_pos": [1024, 9], "def_end_pos": [1024, 31]}]], "state_before": "J : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\nC : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} C\nF : J \u2964 C\nt : Cocone F\nX : C\nh : coyoneda.obj { unop := X } \u22d9 uliftFunctor.{u\u2081, v\u2083} \u2245 F.cocones\ns : Cocone F\n\u22a2 (colimitCocone h).extend (homOfCocone h (coconeOfHom h (homOfCocone h s))) = coconeOfHom h (homOfCocone h s)", "state_after": "J : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\nC : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} C\nF : J \u2964 C\nt : Cocone F\nX : C\nh : coyoneda.obj { unop := X } \u22d9 uliftFunctor.{u\u2081, v\u2083} \u2245 F.cocones\ns : Cocone F\n\u22a2 (colimitCocone h).extend (homOfCocone h s) = coconeOfHom h (homOfCocone h s)"}, {"tactic": "apply (coconeOfHom_fac _ _).symm", "annotated_tactic": ["apply (<a>coconeOfHom_fac</a> _ _).<a>symm</a>", [{"full_name": "CategoryTheory.Limits.IsColimit.OfNatIso.coconeOfHom_fac", "def_path": "Mathlib/CategoryTheory/Limits/IsLimit.lean", "def_pos": [1036, 9], "def_end_pos": [1036, 24]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "J : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\nC : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} C\nF : J \u2964 C\nt : Cocone F\nX : C\nh : coyoneda.obj { unop := X } \u22d9 uliftFunctor.{u\u2081, v\u2083} \u2245 F.cocones\ns : Cocone F\n\u22a2 (colimitCocone h).extend (homOfCocone h s) = coconeOfHom h (homOfCocone h s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/NNRat/Defs.lean", "full_name": "NNRat.den_mul_eq_num", "start": [432, 1], "end": [432, 95], "traced_tactics": [{"tactic": "rw [mul_comm, mul_den_eq_num]", "annotated_tactic": ["rw [<a>mul_comm</a>, <a>mul_den_eq_num</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "NNRat.mul_den_eq_num", "def_path": "Mathlib/Data/NNRat/Defs.lean", "def_pos": [426, 15], "def_end_pos": [426, 29]}]], "state_before": "p q\u271d : \u211a\u22650\nn\u2081 n\u2082 d\u2081 d\u2082 d : \u2115\nq : \u211a\u22650\n\u22a2 \u2191q.den * q = \u2191q.num", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "full_name": "isOpen_extChartAt_preimage", "start": [1289, 1], "end": [1292, 43], "traced_tactics": [{"tactic": "rw [\u2190 extChartAt_source I]", "annotated_tactic": ["rw [\u2190 <a>extChartAt_source</a> I]", [{"full_name": "extChartAt_source", "def_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "def_pos": [1136, 9], "def_end_pos": [1136, 26]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners \ud835\udd5c E H\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\nI' : ModelWithCorners \ud835\udd5c E' H'\ns\u271d t : Set M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : ChartedSpace H' M'\nx : M\ns : Set E\nhs : IsOpen s\n\u22a2 IsOpen ((chartAt H x).source \u2229 \u2191(extChartAt I x) \u207b\u00b9' s)", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners \ud835\udd5c E H\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\nI' : ModelWithCorners \ud835\udd5c E' H'\ns\u271d t : Set M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : ChartedSpace H' M'\nx : M\ns : Set E\nhs : IsOpen s\n\u22a2 IsOpen ((extChartAt I x).source \u2229 \u2191(extChartAt I x) \u207b\u00b9' s)"}, {"tactic": "exact isOpen_extChartAt_preimage' I x hs", "annotated_tactic": ["exact <a>isOpen_extChartAt_preimage'</a> I x hs", [{"full_name": "isOpen_extChartAt_preimage'", "def_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "def_pos": [1284, 9], "def_end_pos": [1284, 36]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners \ud835\udd5c E H\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\nI' : ModelWithCorners \ud835\udd5c E' H'\ns\u271d t : Set M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : ChartedSpace H' M'\nx : M\ns : Set E\nhs : IsOpen s\n\u22a2 IsOpen ((extChartAt I x).source \u2229 \u2191(extChartAt I x) \u207b\u00b9' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Quaternion.lean", "full_name": "Quaternion.intCast_imJ", "start": [1097, 1], "end": [1097, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "full_name": "ContinuousLinearMap.ext_iff", "start": [474, 1], "end": [475, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.Integrable.apply_continuousLinearMap", "start": [1618, 1], "end": [1620, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Factorial/Basic.lean", "full_name": "Nat.add_factorial_succ_lt_factorial_add_succ", "start": [150, 1], "end": [155, 32], "traced_tactics": [{"tactic": "rw [factorial_succ (i + _), Nat.add_mul, Nat.one_mul]", "annotated_tactic": ["rw [<a>factorial_succ</a> (i + _), <a>Nat.add_mul</a>, <a>Nat.one_mul</a>]", [{"full_name": "Nat.factorial_succ", "def_path": "Mathlib/Data/Nat/Factorial/Basic.lean", "def_pos": [47, 9], "def_end_pos": [47, 23]}, {"full_name": "Nat.add_mul", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [242, 19], "def_end_pos": [242, 26]}, {"full_name": "Nat.one_mul", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [225, 27], "def_end_pos": [225, 34]}]], "state_before": "m n\u271d i n : \u2115\nhi : 2 \u2264 i\n\u22a2 i + (n + 1)! < (i + n + 1)!", "state_after": "m n\u271d i n : \u2115\nhi : 2 \u2264 i\n\u22a2 i + (n + 1)! < (i + n) * (i + n)! + (i + n)!"}, {"tactic": "have := (i + n).self_le_factorial", "annotated_tactic": ["have := (i + n).<a>self_le_factorial</a>", [{"full_name": "Nat.self_le_factorial", "def_path": "Mathlib/Data/Nat/Factorial/Basic.lean", "def_pos": [137, 9], "def_end_pos": [137, 26]}]], "state_before": "m n\u271d i n : \u2115\nhi : 2 \u2264 i\n\u22a2 i + (n + 1)! < (i + n) * (i + n)! + (i + n)!", "state_after": "m n\u271d i n : \u2115\nhi : 2 \u2264 i\nthis : i + n \u2264 (i + n)!\n\u22a2 i + (n + 1)! < (i + n) * (i + n)! + (i + n)!"}, {"tactic": "refine Nat.add_lt_add_of_lt_of_le (Nat.lt_of_le_of_lt ?_ ((Nat.lt_mul_iff_one_lt_right ?_).2 ?_))\n  (factorial_le ?_) <;> omega", "annotated_tactic": ["refine <a>Nat.add_lt_add_of_lt_of_le</a> (<a>Nat.lt_of_le_of_lt</a> ?_ ((<a>Nat.lt_mul_iff_one_lt_right</a> ?_).2 ?_))\n    (<a>factorial_le</a> ?_) <;> omega", [{"full_name": "Nat.add_lt_add_of_lt_of_le", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean", "def_pos": [65, 19], "def_end_pos": [65, 41]}, {"full_name": "Nat.lt_of_le_of_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1747, 19], "def_end_pos": [1747, 37]}, {"full_name": "Nat.lt_mul_iff_one_lt_right", "def_path": "Mathlib/Data/Nat/Defs.lean", "def_pos": [455, 25], "def_end_pos": [455, 48]}, {"full_name": "Nat.factorial_le", "def_path": "Mathlib/Data/Nat/Factorial/Basic.lean", "def_pos": [84, 9], "def_end_pos": [84, 21]}]], "state_before": "m n\u271d i n : \u2115\nhi : 2 \u2264 i\nthis : i + n \u2264 (i + n)!\n\u22a2 i + (n + 1)! < (i + n) * (i + n)! + (i + n)!", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Complex.cosh_zero", "start": [309, 1], "end": [309, 49], "traced_tactics": [{"tactic": "simp [cosh]", "annotated_tactic": ["simp [<a>cosh</a>]", [{"full_name": "Complex.cosh", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [89, 5], "def_end_pos": [89, 9]}]], "state_before": "x y : \u2102\n\u22a2 cosh 0 = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "full_name": "Subgroup.map_iSup", "start": [1519, 1], "end": [1521, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean", "full_name": "Real.Gamma_pos_of_pos", "start": [563, 1], "end": [575, 38], "traced_tactics": [{"tactic": "rw [Gamma_eq_integral hs]", "annotated_tactic": ["rw [<a>Gamma_eq_integral</a> hs]", [{"full_name": "Real.Gamma_eq_integral", "def_path": "Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean", "def_pos": [508, 9], "def_end_pos": [508, 26]}]], "state_before": "s : \u211d\nhs : 0 < s\n\u22a2 0 < Gamma s", "state_after": "s : \u211d\nhs : 0 < s\n\u22a2 0 < \u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (s - 1)"}, {"tactic": "have : (Function.support fun x : \u211d => exp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0 := by\n  rw [inter_eq_right]\n  intro x hx\n  rw [Function.mem_support]\n  exact mul_ne_zero (exp_pos _).ne' (rpow_pos_of_pos hx _).ne'", "annotated_tactic": ["have : (<a>Function.support</a> fun x : \u211d => <a>exp</a> (-x) * x ^ (s - 1)) \u2229 <a>Ioi</a> 0 = <a>Ioi</a> 0 := by\n    rw [<a>inter_eq_right</a>]\n    intro x hx\n    rw [<a>Function.mem_support</a>]\n    exact <a>mul_ne_zero</a> (<a>exp_pos</a> _).<a>ne'</a> (<a>rpow_pos_of_pos</a> hx _).<a>ne'</a>", [{"full_name": "Function.support", "def_path": "Mathlib/Algebra/Group/Support.lean", "def_pos": [30, 3], "def_end_pos": [30, 14]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "Set.inter_eq_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [949, 15], "def_end_pos": [949, 29]}, {"full_name": "Function.mem_support", "def_path": "Mathlib/Algebra/Group/Support.lean", "def_pos": [53, 3], "def_end_pos": [53, 14]}, {"full_name": "mul_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 20]}, {"full_name": "Real.exp_pos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1197, 9], "def_end_pos": [1197, 16]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [120, 9], "def_end_pos": [120, 24]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}]], "state_before": "s : \u211d\nhs : 0 < s\n\u22a2 0 < \u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (s - 1)", "state_after": "s : \u211d\nhs : 0 < s\nthis : (Function.support fun x => rexp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0\n\u22a2 0 < \u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (s - 1)"}, {"tactic": "rw [setIntegral_pos_iff_support_of_nonneg_ae]", "annotated_tactic": ["rw [<a>setIntegral_pos_iff_support_of_nonneg_ae</a>]", [{"full_name": "MeasureTheory.setIntegral_pos_iff_support_of_nonneg_ae", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [681, 9], "def_end_pos": [681, 49]}]], "state_before": "s : \u211d\nhs : 0 < s\nthis : (Function.support fun x => rexp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0\n\u22a2 0 < \u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (s - 1)", "state_after": "s : \u211d\nhs : 0 < s\nthis : (Function.support fun x => rexp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0\n\u22a2 0 < volume ((Function.support fun x => rexp (-x) * x ^ (s - 1)) \u2229 Ioi 0)\n\ncase hf\ns : \u211d\nhs : 0 < s\nthis : (Function.support fun x => rexp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0\n\u22a2 0 \u2264\u1da0[ae (volume.restrict (Ioi 0))] fun x => rexp (-x) * x ^ (s - 1)\n\ncase hfi\ns : \u211d\nhs : 0 < s\nthis : (Function.support fun x => rexp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0\n\u22a2 IntegrableOn (fun x => rexp (-x) * x ^ (s - 1)) (Ioi 0) volume"}, {"tactic": "rw [inter_eq_right]", "annotated_tactic": ["rw [<a>inter_eq_right</a>]", [{"full_name": "Set.inter_eq_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [949, 15], "def_end_pos": [949, 29]}]], "state_before": "s : \u211d\nhs : 0 < s\n\u22a2 (Function.support fun x => rexp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0", "state_after": "s : \u211d\nhs : 0 < s\n\u22a2 Ioi 0 \u2286 Function.support fun x => rexp (-x) * x ^ (s - 1)"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "s : \u211d\nhs : 0 < s\n\u22a2 Ioi 0 \u2286 Function.support fun x => rexp (-x) * x ^ (s - 1)", "state_after": "s : \u211d\nhs : 0 < s\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 x \u2208 Function.support fun x => rexp (-x) * x ^ (s - 1)"}, {"tactic": "rw [Function.mem_support]", "annotated_tactic": ["rw [<a>Function.mem_support</a>]", [{"full_name": "Function.mem_support", "def_path": "Mathlib/Algebra/Group/Support.lean", "def_pos": [53, 3], "def_end_pos": [53, 14]}]], "state_before": "s : \u211d\nhs : 0 < s\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 x \u2208 Function.support fun x => rexp (-x) * x ^ (s - 1)", "state_after": "s : \u211d\nhs : 0 < s\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 rexp (-x) * x ^ (s - 1) \u2260 0"}, {"tactic": "exact mul_ne_zero (exp_pos _).ne' (rpow_pos_of_pos hx _).ne'", "annotated_tactic": ["exact <a>mul_ne_zero</a> (<a>exp_pos</a> _).<a>ne'</a> (<a>rpow_pos_of_pos</a> hx _).<a>ne'</a>", [{"full_name": "mul_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 20]}, {"full_name": "Real.exp_pos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1197, 9], "def_end_pos": [1197, 16]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [120, 9], "def_end_pos": [120, 24]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}]], "state_before": "s : \u211d\nhs : 0 < s\nx : \u211d\nhx : x \u2208 Ioi 0\n\u22a2 rexp (-x) * x ^ (s - 1) \u2260 0", "state_after": "no goals"}, {"tactic": "rw [this, volume_Ioi, \u2190 ENNReal.ofReal_zero]", "annotated_tactic": ["rw [this, <a>volume_Ioi</a>, \u2190 <a>ENNReal.ofReal_zero</a>]", [{"full_name": "Real.volume_Ioi", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [145, 9], "def_end_pos": [145, 19]}, {"full_name": "ENNReal.ofReal_zero", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [271, 17], "def_end_pos": [271, 28]}]], "state_before": "s : \u211d\nhs : 0 < s\nthis : (Function.support fun x => rexp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0\n\u22a2 0 < volume ((Function.support fun x => rexp (-x) * x ^ (s - 1)) \u2229 Ioi 0)", "state_after": "s : \u211d\nhs : 0 < s\nthis : (Function.support fun x => rexp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0\n\u22a2 ENNReal.ofReal 0 < \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/ENNReal/Basic.lean", "def_pos": [337, 17], "def_end_pos": [337, 30]}]], "state_before": "s : \u211d\nhs : 0 < s\nthis : (Function.support fun x => rexp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0\n\u22a2 ENNReal.ofReal 0 < \u22a4", "state_after": "no goals"}, {"tactic": "refine eventually_of_mem (self_mem_ae_restrict measurableSet_Ioi) ?_", "annotated_tactic": ["refine <a>eventually_of_mem</a> (<a>self_mem_ae_restrict</a> <a>measurableSet_Ioi</a>) ?_", [{"full_name": "Filter.eventually_of_mem", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1116, 9], "def_end_pos": [1116, 26]}, {"full_name": "MeasureTheory.self_mem_ae_restrict", "def_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "def_pos": [743, 9], "def_end_pos": [743, 29]}, {"full_name": "measurableSet_Ioi", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean", "def_pos": [187, 9], "def_end_pos": [187, 26]}]], "state_before": "case hf\ns : \u211d\nhs : 0 < s\nthis : (Function.support fun x => rexp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0\n\u22a2 0 \u2264\u1da0[ae (volume.restrict (Ioi 0))] fun x => rexp (-x) * x ^ (s - 1)", "state_after": "case hf\ns : \u211d\nhs : 0 < s\nthis : (Function.support fun x => rexp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0\n\u22a2 \u2200 x \u2208 Ioi 0, 0 x \u2264 (fun x => rexp (-x) * x ^ (s - 1)) x"}, {"tactic": "exact fun x hx => (mul_pos (exp_pos _) (rpow_pos_of_pos hx _)).le", "annotated_tactic": ["exact fun x hx => (<a>mul_pos</a> (<a>exp_pos</a> _) (<a>rpow_pos_of_pos</a> hx _)).<a>le</a>", [{"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [406, 7], "def_end_pos": [406, 14]}, {"full_name": "Real.exp_pos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1197, 9], "def_end_pos": [1197, 16]}, {"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [120, 9], "def_end_pos": [120, 24]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}]], "state_before": "case hf\ns : \u211d\nhs : 0 < s\nthis : (Function.support fun x => rexp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0\n\u22a2 \u2200 x \u2208 Ioi 0, 0 x \u2264 (fun x => rexp (-x) * x ^ (s - 1)) x", "state_after": "no goals"}, {"tactic": "exact GammaIntegral_convergent hs", "annotated_tactic": ["exact <a>GammaIntegral_convergent</a> hs", [{"full_name": "Real.GammaIntegral_convergent", "def_path": "Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean", "def_pos": [71, 9], "def_end_pos": [71, 33]}]], "state_before": "case hfi\ns : \u211d\nhs : 0 < s\nthis : (Function.support fun x => rexp (-x) * x ^ (s - 1)) \u2229 Ioi 0 = Ioi 0\n\u22a2 IntegrableOn (fun x => rexp (-x) * x ^ (s - 1)) (Ioi 0) volume", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/Star/Spectrum.lean", "full_name": "AlgHomClass.instStarAlgHomClass", "start": [202, 1], "end": [205, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "full_name": "add_tsub_cancel_right", "start": [356, 1], "end": [357, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/GradedAlgebra/Basic.lean", "full_name": "GradedRing.mem_support_iff", "start": [120, 1], "end": [122, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "full_name": "le_csInf_iff", "start": [601, 1], "end": [602, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/SuccPred/Basic.lean", "full_name": "succ_min", "start": [1479, 1], "end": [1479, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "full_name": "mem_parallelepiped_iff", "start": [52, 1], "end": [54, 33], "traced_tactics": [{"tactic": "simp [parallelepiped, eq_comm]", "annotated_tactic": ["simp [<a>parallelepiped</a>, <a>eq_comm</a>]", [{"full_name": "parallelepiped", "def_path": "Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean", "def_pos": [48, 5], "def_end_pos": [48, 19]}, {"full_name": "eq_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}]], "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\nx : E\n\u22a2 x \u2208 parallelepiped v \u2194 \u2203 t \u2208 Icc 0 1, x = \u2211 i : \u03b9, t i \u2022 v i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Smeval.lean", "full_name": "Polynomial.smeval_X_pow", "start": [93, 1], "end": [95, 98], "traced_tactics": [{"tactic": "simp only [smeval_eq_sum, smul_pow, X_pow_eq_monomial, zero_smul, sum_monomial_index, one_smul]", "annotated_tactic": ["simp only [<a>smeval_eq_sum</a>, <a>smul_pow</a>, <a>X_pow_eq_monomial</a>, <a>zero_smul</a>, <a>sum_monomial_index</a>, <a>one_smul</a>]", [{"full_name": "Polynomial.smeval_eq_sum", "def_path": "Mathlib/Algebra/Polynomial/Smeval.lean", "def_pos": [54, 9], "def_end_pos": [54, 22]}, {"full_name": "Polynomial.smul_pow", "def_path": "Mathlib/Algebra/Polynomial/Smeval.lean", "def_pos": [48, 5], "def_end_pos": [48, 13]}, {"full_name": "Polynomial.X_pow_eq_monomial", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [923, 9], "def_end_pos": [923, 26]}, {"full_name": "zero_smul", "def_path": "Mathlib/Algebra/SMulWithZero.lean", "def_pos": [70, 9], "def_end_pos": [70, 18]}, {"full_name": "Polynomial.sum_monomial_index", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [1001, 9], "def_end_pos": [1001, 27]}, {"full_name": "one_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [453, 7], "def_end_pos": [453, 15]}]], "state_before": "R : Type u_1\ninst\u271d\u00b3 : Semiring R\nr : R\np : R[X]\nS : Type u_2\ninst\u271d\u00b2 : AddCommMonoid S\ninst\u271d\u00b9 : Pow S \u2115\ninst\u271d : MulActionWithZero R S\nx : S\nn : \u2115\n\u22a2 (X ^ n).smeval x = x ^ n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Cone/Basic.lean", "full_name": "ConvexCone.ext", "start": [96, 1], "end": [97, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/MulAction.lean", "full_name": "ContinuousOn.smul", "start": [118, 1], "end": [119, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "full_name": "ENNReal.essSup_eq_zero_iff", "start": [277, 1], "end": [278, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Connected/Basic.lean", "full_name": "frontier_eq_empty_iff", "start": [936, 1], "end": [938, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Infsep.lean", "full_name": "Set.infsep_pos", "start": [340, 1], "end": [341, 43], "traced_tactics": [{"tactic": "simp_rw [infsep, ENNReal.toReal_pos_iff]", "annotated_tactic": ["simp_rw [<a>infsep</a>, <a>ENNReal.toReal_pos_iff</a>]", [{"full_name": "Set.infsep", "def_path": "Mathlib/Topology/MetricSpace/Infsep.lean", "def_pos": [324, 19], "def_end_pos": [324, 25]}, {"full_name": "ENNReal.toReal_pos_iff", "def_path": "Mathlib/Data/ENNReal/Real.lean", "def_pos": [167, 9], "def_end_pos": [167, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : EDist \u03b1\nx y : \u03b1\ns : Set \u03b1\n\u22a2 0 < s.infsep \u2194 0 < s.einfsep \u2227 s.einfsep < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.bsup_eq_blsub_of_lt_succ_limit", "start": [1866, 1], "end": [1870, 55], "traced_tactics": [{"tactic": "rw [bsup_eq_blsub_iff_lt_bsup]", "annotated_tactic": ["rw [<a>bsup_eq_blsub_iff_lt_bsup</a>]", [{"full_name": "Ordinal.bsup_eq_blsub_iff_lt_bsup", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [1859, 9], "def_end_pos": [1859, 34]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u}\nho : o.IsLimit\nf : (a : Ordinal.{u}) \u2192 a < o \u2192 Ordinal.{max u v}\nhf : \u2200 (a : Ordinal.{u}) (ha : a < o), f a ha < f (succ a) \u22ef\n\u22a2 o.bsup f = o.blsub f", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u}\nho : o.IsLimit\nf : (a : Ordinal.{u}) \u2192 a < o \u2192 Ordinal.{max u v}\nhf : \u2200 (a : Ordinal.{u}) (ha : a < o), f a ha < f (succ a) \u22ef\n\u22a2 \u2200 (i : Ordinal.{u}) (hi : i < o), f i hi < o.bsup f"}, {"tactic": "exact fun i hi => (hf i hi).trans_le (le_bsup f _ _)", "annotated_tactic": ["exact fun i hi => (hf i hi).<a>trans_le</a> (<a>le_bsup</a> f _ _)", [{"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [143, 7], "def_end_pos": [143, 21]}, {"full_name": "Ordinal.le_bsup", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [1496, 9], "def_end_pos": [1496, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u}\nho : o.IsLimit\nf : (a : Ordinal.{u}) \u2192 a < o \u2192 Ordinal.{max u v}\nhf : \u2200 (a : Ordinal.{u}) (ha : a < o), f a ha < f (succ a) \u22ef\n\u22a2 \u2200 (i : Ordinal.{u}) (hi : i < o), f i hi < o.bsup f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Monovary.lean", "full_name": "AntivaryOn.mul_left", "start": [81, 1], "end": [82, 96], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/Partrec.lean", "full_name": "Computable.vector_head", "start": [382, 1], "end": [383, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Associated.lean", "full_name": "Associated.map", "start": [445, 1], "end": [448, 81], "traced_tactics": [{"tactic": "obtain \u27e8u, ha\u27e9 := ha", "annotated_tactic": ["obtain \u27e8u, ha\u27e9 := ha", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Monoid N\nF : Type u_7\ninst\u271d\u00b9 : FunLike F M N\ninst\u271d : MonoidHomClass F M N\nf : F\nx y : M\nha : x ~\u1d64 y\n\u22a2 f x ~\u1d64 f y", "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 : Type u_5\nN : Type u_6\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Monoid N\nF : Type u_7\ninst\u271d\u00b9 : FunLike F M N\ninst\u271d : MonoidHomClass F M N\nf : F\nx y : M\nu : M\u02e3\nha : x * \u2191u = y\n\u22a2 f x ~\u1d64 f y"}, {"tactic": "exact \u27e8Units.map f u, by rw [\u2190 ha, map_mul, Units.coe_map, MonoidHom.coe_coe]\u27e9", "annotated_tactic": ["exact \u27e8<a>Units.map</a> f u, by rw [\u2190 ha, <a>map_mul</a>, <a>Units.coe_map</a>, <a>MonoidHom.coe_coe</a>]\u27e9", [{"full_name": "Units.map", "def_path": "Mathlib/Algebra/Group/Units/Hom.lean", "def_pos": [63, 5], "def_end_pos": [63, 8]}, {"full_name": "map_mul", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [309, 9], "def_end_pos": [309, 16]}, {"full_name": "Units.coe_map", "def_path": "Mathlib/Algebra/Group/Units/Hom.lean", "def_pos": [73, 9], "def_end_pos": [73, 16]}, {"full_name": "MonoidHom.coe_coe", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [411, 9], "def_end_pos": [411, 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 : Type u_5\nN : Type u_6\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Monoid N\nF : Type u_7\ninst\u271d\u00b9 : FunLike F M N\ninst\u271d : MonoidHomClass F M N\nf : F\nx y : M\nu : M\u02e3\nha : x * \u2191u = y\n\u22a2 f x ~\u1d64 f y", "state_after": "no goals"}, {"tactic": "rw [\u2190 ha, map_mul, Units.coe_map, MonoidHom.coe_coe]", "annotated_tactic": ["rw [\u2190 ha, <a>map_mul</a>, <a>Units.coe_map</a>, <a>MonoidHom.coe_coe</a>]", [{"full_name": "map_mul", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [309, 9], "def_end_pos": [309, 16]}, {"full_name": "Units.coe_map", "def_path": "Mathlib/Algebra/Group/Units/Hom.lean", "def_pos": [73, 9], "def_end_pos": [73, 16]}, {"full_name": "MonoidHom.coe_coe", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [411, 9], "def_end_pos": [411, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nM : Type u_5\nN : Type u_6\ninst\u271d\u00b3 : Monoid M\ninst\u271d\u00b2 : Monoid N\nF : Type u_7\ninst\u271d\u00b9 : FunLike F M N\ninst\u271d : MonoidHomClass F M N\nf : F\nx y : M\nu : M\u02e3\nha : x * \u2191u = y\n\u22a2 f x * \u2191((Units.map \u2191f) u) = f y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Equivalence.lean", "full_name": "CategoryTheory.Equivalence.Equivalence_mk'_counitInv", "start": [147, 1], "end": [149, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean", "full_name": "CategoryTheory.regularTopology.isSheafFor_regular_of_projective", "start": [224, 1], "end": [231, 42], "traced_tactics": [{"tactic": "obtain \u27e8Y, f, rfl, hf\u27e9 := Presieve.regular.single_epi (R := S)", "annotated_tactic": ["obtain \u27e8Y, f, rfl, hf\u27e9 := <a>Presieve.regular.single_epi</a> (R := S)", [{"full_name": "CategoryTheory.Presieve.regular.single_epi", "def_path": "Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean", "def_pos": [39, 3], "def_end_pos": [39, 13]}]], "state_before": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u2074 : Category.{u_5, u_1} C\ninst\u271d\u00b3 : Category.{?u.128896, u_2} D\ninst\u271d\u00b2 : Category.{?u.128900, u_3} E\nX : C\nS : Presieve X\ninst\u271d\u00b9 : S.regular\ninst\u271d : Projective X\nF : C\u1d52\u1d56 \u2964 Type u_4\n\u22a2 IsSheafFor F S", "state_after": "case intro.intro.intro\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u2074 : Category.{u_5, u_1} C\ninst\u271d\u00b3 : Category.{?u.128896, u_2} D\ninst\u271d\u00b2 : Category.{?u.128900, u_3} E\nX : C\ninst\u271d\u00b9 : Projective X\nF : C\u1d52\u1d56 \u2964 Type u_4\nY : C\nf : Y \u27f6 X\nhf : EffectiveEpi f\ninst\u271d : (ofArrows (fun x => Y) fun x => f).regular\n\u22a2 IsSheafFor F (ofArrows (fun x => Y) fun x => f)"}, {"tactic": "rw [isSheafFor_arrows_iff]", "annotated_tactic": ["rw [<a>isSheafFor_arrows_iff</a>]", [{"full_name": "CategoryTheory.Presieve.isSheafFor_arrows_iff", "def_path": "Mathlib/CategoryTheory/Sites/IsSheafFor.lean", "def_pos": [763, 9], "def_end_pos": [763, 30]}]], "state_before": "case intro.intro.intro\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u2074 : Category.{u_5, u_1} C\ninst\u271d\u00b3 : Category.{?u.128896, u_2} D\ninst\u271d\u00b2 : Category.{?u.128900, u_3} E\nX : C\ninst\u271d\u00b9 : Projective X\nF : C\u1d52\u1d56 \u2964 Type u_4\nY : C\nf : Y \u27f6 X\nhf : EffectiveEpi f\ninst\u271d : (ofArrows (fun x => Y) fun x => f).regular\n\u22a2 IsSheafFor F (ofArrows (fun x => Y) fun x => f)", "state_after": "case intro.intro.intro\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u2074 : Category.{u_5, u_1} C\ninst\u271d\u00b3 : Category.{?u.128896, u_2} D\ninst\u271d\u00b2 : Category.{?u.128900, u_3} E\nX : C\ninst\u271d\u00b9 : Projective X\nF : C\u1d52\u1d56 \u2964 Type u_4\nY : C\nf : Y \u27f6 X\nhf : EffectiveEpi f\ninst\u271d : (ofArrows (fun x => Y) fun x => f).regular\n\u22a2 \u2200 (x : Unit \u2192 F.obj { unop := Y }), Arrows.Compatible F (fun x => f) x \u2192 \u2203! t, \u2200 (i : Unit), F.map f.op t = x i"}, {"tactic": "refine fun x hx \u21a6 \u27e8F.map (Projective.factorThru (\ud835\udfd9 _) f).op <| x (), fun _ \u21a6 ?_, fun y h \u21a6 ?_\u27e9", "annotated_tactic": ["refine fun x hx \u21a6 \u27e8F.map (<a>Projective.factorThru</a> (\ud835\udfd9 _) f).<a>op</a> <| x (), fun _ \u21a6 ?_, fun y h \u21a6 ?_\u27e9", [{"full_name": "CategoryTheory.Projective.factorThru", "def_path": "Mathlib/CategoryTheory/Preadditive/Projective.lean", "def_pos": [82, 5], "def_end_pos": [82, 15]}, {"full_name": "Quiver.Hom.op", "def_path": "Mathlib/Combinatorics/Quiver/Basic.lean", "def_pos": [152, 5], "def_end_pos": [152, 11]}]], "state_before": "case intro.intro.intro\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u2074 : Category.{u_5, u_1} C\ninst\u271d\u00b3 : Category.{?u.128896, u_2} D\ninst\u271d\u00b2 : Category.{?u.128900, u_3} E\nX : C\ninst\u271d\u00b9 : Projective X\nF : C\u1d52\u1d56 \u2964 Type u_4\nY : C\nf : Y \u27f6 X\nhf : EffectiveEpi f\ninst\u271d : (ofArrows (fun x => Y) fun x => f).regular\n\u22a2 \u2200 (x : Unit \u2192 F.obj { unop := Y }), Arrows.Compatible F (fun x => f) x \u2192 \u2203! t, \u2200 (i : Unit), F.map f.op t = x i", "state_after": "case intro.intro.intro.refine_1\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u2074 : Category.{u_5, u_1} C\ninst\u271d\u00b3 : Category.{?u.128896, u_2} D\ninst\u271d\u00b2 : Category.{?u.128900, u_3} E\nX : C\ninst\u271d\u00b9 : Projective X\nF : C\u1d52\u1d56 \u2964 Type u_4\nY : C\nf : Y \u27f6 X\nhf : EffectiveEpi f\ninst\u271d : (ofArrows (fun x => Y) fun x => f).regular\nx : Unit \u2192 F.obj { unop := Y }\nhx : Arrows.Compatible F (fun x => f) x\nx\u271d : Unit\n\u22a2 F.map f.op (F.map (Projective.factorThru (\ud835\udfd9 X) f).op (x ())) = x x\u271d\n\ncase intro.intro.intro.refine_2\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u2074 : Category.{u_5, u_1} C\ninst\u271d\u00b3 : Category.{?u.128896, u_2} D\ninst\u271d\u00b2 : Category.{?u.128900, u_3} E\nX : C\ninst\u271d\u00b9 : Projective X\nF : C\u1d52\u1d56 \u2964 Type u_4\nY : C\nf : Y \u27f6 X\nhf : EffectiveEpi f\ninst\u271d : (ofArrows (fun x => Y) fun x => f).regular\nx : Unit \u2192 F.obj { unop := Y }\nhx : Arrows.Compatible F (fun x => f) x\ny : F.obj { unop := X }\nh : (fun t => \u2200 (i : Unit), F.map f.op t = x i) y\n\u22a2 y = F.map (Projective.factorThru (\ud835\udfd9 X) f).op (x ())"}, {"tactic": "simpa using (hx () () Y (\ud835\udfd9 Y) (f \u226b (Projective.factorThru (\ud835\udfd9 _) f)) (by simp)).symm", "annotated_tactic": ["simpa using (hx () () Y (\ud835\udfd9 Y) (f \u226b (<a>Projective.factorThru</a> (\ud835\udfd9 _) f)) (by simp)).<a>symm</a>", [{"full_name": "CategoryTheory.Projective.factorThru", "def_path": "Mathlib/CategoryTheory/Preadditive/Projective.lean", "def_pos": [82, 5], "def_end_pos": [82, 15]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case intro.intro.intro.refine_1\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u2074 : Category.{u_5, u_1} C\ninst\u271d\u00b3 : Category.{?u.128896, u_2} D\ninst\u271d\u00b2 : Category.{?u.128900, u_3} E\nX : C\ninst\u271d\u00b9 : Projective X\nF : C\u1d52\u1d56 \u2964 Type u_4\nY : C\nf : Y \u27f6 X\nhf : EffectiveEpi f\ninst\u271d : (ofArrows (fun x => Y) fun x => f).regular\nx : Unit \u2192 F.obj { unop := Y }\nhx : Arrows.Compatible F (fun x => f) x\nx\u271d : Unit\n\u22a2 F.map f.op (F.map (Projective.factorThru (\ud835\udfd9 X) f).op (x ())) = x x\u271d", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u2074 : Category.{u_5, u_1} C\ninst\u271d\u00b3 : Category.{?u.128896, u_2} D\ninst\u271d\u00b2 : Category.{?u.128900, u_3} E\nX : C\ninst\u271d\u00b9 : Projective X\nF : C\u1d52\u1d56 \u2964 Type u_4\nY : C\nf : Y \u27f6 X\nhf : EffectiveEpi f\ninst\u271d : (ofArrows (fun x => Y) fun x => f).regular\nx : Unit \u2192 F.obj { unop := Y }\nhx : Arrows.Compatible F (fun x => f) x\nx\u271d : Unit\n\u22a2 \ud835\udfd9 Y \u226b (fun x => f) () = (f \u226b Projective.factorThru (\ud835\udfd9 X) f) \u226b (fun x => f) ()", "state_after": "no goals"}, {"tactic": "simp only [\u2190 h (), \u2190 FunctorToTypes.map_comp_apply, \u2190 op_comp, Projective.factorThru_comp,\n  op_id, FunctorToTypes.map_id_apply]", "annotated_tactic": ["simp only [\u2190 h (), \u2190 <a>FunctorToTypes.map_comp_apply</a>, \u2190 <a>op_comp</a>, <a>Projective.factorThru_comp</a>,\n      <a>op_id</a>, <a>FunctorToTypes.map_id_apply</a>]", [{"full_name": "CategoryTheory.FunctorToTypes.map_comp_apply", "def_path": "Mathlib/CategoryTheory/Types.lean", "def_pos": [151, 9], "def_end_pos": [151, 23]}, {"full_name": "CategoryTheory.op_comp", "def_path": "Mathlib/CategoryTheory/Opposites.lean", "def_pos": [79, 9], "def_end_pos": [79, 16]}, {"full_name": "CategoryTheory.Projective.factorThru_comp", "def_path": "Mathlib/CategoryTheory/Preadditive/Projective.lean", "def_pos": [87, 9], "def_end_pos": [87, 24]}, {"full_name": "CategoryTheory.op_id", "def_path": "Mathlib/CategoryTheory/Opposites.lean", "def_pos": [84, 9], "def_end_pos": [84, 14]}, {"full_name": "CategoryTheory.FunctorToTypes.map_id_apply", "def_path": "Mathlib/CategoryTheory/Types.lean", "def_pos": [156, 9], "def_end_pos": [156, 21]}]], "state_before": "case intro.intro.intro.refine_2\nC : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u2074 : Category.{u_5, u_1} C\ninst\u271d\u00b3 : Category.{?u.128896, u_2} D\ninst\u271d\u00b2 : Category.{?u.128900, u_3} E\nX : C\ninst\u271d\u00b9 : Projective X\nF : C\u1d52\u1d56 \u2964 Type u_4\nY : C\nf : Y \u27f6 X\nhf : EffectiveEpi f\ninst\u271d : (ofArrows (fun x => Y) fun x => f).regular\nx : Unit \u2192 F.obj { unop := Y }\nhx : Arrows.Compatible F (fun x => f) x\ny : F.obj { unop := X }\nh : (fun t => \u2200 (i : Unit), F.map f.op t = x i) y\n\u22a2 y = F.map (Projective.factorThru (\ud835\udfd9 X) f).op (x ())", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Rat/Defs.lean", "full_name": "Rat.divInt_ne_zero", "start": [131, 1], "end": [132, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.L1.setToL1_congr_left'", "start": [1065, 1], "end": [1074, 50], "traced_tactics": [{"tactic": "suffices setToL1 hT = setToL1 hT' by rw [this]", "annotated_tactic": ["suffices <a>setToL1</a> hT = <a>setToL1</a> hT' by rw [this]", [{"full_name": "MeasureTheory.L1.setToL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1019, 5], "def_end_pos": [1019, 12]}, {"full_name": "MeasureTheory.L1.setToL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1019, 5], "def_end_pos": [1019, 12]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s < \u22a4 \u2192 T s = T' s\nf : \u21a5(Lp E 1 \u03bc)\n\u22a2 (setToL1 hT) f = (setToL1 hT') f", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s < \u22a4 \u2192 T s = T' s\nf : \u21a5(Lp E 1 \u03bc)\n\u22a2 setToL1 hT = setToL1 hT'"}, {"tactic": "refine ContinuousLinearMap.extend_unique (setToL1SCLM \u03b1 E \u03bc hT) _ _ _ _ ?_", "annotated_tactic": ["refine <a>ContinuousLinearMap.extend_unique</a> (<a>setToL1SCLM</a> \u03b1 E \u03bc hT) _ _ _ _ ?_", [{"full_name": "ContinuousLinearMap.extend_unique", "def_path": "Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean", "def_pos": [228, 9], "def_end_pos": [228, 22]}, {"full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [878, 5], "def_end_pos": [878, 16]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s < \u22a4 \u2192 T s = T' s\nf : \u21a5(Lp E 1 \u03bc)\n\u22a2 setToL1 hT = setToL1 hT'", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s < \u22a4 \u2192 T s = T' s\nf : \u21a5(Lp E 1 \u03bc)\n\u22a2 (setToL1 hT').comp (coeToLp \u03b1 E \u211d) = setToL1SCLM \u03b1 E \u03bc hT"}, {"tactic": "ext1 f", "annotated_tactic": ["ext1 f", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s < \u22a4 \u2192 T s = T' s\nf : \u21a5(Lp E 1 \u03bc)\n\u22a2 (setToL1 hT').comp (coeToLp \u03b1 E \u211d) = setToL1SCLM \u03b1 E \u03bc hT", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : \u21a5(Lp E 1 \u03bc)\nf : \u21a5(simpleFunc E 1 \u03bc)\n\u22a2 ((setToL1 hT').comp (coeToLp \u03b1 E \u211d)) f = (setToL1SCLM \u03b1 E \u03bc hT) f"}, {"tactic": "suffices setToL1 hT' f = setToL1SCLM \u03b1 E \u03bc hT f by rw [\u2190 this]; rfl", "annotated_tactic": ["suffices <a>setToL1</a> hT' f = <a>setToL1SCLM</a> \u03b1 E \u03bc hT f by rw [\u2190 this]; rfl", [{"full_name": "MeasureTheory.L1.setToL1", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1019, 5], "def_end_pos": [1019, 12]}, {"full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [878, 5], "def_end_pos": [878, 16]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : \u21a5(Lp E 1 \u03bc)\nf : \u21a5(simpleFunc E 1 \u03bc)\n\u22a2 ((setToL1 hT').comp (coeToLp \u03b1 E \u211d)) f = (setToL1SCLM \u03b1 E \u03bc hT) f", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : \u21a5(Lp E 1 \u03bc)\nf : \u21a5(simpleFunc E 1 \u03bc)\n\u22a2 (setToL1 hT') \u2191f = (setToL1SCLM \u03b1 E \u03bc hT) f"}, {"tactic": "rw [setToL1_eq_setToL1SCLM]", "annotated_tactic": ["rw [<a>setToL1_eq_setToL1SCLM</a>]", [{"full_name": "MeasureTheory.L1.setToL1_eq_setToL1SCLM", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [1024, 9], "def_end_pos": [1024, 31]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : \u21a5(Lp E 1 \u03bc)\nf : \u21a5(simpleFunc E 1 \u03bc)\n\u22a2 (setToL1 hT') \u2191f = (setToL1SCLM \u03b1 E \u03bc hT) f", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : \u21a5(Lp E 1 \u03bc)\nf : \u21a5(simpleFunc E 1 \u03bc)\n\u22a2 (setToL1SCLM \u03b1 E \u03bc hT') f = (setToL1SCLM \u03b1 E \u03bc hT) f"}, {"tactic": "exact (setToL1SCLM_congr_left' hT hT' h f).symm", "annotated_tactic": ["exact (<a>setToL1SCLM_congr_left'</a> hT hT' h f).<a>symm</a>", [{"full_name": "MeasureTheory.L1.SimpleFunc.setToL1SCLM_congr_left'", "def_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "def_pos": [907, 9], "def_end_pos": [907, 32]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : \u21a5(Lp E 1 \u03bc)\nf : \u21a5(simpleFunc E 1 \u03bc)\n\u22a2 (setToL1SCLM \u03b1 E \u03bc hT') f = (setToL1SCLM \u03b1 E \u03bc hT) f", "state_after": "no goals"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s < \u22a4 \u2192 T s = T' s\nf : \u21a5(Lp E 1 \u03bc)\nthis : setToL1 hT = setToL1 hT'\n\u22a2 (setToL1 hT) f = (setToL1 hT') f", "state_after": "no goals"}, {"tactic": "rw [\u2190 this]", "annotated_tactic": ["rw [\u2190 this]", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : \u21a5(Lp E 1 \u03bc)\nf : \u21a5(simpleFunc E 1 \u03bc)\nthis : (setToL1 hT') \u2191f = (setToL1SCLM \u03b1 E \u03bc hT) f\n\u22a2 ((setToL1 hT').comp (coeToLp \u03b1 E \u211d)) f = (setToL1SCLM \u03b1 E \u03bc hT) f", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : \u21a5(Lp E 1 \u03bc)\nf : \u21a5(simpleFunc E 1 \u03bc)\nthis : (setToL1 hT') \u2191f = (setToL1SCLM \u03b1 E \u03bc hT) f\n\u22a2 ((setToL1 hT').comp (coeToLp \u03b1 E \u211d)) f = (setToL1 hT') \u2191f"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u2070 : NormedAddCommGroup E\ninst\u271d\u2079 : NormedSpace \u211d E\ninst\u271d\u2078 : NormedAddCommGroup F\ninst\u271d\u2077 : NormedSpace \u211d F\ninst\u271d\u2076 : NormedAddCommGroup F'\ninst\u271d\u2075 : NormedSpace \u211d F'\ninst\u271d\u2074 : NormedAddCommGroup G\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c F\ninst\u271d : CompleteSpace F\nT\u271d T'\u271d T'' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC\u271d C'\u271d C'' : \u211d\nT T' : Set \u03b1 \u2192 E \u2192L[\u211d] F\nC C' : \u211d\nhT : DominatedFinMeasAdditive \u03bc T C\nhT' : DominatedFinMeasAdditive \u03bc T' C'\nh : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s < \u22a4 \u2192 T s = T' s\nf\u271d : \u21a5(Lp E 1 \u03bc)\nf : \u21a5(simpleFunc E 1 \u03bc)\nthis : (setToL1 hT') \u2191f = (setToL1SCLM \u03b1 E \u03bc hT) f\n\u22a2 ((setToL1 hT').comp (coeToLp \u03b1 E \u211d)) f = (setToL1 hT') \u2191f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/Basic.lean", "full_name": "Uniform.continuousWithinAt_iff'_right", "start": [1950, 1], "end": [1952, 46], "traced_tactics": [{"tactic": "rw [ContinuousWithinAt, tendsto_nhds_right]", "annotated_tactic": ["rw [<a>ContinuousWithinAt</a>, <a>tendsto_nhds_right</a>]", [{"full_name": "ContinuousWithinAt", "def_path": "Mathlib/Topology/Defs/Filter.lean", "def_pos": [163, 5], "def_end_pos": [163, 23]}, {"full_name": "Uniform.tendsto_nhds_right", "def_path": "Mathlib/Topology/UniformSpace/Basic.lean", "def_pos": [1924, 9], "def_end_pos": [1924, 27]}]], "state_before": "\u03b1 : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort u_1\ninst\u271d\u00b9 : UniformSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b2 \u2192 \u03b1\nb : \u03b2\ns : Set \u03b2\n\u22a2 ContinuousWithinAt f s b \u2194 Tendsto (fun x => (f b, f x)) (\ud835\udcdd[s] b) (\ud835\udce4 \u03b1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/ModelTheory/FinitelyGenerated.lean", "full_name": "FirstOrder.Language.Equiv.cg_iff", "start": [278, 1], "end": [281, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "full_name": "AEMeasurable.comp_aemeasurable'", "start": [174, 1], "end": [175, 90], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "full_name": "Orthonormal.inner_sum", "start": [836, 1], "end": [840, 32], "traced_tactics": [{"tactic": "simp_rw [sum_inner, inner_smul_left]", "annotated_tactic": ["simp_rw [<a>sum_inner</a>, <a>inner_smul_left</a>]", [{"full_name": "sum_inner", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [508, 9], "def_end_pos": [508, 18]}, {"full_name": "inner_smul_left", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [464, 9], "def_end_pos": [464, 24]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\n\u03b9 : Type u_4\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl\u2081 l\u2082 : \u03b9 \u2192 \ud835\udd5c\ns : Finset \u03b9\n\u22a2 \u27ea\u2211 i \u2208 s, l\u2081 i \u2022 v i, \u2211 i \u2208 s, l\u2082 i \u2022 v i\u27eb_\ud835\udd5c = \u2211 i \u2208 s, (starRingEnd \ud835\udd5c) (l\u2081 i) * l\u2082 i", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\n\u03b9 : Type u_4\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl\u2081 l\u2082 : \u03b9 \u2192 \ud835\udd5c\ns : Finset \u03b9\n\u22a2 \u2211 x \u2208 s, (starRingEnd \ud835\udd5c) (l\u2081 x) * \u27eav x, \u2211 i \u2208 s, l\u2082 i \u2022 v i\u27eb_\ud835\udd5c = \u2211 i \u2208 s, (starRingEnd \ud835\udd5c) (l\u2081 i) * l\u2082 i"}, {"tactic": "refine Finset.sum_congr rfl fun i hi => ?_", "annotated_tactic": ["refine <a>Finset.sum_congr</a> <a>rfl</a> fun i hi => ?_", [{"full_name": "Finset.sum_congr", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [420, 3], "def_end_pos": [420, 14]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\n\u03b9 : Type u_4\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl\u2081 l\u2082 : \u03b9 \u2192 \ud835\udd5c\ns : Finset \u03b9\n\u22a2 \u2211 x \u2208 s, (starRingEnd \ud835\udd5c) (l\u2081 x) * \u27eav x, \u2211 i \u2208 s, l\u2082 i \u2022 v i\u27eb_\ud835\udd5c = \u2211 i \u2208 s, (starRingEnd \ud835\udd5c) (l\u2081 i) * l\u2082 i", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\n\u03b9 : Type u_4\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl\u2081 l\u2082 : \u03b9 \u2192 \ud835\udd5c\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\n\u22a2 (starRingEnd \ud835\udd5c) (l\u2081 i) * \u27eav i, \u2211 i \u2208 s, l\u2082 i \u2022 v i\u27eb_\ud835\udd5c = (starRingEnd \ud835\udd5c) (l\u2081 i) * l\u2082 i"}, {"tactic": "rw [hv.inner_right_sum l\u2082 hi]", "annotated_tactic": ["rw [hv.inner_right_sum l\u2082 hi]", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\ninst\u271d\u2074 : _root_.RCLike \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : InnerProductSpace \u211d F\n\u03b9 : Type u_4\nv : \u03b9 \u2192 E\nhv : Orthonormal \ud835\udd5c v\nl\u2081 l\u2082 : \u03b9 \u2192 \ud835\udd5c\ns : Finset \u03b9\ni : \u03b9\nhi : i \u2208 s\n\u22a2 (starRingEnd \ud835\udd5c) (l\u2081 i) * \u27eav i, \u2211 i \u2208 s, l\u2082 i \u2022 v i\u27eb_\ud835\udd5c = (starRingEnd \ud835\udd5c) (l\u2081 i) * l\u2082 i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Deriv.lean", "full_name": "convexOn_of_deriv2_nonneg'", "start": [305, 1], "end": [309, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "full_name": "Real.Angle.toReal_coe_eq_self_sub_two_pi_iff", "start": [688, 1], "end": [690, 74], "traced_tactics": [{"tactic": "convert @toReal_coe_eq_self_sub_two_mul_int_mul_pi_iff \u03b8 1 <;> norm_num", "annotated_tactic": ["convert @<a>toReal_coe_eq_self_sub_two_mul_int_mul_pi_iff</a> \u03b8 1 <;> norm_num", [{"full_name": "Real.Angle.toReal_coe_eq_self_sub_two_mul_int_mul_pi_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [681, 9], "def_end_pos": [681, 54]}]], "state_before": "\u03b8 : \u211d\n\u22a2 (\u2191\u03b8).toReal = \u03b8 - 2 * \u03c0 \u2194 \u03b8 \u2208 Set.Ioc \u03c0 (3 * \u03c0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Compactness/Compact.lean", "full_name": "not_compactSpace_iff", "start": [876, 1], "end": [877, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Principal.lean", "full_name": "Ordinal.principal_zero", "start": [57, 1], "end": [58, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Divisors.lean", "full_name": "Nat.mem_divisorsAntidiagonal", "start": [116, 1], "end": [131, 59], "traced_tactics": [{"tactic": "simp only [divisorsAntidiagonal, Finset.mem_Ico, Ne, Finset.mem_filter, Finset.mem_product]", "annotated_tactic": ["simp only [<a>divisorsAntidiagonal</a>, <a>Finset.mem_Ico</a>, <a>Ne</a>, <a>Finset.mem_filter</a>, <a>Finset.mem_product</a>]", [{"full_name": "Nat.divisorsAntidiagonal", "def_path": "Mathlib/NumberTheory/Divisors.lean", "def_pos": [54, 5], "def_end_pos": [54, 25]}, {"full_name": "Finset.mem_Ico", "def_path": "Mathlib/Order/Interval/Finset/Defs.lean", "def_pos": [318, 9], "def_end_pos": [318, 16]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2593, 9], "def_end_pos": [2593, 19]}, {"full_name": "Finset.mem_product", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [54, 9], "def_end_pos": [54, 20]}]], "state_before": "n : \u2115\nx : \u2115 \u00d7 \u2115\n\u22a2 x \u2208 n.divisorsAntidiagonal \u2194 x.1 * x.2 = n \u2227 n \u2260 0", "state_after": "n : \u2115\nx : \u2115 \u00d7 \u2115\n\u22a2 ((1 \u2264 x.1 \u2227 x.1 < n + 1) \u2227 1 \u2264 x.2 \u2227 x.2 < n + 1) \u2227 x.1 * x.2 = n \u2194 x.1 * x.2 = n \u2227 \u00acn = 0"}, {"tactic": "rw [and_comm]", "annotated_tactic": ["rw [<a>and_comm</a>]", [{"full_name": "and_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [819, 9], "def_end_pos": [819, 17]}]], "state_before": "n : \u2115\nx : \u2115 \u00d7 \u2115\n\u22a2 ((1 \u2264 x.1 \u2227 x.1 < n + 1) \u2227 1 \u2264 x.2 \u2227 x.2 < n + 1) \u2227 x.1 * x.2 = n \u2194 x.1 * x.2 = n \u2227 \u00acn = 0", "state_after": "n : \u2115\nx : \u2115 \u00d7 \u2115\n\u22a2 x.1 * x.2 = n \u2227 (1 \u2264 x.1 \u2227 x.1 < n + 1) \u2227 1 \u2264 x.2 \u2227 x.2 < n + 1 \u2194 x.1 * x.2 = n \u2227 \u00acn = 0"}, {"tactic": "apply and_congr_right", "annotated_tactic": ["apply <a>and_congr_right</a>", [{"full_name": "and_congr_right", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [142, 9], "def_end_pos": [142, 24]}]], "state_before": "n : \u2115\nx : \u2115 \u00d7 \u2115\n\u22a2 x.1 * x.2 = n \u2227 (1 \u2264 x.1 \u2227 x.1 < n + 1) \u2227 1 \u2264 x.2 \u2227 x.2 < n + 1 \u2194 x.1 * x.2 = n \u2227 \u00acn = 0", "state_after": "case h\nn : \u2115\nx : \u2115 \u00d7 \u2115\n\u22a2 x.1 * x.2 = n \u2192 ((1 \u2264 x.1 \u2227 x.1 < n + 1) \u2227 1 \u2264 x.2 \u2227 x.2 < n + 1 \u2194 \u00acn = 0)"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "case h\nn : \u2115\nx : \u2115 \u00d7 \u2115\n\u22a2 x.1 * x.2 = n \u2192 ((1 \u2264 x.1 \u2227 x.1 < n + 1) \u2227 1 \u2264 x.2 \u2227 x.2 < n + 1 \u2194 \u00acn = 0)", "state_after": "case h\nx : \u2115 \u00d7 \u2115\n\u22a2 (1 \u2264 x.1 \u2227 x.1 < x.1 * x.2 + 1) \u2227 1 \u2264 x.2 \u2227 x.2 < x.1 * x.2 + 1 \u2194 \u00acx.1 * x.2 = 0"}, {"tactic": "constructor <;> intro h", "annotated_tactic": ["constructor <;> intro h", []], "state_before": "case h\nx : \u2115 \u00d7 \u2115\n\u22a2 (1 \u2264 x.1 \u2227 x.1 < x.1 * x.2 + 1) \u2227 1 \u2264 x.2 \u2227 x.2 < x.1 * x.2 + 1 \u2194 \u00acx.1 * x.2 = 0", "state_after": "case h.mp\nx : \u2115 \u00d7 \u2115\nh : (1 \u2264 x.1 \u2227 x.1 < x.1 * x.2 + 1) \u2227 1 \u2264 x.2 \u2227 x.2 < x.1 * x.2 + 1\n\u22a2 \u00acx.1 * x.2 = 0\n\ncase h.mpr\nx : \u2115 \u00d7 \u2115\nh : \u00acx.1 * x.2 = 0\n\u22a2 (1 \u2264 x.1 \u2227 x.1 < x.1 * x.2 + 1) \u2227 1 \u2264 x.2 \u2227 x.2 < x.1 * x.2 + 1"}, {"tactic": "contrapose! h", "annotated_tactic": ["contrapose! h", []], "state_before": "case h.mp\nx : \u2115 \u00d7 \u2115\nh : (1 \u2264 x.1 \u2227 x.1 < x.1 * x.2 + 1) \u2227 1 \u2264 x.2 \u2227 x.2 < x.1 * x.2 + 1\n\u22a2 \u00acx.1 * x.2 = 0", "state_after": "case h.mp\nx : \u2115 \u00d7 \u2115\nh : x.1 * x.2 = 0\n\u22a2 1 \u2264 x.1 \u2227 x.1 < x.1 * x.2 + 1 \u2192 1 \u2264 x.2 \u2192 x.1 * x.2 + 1 \u2264 x.2"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "case h.mp\nx : \u2115 \u00d7 \u2115\nh : x.1 * x.2 = 0\n\u22a2 1 \u2264 x.1 \u2227 x.1 < x.1 * x.2 + 1 \u2192 1 \u2264 x.2 \u2192 x.1 * x.2 + 1 \u2264 x.2", "state_after": "no goals"}, {"tactic": "rw [Nat.lt_add_one_iff, Nat.lt_add_one_iff]", "annotated_tactic": ["rw [<a>Nat.lt_add_one_iff</a>, <a>Nat.lt_add_one_iff</a>]", [{"full_name": "Nat.lt_add_one_iff", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [575, 19], "def_end_pos": [575, 33]}, {"full_name": "Nat.lt_add_one_iff", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [575, 19], "def_end_pos": [575, 33]}]], "state_before": "case h.mpr\nx : \u2115 \u00d7 \u2115\nh : \u00acx.1 * x.2 = 0\n\u22a2 (1 \u2264 x.1 \u2227 x.1 < x.1 * x.2 + 1) \u2227 1 \u2264 x.2 \u2227 x.2 < x.1 * x.2 + 1", "state_after": "case h.mpr\nx : \u2115 \u00d7 \u2115\nh : \u00acx.1 * x.2 = 0\n\u22a2 (1 \u2264 x.1 \u2227 x.1 \u2264 x.1 * x.2) \u2227 1 \u2264 x.2 \u2227 x.2 \u2264 x.1 * x.2"}, {"tactic": "rw [mul_eq_zero, not_or] at h", "annotated_tactic": ["rw [<a>mul_eq_zero</a>, <a>not_or</a>] at h", [{"full_name": "Nat.mul_eq_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean", "def_pos": [430, 9], "def_end_pos": [430, 20]}, {"full_name": "not_or", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [134, 17], "def_end_pos": [134, 23]}]], "state_before": "case h.mpr\nx : \u2115 \u00d7 \u2115\nh : \u00acx.1 * x.2 = 0\n\u22a2 (1 \u2264 x.1 \u2227 x.1 \u2264 x.1 * x.2) \u2227 1 \u2264 x.2 \u2227 x.2 \u2264 x.1 * x.2", "state_after": "case h.mpr\nx : \u2115 \u00d7 \u2115\nh : \u00acx.1 = 0 \u2227 \u00acx.2 = 0\n\u22a2 (1 \u2264 x.1 \u2227 x.1 \u2264 x.1 * x.2) \u2227 1 \u2264 x.2 \u2227 x.2 \u2264 x.1 * x.2"}, {"tactic": "simp only [succ_le_of_lt (Nat.pos_of_ne_zero h.1), succ_le_of_lt (Nat.pos_of_ne_zero h.2),\n  true_and_iff]", "annotated_tactic": ["simp only [<a>succ_le_of_lt</a> (<a>Nat.pos_of_ne_zero</a> h.1), <a>succ_le_of_lt</a> (<a>Nat.pos_of_ne_zero</a> h.2),\n      <a>true_and_iff</a>]", [{"full_name": "Nat.succ_le_of_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [348, 9], "def_end_pos": [348, 22]}, {"full_name": "Nat.pos_of_ne_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [354, 19], "def_end_pos": [354, 33]}, {"full_name": "Nat.succ_le_of_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [348, 9], "def_end_pos": [348, 22]}, {"full_name": "Nat.pos_of_ne_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [354, 19], "def_end_pos": [354, 33]}, {"full_name": "true_and_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [118, 9], "def_end_pos": [118, 21]}]], "state_before": "case h.mpr\nx : \u2115 \u00d7 \u2115\nh : \u00acx.1 = 0 \u2227 \u00acx.2 = 0\n\u22a2 (1 \u2264 x.1 \u2227 x.1 \u2264 x.1 * x.2) \u2227 1 \u2264 x.2 \u2227 x.2 \u2264 x.1 * x.2", "state_after": "case h.mpr\nx : \u2115 \u00d7 \u2115\nh : \u00acx.1 = 0 \u2227 \u00acx.2 = 0\n\u22a2 x.1 \u2264 x.1 * x.2 \u2227 x.2 \u2264 x.1 * x.2"}, {"tactic": "exact\n  \u27e8Nat.le_mul_of_pos_right _ (Nat.pos_of_ne_zero h.2),\n    Nat.le_mul_of_pos_left _ (Nat.pos_of_ne_zero h.1)\u27e9", "annotated_tactic": ["exact\n      \u27e8<a>Nat.le_mul_of_pos_right</a> _ (<a>Nat.pos_of_ne_zero</a> h.2),\n        <a>Nat.le_mul_of_pos_left</a> _ (<a>Nat.pos_of_ne_zero</a> h.1)\u27e9", [{"full_name": "Nat.le_mul_of_pos_right", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean", "def_pos": [468, 19], "def_end_pos": [468, 38]}, {"full_name": "Nat.pos_of_ne_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [354, 19], "def_end_pos": [354, 33]}, {"full_name": "Nat.le_mul_of_pos_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean", "def_pos": [465, 19], "def_end_pos": [465, 37]}, {"full_name": "Nat.pos_of_ne_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [354, 19], "def_end_pos": [354, 33]}]], "state_before": "case h.mpr\nx : \u2115 \u00d7 \u2115\nh : \u00acx.1 = 0 \u2227 \u00acx.2 = 0\n\u22a2 x.1 \u2264 x.1 * x.2 \u2227 x.2 \u2264 x.1 * x.2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Nilpotent.lean", "full_name": "Group.IsNilpotent.nilpotent", "start": [173, 1], "end": [174, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Exp.lean", "full_name": "Continuous.rexp", "start": [192, 1], "end": [193, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Filter.lean", "full_name": "Filter.nhds_nhds", "start": [212, 1], "end": [214, 81], "traced_tactics": [{"tactic": "simp only [(nhds_basis_opens x).nhds.eq_biInf, iInf_and, @iInf_comm _ (_ \u2208 _)]", "annotated_tactic": ["simp only [(<a>nhds_basis_opens</a> x).nhds.eq_biInf, <a>iInf_and</a>, @<a>iInf_comm</a> _ (_ \u2208 _)]", [{"full_name": "nhds_basis_opens", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [808, 9], "def_end_pos": [808, 25]}, {"full_name": "iInf_and", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1281, 9], "def_end_pos": [1281, 17]}, {"full_name": "iInf_comm", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1086, 9], "def_end_pos": [1086, 18]}]], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nX : Type u_4\nY : Type u_5\ninst\u271d : TopologicalSpace X\nx : X\n\u22a2 \ud835\udcdd (\ud835\udcdd x) = \u2a05 s, \u2a05 (_ : IsOpen s), \u2a05 (_ : x \u2208 s), \ud835\udcdf (Iic (\ud835\udcdf s))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Idempotents/Karoubi.lean", "full_name": "CategoryTheory.Idempotents.Karoubi.p_comm", "start": [94, 1], "end": [94, 97], "traced_tactics": [{"tactic": "rw [p_comp, comp_p]", "annotated_tactic": ["rw [<a>p_comp</a>, <a>comp_p</a>]", [{"full_name": "CategoryTheory.Idempotents.Karoubi.p_comp", "def_path": "Mathlib/CategoryTheory/Idempotents/Karoubi.lean", "def_pos": [85, 9], "def_end_pos": [85, 15]}, {"full_name": "CategoryTheory.Idempotents.Karoubi.comp_p", "def_path": "Mathlib/CategoryTheory/Idempotents/Karoubi.lean", "def_pos": [89, 9], "def_end_pos": [89, 15]}]], "state_before": "C : Type u_1\ninst\u271d : Category.{u_2, u_1} C\nP Q : Karoubi C\nf : P.Hom Q\n\u22a2 P.p \u226b f.f = f.f \u226b Q.p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/QuadraticForm/Basic.lean", "full_name": "QuadraticForm.IsOrtho.zero_left", "start": [960, 1], "end": [960, 81], "traced_tactics": [{"tactic": "simp [isOrtho_def]", "annotated_tactic": ["simp [<a>isOrtho_def</a>]", [{"full_name": "QuadraticForm.isOrtho_def", "def_path": "Mathlib/LinearAlgebra/QuadraticForm/Basic.lean", "def_pos": [955, 9], "def_end_pos": [955, 20]}]], "state_before": "S : Type u_1\nT : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nx : M\n\u22a2 Q.IsOrtho 0 x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/InfiniteSum/Group.lean", "full_name": "prod_mul_tprod_subtype_compl", "start": [312, 1], "end": [316, 6], "traced_tactics": [{"tactic": "rw [\u2190 tprod_subtype_mul_tprod_subtype_compl hf s]", "annotated_tactic": ["rw [\u2190 <a>tprod_subtype_mul_tprod_subtype_compl</a> hf s]", [{"full_name": "tprod_subtype_mul_tprod_subtype_compl", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Group.lean", "def_pos": [306, 9], 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{ x // x \u2209 s }), f \u2191x = (\u220f' (x : \u2191\u2191s), f \u2191x) * \u220f' (x : \u2191(\u2191s)\u1d9c), f \u2191x"}, {"tactic": "simp only [Finset.tprod_subtype', mul_right_inj]", "annotated_tactic": ["simp only [<a>Finset.tprod_subtype'</a>, <a>mul_right_inj</a>]", [{"full_name": "Finset.tprod_subtype'", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [479, 9], "def_end_pos": [479, 30]}, {"full_name": "mul_right_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [68, 9], "def_end_pos": [68, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u2074 : CommGroup \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b1\ninst\u271d\u00b2 : UniformGroup \u03b1\nf\u271d g : \u03b2 \u2192 \u03b1\na a\u2081 a\u2082 : \u03b1\ninst\u271d\u00b9 : CompleteSpace \u03b1\ninst\u271d : T2Space \u03b1\nf : \u03b2 \u2192 \u03b1\nhf : Multipliable f\ns : Finset \u03b2\n\u22a2 (\u220f x \u2208 s, f x) * 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AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 (FreeAddMonoid.lift fun p => p.1 \u2022 \u03c6 p.2) (FreeAddMonoid.ofList [(1, f)]) = \u03c6 f", "state_after": "\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 (FreeAddMonoid.lift fun p => p.1 \u2022 \u03c6 p.2) ((Equiv.refl (List (R \u00d7 ((i : \u03b9) \u2192 s i)))) [(1, f)]) = \u03c6 f"}, {"tactic": "rw [\u2190 one_smul R (\u03c6 f)]", "annotated_tactic": ["rw [\u2190 <a>one_smul</a> R (\u03c6 f)]", [{"full_name": "one_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [453, 7], "def_end_pos": [453, 15]}]], "state_before": "\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 (FreeAddMonoid.lift fun p => p.1 \u2022 \u03c6 p.2) ((Equiv.refl (List (R \u00d7 ((i : \u03b9) \u2192 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i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 (FreeAddMonoid.sumAux.match_1 (fun x => E)\n      (List.map (fun p => p.1 \u2022 \u03c6 p.2) (FreeAddMonoid.toList ((Equiv.refl (List (R \u00d7 ((i : \u03b9) \u2192 s i)))) [(1, f)])))\n      (fun _ => 0) fun x xs => List.foldl (fun x x_1 => x + x_1) x xs) =\n    1 \u2022 \u03c6 f"}, {"tactic": "convert one_smul R (\u03c6 f)", "annotated_tactic": ["convert <a>one_smul</a> R (\u03c6 f)", [{"full_name": "one_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [453, 7], "def_end_pos": [453, 15]}]], "state_before": "\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 (FreeAddMonoid.sumAux.match_1 (fun x => E)\n      (List.map (fun p => p.1 \u2022 \u03c6 p.2) (FreeAddMonoid.toList ((Equiv.refl (List (R \u00d7 ((i : \u03b9) \u2192 s i)))) [(1, f)])))\n      (fun _ => 0) fun x xs => List.foldl (fun x x_1 => x + x_1) x xs) =\n    1 \u2022 \u03c6 f", "state_after": "case h.e'_3\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 1 \u2022 \u03c6 f = \u03c6 f"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_3\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\n\u03c6 : MultilinearMap R s E\nf : (i : \u03b9) \u2192 s i\n\u22a2 1 \u2022 \u03c6 f = \u03c6 f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Kernel/Composition.lean", "full_name": "ProbabilityTheory.kernel.map_apply", "start": [632, 1], "end": [633, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Complex/Basic.lean", "full_name": "Complex.comap_abs_nhds_zero", "start": [157, 1], "end": [158, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Heyting/Hom.lean", "full_name": "HeytingHom.toFun_eq_coe", "start": [278, 1], "end": [279, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.lean", "full_name": "aemeasurable_of_re_im", "start": [80, 1], "end": [84, 34], "traced_tactics": [{"tactic": "convert AEMeasurable.add (M := \ud835\udd5c) (RCLike.measurable_ofReal.comp_aemeasurable hre)\n    ((RCLike.measurable_ofReal.comp_aemeasurable him).mul_const RCLike.I)", "annotated_tactic": ["convert <a>AEMeasurable.add</a> (M := \ud835\udd5c) (RCLike.measurable_ofReal.comp_aemeasurable hre)\n      ((RCLike.measurable_ofReal.comp_aemeasurable him).<a>mul_const</a> <a>RCLike.I</a>)", [{"full_name": "AEMeasurable.add", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [153, 3], "def_end_pos": [153, 14]}, {"full_name": "AEMeasurable.mul_const", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [126, 9], "def_end_pos": [126, 31]}, {"full_name": "RCLike.I", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [58, 3], "def_end_pos": [58, 4]}]], "state_before": "\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9 : RCLike \ud835\udd5c\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \ud835\udd5c\n\u03bc : MeasureTheory.Measure \u03b1\nhre : AEMeasurable (fun x => RCLike.re (f x)) \u03bc\nhim : AEMeasurable (fun x => RCLike.im (f x)) \u03bc\n\u22a2 AEMeasurable f \u03bc", "state_after": "case h.e'_5.h\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9 : RCLike \ud835\udd5c\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \ud835\udd5c\n\u03bc : MeasureTheory.Measure \u03b1\nhre : AEMeasurable (fun x => RCLike.re (f x)) \u03bc\nhim : AEMeasurable (fun x => RCLike.im (f x)) \u03bc\nx\u271d : \u03b1\n\u22a2 f x\u271d = (RCLike.ofReal \u2218 fun x => RCLike.re (f x)) x\u271d + (RCLike.ofReal \u2218 fun x => RCLike.im (f x)) x\u271d * RCLike.I"}, {"tactic": "exact (RCLike.re_add_im _).symm", "annotated_tactic": ["exact (<a>RCLike.re_add_im</a> _).<a>symm</a>", [{"full_name": "RCLike.re_add_im", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [115, 9], "def_end_pos": [115, 18]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case h.e'_5.h\n\u03b1 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u00b9 : RCLike \ud835\udd5c\ninst\u271d : MeasurableSpace \u03b1\nf : \u03b1 \u2192 \ud835\udd5c\n\u03bc : MeasureTheory.Measure \u03b1\nhre : AEMeasurable (fun x => RCLike.re (f x)) \u03bc\nhim : AEMeasurable (fun x => RCLike.im (f x)) \u03bc\nx\u271d : \u03b1\n\u22a2 f x\u271d = (RCLike.ofReal \u2218 fun x => RCLike.re (f x)) x\u271d + (RCLike.ofReal \u2218 fun x => RCLike.im (f x)) x\u271d * RCLike.I", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean", "full_name": "HomologicalComplex.homologyMap_neg", "start": [657, 1], "end": [661, 6], "traced_tactics": [{"tactic": "dsimp [homologyMap]", "annotated_tactic": ["dsimp [<a>homologyMap</a>]", [{"full_name": "HomologicalComplex.homologyMap", "def_path": "Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean", "def_pos": [293, 19], "def_end_pos": [293, 30]}]], "state_before": "C : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : Category.{u_3, u_1} C\ninst\u271d\u00b2 : Preadditive C\nc : ComplexShape \u03b9\nK L : HomologicalComplex C c\nf g \u03c6 \u03c8 : K \u27f6 L\ni : \u03b9\ninst\u271d\u00b9 : K.HasHomology i\ninst\u271d : L.HasHomology i\n\u22a2 homologyMap (-\u03c6) i = -homologyMap \u03c6 i", "state_after": "C : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : Category.{u_3, u_1} C\ninst\u271d\u00b2 : Preadditive C\nc : ComplexShape \u03b9\nK L : HomologicalComplex C c\nf g \u03c6 \u03c8 : K \u27f6 L\ni : \u03b9\ninst\u271d\u00b9 : K.HasHomology i\ninst\u271d : L.HasHomology i\n\u22a2 ShortComplex.homologyMap ((shortComplexFunctor C c i).map (-\u03c6)) =\n    -ShortComplex.homologyMap ((shortComplexFunctor C c i).map \u03c6)"}, {"tactic": "rw [\u2190 ShortComplex.homologyMap_neg]", "annotated_tactic": ["rw [\u2190 <a>ShortComplex.homologyMap_neg</a>]", [{"full_name": "CategoryTheory.ShortComplex.homologyMap_neg", "def_path": "Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean", "def_pos": [337, 7], "def_end_pos": [337, 22]}]], "state_before": "C : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : Category.{u_3, u_1} C\ninst\u271d\u00b2 : Preadditive C\nc : ComplexShape \u03b9\nK L : HomologicalComplex C c\nf g \u03c6 \u03c8 : K \u27f6 L\ni : \u03b9\ninst\u271d\u00b9 : K.HasHomology i\ninst\u271d : L.HasHomology i\n\u22a2 ShortComplex.homologyMap ((shortComplexFunctor C c i).map (-\u03c6)) =\n    -ShortComplex.homologyMap ((shortComplexFunctor C c i).map \u03c6)", "state_after": "C : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : Category.{u_3, u_1} C\ninst\u271d\u00b2 : Preadditive C\nc : ComplexShape \u03b9\nK L : HomologicalComplex C c\nf g \u03c6 \u03c8 : K \u27f6 L\ni : \u03b9\ninst\u271d\u00b9 : K.HasHomology i\ninst\u271d : L.HasHomology i\n\u22a2 ShortComplex.homologyMap ((shortComplexFunctor C c i).map (-\u03c6)) =\n    ShortComplex.homologyMap (-(shortComplexFunctor C c i).map \u03c6)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "C : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : Category.{u_3, u_1} C\ninst\u271d\u00b2 : Preadditive C\nc : ComplexShape \u03b9\nK L : HomologicalComplex C c\nf g \u03c6 \u03c8 : K \u27f6 L\ni : \u03b9\ninst\u271d\u00b9 : K.HasHomology i\ninst\u271d : L.HasHomology i\n\u22a2 ShortComplex.homologyMap ((shortComplexFunctor C c i).map (-\u03c6)) =\n    ShortComplex.homologyMap (-(shortComplexFunctor C c i).map \u03c6)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/Instances/Real.lean", "full_name": "EuclideanHalfSpace.ext", "start": [88, 1], "end": [90, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Types.lean", "full_name": "CategoryTheory.types_id", "start": [61, 1], "end": [62, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/Basic.lean", "full_name": "Equiv.Perm.subtypePerm_zpow", "start": [428, 1], "end": [432, 63], "traced_tactics": [{"tactic": "induction' n with n ih", "annotated_tactic": ["induction' n with n ih", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d f : Perm \u03b1\nn : \u2124\nhf : \u2200 (x : \u03b1), p x \u2194 p (f x)\n\u22a2 f.subtypePerm hf ^ n = (f ^ n).subtypePerm \u22ef", "state_after": "case ofNat\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d f : Perm \u03b1\nhf : \u2200 (x : \u03b1), p x \u2194 p (f x)\nn : \u2115\n\u22a2 f.subtypePerm hf ^ Int.ofNat n = (f ^ Int.ofNat n).subtypePerm \u22ef\n\ncase negSucc\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d f : Perm \u03b1\nhf : \u2200 (x : \u03b1), p x \u2194 p (f x)\nih : \u2115\n\u22a2 f.subtypePerm hf ^ Int.negSucc ih = (f ^ Int.negSucc ih).subtypePerm \u22ef"}, {"tactic": "exact subtypePerm_pow _ _ _", "annotated_tactic": ["exact <a>subtypePerm_pow</a> _ _ _", [{"full_name": "Equiv.Perm.subtypePerm_pow", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [414, 9], "def_end_pos": [414, 24]}]], "state_before": "case ofNat\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d f : Perm \u03b1\nhf : \u2200 (x : \u03b1), p x \u2194 p (f x)\nn : \u2115\n\u22a2 f.subtypePerm hf ^ Int.ofNat n = (f ^ Int.ofNat n).subtypePerm \u22ef", "state_after": "no goals"}, {"tactic": "simp only [zpow_negSucc, subtypePerm_pow, subtypePerm_inv]", "annotated_tactic": ["simp only [<a>zpow_negSucc</a>, <a>subtypePerm_pow</a>, <a>subtypePerm_inv</a>]", [{"full_name": "zpow_negSucc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1040, 9], "def_end_pos": [1040, 21]}, {"full_name": "Equiv.Perm.subtypePerm_pow", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [414, 9], "def_end_pos": [414, 24]}, {"full_name": "Equiv.Perm.subtypePerm_inv", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [397, 9], "def_end_pos": [397, 24]}]], "state_before": "case negSucc\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Prop\nf\u271d f : Perm \u03b1\nhf : \u2200 (x : \u03b1), p x \u2194 p (f x)\nih : \u2115\n\u22a2 f.subtypePerm hf ^ Int.negSucc ih = (f ^ Int.negSucc ih).subtypePerm \u22ef", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Tower.lean", "full_name": "AlgHom.restrictScalars_apply", "start": [212, 1], "end": [212, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GroupWithZero/Invertible.lean", "full_name": "mul_inv_cancel_of_invertible", "start": [66, 1], "end": [67, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/Adjoint.lean", "full_name": "ContinuousLinearMap.star_eq_adjoint", "start": [214, 1], "end": [215, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Submonoid/Operations.lean", "full_name": "Submonoid.map_inf_comap_of_surjective", "start": [469, 1], "end": [470, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Etale/Basic.lean", "full_name": "Algebra.Etale.of_equiv", "start": [199, 1], "end": [201, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Multilinear/Basic.lean", "full_name": "MultilinearMap.congr_arg", "start": [133, 8], "end": [134, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SumOverResidueClass.lean", "full_name": "summable_indicator_mod_iff_summable", "start": [29, 1], "end": [46, 90], "traced_tactics": [{"tactic": "trans Summable ({n : \u2115 | (n : ZMod m) = k \u2227 k \u2264 n}.indicator f)", "annotated_tactic": ["trans <a>Summable</a> ({n : \u2115 | (n : <a>ZMod</a> m) = k \u2227 k \u2264 n}.<a>indicator</a> f)", [{"full_name": "Summable", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Defs.lean", "def_pos": [91, 3], "def_end_pos": [91, 14]}, {"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [95, 5], "def_end_pos": [95, 9]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/Group/Indicator.lean", "def_pos": [47, 3], "def_end_pos": [47, 14]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : AddCommGroup R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalAddGroup R\nm : \u2115\nhm : NeZero m\nk : \u2115\nf : \u2115 \u2192 R\n\u22a2 Summable ({n | \u2191n = \u2191k}.indicator f) \u2194 Summable fun n => f (m * n + k)", "state_after": "R : Type u_1\ninst\u271d\u00b2 : AddCommGroup R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalAddGroup R\nm : \u2115\nhm : NeZero m\nk : \u2115\nf : \u2115 \u2192 R\n\u22a2 Summable ({n | \u2191n = \u2191k}.indicator f) \u2194 Summable ({n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f)\n\nR : Type u_1\ninst\u271d\u00b2 : AddCommGroup R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalAddGroup R\nm : \u2115\nhm : NeZero m\nk : \u2115\nf : \u2115 \u2192 R\n\u22a2 Summable ({n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f) \u2194 Summable fun n => f (m * n + k)"}, {"tactic": "rw [\u2190 (finite_lt_nat k).summable_compl_iff (f := {n : \u2115 | (n : ZMod m) = k}.indicator f)]", "annotated_tactic": ["rw [\u2190 (<a>finite_lt_nat</a> k).<a>summable_compl_iff</a> (f := {n : \u2115 | (n : <a>ZMod</a> m) = k}.<a>indicator</a> f)]", [{"full_name": "Set.finite_lt_nat", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}, {"full_name": "Set.Finite.summable_compl_iff", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Group.lean", "def_pos": [130, 3], "def_end_pos": [130, 14]}, {"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [95, 5], "def_end_pos": [95, 9]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/Group/Indicator.lean", "def_pos": [47, 3], "def_end_pos": [47, 14]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : AddCommGroup R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalAddGroup R\nm : \u2115\nhm : NeZero m\nk : \u2115\nf : \u2115 \u2192 R\n\u22a2 Summable ({n | \u2191n = \u2191k}.indicator f) \u2194 Summable ({n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f)", "state_after": "R : Type u_1\ninst\u271d\u00b2 : AddCommGroup R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalAddGroup R\nm : \u2115\nhm : NeZero m\nk : \u2115\nf : \u2115 \u2192 R\n\u22a2 Summable ({n | \u2191n = \u2191k}.indicator f \u2218 Subtype.val) \u2194 Summable ({n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f)"}, {"tactic": "simp only [summable_subtype_iff_indicator, indicator_indicator, inter_comm, setOf_and,\n  compl_setOf, not_lt]", "annotated_tactic": ["simp only [<a>summable_subtype_iff_indicator</a>, <a>indicator_indicator</a>, <a>inter_comm</a>, <a>setOf_and</a>,\n      <a>compl_setOf</a>, <a>not_lt</a>]", [{"full_name": "summable_subtype_iff_indicator", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [107, 3], "def_end_pos": [107, 14]}, {"full_name": "Set.indicator_indicator", "def_path": "Mathlib/Algebra/Group/Indicator.lean", "def_pos": [222, 3], "def_end_pos": [222, 14]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [905, 9], "def_end_pos": [905, 19]}, {"full_name": "Set.setOf_and", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [277, 9], "def_end_pos": [277, 18]}, {"full_name": "Set.compl_setOf", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1605, 9], "def_end_pos": [1605, 20]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [370, 9], "def_end_pos": [370, 15]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : AddCommGroup R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalAddGroup R\nm : \u2115\nhm : NeZero m\nk : \u2115\nf : \u2115 \u2192 R\n\u22a2 Summable ({n | \u2191n = \u2191k}.indicator f \u2218 Subtype.val) \u2194 Summable ({n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f)", "state_after": "no goals"}, {"tactic": "let g : \u2115 \u2192 \u2115 := fun n \u21a6 m * n + k", "annotated_tactic": ["let g : \u2115 \u2192 \u2115 := fun n \u21a6 m * n + k", []], "state_before": "R : Type u_1\ninst\u271d\u00b2 : AddCommGroup R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalAddGroup R\nm : \u2115\nhm : NeZero m\nk : \u2115\nf : \u2115 \u2192 R\n\u22a2 Summable ({n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f) \u2194 Summable fun n => f (m * n + k)", "state_after": "R : Type u_1\ninst\u271d\u00b2 : AddCommGroup R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalAddGroup R\nm : \u2115\nhm : NeZero m\nk : \u2115\nf : \u2115 \u2192 R\ng : \u2115 \u2192 \u2115 := fun n => m * n + k\n\u22a2 Summable ({n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f) \u2194 Summable fun n => f (m * n + k)"}, {"tactic": "have hg : Function.Injective g := fun m n hmn \u21a6 by simpa [g, hm.ne] using hmn", "annotated_tactic": ["have hg : <a>Function.Injective</a> g := fun m n hmn \u21a6 by simpa [g, hm.ne] using hmn", [{"full_name": "Function.Injective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [123, 5], "def_end_pos": [123, 14]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : AddCommGroup R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalAddGroup R\nm : \u2115\nhm : NeZero m\nk : \u2115\nf : \u2115 \u2192 R\ng : \u2115 \u2192 \u2115 := fun n => m * n + k\n\u22a2 Summable ({n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f) \u2194 Summable fun n => f (m * n + k)", "state_after": "R : Type u_1\ninst\u271d\u00b2 : AddCommGroup R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalAddGroup R\nm : \u2115\nhm : NeZero m\nk : \u2115\nf : \u2115 \u2192 R\ng : \u2115 \u2192 \u2115 := fun n => m * n + k\nhg : Function.Injective g\n\u22a2 Summable ({n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f) \u2194 Summable fun n => f (m * n + k)"}, {"tactic": "have hg' : \u2200 n \u2209 range g, {n : \u2115 | (n : ZMod m) = k \u2227 k \u2264 n}.indicator f n = 0 := by\n  intro n hn\n  contrapose! hn\n  exact (Nat.range_mul_add m k).symm \u25b8 mem_of_indicator_ne_zero hn", "annotated_tactic": ["have hg' : \u2200 n \u2209 <a>range</a> g, {n : \u2115 | (n : <a>ZMod</a> m) = k \u2227 k \u2264 n}.<a>indicator</a> f n = 0 := by\n      intro n hn\n      contrapose! hn\n      exact (<a>Nat.range_mul_add</a> m k).<a>symm</a> \u25b8 <a>mem_of_indicator_ne_zero</a> hn", [{"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [157, 5], "def_end_pos": [157, 10]}, {"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [95, 5], "def_end_pos": [95, 9]}, {"full_name": "Set.indicator", "def_path": "Mathlib/Algebra/Group/Indicator.lean", "def_pos": [47, 3], "def_end_pos": [47, 14]}, {"full_name": "Nat.range_mul_add", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [1469, 7], "def_end_pos": [1469, 24]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}, {"full_name": "Set.mem_of_indicator_ne_zero", "def_path": "Mathlib/Algebra/Group/Indicator.lean", "def_pos": [145, 3], "def_end_pos": [145, 14]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : AddCommGroup R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalAddGroup R\nm : \u2115\nhm : NeZero m\nk : \u2115\nf : \u2115 \u2192 R\ng : \u2115 \u2192 \u2115 := fun n => m * n + k\nhg : Function.Injective g\n\u22a2 Summable ({n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f) \u2194 Summable fun n => f (m * n + k)", "state_after": "R : Type u_1\ninst\u271d\u00b2 : AddCommGroup R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalAddGroup R\nm : \u2115\nhm : NeZero m\nk : \u2115\nf : \u2115 \u2192 R\ng : \u2115 \u2192 \u2115 := fun n => m * n + k\nhg : Function.Injective g\nhg' : \u2200 n \u2209 range g, {n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f n = 0\n\u22a2 Summable ({n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f) \u2194 Summable fun n => f (m * n + k)"}, {"tactic": "convert (Function.Injective.summable_iff hg hg').symm using 3", "annotated_tactic": ["convert (<a>Function.Injective.summable_iff</a> hg hg').<a>symm</a> using 3", [{"full_name": "Function.Injective.summable_iff", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [85, 3], "def_end_pos": [85, 14]}, {"full_name": "Iff.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [813, 9], "def_end_pos": [813, 17]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : AddCommGroup R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalAddGroup R\nm : \u2115\nhm : NeZero m\nk : \u2115\nf : \u2115 \u2192 R\ng : \u2115 \u2192 \u2115 := fun n => m * n + k\nhg : Function.Injective g\nhg' : \u2200 n \u2209 range g, {n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f n = 0\n\u22a2 Summable ({n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f) \u2194 Summable fun n => f (m * n + k)", "state_after": "case h.e'_2.h.e'_5.h\nR : Type u_1\ninst\u271d\u00b2 : AddCommGroup R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalAddGroup R\nm : \u2115\nhm : NeZero m\nk : \u2115\nf : \u2115 \u2192 R\ng : \u2115 \u2192 \u2115 := fun n => m * n + k\nhg : Function.Injective g\nhg' : \u2200 n \u2209 range g, {n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f n = 0\nx\u271d : \u2115\n\u22a2 f (m * x\u271d + k) = ({n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f \u2218 g) x\u271d"}, {"tactic": "simp only [Function.comp_apply, mem_setOf_eq, Nat.cast_add, Nat.cast_mul, CharP.cast_eq_zero,\n  zero_mul, zero_add, le_add_iff_nonneg_left, zero_le, and_self, indicator_of_mem, g]", "annotated_tactic": ["simp only [<a>Function.comp_apply</a>, <a>mem_setOf_eq</a>, <a>Nat.cast_add</a>, <a>Nat.cast_mul</a>, <a>CharP.cast_eq_zero</a>,\n      <a>zero_mul</a>, <a>zero_add</a>, <a>le_add_iff_nonneg_left</a>, <a>zero_le</a>, <a>and_self</a>, <a>indicator_of_mem</a>, g]", [{"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [79, 29], "def_end_pos": [79, 41]}, {"full_name": "Nat.cast_add", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 17]}, {"full_name": "Nat.cast_mul", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [62, 26], "def_end_pos": [62, 34]}, {"full_name": "CharP.cast_eq_zero", "def_path": "Mathlib/Algebra/CharP/Defs.lean", "def_pos": [60, 15], "def_end_pos": [60, 27]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [476, 3], "def_end_pos": [476, 14]}, {"full_name": "le_add_iff_nonneg_left", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [465, 30], "def_end_pos": [465, 52]}, {"full_name": "zero_le", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [204, 30], "def_end_pos": [204, 37]}, {"full_name": "and_self", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [111, 17], "def_end_pos": [111, 25]}, {"full_name": "Set.indicator_of_mem", "def_path": "Mathlib/Algebra/Group/Indicator.lean", "def_pos": [68, 3], "def_end_pos": [68, 14]}]], "state_before": "case h.e'_2.h.e'_5.h\nR : Type u_1\ninst\u271d\u00b2 : AddCommGroup R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalAddGroup R\nm : \u2115\nhm : NeZero m\nk : \u2115\nf : \u2115 \u2192 R\ng : \u2115 \u2192 \u2115 := fun n => m * n + k\nhg : Function.Injective g\nhg' : \u2200 n \u2209 range g, {n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f n = 0\nx\u271d : \u2115\n\u22a2 f (m * x\u271d + k) = ({n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f \u2218 g) x\u271d", "state_after": "no goals"}, {"tactic": "simpa [g, hm.ne] using hmn", "annotated_tactic": ["simpa [g, hm.ne] using hmn", []], "state_before": "R : Type u_1\ninst\u271d\u00b2 : AddCommGroup R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalAddGroup R\nm\u271d : \u2115\nhm : NeZero m\u271d\nk : \u2115\nf : \u2115 \u2192 R\ng : \u2115 \u2192 \u2115 := fun n => m\u271d * n + k\nm n : \u2115\nhmn : g m = g n\n\u22a2 m = n", "state_after": "no goals"}, {"tactic": "intro n hn", "annotated_tactic": ["intro n hn", []], "state_before": "R : Type u_1\ninst\u271d\u00b2 : AddCommGroup R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalAddGroup R\nm : \u2115\nhm : NeZero m\nk : \u2115\nf : \u2115 \u2192 R\ng : \u2115 \u2192 \u2115 := fun n => m * n + k\nhg : Function.Injective g\n\u22a2 \u2200 n \u2209 range g, {n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f n = 0", "state_after": "R : Type u_1\ninst\u271d\u00b2 : AddCommGroup R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalAddGroup R\nm : \u2115\nhm : NeZero m\nk : \u2115\nf : \u2115 \u2192 R\ng : \u2115 \u2192 \u2115 := fun n => m * n + k\nhg : Function.Injective g\nn : \u2115\nhn : n \u2209 range g\n\u22a2 {n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f n = 0"}, {"tactic": "contrapose! hn", "annotated_tactic": ["contrapose! hn", []], "state_before": "R : Type u_1\ninst\u271d\u00b2 : AddCommGroup R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalAddGroup R\nm : \u2115\nhm : NeZero m\nk : \u2115\nf : \u2115 \u2192 R\ng : \u2115 \u2192 \u2115 := fun n => m * n + k\nhg : Function.Injective g\nn : \u2115\nhn : n \u2209 range g\n\u22a2 {n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f n = 0", "state_after": "R : Type u_1\ninst\u271d\u00b2 : AddCommGroup R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalAddGroup R\nm : \u2115\nhm : NeZero m\nk : \u2115\nf : \u2115 \u2192 R\ng : \u2115 \u2192 \u2115 := fun n => m * n + k\nhg : Function.Injective g\nn : \u2115\nhn : {n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f n \u2260 0\n\u22a2 n \u2208 range g"}, {"tactic": "exact (Nat.range_mul_add m k).symm \u25b8 mem_of_indicator_ne_zero hn", "annotated_tactic": ["exact (<a>Nat.range_mul_add</a> m k).<a>symm</a> \u25b8 <a>mem_of_indicator_ne_zero</a> hn", [{"full_name": "Nat.range_mul_add", "def_path": "Mathlib/Data/ZMod/Basic.lean", "def_pos": [1469, 7], "def_end_pos": [1469, 24]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}, {"full_name": "Set.mem_of_indicator_ne_zero", "def_path": "Mathlib/Algebra/Group/Indicator.lean", "def_pos": [145, 3], "def_end_pos": [145, 14]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : AddCommGroup R\ninst\u271d\u00b9 : TopologicalSpace R\ninst\u271d : TopologicalAddGroup R\nm : \u2115\nhm : NeZero m\nk : \u2115\nf : \u2115 \u2192 R\ng : \u2115 \u2192 \u2115 := fun n => m * n + k\nhg : Function.Injective g\nn : \u2115\nhn : {n | \u2191n = \u2191k \u2227 k \u2264 n}.indicator f n \u2260 0\n\u22a2 n \u2208 range g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/MeanValue.lean", "full_name": "eq_of_fderiv_eq", "start": [627, 1], "end": [634, 74], "traced_tactics": [{"tactic": "let A : NormedSpace \u211d E := RestrictScalars.normedSpace \u211d \ud835\udd5c E", "annotated_tactic": ["let A : <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": [455, 10], "def_end_pos": [455, 37]}]], "state_before": "E\u271d : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\u271d\ninst\u271d\u2078 : NormedSpace \u211d E\u271d\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u2075 : RCLike \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\u271d\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nf\u271d g\u271d : E\u271d \u2192 G\nC : \u211d\ns : Set E\u271d\nx\u271d y : E\u271d\nf' g' : E\u271d \u2192 E\u271d \u2192L[\ud835\udd5c] G\n\u03c6 : E\u271d \u2192L[\ud835\udd5c] G\nE : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf g : E \u2192 G\nhf : Differentiable \ud835\udd5c f\nhg : Differentiable \ud835\udd5c g\nhf' : \u2200 (x : E), fderiv \ud835\udd5c f x = fderiv \ud835\udd5c g x\nx : E\nhfgx : f x = g x\n\u22a2 f = g", "state_after": "E\u271d : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\u271d\ninst\u271d\u2078 : NormedSpace \u211d E\u271d\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u2075 : RCLike \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\u271d\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nf\u271d g\u271d : E\u271d \u2192 G\nC : \u211d\ns : Set E\u271d\nx\u271d y : E\u271d\nf' g' : E\u271d \u2192 E\u271d \u2192L[\ud835\udd5c] G\n\u03c6 : E\u271d \u2192L[\ud835\udd5c] G\nE : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf g : E \u2192 G\nhf : Differentiable \ud835\udd5c f\nhg : Differentiable \ud835\udd5c g\nhf' : \u2200 (x : E), fderiv \ud835\udd5c f x = fderiv \ud835\udd5c g x\nx : E\nhfgx : f x = g x\nA : NormedSpace \u211d E := RestrictScalars.normedSpace \u211d \ud835\udd5c E\n\u22a2 f = g"}, {"tactic": "suffices Set.univ.EqOn f g from funext fun x => this <| mem_univ x", "annotated_tactic": ["suffices Set.univ.EqOn f g from <a>funext</a> fun x => this <| <a>mem_univ</a> x", [{"full_name": "funext", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1817, 9], "def_end_pos": [1817, 15]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [82, 29], "def_end_pos": [82, 37]}]], "state_before": "E\u271d : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\u271d\ninst\u271d\u2078 : NormedSpace \u211d E\u271d\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u2075 : RCLike \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\u271d\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nf\u271d g\u271d : E\u271d \u2192 G\nC : \u211d\ns : Set E\u271d\nx\u271d y : E\u271d\nf' g' : E\u271d \u2192 E\u271d \u2192L[\ud835\udd5c] G\n\u03c6 : E\u271d \u2192L[\ud835\udd5c] G\nE : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf g : E \u2192 G\nhf : Differentiable \ud835\udd5c f\nhg : Differentiable \ud835\udd5c g\nhf' : \u2200 (x : E), fderiv \ud835\udd5c f x = fderiv \ud835\udd5c g x\nx : E\nhfgx : f x = g x\nA : NormedSpace \u211d E := RestrictScalars.normedSpace \u211d \ud835\udd5c E\n\u22a2 f = g", "state_after": "E\u271d : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\u271d\ninst\u271d\u2078 : NormedSpace \u211d E\u271d\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u2075 : RCLike \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\u271d\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nf\u271d g\u271d : E\u271d \u2192 G\nC : \u211d\ns : Set E\u271d\nx\u271d y : E\u271d\nf' g' : E\u271d \u2192 E\u271d \u2192L[\ud835\udd5c] G\n\u03c6 : E\u271d \u2192L[\ud835\udd5c] G\nE : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf g : E \u2192 G\nhf : Differentiable \ud835\udd5c f\nhg : Differentiable \ud835\udd5c g\nhf' : \u2200 (x : E), fderiv \ud835\udd5c f x = fderiv \ud835\udd5c g x\nx : E\nhfgx : f x = g x\nA : NormedSpace \u211d E := RestrictScalars.normedSpace \u211d \ud835\udd5c E\n\u22a2 EqOn f g univ"}, {"tactic": "exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn\n  uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx", "annotated_tactic": ["exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn\n    <a>uniqueDiffOn_univ</a> (fun x _ => by simpa using hf' _) (<a>mem_univ</a> _) hfgx", [{"full_name": "uniqueDiffOn_univ", "def_path": "Mathlib/Analysis/Calculus/TangentCone.lean", "def_pos": [244, 9], "def_end_pos": [244, 26]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [82, 29], "def_end_pos": [82, 37]}]], "state_before": "E\u271d : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\u271d\ninst\u271d\u2078 : NormedSpace \u211d E\u271d\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u2075 : RCLike \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\u271d\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nf\u271d g\u271d : E\u271d \u2192 G\nC : \u211d\ns : Set E\u271d\nx\u271d y : E\u271d\nf' g' : E\u271d \u2192 E\u271d \u2192L[\ud835\udd5c] G\n\u03c6 : E\u271d \u2192L[\ud835\udd5c] G\nE : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf g : E \u2192 G\nhf : Differentiable \ud835\udd5c f\nhg : Differentiable \ud835\udd5c g\nhf' : \u2200 (x : E), fderiv \ud835\udd5c f x = fderiv \ud835\udd5c g x\nx : E\nhfgx : f x = g x\nA : NormedSpace \u211d E := RestrictScalars.normedSpace \u211d \ud835\udd5c E\n\u22a2 EqOn f g univ", "state_after": "no goals"}, {"tactic": "simpa using hf' _", "annotated_tactic": ["simpa using hf' _", []], "state_before": "E\u271d : Type u_1\ninst\u271d\u2079 : NormedAddCommGroup E\u271d\ninst\u271d\u2078 : NormedSpace \u211d E\u271d\nF : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedSpace \u211d F\n\ud835\udd5c : Type u_3\nG : Type u_4\ninst\u271d\u2075 : RCLike \ud835\udd5c\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\u271d\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nf\u271d g\u271d : E\u271d \u2192 G\nC : \u211d\ns : Set E\u271d\nx\u271d\u00b2 y : E\u271d\nf' g' : E\u271d \u2192 E\u271d \u2192L[\ud835\udd5c] G\n\u03c6 : E\u271d \u2192L[\ud835\udd5c] G\nE : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \ud835\udd5c E\nf g : E \u2192 G\nhf : Differentiable \ud835\udd5c f\nhg : Differentiable \ud835\udd5c g\nhf' : \u2200 (x : E), fderiv \ud835\udd5c f x = fderiv \ud835\udd5c g x\nx\u271d\u00b9 : E\nhfgx : f x\u271d\u00b9 = g x\u271d\u00b9\nA : NormedSpace \u211d E := RestrictScalars.normedSpace \u211d \ud835\udd5c E\nx : E\nx\u271d : x \u2208 univ\n\u22a2 fderivWithin \ud835\udd5c f univ x = fderivWithin \ud835\udd5c g univ x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/FinEnum.lean", "full_name": "FinEnum.mem_toList", "start": [69, 1], "end": [70, 38], "traced_tactics": [{"tactic": "simp [toList]", "annotated_tactic": ["simp [<a>toList</a>]", [{"full_name": "FinEnum.toList", "def_path": "Mathlib/Data/FinEnum.lean", "def_pos": [62, 5], "def_end_pos": [62, 11]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : FinEnum \u03b1\nx : \u03b1\n\u22a2 x \u2208 toList \u03b1", "state_after": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : FinEnum \u03b1\nx : \u03b1\n\u22a2 \u2203 a, equiv.symm a = x"}, {"tactic": "exists equiv x", "annotated_tactic": ["exists <a>equiv</a> x", [{"full_name": "FinEnum.equiv", "def_path": "Mathlib/Data/FinEnum.lean", "def_pos": [30, 3], "def_end_pos": [30, 8]}]], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : FinEnum \u03b1\nx : \u03b1\n\u22a2 \u2203 a, equiv.symm a = x", "state_after": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : FinEnum \u03b1\nx : \u03b1\n\u22a2 equiv.symm (equiv x) = x"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : FinEnum \u03b1\nx : \u03b1\n\u22a2 equiv.symm (equiv x) = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monoidal/End.lean", "full_name": "CategoryTheory.\u03b5_inv_hom_app", "start": [119, 1], "end": [120, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Init/Algebra/Classes.lean", "full_name": "trichotomous_of", "start": [364, 1], "end": [365, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Basic.lean", "full_name": "Polynomial.erase_ne", "start": [1103, 1], "end": [1104, 24], "traced_tactics": [{"tactic": "simp [coeff_erase, h]", "annotated_tactic": ["simp [<a>coeff_erase</a>, h]", [{"full_name": "Polynomial.coeff_erase", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [1080, 9], "def_end_pos": [1080, 20]}]], "state_before": "R : Type u\na b : R\nm n\u271d : \u2115\ninst\u271d : Semiring R\np\u271d q p : R[X]\nn i : \u2115\nh : i \u2260 n\n\u22a2 (erase n p).coeff i = p.coeff i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "full_name": "MeasureTheory.Measure.haar.index_empty", "start": [102, 1], "end": [104, 92], "traced_tactics": [{"tactic": "simp only [index, Nat.sInf_eq_zero]", "annotated_tactic": ["simp only [<a>index</a>, <a>Nat.sInf_eq_zero</a>]", [{"full_name": "MeasureTheory.Measure.haar.index", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Basic.lean", "def_pos": [96, 19], "def_end_pos": [96, 24]}, {"full_name": "Nat.sInf_eq_zero", "def_path": "Mathlib/Data/Nat/Lattice.lean", "def_pos": [50, 9], "def_end_pos": [50, 21]}]], "state_before": "G : Type u_1\ninst\u271d : Group G\nV : Set G\n\u22a2 index \u2205 V = 0", "state_after": "G : Type u_1\ninst\u271d : Group G\nV : Set G\n\u22a2 0 \u2208 Finset.card '' {t | \u2205 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V} \u2228\n    Finset.card '' {t | \u2205 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V} = \u2205"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "G : Type u_1\ninst\u271d : Group G\nV : Set G\n\u22a2 0 \u2208 Finset.card '' {t | \u2205 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V} \u2228\n    Finset.card '' {t | \u2205 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V} = \u2205", "state_after": "case h\nG : Type u_1\ninst\u271d : Group G\nV : Set G\n\u22a2 0 \u2208 Finset.card '' {t | \u2205 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}"}, {"tactic": "use \u2205", "annotated_tactic": ["use \u2205", []], "state_before": "case h\nG : Type u_1\ninst\u271d : Group G\nV : Set G\n\u22a2 0 \u2208 Finset.card '' {t | \u2205 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V}", "state_after": "case h\nG : Type u_1\ninst\u271d : Group G\nV : Set G\n\u22a2 \u2205 \u2208 {t | \u2205 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V} \u2227 \u2205.card = 0"}, {"tactic": "simp only [Finset.card_empty, empty_subset, mem_setOf_eq, eq_self_iff_true, and_self_iff]", "annotated_tactic": ["simp only [<a>Finset.card_empty</a>, <a>empty_subset</a>, <a>mem_setOf_eq</a>, <a>eq_self_iff_true</a>, <a>and_self_iff</a>]", [{"full_name": "Finset.card_empty", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [59, 9], "def_end_pos": [59, 19]}, {"full_name": "Set.empty_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [544, 9], "def_end_pos": [544, 21]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [79, 29], "def_end_pos": [79, 41]}, {"full_name": "eq_self_iff_true", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1380, 9], "def_end_pos": [1380, 25]}, {"full_name": "and_self_iff", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [35, 9], "def_end_pos": [35, 21]}]], "state_before": "case h\nG : Type u_1\ninst\u271d : Group G\nV : Set G\n\u22a2 \u2205 \u2208 {t | \u2205 \u2286 \u22c3 g \u2208 t, (fun h => g * h) \u207b\u00b9' V} \u2227 \u2205.card = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Finset/Defs.lean", "full_name": "Finset.map_subtype_embedding_Iio", "start": [1224, 1], "end": [1226, 74], "traced_tactics": [{"tactic": "rw [subtype_Iio_eq]", "annotated_tactic": ["rw [<a>subtype_Iio_eq</a>]", [{"full_name": "Finset.subtype_Iio_eq", "def_path": "Mathlib/Order/Interval/Finset/Defs.lean", "def_pos": [1213, 9], "def_end_pos": [1213, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrderBot \u03b1\na : Subtype p\nhp : \u2200 \u2983a x : \u03b1\u2984, x \u2264 a \u2192 p a \u2192 p x\n\u22a2 map (Embedding.subtype p) (Iio a) = Iio \u2191a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrderBot \u03b1\na : Subtype p\nhp : \u2200 \u2983a x : \u03b1\u2984, x \u2264 a \u2192 p a \u2192 p x\n\u22a2 map (Embedding.subtype p) (Finset.subtype p (Iio \u2191a)) = Iio \u2191a"}, {"tactic": "exact Finset.subtype_map_of_mem fun x hx => hp (mem_Iio.1 hx).le a.prop", "annotated_tactic": ["exact <a>Finset.subtype_map_of_mem</a> fun x hx => hp (<a>mem_Iio</a>.1 hx).<a>le</a> a.prop", [{"full_name": "Finset.subtype_map_of_mem", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [759, 9], "def_end_pos": [759, 27]}, {"full_name": "Finset.mem_Iio", "def_path": "Mathlib/Order/Interval/Finset/Defs.lean", "def_pos": [410, 9], "def_end_pos": [410, 16]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : LocallyFiniteOrderBot \u03b1\na : Subtype p\nhp : \u2200 \u2983a x : \u03b1\u2984, x \u2264 a \u2192 p a \u2192 p x\n\u22a2 map (Embedding.subtype p) (Finset.subtype p (Iio \u2191a)) = Iio \u2191a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.mk_singleton", "start": [1885, 1], "end": [1886, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.VectorMeasure.MutuallySingular.smul_right", "start": [1199, 1], "end": [1202, 91], "traced_tactics": [{"tactic": "simp only [coe_smul, Pi.smul_apply, hs\u2082 t ht, smul_zero]", "annotated_tactic": ["simp only [<a>coe_smul</a>, <a>Pi.smul_apply</a>, hs\u2082 t ht, <a>smul_zero</a>]", [{"full_name": "MeasureTheory.VectorMeasure.coe_smul", "def_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "def_pos": [269, 9], "def_end_pos": [269, 17]}, {"full_name": "Pi.smul_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [129, 60], "def_end_pos": [129, 70]}, {"full_name": "smul_zero", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [66, 9], "def_end_pos": [66, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nm : MeasurableSpace \u03b1\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u2078 : AddCommMonoid L\ninst\u271d\u2077 : TopologicalSpace L\ninst\u271d\u2076 : AddCommMonoid M\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : AddCommMonoid N\ninst\u271d\u00b3 : TopologicalSpace N\nv v\u2081 v\u2082 : VectorMeasure \u03b1 M\nw w\u2081 w\u2082 : VectorMeasure \u03b1 N\nR : Type u_6\ninst\u271d\u00b2 : Semiring R\ninst\u271d\u00b9 : DistribMulAction R N\ninst\u271d : ContinuousConstSMul R N\nr : R\nh : v \u27c2\u1d65 w\ns : Set \u03b1\nhmeas : MeasurableSet s\nhs\u2081 : \u2200 t \u2286 s, \u2191v t = 0\nhs\u2082 : \u2200 t \u2286 s\u1d9c, \u2191w t = 0\nt : Set \u03b1\nht : t \u2286 s\u1d9c\n\u22a2 \u2191(r \u2022 w) t = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Filtered/Final.lean", "full_name": "CategoryTheory.isCofiltered_costructuredArrow_of_isCofiltered_of_exists", "start": [74, 1], "end": [87, 59], "traced_tactics": [{"tactic": "suffices IsFiltered (CostructuredArrow F d)\u1d52\u1d56 from isCofiltered_of_isFiltered_op _", "annotated_tactic": ["suffices <a>IsFiltered</a> (<a>CostructuredArrow</a> F d)\u1d52\u1d56 from <a>isCofiltered_of_isFiltered_op</a> _", [{"full_name": "CategoryTheory.IsFiltered", "def_path": "Mathlib/CategoryTheory/Filtered/Basic.lean", "def_pos": [92, 7], "def_end_pos": [92, 17]}, {"full_name": "CategoryTheory.CostructuredArrow", "def_path": "Mathlib/CategoryTheory/Comma/StructuredArrow.lean", "def_pos": [419, 5], "def_end_pos": [419, 22]}, {"full_name": "CategoryTheory.isCofiltered_of_isFiltered_op", "def_path": "Mathlib/CategoryTheory/Filtered/Basic.lean", "def_pos": [944, 7], "def_end_pos": [944, 36]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d : IsCofilteredOrEmpty C\nh\u2081 : \u2200 (d : D), \u2203 c, Nonempty (F.obj c \u27f6 d)\nh\u2082 : \u2200 {d : D} {c : C} (s s' : F.obj c \u27f6 d), \u2203 c' t, F.map t \u226b s = F.map t \u226b s'\nd : D\n\u22a2 IsCofiltered (CostructuredArrow F d)", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d : IsCofilteredOrEmpty C\nh\u2081 : \u2200 (d : D), \u2203 c, Nonempty (F.obj c \u27f6 d)\nh\u2082 : \u2200 {d : D} {c : C} (s s' : F.obj c \u27f6 d), \u2203 c' t, F.map t \u226b s = F.map t \u226b s'\nd : D\n\u22a2 IsFiltered (CostructuredArrow F d)\u1d52\u1d56"}, {"tactic": "suffices IsFiltered (StructuredArrow (op d) F.op) from\n  IsFiltered.of_equivalence (costructuredArrowOpEquivalence _ _).symm", "annotated_tactic": ["suffices <a>IsFiltered</a> (<a>StructuredArrow</a> (<a>op</a> d) F.op) from\n    <a>IsFiltered.of_equivalence</a> (<a>costructuredArrowOpEquivalence</a> _ _).<a>symm</a>", [{"full_name": "CategoryTheory.IsFiltered", "def_path": "Mathlib/CategoryTheory/Filtered/Basic.lean", "def_pos": [92, 7], "def_end_pos": [92, 17]}, {"full_name": "CategoryTheory.StructuredArrow", "def_path": "Mathlib/CategoryTheory/Comma/StructuredArrow.lean", "def_pos": [41, 5], "def_end_pos": [41, 20]}, {"full_name": "Opposite.op", "def_path": "Mathlib/Data/Opposite.lean", "def_pos": [37, 3], "def_end_pos": [37, 5]}, {"full_name": "CategoryTheory.IsFiltered.of_equivalence", "def_path": "Mathlib/CategoryTheory/Filtered/Basic.lean", "def_pos": [360, 9], "def_end_pos": [360, 23]}, {"full_name": "CategoryTheory.costructuredArrowOpEquivalence", "def_path": "Mathlib/CategoryTheory/Comma/StructuredArrow.lean", "def_pos": [939, 5], "def_end_pos": [939, 35]}, {"full_name": "CategoryTheory.Equivalence.symm", "def_path": "Mathlib/CategoryTheory/Equivalence.lean", "def_pos": [290, 5], "def_end_pos": [290, 9]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d : IsCofilteredOrEmpty C\nh\u2081 : \u2200 (d : D), \u2203 c, Nonempty (F.obj c \u27f6 d)\nh\u2082 : \u2200 {d : D} {c : C} (s s' : F.obj c \u27f6 d), \u2203 c' t, F.map t \u226b s = F.map t \u226b s'\nd : D\n\u22a2 IsFiltered (CostructuredArrow F d)\u1d52\u1d56", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d : IsCofilteredOrEmpty C\nh\u2081 : \u2200 (d : D), \u2203 c, Nonempty (F.obj c \u27f6 d)\nh\u2082 : \u2200 {d : D} {c : C} (s s' : F.obj c \u27f6 d), \u2203 c' t, F.map t \u226b s = F.map t \u226b s'\nd : D\n\u22a2 IsFiltered (StructuredArrow { unop := d } F.op)"}, {"tactic": "apply isFiltered_structuredArrow_of_isFiltered_of_exists", "annotated_tactic": ["apply <a>isFiltered_structuredArrow_of_isFiltered_of_exists</a>", [{"full_name": "CategoryTheory.isFiltered_structuredArrow_of_isFiltered_of_exists", "def_path": "Mathlib/CategoryTheory/Filtered/Final.lean", "def_pos": [56, 9], "def_end_pos": [56, 59]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d : IsCofilteredOrEmpty C\nh\u2081 : \u2200 (d : D), \u2203 c, Nonempty (F.obj c \u27f6 d)\nh\u2082 : \u2200 {d : D} {c : C} (s s' : F.obj c \u27f6 d), \u2203 c' t, F.map t \u226b s = F.map t \u226b s'\nd : D\n\u22a2 IsFiltered (StructuredArrow { unop := d } F.op)", "state_after": "case h\u2081\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d : IsCofilteredOrEmpty C\nh\u2081 : \u2200 (d : D), \u2203 c, Nonempty (F.obj c \u27f6 d)\nh\u2082 : \u2200 {d : D} {c : C} (s s' : F.obj c \u27f6 d), \u2203 c' t, F.map t \u226b s = F.map t \u226b s'\nd : D\n\u22a2 \u2200 (d : D\u1d52\u1d56), \u2203 c, Nonempty (d \u27f6 F.op.obj c)\n\ncase h\u2082\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d : IsCofilteredOrEmpty C\nh\u2081 : \u2200 (d : D), \u2203 c, Nonempty (F.obj c \u27f6 d)\nh\u2082 : \u2200 {d : D} {c : C} (s s' : F.obj c \u27f6 d), \u2203 c' t, F.map t \u226b s = F.map t \u226b s'\nd : D\n\u22a2 \u2200 {d : D\u1d52\u1d56} {c : C\u1d52\u1d56} (s s' : d \u27f6 F.op.obj c), \u2203 c' t, s \u226b F.op.map t = s' \u226b F.op.map t"}, {"tactic": "intro d", "annotated_tactic": ["intro d", []], "state_before": "case h\u2081\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d : IsCofilteredOrEmpty C\nh\u2081 : \u2200 (d : D), \u2203 c, Nonempty (F.obj c \u27f6 d)\nh\u2082 : \u2200 {d : D} {c : C} (s s' : F.obj c \u27f6 d), \u2203 c' t, F.map t \u226b s = F.map t \u226b s'\nd : D\n\u22a2 \u2200 (d : D\u1d52\u1d56), \u2203 c, Nonempty (d \u27f6 F.op.obj c)", "state_after": "case h\u2081\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d : IsCofilteredOrEmpty C\nh\u2081 : \u2200 (d : D), \u2203 c, Nonempty (F.obj c \u27f6 d)\nh\u2082 : \u2200 {d : D} {c : C} (s s' : F.obj c \u27f6 d), \u2203 c' t, F.map t \u226b s = F.map t \u226b s'\nd\u271d : D\nd : D\u1d52\u1d56\n\u22a2 \u2203 c, Nonempty (d \u27f6 F.op.obj c)"}, {"tactic": "obtain \u27e8c, \u27e8t\u27e9\u27e9 := h\u2081 d.unop", "annotated_tactic": ["obtain \u27e8c, \u27e8t\u27e9\u27e9 := h\u2081 d.unop", []], "state_before": "case h\u2081\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d : IsCofilteredOrEmpty C\nh\u2081 : \u2200 (d : D), \u2203 c, Nonempty (F.obj c \u27f6 d)\nh\u2082 : \u2200 {d : D} {c : C} (s s' : F.obj c \u27f6 d), \u2203 c' t, F.map t \u226b s = F.map t \u226b s'\nd\u271d : D\nd : D\u1d52\u1d56\n\u22a2 \u2203 c, Nonempty (d \u27f6 F.op.obj c)", "state_after": "case h\u2081.intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d : IsCofilteredOrEmpty C\nh\u2081 : \u2200 (d : D), \u2203 c, Nonempty (F.obj c \u27f6 d)\nh\u2082 : \u2200 {d : D} {c : C} (s s' : F.obj c \u27f6 d), \u2203 c' t, F.map t \u226b s = F.map t \u226b s'\nd\u271d : D\nd : D\u1d52\u1d56\nc : C\nt : F.obj c \u27f6 d.unop\n\u22a2 \u2203 c, Nonempty (d \u27f6 F.op.obj c)"}, {"tactic": "exact \u27e8op c, \u27e8Quiver.Hom.op t\u27e9\u27e9", "annotated_tactic": ["exact \u27e8<a>op</a> c, \u27e8<a>Quiver.Hom.op</a> t\u27e9\u27e9", [{"full_name": "Opposite.op", "def_path": "Mathlib/Data/Opposite.lean", "def_pos": [37, 3], "def_end_pos": [37, 5]}, {"full_name": "Quiver.Hom.op", "def_path": "Mathlib/Combinatorics/Quiver/Basic.lean", "def_pos": [152, 5], "def_end_pos": [152, 11]}]], "state_before": "case h\u2081.intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d : IsCofilteredOrEmpty C\nh\u2081 : \u2200 (d : D), \u2203 c, Nonempty (F.obj c \u27f6 d)\nh\u2082 : \u2200 {d : D} {c : C} (s s' : F.obj c \u27f6 d), \u2203 c' t, F.map t \u226b s = F.map t \u226b s'\nd\u271d : D\nd : D\u1d52\u1d56\nc : C\nt : F.obj c \u27f6 d.unop\n\u22a2 \u2203 c, Nonempty (d \u27f6 F.op.obj c)", "state_after": "no goals"}, {"tactic": "intro d c s s'", "annotated_tactic": ["intro d c s s'", []], "state_before": "case h\u2082\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d : IsCofilteredOrEmpty C\nh\u2081 : \u2200 (d : D), \u2203 c, Nonempty (F.obj c \u27f6 d)\nh\u2082 : \u2200 {d : D} {c : C} (s s' : F.obj c \u27f6 d), \u2203 c' t, F.map t \u226b s = F.map t \u226b s'\nd : D\n\u22a2 \u2200 {d : D\u1d52\u1d56} {c : C\u1d52\u1d56} (s s' : d \u27f6 F.op.obj c), \u2203 c' t, s \u226b F.op.map t = s' \u226b F.op.map t", "state_after": "case h\u2082\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d : IsCofilteredOrEmpty C\nh\u2081 : \u2200 (d : D), \u2203 c, Nonempty (F.obj c \u27f6 d)\nh\u2082 : \u2200 {d : D} {c : C} (s s' : F.obj c \u27f6 d), \u2203 c' t, F.map t \u226b s = F.map t \u226b s'\nd\u271d : D\nd : D\u1d52\u1d56\nc : C\u1d52\u1d56\ns s' : d \u27f6 F.op.obj c\n\u22a2 \u2203 c' t, s \u226b F.op.map t = s' \u226b F.op.map t"}, {"tactic": "obtain \u27e8c', t, ht\u27e9 := h\u2082 s.unop s'.unop", "annotated_tactic": ["obtain \u27e8c', t, ht\u27e9 := h\u2082 s.unop s'.unop", []], "state_before": "case h\u2082\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d : IsCofilteredOrEmpty C\nh\u2081 : \u2200 (d : D), \u2203 c, Nonempty (F.obj c \u27f6 d)\nh\u2082 : \u2200 {d : D} {c : C} (s s' : F.obj c \u27f6 d), \u2203 c' t, F.map t \u226b s = F.map t \u226b s'\nd\u271d : D\nd : D\u1d52\u1d56\nc : C\u1d52\u1d56\ns s' : d \u27f6 F.op.obj c\n\u22a2 \u2203 c' t, s \u226b F.op.map t = s' \u226b F.op.map t", "state_after": "case h\u2082.intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d : IsCofilteredOrEmpty C\nh\u2081 : \u2200 (d : D), \u2203 c, Nonempty (F.obj c \u27f6 d)\nh\u2082 : \u2200 {d : D} {c : C} (s s' : F.obj c \u27f6 d), \u2203 c' t, F.map t \u226b s = F.map t \u226b s'\nd\u271d : D\nd : D\u1d52\u1d56\nc : C\u1d52\u1d56\ns s' : d \u27f6 F.op.obj c\nc' : C\nt : c' \u27f6 c.unop\nht : F.map t \u226b s.unop = F.map t \u226b s'.unop\n\u22a2 \u2203 c' t, s \u226b F.op.map t = s' \u226b F.op.map t"}, {"tactic": "exact \u27e8op c', Quiver.Hom.op t, Quiver.Hom.unop_inj ht\u27e9", "annotated_tactic": ["exact \u27e8<a>op</a> c', <a>Quiver.Hom.op</a> t, <a>Quiver.Hom.unop_inj</a> ht\u27e9", [{"full_name": "Opposite.op", "def_path": "Mathlib/Data/Opposite.lean", "def_pos": [37, 3], "def_end_pos": [37, 5]}, {"full_name": "Quiver.Hom.op", "def_path": "Mathlib/Combinatorics/Quiver/Basic.lean", "def_pos": [152, 5], "def_end_pos": [152, 11]}, {"full_name": "Quiver.Hom.unop_inj", "def_path": "Mathlib/CategoryTheory/Opposites.lean", "def_pos": [42, 9], "def_end_pos": [42, 28]}]], "state_before": "case h\u2082.intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d : IsCofilteredOrEmpty C\nh\u2081 : \u2200 (d : D), \u2203 c, Nonempty (F.obj c \u27f6 d)\nh\u2082 : \u2200 {d : D} {c : C} (s s' : F.obj c \u27f6 d), \u2203 c' t, F.map t \u226b s = F.map t \u226b s'\nd\u271d : D\nd : D\u1d52\u1d56\nc : C\u1d52\u1d56\ns s' : d \u27f6 F.op.obj c\nc' : C\nt : c' \u27f6 c.unop\nht : F.map t \u226b s.unop = F.map t \u226b s'.unop\n\u22a2 \u2203 c' t, s \u226b F.op.map t = s' \u226b F.op.map t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/RingedSpace/OpenImmersion.lean", "full_name": "AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.isoRestrict_hom_ofRestrict", "start": [1275, 1], "end": [1280, 68], "traced_tactics": [{"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nH : IsOpenImmersion f\n\u22a2 (isoRestrict f).hom \u226b Y.ofRestrict \u22ef = f", "state_after": "case h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nH : IsOpenImmersion f\n\u22a2 ((isoRestrict f).hom \u226b Y.ofRestrict \u22ef).val = f.val"}, {"tactic": "dsimp [isoRestrict, isoOfSheafedSpaceIso]", "annotated_tactic": ["dsimp [<a>isoRestrict</a>, <a>isoOfSheafedSpaceIso</a>]", [{"full_name": "AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.isoRestrict", "def_path": "Mathlib/Geometry/RingedSpace/OpenImmersion.lean", "def_pos": [1243, 19], "def_end_pos": [1243, 30]}, {"full_name": "AlgebraicGeometry.LocallyRingedSpace.isoOfSheafedSpaceIso", "def_path": "Mathlib/Geometry/RingedSpace/LocallyRingedSpace.lean", "def_pos": [220, 5], "def_end_pos": [220, 25]}]], "state_before": "case h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nH : IsOpenImmersion f\n\u22a2 ((isoRestrict f).hom \u226b Y.ofRestrict \u22ef).val = f.val", "state_after": "case h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nH : IsOpenImmersion f\n\u22a2 SheafedSpace.forgetToPresheafedSpace.preimage (PresheafedSpace.IsOpenImmersion.isoRestrict f.val).hom \u226b\n      (Y.ofRestrict \u22ef).val =\n    f.val"}, {"tactic": "apply SheafedSpace.forgetToPresheafedSpace.map_injective", "annotated_tactic": ["apply SheafedSpace.forgetToPresheafedSpace.map_injective", []], "state_before": "case h\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nH : IsOpenImmersion f\n\u22a2 SheafedSpace.forgetToPresheafedSpace.preimage (PresheafedSpace.IsOpenImmersion.isoRestrict f.val).hom \u226b\n      (Y.ofRestrict \u22ef).val =\n    f.val", "state_after": "case h.a\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nH : IsOpenImmersion f\n\u22a2 SheafedSpace.forgetToPresheafedSpace.map\n      (SheafedSpace.forgetToPresheafedSpace.preimage (PresheafedSpace.IsOpenImmersion.isoRestrict f.val).hom \u226b\n        (Y.ofRestrict \u22ef).val) =\n    SheafedSpace.forgetToPresheafedSpace.map f.val"}, {"tactic": "rw [Functor.map_comp, SheafedSpace.forgetToPresheafedSpace.map_preimage]", "annotated_tactic": ["rw [<a>Functor.map_comp</a>, SheafedSpace.forgetToPresheafedSpace.map_preimage]", [{"full_name": "CategoryTheory.Functor.map_comp", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [43, 3], "def_end_pos": [43, 11]}]], "state_before": "case h.a\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nH : IsOpenImmersion f\n\u22a2 SheafedSpace.forgetToPresheafedSpace.map\n      (SheafedSpace.forgetToPresheafedSpace.preimage (PresheafedSpace.IsOpenImmersion.isoRestrict f.val).hom \u226b\n        (Y.ofRestrict \u22ef).val) =\n    SheafedSpace.forgetToPresheafedSpace.map f.val", "state_after": "case h.a\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nH : IsOpenImmersion f\n\u22a2 (PresheafedSpace.IsOpenImmersion.isoRestrict f.val).hom \u226b\n      SheafedSpace.forgetToPresheafedSpace.map (Y.ofRestrict \u22ef).val =\n    SheafedSpace.forgetToPresheafedSpace.map f.val"}, {"tactic": "exact SheafedSpace.IsOpenImmersion.isoRestrict_hom_ofRestrict f.1", "annotated_tactic": ["exact <a>SheafedSpace.IsOpenImmersion.isoRestrict_hom_ofRestrict</a> f.1", [{"full_name": "AlgebraicGeometry.SheafedSpace.IsOpenImmersion.isoRestrict_hom_ofRestrict", "def_path": "Mathlib/Geometry/RingedSpace/OpenImmersion.lean", "def_pos": [841, 9], "def_end_pos": [841, 35]}]], "state_before": "case h.a\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : LocallyRingedSpace\nf : X \u27f6 Y\nH : IsOpenImmersion f\n\u22a2 (PresheafedSpace.IsOpenImmersion.isoRestrict f.val).hom \u226b\n      SheafedSpace.forgetToPresheafedSpace.map (Y.ofRestrict \u22ef).val =\n    SheafedSpace.forgetToPresheafedSpace.map f.val", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Lagrange.lean", "full_name": "Lagrange.basis_eq_prod_sub_inv_mul_nodal_div", "start": [655, 1], "end": [658, 40], "traced_tactics": [{"tactic": "simp_rw [Lagrange.basis, basisDivisor, nodalWeight, prod_mul_distrib, map_prod, \u2190\n  nodal_erase_eq_nodal_div hi, nodal]", "annotated_tactic": ["simp_rw [<a>Lagrange.basis</a>, <a>basisDivisor</a>, <a>nodalWeight</a>, <a>prod_mul_distrib</a>, <a>map_prod</a>, \u2190\n    <a>nodal_erase_eq_nodal_div</a> hi, <a>nodal</a>]", [{"full_name": "Lagrange.basis", "def_path": "Mathlib/LinearAlgebra/Lagrange.lean", "def_pos": [212, 15], "def_end_pos": [212, 20]}, {"full_name": "Lagrange.basisDivisor", "def_path": "Mathlib/LinearAlgebra/Lagrange.lean", "def_pos": [149, 5], "def_end_pos": [149, 17]}, {"full_name": "Lagrange.nodalWeight", "def_path": "Mathlib/LinearAlgebra/Lagrange.lean", "def_pos": [619, 5], "def_end_pos": [619, 16]}, {"full_name": "Finset.prod_mul_distrib", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [877, 9], "def_end_pos": [877, 25]}, {"full_name": "map_prod", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [296, 9], "def_end_pos": [296, 17]}, {"full_name": "Lagrange.nodal_erase_eq_nodal_div", "def_path": "Mathlib/LinearAlgebra/Lagrange.lean", "def_pos": [628, 9], "def_end_pos": [628, 33]}, {"full_name": "Lagrange.nodal", "def_path": "Mathlib/LinearAlgebra/Lagrange.lean", "def_pos": [511, 5], "def_end_pos": [511, 10]}]], "state_before": "F : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv r : \u03b9 \u2192 F\ni : \u03b9\nx : F\nhi : i \u2208 s\n\u22a2 Lagrange.basis s v i = C (nodalWeight s v i) * (nodal s v / (X - C (v i)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/EMetricSpace/Paracompact.lean", "full_name": "EMetric.t4Space", "start": [169, 1], "end": [169, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/Diffeomorph.lean", "full_name": "Diffeomorph.refl_toEquiv", "start": [189, 1], "end": [190, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Rat.lean", "full_name": "Rat.divInt_div_divInt_cancel_right", "start": [76, 1], "end": [78, 74], "traced_tactics": [{"tactic": "rw [div_eq_mul_inv, inv_divInt', mul_comm, divInt_mul_divInt_cancel hx]", "annotated_tactic": ["rw [<a>div_eq_mul_inv</a>, <a>inv_divInt'</a>, <a>mul_comm</a>, <a>divInt_mul_divInt_cancel</a> hx]", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1059, 9], "def_end_pos": [1059, 23]}, {"full_name": "Rat.inv_divInt'", "def_path": "Mathlib/Data/Rat/Defs.lean", "def_pos": [286, 15], "def_end_pos": [286, 26]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "Rat.divInt_mul_divInt_cancel", "def_path": "Mathlib/Data/Rat/Defs.lean", "def_pos": [530, 9], "def_end_pos": [530, 33]}]], "state_before": "x : \u2124\nhx : x \u2260 0\nn d : \u2124\n\u22a2 x /. n / (x /. d) = d /. n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Complex/Basic.lean", "full_name": "RCLike.re_eq_complex_re", "start": [452, 1], "end": [453, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "full_name": "Matrix.TransvectionStruct.mul_inv", "start": [210, 1], "end": [212, 61], "traced_tactics": [{"tactic": "rcases t with \u27e8_, _, t_hij\u27e9", "annotated_tactic": ["rcases t with \u27e8_, _, t_hij\u27e9", []], "state_before": "n : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\nt : TransvectionStruct n R\n\u22a2 t.toMatrix * t.inv.toMatrix = 1", "state_after": "case mk\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\ni\u271d j\u271d : n\nt_hij : i\u271d \u2260 j\u271d\nc\u271d : R\n\u22a2 { i := i\u271d, j := j\u271d, hij := t_hij, c := c\u271d }.toMatrix * { i := i\u271d, j := j\u271d, hij := t_hij, c := c\u271d }.inv.toMatrix = 1"}, {"tactic": "simp [toMatrix, transvection_mul_transvection_same, t_hij]", "annotated_tactic": ["simp [<a>toMatrix</a>, <a>transvection_mul_transvection_same</a>, t_hij]", [{"full_name": "Matrix.TransvectionStruct.toMatrix", "def_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "def_pos": [168, 5], "def_end_pos": [168, 13]}, {"full_name": "Matrix.transvection_mul_transvection_same", "def_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "def_pos": [113, 9], "def_end_pos": [113, 43]}]], "state_before": "case mk\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\ni\u271d j\u271d : n\nt_hij : i\u271d \u2260 j\u271d\nc\u271d : R\n\u22a2 { i := i\u271d, j := j\u271d, hij := t_hij, c := c\u271d }.toMatrix * { i := i\u271d, j := j\u271d, hij := t_hij, c := c\u271d }.inv.toMatrix = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/WF.lean", "full_name": "Acc.recC_intro", "start": [58, 9], "end": [63, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Ext.lean", "full_name": "AddGroupWithOne.ext", "start": [249, 8], "end": [264, 8], "traced_tactics": [{"tactic": "have : inst\u2081.toAddMonoidWithOne = inst\u2082.toAddMonoidWithOne :=\n  AddMonoidWithOne.ext h_add h_one", "annotated_tactic": ["have : inst\u2081.toAddMonoidWithOne = inst\u2082.toAddMonoidWithOne :=\n    <a>AddMonoidWithOne.ext</a> h_add h_one", [{"full_name": "AddMonoidWithOne.ext", "def_path": "Mathlib/Algebra/Ring/Ext.lean", "def_pos": [116, 16], "def_end_pos": [116, 36]}]], "state_before": "R : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\n\u22a2 inst\u2081 = inst\u2082", "state_after": "R : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis : toAddMonoidWithOne = toAddMonoidWithOne\n\u22a2 inst\u2081 = inst\u2082"}, {"tactic": "have h_group : inst\u2081.toAddGroup = inst\u2082.toAddGroup := by ext : 1; exact h_add", "annotated_tactic": ["have h_group : inst\u2081.toAddGroup = inst\u2082.toAddGroup := by ext : 1; exact h_add", []], "state_before": "R : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d : toAddMonoidWithOne = toAddMonoidWithOne\nthis : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\n\u22a2 inst\u2081 = inst\u2082", "state_after": "R : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d : toAddMonoidWithOne = toAddMonoidWithOne\nthis : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\nh_group : toAddGroup = toAddGroup\n\u22a2 inst\u2081 = inst\u2082"}, {"tactic": "injection h_group with h_group", "annotated_tactic": ["injection h_group with h_group", []], "state_before": "R : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d : toAddMonoidWithOne = toAddMonoidWithOne\nthis : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\nh_group : toAddGroup = toAddGroup\n\u22a2 inst\u2081 = inst\u2082", "state_after": "R : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d : toAddMonoidWithOne = toAddMonoidWithOne\nthis : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\nh_group : SubNegMonoid.mk \u22ef AddGroupWithOne.zsmul \u22ef \u22ef \u22ef = SubNegMonoid.mk \u22ef AddGroupWithOne.zsmul \u22ef \u22ef \u22ef\n\u22a2 inst\u2081 = inst\u2082"}, {"tactic": "injection h_group", "annotated_tactic": ["injection h_group", []], "state_before": "R : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d : toAddMonoidWithOne = toAddMonoidWithOne\nthis : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\nh_group : SubNegMonoid.mk \u22ef AddGroupWithOne.zsmul \u22ef \u22ef \u22ef = SubNegMonoid.mk \u22ef AddGroupWithOne.zsmul \u22ef \u22ef \u22ef\n\u22a2 inst\u2081 = inst\u2082", "state_after": "R : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d : toAddMonoidWithOne = toAddMonoidWithOne\nthis : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\ntoAddMonoid_eq\u271d : AddMonoidWithOne.toAddMonoid = AddMonoidWithOne.toAddMonoid\ntoNeg_eq\u271d : toNeg = toNeg\ntoSub_eq\u271d : toSub = toSub\nzsmul_eq\u271d : AddGroupWithOne.zsmul = AddGroupWithOne.zsmul\n\u22a2 inst\u2081 = inst\u2082"}, {"tactic": "have : inst\u2081.toIntCast.intCast = inst\u2082.toIntCast.intCast := by\n  funext n; cases n with\n  | ofNat n   => rewrite [Int.ofNat_eq_coe, inst\u2081.intCast_ofNat, inst\u2082.intCast_ofNat]; congr\n  | negSucc n => rewrite [inst\u2081.intCast_negSucc, inst\u2082.intCast_negSucc]; congr", "annotated_tactic": ["have : inst\u2081.toIntCast.intCast = inst\u2082.toIntCast.intCast := by\n    funext n; cases n with\n    | <a>ofNat</a> n   => rewrite [<a>Int.ofNat_eq_coe</a>, inst\u2081.intCast_ofNat, inst\u2082.intCast_ofNat]; congr\n    | <a>negSucc</a> n => rewrite [inst\u2081.intCast_negSucc, inst\u2082.intCast_negSucc]; congr", [{"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.ofNat_eq_coe", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [71, 17], "def_end_pos": [71, 29]}, {"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": "R : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d : toAddMonoidWithOne = toAddMonoidWithOne\nthis : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\ntoAddMonoid_eq\u271d : AddMonoidWithOne.toAddMonoid = AddMonoidWithOne.toAddMonoid\ntoNeg_eq\u271d : toNeg = toNeg\ntoSub_eq\u271d : toSub = toSub\nzsmul_eq\u271d : AddGroupWithOne.zsmul = AddGroupWithOne.zsmul\n\u22a2 inst\u2081 = inst\u2082", "state_after": "R : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d\u00b9 : toAddMonoidWithOne = toAddMonoidWithOne\nthis\u271d : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\ntoAddMonoid_eq\u271d : AddMonoidWithOne.toAddMonoid = AddMonoidWithOne.toAddMonoid\ntoNeg_eq\u271d : toNeg = toNeg\ntoSub_eq\u271d : toSub = toSub\nzsmul_eq\u271d : AddGroupWithOne.zsmul = AddGroupWithOne.zsmul\nthis : IntCast.intCast = IntCast.intCast\n\u22a2 inst\u2081 = inst\u2082"}, {"tactic": "rcases inst\u2081 with @\u27e8\u27e8\u27e9\u27e9", "annotated_tactic": ["rcases inst\u2081 with @\u27e8\u27e8\u27e9\u27e9", []], "state_before": "R : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d\u00b9 : toAddMonoidWithOne = toAddMonoidWithOne\nthis\u271d : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\ntoAddMonoid_eq\u271d : AddMonoidWithOne.toAddMonoid = AddMonoidWithOne.toAddMonoid\ntoNeg_eq\u271d : toNeg = toNeg\ntoSub_eq\u271d : toSub = toSub\nzsmul_eq\u271d : AddGroupWithOne.zsmul = AddGroupWithOne.zsmul\nthis : IntCast.intCast = IntCast.intCast\n\u22a2 inst\u2081 = inst\u2082", "state_after": "case mk.mk\nR : Type u\ninst\u2082 : AddGroupWithOne R\ntoAddMonoidWithOne\u271d : AddMonoidWithOne R\ntoNeg\u271d : Neg R\ntoSub\u271d : Sub R\nsub_eq_add_neg\u271d : \u2200 (a b : R), a - b = a + -b\nzsmul\u271d : \u2124 \u2192 R \u2192 R\nzsmul_zero'\u271d : \u2200 (a : R), zsmul\u271d 0 a = 0\nzsmul_succ'\u271d : \u2200 (n : \u2115) (a : R), zsmul\u271d (Int.ofNat n.succ) a = zsmul\u271d (Int.ofNat n) a + a\nzsmul_neg'\u271d : \u2200 (n : \u2115) (a : R), zsmul\u271d (Int.negSucc n) a = -zsmul\u271d (\u2191n.succ) a\nadd_left_neg\u271d : \u2200 (a : R), -a + a = 0\nintCast\u271d : \u2124 \u2192 R\nintCast_ofNat\u271d : \u2200 (n : \u2115), IntCast.intCast \u2191n = \u2191n\nintCast_negSucc\u271d : \u2200 (n : \u2115), IntCast.intCast (Int.negSucc n) = -\u2191(n + 1)\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d\u00b9 : toAddMonoidWithOne = toAddMonoidWithOne\nthis\u271d : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\ntoAddMonoid_eq\u271d : AddMonoidWithOne.toAddMonoid = AddMonoidWithOne.toAddMonoid\ntoNeg_eq\u271d : toNeg = toNeg\ntoSub_eq\u271d : toSub = toSub\nzsmul_eq\u271d : AddGroupWithOne.zsmul = AddGroupWithOne.zsmul\nthis : IntCast.intCast = IntCast.intCast\n\u22a2 mk sub_eq_add_neg\u271d zsmul\u271d zsmul_zero'\u271d zsmul_succ'\u271d zsmul_neg'\u271d add_left_neg\u271d intCast_ofNat\u271d intCast_negSucc\u271d = inst\u2082"}, {"tactic": "rcases inst\u2082 with @\u27e8\u27e8\u27e9\u27e9", "annotated_tactic": ["rcases inst\u2082 with @\u27e8\u27e8\u27e9\u27e9", []], "state_before": "case mk.mk\nR : Type u\ninst\u2082 : AddGroupWithOne R\ntoAddMonoidWithOne\u271d : AddMonoidWithOne R\ntoNeg\u271d : Neg R\ntoSub\u271d : Sub R\nsub_eq_add_neg\u271d : \u2200 (a b : R), a - b = a + -b\nzsmul\u271d : \u2124 \u2192 R \u2192 R\nzsmul_zero'\u271d : \u2200 (a : R), zsmul\u271d 0 a = 0\nzsmul_succ'\u271d : \u2200 (n : \u2115) (a : R), zsmul\u271d (Int.ofNat n.succ) a = zsmul\u271d (Int.ofNat n) a + a\nzsmul_neg'\u271d : \u2200 (n : \u2115) (a : R), zsmul\u271d (Int.negSucc n) a = -zsmul\u271d (\u2191n.succ) a\nadd_left_neg\u271d : \u2200 (a : R), -a + a = 0\nintCast\u271d : \u2124 \u2192 R\nintCast_ofNat\u271d : \u2200 (n : \u2115), IntCast.intCast \u2191n = \u2191n\nintCast_negSucc\u271d : \u2200 (n : \u2115), IntCast.intCast (Int.negSucc n) = -\u2191(n + 1)\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d\u00b9 : toAddMonoidWithOne = toAddMonoidWithOne\nthis\u271d : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\ntoAddMonoid_eq\u271d : AddMonoidWithOne.toAddMonoid = AddMonoidWithOne.toAddMonoid\ntoNeg_eq\u271d : toNeg = toNeg\ntoSub_eq\u271d : toSub = toSub\nzsmul_eq\u271d : AddGroupWithOne.zsmul = AddGroupWithOne.zsmul\nthis : IntCast.intCast = IntCast.intCast\n\u22a2 mk sub_eq_add_neg\u271d zsmul\u271d zsmul_zero'\u271d zsmul_succ'\u271d zsmul_neg'\u271d add_left_neg\u271d intCast_ofNat\u271d intCast_negSucc\u271d = inst\u2082", "state_after": "case mk.mk.mk.mk\nR : Type u\ntoAddMonoidWithOne\u271d\u00b9 : AddMonoidWithOne R\ntoNeg\u271d\u00b9 : Neg R\ntoSub\u271d\u00b9 : Sub R\nsub_eq_add_neg\u271d\u00b9 : \u2200 (a b : R), a - b = a + -b\nzsmul\u271d\u00b9 : \u2124 \u2192 R \u2192 R\nzsmul_zero'\u271d\u00b9 : \u2200 (a : R), zsmul\u271d\u00b9 0 a = 0\nzsmul_succ'\u271d\u00b9 : \u2200 (n : \u2115) (a : R), zsmul\u271d\u00b9 (Int.ofNat n.succ) a = zsmul\u271d\u00b9 (Int.ofNat n) a + a\nzsmul_neg'\u271d\u00b9 : \u2200 (n : \u2115) (a : R), zsmul\u271d\u00b9 (Int.negSucc n) a = -zsmul\u271d\u00b9 (\u2191n.succ) a\nadd_left_neg\u271d\u00b9 : \u2200 (a : R), -a + a = 0\nintCast\u271d\u00b9 : \u2124 \u2192 R\nintCast_ofNat\u271d\u00b9 : \u2200 (n : \u2115), IntCast.intCast \u2191n = \u2191n\nintCast_negSucc\u271d\u00b9 : \u2200 (n : \u2115), IntCast.intCast (Int.negSucc n) = -\u2191(n + 1)\ntoAddMonoidWithOne\u271d : AddMonoidWithOne R\ntoNeg\u271d : Neg R\ntoSub\u271d : Sub R\nsub_eq_add_neg\u271d : \u2200 (a b : R), a - b = a + -b\nzsmul\u271d : \u2124 \u2192 R \u2192 R\nzsmul_zero'\u271d : \u2200 (a : R), zsmul\u271d 0 a = 0\nzsmul_succ'\u271d : \u2200 (n : \u2115) (a : R), zsmul\u271d (Int.ofNat n.succ) a = zsmul\u271d (Int.ofNat n) a + a\nzsmul_neg'\u271d : \u2200 (n : \u2115) (a : R), zsmul\u271d (Int.negSucc n) a = -zsmul\u271d (\u2191n.succ) a\nadd_left_neg\u271d : \u2200 (a : R), -a + a = 0\nintCast\u271d : \u2124 \u2192 R\nintCast_ofNat\u271d : \u2200 (n : \u2115), IntCast.intCast \u2191n = \u2191n\nintCast_negSucc\u271d : \u2200 (n : \u2115), IntCast.intCast (Int.negSucc n) = -\u2191(n + 1)\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d\u00b9 : toAddMonoidWithOne = toAddMonoidWithOne\nthis\u271d : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\ntoAddMonoid_eq\u271d : AddMonoidWithOne.toAddMonoid = AddMonoidWithOne.toAddMonoid\ntoNeg_eq\u271d : toNeg = toNeg\ntoSub_eq\u271d : toSub = toSub\nzsmul_eq\u271d : AddGroupWithOne.zsmul = AddGroupWithOne.zsmul\nthis : IntCast.intCast = IntCast.intCast\n\u22a2 mk sub_eq_add_neg\u271d\u00b9 zsmul\u271d\u00b9 zsmul_zero'\u271d\u00b9 zsmul_succ'\u271d\u00b9 zsmul_neg'\u271d\u00b9 add_left_neg\u271d\u00b9 intCast_ofNat\u271d\u00b9\n      intCast_negSucc\u271d\u00b9 =\n    mk sub_eq_add_neg\u271d zsmul\u271d zsmul_zero'\u271d zsmul_succ'\u271d zsmul_neg'\u271d add_left_neg\u271d intCast_ofNat\u271d intCast_negSucc\u271d"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case mk.mk.mk.mk\nR : Type u\ntoAddMonoidWithOne\u271d\u00b9 : AddMonoidWithOne R\ntoNeg\u271d\u00b9 : Neg R\ntoSub\u271d\u00b9 : Sub R\nsub_eq_add_neg\u271d\u00b9 : \u2200 (a b : R), a - b = a + -b\nzsmul\u271d\u00b9 : \u2124 \u2192 R \u2192 R\nzsmul_zero'\u271d\u00b9 : \u2200 (a : R), zsmul\u271d\u00b9 0 a = 0\nzsmul_succ'\u271d\u00b9 : \u2200 (n : \u2115) (a : R), zsmul\u271d\u00b9 (Int.ofNat n.succ) a = zsmul\u271d\u00b9 (Int.ofNat n) a + a\nzsmul_neg'\u271d\u00b9 : \u2200 (n : \u2115) (a : R), zsmul\u271d\u00b9 (Int.negSucc n) a = -zsmul\u271d\u00b9 (\u2191n.succ) a\nadd_left_neg\u271d\u00b9 : \u2200 (a : R), -a + a = 0\nintCast\u271d\u00b9 : \u2124 \u2192 R\nintCast_ofNat\u271d\u00b9 : \u2200 (n : \u2115), IntCast.intCast \u2191n = \u2191n\nintCast_negSucc\u271d\u00b9 : \u2200 (n : \u2115), IntCast.intCast (Int.negSucc n) = -\u2191(n + 1)\ntoAddMonoidWithOne\u271d : AddMonoidWithOne R\ntoNeg\u271d : Neg R\ntoSub\u271d : Sub R\nsub_eq_add_neg\u271d : \u2200 (a b : R), a - b = a + -b\nzsmul\u271d : \u2124 \u2192 R \u2192 R\nzsmul_zero'\u271d : \u2200 (a : R), zsmul\u271d 0 a = 0\nzsmul_succ'\u271d : \u2200 (n : \u2115) (a : R), zsmul\u271d (Int.ofNat n.succ) a = zsmul\u271d (Int.ofNat n) a + a\nzsmul_neg'\u271d : \u2200 (n : \u2115) (a : R), zsmul\u271d (Int.negSucc n) a = -zsmul\u271d (\u2191n.succ) a\nadd_left_neg\u271d : \u2200 (a : R), -a + a = 0\nintCast\u271d : \u2124 \u2192 R\nintCast_ofNat\u271d : \u2200 (n : \u2115), IntCast.intCast \u2191n = \u2191n\nintCast_negSucc\u271d : \u2200 (n : \u2115), IntCast.intCast (Int.negSucc n) = -\u2191(n + 1)\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d\u00b9 : toAddMonoidWithOne = toAddMonoidWithOne\nthis\u271d : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\ntoAddMonoid_eq\u271d : AddMonoidWithOne.toAddMonoid = AddMonoidWithOne.toAddMonoid\ntoNeg_eq\u271d : toNeg = toNeg\ntoSub_eq\u271d : toSub = toSub\nzsmul_eq\u271d : AddGroupWithOne.zsmul = AddGroupWithOne.zsmul\nthis : IntCast.intCast = IntCast.intCast\n\u22a2 mk sub_eq_add_neg\u271d\u00b9 zsmul\u271d\u00b9 zsmul_zero'\u271d\u00b9 zsmul_succ'\u271d\u00b9 zsmul_neg'\u271d\u00b9 add_left_neg\u271d\u00b9 intCast_ofNat\u271d\u00b9\n      intCast_negSucc\u271d\u00b9 =\n    mk sub_eq_add_neg\u271d zsmul\u271d zsmul_zero'\u271d zsmul_succ'\u271d zsmul_neg'\u271d add_left_neg\u271d intCast_ofNat\u271d intCast_negSucc\u271d", "state_after": "no goals"}, {"tactic": "ext : 1", "annotated_tactic": ["ext : 1", []], "state_before": "R : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d : toAddMonoidWithOne = toAddMonoidWithOne\nthis : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\n\u22a2 toAddGroup = toAddGroup", "state_after": "case h_mul\nR : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d : toAddMonoidWithOne = toAddMonoidWithOne\nthis : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\n\u22a2 Add.add = Add.add"}, {"tactic": "exact h_add", "annotated_tactic": ["exact h_add", []], "state_before": "case h_mul\nR : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d : toAddMonoidWithOne = toAddMonoidWithOne\nthis : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\n\u22a2 Add.add = Add.add", "state_after": "no goals"}, {"tactic": "funext n", "annotated_tactic": ["funext n", []], "state_before": "R : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d : toAddMonoidWithOne = toAddMonoidWithOne\nthis : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\ntoAddMonoid_eq\u271d : AddMonoidWithOne.toAddMonoid = AddMonoidWithOne.toAddMonoid\ntoNeg_eq\u271d : toNeg = toNeg\ntoSub_eq\u271d : toSub = toSub\nzsmul_eq\u271d : AddGroupWithOne.zsmul = AddGroupWithOne.zsmul\n\u22a2 IntCast.intCast = IntCast.intCast", "state_after": "case h\nR : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d : toAddMonoidWithOne = toAddMonoidWithOne\nthis : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\ntoAddMonoid_eq\u271d : AddMonoidWithOne.toAddMonoid = AddMonoidWithOne.toAddMonoid\ntoNeg_eq\u271d : toNeg = toNeg\ntoSub_eq\u271d : toSub = toSub\nzsmul_eq\u271d : AddGroupWithOne.zsmul = AddGroupWithOne.zsmul\nn : \u2124\n\u22a2 IntCast.intCast n = IntCast.intCast n"}, {"tactic": "cases n with\n| ofNat n   => rewrite [Int.ofNat_eq_coe, inst\u2081.intCast_ofNat, inst\u2082.intCast_ofNat]; congr\n| negSucc n => rewrite [inst\u2081.intCast_negSucc, inst\u2082.intCast_negSucc]; congr", "annotated_tactic": ["cases n with\n    | <a>ofNat</a> n   => rewrite [<a>Int.ofNat_eq_coe</a>, inst\u2081.intCast_ofNat, inst\u2082.intCast_ofNat]; congr\n    | <a>negSucc</a> n => rewrite [inst\u2081.intCast_negSucc, inst\u2082.intCast_negSucc]; congr", [{"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.ofNat_eq_coe", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [71, 17], "def_end_pos": [71, 29]}, {"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 h\nR : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d : toAddMonoidWithOne = toAddMonoidWithOne\nthis : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\ntoAddMonoid_eq\u271d : AddMonoidWithOne.toAddMonoid = AddMonoidWithOne.toAddMonoid\ntoNeg_eq\u271d : toNeg = toNeg\ntoSub_eq\u271d : toSub = toSub\nzsmul_eq\u271d : AddGroupWithOne.zsmul = AddGroupWithOne.zsmul\nn : \u2124\n\u22a2 IntCast.intCast n = IntCast.intCast n", "state_after": "no goals"}, {"tactic": "rewrite [Int.ofNat_eq_coe, inst\u2081.intCast_ofNat, inst\u2082.intCast_ofNat]", "annotated_tactic": ["rewrite [<a>Int.ofNat_eq_coe</a>, inst\u2081.intCast_ofNat, inst\u2082.intCast_ofNat]", [{"full_name": "Int.ofNat_eq_coe", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [71, 17], "def_end_pos": [71, 29]}]], "state_before": "case h.ofNat\nR : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d : toAddMonoidWithOne = toAddMonoidWithOne\nthis : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\ntoAddMonoid_eq\u271d : AddMonoidWithOne.toAddMonoid = AddMonoidWithOne.toAddMonoid\ntoNeg_eq\u271d : toNeg = toNeg\ntoSub_eq\u271d : toSub = toSub\nzsmul_eq\u271d : AddGroupWithOne.zsmul = AddGroupWithOne.zsmul\nn : \u2115\n\u22a2 IntCast.intCast (Int.ofNat n) = IntCast.intCast (Int.ofNat n)", "state_after": "case h.ofNat\nR : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d : toAddMonoidWithOne = toAddMonoidWithOne\nthis : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\ntoAddMonoid_eq\u271d : AddMonoidWithOne.toAddMonoid = AddMonoidWithOne.toAddMonoid\ntoNeg_eq\u271d : toNeg = toNeg\ntoSub_eq\u271d : toSub = toSub\nzsmul_eq\u271d : AddGroupWithOne.zsmul = AddGroupWithOne.zsmul\nn : \u2115\n\u22a2 \u2191n = \u2191n"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case h.ofNat\nR : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d : toAddMonoidWithOne = toAddMonoidWithOne\nthis : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\ntoAddMonoid_eq\u271d : AddMonoidWithOne.toAddMonoid = AddMonoidWithOne.toAddMonoid\ntoNeg_eq\u271d : toNeg = toNeg\ntoSub_eq\u271d : toSub = toSub\nzsmul_eq\u271d : AddGroupWithOne.zsmul = AddGroupWithOne.zsmul\nn : \u2115\n\u22a2 \u2191n = \u2191n", "state_after": "no goals"}, {"tactic": "rewrite [inst\u2081.intCast_negSucc, inst\u2082.intCast_negSucc]", "annotated_tactic": ["rewrite [inst\u2081.intCast_negSucc, inst\u2082.intCast_negSucc]", []], "state_before": "case h.negSucc\nR : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d : toAddMonoidWithOne = toAddMonoidWithOne\nthis : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\ntoAddMonoid_eq\u271d : AddMonoidWithOne.toAddMonoid = AddMonoidWithOne.toAddMonoid\ntoNeg_eq\u271d : toNeg = toNeg\ntoSub_eq\u271d : toSub = toSub\nzsmul_eq\u271d : AddGroupWithOne.zsmul = AddGroupWithOne.zsmul\nn : \u2115\n\u22a2 IntCast.intCast (Int.negSucc n) = IntCast.intCast (Int.negSucc n)", "state_after": "case h.negSucc\nR : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d : toAddMonoidWithOne = toAddMonoidWithOne\nthis : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\ntoAddMonoid_eq\u271d : AddMonoidWithOne.toAddMonoid = AddMonoidWithOne.toAddMonoid\ntoNeg_eq\u271d : toNeg = toNeg\ntoSub_eq\u271d : toSub = toSub\nzsmul_eq\u271d : AddGroupWithOne.zsmul = AddGroupWithOne.zsmul\nn : \u2115\n\u22a2 -\u2191(n + 1) = -\u2191(n + 1)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case h.negSucc\nR : Type u\ninst\u2081 inst\u2082 : AddGroupWithOne R\nh_add : HAdd.hAdd = HAdd.hAdd\nh_one : One.one = One.one\nthis\u271d : toAddMonoidWithOne = toAddMonoidWithOne\nthis : AddMonoidWithOne.toNatCast = AddMonoidWithOne.toNatCast\ntoAddMonoid_eq\u271d : AddMonoidWithOne.toAddMonoid = AddMonoidWithOne.toAddMonoid\ntoNeg_eq\u271d : toNeg = toNeg\ntoSub_eq\u271d : toSub = toSub\nzsmul_eq\u271d : AddGroupWithOne.zsmul = AddGroupWithOne.zsmul\nn : \u2115\n\u22a2 -\u2191(n + 1) = -\u2191(n + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/Quiver/Path.lean", "full_name": "Quiver.Path.toList_chain_nonempty", "start": [176, 1], "end": [179, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Card.lean", "full_name": "Finset.case_strong_induction_on", "start": [858, 1], "end": [862, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Equiv.lean", "full_name": "ContinuousLinearEquiv.differentiableWithinAt", "start": [74, 11], "end": [75, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Subring/Basic.lean", "full_name": "RingHom.eqLocus_same", "start": [1196, 1], "end": [1197, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/HausdorffDistance.lean", "full_name": "EMetric.hausdorffEdist_closure", "start": [428, 1], "end": [429, 7], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u\n\u03b2 : Type v\ninst\u271d\u00b9 : PseudoEMetricSpace \u03b1\ninst\u271d : PseudoEMetricSpace \u03b2\nx y : \u03b1\ns t u : Set \u03b1\n\u03a6 : \u03b1 \u2192 \u03b2\n\u22a2 hausdorffEdist (closure s) (closure t) = hausdorffEdist s t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Separation.lean", "full_name": "IsCompact.isCompact_isClosed_basis_nhds", "start": [1327, 1], "end": [1333, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/String/Lemmas.lean", "full_name": "String.drop_eq", "start": [1071, 1], "end": [1072, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/FiberBundle/Basic.lean", "full_name": "FiberBundleCore.mem_trivChange_source", "start": [474, 1], "end": [477, 7], "traced_tactics": [{"tactic": "erw [mem_prod]", "annotated_tactic": ["erw [<a>mem_prod</a>]", [{"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [207, 9], "def_end_pos": [207, 17]}]], "state_before": "\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni j : \u03b9\np : B \u00d7 F\n\u22a2 p \u2208 (Z.trivChange i j).source \u2194 p.1 \u2208 Z.baseSet i \u2229 Z.baseSet j", "state_after": "\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni j : \u03b9\np : B \u00d7 F\n\u22a2 p.1 \u2208 Z.baseSet i \u2229 Z.baseSet j \u2227 p.2 \u2208 univ \u2194 p.1 \u2208 Z.baseSet i \u2229 Z.baseSet j"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nZ : FiberBundleCore \u03b9 B F\ni j : \u03b9\np : B \u00d7 F\n\u22a2 p.1 \u2208 Z.baseSet i \u2229 Z.baseSet j \u2227 p.2 \u2208 univ \u2194 p.1 \u2208 Z.baseSet i \u2229 Z.baseSet j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Category/Profinite/Nobeling.lean", "full_name": "Profinite.NobelingProof.CC_comp_zero", "start": [1259, 1], "end": [1270, 24], "traced_tactics": [{"tactic": "intro y", "annotated_tactic": ["intro y", []], "state_before": "I : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\n\u22a2 \u2200 (y : LocallyConstant \u2191(\u03c0 C fun x => ord I x < o) \u2124), (Linear_CC' C hsC ho) ((\u03c0s C o) y) = 0", "state_after": "I : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\ny : LocallyConstant \u2191(\u03c0 C fun x => ord I x < o) \u2124\n\u22a2 (Linear_CC' C hsC ho) ((\u03c0s C o) y) = 0"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "I : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\ny : LocallyConstant \u2191(\u03c0 C fun x => ord I x < o) \u2124\n\u22a2 (Linear_CC' C hsC ho) ((\u03c0s C o) y) = 0", "state_after": "case h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\ny : LocallyConstant \u2191(\u03c0 C fun x => ord I x < o) \u2124\nx : \u2191(C' C ho)\n\u22a2 ((Linear_CC' C hsC ho) ((\u03c0s C o) y)) x = 0 x"}, {"tactic": "dsimp [Linear_CC', Linear_CC'\u2080, Linear_CC'\u2081, LocallyConstant.sub_apply]", "annotated_tactic": ["dsimp [<a>Linear_CC'</a>, <a>Linear_CC'\u2080</a>, <a>Linear_CC'\u2081</a>, <a>LocallyConstant.sub_apply</a>]", [{"full_name": "Profinite.NobelingProof.Linear_CC'", "def_path": "Mathlib/Topology/Category/Profinite/Nobeling.lean", "def_pos": [1256, 5], "def_end_pos": [1256, 15]}, {"full_name": "Profinite.NobelingProof.Linear_CC'\u2080", "def_path": "Mathlib/Topology/Category/Profinite/Nobeling.lean", "def_pos": [1246, 5], "def_end_pos": [1246, 16]}, {"full_name": "Profinite.NobelingProof.Linear_CC'\u2081", "def_path": "Mathlib/Topology/Category/Profinite/Nobeling.lean", "def_pos": [1251, 5], "def_end_pos": [1251, 16]}, {"full_name": "LocallyConstant.sub_apply", "def_path": "Mathlib/Topology/LocallyConstant/Algebra.lean", "def_pos": [137, 3], "def_end_pos": [137, 14]}]], "state_before": "case h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\ny : LocallyConstant \u2191(\u03c0 C fun x => ord I x < o) \u2124\nx : \u2191(C' C ho)\n\u22a2 ((Linear_CC' C hsC ho) ((\u03c0s C o) y)) x = 0 x", "state_after": "case h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\ny : LocallyConstant \u2191(\u03c0 C fun x => ord I x < o) \u2124\nx : \u2191(C' C ho)\n\u22a2 y (ProjRestrict C (fun x => ord I x < o) (CC'\u2081 C hsC ho x)) -\n      y (ProjRestrict C (fun x => ord I x < o) (CC'\u2080 C ho x)) =\n    0"}, {"tactic": "simp only [Pi.zero_apply, sub_eq_zero]", "annotated_tactic": ["simp only [<a>Pi.zero_apply</a>, <a>sub_eq_zero</a>]", [{"full_name": "Pi.zero_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [62, 3], "def_end_pos": [62, 14]}, {"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1070, 3], "def_end_pos": [1070, 14]}]], "state_before": "case h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\ny : LocallyConstant \u2191(\u03c0 C fun x => ord I x < o) \u2124\nx : \u2191(C' C ho)\n\u22a2 y (ProjRestrict C (fun x => ord I x < o) (CC'\u2081 C hsC ho x)) -\n      y (ProjRestrict C (fun x => ord I x < o) (CC'\u2080 C ho x)) =\n    0", "state_after": "case h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\ny : LocallyConstant \u2191(\u03c0 C fun x => ord I x < o) \u2124\nx : \u2191(C' C ho)\n\u22a2 y (ProjRestrict C (fun x => ord I x < o) (CC'\u2081 C hsC ho x)) = y (ProjRestrict C (fun x => ord I x < o) (CC'\u2080 C ho x))"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "case h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\ny : LocallyConstant \u2191(\u03c0 C fun x => ord I x < o) \u2124\nx : \u2191(C' C ho)\n\u22a2 y (ProjRestrict C (fun x => ord I x < o) (CC'\u2081 C hsC ho x)) = y (ProjRestrict C (fun x => ord I x < o) (CC'\u2080 C ho x))", "state_after": "case h.h.e_6.h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\ny : LocallyConstant \u2191(\u03c0 C fun x => ord I x < o) \u2124\nx : \u2191(C' C ho)\n\u22a2 ProjRestrict C (fun x => ord I x < o) (CC'\u2081 C hsC ho x) = ProjRestrict C (fun x => ord I x < o) (CC'\u2080 C ho x)"}, {"tactic": "ext i", "annotated_tactic": ["ext i", []], "state_before": "case h.h.e_6.h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\ny : LocallyConstant \u2191(\u03c0 C fun x => ord I x < o) \u2124\nx : \u2191(C' C ho)\n\u22a2 ProjRestrict C (fun x => ord I x < o) (CC'\u2081 C hsC ho x) = ProjRestrict C (fun x => ord I x < o) (CC'\u2080 C ho x)", "state_after": "case h.h.e_6.h.a.h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\ny : LocallyConstant \u2191(\u03c0 C fun x => ord I x < o) \u2124\nx : \u2191(C' C ho)\ni : I\n\u22a2 \u2191(ProjRestrict C (fun x => ord I x < o) (CC'\u2081 C hsC ho x)) i =\n    \u2191(ProjRestrict C (fun x => ord I x < o) (CC'\u2080 C ho x)) i"}, {"tactic": "dsimp [CC'\u2080, CC'\u2081, ProjRestrict, Proj]", "annotated_tactic": ["dsimp [<a>CC'\u2080</a>, <a>CC'\u2081</a>, <a>ProjRestrict</a>, <a>Proj</a>]", [{"full_name": "Profinite.NobelingProof.CC'\u2080", "def_path": "Mathlib/Topology/Category/Profinite/Nobeling.lean", "def_pos": [1232, 5], "def_end_pos": [1232, 9]}, {"full_name": "Profinite.NobelingProof.CC'\u2081", "def_path": "Mathlib/Topology/Category/Profinite/Nobeling.lean", "def_pos": [1236, 5], "def_end_pos": [1236, 9]}, {"full_name": "Profinite.NobelingProof.ProjRestrict", "def_path": "Mathlib/Topology/Category/Profinite/Nobeling.lean", "def_pos": [106, 5], "def_end_pos": [106, 17]}, {"full_name": "Profinite.NobelingProof.Proj", "def_path": "Mathlib/Topology/Category/Profinite/Nobeling.lean", "def_pos": [88, 5], "def_end_pos": [88, 9]}]], "state_before": "case h.h.e_6.h.a.h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\ny : LocallyConstant \u2191(\u03c0 C fun x => ord I x < o) \u2124\nx : \u2191(C' C ho)\ni : I\n\u22a2 \u2191(ProjRestrict C (fun x => ord I x < o) (CC'\u2081 C hsC ho x)) i =\n    \u2191(ProjRestrict C (fun x => ord I x < o) (CC'\u2080 C ho x)) i", "state_after": "case h.h.e_6.h.a.h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\ny : LocallyConstant \u2191(\u03c0 C fun x => ord I x < o) \u2124\nx : \u2191(C' C ho)\ni : I\n\u22a2 (if ord I i < o then SwapTrue o (\u2191x) i else false) = if ord I i < o then \u2191x i else false"}, {"tactic": "apply if_ctx_congr Iff.rfl _ (fun _ \u21a6 rfl)", "annotated_tactic": ["apply <a>if_ctx_congr</a> <a>Iff.rfl</a> _ (fun _ \u21a6 <a>rfl</a>)", [{"full_name": "if_ctx_congr", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [372, 9], "def_end_pos": [372, 21]}, {"full_name": "Iff.rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [796, 19], "def_end_pos": [796, 26]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case h.h.e_6.h.a.h\nI : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\ny : LocallyConstant \u2191(\u03c0 C fun x => ord I x < o) \u2124\nx : \u2191(C' C ho)\ni : I\n\u22a2 (if ord I i < o then SwapTrue o (\u2191x) i else false) = if ord I i < o then \u2191x i else false", "state_after": "I : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\ny : LocallyConstant \u2191(\u03c0 C fun x => ord I x < o) \u2124\nx : \u2191(C' C ho)\ni : I\n\u22a2 ord I i < o \u2192 SwapTrue o (\u2191x) i = \u2191x i"}, {"tactic": "simp only [SwapTrue, ite_eq_right_iff]", "annotated_tactic": ["simp only [<a>SwapTrue</a>, <a>ite_eq_right_iff</a>]", [{"full_name": "Profinite.NobelingProof.SwapTrue", "def_path": "Mathlib/Topology/Category/Profinite/Nobeling.lean", "def_pos": [1202, 5], "def_end_pos": [1202, 13]}, {"full_name": "ite_eq_right_iff", "def_path": ".lake/packages/lean4/src/lean/Init/ByCases.lean", "def_pos": [51, 17], "def_end_pos": [51, 33]}]], "state_before": "I : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\ny : LocallyConstant \u2191(\u03c0 C fun x => ord I x < o) \u2124\nx : \u2191(C' C ho)\ni : I\n\u22a2 ord I i < o \u2192 SwapTrue o (\u2191x) i = \u2191x i", "state_after": "I : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\ny : LocallyConstant \u2191(\u03c0 C fun x => ord I x < o) \u2124\nx : \u2191(C' C ho)\ni : I\n\u22a2 ord I i < o \u2192 ord I i = o \u2192 true = \u2191x i"}, {"tactic": "intro h\u2081 h\u2082", "annotated_tactic": ["intro h\u2081 h\u2082", []], "state_before": "I : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\ny : LocallyConstant \u2191(\u03c0 C fun x => ord I x < o) \u2124\nx : \u2191(C' C ho)\ni : I\n\u22a2 ord I i < o \u2192 ord I i = o \u2192 true = \u2191x i", "state_after": "I : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\ny : LocallyConstant \u2191(\u03c0 C fun x => ord I x < o) \u2124\nx : \u2191(C' C ho)\ni : I\nh\u2081 : ord I i < o\nh\u2082 : ord I i = o\n\u22a2 true = \u2191x i"}, {"tactic": "exact (h\u2081.ne h\u2082).elim", "annotated_tactic": ["exact (h\u2081.ne h\u2082).<a>elim</a>", [{"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "I : Type u\ninst\u271d\u00b9 : LinearOrder I\ninst\u271d : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x x_1 => x < x_1\ny : LocallyConstant \u2191(\u03c0 C fun x => ord I x < o) \u2124\nx : \u2191(C' C ho)\ni : I\nh\u2081 : ord I i < o\nh\u2082 : ord I i = o\n\u22a2 true = \u2191x i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Hyperoperation.lean", "full_name": "hyperoperation_ge_two_eq_self", "start": [91, 1], "end": [95, 71], "traced_tactics": [{"tactic": "induction' n with nn nih", "annotated_tactic": ["induction' n with nn nih", []], "state_before": "n m : \u2115\n\u22a2 hyperoperation (n + 2) m 1 = m", "state_after": "case zero\nm : \u2115\n\u22a2 hyperoperation (0 + 2) m 1 = m\n\ncase succ\nm nn : \u2115\nnih : hyperoperation (nn + 2) m 1 = m\n\u22a2 hyperoperation (nn + 1 + 2) m 1 = m"}, {"tactic": "rw [hyperoperation_two]", "annotated_tactic": ["rw [<a>hyperoperation_two</a>]", [{"full_name": "hyperoperation_two", "def_path": "Mathlib/Data/Nat/Hyperoperation.lean", "def_pos": [69, 9], "def_end_pos": [69, 27]}]], "state_before": "case zero\nm : \u2115\n\u22a2 hyperoperation (0 + 2) m 1 = m", "state_after": "case zero\nm : \u2115\n\u22a2 (fun x x_1 => x * x_1) m 1 = m"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case zero\nm : \u2115\n\u22a2 (fun x x_1 => x * x_1) m 1 = m", "state_after": "no goals"}, {"tactic": "rw [hyperoperation_recursion, hyperoperation_ge_three_eq_one, nih]", "annotated_tactic": ["rw [<a>hyperoperation_recursion</a>, <a>hyperoperation_ge_three_eq_one</a>, nih]", [{"full_name": "hyperoperation_recursion", "def_path": "Mathlib/Data/Nat/Hyperoperation.lean", "def_pos": [53, 9], "def_end_pos": [53, 33]}, {"full_name": "hyperoperation_ge_three_eq_one", "def_path": "Mathlib/Data/Nat/Hyperoperation.lean", "def_pos": [49, 9], "def_end_pos": [49, 39]}]], "state_before": "case succ\nm nn : \u2115\nnih : hyperoperation (nn + 2) m 1 = m\n\u22a2 hyperoperation (nn + 1 + 2) m 1 = m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Log.lean", "full_name": "Nat.clog_zero_left", "start": [259, 1], "end": [259, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Game/PGame.lean", "full_name": "SetTheory.PGame.le_zero", "start": [675, 1], "end": [677, 7], "traced_tactics": [{"tactic": "rw [le_def]", "annotated_tactic": ["rw [<a>le_def</a>]", [{"full_name": "SetTheory.PGame.le_def", "def_path": "Mathlib/SetTheory/Game/PGame.lean", "def_pos": [623, 9], "def_end_pos": [623, 15]}]], "state_before": "xl xr : Type u\nx : PGame\n\u22a2 x \u2264 0 \u2194 \u2200 (i : x.LeftMoves), \u2203 j, (x.moveLeft i).moveRight j \u2264 0", "state_after": "xl xr : Type u\nx : PGame\n\u22a2 ((\u2200 (i : x.LeftMoves), (\u2203 i', x.moveLeft i \u2264 moveLeft 0 i') \u2228 \u2203 j, (x.moveLeft i).moveRight j \u2264 0) \u2227\n      \u2200 (j : RightMoves 0), (\u2203 i, x \u2264 (moveRight 0 j).moveLeft i) \u2228 \u2203 j', x.moveRight j' \u2264 moveRight 0 j) \u2194\n    \u2200 (i : x.LeftMoves), \u2203 j, (x.moveLeft i).moveRight j \u2264 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "xl xr : Type u\nx : PGame\n\u22a2 ((\u2200 (i : x.LeftMoves), (\u2203 i', x.moveLeft i \u2264 moveLeft 0 i') \u2228 \u2203 j, (x.moveLeft i).moveRight j \u2264 0) \u2227\n      \u2200 (j : RightMoves 0), (\u2203 i, x \u2264 (moveRight 0 j).moveLeft i) \u2228 \u2203 j', x.moveRight j' \u2264 moveRight 0 j) \u2194\n    \u2200 (i : x.LeftMoves), \u2203 j, (x.moveLeft i).moveRight j \u2264 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Subsemiring/Basic.lean", "full_name": "Subsemiring.sInf_toAddSubmonoid", "start": [600, 1], "end": [602, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Valuation/Basic.lean", "full_name": "AddValuation.IsEquiv.trans", "start": [829, 1], "end": [830, 34], "traced_tactics": []}, 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MeasurableEmbedding f\nx : \u03b1\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 MeasurableSet ((fun y => (f y, x)) '' s)"}, {"tactic": "convert (hf.measurableSet_image.mpr hs).prod (MeasurableSet.singleton x)", "annotated_tactic": ["convert (hf.measurableSet_image.mpr hs).<a>prod</a> (<a>MeasurableSet.singleton</a> x)", [{"full_name": "MeasurableSet.prod", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [786, 19], "def_end_pos": [786, 37]}, {"full_name": "MeasurableSet.singleton", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [281, 7], "def_end_pos": [281, 30]}]], "state_before": "\u03b1 : 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\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d\u00b9 : NormedAddCommGroup E\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : MeasurableSingletonClass \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\nhf : MeasurableEmbedding f\nx : \u03b1\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 MeasurableSet ((fun y => (f y, x)) '' s)", "state_after": "case h.e'_3\n\u03b1 : 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\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d\u00b9 : NormedAddCommGroup E\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : MeasurableSingletonClass \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\nhf : MeasurableEmbedding f\nx : \u03b1\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 (fun y => (f y, x)) '' s = (f '' s) \u00d7\u02e2 {x}"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "case h.e'_3\n\u03b1 : 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\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d\u00b9 : NormedAddCommGroup E\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : MeasurableSingletonClass \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\nhf : MeasurableEmbedding f\nx : \u03b1\ns : Set \u03b3\nhs : MeasurableSet s\n\u22a2 (fun y => (f y, x)) '' s = (f '' s) \u00d7\u02e2 {x}", "state_after": "case h.e'_3.h\n\u03b1 : 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\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d\u00b9 : NormedAddCommGroup E\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : MeasurableSingletonClass \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\nhf : MeasurableEmbedding f\nx\u271d : \u03b1\ns : Set \u03b3\nhs : MeasurableSet s\nx : \u03b2 \u00d7 \u03b1\n\u22a2 x \u2208 (fun y => (f y, x\u271d)) '' s \u2194 x \u2208 (f '' s) \u00d7\u02e2 {x\u271d}"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_3.h\n\u03b1 : 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\u2076 : MeasurableSpace \u03b1\ninst\u271d\u2075 : MeasurableSpace \u03b1'\ninst\u271d\u2074 : MeasurableSpace \u03b2\u271d\ninst\u271d\u00b3 : MeasurableSpace \u03b2'\ninst\u271d\u00b2 : MeasurableSpace \u03b3\u271d\n\u03bc \u03bc' : Measure \u03b1\n\u03bd \u03bd' : Measure \u03b2\u271d\n\u03c4 : Measure \u03b3\u271d\ninst\u271d\u00b9 : NormedAddCommGroup E\n\u03b2 : Type u_7\n\u03b3 : Type u_8\ninst\u271d : MeasurableSingletonClass \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\nf : \u03b3 \u2192 \u03b2\nhf : MeasurableEmbedding f\nx\u271d : \u03b1\ns : Set \u03b3\nhs : MeasurableSet s\nx : \u03b2 \u00d7 \u03b1\n\u22a2 x \u2208 (fun y => (f y, x\u271d)) '' s \u2194 x \u2208 (f '' s) \u00d7\u02e2 {x\u271d}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Valuation/ValuationSubring.lean", "full_name": "ValuationSubring.coe_mem_principalUnitGroup_iff", "start": [687, 1], "end": [693, 36], "traced_tactics": [{"tactic": "rw [MonoidHom.mem_ker, Units.ext_iff]", "annotated_tactic": ["rw [<a>MonoidHom.mem_ker</a>, <a>Units.ext_iff</a>]", [{"full_name": "MonoidHom.mem_ker", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [2690, 9], "def_end_pos": [2690, 16]}, {"full_name": "Units.ext_iff", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [160, 9], "def_end_pos": [160, 16]}]], "state_before": "K : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx : \u21a5A.unitGroup\n\u22a2 \u2191x \u2208 A.principalUnitGroup \u2194 A.unitGroupMulEquiv x \u2208 (Units.map \u2191(LocalRing.residue \u21a5A)).ker", "state_after": "K : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx : \u21a5A.unitGroup\n\u22a2 \u2191x \u2208 A.principalUnitGroup \u2194 \u2191((Units.map \u2191(LocalRing.residue \u21a5A)) (A.unitGroupMulEquiv x)) = \u21911"}, {"tactic": "let \u03c0 := Ideal.Quotient.mk (LocalRing.maximalIdeal A)", "annotated_tactic": ["let \u03c0 := <a>Ideal.Quotient.mk</a> (<a>LocalRing.maximalIdeal</a> A)", [{"full_name": "Ideal.Quotient.mk", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [92, 5], "def_end_pos": [92, 7]}, {"full_name": "LocalRing.maximalIdeal", "def_path": "Mathlib/RingTheory/Ideal/LocalRing.lean", "def_pos": [106, 5], "def_end_pos": [106, 17]}]], "state_before": "K : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx : \u21a5A.unitGroup\n\u22a2 \u2191x \u2208 A.principalUnitGroup \u2194 \u2191((Units.map \u2191(LocalRing.residue \u21a5A)) (A.unitGroupMulEquiv x)) = \u21911", "state_after": "K : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx : \u21a5A.unitGroup\n\u03c0 : \u21a5A \u2192+* \u21a5A \u29f8 LocalRing.maximalIdeal \u21a5A := Ideal.Quotient.mk (LocalRing.maximalIdeal \u21a5A)\n\u22a2 \u2191x \u2208 A.principalUnitGroup \u2194 \u2191((Units.map \u2191(LocalRing.residue \u21a5A)) (A.unitGroupMulEquiv x)) = \u21911"}, {"tactic": "convert_to _ \u2194 \u03c0 _ = 1", "annotated_tactic": ["convert_to _ \u2194 \u03c0 _ = 1", []], "state_before": "K : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx : \u21a5A.unitGroup\n\u03c0 : \u21a5A \u2192+* \u21a5A \u29f8 LocalRing.maximalIdeal \u21a5A := Ideal.Quotient.mk (LocalRing.maximalIdeal \u21a5A)\n\u22a2 \u2191x \u2208 A.principalUnitGroup \u2194 \u2191((Units.map \u2191(LocalRing.residue \u21a5A)) (A.unitGroupMulEquiv x)) = \u21911", "state_after": "case convert_3\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx : \u21a5A.unitGroup\n\u03c0 : \u21a5A \u2192+* \u21a5A \u29f8 LocalRing.maximalIdeal \u21a5A := Ideal.Quotient.mk (LocalRing.maximalIdeal \u21a5A)\n\u22a2 \u2191x \u2208 A.principalUnitGroup \u2194 \u03c0 \u2191(A.unitGroupMulEquiv x) = 1"}, {"tactic": "rw [\u2190 \u03c0.map_one, \u2190 sub_eq_zero, \u2190 \u03c0.map_sub, Ideal.Quotient.eq_zero_iff_mem, valuation_lt_one_iff]", "annotated_tactic": ["rw [\u2190 \u03c0.map_one, \u2190 <a>sub_eq_zero</a>, \u2190 \u03c0.map_sub, <a>Ideal.Quotient.eq_zero_iff_mem</a>, <a>valuation_lt_one_iff</a>]", [{"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1070, 3], "def_end_pos": [1070, 14]}, {"full_name": "Ideal.Quotient.eq_zero_iff_mem", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [125, 9], "def_end_pos": [125, 24]}, {"full_name": "ValuationSubring.valuation_lt_one_iff", "def_path": "Mathlib/RingTheory/Valuation/ValuationSubring.lean", "def_pos": [218, 9], "def_end_pos": [218, 29]}]], "state_before": "case convert_3\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx : \u21a5A.unitGroup\n\u03c0 : \u21a5A \u2192+* \u21a5A \u29f8 LocalRing.maximalIdeal \u21a5A := Ideal.Quotient.mk (LocalRing.maximalIdeal \u21a5A)\n\u22a2 \u2191x \u2208 A.principalUnitGroup \u2194 \u03c0 \u2191(A.unitGroupMulEquiv x) = 1", "state_after": "case convert_3\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx : \u21a5A.unitGroup\n\u03c0 : \u21a5A \u2192+* \u21a5A \u29f8 LocalRing.maximalIdeal \u21a5A := Ideal.Quotient.mk (LocalRing.maximalIdeal \u21a5A)\n\u22a2 \u2191x \u2208 A.principalUnitGroup \u2194 A.valuation \u2191(\u2191(A.unitGroupMulEquiv x) - 1) < 1"}, {"tactic": "simp [mem_principalUnitGroup_iff]", "annotated_tactic": ["simp [<a>mem_principalUnitGroup_iff</a>]", [{"full_name": "ValuationSubring.mem_principalUnitGroup_iff", "def_path": "Mathlib/RingTheory/Valuation/ValuationSubring.lean", "def_pos": [650, 9], "def_end_pos": [650, 35]}]], "state_before": "case convert_3\nK : Type u\ninst\u271d : Field K\nA : ValuationSubring K\nx : \u21a5A.unitGroup\n\u03c0 : \u21a5A \u2192+* \u21a5A \u29f8 LocalRing.maximalIdeal \u21a5A := Ideal.Quotient.mk (LocalRing.maximalIdeal \u21a5A)\n\u22a2 \u2191x \u2208 A.principalUnitGroup \u2194 A.valuation \u2191(\u2191(A.unitGroupMulEquiv x) - 1) < 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Mirror.lean", "full_name": "Polynomial.mirror_eq_zero", "start": [144, 1], "end": [145, 88], "traced_tactics": [{"tactic": "rw [\u2190 p.mirror_mirror, h, mirror_zero]", "annotated_tactic": ["rw [\u2190 p.mirror_mirror, h, <a>mirror_zero</a>]", [{"full_name": "Polynomial.mirror_zero", "def_path": "Mathlib/Algebra/Polynomial/Mirror.lean", "def_pos": [44, 9], "def_end_pos": [44, 20]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\np q : R[X]\nh : p.mirror = 0\n\u22a2 p = 0", "state_after": "no goals"}, {"tactic": "rw [h, mirror_zero]", "annotated_tactic": ["rw [h, <a>mirror_zero</a>]", [{"full_name": "Polynomial.mirror_zero", "def_path": "Mathlib/Algebra/Polynomial/Mirror.lean", "def_pos": [44, 9], "def_end_pos": [44, 20]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\np q : R[X]\nh : p = 0\n\u22a2 p.mirror = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Module/Pointwise.lean", "full_name": "BddBelow.smul_of_nonneg", "start": [33, 1], "end": [34, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/PiSystem.lean", "full_name": "piiUnionInter_mono_left", "start": [462, 1], "end": [464, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/UpperLower/Basic.lean", "full_name": "UpperSet.le_Ici", "start": [1228, 1], "end": [1228, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Prod.lean", "full_name": "LinearMap.prod_eq_sup_map", "start": [486, 1], "end": [488, 49], "traced_tactics": [{"tactic": "rw [\u2190 map_coprod_prod, coprod_inl_inr, map_id]", "annotated_tactic": ["rw [\u2190 <a>map_coprod_prod</a>, <a>coprod_inl_inr</a>, <a>map_id</a>]", [{"full_name": "LinearMap.map_coprod_prod", "def_path": "Mathlib/LinearAlgebra/Prod.lean", "def_pos": [466, 9], "def_end_pos": [466, 24]}, {"full_name": "LinearMap.coprod_inl_inr", "def_path": "Mathlib/LinearAlgebra/Prod.lean", "def_pos": [239, 9], "def_end_pos": [239, 23]}, {"full_name": "Submodule.map_id", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [102, 9], "def_end_pos": [102, 15]}]], "state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\np : Submodule R M\nq : Submodule R M\u2082\n\u22a2 p.prod q = map (inl R M M\u2082) p \u2294 map (inr R M M\u2082) q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Fintype.lean", "full_name": "Multiset.card_coe", "start": [241, 1], "end": [243, 50], "traced_tactics": [{"tactic": "rw [Fintype.card_congr m.coeEquiv]", "annotated_tactic": ["rw [<a>Fintype.card_congr</a> m.coeEquiv]", [{"full_name": "Fintype.card_congr", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [151, 9], "def_end_pos": [151, 19]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nm\u271d m : Multiset \u03b1\n\u22a2 Fintype.card m.ToType = card m", "state_after": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nm\u271d m : Multiset \u03b1\n\u22a2 Fintype.card { x // x \u2208 m.toEnumFinset } = card m"}, {"tactic": "simp only [Fintype.card_coe, card_toEnumFinset]", "annotated_tactic": ["simp only [<a>Fintype.card_coe</a>, <a>card_toEnumFinset</a>]", [{"full_name": "Fintype.card_coe", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [754, 9], "def_end_pos": [754, 25]}, {"full_name": "Multiset.card_toEnumFinset", "def_path": "Mathlib/Data/Multiset/Fintype.lean", "def_pos": [234, 9], "def_end_pos": [234, 35]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nm\u271d m : Multiset \u03b1\n\u22a2 Fintype.card { x // x \u2208 m.toEnumFinset } = card m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Bases.lean", "full_name": "TopologicalSpace.IsSeparable.mono", "start": [454, 1], "end": [456, 34], "traced_tactics": [{"tactic": "rcases hs with \u27e8c, c_count, hs\u27e9", "annotated_tactic": ["rcases hs with \u27e8c, c_count, hs\u27e9", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\nt : TopologicalSpace \u03b1\nB : Set (Set \u03b1)\ns\u271d s u : Set \u03b1\nhs : IsSeparable s\nhu : u \u2286 s\n\u22a2 IsSeparable u", "state_after": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type u_1\nt : TopologicalSpace \u03b1\nB : Set (Set \u03b1)\ns\u271d s u : Set \u03b1\nhu : u \u2286 s\nc : Set \u03b1\nc_count : c.Countable\nhs : s \u2286 closure c\n\u22a2 IsSeparable u"}, {"tactic": "exact \u27e8c, c_count, hu.trans hs\u27e9", "annotated_tactic": ["exact \u27e8c, c_count, hu.trans hs\u27e9", []], "state_before": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type u_1\nt : TopologicalSpace \u03b1\nB : Set (Set \u03b1)\ns\u271d s u : Set \u03b1\nhu : u \u2286 s\nc : Set \u03b1\nc_count : c.Countable\nhs : s \u2286 closure c\n\u22a2 IsSeparable u", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Final.lean", "full_name": "CategoryTheory.Functor.initial_iff_comp_equivalence", "start": [717, 1], "end": [718, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Reverse.lean", "full_name": "Polynomial.trailingCoeff_mul", "start": [329, 1], "end": [332, 26], "traced_tactics": [{"tactic": "rw [\u2190 reverse_leadingCoeff, reverse_mul_of_domain, leadingCoeff_mul, reverse_leadingCoeff,\n  reverse_leadingCoeff]", "annotated_tactic": ["rw [\u2190 <a>reverse_leadingCoeff</a>, <a>reverse_mul_of_domain</a>, <a>leadingCoeff_mul</a>, <a>reverse_leadingCoeff</a>,\n    <a>reverse_leadingCoeff</a>]", [{"full_name": "Polynomial.reverse_leadingCoeff", "def_path": "Mathlib/Algebra/Polynomial/Reverse.lean", "def_pos": [299, 9], "def_end_pos": [299, 29]}, {"full_name": "Polynomial.reverse_mul_of_domain", "def_path": "Mathlib/Algebra/Polynomial/Reverse.lean", "def_pos": [320, 9], "def_end_pos": [320, 30]}, {"full_name": "Polynomial.leadingCoeff_mul", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [1673, 9], "def_end_pos": [1673, 25]}, {"full_name": "Polynomial.reverse_leadingCoeff", "def_path": "Mathlib/Algebra/Polynomial/Reverse.lean", "def_pos": [299, 9], "def_end_pos": [299, 29]}, {"full_name": "Polynomial.reverse_leadingCoeff", "def_path": "Mathlib/Algebra/Polynomial/Reverse.lean", "def_pos": [299, 9], "def_end_pos": [299, 29]}]], "state_before": "R\u271d : Type u_1\ninst\u271d\u00b2 : Semiring R\u271d\nf : R\u271d[X]\nR : Type u_2\ninst\u271d\u00b9 : Ring R\ninst\u271d : NoZeroDivisors R\np q : R[X]\n\u22a2 (p * q).trailingCoeff = p.trailingCoeff * q.trailingCoeff", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/ModelTheory/LanguageMap.lean", "full_name": "FirstOrder.Language.constantsOnMap_isExpansionOn", "start": [426, 1], "end": [433, 21], "traced_tactics": [{"tactic": "letI := constantsOn.structure f\u03b1", "annotated_tactic": ["letI := <a>constantsOn.structure</a> f\u03b1", [{"full_name": "FirstOrder.Language.constantsOn.structure", "def_path": "Mathlib/ModelTheory/LanguageMap.lean", "def_pos": [415, 5], "def_end_pos": [415, 26]}]], "state_before": "L : Language\nL' : Language\nM : Type w\ninst\u271d : L.Structure M\n\u03b1 : Type u'\n\u03b2 : Type v'\nf : \u03b1 \u2192 \u03b2\nf\u03b1 : \u03b1 \u2192 M\nf\u03b2 : \u03b2 \u2192 M\nh : f\u03b2 \u2218 f = f\u03b1\n\u22a2 (LHom.constantsOnMap f).IsExpansionOn M", "state_after": "L : Language\nL' : Language\nM : Type w\ninst\u271d : L.Structure M\n\u03b1 : Type u'\n\u03b2 : Type v'\nf : \u03b1 \u2192 \u03b2\nf\u03b1 : \u03b1 \u2192 M\nf\u03b2 : \u03b2 \u2192 M\nh : f\u03b2 \u2218 f = f\u03b1\nthis : (constantsOn \u03b1).Structure M := constantsOn.structure f\u03b1\n\u22a2 (LHom.constantsOnMap f).IsExpansionOn M"}, {"tactic": "letI := constantsOn.structure f\u03b2", "annotated_tactic": ["letI := <a>constantsOn.structure</a> f\u03b2", [{"full_name": "FirstOrder.Language.constantsOn.structure", "def_path": "Mathlib/ModelTheory/LanguageMap.lean", "def_pos": [415, 5], "def_end_pos": [415, 26]}]], "state_before": "L : Language\nL' : Language\nM : Type w\ninst\u271d : L.Structure M\n\u03b1 : Type u'\n\u03b2 : Type v'\nf : \u03b1 \u2192 \u03b2\nf\u03b1 : \u03b1 \u2192 M\nf\u03b2 : \u03b2 \u2192 M\nh : f\u03b2 \u2218 f = f\u03b1\nthis : (constantsOn \u03b1).Structure M := constantsOn.structure f\u03b1\n\u22a2 (LHom.constantsOnMap f).IsExpansionOn M", "state_after": "L : Language\nL' : Language\nM : Type w\ninst\u271d : L.Structure M\n\u03b1 : Type u'\n\u03b2 : Type v'\nf : \u03b1 \u2192 \u03b2\nf\u03b1 : \u03b1 \u2192 M\nf\u03b2 : \u03b2 \u2192 M\nh : f\u03b2 \u2218 f = f\u03b1\nthis\u271d : (constantsOn \u03b1).Structure M := constantsOn.structure f\u03b1\nthis : (constantsOn \u03b2).Structure M := constantsOn.structure f\u03b2\n\u22a2 (LHom.constantsOnMap f).IsExpansionOn M"}, {"tactic": "exact\n  \u27e8fun {n} => Nat.casesOn n (fun F _x => (congr_fun h F : _)) fun n F => isEmptyElim F, fun R =>\n    isEmptyElim R\u27e9", "annotated_tactic": ["exact\n    \u27e8fun {n} => <a>Nat.casesOn</a> n (fun F _x => (<a>congr_fun</a> h F : _)) fun n F => <a>isEmptyElim</a> F, fun R =>\n      <a>isEmptyElim</a> R\u27e9", [{"full_name": "Nat.casesOn", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1073, 11], "def_end_pos": [1073, 14]}, {"full_name": "congr_fun", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [78, 7], "def_end_pos": [78, 16]}, {"full_name": "isEmptyElim", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [96, 5], "def_end_pos": [96, 16]}, {"full_name": "isEmptyElim", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [96, 5], "def_end_pos": [96, 16]}]], "state_before": "L : Language\nL' : Language\nM : Type w\ninst\u271d : L.Structure M\n\u03b1 : Type u'\n\u03b2 : Type v'\nf : \u03b1 \u2192 \u03b2\nf\u03b1 : \u03b1 \u2192 M\nf\u03b2 : \u03b2 \u2192 M\nh : f\u03b2 \u2218 f = f\u03b1\nthis\u271d : (constantsOn \u03b1).Structure M := constantsOn.structure f\u03b1\nthis : (constantsOn \u03b2).Structure M := constantsOn.structure f\u03b2\n\u22a2 (LHom.constantsOnMap f).IsExpansionOn M", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Nth.lean", "full_name": "Nat.le_nth_of_count_le", "start": [370, 1], "end": [371, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean", "full_name": "Real.arccos_le_pi", "start": [349, 1], "end": [350, 55], "traced_tactics": [{"tactic": "unfold arccos", "annotated_tactic": ["unfold <a>arccos</a>", [{"full_name": "Real.arccos", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean", "def_pos": [338, 19], "def_end_pos": [338, 25]}]], "state_before": "x\u271d y x : \u211d\n\u22a2 arccos x \u2264 \u03c0", "state_after": "x\u271d y x : \u211d\n\u22a2 \u03c0 / 2 - arcsin x \u2264 \u03c0"}, {"tactic": "linarith [neg_pi_div_two_le_arcsin x]", "annotated_tactic": ["linarith [<a>neg_pi_div_two_le_arcsin</a> x]", [{"full_name": "Real.neg_pi_div_two_le_arcsin", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean", "def_pos": [54, 9], "def_end_pos": [54, 33]}]], "state_before": "x\u271d y x : \u211d\n\u22a2 \u03c0 / 2 - arcsin x \u2264 \u03c0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matroid/Constructions.lean", "full_name": "Matroid.emptyOn_ground", "start": [51, 9], "end": [51, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Cast/Defs.lean", "full_name": "Nat.cast_ofNat", "start": [76, 20], "end": [77, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "MeasureTheory.NullMeasurableSet.right_of_prod", "start": [546, 1], "end": [553, 41], "traced_tactics": [{"tactic": "rcases h with \u27e8u, hum, hu\u27e9", "annotated_tactic": ["rcases h with \u27e8u, hum, hu\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\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 : SFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\nh : NullMeasurableSet (s \u00d7\u02e2 t) (\u03bc.prod \u03bd)\nhs : \u03bc s \u2260 0\n\u22a2 NullMeasurableSet t \u03bd", "state_after": "case 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\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 : SFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\nhs : \u03bc s \u2260 0\nu : Set (\u03b1 \u00d7 \u03b2)\nhum : MeasurableSet u\nhu : s \u00d7\u02e2 t =\u1da0[ae (\u03bc.prod \u03bd)] u\n\u22a2 NullMeasurableSet t \u03bd"}, {"tactic": "obtain \u27e8x, hxs, hx\u27e9 : \u2203 x \u2208 s, (Prod.mk x \u207b\u00b9' (s \u00d7\u02e2 t)) =\u1d50[\u03bd] (Prod.mk x \u207b\u00b9' u) :=\n  ((frequently_ae_iff.2 hs).and_eventually (ae_ae_eq_curry_of_prod hu)).exists", "annotated_tactic": ["obtain \u27e8x, hxs, hx\u27e9 : \u2203 x \u2208 s, (<a>Prod.mk</a> x \u207b\u00b9' (s \u00d7\u02e2 t)) =\u1d50[\u03bd] (<a>Prod.mk</a> x \u207b\u00b9' u) :=\n    ((<a>frequently_ae_iff</a>.2 hs).<a>and_eventually</a> (<a>ae_ae_eq_curry_of_prod</a> hu)).<a>exists</a>", [{"full_name": "Prod.mk", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [481, 3], "def_end_pos": [481, 5]}, {"full_name": "Prod.mk", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [481, 3], "def_end_pos": [481, 5]}, {"full_name": "MeasureTheory.frequently_ae_iff", "def_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean", "def_pos": [82, 9], "def_end_pos": [82, 26]}, {"full_name": "Filter.Frequently.and_eventually", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1311, 9], "def_end_pos": [1311, 34]}, {"full_name": "MeasureTheory.Measure.ae_ae_eq_curry_of_prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [506, 9], "def_end_pos": [506, 31]}, {"full_name": "Filter.Frequently.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1322, 9], "def_end_pos": [1322, 26]}]], "state_before": "case 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\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 : SFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\nhs : \u03bc s \u2260 0\nu : Set (\u03b1 \u00d7 \u03b2)\nhum : MeasurableSet u\nhu : s \u00d7\u02e2 t =\u1da0[ae (\u03bc.prod \u03bd)] u\n\u22a2 NullMeasurableSet t \u03bd", "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\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 : SFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\nhs : \u03bc s \u2260 0\nu : Set (\u03b1 \u00d7 \u03b2)\nhum : MeasurableSet u\nhu : s \u00d7\u02e2 t =\u1da0[ae (\u03bc.prod \u03bd)] u\nx : \u03b1\nhxs : x \u2208 s\nhx : Prod.mk x \u207b\u00b9' s \u00d7\u02e2 t =\u1da0[ae \u03bd] Prod.mk x \u207b\u00b9' u\n\u22a2 NullMeasurableSet t \u03bd"}, {"tactic": "refine \u27e8Prod.mk x \u207b\u00b9' u, measurable_prod_mk_left hum, ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>Prod.mk</a> x \u207b\u00b9' u, <a>measurable_prod_mk_left</a> hum, ?_\u27e9", [{"full_name": "Prod.mk", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [481, 3], "def_end_pos": [481, 5]}, {"full_name": "measurable_prod_mk_left", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [752, 9], "def_end_pos": [752, 32]}]], "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\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 : SFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\nhs : \u03bc s \u2260 0\nu : Set (\u03b1 \u00d7 \u03b2)\nhum : MeasurableSet u\nhu : s \u00d7\u02e2 t =\u1da0[ae (\u03bc.prod \u03bd)] u\nx : \u03b1\nhxs : x \u2208 s\nhx : Prod.mk x \u207b\u00b9' s \u00d7\u02e2 t =\u1da0[ae \u03bd] Prod.mk x \u207b\u00b9' u\n\u22a2 NullMeasurableSet t \u03bd", "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\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 : SFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\nhs : \u03bc s \u2260 0\nu : Set (\u03b1 \u00d7 \u03b2)\nhum : MeasurableSet u\nhu : s \u00d7\u02e2 t =\u1da0[ae (\u03bc.prod \u03bd)] u\nx : \u03b1\nhxs : x \u2208 s\nhx : Prod.mk x \u207b\u00b9' s \u00d7\u02e2 t =\u1da0[ae \u03bd] Prod.mk x \u207b\u00b9' u\n\u22a2 t =\u1da0[ae \u03bd] Prod.mk x \u207b\u00b9' u"}, {"tactic": "rwa [mk_preimage_prod_right hxs] at hx", "annotated_tactic": ["rwa [<a>mk_preimage_prod_right</a> hxs] at hx", [{"full_name": "Set.mk_preimage_prod_right", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [231, 9], "def_end_pos": [231, 31]}]], "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\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 : SFinite \u03bd\ns : Set \u03b1\nt : Set \u03b2\nhs : \u03bc s \u2260 0\nu : Set (\u03b1 \u00d7 \u03b2)\nhum : MeasurableSet u\nhu : s \u00d7\u02e2 t =\u1da0[ae (\u03bc.prod \u03bd)] u\nx : \u03b1\nhxs : x \u2208 s\nhx : Prod.mk x \u207b\u00b9' s \u00d7\u02e2 t =\u1da0[ae \u03bd] Prod.mk x \u207b\u00b9' u\n\u22a2 t =\u1da0[ae \u03bd] Prod.mk x \u207b\u00b9' u", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GroupWithZero/WithZero.lean", "full_name": "WithZero.map'_map'", "start": [135, 1], "end": [136, 22], "traced_tactics": [{"tactic": "induction x <;> rfl", "annotated_tactic": ["induction x <;> rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : MulOneClass \u03b2\ninst\u271d : MulOneClass \u03b3\nf : \u03b1 \u2192* \u03b2\ng : \u03b2 \u2192* \u03b3\nx : WithZero \u03b1\n\u22a2 (map' g) ((map' f) x) = (map' (g.comp f)) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Roots.lean", "full_name": "Polynomial.prod_multiset_X_sub_C_dvd", "start": [658, 1], "end": [665, 62], "traced_tactics": [{"tactic": "rw [\u2190 map_dvd_map _ (IsFractionRing.injective R <| FractionRing R) monic_prod_multiset_X_sub_C]", "annotated_tactic": ["rw [\u2190 <a>map_dvd_map</a> _ (<a>IsFractionRing.injective</a> R <| <a>FractionRing</a> R) <a>monic_prod_multiset_X_sub_C</a>]", [{"full_name": "Polynomial.map_dvd_map", "def_path": "Mathlib/Algebra/Polynomial/Div.lean", "def_pos": [424, 9], "def_end_pos": [424, 20]}, {"full_name": "IsFractionRing.injective", "def_path": "Mathlib/RingTheory/Localization/FractionRing.lean", "def_pos": [82, 19], "def_end_pos": [82, 28]}, {"full_name": "FractionRing", "def_path": "Mathlib/RingTheory/Localization/FractionRing.lean", "def_pos": [289, 8], "def_end_pos": [289, 20]}, {"full_name": "Polynomial.monic_prod_multiset_X_sub_C", "def_path": "Mathlib/Algebra/Polynomial/Roots.lean", "def_pos": [646, 9], "def_end_pos": [646, 36]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q p : R[X]\n\u22a2 (Multiset.map (fun a => X - C a) p.roots).prod \u2223 p", "state_after": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q p : R[X]\n\u22a2 map (algebraMap R (FractionRing R)) (Multiset.map (fun a => X - C a) p.roots).prod \u2223\n    map (algebraMap R (FractionRing R)) p"}, {"tactic": "rw [prod_multiset_root_eq_finset_root, Polynomial.map_prod]", "annotated_tactic": ["rw [<a>prod_multiset_root_eq_finset_root</a>, <a>Polynomial.map_prod</a>]", [{"full_name": "Polynomial.prod_multiset_root_eq_finset_root", "def_path": "Mathlib/Algebra/Polynomial/Roots.lean", "def_pos": [651, 9], "def_end_pos": [651, 42]}, {"full_name": "Polynomial.map_prod", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [1256, 19], "def_end_pos": [1256, 27]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q p : R[X]\n\u22a2 map (algebraMap R (FractionRing R)) (Multiset.map (fun a => X - C a) p.roots).prod \u2223\n    map (algebraMap R (FractionRing R)) p", "state_after": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q p : R[X]\n\u22a2 \u220f i \u2208 p.roots.toFinset, map (algebraMap R (FractionRing R)) ((X - C i) ^ rootMultiplicity i p) \u2223\n    map (algebraMap R (FractionRing R)) p"}, {"tactic": "refine Finset.prod_dvd_of_coprime (fun a _ b _ h => ?_) fun a _ => ?_", "annotated_tactic": ["refine <a>Finset.prod_dvd_of_coprime</a> (fun a _ b _ h => ?_) fun a _ => ?_", [{"full_name": "Finset.prod_dvd_of_coprime", "def_path": "Mathlib/RingTheory/Coprime/Lemmas.lean", "def_pos": [94, 9], "def_end_pos": [94, 35]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q p : R[X]\n\u22a2 \u220f i \u2208 p.roots.toFinset, map (algebraMap R (FractionRing R)) ((X - C i) ^ rootMultiplicity i p) \u2223\n    map (algebraMap R (FractionRing R)) p", "state_after": "case refine_1\nR : Type u\nS : Type v\nT : Type w\na\u271d b\u271d : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q p : R[X]\na : R\nx\u271d\u00b9 : a \u2208 \u2191p.roots.toFinset\nb : R\nx\u271d : b \u2208 \u2191p.roots.toFinset\nh : a \u2260 b\n\u22a2 (IsCoprime on fun i => map (algebraMap R (FractionRing R)) ((X - C i) ^ rootMultiplicity i p)) a b\n\ncase refine_2\nR : Type u\nS : Type v\nT : Type w\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q p : R[X]\na : R\nx\u271d : a \u2208 p.roots.toFinset\n\u22a2 map (algebraMap R (FractionRing R)) ((X - C a) ^ rootMultiplicity a p) \u2223 map (algebraMap R (FractionRing R)) p"}, {"tactic": "simp_rw [Polynomial.map_pow, Polynomial.map_sub, map_C, map_X]", "annotated_tactic": ["simp_rw [<a>Polynomial.map_pow</a>, <a>Polynomial.map_sub</a>, <a>map_C</a>, <a>map_X</a>]", [{"full_name": "Polynomial.map_pow", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [936, 19], "def_end_pos": [936, 26]}, {"full_name": "Polynomial.map_sub", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [1284, 19], "def_end_pos": [1284, 26]}, {"full_name": "Polynomial.map_C", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [718, 9], "def_end_pos": [718, 14]}, {"full_name": "Polynomial.map_X", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [723, 9], "def_end_pos": [723, 14]}]], "state_before": "case refine_1\nR : Type u\nS : Type v\nT : Type w\na\u271d b\u271d : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q p : R[X]\na : R\nx\u271d\u00b9 : a \u2208 \u2191p.roots.toFinset\nb : R\nx\u271d : b \u2208 \u2191p.roots.toFinset\nh : a \u2260 b\n\u22a2 (IsCoprime on fun i => map (algebraMap R (FractionRing R)) ((X - C i) ^ rootMultiplicity i p)) a b", "state_after": "case refine_1\nR : Type u\nS : Type v\nT : Type w\na\u271d b\u271d : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q p : R[X]\na : R\nx\u271d\u00b9 : a \u2208 \u2191p.roots.toFinset\nb : R\nx\u271d : b \u2208 \u2191p.roots.toFinset\nh : a \u2260 b\n\u22a2 (IsCoprime on fun i => (X - C ((algebraMap R (FractionRing R)) i)) ^ rootMultiplicity i p) a b"}, {"tactic": "exact (pairwise_coprime_X_sub_C (IsFractionRing.injective R <| FractionRing R) h).pow", "annotated_tactic": ["exact (<a>pairwise_coprime_X_sub_C</a> (<a>IsFractionRing.injective</a> R <| <a>FractionRing</a> R) h).<a>pow</a>", [{"full_name": "Polynomial.pairwise_coprime_X_sub_C", "def_path": "Mathlib/Algebra/Polynomial/RingDivision.lean", "def_pos": [763, 9], "def_end_pos": [763, 33]}, {"full_name": "IsFractionRing.injective", "def_path": "Mathlib/RingTheory/Localization/FractionRing.lean", "def_pos": [82, 19], "def_end_pos": [82, 28]}, {"full_name": "FractionRing", "def_path": "Mathlib/RingTheory/Localization/FractionRing.lean", "def_pos": [289, 8], "def_end_pos": [289, 20]}, {"full_name": "IsCoprime.pow", "def_path": "Mathlib/RingTheory/Coprime/Lemmas.lean", "def_pos": [209, 9], "def_end_pos": [209, 22]}]], "state_before": "case refine_1\nR : Type u\nS : Type v\nT : Type w\na\u271d b\u271d : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q p : R[X]\na : R\nx\u271d\u00b9 : a \u2208 \u2191p.roots.toFinset\nb : R\nx\u271d : b \u2208 \u2191p.roots.toFinset\nh : a \u2260 b\n\u22a2 (IsCoprime on fun i => (X - C ((algebraMap R (FractionRing R)) i)) ^ rootMultiplicity i p) a b", "state_after": "no goals"}, {"tactic": "exact Polynomial.map_dvd _ (pow_rootMultiplicity_dvd p a)", "annotated_tactic": ["exact <a>Polynomial.map_dvd</a> _ (<a>pow_rootMultiplicity_dvd</a> p a)", [{"full_name": "Polynomial.map_dvd", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [798, 9], "def_end_pos": [798, 16]}, {"full_name": "Polynomial.pow_rootMultiplicity_dvd", "def_path": "Mathlib/Algebra/Polynomial/Div.lean", "def_pos": [570, 9], "def_end_pos": [570, 33]}]], "state_before": "case refine_2\nR : Type u\nS : Type v\nT : Type w\na\u271d b : R\nn : \u2115\ninst\u271d\u00b9 : CommRing R\ninst\u271d : IsDomain R\np\u271d q p : R[X]\na : R\nx\u271d : a \u2208 p.roots.toFinset\n\u22a2 map (algebraMap R (FractionRing R)) ((X - C a) ^ rootMultiplicity a p) \u2223 map (algebraMap R (FractionRing R)) p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Kernel/Disintegration/Density.lean", "full_name": "ProbabilityTheory.kernel.densityProcess_def", "start": [97, 1], "end": [100, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Sigma.lean", "full_name": "List.keys_nil", "start": [46, 1], "end": [47, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Seq/Computation.lean", "full_name": "Computation.map_think", "start": [689, 1], "end": [690, 78], "traced_tactics": [{"tactic": "apply Subtype.eq", "annotated_tactic": ["apply <a>Subtype.eq</a>", [{"full_name": "Subtype.eq", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1110, 19], "def_end_pos": [1110, 21]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, s n = some a \u2192 s (n + 1) = some a\n\u22a2 map f (think \u27e8s, al\u27e9) = (map f \u27e8s, al\u27e9).think", "state_after": "case a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, s n = some a \u2192 s (n + 1) = some a\n\u22a2 \u2191(map f (think \u27e8s, al\u27e9)) = \u2191(map f \u27e8s, al\u27e9).think"}, {"tactic": "dsimp [think, map]", "annotated_tactic": ["dsimp [<a>think</a>, <a>map</a>]", [{"full_name": "Computation.think", "def_path": "Mathlib/Data/Seq/Computation.lean", "def_pos": [53, 5], "def_end_pos": [53, 10]}, {"full_name": "Computation.map", "def_path": "Mathlib/Data/Seq/Computation.lean", "def_pos": [639, 5], "def_end_pos": [639, 8]}]], "state_before": "case a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, s n = some a \u2192 s (n + 1) = some a\n\u22a2 \u2191(map f (think \u27e8s, al\u27e9)) = \u2191(map f \u27e8s, al\u27e9).think", "state_after": "case a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, s n = some a \u2192 s (n + 1) = some a\n\u22a2 Stream'.map (fun o => Option.rec none (fun val => some (f val)) o) (Stream'.cons none s) =\n    Stream'.cons none (Stream'.map (fun o => Option.rec none (fun val => some (f val)) o) s)"}, {"tactic": "rw [Stream'.map_cons]", "annotated_tactic": ["rw [<a>Stream'.map_cons</a>]", [{"full_name": "Stream'.map_cons", "def_path": "Mathlib/Data/Stream/Init.lean", "def_pos": [166, 9], "def_end_pos": [166, 17]}]], "state_before": "case a\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nf : \u03b1 \u2192 \u03b2\ns : Stream' (Option \u03b1)\nal : \u2200 \u2983n : \u2115\u2984 \u2983a : \u03b1\u2984, s n = some a \u2192 s (n + 1) = some a\n\u22a2 Stream'.map (fun o => Option.rec none (fun val => some (f val)) o) (Stream'.cons none s) =\n    Stream'.cons none (Stream'.map (fun o => Option.rec none (fun val => some (f val)) o) s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "full_name": "UniqueFactorizationMonoid.factors_one", "start": [503, 1], "end": [508, 33], "traced_tactics": [{"tactic": "nontriviality \u03b1 using factors", "annotated_tactic": ["nontriviality \u03b1 using <a>factors</a>", [{"full_name": "UniqueFactorizationMonoid.factors", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [470, 19], "def_end_pos": [470, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\n\u22a2 factors 1 = 0", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na\u271d : Nontrivial \u03b1\n\u22a2 factors 1 = 0"}, {"tactic": "rw [\u2190 Multiset.rel_zero_right]", "annotated_tactic": ["rw [\u2190 <a>Multiset.rel_zero_right</a>]", [{"full_name": "Multiset.rel_zero_right", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2822, 9], "def_end_pos": [2822, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na\u271d : Nontrivial \u03b1\n\u22a2 factors 1 = 0", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na\u271d : Nontrivial \u03b1\n\u22a2 Multiset.Rel ?m.115145 (factors 1) 0\n\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na\u271d : Nontrivial \u03b1\n\u22a2 Type ?u.115142\n\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na\u271d : Nontrivial \u03b1\n\u22a2 \u03b1 \u2192 ?m.115144 \u2192 Prop"}, {"tactic": "refine factors_unique irreducible_of_factor (fun x hx => (Multiset.not_mem_zero x hx).elim) ?_", "annotated_tactic": ["refine <a>factors_unique</a> <a>irreducible_of_factor</a> (fun x hx => (<a>Multiset.not_mem_zero</a> x hx).<a>elim</a>) ?_", [{"full_name": "UniqueFactorizationMonoid.factors_unique", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [278, 9], "def_end_pos": [278, 23]}, {"full_name": "UniqueFactorizationMonoid.irreducible_of_factor", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [498, 9], "def_end_pos": [498, 30]}, {"full_name": "Multiset.not_mem_zero", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [274, 9], "def_end_pos": [274, 21]}, {"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na\u271d : Nontrivial \u03b1\n\u22a2 Multiset.Rel ?m.115145 (factors 1) 0\n\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na\u271d : Nontrivial \u03b1\n\u22a2 Type ?u.115142\n\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na\u271d : Nontrivial \u03b1\n\u22a2 \u03b1 \u2192 ?m.115144 \u2192 Prop", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na\u271d : Nontrivial \u03b1\n\u22a2 (factors 1).prod ~\u1d64 Multiset.prod 0"}, {"tactic": "rw [Multiset.prod_zero]", "annotated_tactic": ["rw [<a>Multiset.prod_zero</a>]", [{"full_name": "Multiset.prod_zero", "def_path": "Mathlib/Algebra/BigOperators/Group/Multiset.lean", "def_pos": [73, 9], "def_end_pos": [73, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na\u271d : Nontrivial \u03b1\n\u22a2 (factors 1).prod ~\u1d64 Multiset.prod 0", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na\u271d : Nontrivial \u03b1\n\u22a2 (factors 1).prod ~\u1d64 1"}, {"tactic": "exact factors_prod one_ne_zero", "annotated_tactic": ["exact <a>factors_prod</a> <a>one_ne_zero</a>", [{"full_name": "UniqueFactorizationMonoid.factors_prod", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [474, 9], "def_end_pos": [474, 21]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [58, 15], "def_end_pos": [58, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : UniqueFactorizationMonoid \u03b1\na\u271d : Nontrivial \u03b1\n\u22a2 (factors 1).prod ~\u1d64 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/List/Lemmas.lean", "full_name": "List.eraseP_nil", "start": [483, 9], "end": [483, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/Irrational.lean", "full_name": "Int.not_irrational", "start": [218, 1], "end": [218, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Factors.lean", "full_name": "Nat.eq_prime_pow_of_unique_prime_dvd", "start": [197, 1], "end": [201, 89], "traced_tactics": [{"tactic": "set k := n.factors.length", "annotated_tactic": ["set k := n.factors.length", []], "state_before": "n p : \u2115\nhpos : n \u2260 0\nh : \u2200 {d : \u2115}, Prime d \u2192 d \u2223 n \u2192 d = p\n\u22a2 n = p ^ n.factors.length", "state_after": "n p : \u2115\nhpos : n \u2260 0\nh : \u2200 {d : \u2115}, Prime d \u2192 d \u2223 n \u2192 d = p\nk : \u2115 := n.factors.length\n\u22a2 n = p ^ k"}, {"tactic": "rw [\u2190 prod_factors hpos, \u2190 prod_replicate k p,\n  eq_replicate_of_mem fun d hd => h (prime_of_mem_factors hd) (dvd_of_mem_factors hd)]", "annotated_tactic": ["rw [\u2190 <a>prod_factors</a> hpos, \u2190 <a>prod_replicate</a> k p,\n    <a>eq_replicate_of_mem</a> fun d hd => h (<a>prime_of_mem_factors</a> hd) (<a>dvd_of_mem_factors</a> hd)]", [{"full_name": "Nat.prod_factors", "def_path": "Mathlib/Data/Nat/Factors.lean", "def_pos": [72, 9], "def_end_pos": [72, 21]}, {"full_name": "List.prod_replicate", "def_path": "Mathlib/Algebra/BigOperators/Group/List.lean", "def_pos": [141, 9], "def_end_pos": [141, 23]}, {"full_name": "List.eq_replicate_of_mem", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [1340, 9], "def_end_pos": [1340, 28]}, {"full_name": "Nat.prime_of_mem_factors", "def_path": "Mathlib/Data/Nat/Factors.lean", "def_pos": [55, 9], "def_end_pos": [55, 29]}, {"full_name": "Nat.dvd_of_mem_factors", "def_path": "Mathlib/Data/Nat/Factors.lean", "def_pos": [152, 9], "def_end_pos": [152, 27]}]], "state_before": "n p : \u2115\nhpos : n \u2260 0\nh : \u2200 {d : \u2115}, Prime d \u2192 d \u2223 n \u2192 d = p\nk : \u2115 := n.factors.length\n\u22a2 n = p ^ k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Monovary.lean", "full_name": "antivary_iff_smul_rearrangement", "start": [366, 1], "end": [369, 83], "traced_tactics": [{"tactic": "rw [monovary_iff_smul_rearrangement]", "annotated_tactic": ["rw [<a>monovary_iff_smul_rearrangement</a>]", [{"full_name": "monovary_iff_smul_rearrangement", "def_path": "Mathlib/Algebra/Order/Monovary.lean", "def_pos": [361, 7], "def_end_pos": [361, 38]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\ns : Set \u03b9\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\n\u22a2 Monovary f (\u21d1OrderDual.toDual \u2218 g) \u2194 \u2200 (i j : \u03b9), f i \u2022 g i + f j \u2022 g j \u2264 f i \u2022 g j + f j \u2022 g i", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\ns : Set \u03b9\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\n\u22a2 (\u2200 (i j : \u03b9),\n      f i \u2022 (\u21d1OrderDual.toDual \u2218 g) j + f j \u2022 (\u21d1OrderDual.toDual \u2218 g) i \u2264\n        f i \u2022 (\u21d1OrderDual.toDual \u2218 g) i + f j \u2022 (\u21d1OrderDual.toDual \u2218 g) j) \u2194\n    \u2200 (i j : \u03b9), f i \u2022 g i + f j \u2022 g j \u2264 f i \u2022 g j + f j \u2022 g i"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrderedRing \u03b1\ninst\u271d\u00b2 : LinearOrderedAddCommGroup \u03b2\ninst\u271d\u00b9 : Module \u03b1 \u03b2\ninst\u271d : OrderedSMul \u03b1 \u03b2\nf : \u03b9 \u2192 \u03b1\ng : \u03b9 \u2192 \u03b2\ns : Set \u03b9\na a\u2081 a\u2082 : \u03b1\nb b\u2081 b\u2082 : \u03b2\n\u22a2 (\u2200 (i j : \u03b9),\n      f i \u2022 (\u21d1OrderDual.toDual \u2218 g) j + f j \u2022 (\u21d1OrderDual.toDual \u2218 g) i \u2264\n        f i \u2022 (\u21d1OrderDual.toDual \u2218 g) i + f j \u2022 (\u21d1OrderDual.toDual \u2218 g) j) \u2194\n    \u2200 (i j : \u03b9), f i \u2022 g i + f j \u2022 g j \u2264 f i \u2022 g j + f j \u2022 g i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/PartialHomeomorph.lean", "full_name": "PartialHomeomorph.IsImage.iff_preimage_eq", "start": [565, 1], "end": [566, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.ProgMeasurable.adapted_stoppedProcess", "start": [844, 1], "end": [846, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Cardinal/ENat.lean", "full_name": "Cardinal.ofNat_lt_ofENat", "start": [76, 1], "end": [77, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.disjoint_atBot_principal_Ioi", "start": [79, 1], "end": [80, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Cycle.lean", "full_name": "Cycle.prev_next", "start": [875, 8], "end": [877, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/DFinsupp/Order.lean", "full_name": "DFinsupp.subset_support_tsub", "start": [302, 1], "end": [303, 55], "traced_tactics": [{"tactic": "simp (config := { contextual := true }) [subset_iff]", "annotated_tactic": ["simp (config := { contextual := <a>true</a> }) [<a>subset_iff</a>]", [{"full_name": "Bool.true", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [571, 5], "def_end_pos": [571, 9]}, {"full_name": "Finset.subset_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [371, 9], "def_end_pos": [371, 19]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : (i : \u03b9) \u2192 CanonicallyOrderedAddCommMonoid (\u03b1 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Sub (\u03b1 i)\ninst\u271d\u00b2 : \u2200 (i : \u03b9), OrderedSub (\u03b1 i)\nf g : \u03a0\u2080 (i : \u03b9), \u03b1 i\ni : \u03b9\na b : \u03b1 i\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 (x : \u03b1 i) \u2192 Decidable (x \u2260 0)\n\u22a2 f.support \\ g.support \u2286 (f - g).support", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Eigenspace/Basic.lean", "full_name": "Module.End.exp_ne_zero_of_hasGenEigenvalue", "start": [209, 1], "end": [212, 27], "traced_tactics": [{"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "K R : Type v\nV M : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End R M\n\u03bc : R\nk : \u2115\nh : f.HasGenEigenvalue \u03bc k\n\u22a2 k \u2260 0", "state_after": "K R : Type v\nV M : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End R M\n\u03bc : R\nh : f.HasGenEigenvalue \u03bc 0\n\u22a2 False"}, {"tactic": "exact h LinearMap.ker_id", "annotated_tactic": ["exact h <a>LinearMap.ker_id</a>", [{"full_name": "LinearMap.ker_id", "def_path": "Mathlib/Algebra/Module/Submodule/Ker.lean", "def_pos": [70, 9], "def_end_pos": [70, 15]}]], "state_before": "K R : Type v\nV M : Type w\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : AddCommGroup M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : End R M\n\u03bc : R\nh : f.HasGenEigenvalue \u03bc 0\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Rel.lean", "full_name": "Rel.image_eq_dom_of_codomain_subset", "start": [275, 1], "end": [283, 25], "traced_tactics": [{"tactic": "rw [\u2190 preimage_univ]", "annotated_tactic": ["rw [\u2190 <a>preimage_univ</a>]", [{"full_name": "Rel.preimage_univ", "def_path": "Mathlib/Data/Rel.lean", "def_pos": [258, 9], "def_end_pos": [258, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b2\nh : r.codom \u2286 s\n\u22a2 r.preimage s = r.dom", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b2\nh : r.codom \u2286 s\n\u22a2 r.preimage s = r.preimage Set.univ"}, {"tactic": "apply Set.eq_of_subset_of_subset", "annotated_tactic": ["apply <a>Set.eq_of_subset_of_subset</a>", [{"full_name": "Set.eq_of_subset_of_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [356, 9], "def_end_pos": [356, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b2\nh : r.codom \u2286 s\n\u22a2 r.preimage s = r.preimage Set.univ", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b2\nh : r.codom \u2286 s\n\u22a2 r.preimage s \u2286 r.preimage Set.univ\n\ncase a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b2\nh : r.codom \u2286 s\n\u22a2 r.preimage Set.univ \u2286 r.preimage s"}, {"tactic": "exact image_subset _ (Set.subset_univ _)", "annotated_tactic": ["exact <a>image_subset</a> _ (<a>Set.subset_univ</a> _)", [{"full_name": "Rel.image_subset", "def_path": "Mathlib/Data/Rel.lean", "def_pos": [173, 9], "def_end_pos": [173, 21]}, {"full_name": "Set.subset_univ", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [657, 9], "def_end_pos": [657, 20]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b2\nh : r.codom \u2286 s\n\u22a2 r.preimage s \u2286 r.preimage Set.univ", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b2\nh : r.codom \u2286 s\n\u22a2 r.preimage Set.univ \u2286 r.preimage s", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b2\nh : r.codom \u2286 s\nx : \u03b1\nhx : x \u2208 r.preimage Set.univ\n\u22a2 x \u2208 r.preimage s"}, {"tactic": "simp only [mem_preimage, Set.mem_univ, true_and] at hx", "annotated_tactic": ["simp only [<a>mem_preimage</a>, <a>Set.mem_univ</a>, <a>true_and</a>] at hx", [{"full_name": "Rel.mem_preimage", "def_path": "Mathlib/Data/Rel.lean", "def_pos": [230, 9], "def_end_pos": [230, 21]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [82, 29], "def_end_pos": [82, 37]}, {"full_name": "true_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [105, 17], "def_end_pos": [105, 25]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b2\nh : r.codom \u2286 s\nx : \u03b1\nhx : x \u2208 r.preimage Set.univ\n\u22a2 x \u2208 r.preimage s", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b2\nh : r.codom \u2286 s\nx : \u03b1\nhx : \u2203 y, r x y\n\u22a2 x \u2208 r.preimage s"}, {"tactic": "rcases hx with \u27e8y, ryx\u27e9", "annotated_tactic": ["rcases hx with \u27e8y, ryx\u27e9", []], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b2\nh : r.codom \u2286 s\nx : \u03b1\nhx : \u2203 y, r x y\n\u22a2 x \u2208 r.preimage s", "state_after": "case a.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b2\nh : r.codom \u2286 s\nx : \u03b1\ny : \u03b2\nryx : r x y\n\u22a2 x \u2208 r.preimage s"}, {"tactic": "have hy : y \u2208 s := h \u27e8x, ryx\u27e9", "annotated_tactic": ["have hy : y \u2208 s := h \u27e8x, ryx\u27e9", []], "state_before": "case a.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b2\nh : r.codom \u2286 s\nx : \u03b1\ny : \u03b2\nryx : r x y\n\u22a2 x \u2208 r.preimage s", "state_after": "case a.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b2\nh : r.codom \u2286 s\nx : \u03b1\ny : \u03b2\nryx : r x y\nhy : y \u2208 s\n\u22a2 x \u2208 r.preimage s"}, {"tactic": "exact \u27e8y, \u27e8hy, ryx\u27e9\u27e9", "annotated_tactic": ["exact \u27e8y, \u27e8hy, ryx\u27e9\u27e9", []], "state_before": "case a.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nr : Rel \u03b1 \u03b2\ns : Set \u03b2\nh : r.codom \u2286 s\nx : \u03b1\ny : \u03b2\nryx : r x y\nhy : y \u2208 s\n\u22a2 x \u2208 r.preimage s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "full_name": "Ordinal.lift_umax'", "start": [683, 1], "end": [684, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/IsTensorProduct.lean", "full_name": "Algebra.lift_algHom_comp_left", "start": [470, 1], "end": [473, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Sublattice.lean", "full_name": "Sublattice.mem_map_of_mem", "start": [226, 1], "end": [226, 96], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/Extension.lean", "full_name": "IntermediateField.nonempty_algHom_of_adjoin_splits", "start": [169, 1], "end": [170, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.lintegral_nnnorm_zero", "start": [84, 1], "end": [84, 73], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": 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"29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean", "full_name": "tprod_setProd_singleton_right", "start": [50, 1], "end": [53, 91], "traced_tactics": [{"tactic": "rw [tprod_congr_set_coe _ Set.prod_singleton, tprod_image _ (Prod.mk.inj_right c).injOn]", "annotated_tactic": ["rw [<a>tprod_congr_set_coe</a> _ <a>Set.prod_singleton</a>, <a>tprod_image</a> _ (<a>Prod.mk.inj_right</a> c).<a>injOn</a>]", [{"full_name": "tprod_congr_set_coe", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [391, 9], "def_end_pos": [391, 28]}, {"full_name": "Set.prod_singleton", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [117, 9], "def_end_pos": [117, 23]}, {"full_name": "tprod_image", "def_path": "Mathlib/Topology/Algebra/InfiniteSum/Basic.lean", "def_pos": [538, 9], "def_end_pos": [538, 20]}, {"full_name": "Prod.mk.inj_right", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [112, 9], "def_end_pos": [112, 21]}, {"full_name": "Function.Injective.injOn", "def_path": "Mathlib/Data/Set/Function.lean", "def_pos": [674, 7], "def_end_pos": [674, 38]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : CommMonoid \u03b1\ninst\u271d : TopologicalSpace \u03b1\ns : Set \u03b2\nc : \u03b3\nf : \u03b2 \u00d7 \u03b3 \u2192 \u03b1\n\u22a2 \u220f' (x : \u2191(s \u00d7\u02e2 {c})), f \u2191x = \u220f' (b : \u2191s), f (\u2191b, c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/RelClasses.lean", "full_name": "Set.not_bounded_iff", "start": [552, 1], "end": [553, 88], "traced_tactics": [{"tactic": "simp only [Bounded, Unbounded, not_forall, not_exists, exists_prop, not_and, not_not]", "annotated_tactic": ["simp only [<a>Bounded</a>, <a>Unbounded</a>, <a>not_forall</a>, <a>not_exists</a>, <a>exists_prop</a>, <a>not_and</a>, <a>not_not</a>]", [{"full_name": "Set.Bounded", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [547, 5], "def_end_pos": [547, 12]}, {"full_name": "Set.Unbounded", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [542, 5], "def_end_pos": [542, 14]}, {"full_name": "Classical.not_forall", "def_path": ".lake/packages/lean4/src/lean/Init/Classical.lean", "def_pos": [137, 21], "def_end_pos": [137, 31]}, {"full_name": "not_exists", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [254, 17], "def_end_pos": [254, 27]}, {"full_name": "exists_prop", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [307, 17], "def_end_pos": [307, 28]}, {"full_name": "not_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [116, 17], "def_end_pos": [116, 24]}, {"full_name": "Classical.not_not", "def_path": ".lake/packages/lean4/src/lean/Init/Classical.lean", "def_pos": [135, 17], "def_end_pos": [135, 24]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nr\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop\ns\u271d : \u03b2 \u2192 \u03b2 \u2192 Prop\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b1\n\u22a2 \u00acBounded r s \u2194 Unbounded r s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/Image.lean", "full_name": "Set.image_subtype_val_Ioi_Iic", "start": [325, 1], "end": [327, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/PiTensorProduct.lean", "full_name": "PiTensorProduct.reindex_symm", "start": [787, 1], "end": [792, 29], "traced_tactics": [{"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\ne : \u03b9 \u2243 \u03b9\u2082\n\u22a2 (reindex R (fun x => M) e).symm = reindex R (fun x => M) e.symm", "state_after": "case h\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\ne : \u03b9 \u2243 \u03b9\u2082\nx : \u2a02[R] (i : \u03b9\u2082), M\n\u22a2 (reindex R (fun x => M) e).symm x = (reindex R (fun x => M) e.symm) x"}, {"tactic": "simp only [reindex, domDomCongrLinearEquiv', LinearEquiv.coe_symm_mk, LinearEquiv.coe_mk,\n  LinearEquiv.ofLinear_symm_apply, Equiv.symm_symm_apply, LinearEquiv.ofLinear_apply,\n  Equiv.piCongrLeft'_symm]", "annotated_tactic": ["simp only [<a>reindex</a>, <a>domDomCongrLinearEquiv'</a>, <a>LinearEquiv.coe_symm_mk</a>, <a>LinearEquiv.coe_mk</a>,\n    <a>LinearEquiv.ofLinear_symm_apply</a>, <a>Equiv.symm_symm_apply</a>, <a>LinearEquiv.ofLinear_apply</a>,\n    <a>Equiv.piCongrLeft'_symm</a>]", [{"full_name": "PiTensorProduct.reindex", "def_path": "Mathlib/LinearAlgebra/PiTensorProduct.lean", "def_pos": [720, 5], "def_end_pos": [720, 12]}, {"full_name": "MultilinearMap.domDomCongrLinearEquiv'", "def_path": "Mathlib/LinearAlgebra/Multilinear/Basic.lean", "def_pos": [935, 5], "def_end_pos": [935, 28]}, {"full_name": "LinearEquiv.coe_symm_mk", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [546, 9], "def_end_pos": [546, 20]}, {"full_name": "LinearEquiv.coe_mk", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [201, 9], "def_end_pos": [201, 15]}, {"full_name": "LinearEquiv.ofLinear_symm_apply", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [1079, 9], "def_end_pos": [1079, 28]}, {"full_name": "Equiv.symm_symm_apply", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [313, 32], "def_end_pos": [313, 47]}, {"full_name": "LinearEquiv.ofLinear_apply", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [1074, 9], "def_end_pos": [1074, 23]}, {"full_name": "Equiv.piCongrLeft'_symm", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1847, 9], "def_end_pos": [1847, 26]}]], "state_before": "case h\n\u03b9 : Type u_1\n\u03b9\u2082 : Type u_2\n\u03b9\u2083 : Type u_3\nR : Type u_4\ninst\u271d\u2077 : CommSemiring R\nR\u2081 : Type u_5\nR\u2082 : Type u_6\ns : \u03b9 \u2192 Type u_7\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (s i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module R (s i)\nM : Type u_8\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nE : Type u_9\ninst\u271d\u00b2 : AddCommMonoid E\ninst\u271d\u00b9 : Module R E\nF : Type u_10\ninst\u271d : AddCommMonoid F\ne : \u03b9 \u2243 \u03b9\u2082\nx : \u2a02[R] (i : \u03b9\u2082), M\n\u22a2 (reindex R (fun x => M) e).symm x = (reindex R (fun x => M) e.symm) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "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": [345, 9], "def_end_pos": [345, 44]}, {"full_name": "measurable_id''", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [227, 9], "def_end_pos": [227, 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 \u03bc\n\u22a2 Integrable (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 : NormedAddCommGroup F\nhm : m \u2264 m\u03a9\nhf : Integrable (fun x => f x.2) (Measure.map (fun \u03c9 => (id \u03c9, \u03c9)) \u03bc)\n\u22a2 Integrable (fun x => f x.2) (Measure.map (fun \u03c9 => (id \u03c9, id \u03c9)) \u03bc)"}, {"tactic": "simp_rw [id] at hf \u22a2", "annotated_tactic": ["simp_rw [<a>id</a>] at hf \u22a2", [{"full_name": "id", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "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) (Measure.map (fun \u03c9 => (id \u03c9, \u03c9)) \u03bc)\n\u22a2 Integrable (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 : NormedAddCommGroup F\nhm : m \u2264 m\u03a9\nhf : Integrable (fun x => f x.2) (Measure.map (fun \u03c9 => (\u03c9, \u03c9)) \u03bc)\n\u22a2 Integrable (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 : NormedAddCommGroup F\nhm : m \u2264 m\u03a9\nhf : Integrable (fun x => f x.2) (Measure.map (fun \u03c9 => (\u03c9, \u03c9)) \u03bc)\n\u22a2 Integrable (fun x => f x.2) (Measure.map (fun \u03c9 => (\u03c9, \u03c9)) \u03bc)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fin/Basic.lean", "full_name": "Fin.rev_pred", "start": [1239, 1], "end": [1241, 64], "traced_tactics": [{"tactic": "rw [\u2190 castSucc_inj, castSucc_castPred, \u2190 rev_succ, succ_pred]", "annotated_tactic": ["rw [\u2190 <a>castSucc_inj</a>, <a>castSucc_castPred</a>, \u2190 <a>rev_succ</a>, <a>succ_pred</a>]", [{"full_name": "Fin.castSucc_inj", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean", "def_pos": [400, 9], "def_end_pos": [400, 21]}, {"full_name": "Fin.castSucc_castPred", "def_path": "Mathlib/Data/Fin/Basic.lean", "def_pos": [1182, 9], "def_end_pos": [1182, 26]}, {"full_name": "Fin.rev_succ", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean", "def_pos": [515, 9], "def_end_pos": [515, 17]}, {"full_name": "Fin.succ_pred", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean", "def_pos": [521, 17], "def_end_pos": [521, 26]}]], "state_before": "n m : \u2115\ni : Fin (n + 1)\nh : i \u2260 0\nh' : optParam (i.rev \u2260 last n) \u22ef\n\u22a2 (i.pred h).rev = i.rev.castPred h'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Sheaves/Functors.lean", "full_name": "TopCat.Sheaf.pushforward_obj_val", "start": [76, 1], "end": [77, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean", "full_name": "ProjectiveSpectrum.zeroLocus_univ", "start": [226, 1], "end": [227, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/RCLike/Basic.lean", "full_name": "RCLike.im_sq_le_normSq", "start": [503, 1], "end": [504, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/CharP/ExpChar.lean", "full_name": "iterateFrobenius_mul_apply", "start": [328, 1], "end": [330, 55], "traced_tactics": [{"tactic": "simp_rw [coe_iterateFrobenius, Function.iterate_mul]", "annotated_tactic": ["simp_rw [<a>coe_iterateFrobenius</a>, <a>Function.iterate_mul</a>]", [{"full_name": "coe_iterateFrobenius", "def_path": "Mathlib/Algebra/CharP/ExpChar.lean", "def_pos": [303, 9], "def_end_pos": [303, 29]}, {"full_name": "Function.iterate_mul", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [91, 9], "def_end_pos": [91, 20]}]], "state_before": "R : Type u\ninst\u271d\u00b3 : CommSemiring R\nS : Type u_1\ninst\u271d\u00b2 : CommSemiring S\nf : R \u2192* S\ng : R \u2192+* S\np m n : \u2115\ninst\u271d\u00b9 : ExpChar R p\ninst\u271d : ExpChar S p\nx y : R\n\u22a2 (iterateFrobenius R p (m * n)) x = (\u21d1(iterateFrobenius R p m))^[n] x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Bounded.lean", "full_name": "Metric.ediam_of_unbounded", "start": [498, 1], "end": [498, 101], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "full_name": "Real.cos_pos_of_mem_Ioo", "start": [521, 1], "end": [522, 85], "traced_tactics": [{"tactic": "linarith [hx.1]", "annotated_tactic": ["linarith [hx.1]", []], "state_before": "x : \u211d\nhx : x \u2208 Ioo (-(\u03c0 / 2)) (\u03c0 / 2)\n\u22a2 0 < x + \u03c0 / 2", "state_after": "no goals"}, {"tactic": "linarith [hx.2]", "annotated_tactic": ["linarith [hx.2]", []], "state_before": "x : \u211d\nhx : x \u2208 Ioo (-(\u03c0 / 2)) (\u03c0 / 2)\n\u22a2 x + \u03c0 / 2 < \u03c0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/AdicCompletion/Basic.lean", "full_name": "AdicCompletion.AdicCauchySequence.sub_apply", "start": [384, 1], "end": [385, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Equiv/Defs.lean", "full_name": "Equiv.forall_congr_right", "start": [868, 1], "end": [869, 57], "traced_tactics": [{"tactic": "simpa using h (e.symm a)", "annotated_tactic": ["simpa using h (e.symm a)", []], "state_before": "\u03b1 : Sort u\n\u03b2 : Sort v\n\u03b3 : Sort w\np : \u03b1 \u2192 Prop\nq : \u03b2 \u2192 Prop\ne : \u03b1 \u2243 \u03b2\nh : \u2200 (a : \u03b1), q (e a)\na : \u03b2\n\u22a2 q a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Int/CardIntervalMod.lean", "full_name": "Nat.count_modEq_card", "start": [127, 1], "end": [140, 58], "traced_tactics": [{"tactic": "have hr' : 0 < (r : \u211a) := by positivity", "annotated_tactic": ["have hr' : 0 < (r : \u211a) := by positivity", []], "state_before": "a b r : \u2115\nhr : 0 < r\nv : \u2115\n\u22a2 count (fun x => x \u2261 v [MOD r]) b = b / r + if v % r < b % r then 1 else 0", "state_after": "a b r : \u2115\nhr : 0 < r\nv : \u2115\nhr' : 0 < \u2191r\n\u22a2 count (fun x => x \u2261 v [MOD r]) b = b / r + if v % r < b % r then 1 else 0"}, {"tactic": "rw [\u2190 ofNat_inj, count_modEq_card_eq_ceil _ hr, cast_add]", "annotated_tactic": ["rw [\u2190 <a>ofNat_inj</a>, <a>count_modEq_card_eq_ceil</a> _ hr, <a>cast_add</a>]", [{"full_name": "Int.ofNat_inj", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Lemmas.lean", "def_pos": [56, 22], "def_end_pos": [56, 31]}, {"full_name": "Nat.count_modEq_card_eq_ceil", "def_path": "Mathlib/Data/Int/CardIntervalMod.lean", "def_pos": [112, 9], "def_end_pos": [112, 33]}, {"full_name": "Nat.cast_add", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 17]}]], "state_before": "a b r : \u2115\nhr : 0 < r\nv : \u2115\nhr' : 0 < \u2191r\n\u22a2 count (fun x => x \u2261 v [MOD r]) b = b / r + if v % r < b % r then 1 else 0", "state_after": "a b r : \u2115\nhr : 0 < r\nv : \u2115\nhr' : 0 < \u2191r\n\u22a2 \u2308(\u2191b - \u2191(v % r)) / \u2191r\u2309 = \u2191(b / r) + \u2191(if v % r < b % r then 1 else 0)"}, {"tactic": "conv_lhs => rw [\u2190 div_add_mod b r, cast_add, cast_mul, \u2190 add_sub, _root_.add_div,\n  mul_div_cancel_left\u2080 _ hr'.ne', add_comm, Int.ceil_add_nat, add_comm]", "annotated_tactic": ["conv_lhs => rw [\u2190 <a>div_add_mod</a> b r, <a>cast_add</a>, <a>cast_mul</a>, \u2190 <a>add_sub</a>, <a>_root_.add_div</a>,\n    <a>mul_div_cancel_left\u2080</a> _ hr'.ne', <a>add_comm</a>, <a>Int.ceil_add_nat</a>, <a>add_comm</a>]", [{"full_name": "Nat.div_add_mod", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [186, 9], "def_end_pos": [186, 20]}, {"full_name": "Nat.cast_add", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 17]}, {"full_name": "Nat.cast_mul", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [62, 26], "def_end_pos": [62, 34]}, {"full_name": "add_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [473, 3], "def_end_pos": [473, 14]}, {"full_name": "add_div", "def_path": "Mathlib/Algebra/Field/Basic.lean", "def_pos": [29, 9], "def_end_pos": [29, 16]}, {"full_name": "mul_div_cancel_left\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [198, 15], "def_end_pos": [198, 35]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "Int.ceil_add_nat", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [1260, 9], "def_end_pos": [1260, 21]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "a b r : \u2115\nhr : 0 < r\nv : \u2115\nhr' : 0 < \u2191r\n\u22a2 \u2308(\u2191b - \u2191(v % r)) / \u2191r\u2309 = \u2191(b / r) + \u2191(if v % r < b % r then 1 else 0)", "state_after": "a b r : \u2115\nhr : 0 < r\nv : \u2115\nhr' : 0 < \u2191r\n\u22a2 \u2191(b / r) + \u2308(\u2191(b % r) - \u2191(v % r)) / \u2191r\u2309 = \u2191(b / r) + \u2191(if v % r < b % r then 1 else 0)"}, {"tactic": "rw [add_right_inj]", "annotated_tactic": ["rw [<a>add_right_inj</a>]", [{"full_name": "add_right_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}]], "state_before": "a b r : \u2115\nhr : 0 < r\nv : \u2115\nhr' : 0 < \u2191r\n\u22a2 \u2191(b / r) + \u2308(\u2191(b % r) - \u2191(v % r)) / \u2191r\u2309 = \u2191(b / r) + \u2191(if v % r < b % r then 1 else 0)", "state_after": "a b r : \u2115\nhr : 0 < r\nv : \u2115\nhr' : 0 < \u2191r\n\u22a2 \u2308(\u2191(b % r) - \u2191(v % r)) / \u2191r\u2309 = \u2191(if v % r < b % r then 1 else 0)"}, {"tactic": "split_ifs with h", "annotated_tactic": ["split_ifs with h", []], "state_before": "a b r : \u2115\nhr : 0 < r\nv : \u2115\nhr' : 0 < \u2191r\n\u22a2 \u2308(\u2191(b % r) - \u2191(v % r)) / \u2191r\u2309 = \u2191(if v % r < b % r then 1 else 0)", "state_after": "case pos\na b r : \u2115\nhr : 0 < r\nv : \u2115\nhr' : 0 < \u2191r\nh : v % r < b % r\n\u22a2 \u2308(\u2191(b % r) - \u2191(v % r)) / \u2191r\u2309 = \u21911\n\ncase neg\na b r : \u2115\nhr : 0 < r\nv : \u2115\nhr' : 0 < \u2191r\nh : \u00acv % r < b % r\n\u22a2 \u2308(\u2191(b % r) - \u2191(v % r)) / \u2191r\u2309 = \u21910"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "a b r : \u2115\nhr : 0 < r\nv : \u2115\n\u22a2 0 < \u2191r", "state_after": "no goals"}, {"tactic": "rw [\u2190 cast_sub h.le, Int.ceil_eq_iff, div_le_iff hr', lt_div_iff hr', cast_one, Int.cast_one,\n  sub_self, zero_mul, cast_pos, tsub_pos_iff_lt, one_mul, cast_le, tsub_le_iff_right]", "annotated_tactic": ["rw [\u2190 <a>cast_sub</a> h.le, <a>Int.ceil_eq_iff</a>, <a>div_le_iff</a> hr', <a>lt_div_iff</a> hr', <a>cast_one</a>, <a>Int.cast_one</a>,\n      <a>sub_self</a>, <a>zero_mul</a>, <a>cast_pos</a>, <a>tsub_pos_iff_lt</a>, <a>one_mul</a>, <a>cast_le</a>, <a>tsub_le_iff_right</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": "Int.ceil_eq_iff", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [1329, 9], "def_end_pos": [1329, 20]}, {"full_name": "div_le_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [61, 9], "def_end_pos": [61, 19]}, {"full_name": "lt_div_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [82, 9], "def_end_pos": [82, 19]}, {"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [154, 9], "def_end_pos": [154, 17]}, {"full_name": "Int.cast_one", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [79, 9], "def_end_pos": [79, 17]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1003, 30], "def_end_pos": [1003, 38]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}, {"full_name": "Nat.cast_pos", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [92, 9], "def_end_pos": [92, 17]}, {"full_name": "tsub_pos_iff_lt", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [428, 9], "def_end_pos": [428, 24]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "Nat.cast_le", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [124, 9], "def_end_pos": [124, 16]}, {"full_name": "tsub_le_iff_right", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [62, 9], "def_end_pos": [62, 26]}]], "state_before": "case pos\na b r : \u2115\nhr : 0 < r\nv : \u2115\nhr' : 0 < \u2191r\nh : v % r < b % r\n\u22a2 \u2308(\u2191(b % r) - \u2191(v % r)) / \u2191r\u2309 = \u21911", "state_after": "case pos\na b r : \u2115\nhr : 0 < r\nv : \u2115\nhr' : 0 < \u2191r\nh : v % r < b % r\n\u22a2 v % r < b % r \u2227 b % r \u2264 r + v % r"}, {"tactic": "exact \u27e8h, ((mod_lt _ hr).trans_le (by simp)).le\u27e9", "annotated_tactic": ["exact \u27e8h, ((<a>mod_lt</a> _ hr).<a>trans_le</a> (by simp)).<a>le</a>\u27e9", [{"full_name": "Nat.mod_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [142, 9], "def_end_pos": [142, 15]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [143, 7], "def_end_pos": [143, 21]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}]], "state_before": "case pos\na b r : \u2115\nhr : 0 < r\nv : \u2115\nhr' : 0 < \u2191r\nh : v % r < b % r\n\u22a2 v % r < b % r \u2227 b % r \u2264 r + v % r", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "a b r : \u2115\nhr : 0 < r\nv : \u2115\nhr' : 0 < \u2191r\nh : v % r < b % r\n\u22a2 r \u2264 r + v % r", "state_after": "no goals"}, {"tactic": "rw [cast_zero, ceil_eq_zero_iff, Set.mem_Ioc, div_le_iff hr', lt_div_iff hr', zero_mul,\n  tsub_nonpos, \u2190 neg_eq_neg_one_mul, neg_lt_sub_iff_lt_add, \u2190 cast_add, cast_lt, cast_le]", "annotated_tactic": ["rw [<a>cast_zero</a>, <a>ceil_eq_zero_iff</a>, <a>Set.mem_Ioc</a>, <a>div_le_iff</a> hr', <a>lt_div_iff</a> hr', <a>zero_mul</a>,\n      <a>tsub_nonpos</a>, \u2190 <a>neg_eq_neg_one_mul</a>, <a>neg_lt_sub_iff_lt_add</a>, \u2190 <a>cast_add</a>, <a>cast_lt</a>, <a>cast_le</a>]", [{"full_name": "Nat.cast_zero", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}, {"full_name": "Int.ceil_eq_zero_iff", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [1335, 9], "def_end_pos": [1335, 25]}, {"full_name": "Set.mem_Ioc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [141, 9], "def_end_pos": [141, 16]}, {"full_name": "div_le_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [61, 9], "def_end_pos": [61, 19]}, {"full_name": "lt_div_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [82, 9], "def_end_pos": [82, 19]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}, {"full_name": "tsub_nonpos", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [248, 9], "def_end_pos": [248, 20]}, {"full_name": "neg_eq_neg_one_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [347, 9], "def_end_pos": [347, 27]}, {"full_name": "neg_lt_sub_iff_lt_add", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [943, 3], "def_end_pos": [943, 14]}, {"full_name": "Nat.cast_add", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 17]}, {"full_name": "Nat.cast_lt", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [129, 9], "def_end_pos": [129, 16]}, {"full_name": "Nat.cast_le", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [124, 9], "def_end_pos": [124, 16]}]], "state_before": "case neg\na b r : \u2115\nhr : 0 < r\nv : \u2115\nhr' : 0 < \u2191r\nh : \u00acv % r < b % r\n\u22a2 \u2308(\u2191(b % r) - \u2191(v % r)) / \u2191r\u2309 = \u21910", "state_after": "case neg\na b r : \u2115\nhr : 0 < r\nv : \u2115\nhr' : 0 < \u2191r\nh : \u00acv % r < b % r\n\u22a2 v % r < r + b % r \u2227 b % r \u2264 v % r"}, {"tactic": "exact \u27e8(mod_lt _ hr).trans_le (by simp), not_lt.mp h\u27e9", "annotated_tactic": ["exact \u27e8(<a>mod_lt</a> _ hr).<a>trans_le</a> (by simp), not_lt.mp h\u27e9", [{"full_name": "Nat.mod_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [142, 9], "def_end_pos": [142, 15]}, {"full_name": "LT.lt.trans_le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [143, 7], "def_end_pos": [143, 21]}]], "state_before": "case neg\na b r : \u2115\nhr : 0 < r\nv : \u2115\nhr' : 0 < \u2191r\nh : \u00acv % r < b % r\n\u22a2 v % r < r + b % r \u2227 b % r \u2264 v % r", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "a b r : \u2115\nhr : 0 < r\nv : \u2115\nhr' : 0 < \u2191r\nh : \u00acv % r < b % r\n\u22a2 r \u2264 r + b % r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "full_name": "MeasureTheory.ae_restrict_biUnion_iff", "start": [576, 1], "end": [578, 69], "traced_tactics": [{"tactic": "simp_rw [Filter.Eventually, ae_restrict_biUnion_eq s ht, mem_iSup]", "annotated_tactic": ["simp_rw [<a>Filter.Eventually</a>, <a>ae_restrict_biUnion_eq</a> s ht, <a>mem_iSup</a>]", [{"full_name": "Filter.Eventually", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1090, 15], "def_end_pos": [1090, 25]}, {"full_name": "MeasureTheory.ae_restrict_biUnion_eq", "def_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "def_pos": [557, 9], "def_end_pos": [557, 31]}, {"full_name": "Filter.mem_iSup", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [595, 9], "def_end_pos": [595, 17]}]], "state_before": "R : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b4 : Type u_4\n\u03b3 : Type u_5\n\u03b9 : Type u_6\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\ninst\u271d : MeasurableSpace \u03b3\n\u03bc \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns\u271d s' t\u271d : Set \u03b1\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b9\nht : t.Countable\np : \u03b1 \u2192 Prop\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc.restrict (\u22c3 i \u2208 t, s i), p x) \u2194 \u2200 i \u2208 t, \u2200\u1d50 (x : \u03b1) \u2202\u03bc.restrict (s i), p x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Maps.lean", "full_name": "SimpleGraph.Iso.apply_mem_neighborSet_iff", "start": [550, 1], "end": [551, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/GradedModule.lean", "full_name": "DirectSum.Gmodule.one_smul'", "start": [116, 9], "end": [123, 95], "traced_tactics": [{"tactic": "suffices smulAddMonoidHom A M 1 = AddMonoidHom.id (\u2a01 i, M i) from DFunLike.congr_fun this x", "annotated_tactic": ["suffices <a>smulAddMonoidHom</a> A M 1 = <a>AddMonoidHom.id</a> (\u2a01 i, M i) from <a>DFunLike.congr_fun</a> this x", [{"full_name": "DirectSum.Gmodule.smulAddMonoidHom", "def_path": "Mathlib/Algebra/Module/GradedModule.lean", "def_pos": [77, 5], "def_end_pos": [77, 21]}, {"full_name": "AddMonoidHom.id", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [826, 3], "def_end_pos": [826, 14]}, {"full_name": "DFunLike.congr_fun", "def_path": "Mathlib/Data/FunLike/Basic.lean", "def_pos": [209, 19], "def_end_pos": [209, 28]}]], "state_before": "\u03b9A : Type u_1\n\u03b9B : Type u_2\nA : \u03b9A \u2192 Type u_3\nM : \u03b9B \u2192 Type u_4\ninst\u271d\u2077 : AddMonoid \u03b9A\ninst\u271d\u2076 : VAdd \u03b9A \u03b9B\ninst\u271d\u2075 : (i : \u03b9A) \u2192 AddCommMonoid (A i)\ninst\u271d\u2074 : (i : \u03b9B) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b3 : DecidableEq \u03b9A\ninst\u271d\u00b2 : DecidableEq \u03b9B\ninst\u271d\u00b9 : GMonoid A\ninst\u271d : Gmodule A M\nx : \u2a01 (i : \u03b9B), M i\n\u22a2 1 \u2022 x = x", "state_after": "\u03b9A : Type u_1\n\u03b9B : Type u_2\nA : \u03b9A \u2192 Type u_3\nM : \u03b9B \u2192 Type u_4\ninst\u271d\u2077 : AddMonoid \u03b9A\ninst\u271d\u2076 : VAdd \u03b9A \u03b9B\ninst\u271d\u2075 : (i : \u03b9A) \u2192 AddCommMonoid (A i)\ninst\u271d\u2074 : (i : \u03b9B) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b3 : DecidableEq \u03b9A\ninst\u271d\u00b2 : DecidableEq \u03b9B\ninst\u271d\u00b9 : GMonoid A\ninst\u271d : Gmodule A M\nx : \u2a01 (i : \u03b9B), M i\n\u22a2 (smulAddMonoidHom A M) 1 = AddMonoidHom.id (\u2a01 (i : \u03b9B), M i)"}, {"tactic": "apply DirectSum.addHom_ext", "annotated_tactic": ["apply <a>DirectSum.addHom_ext</a>", [{"full_name": "DirectSum.addHom_ext", "def_path": "Mathlib/Algebra/DirectSum/Basic.lean", "def_pos": [182, 9], "def_end_pos": [182, 19]}]], "state_before": "\u03b9A : Type u_1\n\u03b9B : Type u_2\nA : \u03b9A \u2192 Type u_3\nM : \u03b9B \u2192 Type u_4\ninst\u271d\u2077 : AddMonoid \u03b9A\ninst\u271d\u2076 : VAdd \u03b9A \u03b9B\ninst\u271d\u2075 : (i : \u03b9A) \u2192 AddCommMonoid (A i)\ninst\u271d\u2074 : (i : \u03b9B) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b3 : DecidableEq \u03b9A\ninst\u271d\u00b2 : DecidableEq \u03b9B\ninst\u271d\u00b9 : GMonoid A\ninst\u271d : Gmodule A M\nx : \u2a01 (i : \u03b9B), M i\n\u22a2 (smulAddMonoidHom A M) 1 = AddMonoidHom.id (\u2a01 (i : \u03b9B), M i)", "state_after": "case H\n\u03b9A : Type u_1\n\u03b9B : Type u_2\nA : \u03b9A \u2192 Type u_3\nM : \u03b9B \u2192 Type u_4\ninst\u271d\u2077 : AddMonoid \u03b9A\ninst\u271d\u2076 : VAdd \u03b9A \u03b9B\ninst\u271d\u2075 : (i : \u03b9A) \u2192 AddCommMonoid (A i)\ninst\u271d\u2074 : (i : \u03b9B) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b3 : DecidableEq \u03b9A\ninst\u271d\u00b2 : DecidableEq \u03b9B\ninst\u271d\u00b9 : GMonoid A\ninst\u271d : Gmodule A M\nx : \u2a01 (i : \u03b9B), M i\n\u22a2 \u2200 (i : \u03b9B) (y : M i), ((smulAddMonoidHom A M) 1) ((of M i) y) = (AddMonoidHom.id (\u2a01 (i : \u03b9B), M i)) ((of M i) y)"}, {"tactic": "intro i xi", "annotated_tactic": ["intro i xi", []], "state_before": "case H\n\u03b9A : Type u_1\n\u03b9B : Type u_2\nA : \u03b9A \u2192 Type u_3\nM : \u03b9B \u2192 Type u_4\ninst\u271d\u2077 : AddMonoid \u03b9A\ninst\u271d\u2076 : VAdd \u03b9A \u03b9B\ninst\u271d\u2075 : (i : \u03b9A) \u2192 AddCommMonoid (A i)\ninst\u271d\u2074 : (i : \u03b9B) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b3 : DecidableEq \u03b9A\ninst\u271d\u00b2 : DecidableEq \u03b9B\ninst\u271d\u00b9 : GMonoid A\ninst\u271d : Gmodule A M\nx : \u2a01 (i : \u03b9B), M i\n\u22a2 \u2200 (i : \u03b9B) (y : M i), ((smulAddMonoidHom A M) 1) ((of M i) y) = (AddMonoidHom.id (\u2a01 (i : \u03b9B), M i)) ((of M i) y)", "state_after": "case H\n\u03b9A : Type u_1\n\u03b9B : Type u_2\nA : \u03b9A \u2192 Type u_3\nM : \u03b9B \u2192 Type u_4\ninst\u271d\u2077 : AddMonoid \u03b9A\ninst\u271d\u2076 : VAdd \u03b9A \u03b9B\ninst\u271d\u2075 : (i : \u03b9A) \u2192 AddCommMonoid (A i)\ninst\u271d\u2074 : (i : \u03b9B) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b3 : DecidableEq \u03b9A\ninst\u271d\u00b2 : DecidableEq \u03b9B\ninst\u271d\u00b9 : GMonoid A\ninst\u271d : Gmodule A M\nx : \u2a01 (i : \u03b9B), M i\ni : \u03b9B\nxi : M i\n\u22a2 ((smulAddMonoidHom A M) 1) ((of M i) xi) = (AddMonoidHom.id (\u2a01 (i : \u03b9B), M i)) ((of M i) xi)"}, {"tactic": "rw [show (1 : DirectSum \u03b9A fun i => A i) = (of A 0) GOne.one by rfl]", "annotated_tactic": ["rw [show (1 : <a>DirectSum</a> \u03b9A fun i => A i) = (<a>of</a> A 0) <a>GOne.one</a> by rfl]", [{"full_name": "DirectSum", "def_path": "Mathlib/Algebra/DirectSum/Basic.lean", "def_pos": [35, 5], "def_end_pos": [35, 14]}, {"full_name": "DirectSum.of", "def_path": "Mathlib/Algebra/DirectSum/Basic.lean", "def_pos": [118, 5], "def_end_pos": [118, 7]}, {"full_name": "GradedMonoid.GOne.one", "def_path": "Mathlib/Algebra/GradedMonoid.lean", "def_pos": [157, 3], "def_end_pos": [157, 6]}]], "state_before": "case H\n\u03b9A : Type u_1\n\u03b9B : Type u_2\nA : \u03b9A \u2192 Type u_3\nM : \u03b9B \u2192 Type u_4\ninst\u271d\u2077 : AddMonoid \u03b9A\ninst\u271d\u2076 : VAdd \u03b9A \u03b9B\ninst\u271d\u2075 : (i : \u03b9A) \u2192 AddCommMonoid (A i)\ninst\u271d\u2074 : (i : \u03b9B) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b3 : DecidableEq \u03b9A\ninst\u271d\u00b2 : DecidableEq \u03b9B\ninst\u271d\u00b9 : GMonoid A\ninst\u271d : Gmodule A M\nx : \u2a01 (i : \u03b9B), M i\ni : \u03b9B\nxi : M i\n\u22a2 ((smulAddMonoidHom A M) 1) ((of M i) xi) = (AddMonoidHom.id (\u2a01 (i : \u03b9B), M i)) ((of M i) xi)", "state_after": "case H\n\u03b9A : Type u_1\n\u03b9B : Type u_2\nA : \u03b9A \u2192 Type u_3\nM : \u03b9B \u2192 Type u_4\ninst\u271d\u2077 : AddMonoid \u03b9A\ninst\u271d\u2076 : VAdd \u03b9A \u03b9B\ninst\u271d\u2075 : (i : \u03b9A) \u2192 AddCommMonoid (A i)\ninst\u271d\u2074 : (i : \u03b9B) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b3 : DecidableEq \u03b9A\ninst\u271d\u00b2 : DecidableEq \u03b9B\ninst\u271d\u00b9 : GMonoid A\ninst\u271d : Gmodule A M\nx : \u2a01 (i : \u03b9B), M i\ni : \u03b9B\nxi : M i\n\u22a2 ((smulAddMonoidHom A M) ((of A 0) GOne.one)) ((of M i) xi) = (AddMonoidHom.id (\u2a01 (i : \u03b9B), M i)) ((of M i) xi)"}, {"tactic": "rw [smulAddMonoidHom_apply_of_of]", "annotated_tactic": ["rw [<a>smulAddMonoidHom_apply_of_of</a>]", [{"full_name": "DirectSum.Gmodule.smulAddMonoidHom_apply_of_of", "def_path": "Mathlib/Algebra/Module/GradedModule.lean", "def_pos": [99, 9], "def_end_pos": [99, 37]}]], "state_before": "case H\n\u03b9A : Type u_1\n\u03b9B : Type u_2\nA : \u03b9A \u2192 Type u_3\nM : \u03b9B \u2192 Type u_4\ninst\u271d\u2077 : AddMonoid \u03b9A\ninst\u271d\u2076 : VAdd \u03b9A \u03b9B\ninst\u271d\u2075 : (i : \u03b9A) \u2192 AddCommMonoid (A i)\ninst\u271d\u2074 : (i : \u03b9B) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b3 : DecidableEq \u03b9A\ninst\u271d\u00b2 : DecidableEq \u03b9B\ninst\u271d\u00b9 : GMonoid A\ninst\u271d : Gmodule A M\nx : \u2a01 (i : \u03b9B), M i\ni : \u03b9B\nxi : M i\n\u22a2 ((smulAddMonoidHom A M) ((of A 0) GOne.one)) ((of M i) xi) = (AddMonoidHom.id (\u2a01 (i : \u03b9B), M i)) ((of M i) xi)", "state_after": "case H\n\u03b9A : Type u_1\n\u03b9B : Type u_2\nA : \u03b9A \u2192 Type u_3\nM : \u03b9B \u2192 Type u_4\ninst\u271d\u2077 : AddMonoid \u03b9A\ninst\u271d\u2076 : VAdd \u03b9A \u03b9B\ninst\u271d\u2075 : (i : \u03b9A) \u2192 AddCommMonoid (A i)\ninst\u271d\u2074 : (i : \u03b9B) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b3 : DecidableEq \u03b9A\ninst\u271d\u00b2 : DecidableEq \u03b9B\ninst\u271d\u00b9 : GMonoid A\ninst\u271d : Gmodule A M\nx : \u2a01 (i : \u03b9B), M i\ni : \u03b9B\nxi : M i\n\u22a2 (of M (0 +\u1d65 i)) (GSMul.smul GOne.one xi) = (AddMonoidHom.id (\u2a01 (i : \u03b9B), M i)) ((of M i) xi)"}, {"tactic": "exact DirectSum.of_eq_of_gradedMonoid_eq (one_smul (GradedMonoid A) <| GradedMonoid.mk i xi)", "annotated_tactic": ["exact <a>DirectSum.of_eq_of_gradedMonoid_eq</a> (<a>one_smul</a> (<a>GradedMonoid</a> A) <| <a>GradedMonoid.mk</a> i xi)", [{"full_name": "DirectSum.of_eq_of_gradedMonoid_eq", "def_path": "Mathlib/Algebra/DirectSum/Ring.lean", "def_pos": [147, 9], "def_end_pos": [147, 33]}, {"full_name": "one_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [453, 7], "def_end_pos": [453, 15]}, {"full_name": "GradedMonoid", "def_path": "Mathlib/Algebra/GradedMonoid.lean", "def_pos": [98, 5], "def_end_pos": [98, 17]}, {"full_name": "GradedMonoid.mk", "def_path": "Mathlib/Algebra/GradedMonoid.lean", "def_pos": [108, 5], "def_end_pos": [108, 7]}]], "state_before": "case H\n\u03b9A : Type u_1\n\u03b9B : Type u_2\nA : \u03b9A \u2192 Type u_3\nM : \u03b9B \u2192 Type u_4\ninst\u271d\u2077 : AddMonoid \u03b9A\ninst\u271d\u2076 : VAdd \u03b9A \u03b9B\ninst\u271d\u2075 : (i : \u03b9A) \u2192 AddCommMonoid (A i)\ninst\u271d\u2074 : (i : \u03b9B) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b3 : DecidableEq \u03b9A\ninst\u271d\u00b2 : DecidableEq \u03b9B\ninst\u271d\u00b9 : GMonoid A\ninst\u271d : Gmodule A M\nx : \u2a01 (i : \u03b9B), M i\ni : \u03b9B\nxi : M i\n\u22a2 (of M (0 +\u1d65 i)) (GSMul.smul GOne.one xi) = (AddMonoidHom.id (\u2a01 (i : \u03b9B), M i)) ((of M i) xi)", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b9A : Type u_1\n\u03b9B : Type u_2\nA : \u03b9A \u2192 Type u_3\nM : \u03b9B \u2192 Type u_4\ninst\u271d\u2077 : AddMonoid \u03b9A\ninst\u271d\u2076 : VAdd \u03b9A \u03b9B\ninst\u271d\u2075 : (i : \u03b9A) \u2192 AddCommMonoid (A i)\ninst\u271d\u2074 : (i : \u03b9B) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b3 : DecidableEq \u03b9A\ninst\u271d\u00b2 : DecidableEq \u03b9B\ninst\u271d\u00b9 : GMonoid A\ninst\u271d : Gmodule A M\nx : \u2a01 (i : \u03b9B), M i\ni : \u03b9B\nxi : M i\n\u22a2 1 = (of A 0) GOne.one", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Equiv/PartialEquiv.lean", "full_name": "PartialEquiv.image_source_inter_eq", "start": [495, 1], "end": [497, 79], "traced_tactics": [{"tactic": "rw [inter_comm, e.leftInvOn.image_inter, image_source_eq_target, inter_comm]", "annotated_tactic": ["rw [<a>inter_comm</a>, e.leftInvOn.image_inter, <a>image_source_eq_target</a>, <a>inter_comm</a>]", [{"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [905, 9], "def_end_pos": [905, 19]}, {"full_name": "PartialEquiv.image_source_eq_target", "def_path": "Mathlib/Logic/Equiv/PartialEquiv.lean", "def_pos": [341, 9], "def_end_pos": [341, 31]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [905, 9], "def_end_pos": [905, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ne : PartialEquiv \u03b1 \u03b2\ne' : PartialEquiv \u03b2 \u03b3\ns : Set \u03b1\n\u22a2 \u2191e '' (e.source \u2229 s) = e.target \u2229 \u2191e.symm \u207b\u00b9' (e.source \u2229 s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Sum/Basic.lean", "full_name": "Sum.map_injective", "start": [300, 1], "end": [305, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Complement.lean", "full_name": "Subgroup.IsComplement.equiv_one", "start": [492, 1], "end": [494, 54], "traced_tactics": [{"tactic": "rw [Equiv.apply_eq_iff_eq_symm_apply]", "annotated_tactic": ["rw [<a>Equiv.apply_eq_iff_eq_symm_apply</a>]", [{"full_name": "Equiv.apply_eq_iff_eq_symm_apply", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [319, 9], "def_end_pos": [319, 35]}]], "state_before": "G : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nhST : IsComplement S T\nhHT : IsComplement (\u2191H) T\nhSK : IsComplement S \u2191K\nhs1 : 1 \u2208 S\nht1 : 1 \u2208 T\n\u22a2 hST.equiv 1 = (\u27e81, hs1\u27e9, \u27e81, ht1\u27e9)", "state_after": "G : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nhST : IsComplement S T\nhHT : IsComplement (\u2191H) T\nhSK : IsComplement S \u2191K\nhs1 : 1 \u2208 S\nht1 : 1 \u2208 T\n\u22a2 1 = hST.equiv.symm (\u27e81, hs1\u27e9, \u27e81, ht1\u27e9)"}, {"tactic": "simp [equiv]", "annotated_tactic": ["simp [<a>equiv</a>]", [{"full_name": "Subgroup.IsComplement.equiv", "def_path": "Mathlib/GroupTheory/Complement.lean", "def_pos": [382, 19], "def_end_pos": [382, 24]}]], "state_before": "G : Type u_1\ninst\u271d : Group G\nH K : Subgroup G\nS T : Set G\nhST : IsComplement S T\nhHT : IsComplement (\u2191H) T\nhSK : IsComplement S \u2191K\nhs1 : 1 \u2208 S\nht1 : 1 \u2208 T\n\u22a2 1 = hST.equiv.symm (\u27e81, hs1\u27e9, \u27e81, ht1\u27e9)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Connected/PathConnected.lean", "full_name": "isPathConnected_range", "start": [1188, 1], "end": [1191, 38], "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": [701, 9], "def_end_pos": [701, 19]}]], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\nx y z : X\n\u03b9 : Type u_3\nF : Set X\ninst\u271d : PathConnectedSpace X\nf : X \u2192 Y\nhf : Continuous f\n\u22a2 IsPathConnected (range f)", "state_after": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\nx y z : X\n\u03b9 : Type u_3\nF : Set X\ninst\u271d : PathConnectedSpace X\nf : X \u2192 Y\nhf : Continuous f\n\u22a2 IsPathConnected (f '' univ)"}, {"tactic": "exact isPathConnected_univ.image hf", "annotated_tactic": ["exact isPathConnected_univ.image hf", []], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\nx y z : X\n\u03b9 : Type u_3\nF : Set X\ninst\u271d : PathConnectedSpace X\nf : X \u2192 Y\nhf : Continuous f\n\u22a2 IsPathConnected (f '' univ)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/FilterBasis.lean", "full_name": "GroupFilterBasis.N_one", "start": [149, 1], "end": [150, 34], "traced_tactics": [{"tactic": "simp only [N, one_mul, map_id']", "annotated_tactic": ["simp only [<a>N</a>, <a>one_mul</a>, <a>map_id'</a>]", [{"full_name": "GroupFilterBasis.N", "def_path": "Mathlib/Topology/Algebra/FilterBasis.lean", "def_pos": [141, 5], "def_end_pos": [141, 6]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "Filter.map_id'", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1966, 9], "def_end_pos": [1966, 16]}]], "state_before": "G : Type u\ninst\u271d : Group G\nB\u271d B : GroupFilterBasis G\n\u22a2 B.N 1 = toFilterBasis.filter", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "full_name": "Real.log_nat_eq_sum_factorization", "start": [397, 1], "end": [404, 75], "traced_tactics": [{"tactic": "rcases eq_or_ne n 0 with (rfl | hn)", "annotated_tactic": ["rcases <a>eq_or_ne</a> n 0 with (rfl | hn)", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 17]}]], "state_before": "x y : \u211d\nn : \u2115\n\u22a2 log \u2191n = n.factorization.sum fun p t => \u2191t * log \u2191p", "state_after": "case inl\nx y : \u211d\n\u22a2 log \u21910 = (Nat.factorization 0).sum fun p t => \u2191t * log \u2191p\n\ncase inr\nx y : \u211d\nn : \u2115\nhn : n \u2260 0\n\u22a2 log \u2191n = n.factorization.sum fun p t => \u2191t * log \u2191p"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case inl\nx y : \u211d\n\u22a2 log \u21910 = (Nat.factorization 0).sum fun p t => \u2191t * log \u2191p", "state_after": "no goals"}, {"tactic": "simp only [\u2190 log_pow, \u2190 Nat.cast_pow]", "annotated_tactic": ["simp only [\u2190 <a>log_pow</a>, \u2190 <a>Nat.cast_pow</a>]", [{"full_name": "Real.log_pow", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [300, 9], "def_end_pos": [300, 16]}, {"full_name": "Nat.cast_pow", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [93, 7], "def_end_pos": [93, 15]}]], "state_before": "case inr\nx y : \u211d\nn : \u2115\nhn : n \u2260 0\n\u22a2 log \u2191n = n.factorization.sum fun p t => \u2191t * log \u2191p", "state_after": "case inr\nx y : \u211d\nn : \u2115\nhn : n \u2260 0\n\u22a2 log \u2191n = n.factorization.sum fun p t => log \u2191(p ^ t)"}, {"tactic": "rw [\u2190 Finsupp.log_prod, \u2190 Nat.cast_finsupp_prod, Nat.factorization_prod_pow_eq_self hn]", "annotated_tactic": ["rw [\u2190 <a>Finsupp.log_prod</a>, \u2190 <a>Nat.cast_finsupp_prod</a>, <a>Nat.factorization_prod_pow_eq_self</a> hn]", [{"full_name": "Finsupp.log_prod", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [393, 19], "def_end_pos": [393, 42]}, {"full_name": "Nat.cast_finsupp_prod", "def_path": "Mathlib/Data/Finsupp/Basic.lean", "def_pos": [388, 9], "def_end_pos": [388, 26]}, {"full_name": "Nat.factorization_prod_pow_eq_self", "def_path": "Mathlib/Data/Nat/Factorization/Basic.lean", "def_pos": [99, 9], "def_end_pos": [99, 39]}]], "state_before": "case inr\nx y : \u211d\nn : \u2115\nhn : n \u2260 0\n\u22a2 log \u2191n = n.factorization.sum fun p t => log \u2191(p ^ t)", "state_after": "case inr.hg\nx y : \u211d\nn : \u2115\nhn : n \u2260 0\n\u22a2 \u2200 (a : \u2115), \u2191(a ^ n.factorization a) = 0 \u2192 n.factorization a = 0"}, {"tactic": "intro p hp", "annotated_tactic": ["intro p hp", []], "state_before": "case inr.hg\nx y : \u211d\nn : \u2115\nhn : n \u2260 0\n\u22a2 \u2200 (a : \u2115), \u2191(a ^ n.factorization a) = 0 \u2192 n.factorization a = 0", "state_after": "case inr.hg\nx y : \u211d\nn : \u2115\nhn : n \u2260 0\np : \u2115\nhp : \u2191(p ^ n.factorization p) = 0\n\u22a2 n.factorization p = 0"}, {"tactic": "rw [pow_eq_zero (Nat.cast_eq_zero.1 hp), Nat.factorization_zero_right]", "annotated_tactic": ["rw [<a>pow_eq_zero</a> (<a>Nat.cast_eq_zero</a>.1 hp), <a>Nat.factorization_zero_right</a>]", [{"full_name": "pow_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [186, 7], "def_end_pos": [186, 18]}, {"full_name": "Nat.cast_eq_zero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 21]}, {"full_name": "Nat.factorization_zero_right", "def_path": "Mathlib/Data/Nat/Factorization/Basic.lean", "def_pos": [152, 9], "def_end_pos": [152, 33]}]], "state_before": "case inr.hg\nx y : \u211d\nn : \u2115\nhn : n \u2260 0\np : \u2115\nhp : \u2191(p ^ n.factorization p) = 0\n\u22a2 n.factorization p = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Floor/Prime.lean", "full_name": "FloorRing.exists_prime_mul_pow_lt_factorial", "start": [22, 1], "end": [34, 62], "traced_tactics": [{"tactic": "obtain \u27e8p, pn, pp, h\u27e9 := n.exists_prime_mul_pow_lt_factorial \u2308|a|\u2309.natAbs \u2308|c|\u2309.natAbs", "annotated_tactic": ["obtain \u27e8p, pn, pp, h\u27e9 := n.exists_prime_mul_pow_lt_factorial \u2308|a|\u2309.<a>natAbs</a> \u2308|c|\u2309.<a>natAbs</a>", [{"full_name": "Int.natAbs", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [262, 5], "def_end_pos": [262, 11]}, {"full_name": "Int.natAbs", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [262, 5], "def_end_pos": [262, 11]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : LinearOrderedRing K\ninst\u271d : FloorRing K\nn : \u2115\na c : K\n\u22a2 \u2203 p > n, Nat.Prime p \u2227 a * c ^ p < \u2191(p - 1)!", "state_after": "case intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedRing K\ninst\u271d : FloorRing K\nn : \u2115\na c : K\np : \u2115\npn : p > n\npp : Nat.Prime p\nh : \u2308|a|\u2309.natAbs * \u2308|c|\u2309.natAbs ^ p < (p - 1)!\n\u22a2 \u2203 p > n, Nat.Prime p \u2227 a * c ^ p < \u2191(p - 1)!"}, {"tactic": "use p, pn, pp", "annotated_tactic": ["use p, pn, pp", []], "state_before": "case intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedRing K\ninst\u271d : FloorRing K\nn : \u2115\na c : K\np : \u2115\npn : p > n\npp : Nat.Prime p\nh : \u2308|a|\u2309.natAbs * \u2308|c|\u2309.natAbs ^ p < (p - 1)!\n\u22a2 \u2203 p > n, Nat.Prime p \u2227 a * c ^ p < \u2191(p - 1)!", "state_after": "case right\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedRing K\ninst\u271d : FloorRing K\nn : \u2115\na c : K\np : \u2115\npn : p > n\npp : Nat.Prime p\nh : \u2308|a|\u2309.natAbs * \u2308|c|\u2309.natAbs ^ p < (p - 1)!\n\u22a2 a * c ^ p < \u2191(p - 1)!"}, {"tactic": "calc a * c ^ p\n  _ \u2264 |a * c ^ p| := le_abs_self _\n  _ \u2264 \u2308|a|\u2309 * (\u2308|c|\u2309 : K) ^ p := ?_\n  _ = \u2191(Int.natAbs \u2308|a|\u2309 * Int.natAbs \u2308|c|\u2309 ^ p) := ?_\n  _ < \u2191(p - 1)! := Nat.cast_lt.mpr h", "annotated_tactic": ["calc a * c ^ p\n    _ \u2264 |a * c ^ p| := <a>le_abs_self</a> _\n    _ \u2264 \u2308|a|\u2309 * (\u2308|c|\u2309 : K) ^ p := ?_\n    _ = \u2191(<a>Int.natAbs</a> \u2308|a|\u2309 * <a>Int.natAbs</a> \u2308|c|\u2309 ^ p) := ?_\n    _ < \u2191(p - 1)! := Nat.cast_lt.mpr h", [{"full_name": "le_abs_self", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [73, 3], "def_end_pos": [73, 14]}, {"full_name": "Int.natAbs", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [262, 5], "def_end_pos": [262, 11]}, {"full_name": "Int.natAbs", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/Basic.lean", "def_pos": [262, 5], "def_end_pos": [262, 11]}]], "state_before": "case right\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedRing K\ninst\u271d : FloorRing K\nn : \u2115\na c : K\np : \u2115\npn : p > n\npp : Nat.Prime p\nh : \u2308|a|\u2309.natAbs * \u2308|c|\u2309.natAbs ^ p < (p - 1)!\n\u22a2 a * c ^ p < \u2191(p - 1)!", "state_after": "case right.calc_1\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedRing K\ninst\u271d : FloorRing K\nn : \u2115\na c : K\np : \u2115\npn : p > n\npp : Nat.Prime p\nh : \u2308|a|\u2309.natAbs * \u2308|c|\u2309.natAbs ^ p < (p - 1)!\n\u22a2 |a * c ^ p| \u2264 \u2191\u2308|a|\u2309 * \u2191\u2308|c|\u2309 ^ p\n\ncase right.calc_2\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedRing K\ninst\u271d : FloorRing K\nn : \u2115\na c : K\np : \u2115\npn : p > n\npp : Nat.Prime p\nh : \u2308|a|\u2309.natAbs * \u2308|c|\u2309.natAbs ^ p < (p - 1)!\n\u22a2 \u2191\u2308|a|\u2309 * \u2191\u2308|c|\u2309 ^ p = \u2191(\u2308|a|\u2309.natAbs * \u2308|c|\u2309.natAbs ^ p)"}, {"tactic": "rw [abs_mul, abs_pow]", "annotated_tactic": ["rw [<a>abs_mul</a>, <a>abs_pow</a>]", [{"full_name": "abs_mul", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [48, 7], "def_end_pos": [48, 14]}, {"full_name": "abs_pow", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [64, 7], "def_end_pos": [64, 14]}]], "state_before": "case right.calc_1\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedRing K\ninst\u271d : FloorRing K\nn : \u2115\na c : K\np : \u2115\npn : p > n\npp : Nat.Prime p\nh : \u2308|a|\u2309.natAbs * \u2308|c|\u2309.natAbs ^ p < (p - 1)!\n\u22a2 |a * c ^ p| \u2264 \u2191\u2308|a|\u2309 * \u2191\u2308|c|\u2309 ^ p", "state_after": "case right.calc_1\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedRing K\ninst\u271d : FloorRing K\nn : \u2115\na c : K\np : \u2115\npn : p > n\npp : Nat.Prime p\nh : \u2308|a|\u2309.natAbs * \u2308|c|\u2309.natAbs ^ p < (p - 1)!\n\u22a2 |a| * |c| ^ p \u2264 \u2191\u2308|a|\u2309 * \u2191\u2308|c|\u2309 ^ p"}, {"tactic": "gcongr <;> try first | positivity | apply Int.le_ceil", "annotated_tactic": ["gcongr <;> try first | positivity | apply <a>Int.le_ceil</a>", [{"full_name": "Int.le_ceil", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [1229, 9], "def_end_pos": [1229, 16]}]], "state_before": "case right.calc_1\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedRing K\ninst\u271d : FloorRing K\nn : \u2115\na c : K\np : \u2115\npn : p > n\npp : Nat.Prime p\nh : \u2308|a|\u2309.natAbs * \u2308|c|\u2309.natAbs ^ p < (p - 1)!\n\u22a2 |a| * |c| ^ p \u2264 \u2191\u2308|a|\u2309 * \u2191\u2308|c|\u2309 ^ p", "state_after": "no goals"}, {"tactic": "first | positivity | apply Int.le_ceil", "annotated_tactic": ["first | positivity | apply <a>Int.le_ceil</a>", [{"full_name": "Int.le_ceil", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [1229, 9], "def_end_pos": [1229, 16]}]], "state_before": "case right.calc_1.h\u2082.hab\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedRing K\ninst\u271d : FloorRing K\nn : \u2115\na c : K\np : \u2115\npn : p > n\npp : Nat.Prime p\nh : \u2308|a|\u2309.natAbs * \u2308|c|\u2309.natAbs ^ p < (p - 1)!\n\u22a2 |c| \u2264 \u2191\u2308|c|\u2309", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "case right.calc_1.h\u2082.ha\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedRing K\ninst\u271d : FloorRing K\nn : \u2115\na c : K\np : \u2115\npn : p > n\npp : Nat.Prime p\nh : \u2308|a|\u2309.natAbs * \u2308|c|\u2309.natAbs ^ p < (p - 1)!\n\u22a2 0 \u2264 |c|", "state_after": "no goals"}, {"tactic": "apply Int.le_ceil", "annotated_tactic": ["apply <a>Int.le_ceil</a>", [{"full_name": "Int.le_ceil", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [1229, 9], "def_end_pos": [1229, 16]}]], "state_before": "case right.calc_1.h\u2082.hab\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedRing K\ninst\u271d : FloorRing K\nn : \u2115\na c : K\np : \u2115\npn : p > n\npp : Nat.Prime p\nh : \u2308|a|\u2309.natAbs * \u2308|c|\u2309.natAbs ^ p < (p - 1)!\n\u22a2 |c| \u2264 \u2191\u2308|c|\u2309", "state_after": "no goals"}, {"tactic": "simp_rw [Nat.cast_mul, Nat.cast_pow, Int.cast_natAbs,\n  abs_eq_self.mpr (Int.ceil_nonneg (abs_nonneg (_ : K)))]", "annotated_tactic": ["simp_rw [<a>Nat.cast_mul</a>, <a>Nat.cast_pow</a>, <a>Int.cast_natAbs</a>,\n      abs_eq_self.mpr (<a>Int.ceil_nonneg</a> (<a>abs_nonneg</a> (_ : K)))]", [{"full_name": "Nat.cast_mul", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [62, 26], "def_end_pos": [62, 34]}, {"full_name": "Nat.cast_pow", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [93, 7], "def_end_pos": [93, 15]}, {"full_name": "Int.cast_natAbs", "def_path": "Mathlib/Algebra/Order/Ring/Cast.lean", "def_pos": [113, 7], "def_end_pos": [113, 18]}, {"full_name": "Int.ceil_nonneg", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [1326, 9], "def_end_pos": [1326, 20]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [122, 30], "def_end_pos": [122, 40]}]], "state_before": "case right.calc_2\nK : Type u_1\ninst\u271d\u00b9 : LinearOrderedRing K\ninst\u271d : FloorRing K\nn : \u2115\na c : K\np : \u2115\npn : p > n\npp : Nat.Prime p\nh : \u2308|a|\u2309.natAbs * \u2308|c|\u2309.natAbs ^ p < (p - 1)!\n\u22a2 \u2191\u2308|a|\u2309 * \u2191\u2308|c|\u2309 ^ p = \u2191(\u2308|a|\u2309.natAbs * \u2308|c|\u2309.natAbs ^ p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/LinearMap/End.lean", "full_name": "LinearMap.compRight_apply", "start": [425, 1], "end": [426, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Real.tan_div_sqrt_one_add_tan_sq", "start": [1024, 1], "end": [1026, 74], "traced_tactics": [{"tactic": "rw [\u2190 tan_mul_cos hx.ne', \u2190 inv_sqrt_one_add_tan_sq hx, div_eq_mul_inv]", "annotated_tactic": ["rw [\u2190 <a>tan_mul_cos</a> hx.ne', \u2190 <a>inv_sqrt_one_add_tan_sq</a> hx, <a>div_eq_mul_inv</a>]", [{"full_name": "Real.tan_mul_cos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [923, 9], "def_end_pos": [923, 20]}, {"full_name": "Real.inv_sqrt_one_add_tan_sq", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1020, 9], "def_end_pos": [1020, 32]}, {"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1059, 9], "def_end_pos": [1059, 23]}]], "state_before": "x\u271d y x : \u211d\nhx : 0 < cos x\n\u22a2 tan x / \u221a(1 + tan x ^ 2) = sin x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Prod.lean", "full_name": "LinearMap.isCompl_range_inl_inr", "start": [445, 1], "end": [455, 9], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\n\u22a2 IsCompl (range (inl R M M\u2082)) (range (inr R M M\u2082))", "state_after": "case disjoint\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\n\u22a2 Disjoint (range (inl R M M\u2082)) (range (inr R M M\u2082))\n\ncase codisjoint\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\n\u22a2 Codisjoint (range (inl R M M\u2082)) (range (inr R M M\u2082))"}, {"tactic": "rw [disjoint_def]", "annotated_tactic": ["rw [<a>disjoint_def</a>]", [{"full_name": "Submodule.disjoint_def", "def_path": "Mathlib/Algebra/Module/Submodule/Lattice.lean", "def_pos": [374, 9], "def_end_pos": [374, 21]}]], "state_before": "case disjoint\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\n\u22a2 Disjoint (range (inl R M M\u2082)) (range (inr R M M\u2082))", "state_after": "case disjoint\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\n\u22a2 \u2200 x \u2208 range (inl R M M\u2082), x \u2208 range (inr R M M\u2082) \u2192 x = 0"}, {"tactic": "rintro \u27e8_, _\u27e9 \u27e8x, hx\u27e9 \u27e8y, hy\u27e9", "annotated_tactic": ["rintro \u27e8_, _\u27e9 \u27e8x, hx\u27e9 \u27e8y, hy\u27e9", []], "state_before": "case disjoint\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\n\u22a2 \u2200 x \u2208 range (inl R M M\u2082), x \u2208 range (inr R M M\u2082) \u2192 x = 0", "state_after": "case disjoint.mk.intro.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nfst\u271d : M\nsnd\u271d : M\u2082\nx : M\nhx : (inl R M M\u2082) x = (fst\u271d, snd\u271d)\ny : M\u2082\nhy : (inr R M M\u2082) y = (fst\u271d, snd\u271d)\n\u22a2 (fst\u271d, snd\u271d) = 0"}, {"tactic": "simp only [Prod.ext_iff, inl_apply, inr_apply, mem_bot] at hx hy \u22a2", "annotated_tactic": ["simp only [<a>Prod.ext_iff</a>, <a>inl_apply</a>, <a>inr_apply</a>, <a>mem_bot</a>] at hx hy \u22a2", [{"full_name": "Prod.ext_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [124, 9], "def_end_pos": [124, 16]}, {"full_name": "LinearMap.inl_apply", "def_path": "Mathlib/LinearAlgebra/Prod.lean", "def_pos": [190, 9], "def_end_pos": [190, 18]}, {"full_name": "LinearMap.inr_apply", "def_path": "Mathlib/LinearAlgebra/Prod.lean", "def_pos": [199, 9], "def_end_pos": [199, 18]}, {"full_name": "Submodule.mem_bot", "def_path": "Mathlib/Algebra/Module/Submodule/Lattice.lean", "def_pos": [73, 9], "def_end_pos": [73, 16]}]], "state_before": "case disjoint.mk.intro.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nfst\u271d : M\nsnd\u271d : M\u2082\nx : M\nhx : (inl R M M\u2082) x = (fst\u271d, snd\u271d)\ny : M\u2082\nhy : (inr R M M\u2082) y = (fst\u271d, snd\u271d)\n\u22a2 (fst\u271d, snd\u271d) = 0", "state_after": "case disjoint.mk.intro.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nfst\u271d : M\nsnd\u271d : M\u2082\nx : M\ny : M\u2082\nhx : x = fst\u271d \u2227 0 = snd\u271d\nhy : 0 = fst\u271d \u2227 y = snd\u271d\n\u22a2 fst\u271d = 0.1 \u2227 snd\u271d = 0.2"}, {"tactic": "exact \u27e8hy.1.symm, hx.2.symm\u27e9", "annotated_tactic": ["exact \u27e8hy.1.<a>symm</a>, hx.2.<a>symm</a>\u27e9", [{"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case disjoint.mk.intro.intro\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nfst\u271d : M\nsnd\u271d : M\u2082\nx : M\ny : M\u2082\nhx : x = fst\u271d \u2227 0 = snd\u271d\nhy : 0 = fst\u271d \u2227 y = snd\u271d\n\u22a2 fst\u271d = 0.1 \u2227 snd\u271d = 0.2", "state_after": "no goals"}, {"tactic": "rw [codisjoint_iff_le_sup]", "annotated_tactic": ["rw [<a>codisjoint_iff_le_sup</a>]", [{"full_name": "codisjoint_iff_le_sup", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [325, 9], "def_end_pos": [325, 30]}]], "state_before": "case codisjoint\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\n\u22a2 Codisjoint (range (inl R M M\u2082)) (range (inr R M M\u2082))", "state_after": "case codisjoint\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\n\u22a2 \u22a4 \u2264 range (inl R M M\u2082) \u2294 range (inr R M M\u2082)"}, {"tactic": "rintro \u27e8x, y\u27e9 -", "annotated_tactic": ["rintro \u27e8x, y\u27e9 -", []], "state_before": "case codisjoint\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\n\u22a2 \u22a4 \u2264 range (inl R M M\u2082) \u2294 range (inr R M M\u2082)", "state_after": "case codisjoint.mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nx : M\ny : M\u2082\n\u22a2 (x, y) \u2208 range (inl R M M\u2082) \u2294 range (inr R M M\u2082)"}, {"tactic": "simp only [mem_sup, mem_range, exists_prop]", "annotated_tactic": ["simp only [<a>mem_sup</a>, <a>mem_range</a>, <a>exists_prop</a>]", [{"full_name": "Submodule.mem_sup", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [425, 9], "def_end_pos": [425, 16]}, {"full_name": "LinearMap.mem_range", "def_path": "Mathlib/Algebra/Module/Submodule/Range.lean", "def_pos": [72, 9], "def_end_pos": [72, 18]}, {"full_name": 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"state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c F\ns\u271d : Set E\nx\u271d : E\n\u03b9 : Sort u_5\ns : \u03b9 \u2192 Set E\nhdir : Directed (fun x x_1 => x \u2286 x_1) s\nhc : \u2200 \u2983i : \u03b9\u2984, Convex \ud835\udd5c (s i)\nx : E\nhx : \u2203 i, x \u2208 s i\ny : E\nhy : \u2203 i, y \u2208 s i\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\n\u22a2 \u2203 i, a \u2022 x + b \u2022 y \u2208 s i"}, {"tactic": "obtain \u27e8i, hx\u27e9 := hx", "annotated_tactic": ["obtain \u27e8i, hx\u27e9 := hx", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : 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\u2264 b\nhab : a + b = 1\ni : \u03b9\nhx : x \u2208 s i\n\u22a2 \u2203 i, a \u2022 x + b \u2022 y \u2208 s i"}, {"tactic": "obtain \u27e8j, hy\u27e9 := hy", "annotated_tactic": ["obtain \u27e8j, hy\u27e9 := hy", []], "state_before": "case intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c F\ns\u271d : Set E\nx\u271d : E\n\u03b9 : Sort u_5\ns : \u03b9 \u2192 Set E\nhdir : Directed (fun x x_1 => x \u2286 x_1) s\nhc : \u2200 \u2983i : \u03b9\u2984, Convex \ud835\udd5c (s i)\nx y : E\nhy : \u2203 i, y \u2208 s i\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\ni : \u03b9\nhx : x \u2208 s i\n\u22a2 \u2203 i, a \u2022 x + b \u2022 y \u2208 s i", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c F\ns\u271d : Set E\nx\u271d : E\n\u03b9 : Sort u_5\ns : \u03b9 \u2192 Set E\nhdir : Directed (fun x x_1 => x \u2286 x_1) s\nhc : \u2200 \u2983i : \u03b9\u2984, Convex \ud835\udd5c (s i)\nx y : E\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\ni : \u03b9\nhx : x \u2208 s i\nj : \u03b9\nhy : y \u2208 s j\n\u22a2 \u2203 i, a \u2022 x + b \u2022 y \u2208 s i"}, {"tactic": "obtain \u27e8k, hik, hjk\u27e9 := hdir i j", "annotated_tactic": ["obtain \u27e8k, hik, hjk\u27e9 := hdir i j", []], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c F\ns\u271d : Set E\nx\u271d : E\n\u03b9 : Sort u_5\ns : \u03b9 \u2192 Set E\nhdir : Directed (fun x x_1 => x \u2286 x_1) s\nhc : \u2200 \u2983i : \u03b9\u2984, Convex \ud835\udd5c (s i)\nx y : E\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\ni : \u03b9\nhx : x \u2208 s i\nj : \u03b9\nhy : y \u2208 s j\n\u22a2 \u2203 i, a \u2022 x + b \u2022 y \u2208 s i", "state_after": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c F\ns\u271d : Set E\nx\u271d : E\n\u03b9 : Sort u_5\ns : \u03b9 \u2192 Set E\nhdir : Directed (fun x x_1 => x \u2286 x_1) s\nhc : \u2200 \u2983i : \u03b9\u2984, Convex \ud835\udd5c (s i)\nx y : E\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\ni : \u03b9\nhx : x \u2208 s i\nj : \u03b9\nhy : y \u2208 s j\nk : \u03b9\nhik : s i \u2286 s k\nhjk : s j \u2286 s k\n\u22a2 \u2203 i, a \u2022 x + b \u2022 y \u2208 s i"}, {"tactic": "exact \u27e8k, hc (hik hx) (hjk hy) ha hb hab\u27e9", "annotated_tactic": ["exact \u27e8k, hc (hik hx) (hjk hy) ha hb hab\u27e9", []], "state_before": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2074 : OrderedSemiring \ud835\udd5c\ninst\u271d\u00b3 : AddCommMonoid E\ninst\u271d\u00b2 : AddCommMonoid F\ninst\u271d\u00b9 : SMul \ud835\udd5c E\ninst\u271d : SMul \ud835\udd5c F\ns\u271d : Set E\nx\u271d : E\n\u03b9 : Sort u_5\ns : \u03b9 \u2192 Set E\nhdir : Directed (fun x x_1 => x \u2286 x_1) s\nhc : \u2200 \u2983i : \u03b9\u2984, Convex \ud835\udd5c (s i)\nx y : E\na b : \ud835\udd5c\nha : 0 \u2264 a\nhb : 0 \u2264 b\nhab : a + b = 1\ni : \u03b9\nhx : x \u2208 s i\nj : \u03b9\nhy : y \u2208 s j\nk : \u03b9\nhik : s i \u2286 s k\nhjk : s j \u2286 s k\n\u22a2 \u2203 i, a \u2022 x + b \u2022 y \u2208 s i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Analysis/Topology.lean", "full_name": "Ctop.mem_nhds_toTopsp", "start": [98, 1], "end": [102, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/PurelyInseparable.lean", "full_name": "Field.lift_rank_mul_lift_sepDegree_of_isSeparable", "start": [943, 1], "end": [951, 59], "traced_tactics": [{"tactic": "rw [sepDegree, sepDegree, separableClosure.eq_restrictScalars_of_isSeparable F E K]", "annotated_tactic": ["rw [<a>sepDegree</a>, <a>sepDegree</a>, <a>separableClosure.eq_restrictScalars_of_isSeparable</a> F E K]", [{"full_name": "Field.sepDegree", "def_path": "Mathlib/FieldTheory/SeparableClosure.lean", "def_pos": [254, 5], "def_end_pos": [254, 14]}, {"full_name": "Field.sepDegree", "def_path": "Mathlib/FieldTheory/SeparableClosure.lean", "def_pos": [254, 5], "def_end_pos": [254, 14]}, {"full_name": "separableClosure.eq_restrictScalars_of_isSeparable", "def_path": "Mathlib/FieldTheory/SeparableClosure.lean", "def_pos": [225, 9], "def_end_pos": [225, 59]}]], "state_before": "F : Type u\nE : Type v\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra F E\nK : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra E K\ninst\u271d\u00b9 : IsScalarTower F E K\ninst\u271d : IsSeparable F E\n\u22a2 Cardinal.lift.{w, v} (Module.rank F E) * Cardinal.lift.{v, w} (sepDegree E K) = Cardinal.lift.{v, w} (sepDegree F K)", "state_after": "F : Type u\nE : Type v\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra F E\nK : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra E K\ninst\u271d\u00b9 : IsScalarTower F E K\ninst\u271d : IsSeparable F E\n\u22a2 Cardinal.lift.{w, v} (Module.rank F E) * Cardinal.lift.{v, w} (Module.rank E \u21a5(separableClosure E K)) =\n    Cardinal.lift.{v, w} (Module.rank F \u21a5(restrictScalars F (separableClosure E K)))"}, {"tactic": "exact lift_rank_mul_lift_rank F E (separableClosure E K)", "annotated_tactic": ["exact <a>lift_rank_mul_lift_rank</a> F E (<a>separableClosure</a> E K)", [{"full_name": "lift_rank_mul_lift_rank", "def_path": "Mathlib/LinearAlgebra/Dimension/Free.lean", "def_pos": [41, 9], "def_end_pos": [41, 32]}, {"full_name": "separableClosure", "def_path": "Mathlib/FieldTheory/SeparableClosure.lean", "def_pos": [78, 5], "def_end_pos": [78, 21]}]], "state_before": "F : Type u\nE : Type v\ninst\u271d\u2077 : Field F\ninst\u271d\u2076 : Field E\ninst\u271d\u2075 : Algebra F E\nK : Type w\ninst\u271d\u2074 : Field K\ninst\u271d\u00b3 : Algebra F K\ninst\u271d\u00b2 : Algebra E K\ninst\u271d\u00b9 : IsScalarTower F E K\ninst\u271d : IsSeparable F E\n\u22a2 Cardinal.lift.{w, v} (Module.rank F E) * Cardinal.lift.{v, w} (Module.rank E \u21a5(separableClosure E K)) =\n    Cardinal.lift.{v, w} (Module.rank F \u21a5(restrictScalars F (separableClosure E K)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.iIndepFun.indepFun_finset_prod_of_not_mem", "start": [785, 1], "end": [790, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "full_name": "MeasureTheory.AEEqFun.toGerm_injective", "start": [450, 1], "end": [451, 60], "traced_tactics": [{"tactic": "rwa [\u2190 toGerm_eq, \u2190 toGerm_eq]", "annotated_tactic": ["rwa [\u2190 <a>toGerm_eq</a>, \u2190 <a>toGerm_eq</a>]", [{"full_name": "MeasureTheory.AEEqFun.toGerm_eq", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [447, 9], "def_end_pos": [447, 18]}, {"full_name": "MeasureTheory.AEEqFun.toGerm_eq", "def_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "def_pos": [447, 9], "def_end_pos": [447, 18]}]], "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 g : \u03b1 \u2192\u2098[\u03bc] \u03b2\nH : f.toGerm = g.toGerm\n\u22a2 \u2191\u2191f = \u2191\u2191g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "full_name": "MeasureTheory.ProbabilityMeasure.coe_toWeakDualBCNN", "start": [266, 1], "end": [268, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Finset/Basic.lean", "full_name": "Finset.Icc_subset_Icc_iff", "start": [259, 1], "end": [260, 65], "traced_tactics": [{"tactic": "rw [\u2190 coe_subset, coe_Icc, coe_Icc, Set.Icc_subset_Icc_iff h\u2081]", "annotated_tactic": ["rw [\u2190 <a>coe_subset</a>, <a>coe_Icc</a>, <a>coe_Icc</a>, <a>Set.Icc_subset_Icc_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/Interval/Finset/Defs.lean", "def_pos": [333, 9], "def_end_pos": [333, 16]}, {"full_name": "Finset.coe_Icc", "def_path": "Mathlib/Order/Interval/Finset/Defs.lean", "def_pos": [333, 9], "def_end_pos": [333, 16]}, {"full_name": "Set.Icc_subset_Icc_iff", "def_path": 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{"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/IsLUB.lean", "full_name": "IsGLB.mem_upperBounds_of_tendsto", "start": [140, 1], "end": [143, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean", "full_name": "EuclideanGeometry.oangle_rotate_sign", "start": [470, 1], "end": [471, 48], "traced_tactics": [{"tactic": "rw [\u2190 oangle_swap\u2081\u2082_sign, oangle_swap\u2081\u2083_sign]", "annotated_tactic": ["rw [\u2190 <a>oangle_swap\u2081\u2082_sign</a>, <a>oangle_swap\u2081\u2083_sign</a>]", [{"full_name": "EuclideanGeometry.oangle_swap\u2081\u2082_sign", "def_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean", "def_pos": [450, 9], "def_end_pos": [450, 27]}, {"full_name": "EuclideanGeometry.oangle_swap\u2081\u2083_sign", "def_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean", "def_pos": [460, 9], "def_end_pos": [460, 27]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 : P\n\u22a2 (\u2221 p\u2082 p\u2083 p\u2081).sign = (\u2221 p\u2081 p\u2082 p\u2083).sign", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean", "full_name": "CategoryTheory.Limits.HasBinaryBiproduct.mk", "start": [1479, 1], "end": [1480, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "full_name": "MeasureTheory.unifIntegrable_subsingleton", "start": [411, 1], "end": [419, 74], "traced_tactics": [{"tactic": "intro \u03b5 h\u03b5", "annotated_tactic": ["intro \u03b5 h\u03b5", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Subsingleton \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p \u03bc\n\u22a2 UnifIntegrable f p \u03bc", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Subsingleton \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p \u03bc\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 \u03b4,\n    \u2203 (_ : 0 < \u03b4),\n      \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (s.indicator (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": [709, 17], "def_end_pos": [709, 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\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Subsingleton \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p \u03bc\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\n\u22a2 \u2203 \u03b4,\n    \u2203 (_ : 0 < \u03b4),\n      \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (s.indicator (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\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Subsingleton \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p \u03bc\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\n\u22a2 \u2203 \u03b4,\n    \u2203 (_ : 0 < \u03b4),\n      \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (s.indicator (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\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Subsingleton \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p \u03bc\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : \u00acNonempty \u03b9\n\u22a2 \u2203 \u03b4,\n    \u2203 (_ : 0 < \u03b4),\n      \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (s.indicator (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "cases' h\u03b9 with i", "annotated_tactic": ["cases' h\u03b9 with i", []], "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\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Subsingleton \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p \u03bc\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : Nonempty \u03b9\n\u22a2 \u2203 \u03b4,\n    \u2203 (_ : 0 < \u03b4),\n      \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (s.indicator (f i)) p \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\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Subsingleton \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p \u03bc\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ni : \u03b9\n\u22a2 \u2203 \u03b4,\n    \u2203 (_ : 0 < \u03b4),\n      \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (s.indicator (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "obtain \u27e8\u03b4, h\u03b4pos, h\u03b4\u27e9 := (hf i).snorm_indicator_le hp_one hp_top h\u03b5", "annotated_tactic": ["obtain \u27e8\u03b4, h\u03b4pos, h\u03b4\u27e9 := (hf i).<a>snorm_indicator_le</a> hp_one hp_top h\u03b5", [{"full_name": "MeasureTheory.Mem\u2112p.snorm_indicator_le", "def_path": "Mathlib/MeasureTheory/Function/UniformIntegrable.lean", "def_pos": [389, 9], "def_end_pos": [389, 33]}]], "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\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Subsingleton \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p \u03bc\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ni : \u03b9\n\u22a2 \u2203 \u03b4,\n    \u2203 (_ : 0 < \u03b4),\n      \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (s.indicator (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Subsingleton \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p \u03bc\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ni : \u03b9\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (s.indicator (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 \u03b4,\n    \u2203 (_ : 0 < \u03b4),\n      \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (s.indicator (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "refine \u27e8\u03b4, h\u03b4pos, fun j s hs h\u03bcs => ?_\u27e9", "annotated_tactic": ["refine \u27e8\u03b4, h\u03b4pos, fun j s hs h\u03bcs => ?_\u27e9", []], "state_before": "case pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Subsingleton \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p \u03bc\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ni : \u03b9\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (s.indicator (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\n\u22a2 \u2203 \u03b4,\n    \u2203 (_ : 0 < \u03b4),\n      \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (s.indicator (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "case pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Subsingleton \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p \u03bc\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ni : \u03b9\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (s.indicator (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nj : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u03bc s \u2264 ENNReal.ofReal \u03b4\n\u22a2 snorm (s.indicator (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5"}, {"tactic": "convert h\u03b4 s hs h\u03bcs", "annotated_tactic": ["convert h\u03b4 s hs h\u03bcs", []], "state_before": "case pos.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Subsingleton \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p \u03bc\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\ni : \u03b9\n\u03b4 : \u211d\nh\u03b4pos : 0 < \u03b4\nh\u03b4 : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (s.indicator (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5\nj : \u03b9\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u03bc s \u2264 ENNReal.ofReal \u03b4\n\u22a2 snorm (s.indicator (f j)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "no goals"}, {"tactic": "exact \u27e81, zero_lt_one, fun i => False.elim <| h\u03b9 <| Nonempty.intro i\u27e9", "annotated_tactic": ["exact \u27e81, <a>zero_lt_one</a>, fun i => <a>False.elim</a> <| h\u03b9 <| <a>Nonempty.intro</a> i\u27e9", [{"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}, {"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}, {"full_name": "Nonempty.intro", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [711, 5], "def_end_pos": [711, 10]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\np : \u211d\u22650\u221e\nf\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : Subsingleton \u03b9\nhp_one : 1 \u2264 p\nhp_top : p \u2260 \u22a4\nf : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (i : \u03b9), Mem\u2112p (f i) p \u03bc\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b9 : \u00acNonempty \u03b9\n\u22a2 \u2203 \u03b4,\n    \u2203 (_ : 0 < \u03b4),\n      \u2200 (i : \u03b9) (s : Set \u03b1), MeasurableSet s \u2192 \u03bc s \u2264 ENNReal.ofReal \u03b4 \u2192 snorm (s.indicator (f i)) p \u03bc \u2264 ENNReal.ofReal \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matrix/Basic.lean", "full_name": "Matrix.zero_vecMul", "start": [1758, 1], "end": [1760, 16], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o \u2192 Type u_5\nn' : o \u2192 Type u_6\nR : Type u_7\nS : Type u_8\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type u_9\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring \u03b1\ninst\u271d : Fintype m\nA : Matrix m n \u03b1\n\u22a2 0 \u1d65* A = 0", "state_after": "case h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o \u2192 Type u_5\nn' : o \u2192 Type u_6\nR : Type u_7\nS : Type u_8\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type u_9\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring \u03b1\ninst\u271d : Fintype m\nA : Matrix m n \u03b1\nx\u271d : n\n\u22a2 (0 \u1d65* A) x\u271d = 0 x\u271d"}, {"tactic": "simp [vecMul]", "annotated_tactic": ["simp [<a>vecMul</a>]", [{"full_name": "Matrix.vecMul", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [1710, 5], "def_end_pos": [1710, 11]}]], "state_before": "case h\nl : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nm' : o \u2192 Type u_5\nn' : o \u2192 Type u_6\nR : Type u_7\nS : Type u_8\n\u03b1 : Type v\n\u03b2 : Type w\n\u03b3 : Type u_9\ninst\u271d\u00b9 : NonUnitalNonAssocSemiring \u03b1\ninst\u271d : Fintype m\nA : Matrix m n \u03b1\nx\u271d : n\n\u22a2 (0 \u1d65* A) x\u271d = 0 x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.pi_piecewise", "start": [1736, 1], "end": [1738, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "full_name": "Subgroup.normalClosure_le_normal", "start": [2391, 1], "end": [2397, 22], "traced_tactics": [{"tactic": "intro a w", "annotated_tactic": ["intro a w", []], "state_before": "G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : Group G''\nA : Type u_4\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : N.Normal\nh : s \u2286 \u2191N\n\u22a2 normalClosure s \u2264 N", "state_after": "G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : Group G''\nA : Type u_4\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : N.Normal\nh : s \u2286 \u2191N\na : G\nw : a \u2208 normalClosure s\n\u22a2 a \u2208 N"}, {"tactic": "refine closure_induction w (fun x hx => ?_) ?_ (fun x y ihx ihy => ?_) fun x ihx => ?_", "annotated_tactic": ["refine <a>closure_induction</a> w (fun x hx => ?_) ?_ (fun x y ihx ihy => ?_) fun x ihx => ?_", [{"full_name": "Subgroup.closure_induction", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [1122, 9], "def_end_pos": [1122, 26]}]], "state_before": "G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : Group G''\nA : Type u_4\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : N.Normal\nh : s \u2286 \u2191N\na : G\nw : a \u2208 normalClosure s\n\u22a2 a \u2208 N", "state_after": "case refine_1\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : Group G''\nA : Type u_4\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : N.Normal\nh : s \u2286 \u2191N\na : G\nw : a \u2208 normalClosure s\nx : G\nhx : x \u2208 conjugatesOfSet s\n\u22a2 x \u2208 N\n\ncase refine_2\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : Group G''\nA : Type u_4\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : N.Normal\nh : s \u2286 \u2191N\na : G\nw : a \u2208 normalClosure s\n\u22a2 1 \u2208 N\n\ncase refine_3\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : Group G''\nA : Type u_4\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : N.Normal\nh : s \u2286 \u2191N\na : G\nw : a \u2208 normalClosure s\nx y : G\nihx : x \u2208 N\nihy : y \u2208 N\n\u22a2 x * y \u2208 N\n\ncase refine_4\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : Group G''\nA : Type u_4\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : N.Normal\nh : s \u2286 \u2191N\na : G\nw : a \u2208 normalClosure s\nx : G\nihx : x \u2208 N\n\u22a2 x\u207b\u00b9 \u2208 N"}, {"tactic": "exact conjugatesOfSet_subset h hx", "annotated_tactic": ["exact <a>conjugatesOfSet_subset</a> h hx", [{"full_name": "Group.conjugatesOfSet_subset", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [2340, 9], "def_end_pos": [2340, 31]}]], "state_before": "case refine_1\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : Group G''\nA : Type u_4\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : N.Normal\nh : s \u2286 \u2191N\na : G\nw : a \u2208 normalClosure s\nx : G\nhx : x \u2208 conjugatesOfSet s\n\u22a2 x \u2208 N", "state_after": "no goals"}, {"tactic": "exact one_mem _", "annotated_tactic": ["exact <a>one_mem</a> _", [{"full_name": "OneMemClass.one_mem", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [72, 3], "def_end_pos": [72, 10]}]], "state_before": "case refine_2\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : Group G''\nA : Type u_4\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : N.Normal\nh : s \u2286 \u2191N\na : G\nw : a \u2208 normalClosure s\n\u22a2 1 \u2208 N", "state_after": "no goals"}, {"tactic": "exact mul_mem ihx ihy", "annotated_tactic": ["exact <a>mul_mem</a> ihx ihy", [{"full_name": "MulMemClass.mul_mem", "def_path": "Mathlib/Algebra/Group/Subsemigroup/Basic.lean", "def_pos": [64, 3], "def_end_pos": [64, 10]}]], "state_before": "case refine_3\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : Group G''\nA : Type u_4\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : N.Normal\nh : s \u2286 \u2191N\na : G\nw : a \u2208 normalClosure s\nx y : G\nihx : x \u2208 N\nihy : y \u2208 N\n\u22a2 x * y \u2208 N", "state_after": "no goals"}, {"tactic": "exact inv_mem ihx", "annotated_tactic": ["exact <a>inv_mem</a> ihx", [{"full_name": "InvMemClass.inv_mem", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [100, 3], "def_end_pos": [100, 10]}]], "state_before": "case refine_4\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2074 : Group G\ninst\u271d\u00b3 : Group G'\ninst\u271d\u00b2 : Group G''\nA : Type u_4\ninst\u271d\u00b9 : AddGroup A\ns : Set G\nN : Subgroup G\ninst\u271d : N.Normal\nh : s \u2286 \u2191N\na : G\nw : a \u2208 normalClosure s\nx : G\nihx : x \u2208 N\n\u22a2 x\u207b\u00b9 \u2208 N", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Int.lean", "full_name": "Int.two_dvd_ne_zero", "start": [206, 1], "end": [207, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Birkhoff.lean", "full_name": "OrderEmbedding.birkhoffSet_apply", "start": [250, 1], "end": [252, 85], "traced_tactics": [{"tactic": "simp [birkhoffSet]", "annotated_tactic": ["simp [<a>birkhoffSet</a>]", [{"full_name": "OrderEmbedding.birkhoffSet", "def_path": "Mathlib/Order/Birkhoff.lean", "def_pos": [210, 19], "def_end_pos": [210, 30]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : DistribLattice \u03b1\ninst\u271d\u00b3 : OrderBot \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidablePred SupIrred\ninst\u271d : DecidableEq \u03b1\na : \u03b1\n\u22a2 birkhoffSet a = \u2191(OrderIso.lowerSetSupIrred a)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : DistribLattice \u03b1\ninst\u271d\u00b3 : OrderBot \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidablePred SupIrred\ninst\u271d : DecidableEq \u03b1\na : \u03b1\n\u22a2 OrderIso.lowerSetSupIrred a = OrderIso.lowerSetSupIrred a"}, {"tactic": "have : Subsingleton (OrderBot \u03b1) := inferInstance", "annotated_tactic": ["have : <a>Subsingleton</a> (<a>OrderBot</a> \u03b1) := <a>inferInstance</a>", [{"full_name": "Subsingleton", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1009, 7], "def_end_pos": [1009, 19]}, {"full_name": "OrderBot", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [202, 7], "def_end_pos": [202, 15]}, {"full_name": "inferInstance", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [99, 8], "def_end_pos": [99, 21]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : DistribLattice \u03b1\ninst\u271d\u00b3 : OrderBot \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidablePred SupIrred\ninst\u271d : DecidableEq \u03b1\na : \u03b1\n\u22a2 OrderIso.lowerSetSupIrred a = OrderIso.lowerSetSupIrred a", "state_after": "\u03b1 : Type u_1\ninst\u271d\u2074 : DistribLattice \u03b1\ninst\u271d\u00b3 : OrderBot \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidablePred SupIrred\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nthis : Subsingleton (OrderBot \u03b1)\n\u22a2 OrderIso.lowerSetSupIrred a = OrderIso.lowerSetSupIrred a"}, {"tactic": "convert rfl", "annotated_tactic": ["convert <a>rfl</a>", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u2074 : DistribLattice \u03b1\ninst\u271d\u00b3 : OrderBot \u03b1\ninst\u271d\u00b2 : Fintype \u03b1\ninst\u271d\u00b9 : DecidablePred SupIrred\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nthis : Subsingleton (OrderBot \u03b1)\n\u22a2 OrderIso.lowerSetSupIrred a = OrderIso.lowerSetSupIrred a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean", "full_name": "Batteries.RBNode.Path.zoom_del", "start": [711, 1], "end": [727, 25], "traced_tactics": [{"tactic": "unfold RBNode.del", "annotated_tactic": ["unfold <a>RBNode.del</a>", [{"full_name": "Batteries.RBNode.del", "def_path": ".lake/packages/batteries/Batteries/Data/RBMap/Basic.lean", "def_pos": [379, 19], "def_end_pos": [379, 22]}]], "state_before": "\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nt : RBNode \u03b1\n\u22a2 zoom cut t path = (t', path') \u2192\n    path.del (RBNode.del cut t)\n        (match t with\n        | node c l v r => c\n        | x => red) =\n      path'.del t'.delRoot\n        (match t' with\n        | node c l v r => c\n        | x => red)", "state_after": "\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nt : RBNode \u03b1\n\u22a2 zoom cut t path = (t', path') \u2192\n    path.del\n        (match t with\n        | nil => nil\n        | node c a y b =>\n          match cut y with\n          | Ordering.lt =>\n            match a.isBlack with\n            | black => (RBNode.del cut a).balLeft y b\n            | red => node red (RBNode.del cut a) y b\n          | Ordering.gt =>\n            match b.isBlack with\n            | black => a.balRight y (RBNode.del cut b)\n            | red => node red a y (RBNode.del cut b)\n          | Ordering.eq => a.append b)\n        (match t with\n        | node c l v r => c\n        | x => red) =\n      path'.del t'.delRoot\n        (match t' with\n        | node c l v r => c\n        | x => red)"}, {"tactic": "split <;> simp [zoom]", "annotated_tactic": ["split <;> simp [<a>zoom</a>]", [{"full_name": "Batteries.RBNode.zoom", "def_path": ".lake/packages/batteries/Batteries/Data/RBMap/Basic.lean", "def_pos": [468, 19], "def_end_pos": [468, 23]}]], "state_before": "\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nt : RBNode \u03b1\n\u22a2 zoom cut t path = (t', path') \u2192\n    path.del\n        (match t with\n        | nil => nil\n        | node c a y b =>\n          match cut y with\n          | Ordering.lt =>\n            match a.isBlack with\n            | black => (RBNode.del cut a).balLeft y b\n            | red => node red (RBNode.del cut a) y b\n          | Ordering.gt =>\n            match b.isBlack with\n            | black => a.balRight y (RBNode.del cut b)\n            | red => node red a y (RBNode.del cut b)\n          | Ordering.eq => a.append b)\n        (match t with\n        | node c l v r => c\n        | x => red) =\n      path'.del t'.delRoot\n        (match t' with\n        | node c l v r => c\n        | x => red)", "state_after": "case h_1\n\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d : RBNode \u03b1\n\u22a2 nil = t' \u2192\n    path = path' \u2192\n      path.del nil red =\n        path'.del t'.delRoot\n          (match t' with\n          | node c l v r => c\n          | x => red)\n\ncase h_2\n\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d : RBNode \u03b1\na\u271d\u00b3 : RBColor\na\u271d\u00b2 : RBNode \u03b1\na\u271d\u00b9 : \u03b1\na\u271d : RBNode \u03b1\n\u22a2 (match cut a\u271d\u00b9 with\n      | Ordering.lt => zoom cut a\u271d\u00b2 (left a\u271d\u00b3 path a\u271d\u00b9 a\u271d)\n      | Ordering.gt => zoom cut a\u271d (right a\u271d\u00b3 a\u271d\u00b2 a\u271d\u00b9 path)\n      | Ordering.eq => (node a\u271d\u00b3 a\u271d\u00b2 a\u271d\u00b9 a\u271d, path)) =\n      (t', path') \u2192\n    path.del\n        (match cut a\u271d\u00b9 with\n        | Ordering.lt =>\n          match a\u271d\u00b2.isBlack with\n          | black => (RBNode.del cut a\u271d\u00b2).balLeft a\u271d\u00b9 a\u271d\n          | red => node red (RBNode.del cut a\u271d\u00b2) a\u271d\u00b9 a\u271d\n        | Ordering.gt =>\n          match a\u271d.isBlack with\n          | black => a\u271d\u00b2.balRight a\u271d\u00b9 (RBNode.del cut a\u271d)\n          | red => node red a\u271d\u00b2 a\u271d\u00b9 (RBNode.del cut a\u271d)\n        | Ordering.eq => a\u271d\u00b2.append a\u271d)\n        a\u271d\u00b3 =\n      path'.del t'.delRoot\n        (match t' with\n        | node c l v r => c\n        | x => red)"}, {"tactic": "intro | rfl, rfl => rfl", "annotated_tactic": ["intro | <a>rfl</a>, <a>rfl</a> => rfl", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case h_1\n\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d : RBNode \u03b1\n\u22a2 nil = t' \u2192\n    path = path' \u2192\n      path.del nil red =\n        path'.del t'.delRoot\n          (match t' with\n          | node c l v r => c\n          | x => red)", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d\u00b2 : RBNode \u03b1\nx\u271d\u00b9 : nil = t'\nx\u271d : path = path'\n\u22a2 path.del nil red =\n    path.del nil.delRoot\n      (match nil with\n      | node c l v r => c\n      | x => red)", "state_after": "no goals"}, {"tactic": "split", "annotated_tactic": ["split", []], "state_before": "\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d : RBNode \u03b1\nc : RBColor\na : RBNode \u03b1\ny : \u03b1\nb : RBNode \u03b1\n\u22a2 (match cut y with\n      | Ordering.lt => zoom cut a (left c path y b)\n      | Ordering.gt => zoom cut b (right c a y path)\n      | Ordering.eq => (node c a y b, path)) =\n      (t', path') \u2192\n    path.del\n        (match cut y with\n        | Ordering.lt =>\n          match a.isBlack with\n          | black => (RBNode.del cut a).balLeft y b\n          | red => node red (RBNode.del cut a) y b\n        | Ordering.gt =>\n          match b.isBlack with\n          | black => a.balRight y (RBNode.del cut b)\n          | red => node red a y (RBNode.del cut b)\n        | Ordering.eq => a.append b)\n        c =\n      path'.del t'.delRoot\n        (match t' with\n        | node c l v r => c\n        | x => red)", "state_after": "case h_1\n\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d\u00b9 : RBNode \u03b1\nc : RBColor\na : RBNode \u03b1\ny : \u03b1\nb : RBNode \u03b1\nx\u271d : Ordering\nheq\u271d : cut y = Ordering.lt\n\u22a2 zoom cut a (left c path y b) = (t', path') \u2192\n    path.del\n        (match a.isBlack with\n        | black => (RBNode.del cut a).balLeft y b\n        | red => node red (RBNode.del cut a) y b)\n        c =\n      path'.del t'.delRoot\n        (match t' with\n        | node c l v r => c\n        | x => red)\n\ncase h_2\n\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d\u00b9 : RBNode \u03b1\nc : RBColor\na : RBNode \u03b1\ny : \u03b1\nb : RBNode \u03b1\nx\u271d : Ordering\nheq\u271d : cut y = Ordering.gt\n\u22a2 zoom cut b (right c a y path) = (t', path') \u2192\n    path.del\n        (match b.isBlack with\n        | black => a.balRight y (RBNode.del cut b)\n        | red => node red a y (RBNode.del cut b))\n        c =\n      path'.del t'.delRoot\n        (match t' with\n        | node c l v r => c\n        | x => red)\n\ncase h_3\n\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d\u00b9 : RBNode \u03b1\nc : RBColor\na : RBNode \u03b1\ny : \u03b1\nb : RBNode \u03b1\nx\u271d : Ordering\nheq\u271d : cut y = Ordering.eq\n\u22a2 (node c a y b, path) = (t', path') \u2192\n    path.del (a.append b) c =\n      path'.del t'.delRoot\n        (match t' with\n        | node c l v r => c\n        | x => red)"}, {"tactic": "have IH := @zoom_del (t := a)", "annotated_tactic": ["have IH := @zoom_del (t := a)", []], "state_before": "case h_1\n\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d\u00b9 : RBNode \u03b1\nc : RBColor\na : RBNode \u03b1\ny : \u03b1\nb : RBNode \u03b1\nx\u271d : Ordering\nheq\u271d : cut y = Ordering.lt\n\u22a2 zoom cut a (left c path y b) = (t', path') \u2192\n    path.del\n        (match a.isBlack with\n        | black => (RBNode.del cut a).balLeft y b\n        | red => node red (RBNode.del cut a) y b)\n        c =\n      path'.del t'.delRoot\n        (match t' with\n        | node c l v r => c\n        | x => red)", "state_after": "case h_1\n\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d\u00b9 : RBNode \u03b1\nc : RBColor\na : RBNode \u03b1\ny : \u03b1\nb : RBNode \u03b1\nx\u271d : Ordering\nheq\u271d : cut y = Ordering.lt\nIH :\n  \u2200 (cut : \u03b1 \u2192 Ordering) (path : Path \u03b1) (t' : RBNode \u03b1) (path' : Path \u03b1),\n    zoom cut a path = (t', path') \u2192\n      path.del (RBNode.del cut a)\n          (match a with\n          | node c l v r => c\n          | x => red) =\n        path'.del t'.delRoot\n          (match t' with\n          | node c l v r => c\n          | x => red)\n\u22a2 zoom cut a (left c path y b) = (t', path') \u2192\n    path.del\n        (match a.isBlack with\n        | black => (RBNode.del cut a).balLeft y b\n        | red => node red (RBNode.del cut a) y b)\n        c =\n      path'.del t'.delRoot\n        (match t' with\n        | node c l v r => c\n        | x => red)"}, {"tactic": "match a with\n| nil => intro | rfl => rfl\n| node black .. | node red .. => apply IH", "annotated_tactic": ["match a with\n      | <a>nil</a> => intro | <a>rfl</a> => rfl\n      | <a>node</a> <a>black</a> .. | <a>node</a> <a>red</a> .. => apply IH", [{"full_name": "Batteries.RBNode.nil", "def_path": ".lake/packages/batteries/Batteries/Data/RBMap/Basic.lean", "def_pos": [46, 5], "def_end_pos": [46, 8]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "Batteries.RBNode.node", "def_path": ".lake/packages/batteries/Batteries/Data/RBMap/Basic.lean", "def_pos": [50, 5], "def_end_pos": [50, 9]}, {"full_name": "Batteries.RBColor.black", "def_path": ".lake/packages/batteries/Batteries/Data/RBMap/Basic.lean", "def_pos": [34, 5], "def_end_pos": [34, 10]}, {"full_name": "Batteries.RBNode.node", "def_path": ".lake/packages/batteries/Batteries/Data/RBMap/Basic.lean", "def_pos": [50, 5], "def_end_pos": [50, 9]}, {"full_name": "Batteries.RBColor.red", "def_path": ".lake/packages/batteries/Batteries/Data/RBMap/Basic.lean", "def_pos": [32, 5], "def_end_pos": [32, 8]}]], "state_before": "case h_1\n\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d\u00b9 : RBNode \u03b1\nc : RBColor\na : RBNode \u03b1\ny : \u03b1\nb : RBNode \u03b1\nx\u271d : Ordering\nheq\u271d : cut y = Ordering.lt\nIH :\n  \u2200 (cut : \u03b1 \u2192 Ordering) (path : Path \u03b1) (t' : RBNode \u03b1) (path' : Path \u03b1),\n    zoom cut a path = (t', path') \u2192\n      path.del (RBNode.del cut a)\n          (match a with\n          | node c l v r => c\n          | x => red) =\n        path'.del t'.delRoot\n          (match t' with\n          | node c l v r => c\n          | x => red)\n\u22a2 zoom cut a (left c path y b) = (t', path') \u2192\n    path.del\n        (match a.isBlack with\n        | black => (RBNode.del cut a).balLeft y b\n        | red => node red (RBNode.del cut a) y b)\n        c =\n      path'.del t'.delRoot\n        (match t' with\n        | node c l v r => c\n        | x => red)", "state_after": "no goals"}, {"tactic": "intro | rfl => rfl", "annotated_tactic": ["intro | <a>rfl</a> => rfl", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d\u00b9 : RBNode \u03b1\nc : RBColor\na : RBNode \u03b1\ny : \u03b1\nb : RBNode \u03b1\nx\u271d : Ordering\nheq\u271d : cut y = Ordering.lt\nIH :\n  \u2200 (cut : \u03b1 \u2192 Ordering) (path : Path \u03b1) (t' : RBNode \u03b1) (path' : Path \u03b1),\n    zoom cut nil path = (t', path') \u2192\n      path.del (RBNode.del cut nil)\n          (match nil with\n          | node c l v r => c\n          | x => red) =\n        path'.del t'.delRoot\n          (match t' with\n          | node c l v r => c\n          | x => red)\n\u22a2 zoom cut nil (left c path y b) = (t', path') \u2192\n    path.del\n        (match nil.isBlack with\n        | black => (RBNode.del cut nil).balLeft y b\n        | red => node red (RBNode.del cut nil) y b)\n        c =\n      path'.del t'.delRoot\n        (match t' with\n        | node c l v r => c\n        | x => red)", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d\u00b2 : RBNode \u03b1\nc : RBColor\na : RBNode \u03b1\ny : \u03b1\nb : RBNode \u03b1\nx\u271d\u00b9 : Ordering\nheq\u271d : cut y = Ordering.lt\nIH :\n  \u2200 (cut : \u03b1 \u2192 Ordering) (path : Path \u03b1) (t' : RBNode \u03b1) (path' : Path \u03b1),\n    zoom cut nil path = (t', path') \u2192\n      path.del (RBNode.del cut nil)\n          (match nil with\n          | node c l v r => c\n          | x => red) =\n        path'.del t'.delRoot\n          (match t' with\n          | node c l v r => c\n          | x => red)\nx\u271d : zoom cut nil (left c path y b) = (t', path')\n\u22a2 path.del\n      (match nil.isBlack with\n      | black => (RBNode.del cut nil).balLeft y b\n      | red => node red (RBNode.del cut nil) y b)\n      c =\n    (left c path y b).del nil.delRoot\n      (match nil with\n      | node c l v r => c\n      | x => red)", "state_after": "no goals"}, {"tactic": "apply IH", "annotated_tactic": ["apply IH", []], "state_before": "\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d\u00b9 : RBNode \u03b1\nc : RBColor\na : RBNode \u03b1\ny : \u03b1\nb : RBNode \u03b1\nx\u271d : Ordering\nheq\u271d : cut y = Ordering.lt\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nIH :\n  \u2200 (cut : \u03b1 \u2192 Ordering) (path : Path \u03b1) (t' : RBNode \u03b1) (path' : Path \u03b1),\n    zoom cut (node red l\u271d v\u271d r\u271d) path = (t', path') \u2192\n      path.del (RBNode.del cut (node red l\u271d v\u271d r\u271d))\n          (match node red l\u271d v\u271d r\u271d with\n          | node c l v r => c\n          | x => red) =\n        path'.del t'.delRoot\n          (match t' with\n          | node c l v r => c\n          | x => red)\n\u22a2 zoom cut (node red l\u271d v\u271d r\u271d) (left c path y b) = (t', path') \u2192\n    path.del\n        (match (node red l\u271d v\u271d r\u271d).isBlack with\n        | black => (RBNode.del cut (node red l\u271d v\u271d r\u271d)).balLeft y b\n        | red => node red (RBNode.del cut (node red l\u271d v\u271d r\u271d)) y b)\n        c =\n      path'.del t'.delRoot\n        (match t' with\n        | node c l v r => c\n        | x => red)", "state_after": "no goals"}, {"tactic": "have IH := @zoom_del (t := b)", "annotated_tactic": ["have IH := @zoom_del (t := b)", []], "state_before": "case h_2\n\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d\u00b9 : RBNode \u03b1\nc : RBColor\na : RBNode \u03b1\ny : \u03b1\nb : RBNode \u03b1\nx\u271d : Ordering\nheq\u271d : cut y = Ordering.gt\n\u22a2 zoom cut b (right c a y path) = (t', path') \u2192\n    path.del\n        (match b.isBlack with\n        | black => a.balRight y (RBNode.del cut b)\n        | red => node red a y (RBNode.del cut b))\n        c =\n      path'.del t'.delRoot\n        (match t' with\n        | node c l v r => c\n        | x => red)", "state_after": "case h_2\n\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d\u00b9 : RBNode \u03b1\nc : RBColor\na : RBNode \u03b1\ny : \u03b1\nb : RBNode \u03b1\nx\u271d : Ordering\nheq\u271d : cut y = Ordering.gt\nIH :\n  \u2200 (cut : \u03b1 \u2192 Ordering) (path : Path \u03b1) (t' : RBNode \u03b1) (path' : Path \u03b1),\n    zoom cut b path = (t', path') \u2192\n      path.del (RBNode.del cut b)\n          (match b with\n          | node c l v r => c\n          | x => red) =\n        path'.del t'.delRoot\n          (match t' with\n          | node c l v r => c\n          | x => red)\n\u22a2 zoom cut b (right c a y path) = (t', path') \u2192\n    path.del\n        (match b.isBlack with\n        | black => a.balRight y (RBNode.del cut b)\n        | red => node red a y (RBNode.del cut b))\n        c =\n      path'.del t'.delRoot\n        (match t' with\n        | node c l v r => c\n        | x => red)"}, {"tactic": "match b with\n| nil => intro | rfl => rfl\n| node black .. | node red .. => apply IH", "annotated_tactic": ["match b with\n      | <a>nil</a> => intro | <a>rfl</a> => rfl\n      | <a>node</a> <a>black</a> .. | <a>node</a> <a>red</a> .. => apply IH", [{"full_name": "Batteries.RBNode.nil", "def_path": ".lake/packages/batteries/Batteries/Data/RBMap/Basic.lean", "def_pos": [46, 5], "def_end_pos": [46, 8]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "Batteries.RBNode.node", "def_path": ".lake/packages/batteries/Batteries/Data/RBMap/Basic.lean", "def_pos": [50, 5], "def_end_pos": [50, 9]}, {"full_name": "Batteries.RBColor.black", "def_path": ".lake/packages/batteries/Batteries/Data/RBMap/Basic.lean", "def_pos": [34, 5], "def_end_pos": [34, 10]}, {"full_name": "Batteries.RBNode.node", "def_path": ".lake/packages/batteries/Batteries/Data/RBMap/Basic.lean", "def_pos": [50, 5], "def_end_pos": [50, 9]}, {"full_name": "Batteries.RBColor.red", "def_path": ".lake/packages/batteries/Batteries/Data/RBMap/Basic.lean", "def_pos": [32, 5], "def_end_pos": [32, 8]}]], "state_before": "case h_2\n\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d\u00b9 : RBNode \u03b1\nc : RBColor\na : RBNode \u03b1\ny : \u03b1\nb : RBNode \u03b1\nx\u271d : Ordering\nheq\u271d : cut y = Ordering.gt\nIH :\n  \u2200 (cut : \u03b1 \u2192 Ordering) (path : Path \u03b1) (t' : RBNode \u03b1) (path' : Path \u03b1),\n    zoom cut b path = (t', path') \u2192\n      path.del (RBNode.del cut b)\n          (match b with\n          | node c l v r => c\n          | x => red) =\n        path'.del t'.delRoot\n          (match t' with\n          | node c l v r => c\n          | x => red)\n\u22a2 zoom cut b (right c a y path) = (t', path') \u2192\n    path.del\n        (match b.isBlack with\n        | black => a.balRight y (RBNode.del cut b)\n        | red => node red a y (RBNode.del cut b))\n        c =\n      path'.del t'.delRoot\n        (match t' with\n        | node c l v r => c\n        | x => red)", "state_after": "no goals"}, {"tactic": "intro | rfl => rfl", "annotated_tactic": ["intro | <a>rfl</a> => rfl", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d\u00b9 : RBNode \u03b1\nc : RBColor\na : RBNode \u03b1\ny : \u03b1\nb : RBNode \u03b1\nx\u271d : Ordering\nheq\u271d : cut y = Ordering.gt\nIH :\n  \u2200 (cut : \u03b1 \u2192 Ordering) (path : Path \u03b1) (t' : RBNode \u03b1) (path' : Path \u03b1),\n    zoom cut nil path = (t', path') \u2192\n      path.del (RBNode.del cut nil)\n          (match nil with\n          | node c l v r => c\n          | x => red) =\n        path'.del t'.delRoot\n          (match t' with\n          | node c l v r => c\n          | x => red)\n\u22a2 zoom cut nil (right c a y path) = (t', path') \u2192\n    path.del\n        (match nil.isBlack with\n        | black => a.balRight y (RBNode.del cut nil)\n        | red => node red a y (RBNode.del cut nil))\n        c =\n      path'.del t'.delRoot\n        (match t' with\n        | node c l v r => c\n        | x => red)", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d\u00b2 : RBNode \u03b1\nc : RBColor\na : RBNode \u03b1\ny : \u03b1\nb : RBNode \u03b1\nx\u271d\u00b9 : Ordering\nheq\u271d : cut y = Ordering.gt\nIH :\n  \u2200 (cut : \u03b1 \u2192 Ordering) (path : Path \u03b1) (t' : RBNode \u03b1) (path' : Path \u03b1),\n    zoom cut nil path = (t', path') \u2192\n      path.del (RBNode.del cut nil)\n          (match nil with\n          | node c l v r => c\n          | x => red) =\n        path'.del t'.delRoot\n          (match t' with\n          | node c l v r => c\n          | x => red)\nx\u271d : zoom cut nil (right c a y path) = (t', path')\n\u22a2 path.del\n      (match nil.isBlack with\n      | black => a.balRight y (RBNode.del cut nil)\n      | red => node red a y (RBNode.del cut nil))\n      c =\n    (right c a y path).del nil.delRoot\n      (match nil with\n      | node c l v r => c\n      | x => red)", "state_after": "no goals"}, {"tactic": "apply IH", "annotated_tactic": ["apply IH", []], "state_before": "\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d\u00b9 : RBNode \u03b1\nc : RBColor\na : RBNode \u03b1\ny : \u03b1\nb : RBNode \u03b1\nx\u271d : Ordering\nheq\u271d : cut y = Ordering.gt\nl\u271d : RBNode \u03b1\nv\u271d : \u03b1\nr\u271d : RBNode \u03b1\nIH :\n  \u2200 (cut : \u03b1 \u2192 Ordering) (path : Path \u03b1) (t' : RBNode \u03b1) (path' : Path \u03b1),\n    zoom cut (node red l\u271d v\u271d r\u271d) path = (t', path') \u2192\n      path.del (RBNode.del cut (node red l\u271d v\u271d r\u271d))\n          (match node red l\u271d v\u271d r\u271d with\n          | node c l v r => c\n          | x => red) =\n        path'.del t'.delRoot\n          (match t' with\n          | node c l v r => c\n          | x => red)\n\u22a2 zoom cut (node red l\u271d v\u271d r\u271d) (right c a y path) = (t', path') \u2192\n    path.del\n        (match (node red l\u271d v\u271d r\u271d).isBlack with\n        | black => a.balRight y (RBNode.del cut (node red l\u271d v\u271d r\u271d))\n        | red => node red a y (RBNode.del cut (node red l\u271d v\u271d r\u271d)))\n        c =\n      path'.del t'.delRoot\n        (match t' with\n        | node c l v r => c\n        | x => red)", "state_after": "no goals"}, {"tactic": "intro | rfl => rfl", "annotated_tactic": ["intro | <a>rfl</a> => rfl", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case h_3\n\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d\u00b9 : RBNode \u03b1\nc : RBColor\na : RBNode \u03b1\ny : \u03b1\nb : RBNode \u03b1\nx\u271d : Ordering\nheq\u271d : cut y = Ordering.eq\n\u22a2 (node c a y b, path) = (t', path') \u2192\n    path.del (a.append b) c =\n      path'.del t'.delRoot\n        (match t' with\n        | node c l v r => c\n        | x => red)", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\ncut : \u03b1 \u2192 Ordering\npath : Path \u03b1\nt' : RBNode \u03b1\npath' : Path \u03b1\nx\u271d\u00b2 : RBNode \u03b1\nc : RBColor\na : RBNode \u03b1\ny : \u03b1\nb : RBNode \u03b1\nx\u271d\u00b9 : Ordering\nheq\u271d : cut y = Ordering.eq\nx\u271d : (node c a y b, path) = (t', path')\n\u22a2 path.del (a.append b) c =\n    path.del (node c a y b).delRoot\n      (match node c a y b with\n      | node c l v r => c\n      | x => red)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Types.lean", "full_name": "CategoryTheory.Limits.Types.Pushout.condition", "start": [766, 1], "end": [768, 35], "traced_tactics": [{"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "S X\u2081 X\u2082 : Type u\nf : S \u27f6 X\u2081\ng : S \u27f6 X\u2082\n\u22a2 f \u226b inl f g = g \u226b inr f g", "state_after": "case h\nS X\u2081 X\u2082 : Type u\nf : S \u27f6 X\u2081\ng : S \u27f6 X\u2082\nx : S\n\u22a2 (f \u226b inl f g) x = (g \u226b inr f g) x"}, {"tactic": "exact Quot.sound (Rel.inl_inr x)", "annotated_tactic": ["exact <a>Quot.sound</a> (<a>Rel.inl_inr</a> x)", [{"full_name": "Quot.sound", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1471, 7], "def_end_pos": [1471, 12]}, {"full_name": "CategoryTheory.Limits.Types.Pushout.Rel.inl_inr", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Types.lean", "def_pos": [740, 5], "def_end_pos": [740, 12]}]], "state_before": "case h\nS X\u2081 X\u2082 : Type u\nf : S \u27f6 X\u2081\ng : S \u27f6 X\u2082\nx : S\n\u22a2 (f \u226b inl f g) x = (g \u226b inr f g) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Constructions/Over/Products.lean", "full_name": "CategoryTheory.Over.ConstructProducts.has_over_limit_discrete_of_widePullback_limit", "start": [129, 1], "end": [134, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/ZFC/Basic.lean", "full_name": "ZFSet.mem_diff", "start": [1138, 1], "end": [1139, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "full_name": "circleIntegral.integral_sub_inv_smul_sub_smul", "start": [370, 1], "end": [376, 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": [212, 9], "def_end_pos": [212, 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": [87, 5], "def_end_pos": [87, 14]}, {"full_name": "Set.Countable", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [45, 15], "def_end_pos": [45, 24]}, {"full_name": "Set.countable_singleton", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [167, 17], "def_end_pos": [167, 36]}, {"full_name": "Set.Countable.preimage_circleMap", "def_path": "Mathlib/MeasureTheory/Integral/CircleIntegral.lean", "def_pos": [95, 9], "def_end_pos": [95, 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 : (circleMap c R \u207b\u00b9' {w}).Countable\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": [1032, 9], "def_end_pos": [1032, 26]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1158, 9], "def_end_pos": [1158, 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 : (circleMap c R \u207b\u00b9' {w}).Countable\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 : (circleMap c R \u207b\u00b9' {w}).Countable\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2209 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": [87, 5], "def_end_pos": [87, 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 : (circleMap c R \u207b\u00b9' {w}).Countable\n\u03b8 : \u211d\nh\u03b8 : \u03b8 \u2209 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 : (circleMap c R \u207b\u00b9' {w}).Countable\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": [207, 9], "def_end_pos": [207, 23]}, {"full_name": "sub_ne_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1082, 3], "def_end_pos": [1082, 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 : (circleMap c R \u207b\u00b9' {w}).Countable\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": [361, 9], "def_end_pos": [361, 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": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Countable.lean", "full_name": "Set.exists_seq_iSup_eq_top_iff_countable", "start": [193, 1], "end": [208, 42], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\n\u22a2 (\u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4) \u2194 \u2203 S, S.Countable \u2227 (\u2200 s \u2208 S, p s) \u2227 sSup S = \u22a4", "state_after": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\n\u22a2 (\u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4) \u2192 \u2203 S, S.Countable \u2227 (\u2200 s \u2208 S, p s) \u2227 sSup S = \u22a4\n\ncase mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\n\u22a2 (\u2203 S, S.Countable \u2227 (\u2200 s \u2208 S, p s) \u2227 sSup S = \u22a4) \u2192 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4"}, {"tactic": "rintro \u27e8s, hps, hs\u27e9", "annotated_tactic": ["rintro \u27e8s, hps, hs\u27e9", []], "state_before": "case mp\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\n\u22a2 (\u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4) \u2192 \u2203 S, S.Countable \u2227 (\u2200 s \u2208 S, p s) \u2227 sSup S = \u22a4", "state_after": "case mp.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\ns : \u2115 \u2192 \u03b1\nhps : \u2200 (n : \u2115), p (s n)\nhs : \u2a06 n, s n = \u22a4\n\u22a2 \u2203 S, S.Countable \u2227 (\u2200 s \u2208 S, p s) \u2227 sSup S = \u22a4"}, {"tactic": "refine \u27e8range s, countable_range s, forall_mem_range.2 hps, ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>range</a> s, <a>countable_range</a> s, <a>forall_mem_range</a>.2 hps, ?_\u27e9", [{"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [157, 5], "def_end_pos": [157, 10]}, {"full_name": "Set.countable_range", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [125, 9], "def_end_pos": [125, 24]}, {"full_name": "Set.forall_mem_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [662, 9], "def_end_pos": [662, 25]}]], "state_before": "case mp.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\ns : \u2115 \u2192 \u03b1\nhps : \u2200 (n : \u2115), p (s n)\nhs : \u2a06 n, s n = \u22a4\n\u22a2 \u2203 S, S.Countable \u2227 (\u2200 s \u2208 S, p s) \u2227 sSup S = \u22a4", "state_after": "case mp.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\ns : \u2115 \u2192 \u03b1\nhps : \u2200 (n : \u2115), p (s n)\nhs : \u2a06 n, s n = \u22a4\n\u22a2 sSup (range s) = \u22a4"}, {"tactic": "rwa [sSup_range]", "annotated_tactic": ["rwa [<a>sSup_range</a>]", [{"full_name": "sSup_range", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [561, 9], "def_end_pos": [561, 19]}]], "state_before": "case mp.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\ns : \u2115 \u2192 \u03b1\nhps : \u2200 (n : \u2115), p (s n)\nhs : \u2a06 n, s n = \u22a4\n\u22a2 sSup (range s) = \u22a4", "state_after": "no goals"}, {"tactic": "rintro \u27e8S, hSc, hps, hS\u27e9", "annotated_tactic": ["rintro \u27e8S, hSc, hps, hS\u27e9", []], "state_before": "case mpr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\n\u22a2 (\u2203 S, S.Countable \u2227 (\u2200 s \u2208 S, p s) \u2227 sSup S = \u22a4) \u2192 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4", "state_after": "case mpr.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nS : Set \u03b1\nhSc : S.Countable\nhps : \u2200 s \u2208 S, p s\nhS : sSup S = \u22a4\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4"}, {"tactic": "rcases eq_empty_or_nonempty S with (rfl | hne)", "annotated_tactic": ["rcases <a>eq_empty_or_nonempty</a> S with (rfl | hne)", [{"full_name": "Set.eq_empty_or_nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [605, 9], "def_end_pos": [605, 29]}]], "state_before": "case mpr.intro.intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nS : Set \u03b1\nhSc : S.Countable\nhps : \u2200 s \u2208 S, p s\nhS : sSup S = \u22a4\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4", "state_after": "case mpr.intro.intro.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nhSc : \u2205.Countable\nhps : \u2200 s \u2208 \u2205, p s\nhS : sSup \u2205 = \u22a4\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4\n\ncase mpr.intro.intro.intro.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nS : Set \u03b1\nhSc : S.Countable\nhps : \u2200 s \u2208 S, p s\nhS : sSup S = \u22a4\nhne : S.Nonempty\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4"}, {"tactic": "rw [sSup_empty] at hS", "annotated_tactic": ["rw [<a>sSup_empty</a>] at hS", [{"full_name": "sSup_empty", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [419, 9], "def_end_pos": [419, 19]}]], "state_before": "case mpr.intro.intro.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nhSc : \u2205.Countable\nhps : \u2200 s \u2208 \u2205, p s\nhS : sSup \u2205 = \u22a4\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4", "state_after": "case mpr.intro.intro.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nhSc : \u2205.Countable\nhps : \u2200 s \u2208 \u2205, p s\nhS : \u22a5 = \u22a4\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4"}, {"tactic": "haveI := subsingleton_of_bot_eq_top hS", "annotated_tactic": ["haveI := <a>subsingleton_of_bot_eq_top</a> hS", [{"full_name": "subsingleton_of_bot_eq_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [651, 9], "def_end_pos": [651, 35]}]], "state_before": "case mpr.intro.intro.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nhSc : \u2205.Countable\nhps : \u2200 s \u2208 \u2205, p s\nhS : \u22a5 = \u22a4\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4", "state_after": "case mpr.intro.intro.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nhSc : \u2205.Countable\nhps : \u2200 s \u2208 \u2205, p s\nhS : \u22a5 = \u22a4\nthis : Subsingleton \u03b1\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4"}, {"tactic": "rcases h with \u27e8x, hx\u27e9", "annotated_tactic": ["rcases h with \u27e8x, hx\u27e9", []], "state_before": "case mpr.intro.intro.intro.inl\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nhSc : \u2205.Countable\nhps : \u2200 s \u2208 \u2205, p s\nhS : \u22a5 = \u22a4\nthis : Subsingleton \u03b1\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4", "state_after": "case mpr.intro.intro.intro.inl.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nhSc : \u2205.Countable\nhps : \u2200 s \u2208 \u2205, p s\nhS : \u22a5 = \u22a4\nthis : Subsingleton \u03b1\nx : \u03b1\nhx : p x\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4"}, {"tactic": "exact \u27e8fun _ => x, fun _ => hx, Subsingleton.elim _ _\u27e9", "annotated_tactic": ["exact \u27e8fun _ => x, fun _ => hx, <a>Subsingleton.elim</a> _ _\u27e9", [{"full_name": "Subsingleton.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1015, 19], "def_end_pos": [1015, 36]}]], "state_before": "case mpr.intro.intro.intro.inl.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nhSc : \u2205.Countable\nhps : \u2200 s \u2208 \u2205, p s\nhS : \u22a5 = \u22a4\nthis : Subsingleton \u03b1\nx : \u03b1\nhx : p x\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4", "state_after": "no goals"}, {"tactic": "rcases (Set.countable_iff_exists_surjective hne).1 hSc with \u27e8s, hs\u27e9", "annotated_tactic": ["rcases (<a>Set.countable_iff_exists_surjective</a> hne).1 hSc with \u27e8s, hs\u27e9", [{"full_name": "Set.countable_iff_exists_surjective", "def_path": "Mathlib/Data/Set/Countable.lean", "def_pos": [140, 19], "def_end_pos": [140, 50]}]], "state_before": "case mpr.intro.intro.intro.inr\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nS : Set \u03b1\nhSc : S.Countable\nhps : \u2200 s \u2208 S, p s\nhS : sSup S = \u22a4\nhne : S.Nonempty\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4", "state_after": "case mpr.intro.intro.intro.inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nS : Set \u03b1\nhSc : S.Countable\nhps : \u2200 s \u2208 S, p s\nhS : sSup S = \u22a4\nhne : S.Nonempty\ns : \u2115 \u2192 \u2191S\nhs : Surjective s\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4"}, {"tactic": "refine \u27e8fun n => s n, fun n => hps _ (s n).coe_prop, ?_\u27e9", "annotated_tactic": ["refine \u27e8fun n => s n, fun n => hps _ (s n).<a>coe_prop</a>, ?_\u27e9", [{"full_name": "Subtype.coe_prop", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [281, 9], "def_end_pos": [281, 17]}]], "state_before": "case mpr.intro.intro.intro.inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nS : Set \u03b1\nhSc : S.Countable\nhps : \u2200 s \u2208 S, p s\nhS : sSup S = \u22a4\nhne : S.Nonempty\ns : \u2115 \u2192 \u2191S\nhs : Surjective s\n\u22a2 \u2203 s, (\u2200 (n : \u2115), p (s n)) \u2227 \u2a06 n, s n = \u22a4", "state_after": "case mpr.intro.intro.intro.inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nS : Set \u03b1\nhSc : S.Countable\nhps : \u2200 s \u2208 S, p s\nhS : sSup S = \u22a4\nhne : S.Nonempty\ns : \u2115 \u2192 \u2191S\nhs : Surjective s\n\u22a2 \u2a06 n, (fun n => \u2191(s n)) n = \u22a4"}, {"tactic": "rwa [hs.iSup_comp, \u2190 sSup_eq_iSup']", "annotated_tactic": ["rwa [hs.iSup_comp, \u2190 <a>sSup_eq_iSup'</a>]", [{"full_name": "sSup_eq_iSup'", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [565, 9], "def_end_pos": [565, 22]}]], "state_before": "case mpr.intro.intro.intro.inr.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d : CompleteLattice \u03b1\np : \u03b1 \u2192 Prop\nh : \u2203 x, p x\nS : Set \u03b1\nhSc : S.Countable\nhps : \u2200 s \u2208 S, p s\nhS : sSup S = \u22a4\nhne : S.Nonempty\ns : \u2115 \u2192 \u2191S\nhs : Surjective s\n\u22a2 \u2a06 n, (fun n => \u2191(s n)) n = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Prod.lean", "full_name": "HasStrictFDerivAt.prod", "start": [52, 11], "end": [55, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_indicator_const\u2080", "start": [844, 1], "end": [846, 53], "traced_tactics": [{"tactic": "rw [lintegral_indicator\u2080 _ hs, setLIntegral_const]", "annotated_tactic": ["rw [<a>lintegral_indicator\u2080</a> _ hs, <a>setLIntegral_const</a>]", [{"full_name": "MeasureTheory.lintegral_indicator\u2080", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [833, 9], "def_end_pos": [833, 29]}, {"full_name": "MeasureTheory.setLIntegral_const", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [153, 9], "def_end_pos": [153, 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\ns : Set \u03b1\nhs : NullMeasurableSet s \u03bc\nc : \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1), s.indicator (fun x => c) a \u2202\u03bc = c * \u03bc s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/LocallyConstant/Basic.lean", "full_name": "IsLocallyConstant.iff_exists_open", "start": [83, 1], "end": [85, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Field/Subfield.lean", "full_name": "Subfield.coe_set_mk", "start": [202, 1], "end": [203, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Unitization.lean", "full_name": "Unitization.snd_smul", "start": [264, 1], "end": [265, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean", "full_name": "CategoryTheory.Limits.biproduct.whiskerEquiv_inv_eq_lift", "start": [700, 1], "end": [716, 16], "traced_tactics": [{"tactic": "simp only [whiskerEquiv_inv]", "annotated_tactic": ["simp only [<a>whiskerEquiv_inv</a>]", [{"full_name": "CategoryTheory.Limits.biproduct.whiskerEquiv_inv", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean", "def_pos": [678, 3], "def_end_pos": [678, 8]}]], "state_before": "J : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\n\u22a2 (whiskerEquiv e w).inv = lift fun j => \u03c0 g (e j) \u226b (w j).hom", "state_after": "J : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\n\u22a2 (desc fun k => eqToHom \u22ef \u226b (w (e.symm k)).hom \u226b \u03b9 f (e.symm k)) = lift fun j => \u03c0 g (e j) \u226b (w j).hom"}, {"tactic": "ext j k", "annotated_tactic": ["ext j k", []], "state_before": "J : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\n\u22a2 (desc fun k => eqToHom \u22ef \u226b (w (e.symm k)).hom \u226b \u03b9 f (e.symm k)) = lift fun j => \u03c0 g (e j) \u226b (w j).hom", "state_after": "case w.w\nJ : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\nk : K\n\u22a2 \u03b9 g k \u226b (desc fun k => eqToHom \u22ef \u226b (w (e.symm k)).hom \u226b \u03b9 f (e.symm k)) \u226b \u03c0 f j =\n    \u03b9 g k \u226b (lift fun j => \u03c0 g (e j) \u226b (w j).hom) \u226b \u03c0 f j"}, {"tactic": "by_cases h : k = e j", "annotated_tactic": ["by_cases h : k = e j", []], "state_before": "case w.w\nJ : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\nk : K\n\u22a2 \u03b9 g k \u226b (desc fun k => eqToHom \u22ef \u226b (w (e.symm k)).hom \u226b \u03b9 f (e.symm k)) \u226b \u03c0 f j =\n    \u03b9 g k \u226b (lift fun j => \u03c0 g (e j) \u226b (w j).hom) \u226b \u03c0 f j", "state_after": "case pos\nJ : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\nk : K\nh : k = e j\n\u22a2 \u03b9 g k \u226b (desc fun k => eqToHom \u22ef \u226b (w (e.symm k)).hom \u226b \u03b9 f (e.symm k)) \u226b \u03c0 f j =\n    \u03b9 g k \u226b (lift fun j => \u03c0 g (e j) \u226b (w j).hom) \u226b \u03c0 f j\n\ncase neg\nJ : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\nk : K\nh : \u00ack = e j\n\u22a2 \u03b9 g k \u226b (desc fun k => eqToHom \u22ef \u226b (w (e.symm k)).hom \u226b \u03b9 f (e.symm k)) \u226b \u03c0 f j =\n    \u03b9 g k \u226b (lift fun j => \u03c0 g (e j) \u226b (w j).hom) \u226b \u03c0 f j"}, {"tactic": "subst h", "annotated_tactic": ["subst h", []], "state_before": "case pos\nJ : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\nk : K\nh : k = e j\n\u22a2 \u03b9 g k \u226b (desc fun k => eqToHom \u22ef \u226b (w (e.symm k)).hom \u226b \u03b9 f (e.symm k)) \u226b \u03c0 f j =\n    \u03b9 g k \u226b (lift fun j => \u03c0 g (e j) \u226b (w j).hom) \u226b \u03c0 f j", "state_after": "case pos\nJ : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\n\u22a2 \u03b9 g (e j) \u226b (desc fun k => eqToHom \u22ef \u226b (w (e.symm k)).hom \u226b \u03b9 f (e.symm k)) \u226b \u03c0 f j =\n    \u03b9 g (e j) \u226b (lift fun j => \u03c0 g (e j) \u226b (w j).hom) \u226b \u03c0 f j"}, {"tactic": "simp only [\u03b9_desc_assoc, \u2190 eqToHom_iso_hom_naturality_assoc w (e.symm_apply_apply j).symm,\n  Equiv.symm_apply_apply, eqToHom_comp_\u03b9, Category.assoc, bicone_\u03b9_\u03c0_self, Category.comp_id,\n  lift_\u03c0, bicone_\u03b9_\u03c0_self_assoc]", "annotated_tactic": ["simp only [<a>\u03b9_desc_assoc</a>, \u2190 <a>eqToHom_iso_hom_naturality_assoc</a> w (e.symm_apply_apply j).<a>symm</a>,\n      <a>Equiv.symm_apply_apply</a>, <a>eqToHom_comp_\u03b9</a>, <a>Category.assoc</a>, <a>bicone_\u03b9_\u03c0_self</a>, <a>Category.comp_id</a>,\n      <a>lift_\u03c0</a>, <a>bicone_\u03b9_\u03c0_self_assoc</a>]", [{"full_name": "CategoryTheory.Limits.biproduct.\u03b9_desc_assoc", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean", "def_pos": [550, 3], "def_end_pos": [550, 25]}, {"full_name": "CategoryTheory.eqToHom_iso_hom_naturality_assoc", "def_path": "Mathlib/CategoryTheory/EqToHom.lean", "def_pos": [85, 3], "def_end_pos": [85, 40]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}, {"full_name": "Equiv.symm_apply_apply", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [282, 17], "def_end_pos": [282, 33]}, {"full_name": "CategoryTheory.Limits.biproduct.eqToHom_comp_\u03b9", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean", "def_pos": [521, 9], "def_end_pos": [521, 33]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.Limits.bicone_\u03b9_\u03c0_self", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean", "def_pos": [87, 9], "def_end_pos": [87, 24]}, {"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [179, 3], "def_end_pos": [179, 10]}, {"full_name": "CategoryTheory.Limits.biproduct.lift_\u03c0", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean", "def_pos": [546, 9], "def_end_pos": [546, 25]}, {"full_name": "CategoryTheory.Limits.bicone_\u03b9_\u03c0_self_assoc", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean", "def_pos": [86, 3], "def_end_pos": [86, 25]}]], "state_before": "case pos\nJ : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\n\u22a2 \u03b9 g (e j) \u226b (desc fun k => eqToHom \u22ef \u226b (w (e.symm k)).hom \u226b \u03b9 f (e.symm k)) \u226b \u03c0 f j =\n    \u03b9 g (e j) \u226b (lift fun j => \u03c0 g (e j) \u226b (w j).hom) \u226b \u03c0 f j", "state_after": "no goals"}, {"tactic": "simp only [\u03b9_desc_assoc, Category.assoc, ne_eq, lift_\u03c0]", "annotated_tactic": ["simp only [<a>\u03b9_desc_assoc</a>, <a>Category.assoc</a>, <a>ne_eq</a>, <a>lift_\u03c0</a>]", [{"full_name": "CategoryTheory.Limits.biproduct.\u03b9_desc_assoc", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean", "def_pos": [550, 3], "def_end_pos": [550, 25]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "ne_eq", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [89, 17], "def_end_pos": [89, 22]}, {"full_name": "CategoryTheory.Limits.biproduct.lift_\u03c0", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean", "def_pos": [546, 9], "def_end_pos": [546, 25]}]], "state_before": "case neg\nJ : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\nk : K\nh : \u00ack = e j\n\u22a2 \u03b9 g k \u226b (desc fun k => eqToHom \u22ef \u226b (w (e.symm k)).hom \u226b \u03b9 f (e.symm k)) \u226b \u03c0 f j =\n    \u03b9 g k \u226b (lift fun j => \u03c0 g (e j) \u226b (w j).hom) \u226b \u03c0 f j", "state_after": "case neg\nJ : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\nk : K\nh : \u00ack = e j\n\u22a2 eqToHom \u22ef \u226b (w (e.symm k)).hom \u226b \u03b9 f (e.symm k) \u226b \u03c0 f j = \u03b9 g k \u226b \u03c0 g (e j) \u226b (w j).hom"}, {"tactic": "rw [biproduct.\u03b9_\u03c0_ne, biproduct.\u03b9_\u03c0_ne_assoc]", "annotated_tactic": ["rw [<a>biproduct.\u03b9_\u03c0_ne</a>, <a>biproduct.\u03b9_\u03c0_ne_assoc</a>]", [{"full_name": "CategoryTheory.Limits.biproduct.\u03b9_\u03c0_ne", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean", "def_pos": [513, 9], "def_end_pos": [513, 25]}, {"full_name": "CategoryTheory.Limits.biproduct.\u03b9_\u03c0_ne_assoc", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean", "def_pos": [512, 3], "def_end_pos": [512, 25]}]], "state_before": "case neg\nJ : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\nk : K\nh : \u00ack = e j\n\u22a2 eqToHom \u22ef \u226b (w (e.symm k)).hom \u226b \u03b9 f (e.symm k) \u226b \u03c0 f j = \u03b9 g k \u226b \u03c0 g (e j) \u226b (w j).hom", "state_after": "case neg\nJ : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\nk : K\nh : \u00ack = e j\n\u22a2 eqToHom \u22ef \u226b (w (e.symm k)).hom \u226b 0 = 0 \u226b (w j).hom\n\ncase neg.h\nJ : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\nk : K\nh : \u00ack = e j\n\u22a2 k \u2260 e j\n\ncase neg.h\nJ : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\nk : K\nh : \u00ack = e j\n\u22a2 e.symm k \u2260 j"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case neg\nJ : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\nk : K\nh : \u00ack = e j\n\u22a2 eqToHom \u22ef \u226b (w (e.symm k)).hom \u226b 0 = 0 \u226b (w j).hom", "state_after": "no goals"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "case neg.h\nJ : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\nk : K\nh : \u00ack = e j\n\u22a2 k \u2260 e j", "state_after": "no goals"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "case neg.h\nJ : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nj : J\nk : K\nh : \u00ack = e j\n\u22a2 e.symm k \u2260 j", "state_after": "case neg.h\nJ : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nk : K\nh : \u00ack = e (e.symm k)\n\u22a2 False"}, {"tactic": "simp at h", "annotated_tactic": ["simp at h", []], "state_before": "case neg.h\nJ : Type w\nK : Type u_1\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\nf : J \u2192 C\ng : K \u2192 C\ne : J \u2243 K\nw : (j : J) \u2192 g (e j) \u2245 f j\ninst\u271d\u00b9 : HasBiproduct f\ninst\u271d : HasBiproduct g\nk : K\nh : \u00ack = e (e.symm k)\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "full_name": "LinearEquiv.coe_isometryOfInner", "start": [1293, 1], "end": [1294, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "full_name": "finrank_eq_one_iff_of_nonzero", "start": [1037, 1], "end": [1044, 18], "traced_tactics": [{"tactic": "simpa using (basisSingleton Unit h v nz).span_eq", "annotated_tactic": ["simpa using (<a>basisSingleton</a> <a>Unit</a> h v nz).<a>span_eq</a>", [{"full_name": "FiniteDimensional.basisSingleton", "def_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "def_pos": [309, 19], "def_end_pos": [309, 33]}, {"full_name": "Unit", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [142, 8], "def_end_pos": [142, 12]}, {"full_name": "Basis.span_eq", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [571, 19], "def_end_pos": [571, 26]}]], "state_before": "K : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : V\nnz : v \u2260 0\nh : finrank K V = 1\n\u22a2 span K {v} = \u22a4", "state_after": "no goals"}, {"tactic": "convert s.ge", "annotated_tactic": ["convert s.ge", []], "state_before": "K : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : V\nnz : v \u2260 0\ns : span K {v} = \u22a4\n\u22a2 \u22a4 \u2264 span K (Set.range fun x => \u2191x)", "state_after": "case h.e'_4.h.e'_6\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : V\nnz : v \u2260 0\ns : span K {v} = \u22a4\n\u22a2 (Set.range fun x => \u2191x) = {v}"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_4.h.e'_6\nK : Type u\nV : Type v\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nv : V\nnz : v \u2260 0\ns : span K {v} = \u22a4\n\u22a2 (Set.range fun x => \u2191x) = {v}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Derivative.lean", "full_name": "Polynomial.derivative_natCast_mul", "start": [332, 1], "end": [334, 7], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn\u271d : \u2115\ninst\u271d : Semiring R\nn : \u2115\nf : R[X]\n\u22a2 derivative (\u2191n * f) = \u2191n * derivative f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.mk_fintype", "start": [461, 1], "end": [462, 45], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1\u271d \u03b2 \u03b1 : Type u\nh : Fintype \u03b1\n\u22a2 Fintype.card \u03b1 = Fintype.card (ULift.{u, 0} (Fin (Fintype.card \u03b1)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.mk_biUnion_le_lift", "start": [2053, 1], "end": [2056, 26], "traced_tactics": [{"tactic": "rw [biUnion_eq_iUnion]", "annotated_tactic": ["rw [<a>biUnion_eq_iUnion</a>]", [{"full_name": "Set.biUnion_eq_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [862, 9], "def_end_pos": [862, 26]}]], "state_before": "\u03b1\u271d \u03b2 : Type u\nc : Cardinal.{?u.204480}\n\u03b1 : Type u\n\u03b9 : Type v\nA : \u03b9 \u2192 Set \u03b1\ns : Set \u03b9\n\u22a2 lift.{v, u} #\u2191(\u22c3 x \u2208 s, A x) \u2264 lift.{u, v} #\u2191s * \u2a06 x, lift.{v, u} #\u2191(A \u2191x)", "state_after": "\u03b1\u271d \u03b2 : Type u\nc : Cardinal.{?u.204480}\n\u03b1 : Type u\n\u03b9 : Type v\nA : \u03b9 \u2192 Set \u03b1\ns : Set \u03b9\n\u22a2 lift.{v, u} #\u2191(\u22c3 x, A \u2191x) \u2264 lift.{u, v} #\u2191s * \u2a06 x, lift.{v, u} #\u2191(A \u2191x)"}, {"tactic": "apply mk_iUnion_le_lift", "annotated_tactic": ["apply <a>mk_iUnion_le_lift</a>", [{"full_name": "Cardinal.mk_iUnion_le_lift", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [2037, 9], "def_end_pos": [2037, 26]}]], "state_before": "\u03b1\u271d \u03b2 : Type u\nc : Cardinal.{?u.204480}\n\u03b1 : Type u\n\u03b9 : Type v\nA : \u03b9 \u2192 Set \u03b1\ns : Set \u03b9\n\u22a2 lift.{v, u} #\u2191(\u22c3 x, A \u2191x) \u2264 lift.{u, v} #\u2191s * \u2a06 x, lift.{v, u} #\u2191(A \u2191x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Normed/Group/Constructions.lean", "full_name": "nnnorm_toDual", "start": [218, 1], "end": [218, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/BoundedVariation.lean", "full_name": "LipschitzOnWith.locallyBoundedVariationOn", "start": [834, 1], "end": [837, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ENNReal/Basic.lean", "full_name": "ENNReal.range_coe'", "start": [186, 1], "end": [186, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Simple.lean", "full_name": "CategoryTheory.Simple.iff_of_iso", "start": [80, 1], "end": [81, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/MvPolynomial/Homogeneous.lean", "full_name": "MvPolynomial.IsHomogeneous.totalDegree_le", "start": [260, 1], "end": [269, 20], "traced_tactics": [{"tactic": "apply Finset.sup_le", "annotated_tactic": ["apply <a>Finset.sup_le</a>", [{"full_name": "Finset.sup_le", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [112, 21], "def_end_pos": [112, 27]}]], "state_before": "\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u00b2 inst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : \u2115\nh\u03c6 : \u03c6.IsHomogeneous n\n\u22a2 \u03c6.totalDegree \u2264 n", "state_after": "case a\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u00b2 inst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : \u2115\nh\u03c6 : \u03c6.IsHomogeneous n\n\u22a2 \u2200 b \u2208 \u03c6.support, (b.sum fun x e => e) \u2264 n"}, {"tactic": "intro d hd", "annotated_tactic": ["intro d hd", []], "state_before": "case a\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u00b2 inst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : \u2115\nh\u03c6 : \u03c6.IsHomogeneous n\n\u22a2 \u2200 b \u2208 \u03c6.support, (b.sum fun x e => e) \u2264 n", "state_after": "case a\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u00b2 inst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : \u2115\nh\u03c6 : \u03c6.IsHomogeneous n\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 \u03c6.support\n\u22a2 (d.sum fun x e => e) \u2264 n"}, {"tactic": "rw [mem_support_iff] at hd", "annotated_tactic": ["rw [<a>mem_support_iff</a>] at hd", [{"full_name": "MvPolynomial.mem_support_iff", "def_path": "Mathlib/Algebra/MvPolynomial/Basic.lean", "def_pos": [592, 9], "def_end_pos": [592, 24]}]], "state_before": "case a\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u00b2 inst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : \u2115\nh\u03c6 : \u03c6.IsHomogeneous n\nd : \u03c3 \u2192\u2080 \u2115\nhd : d \u2208 \u03c6.support\n\u22a2 (d.sum fun x e => e) \u2264 n", "state_after": "case a\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u00b2 inst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : \u2115\nh\u03c6 : \u03c6.IsHomogeneous n\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d \u03c6 \u2260 0\n\u22a2 (d.sum fun x e => e) \u2264 n"}, {"tactic": "rw [Finsupp.sum, \u2190 h\u03c6 hd, weightedDegree_apply]", "annotated_tactic": ["rw [<a>Finsupp.sum</a>, \u2190 h\u03c6 hd, <a>weightedDegree_apply</a>]", [{"full_name": "Finsupp.sum", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [45, 3], "def_end_pos": [45, 14]}, {"full_name": "MvPolynomial.weightedDegree_apply", "def_path": "Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean", "def_pos": [68, 9], "def_end_pos": [68, 29]}]], "state_before": "case a\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u00b2 inst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : \u2115\nh\u03c6 : \u03c6.IsHomogeneous n\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d \u03c6 \u2260 0\n\u22a2 (d.sum fun x e => e) \u2264 n", "state_after": "case a\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u00b2 inst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : \u2115\nh\u03c6 : \u03c6.IsHomogeneous n\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d \u03c6 \u2260 0\n\u22a2 \u2211 a \u2208 d.support, d a \u2264 d.sum fun i c => c \u2022 1 i"}, {"tactic": "simp only [Pi.one_apply, smul_eq_mul, mul_one]", "annotated_tactic": ["simp only [<a>Pi.one_apply</a>, <a>smul_eq_mul</a>, <a>mul_one</a>]", [{"full_name": "Pi.one_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [63, 9], "def_end_pos": [63, 18]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [89, 7], "def_end_pos": [89, 18]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "case a\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u00b2 inst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : \u2115\nh\u03c6 : \u03c6.IsHomogeneous n\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d \u03c6 \u2260 0\n\u22a2 \u2211 a \u2208 d.support, d a \u2264 d.sum fun i c => c \u2022 1 i", "state_after": "case a\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u00b2 inst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : \u2115\nh\u03c6 : \u03c6.IsHomogeneous n\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d \u03c6 \u2260 0\n\u22a2 \u2211 a \u2208 d.support, d a \u2264 d.sum fun i c => c"}, {"tactic": "exact Nat.le.refl", "annotated_tactic": ["exact <a>Nat.le.refl</a>", [{"full_name": "Nat.le.refl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1642, 5], "def_end_pos": [1642, 9]}]], "state_before": "case a\n\u03c3 : Type u_1\n\u03c4 : Type u_2\nR : Type u_3\nS : Type u_4\ninst\u271d\u00b2 inst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\n\u03c6 \u03c8 : MvPolynomial \u03c3 R\nm n : \u2115\nh\u03c6 : \u03c6.IsHomogeneous n\nd : \u03c3 \u2192\u2080 \u2115\nhd : coeff d \u03c6 \u2260 0\n\u22a2 \u2211 a \u2208 d.support, d a \u2264 d.sum fun i c => c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean", "full_name": "Gamma1_mem", "start": [182, 1], "end": [200, 9], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "N\u271d N : \u2115\nA : SL(2, \u2124)\n\u22a2 A \u2208 Gamma1 N \u2194 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0", "state_after": "case mp\nN\u271d N : \u2115\nA : SL(2, \u2124)\n\u22a2 A \u2208 Gamma1 N \u2192 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\n\ncase mpr\nN\u271d N : \u2115\nA : SL(2, \u2124)\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0 \u2192 A \u2208 Gamma1 N"}, {"tactic": "intro ha", "annotated_tactic": ["intro ha", []], "state_before": "case mp\nN\u271d N : \u2115\nA : SL(2, \u2124)\n\u22a2 A \u2208 Gamma1 N \u2192 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0", "state_after": "case mp\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : A \u2208 Gamma1 N\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0"}, {"tactic": "simp_rw [Gamma1, Subgroup.mem_map] at ha", "annotated_tactic": ["simp_rw [<a>Gamma1</a>, <a>Subgroup.mem_map</a>] at ha", [{"full_name": "Gamma1", "def_path": "Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean", "def_pos": [177, 5], "def_end_pos": [177, 11]}, {"full_name": "Subgroup.mem_map", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [1405, 9], "def_end_pos": [1405, 16]}]], "state_before": "case mp\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : A \u2208 Gamma1 N\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0", "state_after": "case mp\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2203 x \u2208 \u22a4, ((Gamma0 N).subtype.comp (Gamma1' N).subtype) x = A\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0"}, {"tactic": "obtain \u27e8\u27e8x, hx\u27e9, hxx\u27e9 := ha", "annotated_tactic": ["obtain \u27e8\u27e8x, hx\u27e9, hxx\u27e9 := ha", []], "state_before": "case mp\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2203 x \u2208 \u22a4, ((Gamma0 N).subtype.comp (Gamma1' N).subtype) x = A\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0", "state_after": "case mp.intro.mk\nN\u271d N : \u2115\nA : SL(2, \u2124)\nx : \u21a5(Gamma0 N)\nhx : x \u2208 Gamma1' N\nhxx : \u27e8x, hx\u27e9 \u2208 \u22a4 \u2227 ((Gamma0 N).subtype.comp (Gamma1' N).subtype) \u27e8x, hx\u27e9 = A\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0"}, {"tactic": "rw [Gamma1_to_Gamma0_mem] at hx", "annotated_tactic": ["rw [<a>Gamma1_to_Gamma0_mem</a>] at hx", [{"full_name": "Gamma1_to_Gamma0_mem", "def_path": "Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean", "def_pos": [156, 9], "def_end_pos": [156, 29]}]], "state_before": "case mp.intro.mk\nN\u271d N : \u2115\nA : SL(2, \u2124)\nx : \u21a5(Gamma0 N)\nhx : x \u2208 Gamma1' N\nhxx : \u27e8x, hx\u27e9 \u2208 \u22a4 \u2227 ((Gamma0 N).subtype.comp (Gamma1' N).subtype) \u27e8x, hx\u27e9 = A\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0", "state_after": "case mp.intro.mk\nN\u271d N : \u2115\nA : SL(2, \u2124)\nx : \u21a5(Gamma0 N)\nhx\u271d : x \u2208 Gamma1' N\nhx : \u2191(\u2191\u2191x 0 0) = 1 \u2227 \u2191(\u2191\u2191x 1 1) = 1 \u2227 \u2191(\u2191\u2191x 1 0) = 0\nhxx : \u27e8x, hx\u271d\u27e9 \u2208 \u22a4 \u2227 ((Gamma0 N).subtype.comp (Gamma1' N).subtype) \u27e8x, hx\u271d\u27e9 = A\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0"}, {"tactic": "simp only [Subgroup.mem_top, true_and] at hxx", "annotated_tactic": ["simp only [<a>Subgroup.mem_top</a>, <a>true_and</a>] at hxx", [{"full_name": "Subgroup.mem_top", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [840, 9], "def_end_pos": [840, 16]}, {"full_name": "true_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [105, 17], "def_end_pos": [105, 25]}]], "state_before": "case mp.intro.mk\nN\u271d N : \u2115\nA : SL(2, \u2124)\nx : \u21a5(Gamma0 N)\nhx\u271d : x \u2208 Gamma1' N\nhx : \u2191(\u2191\u2191x 0 0) = 1 \u2227 \u2191(\u2191\u2191x 1 1) = 1 \u2227 \u2191(\u2191\u2191x 1 0) = 0\nhxx : \u27e8x, hx\u271d\u27e9 \u2208 \u22a4 \u2227 ((Gamma0 N).subtype.comp (Gamma1' N).subtype) \u27e8x, hx\u271d\u27e9 = A\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0", "state_after": "case mp.intro.mk\nN\u271d N : \u2115\nA : SL(2, \u2124)\nx : \u21a5(Gamma0 N)\nhx\u271d : x \u2208 Gamma1' N\nhx : \u2191(\u2191\u2191x 0 0) = 1 \u2227 \u2191(\u2191\u2191x 1 1) = 1 \u2227 \u2191(\u2191\u2191x 1 0) = 0\nhxx : ((Gamma0 N).subtype.comp (Gamma1' N).subtype) \u27e8x, hx\u271d\u27e9 = A\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0"}, {"tactic": "rw [\u2190 hxx]", "annotated_tactic": ["rw [\u2190 hxx]", []], "state_before": "case mp.intro.mk\nN\u271d N : \u2115\nA : SL(2, \u2124)\nx : \u21a5(Gamma0 N)\nhx\u271d : x \u2208 Gamma1' N\nhx : \u2191(\u2191\u2191x 0 0) = 1 \u2227 \u2191(\u2191\u2191x 1 1) = 1 \u2227 \u2191(\u2191\u2191x 1 0) = 0\nhxx : ((Gamma0 N).subtype.comp (Gamma1' N).subtype) \u27e8x, hx\u271d\u27e9 = A\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0", "state_after": "case mp.intro.mk\nN\u271d N : \u2115\nA : SL(2, \u2124)\nx : \u21a5(Gamma0 N)\nhx\u271d : x \u2208 Gamma1' N\nhx : \u2191(\u2191\u2191x 0 0) = 1 \u2227 \u2191(\u2191\u2191x 1 1) = 1 \u2227 \u2191(\u2191\u2191x 1 0) = 0\nhxx : ((Gamma0 N).subtype.comp (Gamma1' N).subtype) \u27e8x, hx\u271d\u27e9 = A\n\u22a2 \u2191(\u2191(((Gamma0 N).subtype.comp (Gamma1' N).subtype) \u27e8x, hx\u271d\u27e9) 0 0) = 1 \u2227\n    \u2191(\u2191(((Gamma0 N).subtype.comp (Gamma1' N).subtype) \u27e8x, hx\u271d\u27e9) 1 1) = 1 \u2227\n      \u2191(\u2191(((Gamma0 N).subtype.comp (Gamma1' N).subtype) \u27e8x, hx\u271d\u27e9) 1 0) = 0"}, {"tactic": "convert hx", "annotated_tactic": ["convert hx", []], "state_before": "case mp.intro.mk\nN\u271d N : \u2115\nA : SL(2, \u2124)\nx : \u21a5(Gamma0 N)\nhx\u271d : x \u2208 Gamma1' N\nhx : \u2191(\u2191\u2191x 0 0) = 1 \u2227 \u2191(\u2191\u2191x 1 1) = 1 \u2227 \u2191(\u2191\u2191x 1 0) = 0\nhxx : ((Gamma0 N).subtype.comp (Gamma1' N).subtype) \u27e8x, hx\u271d\u27e9 = A\n\u22a2 \u2191(\u2191(((Gamma0 N).subtype.comp (Gamma1' N).subtype) \u27e8x, hx\u271d\u27e9) 0 0) = 1 \u2227\n    \u2191(\u2191(((Gamma0 N).subtype.comp (Gamma1' N).subtype) \u27e8x, hx\u271d\u27e9) 1 1) = 1 \u2227\n      \u2191(\u2191(((Gamma0 N).subtype.comp (Gamma1' N).subtype) \u27e8x, hx\u271d\u27e9) 1 0) = 0", "state_after": "no goals"}, {"tactic": "intro ha", "annotated_tactic": ["intro ha", []], "state_before": "case mpr\nN\u271d N : \u2115\nA : SL(2, \u2124)\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0 \u2192 A \u2208 Gamma1 N", "state_after": "case mpr\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\n\u22a2 A \u2208 Gamma1 N"}, {"tactic": "simp_rw [Gamma1, Subgroup.mem_map]", "annotated_tactic": ["simp_rw [<a>Gamma1</a>, <a>Subgroup.mem_map</a>]", [{"full_name": "Gamma1", "def_path": "Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean", "def_pos": [177, 5], "def_end_pos": [177, 11]}, {"full_name": "Subgroup.mem_map", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [1405, 9], "def_end_pos": [1405, 16]}]], "state_before": "case mpr\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\n\u22a2 A \u2208 Gamma1 N", "state_after": "case mpr\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\n\u22a2 \u2203 x \u2208 \u22a4, ((Gamma0 N).subtype.comp (Gamma1' N).subtype) x = A"}, {"tactic": "have hA : A \u2208 Gamma0 N := by simp [ha.right.right, Gamma0_mem]", "annotated_tactic": ["have hA : A \u2208 <a>Gamma0</a> N := by simp [ha.right.right, <a>Gamma0_mem</a>]", [{"full_name": "Gamma0", "def_path": "Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean", "def_pos": [101, 5], "def_end_pos": [101, 11]}, {"full_name": "Gamma0_mem", "def_path": "Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean", "def_pos": [121, 9], "def_end_pos": [121, 19]}]], "state_before": "case mpr\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\n\u22a2 \u2203 x \u2208 \u22a4, ((Gamma0 N).subtype.comp (Gamma1' N).subtype) x = A", "state_after": "case mpr\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\nhA : A \u2208 Gamma0 N\n\u22a2 \u2203 x \u2208 \u22a4, ((Gamma0 N).subtype.comp (Gamma1' N).subtype) x = A"}, {"tactic": "have HA : (\u27e8A, hA\u27e9 : Gamma0 N) \u2208 Gamma1' N := by\n  simp only [Gamma1_to_Gamma0_mem, Subgroup.coe_mk, coe_matrix_coe,\n    Int.coe_castRingHom, map_apply]\n  exact ha", "annotated_tactic": ["have HA : (\u27e8A, hA\u27e9 : <a>Gamma0</a> N) \u2208 <a>Gamma1'</a> N := by\n      simp only [<a>Gamma1_to_Gamma0_mem</a>, <a>Subgroup.coe_mk</a>, <a>coe_matrix_coe</a>,\n        <a>Int.coe_castRingHom</a>, <a>map_apply</a>]\n      exact ha", [{"full_name": "Gamma0", "def_path": "Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean", "def_pos": [101, 5], "def_end_pos": [101, 11]}, {"full_name": "Gamma1'", "def_path": "Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean", "def_pos": [147, 5], "def_end_pos": [147, 12]}, {"full_name": "Gamma1_to_Gamma0_mem", "def_path": "Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean", "def_pos": [156, 9], "def_end_pos": [156, 29]}, {"full_name": "Subgroup.coe_mk", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [720, 9], "def_end_pos": [720, 15]}, {"full_name": "Matrix.SpecialLinearGroup.coe_matrix_coe", "def_path": "Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean", "def_pos": [341, 9], "def_end_pos": [341, 23]}, {"full_name": "Int.coe_castRingHom", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [106, 15], "def_end_pos": [106, 30]}, {"full_name": "Matrix.map_apply", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}]], "state_before": "case mpr\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\nhA : A \u2208 Gamma0 N\n\u22a2 \u2203 x \u2208 \u22a4, ((Gamma0 N).subtype.comp (Gamma1' N).subtype) x = A", "state_after": "case mpr\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\nhA : A \u2208 Gamma0 N\nHA : \u27e8A, hA\u27e9 \u2208 Gamma1' N\n\u22a2 \u2203 x \u2208 \u22a4, ((Gamma0 N).subtype.comp (Gamma1' N).subtype) x = A"}, {"tactic": "refine \u27e8(\u27e8(\u27e8A, hA\u27e9 : Gamma0 N), HA\u27e9 : (Gamma1' N : Subgroup (Gamma0 N))), ?_\u27e9", "annotated_tactic": ["refine \u27e8(\u27e8(\u27e8A, hA\u27e9 : <a>Gamma0</a> N), HA\u27e9 : (<a>Gamma1'</a> N : <a>Subgroup</a> (<a>Gamma0</a> N))), ?_\u27e9", [{"full_name": "Gamma0", "def_path": "Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean", "def_pos": [101, 5], "def_end_pos": [101, 11]}, {"full_name": "Gamma1'", "def_path": "Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean", "def_pos": [147, 5], "def_end_pos": [147, 12]}, {"full_name": "Subgroup", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [354, 11], "def_end_pos": [354, 19]}, {"full_name": "Gamma0", "def_path": "Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean", "def_pos": [101, 5], "def_end_pos": [101, 11]}]], "state_before": "case mpr\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\nhA : A \u2208 Gamma0 N\nHA : \u27e8A, hA\u27e9 \u2208 Gamma1' N\n\u22a2 \u2203 x \u2208 \u22a4, ((Gamma0 N).subtype.comp (Gamma1' N).subtype) x = A", "state_after": "case mpr\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\nhA : A \u2208 Gamma0 N\nHA : \u27e8A, hA\u27e9 \u2208 Gamma1' N\n\u22a2 \u27e8\u27e8A, hA\u27e9, HA\u27e9 \u2208 \u22a4 \u2227 ((Gamma0 N).subtype.comp (Gamma1' N).subtype) \u27e8\u27e8A, hA\u27e9, HA\u27e9 = A"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case mpr\nN\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\nhA : A \u2208 Gamma0 N\nHA : \u27e8A, hA\u27e9 \u2208 Gamma1' N\n\u22a2 \u27e8\u27e8A, hA\u27e9, HA\u27e9 \u2208 \u22a4 \u2227 ((Gamma0 N).subtype.comp (Gamma1' N).subtype) \u27e8\u27e8A, hA\u27e9, HA\u27e9 = A", "state_after": "no goals"}, {"tactic": "simp [ha.right.right, Gamma0_mem]", "annotated_tactic": ["simp [ha.right.right, <a>Gamma0_mem</a>]", [{"full_name": "Gamma0_mem", "def_path": "Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean", "def_pos": [121, 9], "def_end_pos": [121, 19]}]], "state_before": "N\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\n\u22a2 A \u2208 Gamma0 N", "state_after": "no goals"}, {"tactic": "simp only [Gamma1_to_Gamma0_mem, Subgroup.coe_mk, coe_matrix_coe,\n  Int.coe_castRingHom, map_apply]", "annotated_tactic": ["simp only [<a>Gamma1_to_Gamma0_mem</a>, <a>Subgroup.coe_mk</a>, <a>coe_matrix_coe</a>,\n        <a>Int.coe_castRingHom</a>, <a>map_apply</a>]", [{"full_name": "Gamma1_to_Gamma0_mem", "def_path": "Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean", "def_pos": [156, 9], "def_end_pos": [156, 29]}, {"full_name": "Subgroup.coe_mk", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [720, 9], "def_end_pos": [720, 15]}, {"full_name": "Matrix.SpecialLinearGroup.coe_matrix_coe", "def_path": "Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean", "def_pos": [341, 9], "def_end_pos": [341, 23]}, {"full_name": "Int.coe_castRingHom", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [106, 15], "def_end_pos": [106, 30]}, {"full_name": "Matrix.map_apply", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}]], "state_before": "N\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\nhA : A \u2208 Gamma0 N\n\u22a2 \u27e8A, hA\u27e9 \u2208 Gamma1' N", "state_after": "N\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\nhA : A \u2208 Gamma0 N\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0"}, {"tactic": "exact ha", "annotated_tactic": ["exact ha", []], "state_before": "N\u271d N : \u2115\nA : SL(2, \u2124)\nha : \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0\nhA : A \u2208 Gamma0 N\n\u22a2 \u2191(\u2191A 0 0) = 1 \u2227 \u2191(\u2191A 1 1) = 1 \u2227 \u2191(\u2191A 1 0) = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/MvPolynomial/Basic.lean", "full_name": "MvPolynomial.eval\u2082_eq", "start": [1020, 1], "end": [1022, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/String/Lemmas.lean", "full_name": "String.Iterator.hasNext_cons_addChar", "start": [464, 1], "end": [466, 43], "traced_tactics": [{"tactic": "simp [hasNext, Nat.add_lt_add_iff_right]", "annotated_tactic": ["simp [<a>hasNext</a>, <a>Nat.add_lt_add_iff_right</a>]", [{"full_name": "String.Iterator.hasNext", "def_path": ".lake/packages/lean4/src/lean/Init/Data/String/Basic.lean", "def_pos": [612, 5], "def_end_pos": [612, 12]}, {"full_name": "Nat.add_lt_add_iff_right", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean", "def_pos": [58, 19], "def_end_pos": [58, 39]}]], "state_before": "c : Char\ncs : List Char\ni : Pos\n\u22a2 { s := { data := c :: cs }, i := i + c }.hasNext = { s := { data := cs }, i := i }.hasNext", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean", "full_name": "Real.smoothTransition.contDiffAt", "start": [216, 11], "end": [217, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "MeasureTheory.Measure.snd_apply", "start": [1083, 1], "end": [1084, 48], "traced_tactics": [{"tactic": "rw [snd, Measure.map_apply measurable_snd hs]", "annotated_tactic": ["rw [<a>snd</a>, <a>Measure.map_apply</a> <a>measurable_snd</a> hs]", [{"full_name": "MeasureTheory.Measure.snd", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [1079, 19], "def_end_pos": [1079, 22]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1291, 9], "def_end_pos": [1291, 18]}, {"full_name": "measurable_snd", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Basic.lean", "def_pos": [711, 9], "def_end_pos": [711, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\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 : SFinite \u03bd\n\u03c1 : Measure (\u03b1 \u00d7 \u03b2)\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 \u03c1.snd s = \u03c1 (Prod.snd \u207b\u00b9' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.map_eq_map", "start": [608, 1], "end": [609, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ENNReal/Real.lean", "full_name": "ENNReal.toReal_inf", "start": [155, 1], "end": [156, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Classes/SatisfiesM.lean", "full_name": "SatisfiesM.seq_post", "start": [99, 11], "end": [102, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Star/Pointwise.lean", "full_name": "Set.inter_star", "start": [76, 1], "end": [76, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Multiplicity.lean", "full_name": "multiplicity.finite_mul_iff", "start": [519, 1], "end": [521, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "full_name": "MeasureTheory.integral_fn_integral_add", "start": [356, 1], "end": [362, 30], "traced_tactics": [{"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\n\u03b1' : 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 g : \u03b1 \u00d7 \u03b2 \u2192 E\nF : E \u2192 E'\nhf : Integrable f (\u03bc.prod \u03bd)\nhg : Integrable g (\u03bc.prod \u03bd)\n\u22a2 \u222b (x : \u03b1), F (\u222b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd) \u2202\u03bc =\n    \u222b (x : \u03b1), F (\u222b (y : \u03b2), f (x, y) \u2202\u03bd + \u222b (y : \u03b2), g (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'\nf g : \u03b1 \u00d7 \u03b2 \u2192 E\nF : E \u2192 E'\nhf : Integrable f (\u03bc.prod \u03bd)\nhg : Integrable g (\u03bc.prod \u03bd)\n\u22a2 (fun x => F (\u222b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd)) =\u1da0[ae \u03bc] fun x => F (\u222b (y : \u03b2), f (x, y) \u2202\u03bd + \u222b (y : \u03b2), g (x, y) \u2202\u03bd)"}, {"tactic": "filter_upwards [hf.prod_right_ae, hg.prod_right_ae] with _ h2f h2g", "annotated_tactic": ["filter_upwards [hf.prod_right_ae, hg.prod_right_ae] with _ h2f h2g", []], "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 g : \u03b1 \u00d7 \u03b2 \u2192 E\nF : E \u2192 E'\nhf : Integrable f (\u03bc.prod \u03bd)\nhg : Integrable g (\u03bc.prod \u03bd)\n\u22a2 (fun x => F (\u222b (y : \u03b2), f (x, y) + g (x, y) \u2202\u03bd)) =\u1da0[ae \u03bc] fun x => F (\u222b (y : \u03b2), f (x, y) \u2202\u03bd + \u222b (y : \u03b2), g (x, y) \u2202\u03bd)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\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 g : \u03b1 \u00d7 \u03b2 \u2192 E\nF : E \u2192 E'\nhf : Integrable f (\u03bc.prod \u03bd)\nhg : Integrable g (\u03bc.prod \u03bd)\na\u271d : \u03b1\nh2f : Integrable (fun y => f (a\u271d, y)) \u03bd\nh2g : Integrable (fun y => g (a\u271d, y)) \u03bd\n\u22a2 F (\u222b (y : \u03b2), f (a\u271d, y) + g (a\u271d, y) \u2202\u03bd) = F (\u222b (y : \u03b2), f (a\u271d, y) \u2202\u03bd + \u222b (y : \u03b2), g (a\u271d, y) \u2202\u03bd)"}, {"tactic": "simp [integral_add h2f h2g]", "annotated_tactic": ["simp [<a>integral_add</a> h2f h2g]", [{"full_name": "MeasureTheory.integral_add", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [868, 9], "def_end_pos": [868, 21]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\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 g : \u03b1 \u00d7 \u03b2 \u2192 E\nF : E \u2192 E'\nhf : Integrable f (\u03bc.prod \u03bd)\nhg : Integrable g (\u03bc.prod \u03bd)\na\u271d : \u03b1\nh2f : Integrable (fun y => f (a\u271d, y)) \u03bd\nh2g : Integrable (fun y => g (a\u271d, y)) \u03bd\n\u22a2 F (\u222b (y : \u03b2), f (a\u271d, y) + g (a\u271d, y) \u2202\u03bd) = F (\u222b (y : \u03b2), f (a\u271d, y) \u2202\u03bd + \u222b (y : \u03b2), g (a\u271d, y) \u2202\u03bd)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Padics/PadicNumbers.lean", "full_name": "padicNormE.add_eq_max_of_ne'", "start": [628, 1], "end": [630, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "full_name": "Asymptotics.isBigOWith_of_le'", "start": [574, 1], "end": [575, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Codisjoint.preimage", "start": [1577, 1], "end": [1580, 14], "traced_tactics": [{"tactic": "simp only [codisjoint_iff_le_sup, Set.sup_eq_union, top_le_iff, \u2190 Set.preimage_union] at h \u22a2", "annotated_tactic": ["simp only [<a>codisjoint_iff_le_sup</a>, <a>Set.sup_eq_union</a>, <a>top_le_iff</a>, \u2190 <a>Set.preimage_union</a>] at h \u22a2", [{"full_name": "codisjoint_iff_le_sup", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [325, 9], "def_end_pos": [325, 30]}, {"full_name": "Set.sup_eq_union", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [101, 9], "def_end_pos": [101, 21]}, {"full_name": "top_le_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [125, 9], "def_end_pos": [125, 19]}, {"full_name": "Set.preimage_union", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [77, 9], "def_end_pos": [77, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2\ns\u271d t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b2\nh : Codisjoint s t\n\u22a2 Codisjoint (f \u207b\u00b9' s) (f \u207b\u00b9' t)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2\ns\u271d t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b2\nh : s \u222a t = \u22a4\n\u22a2 f \u207b\u00b9' (s \u222a t) = \u22a4"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2\ns\u271d t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b2\nh : s \u222a t = \u22a4\n\u22a2 f \u207b\u00b9' (s \u222a t) = \u22a4", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2\ns\u271d t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b2\nh : s \u222a t = \u22a4\n\u22a2 f \u207b\u00b9' \u22a4 = \u22a4"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf\u271d : \u03b1 \u2192 \u03b2\ns\u271d t\u271d : Set \u03b1\nf : \u03b1 \u2192 \u03b2\ns t : Set \u03b2\nh : s \u222a t = \u22a4\n\u22a2 f \u207b\u00b9' \u22a4 = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/Charpoly/Minpoly.lean", "full_name": "Matrix.minpoly_toLin'", "start": [38, 1], "end": [39, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Complex/RealDeriv.lean", "full_name": "ContDiffAt.real_of_complex", "start": [84, 1], "end": [89, 30], "traced_tactics": [{"tactic": "have A : ContDiffAt \u211d n ((\u2191) : \u211d \u2192 \u2102) z := ofRealCLM.contDiff.contDiffAt", "annotated_tactic": ["have A : <a>ContDiffAt</a> \u211d n ((\u2191) : \u211d \u2192 \u2102) z := ofRealCLM.contDiff.contDiffAt", [{"full_name": "ContDiffAt", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "def_pos": [1361, 5], "def_end_pos": [1361, 15]}]], "state_before": "e : \u2102 \u2192 \u2102\ne' : \u2102\nz : \u211d\nn : \u2115\u221e\nh : ContDiffAt \u2102 n e \u2191z\n\u22a2 ContDiffAt \u211d n (fun x => (e \u2191x).re) z", "state_after": "e : \u2102 \u2192 \u2102\ne' : \u2102\nz : \u211d\nn : \u2115\u221e\nh : ContDiffAt \u2102 n e \u2191z\nA : ContDiffAt \u211d n ofReal' z\n\u22a2 ContDiffAt \u211d n (fun x => (e \u2191x).re) z"}, {"tactic": "have B : ContDiffAt \u211d n e z := h.restrict_scalars \u211d", "annotated_tactic": ["have B : <a>ContDiffAt</a> \u211d n e z := h.restrict_scalars \u211d", [{"full_name": "ContDiffAt", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "def_pos": [1361, 5], "def_end_pos": [1361, 15]}]], "state_before": "e : \u2102 \u2192 \u2102\ne' : \u2102\nz : \u211d\nn : \u2115\u221e\nh : ContDiffAt \u2102 n e \u2191z\nA : ContDiffAt \u211d n ofReal' z\n\u22a2 ContDiffAt \u211d n (fun x => (e \u2191x).re) z", "state_after": "e : \u2102 \u2192 \u2102\ne' : \u2102\nz : \u211d\nn : \u2115\u221e\nh : ContDiffAt \u2102 n e \u2191z\nA : ContDiffAt \u211d n ofReal' z\nB : ContDiffAt \u211d n e \u2191z\n\u22a2 ContDiffAt \u211d n (fun x => (e \u2191x).re) z"}, {"tactic": "have C : ContDiffAt \u211d n re (e z) := reCLM.contDiff.contDiffAt", "annotated_tactic": ["have C : <a>ContDiffAt</a> \u211d n <a>re</a> (e z) := reCLM.contDiff.contDiffAt", [{"full_name": "ContDiffAt", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "def_pos": [1361, 5], "def_end_pos": [1361, 15]}, {"full_name": "Complex.re", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [31, 3], "def_end_pos": [31, 5]}]], "state_before": "e : \u2102 \u2192 \u2102\ne' : \u2102\nz : \u211d\nn : \u2115\u221e\nh : ContDiffAt \u2102 n e \u2191z\nA : ContDiffAt \u211d n ofReal' z\nB : ContDiffAt \u211d n e \u2191z\n\u22a2 ContDiffAt \u211d n (fun x => (e \u2191x).re) z", "state_after": "e : \u2102 \u2192 \u2102\ne' : \u2102\nz : \u211d\nn : \u2115\u221e\nh : ContDiffAt \u2102 n e \u2191z\nA : ContDiffAt \u211d n ofReal' z\nB : ContDiffAt \u211d n e \u2191z\nC : ContDiffAt \u211d n re (e \u2191z)\n\u22a2 ContDiffAt \u211d n (fun x => (e \u2191x).re) z"}, {"tactic": "exact C.comp z (B.comp z A)", "annotated_tactic": ["exact C.comp z (B.comp z A)", []], "state_before": "e : \u2102 \u2192 \u2102\ne' : \u2102\nz : \u211d\nn : \u2115\u221e\nh : ContDiffAt \u2102 n e \u2191z\nA : ContDiffAt \u211d n ofReal' z\nB : ContDiffAt \u211d n e \u2191z\nC : ContDiffAt \u211d n re (e \u2191z)\n\u22a2 ContDiffAt \u211d n (fun x => (e \u2191x).re) z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "full_name": "MulAction.mem_orbit_iff", "start": [57, 1], "end": [58, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Dist.lean", "full_name": "Nat.dist_mul_right", "start": [99, 1], "end": [100, 57], "traced_tactics": [{"tactic": "rw [dist, dist, right_distrib, tsub_mul n, tsub_mul m]", "annotated_tactic": ["rw [<a>dist</a>, <a>dist</a>, <a>right_distrib</a>, <a>tsub_mul</a> n, <a>tsub_mul</a> m]", [{"full_name": "Nat.dist", "def_path": "Mathlib/Data/Nat/Dist.lean", "def_pos": [20, 5], "def_end_pos": [20, 9]}, {"full_name": "Nat.dist", "def_path": "Mathlib/Data/Nat/Dist.lean", "def_pos": [20, 5], "def_end_pos": [20, 9]}, {"full_name": "right_distrib", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [86, 9], "def_end_pos": [86, 22]}, {"full_name": "tsub_mul", "def_path": "Mathlib/Algebra/Order/Ring/Canonical.lean", "def_pos": [126, 9], "def_end_pos": [126, 17]}, {"full_name": "tsub_mul", "def_path": "Mathlib/Algebra/Order/Ring/Canonical.lean", "def_pos": 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C\nh\u271d : C \u27f6 D\ninst\u271d\u00b9 : HasZeroMorphisms V\ninst\u271d : HasEqualizers V\ni : B \u2245 D\nh : Exact f g\n\u22a2 Epi (imageToKernel (f \u226b i.hom) (i.inv \u226b g) \u22ef)"}, {"tactic": "rw [imageToKernel_comp_hom_inv_comp]", "annotated_tactic": ["rw [<a>imageToKernel_comp_hom_inv_comp</a>]", [{"full_name": "imageToKernel_comp_hom_inv_comp", "def_path": "Mathlib/Algebra/Homology/ImageToKernel.lean", "def_pos": [147, 9], "def_end_pos": [147, 40]}]], "state_before": "V : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasImages V\nA B C D : V\nf : A \u27f6 B\ng : B \u27f6 C\nh\u271d : C \u27f6 D\ninst\u271d\u00b9 : HasZeroMorphisms V\ninst\u271d : HasEqualizers V\ni : B \u2245 D\nh : Exact f g\n\u22a2 Epi (imageToKernel (f \u226b i.hom) (i.inv \u226b g) \u22ef)", "state_after": "V : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasImages V\nA B C D : V\nf : A \u27f6 B\ng : B \u27f6 C\nh\u271d : C \u27f6 D\ninst\u271d\u00b9 : 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NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d : CompleteSpace F\n\u03b5 : \u211d\nx\u271d : 0 < \u03b5\nr : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u22a2 ((\u2200 (c : \u211d\u22650), l.RCond (r c)) \u2227\n      \u2200 x \u2208 {\u03c0 | \u2203 c, l.MemBaseSet I c (r c) \u03c0 \u2227 \u03c0.IsPartition} \u00d7\u02e2 {\u03c0 | \u2203 c, l.MemBaseSet I c (r c) \u03c0 \u2227 \u03c0.IsPartition},\n        (integralSum f vol x.1, integralSum f vol x.2) \u2208 {p | dist p.1 p.2 \u2264 \u03b5}) \u2194\n    (\u2200 (c : \u211d\u22650), l.RCond (r c)) \u2227\n      \u2200 (c\u2081 c\u2082 : \u211d\u22650) (\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I),\n        l.MemBaseSet I c\u2081 (r c\u2081) \u03c0\u2081 \u2192\n          \u03c0\u2081.IsPartition \u2192\n            l.MemBaseSet I c\u2082 (r c\u2082) \u03c0\u2082 \u2192 \u03c0\u2082.IsPartition \u2192 dist (integralSum f vol \u03c0\u2081) (integralSum f vol \u03c0\u2082) \u2264 \u03b5", "state_after": "\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d : CompleteSpace F\n\u03b5 : \u211d\nx\u271d : 0 < \u03b5\nr : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u22a2 ((\u2200 (c : \u211d\u22650), l.RCond (r c)) \u2227\n      \u2200 (a b : TaggedPrepartition I) (x : \u211d\u22650),\n        l.MemBaseSet I x (r x) a \u2192\n          a.IsPartition \u2192\n            \u2200 (x : \u211d\u22650),\n              l.MemBaseSet I x (r x) b \u2192 b.IsPartition \u2192 dist (integralSum f vol a) (integralSum f vol b) \u2264 \u03b5) \u2194\n    (\u2200 (c : \u211d\u22650), l.RCond (r c)) \u2227\n      \u2200 (c\u2081 c\u2082 : \u211d\u22650) (\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I),\n        l.MemBaseSet I c\u2081 (r c\u2081) \u03c0\u2081 \u2192\n          \u03c0\u2081.IsPartition \u2192\n            l.MemBaseSet I c\u2082 (r c\u2082) \u03c0\u2082 \u2192 \u03c0\u2082.IsPartition \u2192 dist (integralSum f vol \u03c0\u2081) (integralSum f vol \u03c0\u2082) \u2264 \u03b5"}, {"tactic": "exact\n  and_congr Iff.rfl\n    \u27e8fun H c\u2081 c\u2082 \u03c0\u2081 \u03c0\u2082 h\u2081 hU\u2081 h\u2082 hU\u2082 => H \u03c0\u2081 \u03c0\u2082 c\u2081 h\u2081 hU\u2081 c\u2082 h\u2082 hU\u2082,\n      fun H \u03c0\u2081 \u03c0\u2082 c\u2081 h\u2081 hU\u2081 c\u2082 h\u2082 hU\u2082 => H c\u2081 c\u2082 \u03c0\u2081 \u03c0\u2082 h\u2081 hU\u2081 h\u2082 hU\u2082\u27e9", "annotated_tactic": ["exact\n    <a>and_congr</a> <a>Iff.rfl</a>\n      \u27e8fun H c\u2081 c\u2082 \u03c0\u2081 \u03c0\u2082 h\u2081 hU\u2081 h\u2082 hU\u2082 => H \u03c0\u2081 \u03c0\u2082 c\u2081 h\u2081 hU\u2081 c\u2082 h\u2082 hU\u2082,\n        fun H \u03c0\u2081 \u03c0\u2082 c\u2081 h\u2081 hU\u2081 c\u2082 h\u2082 hU\u2082 => H c\u2081 c\u2082 \u03c0\u2081 \u03c0\u2082 h\u2081 hU\u2081 h\u2082 hU\u2082\u27e9", [{"full_name": "and_congr", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [43, 9], "def_end_pos": [43, 18]}, {"full_name": "Iff.rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [796, 19], "def_end_pos": [796, 26]}]], "state_before": "\u03b9 : Type u\nE : Type v\nF : Type w\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \u211d E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \u211d F\nI J : Box \u03b9\n\u03c0 : TaggedPrepartition I\ninst\u271d\u00b9 : Fintype \u03b9\nl : IntegrationParams\nf g : (\u03b9 \u2192 \u211d) \u2192 E\nvol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F\ny y' : F\ninst\u271d : CompleteSpace F\n\u03b5 : \u211d\nx\u271d : 0 < \u03b5\nr : \u211d\u22650 \u2192 (\u03b9 \u2192 \u211d) \u2192 \u2191(Set.Ioi 0)\n\u22a2 ((\u2200 (c : \u211d\u22650), l.RCond (r c)) \u2227\n      \u2200 (a b : TaggedPrepartition I) (x : \u211d\u22650),\n        l.MemBaseSet I x (r x) a \u2192\n          a.IsPartition \u2192\n            \u2200 (x : \u211d\u22650),\n              l.MemBaseSet I x (r x) b \u2192 b.IsPartition \u2192 dist (integralSum f vol a) (integralSum f vol b) \u2264 \u03b5) \u2194\n    (\u2200 (c : \u211d\u22650), l.RCond (r c)) \u2227\n      \u2200 (c\u2081 c\u2082 : \u211d\u22650) (\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I),\n        l.MemBaseSet I c\u2081 (r c\u2081) \u03c0\u2081 \u2192\n          \u03c0\u2081.IsPartition \u2192\n            l.MemBaseSet I c\u2082 (r c\u2082) \u03c0\u2082 \u2192 \u03c0\u2082.IsPartition \u2192 dist (integralSum f vol \u03c0\u2081) (integralSum f vol \u03c0\u2082) \u2264 \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/GroupWithZero/Finset.lean", "full_name": "Units.mk0_prod", "start": [73, 1], "end": [76, 63], "traced_tactics": [{"tactic": "classical induction s using Finset.induction_on <;> simp [*]", "annotated_tactic": ["classical induction s using <a>Finset.induction_on</a> <;> simp [*]", [{"full_name": "Finset.induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1278, 19], "def_end_pos": [1278, 31]}]], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d : CommGroupWithZero G\u2080\ns : Finset \u03b9\nf : \u03b9 \u2192 G\u2080\nh : \u220f i \u2208 s, f i \u2260 0\n\u22a2 mk0 (\u220f i \u2208 s, f i) h = \u220f i \u2208 s.attach, mk0 (f \u2191i) \u22ef", "state_after": "no goals"}, {"tactic": "induction s using Finset.induction_on <;> simp [*]", "annotated_tactic": ["induction s using <a>Finset.induction_on</a> <;> simp [*]", [{"full_name": "Finset.induction_on", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1278, 19], "def_end_pos": [1278, 31]}]], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\nG\u2080 : Type u_3\nM\u2080 : Type u_4\ninst\u271d : CommGroupWithZero G\u2080\ns : Finset \u03b9\nf : \u03b9 \u2192 G\u2080\nh : \u220f i \u2208 s, f i \u2260 0\n\u22a2 mk0 (\u220f i \u2208 s, f i) h = \u220f i \u2208 s.attach, mk0 (f \u2191i) \u22ef", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Equivalence.lean", "full_name": "CategoryTheory.Equivalence.cancel_unit_right", "start": [385, 1], "end": [386, 80], "traced_tactics": [{"tactic": "simp only [cancel_mono]", "annotated_tactic": ["simp only [<a>cancel_mono</a>]", [{"full_name": "CategoryTheory.cancel_mono", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [310, 9], "def_end_pos": [310, 20]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\ne : C \u224c D\nX Y : C\nf f' : X \u27f6 Y\n\u22a2 f \u226b e.unit.app Y = f' \u226b e.unit.app Y \u2194 f = f'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Floor/Div.lean", "full_name": "ceilDiv_le_iff_le_smul", "start": [107, 1], "end": [108, 94], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/BigOperators.lean", "full_name": "Polynomial.natDegree_list_sum_le", "start": [48, 1], "end": [49, 63], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "R : Type u\n\u03b9 : Type w\ns : Finset \u03b9\nS : Type u_1\ninst\u271d : Semiring S\nl : List S[X]\n\u22a2 natDegree 0 \u2264 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/DedekindDomain/Ideal.lean", "full_name": "FractionalIdeal.mul_right_le_iff", "start": [548, 1], "end": [555, 38], "traced_tactics": [{"tactic": "intro I I'", "annotated_tactic": ["intro I I'", []], "state_before": "R : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : IsDedekindDomain A\nJ : FractionalIdeal A\u2070 K\nhJ : J \u2260 0\n\u22a2 \u2200 {I I' : FractionalIdeal A\u2070 K}, I * J \u2264 I' * J \u2194 I \u2264 I'", "state_after": "R : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : IsDedekindDomain A\nJ : FractionalIdeal A\u2070 K\nhJ : J \u2260 0\nI I' : FractionalIdeal A\u2070 K\n\u22a2 I * J \u2264 I' * J \u2194 I \u2264 I'"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "R : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : IsDedekindDomain A\nJ : FractionalIdeal A\u2070 K\nhJ : J \u2260 0\nI I' : FractionalIdeal A\u2070 K\n\u22a2 I * J \u2264 I' * J \u2194 I \u2264 I'", "state_after": "case mp\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : IsDedekindDomain A\nJ : FractionalIdeal A\u2070 K\nhJ : J \u2260 0\nI I' : FractionalIdeal A\u2070 K\n\u22a2 I * J \u2264 I' * J \u2192 I \u2264 I'\n\ncase mpr\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : IsDedekindDomain A\nJ : FractionalIdeal A\u2070 K\nhJ : J \u2260 0\nI I' : FractionalIdeal A\u2070 K\n\u22a2 I \u2264 I' \u2192 I * J \u2264 I' * J"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mp\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : IsDedekindDomain A\nJ : FractionalIdeal A\u2070 K\nhJ : J \u2260 0\nI I' : FractionalIdeal A\u2070 K\n\u22a2 I * J \u2264 I' * J \u2192 I \u2264 I'", "state_after": "case mp\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : IsDedekindDomain A\nJ : FractionalIdeal A\u2070 K\nhJ : J \u2260 0\nI I' : FractionalIdeal A\u2070 K\nh : I * J \u2264 I' * J\n\u22a2 I \u2264 I'"}, {"tactic": "convert mul_right_mono J\u207b\u00b9 h <;> dsimp only <;>\nrw [mul_assoc, FractionalIdeal.mul_inv_cancel hJ, mul_one]", "annotated_tactic": ["convert <a>mul_right_mono</a> J\u207b\u00b9 h <;> dsimp only <;>\n    rw [<a>mul_assoc</a>, <a>FractionalIdeal.mul_inv_cancel</a> hJ, <a>mul_one</a>]", [{"full_name": "FractionalIdeal.mul_right_mono", "def_path": "Mathlib/RingTheory/FractionalIdeal/Basic.lean", "def_pos": [600, 9], "def_end_pos": [600, 23]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "FractionalIdeal.mul_inv_cancel", "def_path": "Mathlib/RingTheory/DedekindDomain/Ideal.lean", "def_pos": [533, 19], "def_end_pos": [533, 33]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "case mp\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : IsDedekindDomain A\nJ : FractionalIdeal A\u2070 K\nhJ : J \u2260 0\nI I' : FractionalIdeal A\u2070 K\nh : I * J \u2264 I' * J\n\u22a2 I \u2264 I'", "state_after": "no goals"}, {"tactic": "exact fun h => mul_right_mono J h", "annotated_tactic": ["exact fun h => <a>mul_right_mono</a> J h", [{"full_name": "FractionalIdeal.mul_right_mono", "def_path": "Mathlib/RingTheory/FractionalIdeal/Basic.lean", "def_pos": [600, 9], "def_end_pos": [600, 23]}]], "state_before": "case mpr\nR : Type u_1\nA : Type u_2\nK : Type u_3\ninst\u271d\u2075 : CommRing R\ninst\u271d\u2074 : CommRing A\ninst\u271d\u00b3 : Field K\ninst\u271d\u00b2 : Algebra A K\ninst\u271d\u00b9 : IsFractionRing A K\ninst\u271d : IsDedekindDomain A\nJ : FractionalIdeal A\u2070 K\nhJ : J \u2260 0\nI I' : FractionalIdeal A\u2070 K\n\u22a2 I \u2264 I' \u2192 I * J \u2264 I' * J", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/CondCount.lean", "full_name": "ProbabilityTheory.condCount_add_compl_eq", "start": [193, 1], "end": [202, 33], "traced_tactics": [{"tactic": "have : condCount s t = (condCount (s \u2229 u) t * condCount (s \u2229 u \u222a s \u2229 u\u1d9c) (s \u2229 u) +\n    condCount (s \u2229 u\u1d9c) t * condCount (s \u2229 u \u222a s \u2229 u\u1d9c) (s \u2229 u\u1d9c)) := by\n  rw [condCount_disjoint_union (hs.inter_of_left _) (hs.inter_of_left _)\n    (disjoint_compl_right.mono inf_le_right inf_le_right), Set.inter_union_compl]", "annotated_tactic": ["have : <a>condCount</a> s t = (<a>condCount</a> (s \u2229 u) t * <a>condCount</a> (s \u2229 u \u222a s \u2229 u\u1d9c) (s \u2229 u) +\n      <a>condCount</a> (s \u2229 u\u1d9c) t * <a>condCount</a> (s \u2229 u \u222a s \u2229 u\u1d9c) (s \u2229 u\u1d9c)) := by\n    rw [<a>condCount_disjoint_union</a> (hs.inter_of_left _) (hs.inter_of_left _)\n      (disjoint_compl_right.mono <a>inf_le_right</a> <a>inf_le_right</a>), <a>Set.inter_union_compl</a>]", [{"full_name": "ProbabilityTheory.condCount", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [54, 5], "def_end_pos": [54, 14]}, {"full_name": "ProbabilityTheory.condCount", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [54, 5], "def_end_pos": [54, 14]}, {"full_name": "ProbabilityTheory.condCount", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [54, 5], "def_end_pos": [54, 14]}, {"full_name": "ProbabilityTheory.condCount", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [54, 5], "def_end_pos": [54, 14]}, {"full_name": "ProbabilityTheory.condCount", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [54, 5], "def_end_pos": [54, 14]}, {"full_name": "ProbabilityTheory.condCount_disjoint_union", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [170, 9], "def_end_pos": [170, 33]}, {"full_name": "inf_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [366, 9], "def_end_pos": [366, 21]}, {"full_name": "inf_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [366, 9], "def_end_pos": [366, 21]}, {"full_name": "Set.inter_union_compl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1839, 9], "def_end_pos": [1839, 26]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u\u271d u t : Set \u03a9\nhs : s.Finite\n\u22a2 (condCount (s \u2229 u)) t * (condCount s) u + (condCount (s \u2229 u\u1d9c)) t * (condCount s) u\u1d9c = (condCount s) t", "state_after": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u\u271d u t : Set \u03a9\nhs : s.Finite\nthis :\n  (condCount s) t =\n    (condCount (s \u2229 u)) t * (condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n      (condCount (s \u2229 u\u1d9c)) t * (condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)\n\u22a2 (condCount (s \u2229 u)) t * (condCount s) u + (condCount (s \u2229 u\u1d9c)) t * (condCount s) u\u1d9c = (condCount s) t"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], 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(condCount s) u + (condCount (s \u2229 u\u1d9c)) t * (condCount s) u\u1d9c =\n    (condCount (s \u2229 u)) t * (condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n      (condCount (s \u2229 u\u1d9c)) t * (condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)"}, {"tactic": "simp [condCount_inter_self hs]", "annotated_tactic": ["simp [<a>condCount_inter_self</a> hs]", [{"full_name": "ProbabilityTheory.condCount_inter_self", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [100, 9], "def_end_pos": [100, 29]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u\u271d u t : Set \u03a9\nhs : s.Finite\nthis :\n  (condCount s) t =\n    (condCount (s \u2229 u)) t * (condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n      (condCount (s \u2229 u\u1d9c)) t * (condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)\n\u22a2 (condCount (s \u2229 u)) t * (condCount s) u + (condCount (s \u2229 u\u1d9c)) t * (condCount s) u\u1d9c =\n    (condCount (s \u2229 u)) t * (condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n      (condCount (s \u2229 u\u1d9c)) t * (condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)", "state_after": "no goals"}, {"tactic": "rw [condCount_disjoint_union (hs.inter_of_left _) (hs.inter_of_left _)\n  (disjoint_compl_right.mono inf_le_right inf_le_right), Set.inter_union_compl]", "annotated_tactic": ["rw [<a>condCount_disjoint_union</a> (hs.inter_of_left _) (hs.inter_of_left _)\n      (disjoint_compl_right.mono <a>inf_le_right</a> <a>inf_le_right</a>), <a>Set.inter_union_compl</a>]", [{"full_name": "ProbabilityTheory.condCount_disjoint_union", "def_path": "Mathlib/Probability/CondCount.lean", "def_pos": [170, 9], "def_end_pos": [170, 33]}, {"full_name": "inf_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [366, 9], "def_end_pos": [366, 21]}, {"full_name": "inf_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [366, 9], "def_end_pos": [366, 21]}, {"full_name": "Set.inter_union_compl", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1839, 9], "def_end_pos": [1839, 26]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03a9\ninst\u271d : MeasurableSingletonClass \u03a9\ns t\u271d u\u271d u t : Set \u03a9\nhs : s.Finite\n\u22a2 (condCount s) t =\n    (condCount (s \u2229 u)) t * (condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u) +\n      (condCount (s \u2229 u\u1d9c)) t * (condCount (s \u2229 u \u222a s \u2229 u\u1d9c)) (s \u2229 u\u1d9c)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/DirectSum/Internal.lean", "full_name": "SetLike.GradeZero.coe_ofNat", "start": [379, 35], "end": [380, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GeomSum.lean", "full_name": "geom_sum\u2082_with_one", "start": [98, 1], "end": [100, 52], "traced_tactics": [{"tactic": "rw [one_pow, mul_one]", "annotated_tactic": ["rw [<a>one_pow</a>, <a>mul_one</a>]", [{"full_name": "one_pow", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [696, 39], "def_end_pos": [696, 46]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "\u03b1 : Type u\ninst\u271d : Semiring \u03b1\nx : \u03b1\nn i : \u2115\nx\u271d : i \u2208 range n\n\u22a2 x ^ i * 1 ^ (n - 1 - i) = x ^ i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Bind.lean", "full_name": "Multiset.singleton_join", "start": [66, 1], "end": [67, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GroupWithZero/Divisibility.lean", "full_name": "not_isRelPrime_zero_zero", "start": [89, 1], "end": [90, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "full_name": "eq_one_of_mul_le_one_left", "start": [1208, 1], "end": [1209, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Bounded.lean", "full_name": "Set.unbounded_inter_not", "start": [309, 1], "end": [311, 51], "traced_tactics": [{"tactic": "simp_rw [\u2190 not_bounded_iff, bounded_inter_not H]", "annotated_tactic": ["simp_rw [\u2190 <a>not_bounded_iff</a>, <a>bounded_inter_not</a> H]", [{"full_name": 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"end": [612, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Coxeter/Basic.lean", "full_name": "CoxeterMatrix.reindex_relationsSet", "start": [110, 1], "end": [119, 50], "traced_tactics": [{"tactic": "simp [Set.range_comp]", "annotated_tactic": ["simp [<a>Set.range_comp</a>]", [{"full_name": "Set.range_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [731, 9], "def_end_pos": [731, 19]}]], "state_before": "B : Type u_1\nB' : Type u_2\nM : CoxeterMatrix B\ne : B \u2243 B'\nM' : CoxeterMatrix B' := CoxeterMatrix.reindex e M\n\u22a2 Set.range (uncurry M'.relation) = Set.range (uncurry M'.relation \u2218 Prod.map \u21d1e \u21d1e)", "state_after": "no goals"}, {"tactic": "apply congrArg Set.range", "annotated_tactic": ["apply <a>congrArg</a> <a>Set.range</a>", [{"full_name": "congrArg", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [362, 9], "def_end_pos": [362, 17]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [157, 5], "def_end_pos": [157, 10]}]], "state_before": "B : Type u_1\nB' : Type u_2\nM : CoxeterMatrix B\ne : B \u2243 B'\nM' : CoxeterMatrix B' := CoxeterMatrix.reindex e M\n\u22a2 Set.range (uncurry M'.relation \u2218 Prod.map \u21d1e \u21d1e) = Set.range (\u21d1(FreeGroup.freeGroupCongr e) \u2218 uncurry M.relation)", "state_after": "B : Type u_1\nB' : Type u_2\nM : CoxeterMatrix B\ne : B \u2243 B'\nM' : CoxeterMatrix B' := CoxeterMatrix.reindex e M\n\u22a2 uncurry M'.relation \u2218 Prod.map \u21d1e \u21d1e = \u21d1(FreeGroup.freeGroupCongr e) \u2218 uncurry M.relation"}, {"tactic": "ext \u27e8i, i'\u27e9", "annotated_tactic": ["ext \u27e8i, i'\u27e9", []], "state_before": "B : Type u_1\nB' : Type u_2\nM : CoxeterMatrix B\ne : B \u2243 B'\nM' : CoxeterMatrix B' := CoxeterMatrix.reindex e M\n\u22a2 uncurry M'.relation \u2218 Prod.map \u21d1e \u21d1e = \u21d1(FreeGroup.freeGroupCongr e) \u2218 uncurry M.relation", "state_after": "case h.mk\nB : Type u_1\nB' : Type u_2\nM : CoxeterMatrix B\ne : B \u2243 B'\nM' : CoxeterMatrix B' := CoxeterMatrix.reindex e M\ni i' : B\n\u22a2 (uncurry M'.relation \u2218 Prod.map \u21d1e \u21d1e) (i, i') = (\u21d1(FreeGroup.freeGroupCongr e) \u2218 uncurry M.relation) (i, i')"}, {"tactic": "simp [relation, reindex_apply, M']", "annotated_tactic": ["simp [<a>relation</a>, <a>reindex_apply</a>, M']", [{"full_name": "CoxeterMatrix.relation", "def_path": "Mathlib/GroupTheory/Coxeter/Basic.lean", "def_pos": [96, 5], "def_end_pos": [96, 13]}, {"full_name": "CoxeterMatrix.reindex_apply", "def_path": "Mathlib/GroupTheory/Coxeter/Matrix.lean", "def_pos": [92, 9], "def_end_pos": [92, 22]}]], "state_before": "case h.mk\nB : Type u_1\nB' : Type u_2\nM : CoxeterMatrix B\ne : B \u2243 B'\nM' : CoxeterMatrix B' := CoxeterMatrix.reindex e M\ni i' : B\n\u22a2 (uncurry M'.relation \u2218 Prod.map \u21d1e \u21d1e) (i, i') = (\u21d1(FreeGroup.freeGroupCongr e) \u2218 uncurry M.relation) (i, i')", "state_after": "no goals"}, {"tactic": "simp [Set.range_comp, relationsSet]", "annotated_tactic": ["simp [<a>Set.range_comp</a>, <a>relationsSet</a>]", [{"full_name": "Set.range_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [731, 9], "def_end_pos": [731, 19]}, {"full_name": "CoxeterMatrix.relationsSet", "def_path": "Mathlib/GroupTheory/Coxeter/Basic.lean", "def_pos": [99, 5], "def_end_pos": [99, 17]}]], "state_before": "B : Type u_1\nB' : Type u_2\nM : CoxeterMatrix B\ne : B \u2243 B'\nM' : CoxeterMatrix B' := CoxeterMatrix.reindex e M\n\u22a2 Set.range (\u21d1(FreeGroup.freeGroupCongr e) \u2218 uncurry M.relation) = \u21d1(FreeGroup.freeGroupCongr e) '' M.relationsSet", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Lattice.lean", "full_name": "MonotoneOn.of_map_sup", 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positivity) (fun a \u21a6 ?_)\n    (summable_geometric_of_lt_one (one_div_nonneg.mpr (zero_le_one.trans hm.le))\n      ((one_div_lt (zero_lt_one.trans hm) zero_lt_one).mpr (one_div_one.le.trans_lt hm)))", "annotated_tactic": ["refine .of_nonneg_of_le (fun a \u21a6 by positivity) (fun a \u21a6 ?_)\n      (<a>summable_geometric_of_lt_one</a> (one_div_nonneg.mpr (zero_le_one.trans hm.le))\n        ((<a>one_div_lt</a> (zero_lt_one.trans hm) <a>zero_lt_one</a>).<a>mpr</a> (one_div_one.le.trans_lt hm)))", [{"full_name": "summable_geometric_of_lt_one", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [273, 9], "def_end_pos": [273, 37]}, {"full_name": "one_div_lt", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [378, 9], "def_end_pos": [378, 19]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : \u211d\nf : \u2115 \u2192 \u2115\nhm : 1 < m\nfi : \u2200 (i : \u2115), i \u2264 f i\n\u22a2 Summable fun i => 1 / m ^ f i", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : \u211d\nf : \u2115 \u2192 \u2115\nhm : 1 < m\nfi : \u2200 (i : \u2115), i \u2264 f i\na : \u2115\n\u22a2 1 / m ^ f a \u2264 (1 / m) ^ a"}, {"tactic": "rw [div_pow, one_pow]", "annotated_tactic": ["rw [<a>div_pow</a>, <a>one_pow</a>]", [{"full_name": "div_pow", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [845, 7], "def_end_pos": [845, 14]}, {"full_name": "one_pow", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [696, 39], "def_end_pos": [696, 46]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : \u211d\nf : \u2115 \u2192 \u2115\nhm : 1 < m\nfi : \u2200 (i : \u2115), i \u2264 f i\na : \u2115\n\u22a2 1 / m ^ f a \u2264 (1 / m) ^ a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : \u211d\nf : \u2115 \u2192 \u2115\nhm : 1 < m\nfi : \u2200 (i : \u2115), i \u2264 f i\na : \u2115\n\u22a2 1 / m ^ f a \u2264 1 / m ^ a"}, {"tactic": "refine (one_div_le_one_div ?_ ?_).mpr (pow_le_pow_right hm.le (fi a)) <;>\n  exact pow_pos (zero_lt_one.trans hm) _", "annotated_tactic": ["refine (<a>one_div_le_one_div</a> ?_ ?_).<a>mpr</a> (<a>pow_le_pow_right</a> hm.le (fi a)) <;>\n    exact <a>pow_pos</a> (zero_lt_one.trans hm) _", [{"full_name": "one_div_le_one_div", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [410, 9], "def_end_pos": [410, 27]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}, {"full_name": "pow_le_pow_right", "def_path": "Mathlib/Algebra/Order/Ring/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 25]}, {"full_name": "pow_pos", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [552, 9], "def_end_pos": [552, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : \u211d\nf : \u2115 \u2192 \u2115\nhm : 1 < m\nfi : \u2200 (i : \u2115), i \u2264 f i\na : \u2115\n\u22a2 1 / m ^ f a \u2264 1 / m ^ a", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm : \u211d\nf : \u2115 \u2192 \u2115\nhm : 1 < m\nfi : \u2200 (i : \u2115), i \u2264 f i\na : \u2115\n\u22a2 0 \u2264 1 / m ^ f a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Instances/AddCircle.lean", "full_name": "AddCircle.liftIco_coe_apply", "start": [222, 1], "end": [228, 6], "traced_tactics": [{"tactic": "have : (equivIco p a) x = \u27e8x, hx\u27e9 := by\n  rw [Equiv.apply_eq_iff_eq_symm_apply]\n  rfl", "annotated_tactic": ["have : (<a>equivIco</a> p a) x = \u27e8x, hx\u27e9 := by\n    rw [<a>Equiv.apply_eq_iff_eq_symm_apply</a>]\n    rfl", [{"full_name": "AddCircle.equivIco", "def_path": "Mathlib/Topology/Instances/AddCircle.lean", "def_pos": [189, 5], "def_end_pos": [189, 13]}, {"full_name": "Equiv.apply_eq_iff_eq_symm_apply", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [319, 9], "def_end_pos": [319, 35]}]], "state_before": "\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : LinearOrderedAddCommGroup \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b9 : OrderTopology \ud835\udd5c\np : \ud835\udd5c\nhp : Fact (0 < p)\na : \ud835\udd5c\ninst\u271d : Archimedean \ud835\udd5c\nf : \ud835\udd5c \u2192 B\nx : \ud835\udd5c\nhx : x \u2208 Ico a (a + p)\n\u22a2 liftIco p a f \u2191x = f x", "state_after": "\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : LinearOrderedAddCommGroup \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b9 : OrderTopology \ud835\udd5c\np : \ud835\udd5c\nhp : Fact (0 < p)\na : \ud835\udd5c\ninst\u271d : Archimedean \ud835\udd5c\nf : \ud835\udd5c \u2192 B\nx : \ud835\udd5c\nhx : x \u2208 Ico a (a + p)\nthis : (equivIco p a) \u2191x = \u27e8x, hx\u27e9\n\u22a2 liftIco p a f \u2191x = f x"}, {"tactic": "rw [liftIco, comp_apply, this]", "annotated_tactic": ["rw [<a>liftIco</a>, <a>comp_apply</a>, this]", [{"full_name": "AddCircle.liftIco", "def_path": "Mathlib/Topology/Instances/AddCircle.lean", "def_pos": [201, 5], "def_end_pos": [201, 12]}, {"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}]], "state_before": "\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : LinearOrderedAddCommGroup \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b9 : OrderTopology \ud835\udd5c\np : \ud835\udd5c\nhp : Fact (0 < p)\na : \ud835\udd5c\ninst\u271d : Archimedean \ud835\udd5c\nf : \ud835\udd5c \u2192 B\nx : \ud835\udd5c\nhx : x \u2208 Ico a (a + p)\nthis : (equivIco p a) \u2191x = \u27e8x, hx\u27e9\n\u22a2 liftIco p a f \u2191x = f x", "state_after": "\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : LinearOrderedAddCommGroup \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b9 : OrderTopology \ud835\udd5c\np : \ud835\udd5c\nhp : Fact (0 < p)\na : \ud835\udd5c\ninst\u271d : Archimedean \ud835\udd5c\nf : \ud835\udd5c \u2192 B\nx : \ud835\udd5c\nhx : x \u2208 Ico a (a + p)\nthis : (equivIco p a) \u2191x = \u27e8x, hx\u27e9\n\u22a2 (Ico a (a + p)).restrict f \u27e8x, hx\u27e9 = f x"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : LinearOrderedAddCommGroup \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b9 : OrderTopology \ud835\udd5c\np : \ud835\udd5c\nhp : Fact (0 < p)\na : \ud835\udd5c\ninst\u271d : Archimedean \ud835\udd5c\nf : \ud835\udd5c \u2192 B\nx : \ud835\udd5c\nhx : x \u2208 Ico a (a + p)\nthis : (equivIco p a) \u2191x = \u27e8x, hx\u27e9\n\u22a2 (Ico a (a + p)).restrict f \u27e8x, hx\u27e9 = f x", "state_after": "no goals"}, {"tactic": "rw [Equiv.apply_eq_iff_eq_symm_apply]", "annotated_tactic": ["rw [<a>Equiv.apply_eq_iff_eq_symm_apply</a>]", [{"full_name": "Equiv.apply_eq_iff_eq_symm_apply", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [319, 9], "def_end_pos": [319, 35]}]], "state_before": "\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : LinearOrderedAddCommGroup \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b9 : OrderTopology \ud835\udd5c\np : \ud835\udd5c\nhp : Fact (0 < p)\na : \ud835\udd5c\ninst\u271d : Archimedean \ud835\udd5c\nf : \ud835\udd5c \u2192 B\nx : \ud835\udd5c\nhx : x \u2208 Ico a (a + p)\n\u22a2 (equivIco p a) \u2191x = \u27e8x, hx\u27e9", "state_after": "\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : LinearOrderedAddCommGroup \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b9 : OrderTopology \ud835\udd5c\np : \ud835\udd5c\nhp : Fact (0 < p)\na : \ud835\udd5c\ninst\u271d : Archimedean \ud835\udd5c\nf : \ud835\udd5c \u2192 B\nx : \ud835\udd5c\nhx : x \u2208 Ico a (a + p)\n\u22a2 \u2191x = (equivIco p a).symm \u27e8x, hx\u27e9"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : LinearOrderedAddCommGroup \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b9 : OrderTopology \ud835\udd5c\np : \ud835\udd5c\nhp : Fact (0 < p)\na : \ud835\udd5c\ninst\u271d : Archimedean \ud835\udd5c\nf : \ud835\udd5c \u2192 B\nx : \ud835\udd5c\nhx : x \u2208 Ico a (a + p)\n\u22a2 \u2191x = (equivIco p a).symm \u27e8x, hx\u27e9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean", "full_name": "mfderiv_fst", "start": [271, 1], "end": [274, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/BifunctorHomotopy.lean", "full_name": "HomologicalComplex.mapBifunctorMapHomotopy.\u03b9MapBifunctor_hom\u2081", "start": [45, 1], "end": [52, 14], "traced_tactics": [{"tactic": "subst h'", "annotated_tactic": ["subst h'", []], "state_before": "C\u2081 : Type u_1\nC\u2082 : Type u_2\nD : Type u_3\nI\u2081 : Type u_4\nI\u2082 : Type u_5\nJ : Type u_6\ninst\u271d\u00b9\u00b9 : Category.{u_9, u_1} C\u2081\ninst\u271d\u00b9\u2070 : Category.{u_8, u_2} C\u2082\ninst\u271d\u2079 : Category.{u_7, u_3} D\ninst\u271d\u2078 : Preadditive C\u2081\ninst\u271d\u2077 : Preadditive C\u2082\ninst\u271d\u2076 : Preadditive D\nc\u2081 : ComplexShape I\u2081\nc\u2082 : ComplexShape I\u2082\nK\u2081 L\u2081 : HomologicalComplex C\u2081 c\u2081\nf\u2081 f\u2081' : K\u2081 \u27f6 L\u2081\nh\u2081 : Homotopy f\u2081 f\u2081'\nK\u2082 L\u2082 : HomologicalComplex C\u2082 c\u2082\nf\u2082 : K\u2082 \u27f6 L\u2082\nF : C\u2081 \u2964 C\u2082 \u2964 D\ninst\u271d\u2075 : F.Additive\ninst\u271d\u2074 : \u2200 (X\u2081 : C\u2081), (F.obj X\u2081).Additive\nc : ComplexShape J\ninst\u271d\u00b3 : DecidableEq J\ninst\u271d\u00b2 : TotalComplexShape c\u2081 c\u2082 c\ninst\u271d\u00b9 : K\u2081.HasMapBifunctor K\u2082 F c\ninst\u271d : L\u2081.HasMapBifunctor L\u2082 F c\ni\u2081 i\u2081' : I\u2081\ni\u2082 : I\u2082\nj j' : J\nh : c\u2081.\u03c0 c\u2082 c (i\u2081', i\u2082) = j\nh' : c\u2081.prev i\u2081' = i\u2081\n\u22a2 K\u2081.\u03b9MapBifunctor K\u2082 F c i\u2081' i\u2082 j h \u226b hom\u2081 h\u2081 f\u2082 F c j j' =\n    c\u2081.\u03b5\u2081 c\u2082 c (i\u2081, i\u2082) \u2022\n      (F.map (h\u2081.hom i\u2081' i\u2081)).app (K\u2082.X i\u2082) \u226b (F.obj (L\u2081.X i\u2081)).map (f\u2082.f i\u2082) \u226b L\u2081.\u03b9MapBifunctorOrZero L\u2082 F c i\u2081 i\u2082 j'", "state_after": "C\u2081 : Type u_1\nC\u2082 : Type u_2\nD : Type u_3\nI\u2081 : Type u_4\nI\u2082 : Type u_5\nJ : Type u_6\ninst\u271d\u00b9\u00b9 : Category.{u_9, u_1} C\u2081\ninst\u271d\u00b9\u2070 : Category.{u_8, u_2} C\u2082\ninst\u271d\u2079 : Category.{u_7, u_3} D\ninst\u271d\u2078 : Preadditive C\u2081\ninst\u271d\u2077 : Preadditive C\u2082\ninst\u271d\u2076 : Preadditive D\nc\u2081 : ComplexShape I\u2081\nc\u2082 : ComplexShape I\u2082\nK\u2081 L\u2081 : HomologicalComplex C\u2081 c\u2081\nf\u2081 f\u2081' : K\u2081 \u27f6 L\u2081\nh\u2081 : Homotopy f\u2081 f\u2081'\nK\u2082 L\u2082 : HomologicalComplex C\u2082 c\u2082\nf\u2082 : K\u2082 \u27f6 L\u2082\nF : C\u2081 \u2964 C\u2082 \u2964 D\ninst\u271d\u2075 : F.Additive\ninst\u271d\u2074 : \u2200 (X\u2081 : C\u2081), (F.obj X\u2081).Additive\nc : ComplexShape J\ninst\u271d\u00b3 : DecidableEq J\ninst\u271d\u00b2 : TotalComplexShape c\u2081 c\u2082 c\ninst\u271d\u00b9 : K\u2081.HasMapBifunctor K\u2082 F c\ninst\u271d : L\u2081.HasMapBifunctor L\u2082 F c\ni\u2081' : I\u2081\ni\u2082 : I\u2082\nj j' : J\nh : c\u2081.\u03c0 c\u2082 c (i\u2081', i\u2082) = j\n\u22a2 K\u2081.\u03b9MapBifunctor K\u2082 F c i\u2081' i\u2082 j h \u226b hom\u2081 h\u2081 f\u2082 F c j j' =\n    c\u2081.\u03b5\u2081 c\u2082 c (c\u2081.prev i\u2081', i\u2082) \u2022\n      (F.map (h\u2081.hom i\u2081' (c\u2081.prev i\u2081'))).app (K\u2082.X i\u2082) \u226b\n        (F.obj (L\u2081.X (c\u2081.prev i\u2081'))).map (f\u2082.f i\u2082) \u226b L\u2081.\u03b9MapBifunctorOrZero L\u2082 F c (c\u2081.prev i\u2081') i\u2082 j'"}, {"tactic": "simp [hom\u2081]", "annotated_tactic": ["simp [<a>hom\u2081</a>]", [{"full_name": "HomologicalComplex.mapBifunctorMapHomotopy.hom\u2081", "def_path": "Mathlib/Algebra/Homology/BifunctorHomotopy.lean", "def_pos": [38, 19], "def_end_pos": [38, 23]}]], "state_before": "C\u2081 : Type u_1\nC\u2082 : Type u_2\nD : Type u_3\nI\u2081 : Type u_4\nI\u2082 : Type u_5\nJ : Type u_6\ninst\u271d\u00b9\u00b9 : Category.{u_9, u_1} C\u2081\ninst\u271d\u00b9\u2070 : Category.{u_8, u_2} C\u2082\ninst\u271d\u2079 : Category.{u_7, u_3} D\ninst\u271d\u2078 : Preadditive C\u2081\ninst\u271d\u2077 : Preadditive C\u2082\ninst\u271d\u2076 : Preadditive D\nc\u2081 : ComplexShape I\u2081\nc\u2082 : ComplexShape I\u2082\nK\u2081 L\u2081 : HomologicalComplex C\u2081 c\u2081\nf\u2081 f\u2081' : K\u2081 \u27f6 L\u2081\nh\u2081 : Homotopy f\u2081 f\u2081'\nK\u2082 L\u2082 : HomologicalComplex C\u2082 c\u2082\nf\u2082 : K\u2082 \u27f6 L\u2082\nF : C\u2081 \u2964 C\u2082 \u2964 D\ninst\u271d\u2075 : F.Additive\ninst\u271d\u2074 : \u2200 (X\u2081 : C\u2081), (F.obj X\u2081).Additive\nc : ComplexShape J\ninst\u271d\u00b3 : DecidableEq J\ninst\u271d\u00b2 : TotalComplexShape c\u2081 c\u2082 c\ninst\u271d\u00b9 : K\u2081.HasMapBifunctor K\u2082 F c\ninst\u271d : L\u2081.HasMapBifunctor L\u2082 F c\ni\u2081' : I\u2081\ni\u2082 : I\u2082\nj j' : J\nh : c\u2081.\u03c0 c\u2082 c (i\u2081', i\u2082) = j\n\u22a2 K\u2081.\u03b9MapBifunctor K\u2082 F c i\u2081' i\u2082 j h \u226b hom\u2081 h\u2081 f\u2082 F c j j' =\n    c\u2081.\u03b5\u2081 c\u2082 c (c\u2081.prev i\u2081', i\u2082) \u2022\n      (F.map (h\u2081.hom i\u2081' (c\u2081.prev i\u2081'))).app (K\u2082.X i\u2082) \u226b\n        (F.obj (L\u2081.X (c\u2081.prev i\u2081'))).map (f\u2082.f i\u2082) \u226b L\u2081.\u03b9MapBifunctorOrZero L\u2082 F c (c\u2081.prev i\u2081') i\u2082 j'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/SuccPred/Limit.lean", "full_name": "Order.isSuccLimitRecOn_succ", "start": [228, 1], "end": [230, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Int/Defs.lean", "full_name": "Int.natCast_succ_pos", "start": [136, 1], "end": [136, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean", "full_name": "MeasureTheory.FiniteMeasure.tendsto_normalize_of_tendsto", "start": [508, 1], "end": [513, 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": [296, 9], "def_end_pos": [296, 54]}, {"full_name": "MeasureTheory.FiniteMeasure.tendsto_iff_forall_testAgainstNN_tendsto", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [511, 9], "def_end_pos": [511, 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 => (\u03bcs i).normalize) F (\ud835\udcdd \u03bc.normalize)", "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 => ((ProbabilityMeasure.toFiniteMeasure \u2218 fun i => (\u03bcs i).normalize) i).testAgainstNN f) F\n      (\ud835\udcdd (\u03bc.normalize.toFiniteMeasure.testAgainstNN 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": [470, 9], "def_end_pos": [470, 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 => ((ProbabilityMeasure.toFiniteMeasure \u2218 fun i => (\u03bcs i).normalize) i).testAgainstNN f) F\n      (\ud835\udcdd (\u03bc.normalize.toFiniteMeasure.testAgainstNN f))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/ContMDiff/NormedSpace.lean", "full_name": "ContMDiffOn.cle_arrowCongr", "start": [255, 1], "end": [260, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/NatAntidiagonal.lean", "full_name": "Finset.Nat.antidiagonal_filter_le_fst_of_le", "start": [133, 1], "end": [143, 34], "traced_tactics": [{"tactic": "ext \u27e8i, j\u27e9", "annotated_tactic": ["ext \u27e8i, j\u27e9", []], "state_before": "n k : \u2115\nh : k \u2264 n\n\u22a2 filter (fun a => k \u2264 a.1) (antidiagonal n) =\n    map ({ toFun := fun x => x + k, inj' := \u22ef }.prodMap (Embedding.refl \u2115)) (antidiagonal (n - k))", "state_after": "case a.mk\nn k : \u2115\nh : k \u2264 n\ni j : \u2115\n\u22a2 (i, j) \u2208 filter (fun a => k \u2264 a.1) (antidiagonal n) \u2194\n    (i, j) \u2208 map ({ toFun := fun x => x + k, inj' := \u22ef }.prodMap (Embedding.refl \u2115)) (antidiagonal (n - k))"}, {"tactic": "suffices i + j = n \u2227 k \u2264 i \u2194 \u2203 a, a + j = n - k \u2227 a + k = i by simpa", "annotated_tactic": ["suffices i + j = n \u2227 k \u2264 i \u2194 \u2203 a, a + j = n - k \u2227 a + k = i by simpa", []], "state_before": "case a.mk\nn k : \u2115\nh : k \u2264 n\ni j : \u2115\n\u22a2 (i, j) \u2208 filter (fun a => k \u2264 a.1) (antidiagonal n) \u2194\n    (i, j) \u2208 map ({ toFun := fun x => x + k, inj' := \u22ef }.prodMap (Embedding.refl \u2115)) (antidiagonal (n - k))", "state_after": "case a.mk\nn k : \u2115\nh : k \u2264 n\ni j : \u2115\n\u22a2 i + j = n \u2227 k \u2264 i \u2194 \u2203 a, a + j = n - k \u2227 a + k = i"}, {"tactic": "refine \u27e8fun hi \u21a6 \u27e8i - k, ?_, tsub_add_cancel_of_le hi.2\u27e9, ?_\u27e9", "annotated_tactic": ["refine \u27e8fun hi \u21a6 \u27e8i - k, ?_, <a>tsub_add_cancel_of_le</a> hi.2\u27e9, ?_\u27e9", [{"full_name": "tsub_add_cancel_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [31, 9], "def_end_pos": [31, 30]}]], "state_before": "case a.mk\nn k : \u2115\nh : k \u2264 n\ni j : \u2115\n\u22a2 i + j = n \u2227 k \u2264 i \u2194 \u2203 a, a + j = n - k \u2227 a + k = i", "state_after": "case a.mk.refine_1\nn k : \u2115\nh : k \u2264 n\ni j : \u2115\nhi : i + j = n \u2227 k \u2264 i\n\u22a2 i - k + j = n - k\n\ncase a.mk.refine_2\nn k : \u2115\nh : k \u2264 n\ni j : \u2115\n\u22a2 (\u2203 a, a + j = n - k \u2227 a + k = i) \u2192 i + j = n \u2227 k \u2264 i"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "n k : \u2115\nh : k \u2264 n\ni j : \u2115\nthis : i + j = n \u2227 k \u2264 i \u2194 \u2203 a, a + j = n - k \u2227 a + k = i\n\u22a2 (i, j) \u2208 filter (fun a => k \u2264 a.1) (antidiagonal n) \u2194\n    (i, j) \u2208 map ({ toFun := fun x => x + k, inj' := \u22ef }.prodMap (Embedding.refl \u2115)) (antidiagonal (n - k))", "state_after": "no goals"}, {"tactic": "rw [\u2190 Nat.sub_add_comm hi.2, hi.1]", "annotated_tactic": ["rw [\u2190 <a>Nat.sub_add_comm</a> hi.2, hi.1]", [{"full_name": "Nat.sub_add_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean", "def_pos": [125, 19], "def_end_pos": [125, 31]}]], "state_before": "case a.mk.refine_1\nn k : \u2115\nh : k \u2264 n\ni j : \u2115\nhi : i + j = n \u2227 k \u2264 i\n\u22a2 i - k + j = n - k", "state_after": "no goals"}, {"tactic": "rintro \u27e8l, hl, rfl\u27e9", "annotated_tactic": ["rintro \u27e8l, hl, rfl\u27e9", []], "state_before": "case a.mk.refine_2\nn k : \u2115\nh : k \u2264 n\ni j : \u2115\n\u22a2 (\u2203 a, a + j = n - k \u2227 a + k = i) \u2192 i + j = n \u2227 k \u2264 i", "state_after": "case a.mk.refine_2.intro.intro\nn k : \u2115\nh : k \u2264 n\nj l : \u2115\nhl : l + j = n - k\n\u22a2 l + k + j = n \u2227 k \u2264 l + k"}, {"tactic": "refine \u27e8?_, Nat.le_add_left k l\u27e9", "annotated_tactic": ["refine \u27e8?_, <a>Nat.le_add_left</a> k l\u27e9", [{"full_name": "Nat.le_add_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [399, 9], "def_end_pos": [399, 20]}]], "state_before": "case a.mk.refine_2.intro.intro\nn k : \u2115\nh : k \u2264 n\nj l : \u2115\nhl : l + j = n - k\n\u22a2 l + k + j = n \u2227 k \u2264 l + k", "state_after": "case a.mk.refine_2.intro.intro\nn k : \u2115\nh : k \u2264 n\nj l : \u2115\nhl : l + j = n - k\n\u22a2 l + k + j = n"}, {"tactic": "rw [add_right_comm, hl]", "annotated_tactic": ["rw [<a>add_right_comm</a>, hl]", [{"full_name": "add_right_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [189, 3], "def_end_pos": [189, 14]}]], "state_before": "case a.mk.refine_2.intro.intro\nn k : \u2115\nh : k \u2264 n\nj l : \u2115\nhl : l + j = n - k\n\u22a2 l + k + j = n", "state_after": "case a.mk.refine_2.intro.intro\nn k : \u2115\nh : k \u2264 n\nj l : \u2115\nhl : l + j = n - k\n\u22a2 n - k + k = n"}, {"tactic": "exact tsub_add_cancel_of_le h", "annotated_tactic": ["exact <a>tsub_add_cancel_of_le</a> h", [{"full_name": "tsub_add_cancel_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [31, 9], "def_end_pos": [31, 30]}]], "state_before": "case a.mk.refine_2.intro.intro\nn k : \u2115\nh : k \u2264 n\nj l : \u2115\nhl : l + j = n - k\n\u22a2 n - k + k = n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Deprecated/Subgroup.lean", "full_name": "IsSubgroup.eq_trivial_iff", "start": [249, 1], "end": [251, 95], "traced_tactics": [{"tactic": "simp only [Set.ext_iff, IsSubgroup.mem_trivial]", "annotated_tactic": ["simp only [<a>Set.ext_iff</a>, <a>IsSubgroup.mem_trivial</a>]", [{"full_name": "Set.ext_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [215, 9], "def_end_pos": [215, 16]}, {"full_name": "IsSubgroup.mem_trivial", "def_path": "Mathlib/Deprecated/Subgroup.lean", "def_pos": [238, 9], "def_end_pos": [238, 20]}]], "state_before": "G : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns : Set G\nhs : IsSubgroup s\n\u22a2 s = trivial G \u2194 \u2200 x \u2208 s, x = 1", "state_after": "G : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns : Set G\nhs : IsSubgroup s\n\u22a2 (\u2200 (x : G), x \u2208 s \u2194 x = 1) \u2194 \u2200 x \u2208 s, x = 1"}, {"tactic": "exact \u27e8fun h x => (h x).1, fun h x => \u27e8h x, fun hx => hx.symm \u25b8 hs.toIsSubmonoid.one_mem\u27e9\u27e9", "annotated_tactic": ["exact \u27e8fun h x => (h x).1, fun h x => \u27e8h x, fun hx => hx.symm \u25b8 hs.toIsSubmonoid.one_mem\u27e9\u27e9", []], "state_before": "G : Type u_1\nH : Type u_2\nA : Type u_3\na a\u2081 a\u2082 b c : G\ninst\u271d : Group G\ns : Set G\nhs : IsSubgroup s\n\u22a2 (\u2200 (x : G), x \u2208 s \u2194 x = 1) \u2194 \u2200 x \u2208 s, x = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Galois/Basic.lean", "full_name": "CategoryTheory.PreGaloisCategory.card_fiber_eq_of_iso", "start": [340, 1], "end": [343, 55], "traced_tactics": [{"tactic": "have e : F.obj X \u2243 F.obj Y := Iso.toEquiv (mapIso (F \u22d9 FintypeCat.incl) i)", "annotated_tactic": ["have e : F.obj X \u2243 F.obj Y := <a>Iso.toEquiv</a> (<a>mapIso</a> (F \u22d9 <a>FintypeCat.incl</a>) i)", [{"full_name": "CategoryTheory.Iso.toEquiv", "def_path": "Mathlib/CategoryTheory/Types.lean", "def_pos": [358, 5], "def_end_pos": [358, 12]}, {"full_name": "CategoryTheory.Functor.mapIso", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [623, 5], "def_end_pos": [623, 11]}, {"full_name": "FintypeCat.incl", "def_path": "Mathlib/CategoryTheory/FintypeCat.lean", "def_pos": [59, 5], "def_end_pos": [59, 9]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{u\u2082, u\u2081} C\ninst\u271d\u00b9 : PreGaloisCategory C\nF : C \u2964 FintypeCat\ninst\u271d : FiberFunctor F\nX Y : C\ni : X \u2245 Y\n\u22a2 Nat.card \u2191(F.obj X) = Nat.card \u2191(F.obj Y)", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{u\u2082, u\u2081} C\ninst\u271d\u00b9 : PreGaloisCategory C\nF : C \u2964 FintypeCat\ninst\u271d : FiberFunctor F\nX Y : C\ni : X \u2245 Y\ne : \u2191(F.obj X) \u2243 \u2191(F.obj Y)\n\u22a2 Nat.card \u2191(F.obj X) = Nat.card \u2191(F.obj Y)"}, {"tactic": "exact Nat.card_eq_of_bijective e (Equiv.bijective e)", "annotated_tactic": ["exact <a>Nat.card_eq_of_bijective</a> e (<a>Equiv.bijective</a> e)", [{"full_name": "Nat.card_eq_of_bijective", "def_path": "Mathlib/SetTheory/Cardinal/Finite.lean", "def_pos": [93, 9], "def_end_pos": [93, 29]}, {"full_name": "Equiv.bijective", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [202, 19], "def_end_pos": [202, 28]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{u\u2082, u\u2081} C\ninst\u271d\u00b9 : PreGaloisCategory C\nF : C \u2964 FintypeCat\ninst\u271d : FiberFunctor F\nX Y : C\ni : X \u2245 Y\ne : \u2191(F.obj X) \u2243 \u2191(F.obj Y)\n\u22a2 Nat.card \u2191(F.obj X) = Nat.card \u2191(F.obj Y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean", "full_name": "Matrix.inv_mul_cancel_right_of_invertible", "start": [326, 1], "end": [327, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Integrals.lean", "full_name": "intervalIntegral.inv_mul_integral_comp_div", "start": [289, 1], "end": [290, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "full_name": "mul_neg_of_neg_of_pos", "start": [423, 1], "end": [424, 60], "traced_tactics": [{"tactic": "simpa only [zero_mul] using mul_lt_mul_of_pos_right ha hb", "annotated_tactic": ["simpa only [<a>zero_mul</a>] using <a>mul_lt_mul_of_pos_right</a> ha hb", [{"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}, {"full_name": "mul_lt_mul_of_pos_right", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [220, 9], "def_end_pos": [220, 32]}]], "state_before": "\u03b1 : Type u_1\na b c d : \u03b1\ninst\u271d\u00b2 : MulZeroClass \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : MulPosStrictMono \u03b1\nha : a < 0\nhb : 0 < b\n\u22a2 a * b < 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean", "full_name": "Action.functorCategoryMonoidalEquivalence.\u03bcIso_inv_app", "start": [219, 1], "end": [224, 46], "traced_tactics": [{"tactic": "rw [\u2190 NatIso.app_inv, \u2190 IsIso.Iso.inv_hom]", "annotated_tactic": ["rw [\u2190 <a>NatIso.app_inv</a>, \u2190 <a>IsIso.Iso.inv_hom</a>]", [{"full_name": "CategoryTheory.NatIso.app_inv", "def_path": "Mathlib/CategoryTheory/NatIso.lean", "def_pos": [93, 9], "def_end_pos": [93, 16]}, {"full_name": "CategoryTheory.IsIso.Iso.inv_hom", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [435, 9], "def_end_pos": [435, 20]}]], "state_before": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nA B : Action V G\n\u22a2 ((functorCategoryMonoidalEquivalence V G).\u03bcIso A B).inv.app PUnit.unit =\n    \ud835\udfd9 (((functorCategoryMonoidalEquivalence V G).obj (A \u2297 B)).obj PUnit.unit)", "state_after": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nA B : Action V G\n\u22a2 inv (((functorCategoryMonoidalEquivalence V G).\u03bcIso A B).app PUnit.unit).hom =\n    \ud835\udfd9 (((functorCategoryMonoidalEquivalence V G).obj (A \u2297 B)).obj PUnit.unit)"}, {"tactic": "refine IsIso.inv_eq_of_hom_inv_id ?_", "annotated_tactic": ["refine <a>IsIso.inv_eq_of_hom_inv_id</a> ?_", [{"full_name": "CategoryTheory.IsIso.inv_eq_of_hom_inv_id", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [366, 9], "def_end_pos": [366, 29]}]], "state_before": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nA B : Action V G\n\u22a2 inv (((functorCategoryMonoidalEquivalence V G).\u03bcIso A B).app PUnit.unit).hom =\n    \ud835\udfd9 (((functorCategoryMonoidalEquivalence V G).obj (A \u2297 B)).obj PUnit.unit)", "state_after": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nA B : Action V G\n\u22a2 (((functorCategoryMonoidalEquivalence V G).\u03bcIso A B).app PUnit.unit).hom \u226b\n      \ud835\udfd9 (((functorCategoryMonoidalEquivalence V G).obj (A \u2297 B)).obj PUnit.unit) =\n    \ud835\udfd9 (((functorCategoryMonoidalEquivalence V G).obj A \u2297 (functorCategoryMonoidalEquivalence V G).obj B).obj PUnit.unit)"}, {"tactic": "rw [Category.comp_id, NatIso.app_hom, MonoidalFunctor.\u03bcIso_hom,\n  functorCategoryMonoidalEquivalence.\u03bc_app]", "annotated_tactic": ["rw [<a>Category.comp_id</a>, <a>NatIso.app_hom</a>, <a>MonoidalFunctor.\u03bcIso_hom</a>,\n    <a>functorCategoryMonoidalEquivalence.\u03bc_app</a>]", [{"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [179, 3], "def_end_pos": [179, 10]}, {"full_name": "CategoryTheory.NatIso.app_hom", "def_path": "Mathlib/CategoryTheory/NatIso.lean", "def_pos": [89, 9], "def_end_pos": [89, 16]}, {"full_name": "CategoryTheory.MonoidalFunctor.\u03bcIso_hom", "def_path": "Mathlib/CategoryTheory/Monoidal/Functor.lean", "def_pos": [412, 9], "def_end_pos": [412, 17]}, {"full_name": "Action.functorCategoryMonoidalEquivalence.\u03bc_app", "def_path": "Mathlib/RepresentationTheory/Action/Monoidal.lean", "def_pos": [212, 9], "def_end_pos": [212, 49]}]], "state_before": "V : Type (u + 1)\ninst\u271d\u00b9 : LargeCategory V\nG : MonCat\ninst\u271d : MonoidalCategory V\nA B : Action V G\n\u22a2 (((functorCategoryMonoidalEquivalence V G).\u03bcIso A B).app PUnit.unit).hom \u226b\n      \ud835\udfd9 (((functorCategoryMonoidalEquivalence V G).obj (A \u2297 B)).obj PUnit.unit) =\n    \ud835\udfd9 (((functorCategoryMonoidalEquivalence V G).obj A \u2297 (functorCategoryMonoidalEquivalence V G).obj B).obj PUnit.unit)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Principal.lean", "full_name": "Ordinal.principal_opow_omega", "start": [420, 1], "end": [424, 23], "traced_tactics": [{"tactic": "simp_rw [\u2190 natCast_opow]", "annotated_tactic": ["simp_rw [\u2190 <a>natCast_opow</a>]", [{"full_name": "Ordinal.natCast_opow", "def_path": "Mathlib/SetTheory/Ordinal/Exponential.lean", "def_pos": [456, 9], "def_end_pos": [456, 21]}]], "state_before": "a b : Ordinal.{u_1}\nm n : \u2115\nha : \u2191m < \u03c9\nhb : \u2191n < \u03c9\n\u22a2 (fun x x_1 => x ^ x_1) \u2191m \u2191n < \u03c9", "state_after": "a b : Ordinal.{u_1}\nm n : \u2115\nha : \u2191m < \u03c9\nhb : \u2191n < \u03c9\n\u22a2 \u2191(m ^ n) < \u03c9"}, {"tactic": "apply nat_lt_omega", "annotated_tactic": ["apply <a>nat_lt_omega</a>", [{"full_name": "Ordinal.nat_lt_omega", "def_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "def_pos": [2463, 9], "def_end_pos": [2463, 21]}]], "state_before": "a b : Ordinal.{u_1}\nm n : \u2115\nha : \u2191m < \u03c9\nhb : \u2191n < \u03c9\n\u22a2 \u2191(m ^ n) < \u03c9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "full_name": "continuousAt_extChartAt'", "start": [1195, 1], "end": [1197, 60], "traced_tactics": [{"tactic": "rwa [\u2190 extChartAt_source I]", "annotated_tactic": ["rwa [\u2190 <a>extChartAt_source</a> I]", [{"full_name": "extChartAt_source", "def_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "def_pos": [1136, 9], "def_end_pos": [1136, 26]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2079 : NormedAddCommGroup E\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E\ninst\u271d\u2077 : TopologicalSpace H\ninst\u271d\u2076 : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners \ud835\udd5c E H\ninst\u271d\u2075 : NormedAddCommGroup E'\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\nI' : ModelWithCorners \ud835\udd5c E' H'\ns t : Set M\ninst\u271d\u00b9 : ChartedSpace H M\ninst\u271d : ChartedSpace H' M'\nx x' : M\nh : x' \u2208 (extChartAt I x).source\n\u22a2 x' \u2208 (chartAt H x).source", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matroid/Map.lean", "full_name": "Matroid.Basis.map", "start": [408, 1], "end": [418, 65], "traced_tactics": [{"tactic": "refine (hIX.indep.map f hf).basis_of_forall_insert (image_subset _ hIX.subset) ?_", "annotated_tactic": ["refine (hIX.indep.map f hf).<a>basis_of_forall_insert</a> (<a>image_subset</a> _ hIX.subset) ?_", [{"full_name": "Matroid.Indep.basis_of_forall_insert", "def_path": "Mathlib/Data/Matroid/Basic.lean", "def_pos": [893, 9], "def_end_pos": [893, 37]}, {"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [291, 9], "def_end_pos": [291, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2\nE I s : Set \u03b1\nM : Matroid \u03b1\nN : Matroid \u03b2\nX : Set \u03b1\nhIX : M.Basis I X\nf : \u03b1 \u2192 \u03b2\nhf : InjOn f M.E\n\u22a2 (M.map f hf).Basis (f '' I) (f '' X)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2\nE I s : Set \u03b1\nM : Matroid \u03b1\nN : Matroid \u03b2\nX : Set \u03b1\nhIX : M.Basis I X\nf : \u03b1 \u2192 \u03b2\nhf : InjOn f M.E\n\u22a2 \u2200 e \u2208 f '' X \\ f '' I, (M.map f hf).Dep (insert e (f '' I))"}, {"tactic": "rintro _ \u27e8\u27e8e,he,rfl\u27e9, he'\u27e9", "annotated_tactic": ["rintro _ \u27e8\u27e8e,he,rfl\u27e9, he'\u27e9", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2\nE I s : Set \u03b1\nM : Matroid \u03b1\nN : Matroid \u03b2\nX : Set \u03b1\nhIX : M.Basis I X\nf : \u03b1 \u2192 \u03b2\nhf : InjOn f M.E\n\u22a2 \u2200 e \u2208 f '' X \\ f '' I, (M.map f hf).Dep (insert e (f '' I))", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2\nE I s : Set \u03b1\nM : Matroid \u03b1\nN : Matroid \u03b2\nX : Set \u03b1\nhIX : M.Basis I X\nf : \u03b1 \u2192 \u03b2\nhf : InjOn f M.E\ne : \u03b1\nhe : e \u2208 X\nhe' : f e \u2209 f '' I\n\u22a2 (M.map f hf).Dep (insert (f e) (f '' I))"}, {"tactic": "have hss := insert_subset (hIX.subset_ground he) hIX.indep.subset_ground", "annotated_tactic": ["have hss := <a>insert_subset</a> (hIX.subset_ground he) hIX.indep.subset_ground", [{"full_name": "Set.insert_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1128, 9], "def_end_pos": [1128, 22]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2\nE I s : Set \u03b1\nM : Matroid \u03b1\nN : Matroid \u03b2\nX : Set \u03b1\nhIX : M.Basis I X\nf : \u03b1 \u2192 \u03b2\nhf : InjOn f M.E\ne : \u03b1\nhe : e \u2208 X\nhe' : f e \u2209 f '' I\n\u22a2 (M.map f hf).Dep (insert (f e) (f '' I))", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2\nE I s : Set \u03b1\nM : Matroid \u03b1\nN : Matroid \u03b2\nX : Set \u03b1\nhIX : M.Basis I X\nf : \u03b1 \u2192 \u03b2\nhf : InjOn f M.E\ne : \u03b1\nhe : e \u2208 X\nhe' : f e \u2209 f '' I\nhss : insert e I \u2286 M.E\n\u22a2 (M.map f hf).Dep (insert (f e) (f '' I))"}, {"tactic": "rw [\u2190 not_indep_iff (by simpa [\u2190 image_insert_eq] using image_subset f hss)]", "annotated_tactic": ["rw [\u2190 <a>not_indep_iff</a> (by simpa [\u2190 <a>image_insert_eq</a>] using <a>image_subset</a> f hss)]", [{"full_name": "Matroid.not_indep_iff", "def_path": "Mathlib/Data/Matroid/Basic.lean", "def_pos": [515, 17], "def_end_pos": [515, 30]}, {"full_name": "Set.image_insert_eq", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [389, 9], "def_end_pos": [389, 24]}, {"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [291, 9], "def_end_pos": [291, 21]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2\nE I s : Set \u03b1\nM : Matroid \u03b1\nN : Matroid \u03b2\nX : Set \u03b1\nhIX : M.Basis I X\nf : \u03b1 \u2192 \u03b2\nhf : InjOn f M.E\ne : \u03b1\nhe : e \u2208 X\nhe' : f e \u2209 f '' I\nhss : insert e I \u2286 M.E\n\u22a2 (M.map f hf).Dep (insert (f e) (f '' I))", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2\nE I s : Set \u03b1\nM : Matroid \u03b1\nN : Matroid \u03b2\nX : Set \u03b1\nhIX : M.Basis I X\nf : \u03b1 \u2192 \u03b2\nhf : InjOn f M.E\ne : \u03b1\nhe : e \u2208 X\nhe' : f e \u2209 f '' I\nhss : insert e I \u2286 M.E\n\u22a2 \u00ac(M.map f hf).Indep (insert (f e) (f '' I))"}, {"tactic": "simp only [map_indep_iff, not_exists, not_and]", "annotated_tactic": ["simp only [<a>map_indep_iff</a>, <a>not_exists</a>, <a>not_and</a>]", [{"full_name": "Matroid.map_indep_iff", "def_path": "Mathlib/Data/Matroid/Map.lean", "def_pos": [357, 15], "def_end_pos": [357, 28]}, {"full_name": "not_exists", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [254, 17], "def_end_pos": [254, 27]}, {"full_name": "not_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [116, 17], "def_end_pos": [116, 24]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2\nE I s : Set \u03b1\nM : Matroid \u03b1\nN : Matroid \u03b2\nX : Set \u03b1\nhIX : M.Basis I X\nf : \u03b1 \u2192 \u03b2\nhf : InjOn f M.E\ne : \u03b1\nhe : e \u2208 X\nhe' : f e \u2209 f '' I\nhss : insert e I \u2286 M.E\n\u22a2 \u00ac(M.map f hf).Indep (insert (f e) (f '' I))", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2\nE I s : Set \u03b1\nM : Matroid \u03b1\nN : Matroid \u03b2\nX : Set \u03b1\nhIX : M.Basis I X\nf : \u03b1 \u2192 \u03b2\nhf : InjOn f M.E\ne : \u03b1\nhe : e \u2208 X\nhe' : f e \u2209 f '' I\nhss : insert e I \u2286 M.E\n\u22a2 \u2200 (x : Set \u03b1), M.Indep x \u2192 \u00acinsert (f e) (f '' I) = f '' x"}, {"tactic": "intro J hJ hins", "annotated_tactic": ["intro J hJ hins", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2\nE I s : Set \u03b1\nM : Matroid \u03b1\nN : Matroid \u03b2\nX : Set \u03b1\nhIX : M.Basis I X\nf : \u03b1 \u2192 \u03b2\nhf : InjOn f M.E\ne : \u03b1\nhe : e \u2208 X\nhe' : f e \u2209 f '' I\nhss : insert e I \u2286 M.E\n\u22a2 \u2200 (x : Set \u03b1), M.Indep x \u2192 \u00acinsert (f e) (f '' I) = f '' x", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2\nE I s : Set \u03b1\nM : Matroid \u03b1\nN : Matroid \u03b2\nX : Set \u03b1\nhIX : M.Basis I X\nf : \u03b1 \u2192 \u03b2\nhf : InjOn f M.E\ne : \u03b1\nhe : e \u2208 X\nhe' : f e \u2209 f '' I\nhss : insert e I \u2286 M.E\nJ : Set \u03b1\nhJ : M.Indep J\nhins : insert (f e) (f '' I) = f '' J\n\u22a2 False"}, {"tactic": "rw [\u2190 image_insert_eq, hf.image_eq_image_iff hss hJ.subset_ground] at hins", "annotated_tactic": ["rw [\u2190 <a>image_insert_eq</a>, hf.image_eq_image_iff hss hJ.subset_ground] at hins", [{"full_name": "Set.image_insert_eq", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [389, 9], "def_end_pos": [389, 24]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2\nE I s : Set \u03b1\nM : Matroid \u03b1\nN : Matroid \u03b2\nX : Set \u03b1\nhIX : M.Basis I X\nf : \u03b1 \u2192 \u03b2\nhf : InjOn f M.E\ne : \u03b1\nhe : e \u2208 X\nhe' : f e \u2209 f '' I\nhss : insert e I \u2286 M.E\nJ : Set \u03b1\nhJ : M.Indep J\nhins : insert (f e) (f '' I) = f '' J\n\u22a2 False", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2\nE I s : Set \u03b1\nM : Matroid \u03b1\nN : Matroid \u03b2\nX : Set \u03b1\nhIX : M.Basis I X\nf : \u03b1 \u2192 \u03b2\nhf : InjOn f M.E\ne : \u03b1\nhe : e \u2208 X\nhe' : f e \u2209 f '' I\nhss : insert e I \u2286 M.E\nJ : Set \u03b1\nhJ : M.Indep J\nhins : insert e I = J\n\u22a2 False"}, {"tactic": "obtain rfl := hins", "annotated_tactic": ["obtain rfl := hins", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2\nE I s : Set \u03b1\nM : Matroid \u03b1\nN : Matroid \u03b2\nX : Set \u03b1\nhIX : M.Basis I X\nf : \u03b1 \u2192 \u03b2\nhf : InjOn f M.E\ne : \u03b1\nhe : e \u2208 X\nhe' : f e \u2209 f '' I\nhss : insert e I \u2286 M.E\nJ : Set \u03b1\nhJ : M.Indep J\nhins : insert e I = J\n\u22a2 False", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2\nE I s : Set \u03b1\nM : Matroid \u03b1\nN : Matroid \u03b2\nX : Set \u03b1\nhIX : M.Basis I X\nf : \u03b1 \u2192 \u03b2\nhf : InjOn f M.E\ne : \u03b1\nhe : e \u2208 X\nhe' : f e \u2209 f '' I\nhss : insert e I \u2286 M.E\nhJ : M.Indep (insert e I)\n\u22a2 False"}, {"tactic": "exact he' (mem_image_of_mem f (hIX.mem_of_insert_indep he hJ))", "annotated_tactic": ["exact he' (<a>mem_image_of_mem</a> f (hIX.mem_of_insert_indep he hJ))", [{"full_name": "Set.mem_image_of_mem", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [132, 9], "def_end_pos": [132, 25]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2\nE I s : Set \u03b1\nM : Matroid \u03b1\nN : Matroid \u03b2\nX : Set \u03b1\nhIX : M.Basis I X\nf : \u03b1 \u2192 \u03b2\nhf : InjOn f M.E\ne : \u03b1\nhe : e \u2208 X\nhe' : f e \u2209 f '' I\nhss : insert e I \u2286 M.E\nhJ : M.Indep (insert e I)\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simpa [\u2190 image_insert_eq] using image_subset f hss", "annotated_tactic": ["simpa [\u2190 <a>image_insert_eq</a>] using <a>image_subset</a> f hss", [{"full_name": "Set.image_insert_eq", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [389, 9], "def_end_pos": [389, 24]}, {"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [291, 9], "def_end_pos": [291, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nf\u271d : \u03b1 \u2192 \u03b2\nE I s : Set \u03b1\nM : Matroid \u03b1\nN : Matroid \u03b2\nX : Set \u03b1\nhIX : M.Basis I X\nf : \u03b1 \u2192 \u03b2\nhf : InjOn f M.E\ne : \u03b1\nhe : e \u2208 X\nhe' : f e \u2209 f '' I\nhss : insert e I \u2286 M.E\n\u22a2 insert (f e) (f '' I) \u2286 (M.map f hf).E", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Artinian.lean", "full_name": "IsArtinian.surjective_of_injective_endomorphism", "start": [293, 1], "end": [298, 69], "traced_tactics": [{"tactic": "obtain \u27e8n, hn\u27e9 := eventually_atTop.mp f.eventually_codisjoint_ker_pow_range_pow", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 := eventually_atTop.mp f.eventually_codisjoint_ker_pow_range_pow", []], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\ns : Injective \u21d1f\n\u22a2 Surjective \u21d1f", "state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\ns : Injective \u21d1f\nn : \u2115\nhn : \u2200 b \u2265 n, Codisjoint (LinearMap.ker (f ^ b)) (LinearMap.range (f ^ b))\n\u22a2 Surjective \u21d1f"}, {"tactic": "specialize hn (n + 1) (n.le_add_right 1)", "annotated_tactic": ["specialize hn (n + 1) (n.le_add_right 1)", []], "state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\ns : Injective \u21d1f\nn : \u2115\nhn : \u2200 b \u2265 n, Codisjoint (LinearMap.ker (f ^ b)) (LinearMap.range (f ^ b))\n\u22a2 Surjective \u21d1f", "state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\ns : Injective \u21d1f\nn : \u2115\nhn : Codisjoint (LinearMap.ker (f ^ (n + 1))) (LinearMap.range (f ^ (n + 1)))\n\u22a2 Surjective \u21d1f"}, {"tactic": "rw [codisjoint_iff, LinearMap.ker_eq_bot.mpr (LinearMap.iterate_injective s _), bot_sup_eq,\n  LinearMap.range_eq_top] at hn", "annotated_tactic": ["rw [<a>codisjoint_iff</a>, LinearMap.ker_eq_bot.mpr (<a>LinearMap.iterate_injective</a> s _), <a>bot_sup_eq</a>,\n    <a>LinearMap.range_eq_top</a>] at hn", [{"full_name": "codisjoint_iff", "def_path": "Mathlib/Order/Disjoint.lean", "def_pos": [329, 9], "def_end_pos": [329, 23]}, {"full_name": "LinearMap.iterate_injective", "def_path": "Mathlib/Algebra/Module/LinearMap/End.lean", "def_pos": [185, 9], "def_end_pos": [185, 26]}, {"full_name": "bot_sup_eq", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [422, 9], "def_end_pos": [422, 19]}, {"full_name": "LinearMap.range_eq_top", "def_path": "Mathlib/Algebra/Module/Submodule/Range.lean", "def_pos": [100, 9], "def_end_pos": [100, 21]}]], "state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\ns : Injective \u21d1f\nn : \u2115\nhn : Codisjoint (LinearMap.ker (f ^ (n + 1))) (LinearMap.range (f ^ (n + 1)))\n\u22a2 Surjective \u21d1f", "state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\ns : Injective \u21d1f\nn : \u2115\nhn : Surjective \u21d1(f ^ (n + 1))\n\u22a2 Surjective \u21d1f"}, {"tactic": "exact LinearMap.surjective_of_iterate_surjective n.succ_ne_zero hn", "annotated_tactic": ["exact <a>LinearMap.surjective_of_iterate_surjective</a> n.succ_ne_zero hn", [{"full_name": "LinearMap.surjective_of_iterate_surjective", "def_path": "Mathlib/Algebra/Module/LinearMap/End.lean", "def_pos": [205, 9], "def_end_pos": [205, 41]}]], "state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b3 : Ring R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\ninst\u271d : IsArtinian R M\nf : M \u2192\u2097[R] M\ns : Injective \u21d1f\nn : \u2115\nhn : Surjective \u21d1(f ^ (n + 1))\n\u22a2 Surjective \u21d1f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "full_name": "UniformEquicontinuous.closure'", "start": [897, 1], "end": [901, 58], "traced_tactics": [{"tactic": "rw [\u2190 uniformEquicontinuousOn_univ] at hA \u22a2", "annotated_tactic": ["rw [\u2190 <a>uniformEquicontinuousOn_univ</a>] at hA \u22a2", [{"full_name": "uniformEquicontinuousOn_univ", "def_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "def_pos": [203, 7], "def_end_pos": [203, 35]}]], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\nX : Type u_3\nX' : Type u_4\nY : Type u_5\nZ : Type u_6\n\u03b1 : Type u_7\n\u03b1' : Type u_8\n\u03b2 : Type u_9\n\u03b2' : Type u_10\n\u03b3 : Type u_11\n\ud835\udcd5 : Type u_12\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\ntZ : TopologicalSpace Z\nu\u03b1 : UniformSpace \u03b1\nu\u03b2 : UniformSpace \u03b2\nu\u03b3 : UniformSpace \u03b3\nA : Set Y\nu : Y \u2192 \u03b2 \u2192 \u03b1\nhA : UniformEquicontinuous (u \u2218 Subtype.val)\nhu : Continuous u\n\u22a2 UniformEquicontinuous (u \u2218 Subtype.val)", "state_after": "\u03b9 : Type u_1\n\u03ba : Type u_2\nX : Type u_3\nX' : Type u_4\nY : Type u_5\nZ : Type u_6\n\u03b1 : Type u_7\n\u03b1' : Type u_8\n\u03b2 : Type u_9\n\u03b2' : Type u_10\n\u03b3 : Type u_11\n\ud835\udcd5 : Type u_12\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\ntZ : TopologicalSpace Z\nu\u03b1 : UniformSpace \u03b1\nu\u03b2 : UniformSpace \u03b2\nu\u03b3 : UniformSpace \u03b3\nA : Set Y\nu : Y \u2192 \u03b2 \u2192 \u03b1\nhA : UniformEquicontinuousOn (u \u2218 Subtype.val) univ\nhu : Continuous u\n\u22a2 UniformEquicontinuousOn (u \u2218 Subtype.val) univ"}, {"tactic": "exact hA.closure' (Pi.continuous_restrict _ |>.comp hu)", "annotated_tactic": ["exact hA.closure' (<a>Pi.continuous_restrict</a> _ |>.<a>comp</a> hu)", [{"full_name": "Pi.continuous_restrict", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1361, 7], "def_end_pos": [1361, 29]}, {"full_name": "Continuous.comp", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1619, 9], "def_end_pos": [1619, 24]}]], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\nX : Type u_3\nX' : Type u_4\nY : Type u_5\nZ : Type u_6\n\u03b1 : Type u_7\n\u03b1' : Type u_8\n\u03b2 : Type u_9\n\u03b2' : Type u_10\n\u03b3 : Type u_11\n\ud835\udcd5 : Type u_12\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\ntZ : TopologicalSpace Z\nu\u03b1 : UniformSpace \u03b1\nu\u03b2 : UniformSpace \u03b2\nu\u03b3 : UniformSpace \u03b3\nA : Set Y\nu : Y \u2192 \u03b2 \u2192 \u03b1\nhA : UniformEquicontinuousOn (u \u2218 Subtype.val) univ\nhu : Continuous u\n\u22a2 UniformEquicontinuousOn (u \u2218 Subtype.val) univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Relation.lean", "full_name": "Relation.reflexive_reflGen", "start": [484, 1], "end": [484, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finsupp/Basic.lean", "full_name": "Finsupp.graph_inj", "start": [110, 1], "end": [111, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/ContinuedFractions/Translations.lean", "full_name": "GeneralizedContinuedFraction.part_denom_eq_s_b", "start": [62, 1], "end": [63, 88], "traced_tactics": [{"tactic": "simp [partialDenominators, s_nth_eq]", "annotated_tactic": ["simp [<a>partialDenominators</a>, s_nth_eq]", [{"full_name": "GeneralizedContinuedFraction.partialDenominators", "def_path": "Mathlib/Algebra/ContinuedFractions/Basic.lean", "def_pos": [140, 5], "def_end_pos": [140, 24]}]], "state_before": "\u03b1 : Type u_1\ng : GeneralizedContinuedFraction \u03b1\nn : \u2115\ngp : Pair \u03b1\ns_nth_eq : g.s.get? n = some gp\n\u22a2 g.partialDenominators.get? n = some gp.b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/Basic.lean", "full_name": "nhds_bot_basis_Iic", "start": [346, 1], "end": [348, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Lagrange.lean", "full_name": "Lagrange.derivative_nodal", "start": [574, 1], "end": [582, 55], "traced_tactics": [{"tactic": "refine s.induction_on ?_ fun i t hit IH => ?_", "annotated_tactic": ["refine s.induction_on ?_ fun i t hit IH => ?_", []], "state_before": "R : Type u_1\ninst\u271d\u00b9 : CommRing R\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 R\ninst\u271d : DecidableEq \u03b9\n\u22a2 derivative (nodal s v) = \u2211 i \u2208 s, nodal (s.erase i) v", "state_after": "case refine_1\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 R\ninst\u271d : DecidableEq \u03b9\n\u22a2 derivative (nodal \u2205 v) = \u2211 i \u2208 \u2205, nodal (\u2205.erase i) v\n\ncase refine_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 R\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nt : Finset \u03b9\nhit : i \u2209 t\nIH : derivative (nodal t v) = \u2211 i \u2208 t, nodal (t.erase i) v\n\u22a2 derivative (nodal (insert i t) v) = \u2211 i_1 \u2208 insert i t, nodal ((insert i t).erase i_1) v"}, {"tactic": "rw [nodal_empty, derivative_one, sum_empty]", "annotated_tactic": ["rw [<a>nodal_empty</a>, <a>derivative_one</a>, <a>sum_empty</a>]", [{"full_name": "Lagrange.nodal_empty", "def_path": "Mathlib/LinearAlgebra/Lagrange.lean", "def_pos": [520, 9], "def_end_pos": [520, 20]}, {"full_name": "Polynomial.derivative_one", "def_path": "Mathlib/Algebra/Polynomial/Derivative.lean", "def_pos": [136, 9], "def_end_pos": [136, 23]}, {"full_name": "Finset.sum_empty", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [329, 3], "def_end_pos": [329, 14]}]], "state_before": "case refine_1\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 R\ninst\u271d : DecidableEq \u03b9\n\u22a2 derivative (nodal \u2205 v) = \u2211 i \u2208 \u2205, nodal (\u2205.erase i) v", "state_after": "no goals"}, {"tactic": "rw [nodal_insert_eq_nodal hit, derivative_mul, IH, derivative_sub, derivative_X, derivative_C,\n  sub_zero, one_mul, sum_insert hit, mul_sum, erase_insert hit, add_right_inj]", "annotated_tactic": ["rw [<a>nodal_insert_eq_nodal</a> hit, <a>derivative_mul</a>, IH, <a>derivative_sub</a>, <a>derivative_X</a>, <a>derivative_C</a>,\n      <a>sub_zero</a>, <a>one_mul</a>, <a>sum_insert</a> hit, <a>mul_sum</a>, <a>erase_insert</a> hit, <a>add_right_inj</a>]", [{"full_name": "Lagrange.nodal_insert_eq_nodal", "def_path": "Mathlib/LinearAlgebra/Lagrange.lean", "def_pos": [569, 9], "def_end_pos": [569, 30]}, {"full_name": "Polynomial.derivative_mul", "def_path": "Mathlib/Algebra/Polynomial/Derivative.lean", "def_pos": [288, 9], "def_end_pos": [288, 23]}, {"full_name": "Polynomial.derivative_sub", "def_path": "Mathlib/Algebra/Polynomial/Derivative.lean", "def_pos": [587, 9], "def_end_pos": [587, 23]}, {"full_name": "Polynomial.derivative_X", "def_path": "Mathlib/Algebra/Polynomial/Derivative.lean", "def_pos": [130, 9], "def_end_pos": [130, 21]}, {"full_name": "Polynomial.derivative_C", "def_path": "Mathlib/Algebra/Polynomial/Derivative.lean", "def_pos": [121, 9], "def_end_pos": [121, 21]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [489, 3], "def_end_pos": [489, 14]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "Finset.sum_insert", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [350, 3], "def_end_pos": [350, 14]}, {"full_name": "Finset.mul_sum", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [61, 7], "def_end_pos": [61, 14]}, {"full_name": "Finset.erase_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1958, 9], "def_end_pos": [1958, 21]}, {"full_name": "add_right_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [67, 3], "def_end_pos": [67, 14]}]], "state_before": "case refine_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 R\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nt : Finset \u03b9\nhit : i \u2209 t\nIH : derivative (nodal t v) = \u2211 i \u2208 t, nodal (t.erase i) v\n\u22a2 derivative (nodal (insert i t) v) = \u2211 i_1 \u2208 insert i t, nodal ((insert i t).erase i_1) v", "state_after": "case refine_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 R\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nt : Finset \u03b9\nhit : i \u2209 t\nIH : derivative (nodal t v) = \u2211 i \u2208 t, nodal (t.erase i) v\n\u22a2 \u2211 i_1 \u2208 t, (X - C (v i)) * nodal (t.erase i_1) v = \u2211 x \u2208 t, nodal ((insert i t).erase x) v"}, {"tactic": "refine sum_congr rfl fun j hjt => ?_", "annotated_tactic": ["refine <a>sum_congr</a> <a>rfl</a> fun j hjt => ?_", [{"full_name": "Finset.sum_congr", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [420, 3], "def_end_pos": [420, 14]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case refine_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 R\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nt : Finset \u03b9\nhit : i \u2209 t\nIH : derivative (nodal t v) = \u2211 i \u2208 t, nodal (t.erase i) v\n\u22a2 \u2211 i_1 \u2208 t, (X - C (v i)) * nodal (t.erase i_1) v = \u2211 x \u2208 t, nodal ((insert i t).erase x) v", "state_after": "case refine_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 R\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nt : Finset \u03b9\nhit : i \u2209 t\nIH : derivative (nodal t v) = \u2211 i \u2208 t, nodal (t.erase i) v\nj : \u03b9\nhjt : j \u2208 t\n\u22a2 (X - C (v i)) * nodal (t.erase j) v = nodal ((insert i t).erase j) v"}, {"tactic": "rw [t.erase_insert_of_ne (ne_of_mem_of_not_mem hjt hit).symm,\n  nodal_insert_eq_nodal (mem_of_mem_erase.mt hit)]", "annotated_tactic": ["rw [t.erase_insert_of_ne (<a>ne_of_mem_of_not_mem</a> hjt hit).<a>symm</a>,\n      <a>nodal_insert_eq_nodal</a> (mem_of_mem_erase.mt hit)]", [{"full_name": "ne_of_mem_of_not_mem", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 29]}, {"full_name": "Ne.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [704, 9], "def_end_pos": [704, 16]}, {"full_name": "Lagrange.nodal_insert_eq_nodal", "def_path": "Mathlib/LinearAlgebra/Lagrange.lean", "def_pos": [569, 9], "def_end_pos": [569, 30]}]], "state_before": "case refine_2\nR : Type u_1\ninst\u271d\u00b9 : CommRing R\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 R\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nt : Finset \u03b9\nhit : i \u2209 t\nIH : derivative (nodal t v) = \u2211 i \u2208 t, nodal (t.erase i) v\nj : \u03b9\nhjt : j \u2208 t\n\u22a2 (X - C (v i)) * nodal (t.erase j) v = nodal ((insert i t).erase j) v", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/EuclideanDomain/Basic.lean", "full_name": "EuclideanDomain.gcd_one_left", "start": [183, 1], "end": [184, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "full_name": "mul_le_mul'", "start": [206, 1], "end": [209, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Between.lean", "full_name": "Sbtw.trans_left", "start": [488, 1], "end": [490, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ENNReal/Real.lean", "full_name": "ENNReal.toNNReal_prod", "start": [413, 1], "end": [415, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/FractionalIdeal/Basic.lean", "full_name": "FractionalIdeal.coe_pow", "start": [621, 1], "end": [622, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "full_name": "hasFTaylorSeriesUpToOn_univ_iff", "start": [1270, 1], "end": [1290, 24], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 HasFTaylorSeriesUpToOn n f p univ \u2194 HasFTaylorSeriesUpTo n f p", "state_after": "case mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 HasFTaylorSeriesUpToOn n f p univ \u2192 HasFTaylorSeriesUpTo n f p\n\ncase mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 HasFTaylorSeriesUpTo n f p \u2192 HasFTaylorSeriesUpToOn n f p univ"}, {"tactic": "intro H", "annotated_tactic": ["intro H", []], "state_before": "case mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 HasFTaylorSeriesUpToOn n f p univ \u2192 HasFTaylorSeriesUpTo n f p", "state_after": "case mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 HasFTaylorSeriesUpTo n f p"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case mp\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 HasFTaylorSeriesUpTo n f p", "state_after": "case mp.zero_eq\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 \u2200 (x : E), (p x 0).uncurry0 = f x\n\ncase mp.fderiv\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 \u2200 (m : \u2115), \u2191m < n \u2192 \u2200 (x : E), HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x\n\ncase mp.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 n \u2192 Continuous fun x => p x m"}, {"tactic": "exact fun x => H.zero_eq x (mem_univ x)", "annotated_tactic": ["exact fun x => H.zero_eq x (<a>mem_univ</a> x)", [{"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [82, 29], "def_end_pos": [82, 37]}]], "state_before": "case mp.zero_eq\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 \u2200 (x : E), (p x 0).uncurry0 = f x", "state_after": "no goals"}, {"tactic": "intro m hm x", "annotated_tactic": ["intro m hm x", []], "state_before": "case mp.fderiv\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 \u2200 (m : \u2115), \u2191m < n \u2192 \u2200 (x : E), HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x", "state_after": "case mp.fderiv\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\nm : \u2115\nhm : \u2191m < n\nx : E\n\u22a2 HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x"}, {"tactic": "rw [\u2190 hasFDerivWithinAt_univ]", "annotated_tactic": ["rw [\u2190 <a>hasFDerivWithinAt_univ</a>]", [{"full_name": "hasFDerivWithinAt_univ", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [403, 9], "def_end_pos": [403, 31]}]], "state_before": "case mp.fderiv\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\nm : \u2115\nhm : \u2191m < n\nx : E\n\u22a2 HasFDerivAt (fun y => p y m) (p x m.succ).curryLeft x", "state_after": "case mp.fderiv\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\nm : \u2115\nhm : \u2191m < n\nx : E\n\u22a2 HasFDerivWithinAt (fun y => p y m) (p x m.succ).curryLeft univ x"}, {"tactic": "exact H.fderivWithin m hm x (mem_univ x)", "annotated_tactic": ["exact H.fderivWithin m hm x (<a>mem_univ</a> x)", [{"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [82, 29], "def_end_pos": [82, 37]}]], "state_before": "case mp.fderiv\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\nm : \u2115\nhm : \u2191m < n\nx : E\n\u22a2 HasFDerivWithinAt (fun y => p y m) (p x m.succ).curryLeft univ x", "state_after": "no goals"}, {"tactic": "intro m hm", "annotated_tactic": ["intro m hm", []], "state_before": "case mp.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 n \u2192 Continuous fun x => p x m", "state_after": "case mp.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 Continuous fun x => p x m"}, {"tactic": "rw [continuous_iff_continuousOn_univ]", "annotated_tactic": ["rw [<a>continuous_iff_continuousOn_univ</a>]", [{"full_name": "continuous_iff_continuousOn_univ", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [543, 9], "def_end_pos": [543, 41]}]], "state_before": "case mp.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 Continuous fun x => p x m", "state_after": "case mp.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 ContinuousOn (fun x => p x m) univ"}, {"tactic": "exact H.cont m hm", "annotated_tactic": ["exact H.cont m hm", []], "state_before": "case mp.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpToOn n f p univ\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 ContinuousOn (fun x => p x m) univ", "state_after": "no goals"}, {"tactic": "intro H", "annotated_tactic": ["intro H", []], "state_before": "case mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\u22a2 HasFTaylorSeriesUpTo n f p \u2192 HasFTaylorSeriesUpToOn n f p univ", "state_after": "case mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\n\u22a2 HasFTaylorSeriesUpToOn n f p univ"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case mpr\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\n\u22a2 HasFTaylorSeriesUpToOn n f p univ", "state_after": "case mpr.zero_eq\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\n\u22a2 \u2200 x \u2208 univ, (p x 0).uncurry0 = f x\n\ncase mpr.fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\n\u22a2 \u2200 (m : \u2115), \u2191m < n \u2192 \u2200 x \u2208 univ, HasFDerivWithinAt (fun x => p x m) (p x m.succ).curryLeft univ x\n\ncase mpr.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => p x m) univ"}, {"tactic": "exact fun x _ => H.zero_eq x", "annotated_tactic": ["exact fun x _ => H.zero_eq x", []], "state_before": "case mpr.zero_eq\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\n\u22a2 \u2200 x \u2208 univ, (p x 0).uncurry0 = f x", "state_after": "no goals"}, {"tactic": "intro m hm x _", "annotated_tactic": ["intro m hm x _", []], "state_before": "case mpr.fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\n\u22a2 \u2200 (m : \u2115), \u2191m < n \u2192 \u2200 x \u2208 univ, HasFDerivWithinAt (fun x => p x m) (p x m.succ).curryLeft univ x", "state_after": "case mpr.fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\nm : \u2115\nhm : \u2191m < n\nx : E\na\u271d : x \u2208 univ\n\u22a2 HasFDerivWithinAt (fun x => p x m) (p x m.succ).curryLeft univ x"}, {"tactic": "rw [hasFDerivWithinAt_univ]", "annotated_tactic": ["rw [<a>hasFDerivWithinAt_univ</a>]", [{"full_name": "hasFDerivWithinAt_univ", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [403, 9], "def_end_pos": [403, 31]}]], "state_before": "case mpr.fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\nm : \u2115\nhm : \u2191m < n\nx : E\na\u271d : x \u2208 univ\n\u22a2 HasFDerivWithinAt (fun x => p x m) (p x m.succ).curryLeft univ x", "state_after": "case mpr.fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\nm : \u2115\nhm : \u2191m < n\nx : E\na\u271d : x \u2208 univ\n\u22a2 HasFDerivAt (fun x => p x m) (p x m.succ).curryLeft x"}, {"tactic": "exact H.fderiv m hm x", "annotated_tactic": ["exact H.fderiv m hm x", []], "state_before": "case mpr.fderivWithin\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx\u271d x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\nm : \u2115\nhm : \u2191m < n\nx : E\na\u271d : x \u2208 univ\n\u22a2 HasFDerivAt (fun x => p x m) (p x m.succ).curryLeft x", "state_after": "no goals"}, {"tactic": "intro m hm", "annotated_tactic": ["intro m hm", []], "state_before": "case mpr.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\n\u22a2 \u2200 (m : \u2115), \u2191m \u2264 n \u2192 ContinuousOn (fun x => p x m) univ", "state_after": "case mpr.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 ContinuousOn (fun x => p x m) univ"}, {"tactic": "rw [\u2190 continuous_iff_continuousOn_univ]", "annotated_tactic": ["rw [\u2190 <a>continuous_iff_continuousOn_univ</a>]", [{"full_name": "continuous_iff_continuousOn_univ", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [543, 9], "def_end_pos": [543, 41]}]], "state_before": "case mpr.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 ContinuousOn (fun x => p x m) univ", "state_after": "case mpr.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 Continuous fun x => p x m"}, {"tactic": "exact H.cont m hm", "annotated_tactic": ["exact H.cont m hm", []], "state_before": "case mpr.cont\n\ud835\udd5c : Type u\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type uX\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nm\u271d n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nH : HasFTaylorSeriesUpTo n f p\nm : \u2115\nhm : \u2191m \u2264 n\n\u22a2 Continuous fun x => p x m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Sigma/Basic.lean", "full_name": "sigma_mk_injective", "start": [110, 1], "end": [111, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Ideal/Basic.lean", "full_name": "Ideal.mem_inf", "start": [434, 1], "end": [435, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/InsertNth.lean", "full_name": "List.getElem_insertNth_of_lt", "start": [138, 1], "end": [148, 32], "traced_tactics": [{"tactic": "induction' n with n IH generalizing k l", "annotated_tactic": ["induction' n with n IH generalizing k l", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\na : \u03b1\nl : List \u03b1\nx : \u03b1\nn k : \u2115\nhn : k < n\nhk : k < l.length\nhk' : optParam (k < (insertNth n x l).length) \u22ef\n\u22a2 (insertNth n x l)[k] = l[k]", "state_after": "case zero\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\na x : \u03b1\nl : List \u03b1\nk : \u2115\nhn : k < 0\nhk : k < l.length\nhk' : optParam (k < (insertNth 0 x l).length) \u22ef\n\u22a2 (insertNth 0 x l)[k] = l[k]\n\ncase succ\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\na x : \u03b1\nn : \u2115\nIH :\n  \u2200 (l : List \u03b1) (k : \u2115),\n    k < n \u2192 \u2200 (hk : k < l.length) (hk' : optParam (k < (insertNth n x l).length) \u22ef), (insertNth n x l)[k] = l[k]\nl : List \u03b1\nk : \u2115\nhn : k < n + 1\nhk : k < l.length\nhk' : optParam (k < (insertNth (n + 1) x l).length) \u22ef\n\u22a2 (insertNth (n + 1) x l)[k] = l[k]"}, {"tactic": "simp at hn", "annotated_tactic": ["simp at hn", []], "state_before": "case zero\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\na x : \u03b1\nl : List \u03b1\nk : \u2115\nhn : k < 0\nhk : k < l.length\nhk' : optParam (k < (insertNth 0 x l).length) \u22ef\n\u22a2 (insertNth 0 x l)[k] = l[k]", "state_after": "no goals"}, {"tactic": "cases' l with hd tl", "annotated_tactic": ["cases' l with hd tl", []], "state_before": "case succ\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\na x : \u03b1\nn : \u2115\nIH :\n  \u2200 (l : List \u03b1) (k : \u2115),\n    k < n \u2192 \u2200 (hk : k < l.length) (hk' : optParam (k < (insertNth n x l).length) \u22ef), (insertNth n x l)[k] = l[k]\nl : List \u03b1\nk : \u2115\nhn : k < n + 1\nhk : k < l.length\nhk' : optParam (k < (insertNth (n + 1) x l).length) \u22ef\n\u22a2 (insertNth (n + 1) x l)[k] = l[k]", "state_after": "case succ.nil\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\na x : \u03b1\nn : \u2115\nIH :\n  \u2200 (l : List \u03b1) (k : \u2115),\n    k < n \u2192 \u2200 (hk : k < l.length) (hk' : optParam (k < (insertNth n x l).length) \u22ef), (insertNth n x l)[k] = l[k]\nk : \u2115\nhn : k < n + 1\nhk : k < [].length\nhk' : optParam (k < (insertNth (n + 1) x []).length) \u22ef\n\u22a2 (insertNth (n + 1) x [])[k] = [][k]\n\ncase succ.cons\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\na x : \u03b1\nn : \u2115\nIH :\n  \u2200 (l : List \u03b1) (k : \u2115),\n    k < n \u2192 \u2200 (hk : k < l.length) (hk' : optParam (k < (insertNth n x l).length) \u22ef), (insertNth n x l)[k] = l[k]\nk : \u2115\nhn : k < n + 1\nhd : \u03b1\ntl : List \u03b1\nhk : k < (hd :: tl).length\nhk' : optParam (k < (insertNth (n + 1) x (hd :: tl)).length) \u22ef\n\u22a2 (insertNth (n + 1) x (hd :: tl))[k] = (hd :: tl)[k]"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case succ.nil\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\na x : \u03b1\nn : \u2115\nIH :\n  \u2200 (l : List \u03b1) (k : \u2115),\n    k < n \u2192 \u2200 (hk : k < l.length) (hk' : optParam (k < (insertNth n x l).length) \u22ef), (insertNth n x l)[k] = l[k]\nk : \u2115\nhn : k < n + 1\nhk : k < [].length\nhk' : optParam (k < (insertNth (n + 1) x []).length) \u22ef\n\u22a2 (insertNth (n + 1) x [])[k] = [][k]", "state_after": "no goals"}, {"tactic": "cases k", "annotated_tactic": ["cases k", []], "state_before": "case succ.cons\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\na x : \u03b1\nn : \u2115\nIH :\n  \u2200 (l : List \u03b1) (k : \u2115),\n    k < n \u2192 \u2200 (hk : k < l.length) (hk' : optParam (k < (insertNth n x l).length) \u22ef), (insertNth n x l)[k] = l[k]\nk : \u2115\nhn : k < n + 1\nhd : \u03b1\ntl : List \u03b1\nhk : k < (hd :: tl).length\nhk' : optParam (k < (insertNth (n + 1) x (hd :: tl)).length) \u22ef\n\u22a2 (insertNth (n + 1) x (hd :: tl))[k] = (hd :: tl)[k]", "state_after": "case succ.cons.zero\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\na x : \u03b1\nn : \u2115\nIH :\n  \u2200 (l : List \u03b1) (k : \u2115),\n    k < n \u2192 \u2200 (hk : k < l.length) (hk' : optParam (k < (insertNth n x l).length) \u22ef), (insertNth n x l)[k] = l[k]\nhd : \u03b1\ntl : List \u03b1\nhn : 0 < n + 1\nhk : 0 < (hd :: tl).length\nhk' : optParam (0 < (insertNth (n + 1) x (hd :: tl)).length) \u22ef\n\u22a2 (insertNth (n + 1) x (hd :: tl))[0] = (hd :: tl)[0]\n\ncase succ.cons.succ\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\na x : \u03b1\nn : \u2115\nIH :\n  \u2200 (l : List \u03b1) (k : \u2115),\n    k < n \u2192 \u2200 (hk : k < l.length) (hk' : optParam (k < (insertNth n x l).length) \u22ef), (insertNth n x l)[k] = l[k]\nhd : \u03b1\ntl : List \u03b1\nn\u271d : \u2115\nhn : n\u271d + 1 < n + 1\nhk : n\u271d + 1 < (hd :: tl).length\nhk' : optParam (n\u271d + 1 < (insertNth (n + 1) x (hd :: tl)).length) \u22ef\n\u22a2 (insertNth (n + 1) x (hd :: tl))[n\u271d + 1] = (hd :: tl)[n\u271d + 1]"}, {"tactic": "simp [get]", "annotated_tactic": ["simp [<a>get</a>]", [{"full_name": "List.get", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2337, 5], "def_end_pos": [2337, 13]}]], "state_before": "case succ.cons.zero\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\na x : \u03b1\nn : \u2115\nIH :\n  \u2200 (l : List \u03b1) (k : \u2115),\n    k < n \u2192 \u2200 (hk : k < l.length) (hk' : optParam (k < (insertNth n x l).length) \u22ef), (insertNth n x l)[k] = l[k]\nhd : \u03b1\ntl : List \u03b1\nhn : 0 < n + 1\nhk : 0 < (hd :: tl).length\nhk' : optParam (0 < (insertNth (n + 1) x (hd :: tl)).length) \u22ef\n\u22a2 (insertNth (n + 1) x (hd :: tl))[0] = (hd :: tl)[0]", "state_after": "no goals"}, {"tactic": "rw [Nat.succ_lt_succ_iff] at hn", "annotated_tactic": ["rw [<a>Nat.succ_lt_succ_iff</a>] at hn", [{"full_name": "Nat.succ_lt_succ_iff", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [637, 9], "def_end_pos": [637, 25]}]], "state_before": "case succ.cons.succ\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\na x : \u03b1\nn : \u2115\nIH :\n  \u2200 (l : List \u03b1) (k : \u2115),\n    k < n \u2192 \u2200 (hk : k < l.length) (hk' : optParam (k < (insertNth n x l).length) \u22ef), (insertNth n x l)[k] = l[k]\nhd : \u03b1\ntl : List \u03b1\nn\u271d : \u2115\nhn : n\u271d + 1 < n + 1\nhk : n\u271d + 1 < (hd :: tl).length\nhk' : optParam (n\u271d + 1 < (insertNth (n + 1) x (hd :: tl)).length) \u22ef\n\u22a2 (insertNth (n + 1) x (hd :: tl))[n\u271d + 1] = (hd :: tl)[n\u271d + 1]", "state_after": "case succ.cons.succ\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\na x : \u03b1\nn : \u2115\nIH :\n  \u2200 (l : List \u03b1) (k : \u2115),\n    k < n \u2192 \u2200 (hk : k < l.length) (hk' : optParam (k < (insertNth n x l).length) \u22ef), (insertNth n x l)[k] = l[k]\nhd : \u03b1\ntl : List \u03b1\nn\u271d : \u2115\nhn : n\u271d < n\nhk : n\u271d + 1 < (hd :: tl).length\nhk' : optParam (n\u271d + 1 < (insertNth (n + 1) x (hd :: tl)).length) \u22ef\n\u22a2 (insertNth (n + 1) x (hd :: tl))[n\u271d + 1] = (hd :: tl)[n\u271d + 1]"}, {"tactic": "simpa using IH _ _ hn _", "annotated_tactic": ["simpa using IH _ _ hn _", []], "state_before": "case succ.cons.succ\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\na x : \u03b1\nn : \u2115\nIH :\n  \u2200 (l : List \u03b1) (k : \u2115),\n    k < n \u2192 \u2200 (hk : k < l.length) (hk' : optParam (k < (insertNth n x l).length) \u22ef), (insertNth n x l)[k] = l[k]\nhd : \u03b1\ntl : List \u03b1\nn\u271d : \u2115\nhn : n\u271d < n\nhk : n\u271d + 1 < (hd :: tl).length\nhk' : optParam (n\u271d + 1 < (insertNth (n + 1) x (hd :: tl)).length) \u22ef\n\u22a2 (insertNth (n + 1) x (hd :: tl))[n\u271d + 1] = (hd :: tl)[n\u271d + 1]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/Quiver/Symmetric.lean", "full_name": "Quiver.Path.reverse_comp", "start": [150, 1], "end": [154, 13], "traced_tactics": [{"tactic": "induction' q with _ _ _ _ h", "annotated_tactic": ["induction' q with _ _ _ _ h", []], "state_before": "U : Type u_1\nV : Type u_2\nW : Type u_3\ninst\u271d\u00b3 : Quiver U\ninst\u271d\u00b2 : Quiver V\ninst\u271d\u00b9 : Quiver W\ninst\u271d : HasReverse V\na b c : V\np : Path a b\nq : Path b c\n\u22a2 (p.comp q).reverse = q.reverse.comp p.reverse", "state_after": "case nil\nU : Type u_1\nV : Type u_2\nW : Type u_3\ninst\u271d\u00b3 : Quiver U\ninst\u271d\u00b2 : Quiver V\ninst\u271d\u00b9 : Quiver W\ninst\u271d : HasReverse V\na b c : V\np : Path a b\n\u22a2 (p.comp nil).reverse = nil.reverse.comp p.reverse\n\ncase cons\nU : Type u_1\nV : Type u_2\nW : Type u_3\ninst\u271d\u00b3 : Quiver U\ninst\u271d\u00b2 : Quiver V\ninst\u271d\u00b9 : Quiver W\ninst\u271d : HasReverse V\na b c : V\np : Path a b\nb\u271d c\u271d : V\na\u271d\u00b9 : Path b b\u271d\na\u271d : b\u271d \u27f6 c\u271d\nh : (p.comp a\u271d\u00b9).reverse = a\u271d\u00b9.reverse.comp p.reverse\n\u22a2 (p.comp (a\u271d\u00b9.cons a\u271d)).reverse = (a\u271d\u00b9.cons a\u271d).reverse.comp p.reverse"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case nil\nU : Type u_1\nV : Type u_2\nW : Type u_3\ninst\u271d\u00b3 : Quiver U\ninst\u271d\u00b2 : Quiver V\ninst\u271d\u00b9 : Quiver W\ninst\u271d : HasReverse V\na b c : V\np : Path a b\n\u22a2 (p.comp nil).reverse = nil.reverse.comp p.reverse", "state_after": "no goals"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "case cons\nU : Type u_1\nV : Type u_2\nW : Type u_3\ninst\u271d\u00b3 : Quiver U\ninst\u271d\u00b2 : Quiver V\ninst\u271d\u00b9 : Quiver W\ninst\u271d : HasReverse V\na b c : V\np : Path a b\nb\u271d c\u271d : V\na\u271d\u00b9 : Path b b\u271d\na\u271d : b\u271d \u27f6 c\u271d\nh : (p.comp a\u271d\u00b9).reverse = a\u271d\u00b9.reverse.comp p.reverse\n\u22a2 (p.comp (a\u271d\u00b9.cons a\u271d)).reverse = (a\u271d\u00b9.cons a\u271d).reverse.comp p.reverse", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Subalgebra.lean", "full_name": "LieSubalgebra.add_mem", "start": [130, 11], "end": [131, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Nodup.lean", "full_name": "List.nodup_iff_injective_getElem", "start": [91, 1], "end": [100, 97], "traced_tactics": [{"tactic": "cases' i with i hi", "annotated_tactic": ["cases' i with i hi", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nl\u271d l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nl : List \u03b1\nh : \u2200 (i j : \u2115) (_hi : i < l.length) (_hj : j < l.length), i < j \u2192 l[i] \u2260 l[j]\ni j : Fin l.length\nhg : (fun i => l[\u2191i]) i = (fun i => l[\u2191i]) j\n\u22a2 i = j", "state_after": "case mk\n\u03b1 : Type u\n\u03b2 : Type v\nl\u271d l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nl : List \u03b1\nh : \u2200 (i j : \u2115) (_hi : i < l.length) (_hj : j < l.length), i < j \u2192 l[i] \u2260 l[j]\nj : Fin l.length\ni : \u2115\nhi : i < l.length\nhg : (fun i => l[\u2191i]) \u27e8i, hi\u27e9 = (fun i => l[\u2191i]) j\n\u22a2 \u27e8i, hi\u27e9 = j"}, {"tactic": "cases' j with j hj", "annotated_tactic": ["cases' j with j hj", []], "state_before": "case mk\n\u03b1 : Type u\n\u03b2 : Type v\nl\u271d l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nl : List \u03b1\nh : \u2200 (i j : \u2115) (_hi : i < l.length) (_hj : j < l.length), i < j \u2192 l[i] \u2260 l[j]\nj : Fin l.length\ni : \u2115\nhi : i < l.length\nhg : (fun i => l[\u2191i]) \u27e8i, hi\u27e9 = (fun i => l[\u2191i]) j\n\u22a2 \u27e8i, hi\u27e9 = j", "state_after": "case mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\nl\u271d l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nl : List \u03b1\nh : \u2200 (i j : \u2115) (_hi : i < l.length) (_hj : j < l.length), i < j \u2192 l[i] \u2260 l[j]\ni : \u2115\nhi : i < l.length\nj : \u2115\nhj : j < l.length\nhg : (fun i => l[\u2191i]) \u27e8i, hi\u27e9 = (fun i => l[\u2191i]) \u27e8j, hj\u27e9\n\u22a2 \u27e8i, hi\u27e9 = \u27e8j, hj\u27e9"}, {"tactic": "rcases lt_trichotomy i j with (hij | rfl | hji)", "annotated_tactic": ["rcases <a>lt_trichotomy</a> i j with (hij | rfl | hji)", [{"full_name": "lt_trichotomy", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [313, 9], "def_end_pos": [313, 22]}]], "state_before": "case mk.mk\n\u03b1 : Type u\n\u03b2 : Type v\nl\u271d l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nl : List \u03b1\nh : \u2200 (i j : \u2115) (_hi : i < l.length) (_hj : j < l.length), i < j \u2192 l[i] \u2260 l[j]\ni : \u2115\nhi : i < l.length\nj : \u2115\nhj : j < l.length\nhg : (fun i => l[\u2191i]) \u27e8i, hi\u27e9 = (fun i => l[\u2191i]) \u27e8j, hj\u27e9\n\u22a2 \u27e8i, hi\u27e9 = \u27e8j, hj\u27e9", "state_after": "case mk.mk.inl\n\u03b1 : Type u\n\u03b2 : Type v\nl\u271d l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nl : List \u03b1\nh : \u2200 (i j : \u2115) (_hi : i < l.length) (_hj : j < l.length), i < j \u2192 l[i] \u2260 l[j]\ni : \u2115\nhi : i < l.length\nj : \u2115\nhj : j < l.length\nhg : (fun i => l[\u2191i]) \u27e8i, hi\u27e9 = (fun i => l[\u2191i]) \u27e8j, hj\u27e9\nhij : i < j\n\u22a2 \u27e8i, hi\u27e9 = \u27e8j, hj\u27e9\n\ncase mk.mk.inr.inl\n\u03b1 : Type u\n\u03b2 : Type v\nl\u271d l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nl : List \u03b1\nh : \u2200 (i j : \u2115) (_hi : i < l.length) (_hj : j < l.length), i < j \u2192 l[i] \u2260 l[j]\ni : \u2115\nhi hj : i < l.length\nhg : (fun i => l[\u2191i]) \u27e8i, hi\u27e9 = (fun i => l[\u2191i]) \u27e8i, hj\u27e9\n\u22a2 \u27e8i, hi\u27e9 = \u27e8i, hj\u27e9\n\ncase mk.mk.inr.inr\n\u03b1 : Type u\n\u03b2 : Type v\nl\u271d l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nl : List \u03b1\nh : \u2200 (i j : \u2115) (_hi : i < l.length) (_hj : j < l.length), i < j \u2192 l[i] \u2260 l[j]\ni : \u2115\nhi : i < l.length\nj : \u2115\nhj : j < l.length\nhg : (fun i => l[\u2191i]) \u27e8i, hi\u27e9 = (fun i => l[\u2191i]) \u27e8j, hj\u27e9\nhji : j < i\n\u22a2 \u27e8i, hi\u27e9 = \u27e8j, hj\u27e9"}, {"tactic": "exact (h i j hi hj hij hg).elim", "annotated_tactic": ["exact (h i j hi hj hij hg).<a>elim</a>", [{"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "case mk.mk.inl\n\u03b1 : Type u\n\u03b2 : Type v\nl\u271d l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nl : List \u03b1\nh : \u2200 (i j : \u2115) (_hi : i < l.length) (_hj : j < l.length), i < j \u2192 l[i] \u2260 l[j]\ni : \u2115\nhi : i < l.length\nj : \u2115\nhj : j < l.length\nhg : (fun i => l[\u2191i]) \u27e8i, hi\u27e9 = (fun i => l[\u2191i]) \u27e8j, hj\u27e9\nhij : i < j\n\u22a2 \u27e8i, hi\u27e9 = \u27e8j, hj\u27e9", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case mk.mk.inr.inl\n\u03b1 : Type u\n\u03b2 : Type v\nl\u271d l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nl : List \u03b1\nh : \u2200 (i j : \u2115) (_hi : i < l.length) (_hj : j < l.length), i < j \u2192 l[i] \u2260 l[j]\ni : \u2115\nhi hj : i < l.length\nhg : (fun i => l[\u2191i]) \u27e8i, hi\u27e9 = (fun i => l[\u2191i]) \u27e8i, hj\u27e9\n\u22a2 \u27e8i, hi\u27e9 = \u27e8i, hj\u27e9", "state_after": "no goals"}, {"tactic": "exact (h j i hj hi hji hg.symm).elim", "annotated_tactic": ["exact (h j i hj hi hji hg.symm).<a>elim</a>", [{"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "case mk.mk.inr.inr\n\u03b1 : Type u\n\u03b2 : Type v\nl\u271d l\u2081 l\u2082 : List \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na b : \u03b1\nl : List \u03b1\nh : \u2200 (i j : \u2115) (_hi : i < l.length) (_hj : j < l.length), i < j \u2192 l[i] \u2260 l[j]\ni : \u2115\nhi : i < l.length\nj : \u2115\nhj : j < l.length\nhg : (fun i => l[\u2191i]) \u27e8i, hi\u27e9 = (fun i => l[\u2191i]) \u27e8j, hj\u27e9\nhji : j < i\n\u22a2 \u27e8i, hi\u27e9 = \u27e8j, hj\u27e9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/ModelTheory/Semantics.lean", "full_name": "FirstOrder.Language.BoundedFormula.realize_toPrenex", "start": [554, 1], "end": [565, 35], "traced_tactics": [{"tactic": "induction' \u03c6 with _ _ _ _ _ _ _ _ _ f1 f2 h1 h2 _ _ h", "annotated_tactic": ["induction' \u03c6 with _ _ _ _ _ _ _ _ _ f1 f2 h1 h2 _ _ h", []], "state_before": "L : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\ninst\u271d\u00b9 : L.Structure P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6\u271d \u03c8 : L.BoundedFormula \u03b1 l\n\u03b8 : L.BoundedFormula \u03b1 l.succ\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\n\u03c6 : L.BoundedFormula \u03b1 n\nv : \u03b1 \u2192 M\n\u22a2 \u2200 {xs : Fin n \u2192 M}, \u03c6.toPrenex.Realize v xs \u2194 \u03c6.Realize v xs", "state_after": "case falsum\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\ninst\u271d\u00b9 : L.Structure P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6 \u03c8 : L.BoundedFormula \u03b1 l\n\u03b8 : L.BoundedFormula \u03b1 l.succ\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d : \u2115\n\u22a2 \u2200 {xs : Fin n\u271d \u2192 M}, falsum.toPrenex.Realize v xs \u2194 falsum.Realize v xs\n\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\ninst\u271d\u00b9 : L.Structure P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6 \u03c8 : L.BoundedFormula \u03b1 l\n\u03b8 : L.BoundedFormula \u03b1 l.succ\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d : \u2115\nt\u2081\u271d t\u2082\u271d : L.Term (\u03b1 \u2295 Fin n\u271d)\n\u22a2 \u2200 {xs : Fin n\u271d \u2192 M}, (equal t\u2081\u271d t\u2082\u271d).toPrenex.Realize v xs \u2194 (equal t\u2081\u271d t\u2082\u271d).Realize v xs\n\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\ninst\u271d\u00b9 : L.Structure P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6 \u03c8 : L.BoundedFormula \u03b1 l\n\u03b8 : L.BoundedFormula \u03b1 l.succ\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d l\u271d : \u2115\nR\u271d : L.Relations l\u271d\nts\u271d : Fin l\u271d \u2192 L.Term (\u03b1 \u2295 Fin n\u271d)\n\u22a2 \u2200 {xs : Fin n\u271d \u2192 M}, (rel R\u271d ts\u271d).toPrenex.Realize v xs \u2194 (rel R\u271d ts\u271d).Realize v xs\n\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\ninst\u271d\u00b9 : L.Structure P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6 \u03c8 : L.BoundedFormula \u03b1 l\n\u03b8 : L.BoundedFormula \u03b1 l.succ\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d : \u2115\nf1 f2 : L.BoundedFormula \u03b1 n\u271d\nh1 : \u2200 {xs : Fin n\u271d \u2192 M}, f1.toPrenex.Realize v xs \u2194 f1.Realize v xs\nh2 : \u2200 {xs : Fin n\u271d \u2192 M}, f2.toPrenex.Realize v xs \u2194 f2.Realize v xs\n\u22a2 \u2200 {xs : Fin n\u271d \u2192 M}, (f1 \u27f9 f2).toPrenex.Realize v xs \u2194 (f1 \u27f9 f2).Realize v xs\n\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\ninst\u271d\u00b9 : L.Structure P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6 \u03c8 : L.BoundedFormula \u03b1 l\n\u03b8 : L.BoundedFormula \u03b1 l.succ\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d : \u2115\nf\u271d : L.BoundedFormula \u03b1 (n\u271d + 1)\nh : \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, f\u271d.toPrenex.Realize v xs \u2194 f\u271d.Realize v xs\n\u22a2 \u2200 {xs : Fin n\u271d \u2192 M}, (\u2200'f\u271d).toPrenex.Realize v xs \u2194 (\u2200'f\u271d).Realize v xs"}, {"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": [796, 19], "def_end_pos": [796, 26]}]], "state_before": "case falsum\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\ninst\u271d\u00b9 : L.Structure P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6 \u03c8 : L.BoundedFormula \u03b1 l\n\u03b8 : L.BoundedFormula \u03b1 l.succ\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d : \u2115\n\u22a2 \u2200 {xs : Fin n\u271d \u2192 M}, falsum.toPrenex.Realize v xs \u2194 falsum.Realize v xs", "state_after": "no goals"}, {"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": [796, 19], "def_end_pos": [796, 26]}]], "state_before": "case equal\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\ninst\u271d\u00b9 : L.Structure P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6 \u03c8 : L.BoundedFormula \u03b1 l\n\u03b8 : L.BoundedFormula \u03b1 l.succ\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d : \u2115\nt\u2081\u271d t\u2082\u271d : L.Term (\u03b1 \u2295 Fin n\u271d)\n\u22a2 \u2200 {xs : Fin n\u271d \u2192 M}, (equal t\u2081\u271d t\u2082\u271d).toPrenex.Realize v xs \u2194 (equal t\u2081\u271d t\u2082\u271d).Realize v xs", "state_after": "no goals"}, {"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": [796, 19], "def_end_pos": [796, 26]}]], "state_before": "case rel\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\ninst\u271d\u00b9 : L.Structure P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6 \u03c8 : L.BoundedFormula \u03b1 l\n\u03b8 : L.BoundedFormula \u03b1 l.succ\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d l\u271d : \u2115\nR\u271d : L.Relations l\u271d\nts\u271d : Fin l\u271d \u2192 L.Term (\u03b1 \u2295 Fin n\u271d)\n\u22a2 \u2200 {xs : Fin n\u271d \u2192 M}, (rel R\u271d ts\u271d).toPrenex.Realize v xs \u2194 (rel R\u271d ts\u271d).Realize v xs", "state_after": "no goals"}, {"tactic": "intros", "annotated_tactic": ["intros", []], "state_before": "case imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\ninst\u271d\u00b9 : L.Structure P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6 \u03c8 : L.BoundedFormula \u03b1 l\n\u03b8 : L.BoundedFormula \u03b1 l.succ\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d : \u2115\nf1 f2 : L.BoundedFormula \u03b1 n\u271d\nh1 : \u2200 {xs : Fin n\u271d \u2192 M}, f1.toPrenex.Realize v xs \u2194 f1.Realize v xs\nh2 : \u2200 {xs : Fin n\u271d \u2192 M}, f2.toPrenex.Realize v xs \u2194 f2.Realize v xs\n\u22a2 \u2200 {xs : Fin n\u271d \u2192 M}, (f1 \u27f9 f2).toPrenex.Realize v xs \u2194 (f1 \u27f9 f2).Realize v xs", "state_after": "case imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\ninst\u271d\u00b9 : L.Structure P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6 \u03c8 : L.BoundedFormula \u03b1 l\n\u03b8 : L.BoundedFormula \u03b1 l.succ\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d : \u2115\nf1 f2 : L.BoundedFormula \u03b1 n\u271d\nh1 : \u2200 {xs : Fin n\u271d \u2192 M}, f1.toPrenex.Realize v xs \u2194 f1.Realize v xs\nh2 : \u2200 {xs : Fin n\u271d \u2192 M}, f2.toPrenex.Realize v xs \u2194 f2.Realize v xs\nxs\u271d : Fin n\u271d \u2192 M\n\u22a2 (f1 \u27f9 f2).toPrenex.Realize v xs\u271d \u2194 (f1 \u27f9 f2).Realize v xs\u271d"}, {"tactic": "rw [toPrenex, realize_toPrenexImp f1.toPrenex_isPrenex f2.toPrenex_isPrenex, realize_imp,\n  realize_imp, h1, h2]", "annotated_tactic": ["rw [<a>toPrenex</a>, <a>realize_toPrenexImp</a> f1.toPrenex_isPrenex f2.toPrenex_isPrenex, <a>realize_imp</a>,\n      <a>realize_imp</a>, h1, h2]", [{"full_name": "FirstOrder.Language.BoundedFormula.toPrenex", "def_path": "Mathlib/ModelTheory/Syntax.lean", "def_pos": [854, 5], "def_end_pos": [854, 13]}, {"full_name": "FirstOrder.Language.BoundedFormula.realize_toPrenexImp", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [531, 9], "def_end_pos": [531, 28]}, {"full_name": "FirstOrder.Language.BoundedFormula.realize_imp", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [294, 9], "def_end_pos": [294, 20]}, {"full_name": "FirstOrder.Language.BoundedFormula.realize_imp", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [294, 9], "def_end_pos": [294, 20]}]], "state_before": "case imp\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\ninst\u271d\u00b9 : L.Structure P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6 \u03c8 : L.BoundedFormula \u03b1 l\n\u03b8 : L.BoundedFormula \u03b1 l.succ\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d : \u2115\nf1 f2 : L.BoundedFormula \u03b1 n\u271d\nh1 : \u2200 {xs : Fin n\u271d \u2192 M}, f1.toPrenex.Realize v xs \u2194 f1.Realize v xs\nh2 : \u2200 {xs : Fin n\u271d \u2192 M}, f2.toPrenex.Realize v xs \u2194 f2.Realize v xs\nxs\u271d : Fin n\u271d \u2192 M\n\u22a2 (f1 \u27f9 f2).toPrenex.Realize v xs\u271d \u2194 (f1 \u27f9 f2).Realize v xs\u271d", "state_after": "no goals"}, {"tactic": "intros", "annotated_tactic": ["intros", []], "state_before": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\ninst\u271d\u00b9 : L.Structure P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6 \u03c8 : L.BoundedFormula \u03b1 l\n\u03b8 : L.BoundedFormula \u03b1 l.succ\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d : \u2115\nf\u271d : L.BoundedFormula \u03b1 (n\u271d + 1)\nh : \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, f\u271d.toPrenex.Realize v xs \u2194 f\u271d.Realize v xs\n\u22a2 \u2200 {xs : Fin n\u271d \u2192 M}, (\u2200'f\u271d).toPrenex.Realize v xs \u2194 (\u2200'f\u271d).Realize v xs", "state_after": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\ninst\u271d\u00b9 : L.Structure P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6 \u03c8 : L.BoundedFormula \u03b1 l\n\u03b8 : L.BoundedFormula \u03b1 l.succ\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d : \u2115\nf\u271d : L.BoundedFormula \u03b1 (n\u271d + 1)\nh : \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, f\u271d.toPrenex.Realize v xs \u2194 f\u271d.Realize v xs\nxs\u271d : Fin n\u271d \u2192 M\n\u22a2 (\u2200'f\u271d).toPrenex.Realize v xs\u271d \u2194 (\u2200'f\u271d).Realize v xs\u271d"}, {"tactic": "rw [realize_all, toPrenex, realize_all]", "annotated_tactic": ["rw [<a>realize_all</a>, <a>toPrenex</a>, <a>realize_all</a>]", [{"full_name": "FirstOrder.Language.BoundedFormula.realize_all", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [338, 9], "def_end_pos": [338, 20]}, {"full_name": "FirstOrder.Language.BoundedFormula.toPrenex", "def_path": "Mathlib/ModelTheory/Syntax.lean", "def_pos": [854, 5], "def_end_pos": [854, 13]}, {"full_name": "FirstOrder.Language.BoundedFormula.realize_all", "def_path": "Mathlib/ModelTheory/Semantics.lean", "def_pos": [338, 9], "def_end_pos": [338, 20]}]], "state_before": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\ninst\u271d\u00b9 : L.Structure P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6 \u03c8 : L.BoundedFormula \u03b1 l\n\u03b8 : L.BoundedFormula \u03b1 l.succ\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d : \u2115\nf\u271d : L.BoundedFormula \u03b1 (n\u271d + 1)\nh : \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, f\u271d.toPrenex.Realize v xs \u2194 f\u271d.Realize v xs\nxs\u271d : Fin n\u271d \u2192 M\n\u22a2 (\u2200'f\u271d).toPrenex.Realize v xs\u271d \u2194 (\u2200'f\u271d).Realize v xs\u271d", "state_after": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\ninst\u271d\u00b9 : L.Structure P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6 \u03c8 : L.BoundedFormula \u03b1 l\n\u03b8 : L.BoundedFormula \u03b1 l.succ\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d : \u2115\nf\u271d : L.BoundedFormula \u03b1 (n\u271d + 1)\nh : \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, f\u271d.toPrenex.Realize v xs \u2194 f\u271d.Realize v xs\nxs\u271d : Fin n\u271d \u2192 M\n\u22a2 (\u2200 (a : M), f\u271d.toPrenex.Realize v (snoc xs\u271d a)) \u2194 \u2200 (a : M), f\u271d.Realize v (snoc xs\u271d a)"}, {"tactic": "exact forall_congr' fun a => h", "annotated_tactic": ["exact <a>forall_congr'</a> fun a => h", [{"full_name": "forall_congr'", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [207, 9], "def_end_pos": [207, 22]}]], "state_before": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : L.Structure M\ninst\u271d\u00b2 : L.Structure N\ninst\u271d\u00b9 : L.Structure P\n\u03b1 : Type u'\n\u03b2 : Type v'\n\u03b3 : Type u_3\nn l : \u2115\n\u03c6 \u03c8 : L.BoundedFormula \u03b1 l\n\u03b8 : L.BoundedFormula \u03b1 l.succ\nv\u271d : \u03b1 \u2192 M\nxs : Fin l \u2192 M\ninst\u271d : Nonempty M\nv : \u03b1 \u2192 M\nn\u271d : \u2115\nf\u271d : L.BoundedFormula \u03b1 (n\u271d + 1)\nh : \u2200 {xs : Fin (n\u271d + 1) \u2192 M}, f\u271d.toPrenex.Realize v xs \u2194 f\u271d.Realize v xs\nxs\u271d : Fin n\u271d \u2192 M\n\u22a2 (\u2200 (a : M), f\u271d.toPrenex.Realize v (snoc xs\u271d a)) \u2194 \u2200 (a : M), f\u271d.Realize v (snoc xs\u271d a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/Hindman.lean", "full_name": "Hindman.FP.finset_prod", "start": [281, 1], "end": [296, 28], "traced_tactics": [{"tactic": "refine FP_drop_subset_FP _ (s.min' hs) ?_", "annotated_tactic": ["refine <a>FP_drop_subset_FP</a> _ (s.min' hs) ?_", [{"full_name": "Hindman.FP_drop_subset_FP", "def_path": "Mathlib/Combinatorics/Hindman.lean", "def_pos": [241, 9], "def_end_pos": [241, 26]}]], "state_before": "M : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nhs : s.Nonempty\n\u22a2 \u220f i \u2208 s, a.get i \u2208 FP a", "state_after": "M : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nhs : s.Nonempty\n\u22a2 \u220f i \u2208 s, a.get i \u2208 FP (Stream'.drop (s.min' hs) a)"}, {"tactic": "induction' s using Finset.strongInduction with s ih", "annotated_tactic": ["induction' s using <a>Finset.strongInduction</a> with s ih", [{"full_name": "Finset.strongInduction", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [828, 5], "def_end_pos": [828, 20]}]], "state_before": "M : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nhs : s.Nonempty\n\u22a2 \u220f i \u2208 s, a.get i \u2208 FP (Stream'.drop (s.min' hs) a)", "state_after": "case H\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\n\u22a2 \u220f i \u2208 s, a.get i \u2208 FP (Stream'.drop (s.min' hs) a)"}, {"tactic": "rw [\u2190 Finset.mul_prod_erase _ _ (s.min'_mem hs), \u2190 Stream'.head_drop]", "annotated_tactic": ["rw [\u2190 <a>Finset.mul_prod_erase</a> _ _ (s.min'_mem hs), \u2190 <a>Stream'.head_drop</a>]", [{"full_name": "Finset.mul_prod_erase", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [1961, 9], "def_end_pos": [1961, 23]}, {"full_name": "Stream'.head_drop", "def_path": "Mathlib/Data/Stream/Init.lean", "def_pos": [94, 9], "def_end_pos": [94, 18]}]], "state_before": "case H\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\n\u22a2 \u220f i \u2208 s, a.get i \u2208 FP (Stream'.drop (s.min' hs) a)", "state_after": "case H\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\n\u22a2 (Stream'.drop (s.min' hs) a).head * \u220f x \u2208 s.erase (s.min' hs), a.get x \u2208 FP (Stream'.drop (s.min' hs) a)"}, {"tactic": "rcases (s.erase (s.min' hs)).eq_empty_or_nonempty with h | h", "annotated_tactic": ["rcases (s.erase (s.min' hs)).<a>eq_empty_or_nonempty</a> with h | h", [{"full_name": "Finset.eq_empty_or_nonempty", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [615, 9], "def_end_pos": [615, 29]}]], "state_before": "case H\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\n\u22a2 (Stream'.drop (s.min' hs) a).head * \u220f x \u2208 s.erase (s.min' hs), a.get x \u2208 FP (Stream'.drop (s.min' hs) a)", "state_after": "case H.inl\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\nh : s.erase (s.min' hs) = \u2205\n\u22a2 (Stream'.drop (s.min' hs) a).head * \u220f x \u2208 s.erase (s.min' hs), a.get x \u2208 FP (Stream'.drop (s.min' hs) a)\n\ncase H.inr\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\nh : (s.erase (s.min' hs)).Nonempty\n\u22a2 (Stream'.drop (s.min' hs) a).head * \u220f x \u2208 s.erase (s.min' hs), a.get x \u2208 FP (Stream'.drop (s.min' hs) a)"}, {"tactic": "rw [h, Finset.prod_empty, mul_one]", "annotated_tactic": ["rw [h, <a>Finset.prod_empty</a>, <a>mul_one</a>]", [{"full_name": "Finset.prod_empty", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [330, 9], "def_end_pos": [330, 19]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "case H.inl\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\nh : s.erase (s.min' hs) = \u2205\n\u22a2 (Stream'.drop (s.min' hs) a).head * \u220f x \u2208 s.erase (s.min' hs), a.get x \u2208 FP (Stream'.drop (s.min' hs) a)", "state_after": "case H.inl\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\nh : s.erase (s.min' hs) = \u2205\n\u22a2 (Stream'.drop (s.min' hs) a).head \u2208 FP (Stream'.drop (s.min' hs) a)"}, {"tactic": "exact FP.head _", "annotated_tactic": ["exact <a>FP.head</a> _", [{"full_name": "Hindman.FP.head", "def_path": "Mathlib/Combinatorics/Hindman.lean", "def_pos": [108, 5], "def_end_pos": [108, 9]}]], "state_before": "case H.inl\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\nh : s.erase (s.min' hs) = \u2205\n\u22a2 (Stream'.drop (s.min' hs) a).head \u2208 FP (Stream'.drop (s.min' hs) a)", "state_after": "no goals"}, {"tactic": "apply FP.cons", "annotated_tactic": ["apply <a>FP.cons</a>", [{"full_name": "Hindman.FP.cons", "def_path": "Mathlib/Combinatorics/Hindman.lean", "def_pos": [110, 5], "def_end_pos": [110, 9]}]], "state_before": "case H.inr\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\nh : (s.erase (s.min' hs)).Nonempty\n\u22a2 (Stream'.drop (s.min' hs) a).head * \u220f x \u2208 s.erase (s.min' hs), a.get x \u2208 FP (Stream'.drop (s.min' hs) a)", "state_after": "case H.inr.h\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\nh : (s.erase (s.min' hs)).Nonempty\n\u22a2 FP (Stream'.drop (s.min' hs) a).tail (\u220f x \u2208 s.erase (s.min' hs), a.get x)"}, {"tactic": "rw [Stream'.tail_eq_drop, Stream'.drop_drop, add_comm]", "annotated_tactic": ["rw [<a>Stream'.tail_eq_drop</a>, <a>Stream'.drop_drop</a>, <a>add_comm</a>]", [{"full_name": "Stream'.tail_eq_drop", "def_path": "Mathlib/Data/Stream/Init.lean", "def_pos": [60, 9], "def_end_pos": [60, 21]}, {"full_name": "Stream'.drop_drop", "def_path": "Mathlib/Data/Stream/Init.lean", "def_pos": [65, 9], "def_end_pos": [65, 18]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "case H.inr.h\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\nh : (s.erase (s.min' hs)).Nonempty\n\u22a2 FP (Stream'.drop (s.min' hs) a).tail (\u220f x \u2208 s.erase (s.min' hs), a.get x)", "state_after": "case H.inr.h\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\nh : (s.erase (s.min' hs)).Nonempty\n\u22a2 FP (Stream'.drop (s.min' hs + 1) a) (\u220f x \u2208 s.erase (s.min' hs), a.get x)"}, {"tactic": "refine Set.mem_of_subset_of_mem ?_ (ih _ (Finset.erase_ssubset <| s.min'_mem hs) h)", "annotated_tactic": ["refine <a>Set.mem_of_subset_of_mem</a> ?_ (ih _ (<a>Finset.erase_ssubset</a> <| s.min'_mem hs) h)", [{"full_name": "Set.mem_of_subset_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [360, 9], "def_end_pos": [360, 29]}, {"full_name": "Finset.erase_ssubset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2007, 9], "def_end_pos": [2007, 22]}]], "state_before": "case H.inr.h\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\nh : (s.erase (s.min' hs)).Nonempty\n\u22a2 FP (Stream'.drop (s.min' hs + 1) a) (\u220f x \u2208 s.erase (s.min' hs), a.get x)", "state_after": "case H.inr.h\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\nh : (s.erase (s.min' hs)).Nonempty\n\u22a2 FP (Stream'.drop ((s.erase (s.min' hs)).min' h) a) \u2286 FP (Stream'.drop (s.min' hs + 1) a)"}, {"tactic": "have : s.min' hs + 1 \u2264 (s.erase (s.min' hs)).min' h :=\n  Nat.succ_le_of_lt (Finset.min'_lt_of_mem_erase_min' _ _ <| Finset.min'_mem _ _)", "annotated_tactic": ["have : s.min' hs + 1 \u2264 (s.erase (s.min' hs)).<a>min'</a> h :=\n      <a>Nat.succ_le_of_lt</a> (<a>Finset.min'_lt_of_mem_erase_min'</a> _ _ <| <a>Finset.min'_mem</a> _ _)", [{"full_name": "Finset.min'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1545, 5], "def_end_pos": [1545, 9]}, {"full_name": "Nat.succ_le_of_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [348, 9], "def_end_pos": [348, 22]}, {"full_name": "Finset.min'_lt_of_mem_erase_min'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1723, 9], "def_end_pos": [1723, 34]}, {"full_name": "Finset.min'_mem", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1558, 9], "def_end_pos": [1558, 17]}]], "state_before": "case H.inr.h\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\nh : (s.erase (s.min' hs)).Nonempty\n\u22a2 FP (Stream'.drop ((s.erase (s.min' hs)).min' h) a) \u2286 FP (Stream'.drop (s.min' hs + 1) a)", "state_after": "case H.inr.h\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\nh : (s.erase (s.min' hs)).Nonempty\nthis : s.min' hs + 1 \u2264 (s.erase (s.min' hs)).min' h\n\u22a2 FP (Stream'.drop ((s.erase (s.min' hs)).min' h) a) \u2286 FP (Stream'.drop (s.min' hs + 1) a)"}, {"tactic": "cases' le_iff_exists_add.mp this with d hd", "annotated_tactic": ["cases' le_iff_exists_add.mp this with d hd", []], "state_before": "case H.inr.h\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\nh : (s.erase (s.min' hs)).Nonempty\nthis : s.min' hs + 1 \u2264 (s.erase (s.min' hs)).min' h\n\u22a2 FP (Stream'.drop ((s.erase (s.min' hs)).min' h) a) \u2286 FP (Stream'.drop (s.min' hs + 1) a)", "state_after": "case H.inr.h.intro\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\nh : (s.erase (s.min' hs)).Nonempty\nthis : s.min' hs + 1 \u2264 (s.erase (s.min' hs)).min' h\nd : \u2115\nhd : (s.erase (s.min' hs)).min' h = s.min' hs + 1 + d\n\u22a2 FP (Stream'.drop ((s.erase (s.min' hs)).min' h) a) \u2286 FP (Stream'.drop (s.min' hs + 1) a)"}, {"tactic": "rw [hd, add_comm, \u2190 Stream'.drop_drop]", "annotated_tactic": ["rw [hd, <a>add_comm</a>, \u2190 <a>Stream'.drop_drop</a>]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "Stream'.drop_drop", "def_path": "Mathlib/Data/Stream/Init.lean", "def_pos": [65, 9], "def_end_pos": [65, 18]}]], "state_before": "case H.inr.h.intro\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\nh : (s.erase (s.min' hs)).Nonempty\nthis : s.min' hs + 1 \u2264 (s.erase (s.min' hs)).min' h\nd : \u2115\nhd : (s.erase (s.min' hs)).min' h = s.min' hs + 1 + d\n\u22a2 FP (Stream'.drop ((s.erase (s.min' hs)).min' h) a) \u2286 FP (Stream'.drop (s.min' hs + 1) a)", "state_after": "case H.inr.h.intro\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\nh : (s.erase (s.min' hs)).Nonempty\nthis : s.min' hs + 1 \u2264 (s.erase (s.min' hs)).min' h\nd : \u2115\nhd : (s.erase (s.min' hs)).min' h = s.min' hs + 1 + d\n\u22a2 FP (Stream'.drop d (Stream'.drop (s.min' hs + 1) a)) \u2286 FP (Stream'.drop (s.min' hs + 1) a)"}, {"tactic": "apply FP_drop_subset_FP", "annotated_tactic": ["apply <a>FP_drop_subset_FP</a>", [{"full_name": "Hindman.FP_drop_subset_FP", "def_path": "Mathlib/Combinatorics/Hindman.lean", "def_pos": [241, 9], "def_end_pos": [241, 26]}]], "state_before": "case H.inr.h.intro\nM : Type u_1\ninst\u271d : CommMonoid M\na : Stream' M\ns : Finset \u2115\nih : \u2200 t \u2282 s, \u2200 (hs : t.Nonempty), \u220f i \u2208 t, a.get i \u2208 FP (Stream'.drop (t.min' hs) a)\nhs : s.Nonempty\nh : (s.erase (s.min' hs)).Nonempty\nthis : s.min' hs + 1 \u2264 (s.erase (s.min' hs)).min' h\nd : \u2115\nhd : (s.erase (s.min' hs)).min' h = s.min' hs + 1 + d\n\u22a2 FP (Stream'.drop d (Stream'.drop (s.min' hs + 1) a)) \u2286 FP (Stream'.drop (s.min' hs + 1) a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Compactness/Lindelof.lean", "full_name": "isLindelof_iff_countable_subfamily_closed", "start": [296, 1], "end": [300, 90], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Sites/Sheafification.lean", "full_name": "CategoryTheory.sheafificationAdjunction_counit_app_val", "start": [156, 1], "end": [160, 7], "traced_tactics": [{"tactic": "unfold sheafifyLift", "annotated_tactic": ["unfold <a>sheafifyLift</a>", [{"full_name": "CategoryTheory.sheafifyLift", "def_path": "Mathlib/CategoryTheory/Sites/Sheafification.lean", "def_pos": [151, 19], "def_end_pos": [151, 31]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} A\nD : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} D\ninst\u271d : HasWeakSheafify J D\nP : Sheaf J D\n\u22a2 ((sheafificationAdjunction J D).counit.app P).val = sheafifyLift J (\ud835\udfd9 P.val) \u22ef", "state_after": "C : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} A\nD : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} D\ninst\u271d : HasWeakSheafify J D\nP : Sheaf J D\n\u22a2 ((sheafificationAdjunction J D).counit.app P).val =\n    (((sheafificationAdjunction J D).homEquiv P.val { val := P.val, cond := \u22ef }).symm (\ud835\udfd9 P.val)).val"}, {"tactic": "rw [Adjunction.homEquiv_counit]", "annotated_tactic": ["rw [<a>Adjunction.homEquiv_counit</a>]", [{"full_name": "CategoryTheory.Adjunction.homEquiv_counit", "def_path": "Mathlib/CategoryTheory/Adjunction/Basic.lean", "def_pos": [73, 3], "def_end_pos": [73, 18]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} A\nD : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} D\ninst\u271d : HasWeakSheafify J D\nP : Sheaf J D\n\u22a2 ((sheafificationAdjunction J D).counit.app P).val =\n    (((sheafificationAdjunction J D).homEquiv P.val { val := P.val, cond := \u22ef }).symm (\ud835\udfd9 P.val)).val", "state_after": "C : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} A\nD : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} D\ninst\u271d : HasWeakSheafify J D\nP : Sheaf J D\n\u22a2 ((sheafificationAdjunction J D).counit.app P).val =\n    ((presheafToSheaf J D).map (\ud835\udfd9 P.val) \u226b (sheafificationAdjunction J D).counit.app { val := P.val, cond := \u22ef }).val"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b3 : Category.{v\u2081, u\u2081} C\nJ : GrothendieckTopology C\nA : Type u\u2082\ninst\u271d\u00b2 : Category.{v\u2082, u\u2082} A\nD : Type u_1\ninst\u271d\u00b9 : Category.{u_2, u_1} D\ninst\u271d : HasWeakSheafify J D\nP : Sheaf J D\n\u22a2 ((sheafificationAdjunction J D).counit.app P).val =\n    ((presheafToSheaf J D).map (\ud835\udfd9 P.val) \u226b (sheafificationAdjunction J D).counit.app { val := P.val, cond := \u22ef }).val", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monoidal/Opposite.lean", "full_name": "CategoryTheory.unop_hom_leftUnitor", "start": [235, 1], "end": [235, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean", "full_name": "Complex.arg_coe_angle_toReal_eq_arg", "start": [558, 1], "end": [560, 22], "traced_tactics": [{"tactic": "rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc]", "annotated_tactic": ["rw [<a>Real.Angle.toReal_coe_eq_self_iff_mem_Ioc</a>]", [{"full_name": "Real.Angle.toReal_coe_eq_self_iff_mem_Ioc", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [536, 9], "def_end_pos": [536, 39]}]], "state_before": "a x z\u271d z : \u2102\n\u22a2 (\u2191z.arg).toReal = z.arg", "state_after": "a x z\u271d z : \u2102\n\u22a2 z.arg \u2208 Ioc (-\u03c0) \u03c0"}, {"tactic": "exact arg_mem_Ioc _", "annotated_tactic": ["exact <a>arg_mem_Ioc</a> _", [{"full_name": "Complex.arg_mem_Ioc", "def_path": "Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean", "def_pos": [143, 9], "def_end_pos": [143, 20]}]], "state_before": "a x z\u271d z : \u2102\n\u22a2 z.arg \u2208 Ioc (-\u03c0) \u03c0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/IndicatorFunction.lean", "full_name": "mulIndicator_eventuallyLE_mulIndicator", "start": [55, 1], "end": [57, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "continuousAt_const", "start": [1670, 1], "end": [1671, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup.lean", "full_name": "Matrix.SpecialLinearGroup.toGLPos_injective", "start": [281, 1], "end": [287, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matrix/DMatrix.lean", "full_name": "DMatrix.zero_apply", "start": [125, 1], "end": [125, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Polynomial/RationalRoot.lean", "full_name": "exists_integer_of_is_root_of_monic", "start": [124, 1], "end": [136, 44], "traced_tactics": [{"tactic": "obtain \u27e8inv, h_inv\u27e9 := hp \u25b8 den_dvd_of_is_root hr", "annotated_tactic": ["obtain \u27e8inv, h_inv\u27e9 := hp \u25b8 <a>den_dvd_of_is_root</a> hr", [{"full_name": "den_dvd_of_is_root", "def_path": "Mathlib/RingTheory/Polynomial/RationalRoot.lean", "def_pos": [92, 9], "def_end_pos": [92, 27]}]], "state_before": "A : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nhp : p.Monic\nr : K\nhr : (aeval r) p = 0\n\u22a2 \u2203 r', r = (algebraMap A K) r' \u2227 r' \u2223 p.coeff 0", "state_after": "case intro\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nhp : p.Monic\nr : K\nhr : (aeval r) p = 0\ninv : A\nh_inv : 1 = \u2191(den A r) * inv\n\u22a2 \u2203 r', r = (algebraMap A K) r' \u2227 r' \u2223 p.coeff 0"}, {"tactic": "use num A r * inv, ?_", "annotated_tactic": ["use <a>num</a> A r * inv, ?_", [{"full_name": "IsFractionRing.num", "def_path": "Mathlib/RingTheory/Localization/NumDen.lean", "def_pos": [51, 19], "def_end_pos": [51, 22]}]], "state_before": "case intro\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nhp : p.Monic\nr : K\nhr : (aeval r) p = 0\ninv : A\nh_inv : 1 = \u2191(den A r) * inv\n\u22a2 \u2203 r', r = (algebraMap A K) r' \u2227 r' \u2223 p.coeff 0", "state_after": "case right\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nhp : p.Monic\nr : K\nhr : (aeval r) p = 0\ninv : A\nh_inv : 1 = \u2191(den A r) * inv\n\u22a2 num A r * inv \u2223 p.coeff 0\n\ncase left\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nhp : p.Monic\nr : K\nhr : (aeval r) p = 0\ninv : A\nh_inv : 1 = \u2191(den A r) * inv\n\u22a2 r = (algebraMap A K) (num A r * inv)"}, {"tactic": "have h : inv \u2223 1 := \u27e8den A r, by simpa [mul_comm] using h_inv\u27e9", "annotated_tactic": ["have h : inv \u2223 1 := \u27e8<a>den</a> A r, by simpa [<a>mul_comm</a>] using h_inv\u27e9", [{"full_name": "IsFractionRing.den", "def_path": "Mathlib/RingTheory/Localization/NumDen.lean", "def_pos": [56, 19], "def_end_pos": [56, 22]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "case right\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nhp : p.Monic\nr : K\nhr : (aeval r) p = 0\ninv : A\nh_inv : 1 = \u2191(den A r) * inv\n\u22a2 num A r * inv \u2223 p.coeff 0", "state_after": "case right\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nhp : p.Monic\nr : K\nhr : (aeval r) p = 0\ninv : A\nh_inv : 1 = \u2191(den A r) * inv\nh : inv \u2223 1\n\u22a2 num A r * inv \u2223 p.coeff 0"}, {"tactic": "simpa using mul_dvd_mul (num_dvd_of_is_root hr) h", "annotated_tactic": ["simpa using <a>mul_dvd_mul</a> (<a>num_dvd_of_is_root</a> hr) h", [{"full_name": "mul_dvd_mul", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [217, 9], "def_end_pos": [217, 20]}, {"full_name": "num_dvd_of_is_root", "def_path": "Mathlib/RingTheory/Polynomial/RationalRoot.lean", "def_pos": [69, 9], "def_end_pos": [69, 27]}]], "state_before": "case right\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nhp : p.Monic\nr : K\nhr : (aeval r) p = 0\ninv : A\nh_inv : 1 = \u2191(den A r) * inv\nh : inv \u2223 1\n\u22a2 num A r * inv \u2223 p.coeff 0", "state_after": "no goals"}, {"tactic": "simpa [mul_comm] using h_inv", "annotated_tactic": ["simpa [<a>mul_comm</a>] using h_inv", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "A : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nhp : p.Monic\nr : K\nhr : (aeval r) p = 0\ninv : A\nh_inv : 1 = \u2191(den A r) * inv\n\u22a2 1 = inv * \u2191(den A r)", "state_after": "no goals"}, {"tactic": "have d_ne_zero : algebraMap A K (den A r) \u2260 0 :=\n  IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors (den A r).prop", "annotated_tactic": ["have d_ne_zero : <a>algebraMap</a> A K (<a>den</a> A r) \u2260 0 :=\n      <a>IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors</a> (<a>den</a> A r).<a>prop</a>", [{"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "IsFractionRing.den", "def_path": "Mathlib/RingTheory/Localization/NumDen.lean", "def_pos": [56, 19], "def_end_pos": [56, 22]}, {"full_name": "IsFractionRing.to_map_ne_zero_of_mem_nonZeroDivisors", "def_path": "Mathlib/RingTheory/Localization/FractionRing.lean", "def_pos": [97, 19], "def_end_pos": [97, 56]}, {"full_name": "IsFractionRing.den", "def_path": "Mathlib/RingTheory/Localization/NumDen.lean", "def_pos": [56, 19], "def_end_pos": [56, 22]}, {"full_name": "Subtype.prop", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [47, 9], "def_end_pos": [47, 13]}]], "state_before": "case left\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nhp : p.Monic\nr : K\nhr : (aeval r) p = 0\ninv : A\nh_inv : 1 = \u2191(den A r) * inv\n\u22a2 r = (algebraMap A K) (num A r * inv)", "state_after": "case left\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nhp : p.Monic\nr : K\nhr : (aeval r) p = 0\ninv : A\nh_inv : 1 = \u2191(den A r) * inv\nd_ne_zero : (algebraMap A K) \u2191(den A r) \u2260 0\n\u22a2 r = (algebraMap A K) (num A r * inv)"}, {"tactic": "nth_rw 1 [\u2190 mk'_num_den' A r]", "annotated_tactic": ["nth_rw 1 [\u2190 <a>mk'_num_den'</a> A r]", [{"full_name": "IsFractionRing.mk'_num_den'", "def_path": "Mathlib/RingTheory/Localization/NumDen.lean", "def_pos": [70, 9], "def_end_pos": [70, 21]}]], "state_before": "case left\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nhp : p.Monic\nr : K\nhr : (aeval r) p = 0\ninv : A\nh_inv : 1 = \u2191(den A r) * inv\nd_ne_zero : (algebraMap A K) \u2191(den A r) \u2260 0\n\u22a2 r = (algebraMap A K) (num A r * inv)", "state_after": "case left\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nhp : p.Monic\nr : K\nhr : (aeval r) p = 0\ninv : A\nh_inv : 1 = \u2191(den A r) * inv\nd_ne_zero : (algebraMap A K) \u2191(den A r) \u2260 0\n\u22a2 (algebraMap A K) (num A r) / (algebraMap A K) \u2191(den A r) = (algebraMap A K) (num A r * inv)"}, {"tactic": "rw [div_eq_iff d_ne_zero, map_mul, mul_assoc, mul_comm ((algebraMap A K) inv),\n  \u2190 map_mul, \u2190 h_inv, map_one, mul_one]", "annotated_tactic": ["rw [<a>div_eq_iff</a> d_ne_zero, <a>map_mul</a>, <a>mul_assoc</a>, <a>mul_comm</a> ((<a>algebraMap</a> A K) inv),\n      \u2190 <a>map_mul</a>, \u2190 h_inv, <a>map_one</a>, <a>mul_one</a>]", [{"full_name": "div_eq_iff", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [354, 22], "def_end_pos": [354, 32]}, {"full_name": "map_mul", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [309, 9], "def_end_pos": [309, 16]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "map_mul", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [309, 9], "def_end_pos": [309, 16]}, {"full_name": "map_one", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [204, 9], "def_end_pos": [204, 16]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "case left\nA : Type u_1\nK : Type u_2\ninst\u271d\u2075 : CommRing A\ninst\u271d\u2074 : IsDomain A\ninst\u271d\u00b3 : UniqueFactorizationMonoid A\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra A K\ninst\u271d : IsFractionRing A K\np : A[X]\nhp : p.Monic\nr : K\nhr : (aeval r) p = 0\ninv : A\nh_inv : 1 = \u2191(den A r) * inv\nd_ne_zero : (algebraMap A K) \u2191(den A r) \u2260 0\n\u22a2 (algebraMap A K) (num A r) / (algebraMap A K) \u2191(den A r) = (algebraMap A K) (num A r * inv)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Game/PGame.lean", "full_name": "SetTheory.PGame.le_neg_iff", "start": [1440, 1], "end": [1440, 99], "traced_tactics": [{"tactic": "rw [\u2190 neg_neg x, neg_le_neg_iff, neg_neg]", "annotated_tactic": ["rw [\u2190 <a>neg_neg</a> x, <a>neg_le_neg_iff</a>, <a>neg_neg</a>]", [{"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [869, 3], "def_end_pos": [869, 14]}, {"full_name": "SetTheory.PGame.neg_le_neg_iff", "def_path": "Mathlib/SetTheory/Game/PGame.lean", "def_pos": [1398, 9], "def_end_pos": [1398, 23]}, {"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [869, 3], "def_end_pos": [869, 14]}]], "state_before": "xl xr : Type u\nx y : PGame\n\u22a2 y \u2264 -x \u2194 x \u2264 -y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GCDMonoid/Multiset.lean", "full_name": "Multiset.gcd_map_mul", "start": [185, 1], "end": [190, 62], "traced_tactics": [{"tactic": "refine s.induction_on ?_ fun b s ih \u21a6 ?_", "annotated_tactic": ["refine s.induction_on ?_ fun b s ih \u21a6 ?_", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na : \u03b1\ns : Multiset \u03b1\n\u22a2 (map (fun x => a * x) s).gcd = normalize a * s.gcd", "state_after": "case refine_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na : \u03b1\ns : Multiset \u03b1\n\u22a2 (map (fun x => a * x) 0).gcd = normalize a * gcd 0\n\ncase refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na : \u03b1\ns\u271d : Multiset \u03b1\nb : \u03b1\ns : Multiset \u03b1\nih : (map (fun x => a * x) s).gcd = normalize a * s.gcd\n\u22a2 (map (fun x => a * x) (b ::\u2098 s)).gcd = normalize a * (b ::\u2098 s).gcd"}, {"tactic": "simp_rw [map_zero, gcd_zero, mul_zero]", "annotated_tactic": ["simp_rw [<a>map_zero</a>, <a>gcd_zero</a>, <a>mul_zero</a>]", [{"full_name": "Multiset.map_zero", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1220, 9], "def_end_pos": [1220, 17]}, {"full_name": "Multiset.gcd_zero", "def_path": "Mathlib/Algebra/GCDMonoid/Multiset.lean", "def_pos": [134, 9], "def_end_pos": [134, 17]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na : \u03b1\ns : Multiset \u03b1\n\u22a2 (map (fun x => a * x) 0).gcd = normalize a * gcd 0", "state_after": "no goals"}, {"tactic": "simp_rw [map_cons, gcd_cons, \u2190 gcd_mul_left]", "annotated_tactic": ["simp_rw [<a>map_cons</a>, <a>gcd_cons</a>, \u2190 <a>gcd_mul_left</a>]", [{"full_name": "Multiset.map_cons", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1225, 9], "def_end_pos": [1225, 17]}, {"full_name": "Multiset.gcd_cons", "def_path": "Mathlib/Algebra/GCDMonoid/Multiset.lean", "def_pos": [139, 9], "def_end_pos": [139, 17]}, {"full_name": "gcd_mul_left", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [454, 9], "def_end_pos": [454, 21]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na : \u03b1\ns\u271d : Multiset \u03b1\nb : \u03b1\ns : Multiset \u03b1\nih : (map (fun x => a * x) s).gcd = normalize a * s.gcd\n\u22a2 (map (fun x => a * x) (b ::\u2098 s)).gcd = normalize a * (b ::\u2098 s).gcd", "state_after": "case refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na : \u03b1\ns\u271d : Multiset \u03b1\nb : \u03b1\ns : Multiset \u03b1\nih : (map (fun x => a * x) s).gcd = normalize a * s.gcd\n\u22a2 GCDMonoid.gcd (a * b) (map (fun x => a * x) s).gcd = GCDMonoid.gcd (a * b) (a * s.gcd)"}, {"tactic": "rw [ih]", "annotated_tactic": ["rw [ih]", []], "state_before": "case refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na : \u03b1\ns\u271d : Multiset \u03b1\nb : \u03b1\ns : Multiset \u03b1\nih : (map (fun x => a * x) s).gcd = normalize a * s.gcd\n\u22a2 GCDMonoid.gcd (a * b) (map (fun x => a * x) s).gcd = GCDMonoid.gcd (a * b) (a * s.gcd)", "state_after": "case refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na : \u03b1\ns\u271d : Multiset \u03b1\nb : \u03b1\ns : Multiset \u03b1\nih : (map (fun x => a * x) s).gcd = normalize a * s.gcd\n\u22a2 GCDMonoid.gcd (a * b) (normalize a * s.gcd) = GCDMonoid.gcd (a * b) (a * s.gcd)"}, {"tactic": "apply ((normalize_associated a).mul_right _).gcd_eq_right", "annotated_tactic": ["apply ((<a>normalize_associated</a> a).<a>mul_right</a> _).<a>gcd_eq_right</a>", [{"full_name": "normalize_associated", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [115, 9], "def_end_pos": [115, 29]}, {"full_name": "Associated.mul_right", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [555, 9], "def_end_pos": [555, 29]}, {"full_name": "Associated.gcd_eq_right", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [525, 9], "def_end_pos": [525, 32]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na : \u03b1\ns\u271d : Multiset \u03b1\nb : \u03b1\ns : Multiset \u03b1\nih : (map (fun x => a * x) s).gcd = normalize a * s.gcd\n\u22a2 GCDMonoid.gcd (a * b) (normalize a * s.gcd) = GCDMonoid.gcd (a * b) (a * s.gcd)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "continuousOn_inv_iff", "start": [407, 1], "end": [409, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "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, \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</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": [447, 9], "def_end_pos": [447, 20]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "pos_iff_ne_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [230, 3], "def_end_pos": [230, 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": [580, 9], "def_end_pos": [580, 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 : \u03bc.Regular\nh\u03bc : \u03bc \u2260 0\nhU : IsOpen U\n\u22a2 \u03bc U = 0 \u2194 U = \u2205", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "full_name": "lcm_mul_left", "start": [863, 1], "end": [875, 70], "traced_tactics": [{"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c : \u03b1\n\u22a2 a = 0 \u2192 lcm (a * b) (a * c) = normalize a * lcm b c", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\nb c : \u03b1\n\u22a2 lcm (0 * b) (0 * c) = normalize 0 * lcm b c"}, {"tactic": "simp only [zero_mul, lcm_zero_left, normalize_zero]", "annotated_tactic": ["simp only [<a>zero_mul</a>, <a>lcm_zero_left</a>, <a>normalize_zero</a>]", [{"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}, {"full_name": "GCDMonoid.lcm_zero_left", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [288, 3], "def_end_pos": [288, 16]}, {"full_name": "normalize_zero", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [138, 9], "def_end_pos": [138, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\nb c : \u03b1\n\u22a2 lcm (0 * b) (0 * c) = normalize 0 * lcm b c", "state_after": "no goals"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1\ninst\u271d : NormalizedGCDMonoid \u03b1\na b c : \u03b1\nha : a \u2260 0\nthis : lcm (a * b) (a * c) = normalize (a * lcm b c)\n\u22a2 lcm (a * b) (a * c) = normalize a * lcm b c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Order/Compact.lean", "full_name": "ContinuousOn.exists_isMinOn'", "start": [299, 1], "end": [308, 80], "traced_tactics": [{"tactic": "rcases (hasBasis_cocompact.inf_principal _).eventually_iff.1 hc with \u27e8K, hK, hKf\u27e9", "annotated_tactic": ["rcases (hasBasis_cocompact.inf_principal _).<a>eventually_iff</a>.1 hc with \u27e8K, hK, hKf\u27e9", [{"full_name": "Filter.HasBasis.eventually_iff", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [365, 9], "def_end_pos": [365, 32]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\nhsc : IsClosed s\nx\u2080 : \u03b2\nh\u2080 : x\u2080 \u2208 s\nhc : \u2200\u1da0 (x : \u03b2) in cocompact \u03b2 \u2293 \ud835\udcdf s, f x\u2080 \u2264 f x\n\u22a2 \u2203 x \u2208 s, IsMinOn f s x", "state_after": "case intro.intro\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\nhsc : IsClosed s\nx\u2080 : \u03b2\nh\u2080 : x\u2080 \u2208 s\nhc : \u2200\u1da0 (x : \u03b2) in cocompact \u03b2 \u2293 \ud835\udcdf s, f x\u2080 \u2264 f x\nK : Set \u03b2\nhK : IsCompact K\nhKf : \u2200 \u2983x : \u03b2\u2984, x \u2208 K\u1d9c \u2229 s \u2192 f x\u2080 \u2264 f x\n\u22a2 \u2203 x \u2208 s, IsMinOn f s x"}, {"tactic": "have hsub : insert x\u2080 (K \u2229 s) \u2286 s := insert_subset_iff.2 \u27e8h\u2080, inter_subset_right\u27e9", "annotated_tactic": ["have hsub : <a>insert</a> x\u2080 (K \u2229 s) \u2286 s := <a>insert_subset_iff</a>.2 \u27e8h\u2080, <a>inter_subset_right</a>\u27e9", [{"full_name": "Insert.insert", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [458, 3], "def_end_pos": [458, 9]}, {"full_name": "Set.insert_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1124, 9], "def_end_pos": [1124, 26]}, {"full_name": "Set.inter_subset_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [934, 9], "def_end_pos": [934, 27]}]], "state_before": "case intro.intro\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\nhsc : IsClosed s\nx\u2080 : \u03b2\nh\u2080 : x\u2080 \u2208 s\nhc : \u2200\u1da0 (x : \u03b2) in cocompact \u03b2 \u2293 \ud835\udcdf s, f x\u2080 \u2264 f x\nK : Set \u03b2\nhK : IsCompact K\nhKf : \u2200 \u2983x : \u03b2\u2984, x \u2208 K\u1d9c \u2229 s \u2192 f x\u2080 \u2264 f x\n\u22a2 \u2203 x \u2208 s, IsMinOn f s x", "state_after": "case intro.intro\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\nhsc : IsClosed s\nx\u2080 : \u03b2\nh\u2080 : x\u2080 \u2208 s\nhc : \u2200\u1da0 (x : \u03b2) in cocompact \u03b2 \u2293 \ud835\udcdf s, f x\u2080 \u2264 f x\nK : Set \u03b2\nhK : IsCompact K\nhKf : \u2200 \u2983x : \u03b2\u2984, x \u2208 K\u1d9c \u2229 s \u2192 f x\u2080 \u2264 f x\nhsub : insert x\u2080 (K \u2229 s) \u2286 s\n\u22a2 \u2203 x \u2208 s, IsMinOn f s x"}, {"tactic": "obtain \u27e8x, hx, hxf\u27e9 : \u2203 x \u2208 insert x\u2080 (K \u2229 s), \u2200 y \u2208 insert x\u2080 (K \u2229 s), f x \u2264 f y :=\n  ((hK.inter_right hsc).insert x\u2080).exists_isMinOn (insert_nonempty _ _) (hf.mono hsub)", "annotated_tactic": ["obtain \u27e8x, hx, hxf\u27e9 : \u2203 x \u2208 <a>insert</a> x\u2080 (K \u2229 s), \u2200 y \u2208 <a>insert</a> x\u2080 (K \u2229 s), f x \u2264 f y :=\n    ((hK.inter_right hsc).<a>insert</a> x\u2080).<a>exists_isMinOn</a> (<a>insert_nonempty</a> _ _) (hf.mono hsub)", [{"full_name": "Insert.insert", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [458, 3], "def_end_pos": [458, 9]}, {"full_name": "Insert.insert", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [458, 3], "def_end_pos": [458, 9]}, {"full_name": "IsCompact.insert", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [526, 19], "def_end_pos": [526, 35]}, {"full_name": "IsCompact.exists_isMinOn", "def_path": "Mathlib/Topology/Algebra/Order/Compact.lean", "def_pos": [262, 9], "def_end_pos": [262, 33]}, {"full_name": "Set.insert_nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1172, 9], "def_end_pos": [1172, 24]}]], "state_before": "case intro.intro\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\nhsc : IsClosed s\nx\u2080 : \u03b2\nh\u2080 : x\u2080 \u2208 s\nhc : \u2200\u1da0 (x : \u03b2) in cocompact \u03b2 \u2293 \ud835\udcdf s, f x\u2080 \u2264 f x\nK : Set \u03b2\nhK : IsCompact K\nhKf : \u2200 \u2983x : \u03b2\u2984, x \u2208 K\u1d9c \u2229 s \u2192 f x\u2080 \u2264 f x\nhsub : insert x\u2080 (K \u2229 s) \u2286 s\n\u22a2 \u2203 x \u2208 s, IsMinOn f s x", "state_after": "case intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\nhsc : IsClosed s\nx\u2080 : \u03b2\nh\u2080 : x\u2080 \u2208 s\nhc : \u2200\u1da0 (x : \u03b2) in cocompact \u03b2 \u2293 \ud835\udcdf s, f x\u2080 \u2264 f x\nK : Set \u03b2\nhK : IsCompact K\nhKf : \u2200 \u2983x : \u03b2\u2984, x \u2208 K\u1d9c \u2229 s \u2192 f x\u2080 \u2264 f x\nhsub : insert x\u2080 (K \u2229 s) \u2286 s\nx : \u03b2\nhx : x \u2208 insert x\u2080 (K \u2229 s)\nhxf : \u2200 y \u2208 insert x\u2080 (K \u2229 s), f x \u2264 f y\n\u22a2 \u2203 x \u2208 s, IsMinOn f s x"}, {"tactic": "refine \u27e8x, hsub hx, fun y hy => ?_\u27e9", "annotated_tactic": ["refine \u27e8x, hsub hx, fun y hy => ?_\u27e9", []], "state_before": "case intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\nhsc : IsClosed s\nx\u2080 : \u03b2\nh\u2080 : x\u2080 \u2208 s\nhc : \u2200\u1da0 (x : \u03b2) in cocompact \u03b2 \u2293 \ud835\udcdf s, f x\u2080 \u2264 f x\nK : Set \u03b2\nhK : IsCompact K\nhKf : \u2200 \u2983x : \u03b2\u2984, x \u2208 K\u1d9c \u2229 s \u2192 f x\u2080 \u2264 f x\nhsub : insert x\u2080 (K \u2229 s) \u2286 s\nx : \u03b2\nhx : x \u2208 insert x\u2080 (K \u2229 s)\nhxf : \u2200 y \u2208 insert x\u2080 (K \u2229 s), f x \u2264 f y\n\u22a2 \u2203 x \u2208 s, IsMinOn f s x", "state_after": "case intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\nhsc : IsClosed s\nx\u2080 : \u03b2\nh\u2080 : x\u2080 \u2208 s\nhc : \u2200\u1da0 (x : \u03b2) in cocompact \u03b2 \u2293 \ud835\udcdf s, f x\u2080 \u2264 f x\nK : Set \u03b2\nhK : IsCompact K\nhKf : \u2200 \u2983x : \u03b2\u2984, x \u2208 K\u1d9c \u2229 s \u2192 f x\u2080 \u2264 f x\nhsub : insert x\u2080 (K \u2229 s) \u2286 s\nx : \u03b2\nhx : x \u2208 insert x\u2080 (K \u2229 s)\nhxf : \u2200 y \u2208 insert x\u2080 (K \u2229 s), f x \u2264 f y\ny : \u03b2\nhy : y \u2208 s\n\u22a2 y \u2208 {x_1 | (fun x_2 => f x \u2264 f x_2) x_1}"}, {"tactic": "by_cases hyK : y \u2208 K", "annotated_tactic": ["by_cases hyK : y \u2208 K", []], "state_before": "case intro.intro.intro.intro\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\nhsc : IsClosed s\nx\u2080 : \u03b2\nh\u2080 : x\u2080 \u2208 s\nhc : \u2200\u1da0 (x : \u03b2) in cocompact \u03b2 \u2293 \ud835\udcdf s, f x\u2080 \u2264 f x\nK : Set \u03b2\nhK : IsCompact K\nhKf : \u2200 \u2983x : \u03b2\u2984, x \u2208 K\u1d9c \u2229 s \u2192 f x\u2080 \u2264 f x\nhsub : insert x\u2080 (K \u2229 s) \u2286 s\nx : \u03b2\nhx : x \u2208 insert x\u2080 (K \u2229 s)\nhxf : \u2200 y \u2208 insert x\u2080 (K \u2229 s), f x \u2264 f y\ny : \u03b2\nhy : y \u2208 s\n\u22a2 y \u2208 {x_1 | (fun x_2 => f x \u2264 f x_2) x_1}", "state_after": "case pos\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\nhsc : IsClosed s\nx\u2080 : \u03b2\nh\u2080 : x\u2080 \u2208 s\nhc : \u2200\u1da0 (x : \u03b2) in cocompact \u03b2 \u2293 \ud835\udcdf s, f x\u2080 \u2264 f x\nK : Set \u03b2\nhK : IsCompact K\nhKf : \u2200 \u2983x : \u03b2\u2984, x \u2208 K\u1d9c \u2229 s \u2192 f x\u2080 \u2264 f x\nhsub : insert x\u2080 (K \u2229 s) \u2286 s\nx : \u03b2\nhx : x \u2208 insert x\u2080 (K \u2229 s)\nhxf : \u2200 y \u2208 insert x\u2080 (K \u2229 s), f x \u2264 f y\ny : \u03b2\nhy : y \u2208 s\nhyK : y \u2208 K\n\u22a2 y \u2208 {x_1 | (fun x_2 => f x \u2264 f x_2) x_1}\n\ncase neg\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\nhsc : IsClosed s\nx\u2080 : \u03b2\nh\u2080 : x\u2080 \u2208 s\nhc : \u2200\u1da0 (x : \u03b2) in cocompact \u03b2 \u2293 \ud835\udcdf s, f x\u2080 \u2264 f x\nK : Set \u03b2\nhK : IsCompact K\nhKf : \u2200 \u2983x : \u03b2\u2984, x \u2208 K\u1d9c \u2229 s \u2192 f x\u2080 \u2264 f x\nhsub : insert x\u2080 (K \u2229 s) \u2286 s\nx : \u03b2\nhx : x \u2208 insert x\u2080 (K \u2229 s)\nhxf : \u2200 y \u2208 insert x\u2080 (K \u2229 s), f x \u2264 f y\ny : \u03b2\nhy : y \u2208 s\nhyK : y \u2209 K\n\u22a2 y \u2208 {x_1 | (fun x_2 => f x \u2264 f x_2) x_1}"}, {"tactic": "exacts [hxf _ (Or.inr \u27e8hyK, hy\u27e9), (hxf _ (Or.inl rfl)).trans (hKf \u27e8hyK, hy\u27e9)]", "annotated_tactic": ["exacts [hxf _ (<a>Or.inr</a> \u27e8hyK, hy\u27e9), (hxf _ (<a>Or.inl</a> <a>rfl</a>)).<a>trans</a> (hKf \u27e8hyK, hy\u27e9)]", [{"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}, {"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}]], "state_before": "case pos\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\nhsc : IsClosed s\nx\u2080 : \u03b2\nh\u2080 : x\u2080 \u2208 s\nhc : \u2200\u1da0 (x : \u03b2) in cocompact \u03b2 \u2293 \ud835\udcdf s, f x\u2080 \u2264 f x\nK : Set \u03b2\nhK : IsCompact K\nhKf : \u2200 \u2983x : \u03b2\u2984, x \u2208 K\u1d9c \u2229 s \u2192 f x\u2080 \u2264 f x\nhsub : insert x\u2080 (K \u2229 s) \u2286 s\nx : \u03b2\nhx : x \u2208 insert x\u2080 (K \u2229 s)\nhxf : \u2200 y \u2208 insert x\u2080 (K \u2229 s), f x \u2264 f y\ny : \u03b2\nhy : y \u2208 s\nhyK : y \u2208 K\n\u22a2 y \u2208 {x_1 | (fun x_2 => f x \u2264 f x_2) x_1}\n\ncase neg\n\u03b1\u271d : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\ninst\u271d\u2074 : LinearOrder \u03b1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\ninst\u271d : ClosedIicTopology \u03b1\ns : Set \u03b2\nf : \u03b2 \u2192 \u03b1\nhf : ContinuousOn f s\nhsc : IsClosed s\nx\u2080 : \u03b2\nh\u2080 : x\u2080 \u2208 s\nhc : \u2200\u1da0 (x : \u03b2) in cocompact \u03b2 \u2293 \ud835\udcdf s, f x\u2080 \u2264 f x\nK : Set \u03b2\nhK : IsCompact K\nhKf : \u2200 \u2983x : \u03b2\u2984, x \u2208 K\u1d9c \u2229 s \u2192 f x\u2080 \u2264 f x\nhsub : insert x\u2080 (K \u2229 s) \u2286 s\nx : \u03b2\nhx : x \u2208 insert x\u2080 (K \u2229 s)\nhxf : \u2200 y \u2208 insert x\u2080 (K \u2229 s), f x \u2264 f y\ny : \u03b2\nhy : y \u2208 s\nhyK : y \u2209 K\n\u22a2 y \u2208 {x_1 | (fun x_2 => f x \u2264 f x_2) x_1}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean", "full_name": "CategoryTheory.ShortComplex.toCycles_naturality", "start": [527, 1], "end": [529, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "full_name": "CategoryTheory.Limits.walkingParallelPairOpEquiv_counitIso_one", "start": [199, 1], "end": [201, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.smul_mem_smul_finset_iff", "start": [2091, 1], "end": [2092, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Fourier/FourierTransformDeriv.lean", "full_name": "Real.fourierIntegral_iteratedFDeriv", "start": [696, 1], "end": [701, 75], "traced_tactics": [{"tactic": "rw [\u2190 innerSL_real_flip V]", "annotated_tactic": ["rw [\u2190 <a>innerSL_real_flip</a> V]", [{"full_name": "innerSL_real_flip", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [1828, 15], "def_end_pos": [1828, 32]}]], "state_before": "E : Type u_1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u2102 E\nV : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : MeasurableSpace V\ninst\u271d : BorelSpace V\nf : V \u2192 E\nN : \u2115\u221e\nhf : ContDiff \u211d N f\nh'f : \u2200 (n : \u2115), \u2191n \u2264 N \u2192 Integrable (iteratedFDeriv \u211d n f) volume\nn : \u2115\nhn : \u2191n \u2264 N\n\u22a2 \ud835\udcd5 (iteratedFDeriv \u211d n f) = fun w => fourierPowSMulRight (-innerSL \u211d) (\ud835\udcd5 f) w n", "state_after": "E : Type u_1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u2102 E\nV : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : MeasurableSpace V\ninst\u271d : BorelSpace V\nf : V \u2192 E\nN : \u2115\u221e\nhf : ContDiff \u211d N f\nh'f : \u2200 (n : \u2115), \u2191n \u2264 N \u2192 Integrable (iteratedFDeriv \u211d n f) volume\nn : \u2115\nhn : \u2191n \u2264 N\n\u22a2 \ud835\udcd5 (iteratedFDeriv \u211d n f) = fun w => fourierPowSMulRight (-(innerSL \u211d).flip) (\ud835\udcd5 f) w n"}, {"tactic": "exact VectorFourier.fourierIntegral_iteratedFDeriv (innerSL \u211d) hf h'f hn", "annotated_tactic": ["exact <a>VectorFourier.fourierIntegral_iteratedFDeriv</a> (<a>innerSL</a> \u211d) hf h'f hn", [{"full_name": "VectorFourier.fourierIntegral_iteratedFDeriv", "def_path": "Mathlib/Analysis/Fourier/FourierTransformDeriv.lean", "def_pos": [511, 9], "def_end_pos": [511, 39]}, {"full_name": "innerSL", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [1777, 5], "def_end_pos": [1777, 12]}]], "state_before": "E : Type u_1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u2102 E\nV : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : MeasurableSpace V\ninst\u271d : BorelSpace V\nf : V \u2192 E\nN : \u2115\u221e\nhf : ContDiff \u211d N f\nh'f : \u2200 (n : \u2115), \u2191n \u2264 N \u2192 Integrable (iteratedFDeriv \u211d n f) volume\nn : \u2115\nhn : \u2191n \u2264 N\n\u22a2 \ud835\udcd5 (iteratedFDeriv \u211d n f) = fun w => fourierPowSMulRight (-(innerSL \u211d).flip) (\ud835\udcd5 f) w n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/CauSeq/Basic.lean", "full_name": "CauSeq.mul_limZero_left", "start": [437, 1], "end": [442, 63], "traced_tactics": [{"tactic": "have := mul_lt_mul'' (H _ ij) (hG j) (abv_nonneg abv _) (abv_nonneg abv _)", "annotated_tactic": ["have := <a>mul_lt_mul''</a> (H _ ij) (hG j) (<a>abv_nonneg</a> abv _) (<a>abv_nonneg</a> abv _)", [{"full_name": "mul_lt_mul''", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [582, 9], "def_end_pos": [582, 21]}, {"full_name": "IsAbsoluteValue.abv_nonneg", "def_path": "Mathlib/Algebra/Order/AbsoluteValue.lean", "def_pos": [314, 7], "def_end_pos": [314, 17]}, {"full_name": "IsAbsoluteValue.abv_nonneg", "def_path": "Mathlib/Algebra/Order/AbsoluteValue.lean", "def_pos": [314, 7], "def_end_pos": [314, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhg : f.LimZero\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nG : \u03b1\nG0 : G > 0\nhG : \u2200 (i : \u2115), abv (\u2191g i) < G\ni : \u2115\nH : \u2200 j \u2265 i, abv (\u2191f j) < \u03b5 / G\nj : \u2115\nij : j \u2265 i\n\u22a2 abv (\u2191(f * g) j) < \u03b5", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhg : f.LimZero\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nG : \u03b1\nG0 : G > 0\nhG : \u2200 (i : \u2115), abv (\u2191g i) < G\ni : \u2115\nH : \u2200 j \u2265 i, abv (\u2191f j) < \u03b5 / G\nj : \u2115\nij : j \u2265 i\nthis : abv (\u2191f j) * abv (\u2191g j) < \u03b5 / G * G\n\u22a2 abv (\u2191(f * g) j) < \u03b5"}, {"tactic": "rwa [div_mul_cancel\u2080 _ (ne_of_gt G0), \u2190 abv_mul] at this", "annotated_tactic": ["rwa [<a>div_mul_cancel\u2080</a> _ (<a>ne_of_gt</a> G0), \u2190 <a>abv_mul</a>] at this", [{"full_name": "div_mul_cancel\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [342, 15], "def_end_pos": [342, 30]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}, {"full_name": "IsAbsoluteValue.abv_mul", "def_path": "Mathlib/Algebra/Order/AbsoluteValue.lean", "def_pos": [331, 7], "def_end_pos": [331, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : LinearOrderedField \u03b1\ninst\u271d\u00b9 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d : IsAbsoluteValue abv\nf g : CauSeq \u03b2 abv\nhg : f.LimZero\n\u03b5 : \u03b1\n\u03b50 : \u03b5 > 0\nG : \u03b1\nG0 : G > 0\nhG : \u2200 (i : \u2115), abv (\u2191g i) < G\ni : \u2115\nH : \u2200 j \u2265 i, abv (\u2191f j) < \u03b5 / G\nj : \u2115\nij : j \u2265 i\nthis : abv (\u2191f j) * abv (\u2191g j) < \u03b5 / G * G\n\u22a2 abv (\u2191(f * g) j) < \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Sigma.lean", "full_name": "List.NodupKeys.pairwise_ne", "start": [91, 1], "end": [93, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Unique.lean", "full_name": "unique_iff_exists_unique", "start": [63, 1], "end": [65, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "full_name": "Finsupp.mul_prod_erase'", "start": [186, 1], "end": [191, 83], "traced_tactics": [{"tactic": "by_cases hyf : y \u2208 f.support", "annotated_tactic": ["by_cases hyf : y \u2208 f.support", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\ninst\u271d\u2075 : AddCommMonoid A\ninst\u271d\u2074 : AddCommMonoid B\ninst\u271d\u00b3 : AddCommMonoid C\nt : \u03b9 \u2192 A \u2192 C\nh0 : \u2200 (i : \u03b9), t i 0 = 0\nh1 : \u2200 (i : \u03b9) (x y : A), t i (x + y) = t i x + t i y\ns : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b9 \u2192\u2080 A\ni : \u03b9\ng\u271d : \u03b9 \u2192\u2080 A\nk : \u03b9 \u2192 A \u2192 \u03b3 \u2192 B\nx : \u03b3\n\u03b2 : Type u_7\nM : Type u_8\nM' : Type u_9\nN : Type u_10\nP : Type u_11\nG : Type u_12\nH : Type u_13\nR : Type u_14\nS : Type u_15\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero M'\ninst\u271d : CommMonoid N\nf : \u03b1 \u2192\u2080 M\ny : \u03b1\ng : \u03b1 \u2192 M \u2192 N\nhg : \u2200 (i : \u03b1), g i 0 = 1\n\u22a2 g y (f y) * (erase y f).prod g = f.prod g", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\ninst\u271d\u2075 : AddCommMonoid A\ninst\u271d\u2074 : AddCommMonoid B\ninst\u271d\u00b3 : AddCommMonoid C\nt : \u03b9 \u2192 A \u2192 C\nh0 : \u2200 (i : \u03b9), t i 0 = 0\nh1 : \u2200 (i : \u03b9) (x y : A), t i (x + y) = t i x + t i y\ns : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b9 \u2192\u2080 A\ni : \u03b9\ng\u271d : \u03b9 \u2192\u2080 A\nk : \u03b9 \u2192 A \u2192 \u03b3 \u2192 B\nx : \u03b3\n\u03b2 : Type u_7\nM : Type u_8\nM' : Type u_9\nN : Type u_10\nP : Type u_11\nG : Type u_12\nH : Type u_13\nR : Type u_14\nS : Type u_15\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero M'\ninst\u271d : CommMonoid N\nf : \u03b1 \u2192\u2080 M\ny : \u03b1\ng : \u03b1 \u2192 M \u2192 N\nhg : \u2200 (i : \u03b1), g i 0 = 1\nhyf : y \u2208 f.support\n\u22a2 g y (f y) * (erase y f).prod g = f.prod g\n\ncase neg\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\ninst\u271d\u2075 : AddCommMonoid A\ninst\u271d\u2074 : AddCommMonoid B\ninst\u271d\u00b3 : AddCommMonoid C\nt : \u03b9 \u2192 A \u2192 C\nh0 : \u2200 (i : \u03b9), t i 0 = 0\nh1 : \u2200 (i : \u03b9) (x y : A), t i (x + y) = t i x + t i y\ns : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b9 \u2192\u2080 A\ni : \u03b9\ng\u271d : \u03b9 \u2192\u2080 A\nk : \u03b9 \u2192 A \u2192 \u03b3 \u2192 B\nx : \u03b3\n\u03b2 : Type u_7\nM : Type u_8\nM' : Type u_9\nN : Type u_10\nP : Type u_11\nG : Type u_12\nH : Type u_13\nR : Type u_14\nS : Type u_15\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero M'\ninst\u271d : CommMonoid N\nf : \u03b1 \u2192\u2080 M\ny : \u03b1\ng : \u03b1 \u2192 M \u2192 N\nhg : \u2200 (i : \u03b1), g i 0 = 1\nhyf : y \u2209 f.support\n\u22a2 g y (f y) * (erase y f).prod g = f.prod g"}, {"tactic": "exact Finsupp.mul_prod_erase f y g hyf", "annotated_tactic": ["exact <a>Finsupp.mul_prod_erase</a> f y g hyf", [{"full_name": "Finsupp.mul_prod_erase", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [169, 9], "def_end_pos": [169, 23]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\ninst\u271d\u2075 : AddCommMonoid A\ninst\u271d\u2074 : AddCommMonoid B\ninst\u271d\u00b3 : AddCommMonoid C\nt : \u03b9 \u2192 A \u2192 C\nh0 : \u2200 (i : \u03b9), t i 0 = 0\nh1 : \u2200 (i : \u03b9) (x y : A), t i (x + y) = t i x + t i y\ns : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b9 \u2192\u2080 A\ni : \u03b9\ng\u271d : \u03b9 \u2192\u2080 A\nk : \u03b9 \u2192 A \u2192 \u03b3 \u2192 B\nx : \u03b3\n\u03b2 : Type u_7\nM : Type u_8\nM' : Type u_9\nN : Type u_10\nP : Type u_11\nG : Type u_12\nH : Type u_13\nR : Type u_14\nS : Type u_15\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero M'\ninst\u271d : CommMonoid N\nf : \u03b1 \u2192\u2080 M\ny : \u03b1\ng : \u03b1 \u2192 M \u2192 N\nhg : \u2200 (i : \u03b1), g i 0 = 1\nhyf : y \u2208 f.support\n\u22a2 g y (f y) * (erase y f).prod g = f.prod g", "state_after": "no goals"}, {"tactic": "rw [not_mem_support_iff.mp hyf, hg y, erase_of_not_mem_support hyf, one_mul]", "annotated_tactic": ["rw [not_mem_support_iff.mp hyf, hg y, <a>erase_of_not_mem_support</a> hyf, <a>one_mul</a>]", [{"full_name": "Finsupp.erase_of_not_mem_support", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [675, 9], "def_end_pos": [675, 33]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b3 : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\ninst\u271d\u2075 : AddCommMonoid A\ninst\u271d\u2074 : AddCommMonoid B\ninst\u271d\u00b3 : AddCommMonoid C\nt : \u03b9 \u2192 A \u2192 C\nh0 : \u2200 (i : \u03b9), t i 0 = 0\nh1 : \u2200 (i : \u03b9) (x y : A), t i (x + y) = t i x + t i y\ns : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b9 \u2192\u2080 A\ni : \u03b9\ng\u271d : \u03b9 \u2192\u2080 A\nk : \u03b9 \u2192 A \u2192 \u03b3 \u2192 B\nx : \u03b3\n\u03b2 : Type u_7\nM : Type u_8\nM' : Type u_9\nN : Type u_10\nP : Type u_11\nG : Type u_12\nH : Type u_13\nR : Type u_14\nS : Type u_15\ninst\u271d\u00b2 : Zero M\ninst\u271d\u00b9 : Zero M'\ninst\u271d : CommMonoid N\nf : \u03b1 \u2192\u2080 M\ny : \u03b1\ng : \u03b1 \u2192 M \u2192 N\nhg : \u2200 (i : \u03b1), g i 0 = 1\nhyf : y \u2209 f.support\n\u22a2 g y (f y) * (erase y f).prod g = f.prod g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Hom/Lattice.lean", "full_name": "SupHom.withTop_id", "start": [1650, 1], "end": [1650, 97], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean", "full_name": "tendsto_rpow_atBot_of_base_gt_one", "start": [93, 1], "end": [97, 54], "traced_tactics": [{"tactic": "simp_rw [Real.rpow_def_of_pos (by positivity : 0 < b)]", "annotated_tactic": ["simp_rw [<a>Real.rpow_def_of_pos</a> (by positivity : 0 < b)]", [{"full_name": "Real.rpow_def_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [56, 9], "def_end_pos": [56, 24]}]], "state_before": "b : \u211d\nhb : 1 < b\n\u22a2 Tendsto (fun x => b ^ x) atBot (\ud835\udcdd 0)", "state_after": "b : \u211d\nhb : 1 < b\n\u22a2 Tendsto (fun x => rexp (log b * x)) atBot (\ud835\udcdd 0)"}, {"tactic": "refine tendsto_exp_atBot.comp <| (tendsto_const_mul_atBot_iff_pos <| tendsto_id (\u03b1 := \u211d)).mpr ?_", "annotated_tactic": ["refine tendsto_exp_atBot.comp <| (<a>tendsto_const_mul_atBot_iff_pos</a> <| <a>tendsto_id</a> (\u03b1 := \u211d)).<a>mpr</a> ?_", [{"full_name": "Filter.tendsto_const_mul_atBot_iff_pos", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [1257, 9], "def_end_pos": [1257, 40]}, {"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3094, 9], "def_end_pos": [3094, 19]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}]], "state_before": "b : \u211d\nhb : 1 < b\n\u22a2 Tendsto (fun x => rexp (log b * x)) atBot (\ud835\udcdd 0)", "state_after": "b : \u211d\nhb : 1 < b\n\u22a2 0 < log b"}, {"tactic": "exact (log_pos_iff (by positivity)).mpr <| by aesop", "annotated_tactic": ["exact (<a>log_pos_iff</a> (by positivity)).<a>mpr</a> <| by aesop", [{"full_name": "Real.log_pos_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [171, 9], "def_end_pos": [171, 20]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}]], "state_before": "b : \u211d\nhb : 1 < b\n\u22a2 0 < log b", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "b : \u211d\nhb : 1 < b\n\u22a2 0 < b", "state_after": "no goals"}, {"tactic": "aesop", "annotated_tactic": ["aesop", []], "state_before": "b : \u211d\nhb : 1 < b\n\u22a2 1 < b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/CofilteredSystem.lean", "full_name": "nonempty_sections_of_finite_cofiltered_system", "start": [82, 1], "end": [101, 16], "traced_tactics": [{"tactic": "let J' : Type max w v u := AsSmall.{max w v} J", "annotated_tactic": ["let J' : Type max w v u := <a>AsSmall</a>.{max w v} J", [{"full_name": "CategoryTheory.AsSmall", "def_path": 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J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\n\u22a2 F.sections.Nonempty", "state_after": "J : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\n\u22a2 F.sections.Nonempty"}, {"tactic": "let F' : J' \u2964 Type max u v w := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}", "annotated_tactic": ["let F' : J' \u2964 Type max u v w := down \u22d9 F \u22d9 <a>uliftFunctor</a>.{max u w, v}", [{"full_name": "CategoryTheory.uliftFunctor", "def_path": "Mathlib/CategoryTheory/Types.lean", "def_pos": [214, 5], "def_end_pos": [214, 17]}]], "state_before": "J : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\n\u22a2 F.sections.Nonempty", "state_after": "J : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\n\u22a2 F.sections.Nonempty"}, {"tactic": "haveI : \u2200 i, Nonempty (F'.obj i) := fun i => \u27e8\u27e8Classical.arbitrary (F.obj (down.obj i))\u27e9\u27e9", "annotated_tactic": ["haveI : \u2200 i, <a>Nonempty</a> (F'.obj i) := fun i => \u27e8\u27e8<a>Classical.arbitrary</a> (F.obj (down.obj i))\u27e9\u27e9", [{"full_name": "Nonempty", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [709, 17], "def_end_pos": [709, 25]}, {"full_name": "Classical.arbitrary", "def_path": "Mathlib/Logic/Nonempty.lean", "def_pos": [99, 32], "def_end_pos": [99, 51]}]], "state_before": "J : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\n\u22a2 F.sections.Nonempty", "state_after": "J : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis : \u2200 (i : J'), Nonempty (F'.obj i)\n\u22a2 F.sections.Nonempty"}, {"tactic": "haveI : \u2200 i, Finite (F'.obj i) := fun i => Finite.of_equiv (F.obj (down.obj i)) Equiv.ulift.symm", "annotated_tactic": ["haveI : \u2200 i, <a>Finite</a> (F'.obj i) := fun i => <a>Finite.of_equiv</a> (F.obj (down.obj i)) Equiv.ulift.symm", [{"full_name": "Finite", "def_path": "Mathlib/Data/Finite/Defs.lean", "def_pos": [81, 17], "def_end_pos": [81, 23]}, {"full_name": "Finite.of_equiv", "def_path": "Mathlib/Data/Finite/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 24]}]], "state_before": "J : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis : \u2200 (i : J'), Nonempty (F'.obj i)\n\u22a2 F.sections.Nonempty", "state_after": "J : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d : \u2200 (i : J'), Nonempty (F'.obj i)\nthis : \u2200 (i : J'), Finite (F'.obj i)\n\u22a2 F.sections.Nonempty"}, {"tactic": "cases isEmpty_or_nonempty J", "annotated_tactic": ["cases <a>isEmpty_or_nonempty</a> J", [{"full_name": "isEmpty_or_nonempty", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [221, 9], "def_end_pos": [221, 28]}]], "state_before": "J : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d : \u2200 (i : J'), Nonempty (F'.obj i)\nthis : \u2200 (i : J'), Finite (F'.obj i)\n\u22a2 F.sections.Nonempty", "state_after": "case inl\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d : \u2200 (i : J'), Nonempty (F'.obj i)\nthis : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : IsEmpty J\n\u22a2 F.sections.Nonempty\n\ncase inr\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d : \u2200 (i : J'), Nonempty (F'.obj i)\nthis : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : Nonempty J\n\u22a2 F.sections.Nonempty"}, {"tactic": "haveI : IsCofiltered J := \u27e8\u27e9", "annotated_tactic": ["haveI : <a>IsCofiltered</a> J := \u27e8\u27e9", [{"full_name": "CategoryTheory.IsCofiltered", "def_path": "Mathlib/CategoryTheory/Filtered/Basic.lean", "def_pos": [561, 7], "def_end_pos": [561, 19]}]], "state_before": "case inr\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d : \u2200 (i : J'), Nonempty (F'.obj i)\nthis : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : Nonempty J\n\u22a2 F.sections.Nonempty", "state_after": "case inr\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d\u00b9 : \u2200 (i : J'), Nonempty (F'.obj i)\nthis\u271d : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : Nonempty J\nthis : IsCofiltered J\n\u22a2 F.sections.Nonempty"}, {"tactic": "obtain \u27e8u, hu\u27e9 := nonempty_sections_of_finite_cofiltered_system.init F'", "annotated_tactic": ["obtain \u27e8u, hu\u27e9 := <a>nonempty_sections_of_finite_cofiltered_system.init</a> F'", [{"full_name": "nonempty_sections_of_finite_cofiltered_system.init", "def_path": "Mathlib/CategoryTheory/CofilteredSystem.lean", "def_pos": [68, 9], "def_end_pos": [68, 59]}]], "state_before": "case inr\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d\u00b9 : \u2200 (i : J'), Nonempty (F'.obj i)\nthis\u271d : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : Nonempty J\nthis : IsCofiltered J\n\u22a2 F.sections.Nonempty", "state_after": "case inr.intro\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d\u00b9 : \u2200 (i : J'), Nonempty (F'.obj i)\nthis\u271d : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : Nonempty J\nthis : IsCofiltered J\nu : (j : J') \u2192 F'.obj j\nhu : u \u2208 F'.sections\n\u22a2 F.sections.Nonempty"}, {"tactic": "use fun j => (u \u27e8j\u27e9).down", "annotated_tactic": ["use fun j => (u \u27e8j\u27e9).<a>down</a>", [{"full_name": "ULift.down", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [811, 41], "def_end_pos": [811, 45]}]], "state_before": "case inr.intro\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d\u00b9 : \u2200 (i : J'), Nonempty (F'.obj i)\nthis\u271d : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : Nonempty J\nthis : IsCofiltered J\nu : (j : J') \u2192 F'.obj j\nhu : u \u2208 F'.sections\n\u22a2 F.sections.Nonempty", "state_after": "case h\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d\u00b9 : \u2200 (i : J'), Nonempty (F'.obj i)\nthis\u271d : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : Nonempty J\nthis : IsCofiltered J\nu : (j : J') \u2192 F'.obj j\nhu : u \u2208 F'.sections\n\u22a2 (fun j => (u { down := j }).down) \u2208 F.sections"}, {"tactic": "intro j j' f", "annotated_tactic": ["intro j j' f", []], "state_before": "case h\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d\u00b9 : \u2200 (i : J'), Nonempty (F'.obj i)\nthis\u271d : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : Nonempty J\nthis : IsCofiltered J\nu : (j : J') \u2192 F'.obj j\nhu : u \u2208 F'.sections\n\u22a2 (fun j => (u { down := j }).down) \u2208 F.sections", "state_after": "case h\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d\u00b9 : \u2200 (i : J'), Nonempty (F'.obj i)\nthis\u271d : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : Nonempty J\nthis : IsCofiltered J\nu : (j : J') \u2192 F'.obj j\nhu : u \u2208 F'.sections\nj j' : J\nf : j \u27f6 j'\n\u22a2 F.map f ((fun j => (u { down := j }).down) j) = (fun j => (u { down := j }).down) j'"}, {"tactic": "have h := @hu (\u27e8j\u27e9 : J') (\u27e8j'\u27e9 : J') (ULift.up f)", "annotated_tactic": ["have h := @hu (\u27e8j\u27e9 : J') (\u27e8j'\u27e9 : J') (<a>ULift.up</a> f)", [{"full_name": "ULift.up", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [810, 41], "def_end_pos": [810, 43]}]], "state_before": "case h\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d\u00b9 : \u2200 (i : J'), Nonempty (F'.obj i)\nthis\u271d : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : Nonempty J\nthis : IsCofiltered J\nu : (j : J') \u2192 F'.obj j\nhu : u \u2208 F'.sections\nj j' : J\nf : j \u27f6 j'\n\u22a2 F.map f ((fun j => (u { down := j }).down) j) = (fun j => (u { down := j }).down) j'", "state_after": "case h\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d\u00b9 : \u2200 (i : J'), Nonempty (F'.obj i)\nthis\u271d : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : Nonempty J\nthis : IsCofiltered J\nu : (j : J') \u2192 F'.obj j\nhu : u \u2208 F'.sections\nj j' : J\nf : j \u27f6 j'\nh : F'.map { down := f } (u { down := j }) = u { down := j' }\n\u22a2 F.map f ((fun j => (u { down := j }).down) j) = (fun j => (u { down := j }).down) j'"}, {"tactic": "simp only [F', down, AsSmall.down, Functor.comp_map, uliftFunctor_map, Functor.op_map] at h", "annotated_tactic": ["simp only [F', down, <a>AsSmall.down</a>, <a>Functor.comp_map</a>, <a>uliftFunctor_map</a>, <a>Functor.op_map</a>] at h", [{"full_name": "CategoryTheory.AsSmall.down", "def_path": "Mathlib/CategoryTheory/Category/ULift.lean", "def_pos": [188, 5], "def_end_pos": [188, 17]}, {"full_name": "CategoryTheory.Functor.comp_map", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 17]}, {"full_name": "CategoryTheory.uliftFunctor_map", "def_path": "Mathlib/CategoryTheory/Types.lean", "def_pos": [224, 9], "def_end_pos": [224, 25]}, {"full_name": "CategoryTheory.Functor.op_map", "def_path": "Mathlib/CategoryTheory/Opposites.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}]], "state_before": "case h\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d\u00b9 : \u2200 (i : J'), Nonempty (F'.obj i)\nthis\u271d : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : Nonempty J\nthis : IsCofiltered J\nu : (j : J') \u2192 F'.obj j\nhu : u \u2208 F'.sections\nj j' : J\nf : j \u27f6 j'\nh : F'.map { down := f } (u { down := j }) = u { down := j' }\n\u22a2 F.map f ((fun j => (u { down := j }).down) j) = (fun j => (u { down := j }).down) j'", "state_after": "case h\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d\u00b9 : \u2200 (i : J'), Nonempty (F'.obj i)\nthis\u271d : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : Nonempty J\nthis : IsCofiltered J\nu : (j : J') \u2192 F'.obj j\nhu : u \u2208 F'.sections\nj j' : J\nf : j \u27f6 j'\nh : { down := F.map f (u { down := j }).down } = u { down := j' }\n\u22a2 F.map f ((fun j => (u { down := j }).down) j) = (fun j => (u { down := j }).down) j'"}, {"tactic": "simp_rw [\u2190 h]", "annotated_tactic": ["simp_rw [\u2190 h]", []], "state_before": "case h\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d\u00b9 : \u2200 (i : J'), Nonempty (F'.obj i)\nthis\u271d : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : Nonempty J\nthis : IsCofiltered J\nu : (j : J') \u2192 F'.obj j\nhu : u \u2208 F'.sections\nj j' : J\nf : j \u27f6 j'\nh : { down := F.map f (u { down := j }).down } = u { down := j' }\n\u22a2 F.map f ((fun j => (u { down := j }).down) j) = (fun j => (u { down := j }).down) j'", "state_after": "no goals"}, {"tactic": "fconstructor <;> apply isEmptyElim", "annotated_tactic": ["fconstructor <;> apply <a>isEmptyElim</a>", [{"full_name": "isEmptyElim", "def_path": "Mathlib/Logic/IsEmpty.lean", "def_pos": [96, 5], "def_end_pos": [96, 16]}]], "state_before": "case inl\nJ : Type u\ninst\u271d\u00b3 : Category.{w, u} J\ninst\u271d\u00b2 : IsCofilteredOrEmpty J\nF : J \u2964 Type v\ninst\u271d\u00b9 : \u2200 (j : J), Finite (F.obj j)\ninst\u271d : \u2200 (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' \u2964 J := AsSmall.down\nF' : J' \u2964 Type (max u v w) := down \u22d9 F \u22d9 uliftFunctor.{max u w, v}\nthis\u271d : \u2200 (i : J'), Nonempty (F'.obj i)\nthis : \u2200 (i : J'), Finite (F'.obj i)\nh\u271d : IsEmpty J\n\u22a2 F.sections.Nonempty", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Dynamics/PeriodicPts.lean", "full_name": "MulAction.pow_period_add_smul", "start": [698, 1], "end": [700, 68], "traced_tactics": [{"tactic": "rw [\u2190 pow_mod_period_smul, Nat.add_mod_left, pow_mod_period_smul]", "annotated_tactic": ["rw [\u2190 <a>pow_mod_period_smul</a>, <a>Nat.add_mod_left</a>, <a>pow_mod_period_smul</a>]", [{"full_name": "MulAction.pow_mod_period_smul", "def_path": "Mathlib/Dynamics/PeriodicPts.lean", "def_pos": [681, 9], "def_end_pos": [681, 28]}, {"full_name": "Nat.add_mod_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [271, 17], "def_end_pos": [271, 29]}, {"full_name": "MulAction.pow_mod_period_smul", "def_path": "Mathlib/Dynamics/PeriodicPts.lean", "def_pos": [681, 9], "def_end_pos": [681, 28]}]], "state_before": "\u03b1 : Type v\nG : Type u\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\nM : Type u\ninst\u271d\u00b9 : Monoid M\ninst\u271d : MulAction M \u03b1\nn : \u2115\nm : M\na : \u03b1\n\u22a2 m ^ (period m a + n) \u2022 a = m ^ n \u2022 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "full_name": "LinearMap.comp_sub", "start": [967, 1], "end": [969, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Equiv.lean", "full_name": "ContinuousLinearEquiv.comp_right_fderiv", "start": [267, 1], "end": [270, 32], "traced_tactics": [{"tactic": "rw [\u2190 fderivWithin_univ, \u2190 fderivWithin_univ, \u2190 iso.comp_right_fderivWithin, preimage_univ]", "annotated_tactic": ["rw [\u2190 <a>fderivWithin_univ</a>, \u2190 <a>fderivWithin_univ</a>, \u2190 iso.comp_right_fderivWithin, <a>preimage_univ</a>]", [{"full_name": "fderivWithin_univ", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [701, 9], "def_end_pos": [701, 26]}, {"full_name": "fderivWithin_univ", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [701, 9], "def_end_pos": [701, 26]}, {"full_name": "Set.preimage_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [63, 9], "def_end_pos": [63, 22]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\nx : E\n\u22a2 fderiv \ud835\udd5c (f \u2218 \u21d1iso) x = (fderiv \ud835\udd5c f (iso x)).comp \u2191iso", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\nx : E\n\u22a2 UniqueDiffWithinAt \ud835\udd5c (\u21d1iso \u207b\u00b9' univ) x"}, {"tactic": "exact uniqueDiffWithinAt_univ", "annotated_tactic": ["exact <a>uniqueDiffWithinAt_univ</a>", [{"full_name": "uniqueDiffWithinAt_univ", "def_path": "Mathlib/Analysis/Calculus/TangentCone.lean", "def_pos": [239, 9], "def_end_pos": [239, 32]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type u_3\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type u_4\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nG' : Type u_5\ninst\u271d\u00b9 : NormedAddCommGroup G'\ninst\u271d : NormedSpace \ud835\udd5c G'\nf\u271d f\u2080 f\u2081 g : E \u2192 F\nf' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F\nx\u271d : E\ns t : Set E\nL L\u2081 L\u2082 : Filter E\niso : E \u2243L[\ud835\udd5c] F\nf : F \u2192 G\nx : E\n\u22a2 UniqueDiffWithinAt \ud835\udd5c (\u21d1iso \u207b\u00b9' univ) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "full_name": "MeasureTheory.snorm_le_add_measure_right", "start": [719, 1], "end": [721, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Flat/EquationalCriterion.lean", "full_name": "Module.Flat.iff_forall_exists_factorization", "start": [212, 1], "end": [215, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/IsDiag.lean", "full_name": "Matrix.isDiag_smul_one", "start": [111, 1], "end": [113, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Asymptotics/Theta.lean", "full_name": "Asymptotics.IsTheta.smul", "start": [229, 1], "end": [232, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Symmetrized.lean", "full_name": "SymAlg.unsym_ne_one_iff", "start": [248, 1], "end": [249, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Lattice.lean", "full_name": "ofDual_min", "start": [929, 1], "end": [930, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "full_name": "UniqueFactorizationMonoid.multiplicative_of_coprime", "start": [1161, 1], "end": [1211, 64], "traced_tactics": [{"tactic": "letI := Classical.decEq \u03b1", "annotated_tactic": ["letI := <a>Classical.decEq</a> \u03b1", [{"full_name": "Classical.decEq", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1020, 19], "def_end_pos": [1020, 24]}]], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\n\u22a2 f (a * b) = f a * f b", "state_after": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\n\u22a2 f (a * b) = f a * f b"}, {"tactic": "by_cases ha0 : a = 0", "annotated_tactic": ["by_cases ha0 : a = 0", []], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\n\u22a2 f (a * b) = f a * f b", "state_after": "case pos\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : a = 0\n\u22a2 f (a * b) = f a * f b\n\ncase neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\n\u22a2 f (a * b) = f a * f b"}, {"tactic": "by_cases hb0 : b = 0", "annotated_tactic": ["by_cases hb0 : b = 0", []], "state_before": "case neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\n\u22a2 f (a * b) = f a * f b", "state_after": "case pos\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : b = 0\n\u22a2 f (a * b) = f a * f b\n\ncase neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\n\u22a2 f (a * b) = f a * f b"}, {"tactic": "by_cases hf1 : f 1 = 0", "annotated_tactic": ["by_cases hf1 : f 1 = 0", []], "state_before": "case neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\n\u22a2 f (a * b) = f a * f b", "state_after": "case pos\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : f 1 = 0\n\u22a2 f (a * b) = f a * f b\n\ncase neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\n\u22a2 f (a * b) = f a * f b"}, {"tactic": "haveI : Nontrivial \u03b1 := \u27e8\u27e8_, _, ha0\u27e9\u27e9", "annotated_tactic": ["haveI : <a>Nontrivial</a> \u03b1 := \u27e8\u27e8_, _, ha0\u27e9\u27e9", [{"full_name": "Nontrivial", "def_path": "Mathlib/Logic/Nontrivial/Defs.lean", "def_pos": [31, 7], "def_end_pos": [31, 17]}]], "state_before": "case neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\n\u22a2 f (a * b) = f a * f b", "state_after": "case neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis : Nontrivial \u03b1\n\u22a2 f (a * b) = f a * f b"}, {"tactic": "letI : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid", "annotated_tactic": ["letI : <a>NormalizationMonoid</a> \u03b1 := <a>UniqueFactorizationMonoid.normalizationMonoid</a>", [{"full_name": "NormalizationMonoid", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [73, 7], "def_end_pos": [73, 26]}, {"full_name": "UniqueFactorizationMonoid.normalizationMonoid", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [877, 29], "def_end_pos": [877, 48]}]], "state_before": "case neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis : Nontrivial \u03b1\n\u22a2 f (a * b) = f a * f b", "state_after": "case neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 f (a * b) = f a * f b"}, {"tactic": "refine multiplicative_prime_power _ _ _ ?_ ?_ @h1 @hpr @hcp", "annotated_tactic": ["refine <a>multiplicative_prime_power</a> _ _ _ ?_ ?_ @h1 @hpr @hcp", [{"full_name": "UniqueFactorizationMonoid.multiplicative_prime_power", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 35]}]], "state_before": "case neg\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))", "state_after": "case neg.refine_1\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 \u2200 p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset, Prime p\n\ncase neg.refine_2\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 \u2200 p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n    \u2200 q \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset, p \u2223 q \u2192 p = q"}, {"tactic": "all_goals simp only [Multiset.mem_toFinset, Finset.mem_union]", "annotated_tactic": ["all_goals simp only [<a>Multiset.mem_toFinset</a>, <a>Finset.mem_union</a>]", [{"full_name": "Multiset.mem_toFinset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3100, 9], "def_end_pos": [3100, 21]}, {"full_name": "Finset.mem_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 18]}]], "state_before": "case neg.refine_1\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 \u2200 p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset, Prime p\n\ncase neg.refine_2\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 \u2200 p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n    \u2200 q \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset, p \u2223 q \u2192 p = q", "state_after": "case neg.refine_1\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 \u2200 (p : \u03b1), p \u2208 normalizedFactors a \u2228 p \u2208 normalizedFactors b \u2192 Prime p\n\ncase neg.refine_2\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 \u2200 (p : \u03b1),\n    p \u2208 normalizedFactors a \u2228 p \u2208 normalizedFactors b \u2192\n      \u2200 (q : \u03b1), q \u2208 normalizedFactors a \u2228 q \u2208 normalizedFactors b \u2192 p \u2223 q \u2192 p = q"}, {"tactic": "rw [ha0, zero_mul, h0, zero_mul]", "annotated_tactic": ["rw [ha0, <a>zero_mul</a>, h0, <a>zero_mul</a>]", [{"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [37, 3], "def_end_pos": [37, 11]}]], "state_before": "case pos\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : a = 0\n\u22a2 f (a * b) = f a * f b", "state_after": "no goals"}, {"tactic": "rw [hb0, mul_zero, h0, mul_zero]", "annotated_tactic": ["rw [hb0, <a>mul_zero</a>, h0, <a>mul_zero</a>]", [{"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}]], "state_before": "case pos\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : b = 0\n\u22a2 f (a * b) = f a * f b", "state_after": "no goals"}, {"tactic": "calc\n  f (a * b) = f (a * b * 1) := by rw [mul_one]\n  _ = 0 := by simp only [h1 isUnit_one, hf1, mul_zero]\n  _ = f a * f (b * 1) := by simp only [h1 isUnit_one, hf1, mul_zero]\n  _ = f a * f b := by rw [mul_one]", "annotated_tactic": ["calc\n      f (a * b) = f (a * b * 1) := by rw [<a>mul_one</a>]\n      _ = 0 := by simp only [h1 <a>isUnit_one</a>, hf1, <a>mul_zero</a>]\n      _ = f a * f (b * 1) := by simp only [h1 <a>isUnit_one</a>, hf1, <a>mul_zero</a>]\n      _ = f a * f b := by rw [<a>mul_one</a>]", [{"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "isUnit_one", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [695, 9], "def_end_pos": [695, 19]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}, {"full_name": "isUnit_one", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [695, 9], "def_end_pos": [695, 19]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "case pos\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : f 1 = 0\n\u22a2 f (a * b) = f a * f b", "state_after": "no goals"}, {"tactic": "rw [mul_one]", "annotated_tactic": ["rw [<a>mul_one</a>]", [{"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : f 1 = 0\n\u22a2 f (a * b) = f (a * b * 1)", "state_after": "no goals"}, {"tactic": "simp only [h1 isUnit_one, hf1, mul_zero]", "annotated_tactic": ["simp only [h1 <a>isUnit_one</a>, hf1, <a>mul_zero</a>]", [{"full_name": "isUnit_one", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [695, 9], "def_end_pos": [695, 19]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}]], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : f 1 = 0\n\u22a2 f (a * b * 1) = 0", "state_after": "no goals"}, {"tactic": "simp only [h1 isUnit_one, hf1, mul_zero]", "annotated_tactic": ["simp only [h1 <a>isUnit_one</a>, hf1, <a>mul_zero</a>]", [{"full_name": "isUnit_one", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [695, 9], "def_end_pos": [695, 19]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}]], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : f 1 = 0\n\u22a2 0 = f a * f (b * 1)", "state_after": "no goals"}, {"tactic": "rw [mul_one]", "annotated_tactic": ["rw [<a>mul_one</a>]", [{"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : f 1 = 0\n\u22a2 f a * f (b * 1) = f a * f b", "state_after": "no goals"}, {"tactic": "obtain \u27e8ua, a_eq\u27e9 := normalizedFactors_prod ha0", "annotated_tactic": ["obtain \u27e8ua, a_eq\u27e9 := <a>normalizedFactors_prod</a> ha0", [{"full_name": "UniqueFactorizationMonoid.normalizedFactors_prod", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [604, 9], "def_end_pos": [604, 31]}]], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\n\u22a2 f (a * b) = f a * f b", "state_after": "case intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\n\u22a2 f (a * b) = f a * f b"}, {"tactic": "obtain \u27e8ub, b_eq\u27e9 := normalizedFactors_prod hb0", "annotated_tactic": ["obtain \u27e8ub, b_eq\u27e9 := <a>normalizedFactors_prod</a> hb0", [{"full_name": "UniqueFactorizationMonoid.normalizedFactors_prod", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [604, 9], "def_end_pos": [604, 31]}]], "state_before": "case intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\n\u22a2 f (a * b) = f a * f b", "state_after": "case intro.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\n\u22a2 f (a * b) = f a * f b"}, {"tactic": "rw [\u2190 a_eq, \u2190 b_eq, mul_right_comm (Multiset.prod (normalizedFactors a)) ua\n    (Multiset.prod (normalizedFactors b) * ub), h1 ua.isUnit, h1 ub.isUnit, h1 ua.isUnit, \u2190\n  mul_assoc, h1 ub.isUnit, mul_right_comm _ (f ua), \u2190 mul_assoc]", "annotated_tactic": ["rw [\u2190 a_eq, \u2190 b_eq, <a>mul_right_comm</a> (<a>Multiset.prod</a> (<a>normalizedFactors</a> a)) ua\n        (<a>Multiset.prod</a> (<a>normalizedFactors</a> b) * ub), h1 ua.isUnit, h1 ub.isUnit, h1 ua.isUnit, \u2190\n      <a>mul_assoc</a>, h1 ub.isUnit, <a>mul_right_comm</a> _ (f ua), \u2190 <a>mul_assoc</a>]", [{"full_name": "mul_right_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [190, 9], "def_end_pos": [190, 23]}, {"full_name": "Multiset.prod", "def_path": "Mathlib/Algebra/BigOperators/Group/Multiset.lean", "def_pos": [40, 5], "def_end_pos": [40, 9]}, {"full_name": "UniqueFactorizationMonoid.normalizedFactors", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [589, 19], "def_end_pos": [589, 36]}, {"full_name": "Multiset.prod", "def_path": "Mathlib/Algebra/BigOperators/Group/Multiset.lean", "def_pos": [40, 5], "def_end_pos": [40, 9]}, {"full_name": "UniqueFactorizationMonoid.normalizedFactors", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [589, 19], "def_end_pos": [589, 36]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "mul_right_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [190, 9], "def_end_pos": [190, 23]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\n\u22a2 f (a * b) = f a * f b", "state_after": "case intro.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\n\u22a2 f ((normalizedFactors a).prod * (normalizedFactors b).prod) * f \u2191ub * f \u2191ua =\n    f (normalizedFactors a).prod * f (normalizedFactors b).prod * f \u2191ub * f \u2191ua"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\n\u22a2 f ((normalizedFactors a).prod * (normalizedFactors b).prod) * f \u2191ub * f \u2191ua =\n    f (normalizedFactors a).prod * f (normalizedFactors b).prod * f \u2191ub * f \u2191ua", "state_after": "case intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\n\u22a2 f ((normalizedFactors a).prod * (normalizedFactors b).prod) =\n    f (normalizedFactors a).prod * f (normalizedFactors b).prod"}, {"tactic": "rw [\u2190 (normalizedFactors a).map_id, \u2190 (normalizedFactors b).map_id,\n  Finset.prod_multiset_map_count, Finset.prod_multiset_map_count,\n  Finset.prod_subset (Finset.subset_union_left (s\u2082:=(normalizedFactors b).toFinset)),\n  Finset.prod_subset (Finset.subset_union_right (s\u2082:=(normalizedFactors b).toFinset)), \u2190\n  Finset.prod_mul_distrib]", "annotated_tactic": ["rw [\u2190 (<a>normalizedFactors</a> a).<a>map_id</a>, \u2190 (<a>normalizedFactors</a> b).<a>map_id</a>,\n      <a>Finset.prod_multiset_map_count</a>, <a>Finset.prod_multiset_map_count</a>,\n      <a>Finset.prod_subset</a> (<a>Finset.subset_union_left</a> (s\u2082:=(<a>normalizedFactors</a> b).<a>toFinset</a>)),\n      <a>Finset.prod_subset</a> (<a>Finset.subset_union_right</a> (s\u2082:=(<a>normalizedFactors</a> b).<a>toFinset</a>)), \u2190\n      <a>Finset.prod_mul_distrib</a>]", [{"full_name": "UniqueFactorizationMonoid.normalizedFactors", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [589, 19], "def_end_pos": [589, 36]}, {"full_name": "Multiset.map_id", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1336, 9], "def_end_pos": [1336, 15]}, {"full_name": "UniqueFactorizationMonoid.normalizedFactors", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [589, 19], "def_end_pos": [589, 36]}, {"full_name": "Multiset.map_id", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1336, 9], "def_end_pos": [1336, 15]}, {"full_name": "Finset.prod_multiset_map_count", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [1618, 9], "def_end_pos": [1618, 32]}, {"full_name": "Finset.prod_multiset_map_count", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [1618, 9], "def_end_pos": [1618, 32]}, {"full_name": "Finset.prod_subset", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [595, 9], "def_end_pos": [595, 20]}, {"full_name": "Finset.subset_union_left", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1438, 9], "def_end_pos": [1438, 26]}, {"full_name": "UniqueFactorizationMonoid.normalizedFactors", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [589, 19], "def_end_pos": [589, 36]}, {"full_name": "Multiset.toFinset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3081, 5], "def_end_pos": [3081, 13]}, {"full_name": "Finset.prod_subset", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [595, 9], "def_end_pos": [595, 20]}, {"full_name": "Finset.subset_union_right", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 27]}, {"full_name": "UniqueFactorizationMonoid.normalizedFactors", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [589, 19], "def_end_pos": [589, 36]}, {"full_name": "Multiset.toFinset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3081, 5], "def_end_pos": [3081, 13]}, {"full_name": "Finset.prod_mul_distrib", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [877, 9], "def_end_pos": [877, 25]}]], "state_before": "case intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\n\u22a2 f ((normalizedFactors a).prod * (normalizedFactors b).prod) =\n    f (normalizedFactors a).prod * f (normalizedFactors b).prod", "state_after": "case intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\n\u22a2 f\n      (\u220f x \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        id x ^ Multiset.count x (normalizedFactors a) * id x ^ Multiset.count x (normalizedFactors b)) =\n    f\n        (\u220f x \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          id x ^ Multiset.count x (normalizedFactors a)) *\n      f\n        (\u220f x \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          id x ^ Multiset.count x (normalizedFactors b))\n\ncase intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\n\u22a2 \u2200 x \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n    x \u2209 (normalizedFactors b).toFinset \u2192 id x ^ Multiset.count x (normalizedFactors b) = 1\n\ncase intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\n\u22a2 \u2200 x \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n    x \u2209 (normalizedFactors a).toFinset \u2192 id x ^ Multiset.count x (normalizedFactors a) = 1"}, {"tactic": "all_goals simp only [Multiset.mem_toFinset]", "annotated_tactic": ["all_goals simp only [<a>Multiset.mem_toFinset</a>]", [{"full_name": "Multiset.mem_toFinset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3100, 9], "def_end_pos": [3100, 21]}]], "state_before": "case intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\n\u22a2 \u2200 x \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n    x \u2209 (normalizedFactors b).toFinset \u2192 id x ^ Multiset.count x (normalizedFactors b) = 1\n\ncase intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\n\u22a2 \u2200 x \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n    x \u2209 (normalizedFactors a).toFinset \u2192 id x ^ Multiset.count x (normalizedFactors a) = 1", "state_after": "case intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\n\u22a2 \u2200 x \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n    x \u2209 normalizedFactors b \u2192 id x ^ Multiset.count x (normalizedFactors b) = 1\n\ncase intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\n\u22a2 \u2200 x \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n    x \u2209 normalizedFactors a \u2192 id x ^ Multiset.count x (normalizedFactors a) = 1"}, {"tactic": "simp_rw [id, \u2190 pow_add, this]", "annotated_tactic": ["simp_rw [<a>id</a>, \u2190 <a>pow_add</a>, this]", [{"full_name": "id", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}, {"full_name": "pow_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [703, 7], "def_end_pos": [703, 14]}]], "state_before": "case intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\n\u22a2 f\n      (\u220f x \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        id x ^ Multiset.count x (normalizedFactors a) * id x ^ Multiset.count x (normalizedFactors b)) =\n    f\n        (\u220f x \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          id x ^ Multiset.count x (normalizedFactors a)) *\n      f\n        (\u220f x \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          id x ^ Multiset.count x (normalizedFactors b))", "state_after": "no goals"}, {"tactic": "simp only [Multiset.mem_toFinset]", "annotated_tactic": ["simp only [<a>Multiset.mem_toFinset</a>]", [{"full_name": "Multiset.mem_toFinset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3100, 9], "def_end_pos": [3100, 21]}]], "state_before": "case intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\n\u22a2 \u2200 x \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n    x \u2209 (normalizedFactors a).toFinset \u2192 id x ^ Multiset.count x (normalizedFactors a) = 1", "state_after": "case intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\n\u22a2 \u2200 x \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n    x \u2209 normalizedFactors a \u2192 id x ^ Multiset.count x (normalizedFactors a) = 1"}, {"tactic": "intro p _ hpb", "annotated_tactic": ["intro p _ hpb", []], "state_before": "case intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\n\u22a2 \u2200 x \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n    x \u2209 normalizedFactors b \u2192 id x ^ Multiset.count x (normalizedFactors b) = 1", "state_after": "case intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\np : \u03b1\na\u271d : p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset\nhpb : p \u2209 normalizedFactors b\n\u22a2 id p ^ Multiset.count p (normalizedFactors b) = 1"}, {"tactic": "simp [hpb]", "annotated_tactic": ["simp [hpb]", []], "state_before": "case intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\np : \u03b1\na\u271d : p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset\nhpb : p \u2209 normalizedFactors b\n\u22a2 id p ^ Multiset.count p (normalizedFactors b) = 1", "state_after": "no goals"}, {"tactic": "intro p _ hpa", "annotated_tactic": ["intro p _ hpa", []], "state_before": "case intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\n\u22a2 \u2200 x \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n    x \u2209 normalizedFactors a \u2192 id x ^ Multiset.count x (normalizedFactors a) = 1", "state_after": "case intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\np : \u03b1\na\u271d : p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset\nhpa : p \u2209 normalizedFactors a\n\u22a2 id p ^ Multiset.count p (normalizedFactors a) = 1"}, {"tactic": "simp [hpa]", "annotated_tactic": ["simp [hpa]", []], "state_before": "case intro.intro.e_a.e_a\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b2 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d\u00b9 : Nontrivial \u03b1\nthis\u271d : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\nthis :\n  f\n      (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n        p ^ (Multiset.count p (normalizedFactors a) + Multiset.count p (normalizedFactors b))) =\n    f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors a)) *\n      f\n        (\u220f p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n          p ^ Multiset.count p (normalizedFactors b))\nua : \u03b1\u02e3\na_eq : (normalizedFactors a).prod * \u2191ua = a\nub : \u03b1\u02e3\nb_eq : (normalizedFactors b).prod * \u2191ub = b\np : \u03b1\na\u271d : p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset\nhpa : p \u2209 normalizedFactors a\n\u22a2 id p ^ Multiset.count p (normalizedFactors a) = 1", "state_after": "no goals"}, {"tactic": "simp only [Multiset.mem_toFinset, Finset.mem_union]", "annotated_tactic": ["simp only [<a>Multiset.mem_toFinset</a>, <a>Finset.mem_union</a>]", [{"full_name": "Multiset.mem_toFinset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3100, 9], "def_end_pos": [3100, 21]}, {"full_name": "Finset.mem_union", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 18]}]], "state_before": "case neg.refine_2\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 \u2200 p \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset,\n    \u2200 q \u2208 (normalizedFactors a).toFinset \u222a (normalizedFactors b).toFinset, p \u2223 q \u2192 p = q", "state_after": "case neg.refine_2\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 \u2200 (p : \u03b1),\n    p \u2208 normalizedFactors a \u2228 p \u2208 normalizedFactors b \u2192\n      \u2200 (q : \u03b1), q \u2208 normalizedFactors a \u2228 q \u2208 normalizedFactors b \u2192 p \u2223 q \u2192 p = q"}, {"tactic": "rintro p (hpa | hpb) <;> apply prime_of_normalized_factor <;> assumption", "annotated_tactic": ["rintro p (hpa | hpb) <;> apply <a>prime_of_normalized_factor</a> <;> assumption", [{"full_name": "UniqueFactorizationMonoid.prime_of_normalized_factor", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [615, 9], "def_end_pos": [615, 35]}]], "state_before": "case neg.refine_1\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 \u2200 (p : \u03b1), p \u2208 normalizedFactors a \u2228 p \u2208 normalizedFactors b \u2192 Prime p", "state_after": "no goals"}, {"tactic": "rintro p (hp | hp) q (hq | hq) hdvd <;>\n  rw [\u2190 normalize_normalized_factor _ hp, \u2190 normalize_normalized_factor _ hq] <;>\n  exact\n    normalize_eq_normalize hdvd\n      ((prime_of_normalized_factor _ hp).irreducible.dvd_symm\n        (prime_of_normalized_factor _ hq).irreducible hdvd)", "annotated_tactic": ["rintro p (hp | hp) q (hq | hq) hdvd <;>\n      rw [\u2190 <a>normalize_normalized_factor</a> _ hp, \u2190 <a>normalize_normalized_factor</a> _ hq] <;>\n      exact\n        <a>normalize_eq_normalize</a> hdvd\n          ((<a>prime_of_normalized_factor</a> _ hp).irreducible.dvd_symm\n            (<a>prime_of_normalized_factor</a> _ hq).<a>irreducible</a> hdvd)", [{"full_name": "UniqueFactorizationMonoid.normalize_normalized_factor", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [628, 9], "def_end_pos": [628, 36]}, {"full_name": "UniqueFactorizationMonoid.normalize_normalized_factor", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [628, 9], "def_end_pos": [628, 36]}, {"full_name": "normalize_eq_normalize", "def_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "def_pos": [172, 9], "def_end_pos": [172, 31]}, {"full_name": "UniqueFactorizationMonoid.prime_of_normalized_factor", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [615, 9], "def_end_pos": [615, 35]}, {"full_name": "UniqueFactorizationMonoid.prime_of_normalized_factor", "def_path": "Mathlib/RingTheory/UniqueFactorizationDomain.lean", "def_pos": [615, 9], "def_end_pos": [615, 35]}, {"full_name": "Prime.irreducible", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [357, 19], "def_end_pos": [357, 36]}]], "state_before": "case neg.refine_2\n\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2074 : CancelCommMonoidWithZero R\ninst\u271d\u00b3 : UniqueFactorizationMonoid R\ninst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1\ninst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1\n\u03b2 : Type u_3\ninst\u271d : CancelCommMonoidWithZero \u03b2\nf : \u03b1 \u2192 \u03b2\na b : \u03b1\nh0 : f 0 = 0\nh1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y\nhpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i\nhcp : \u2200 {x y : \u03b1}, IsRelPrime x y \u2192 f (x * y) = f x * f y\nthis\u271d\u00b9 : DecidableEq \u03b1 := Classical.decEq \u03b1\nha0 : \u00aca = 0\nhb0 : \u00acb = 0\nhf1 : \u00acf 1 = 0\nthis\u271d : Nontrivial \u03b1\nthis : NormalizationMonoid \u03b1 := UniqueFactorizationMonoid.normalizationMonoid\n\u22a2 \u2200 (p : \u03b1),\n    p \u2208 normalizedFactors a \u2228 p \u2208 normalizedFactors b \u2192\n      \u2200 (q : \u03b1), q \u2208 normalizedFactors a \u2228 q \u2208 normalizedFactors b \u2192 p \u2223 q \u2192 p = q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean", "full_name": "CircleDeg1Lift.translationNumber_of_map_pow_eq_add_int", "start": [891, 1], "end": [895, 41], "traced_tactics": [{"tactic": "have := (f ^ n).translationNumber_of_eq_add_int h", "annotated_tactic": ["have := (f ^ n).<a>translationNumber_of_eq_add_int</a> h", [{"full_name": "CircleDeg1Lift.translationNumber_of_eq_add_int", "def_path": "Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean", "def_pos": [845, 9], "def_end_pos": [845, 40]}]], "state_before": "f g : CircleDeg1Lift\nx : \u211d\nn : \u2115\nm : \u2124\nh : (f ^ n) x = x + \u2191m\nhn : 0 < n\n\u22a2 \u03c4 f = \u2191m / \u2191n", "state_after": "f g : CircleDeg1Lift\nx : \u211d\nn : \u2115\nm : \u2124\nh : (f ^ n) x = x + \u2191m\nhn : 0 < n\nthis : \u03c4 (f ^ n) = \u2191m\n\u22a2 \u03c4 f = \u2191m / \u2191n"}, {"tactic": "rwa [translationNumber_pow, mul_comm, \u2190 eq_div_iff] at this", "annotated_tactic": ["rwa [<a>translationNumber_pow</a>, <a>mul_comm</a>, \u2190 <a>eq_div_iff</a>] at this", [{"full_name": "CircleDeg1Lift.translationNumber_pow", "def_path": "Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean", "def_pos": [743, 9], "def_end_pos": [743, 30]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "eq_div_iff", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [357, 22], "def_end_pos": [357, 32]}]], "state_before": "f g : CircleDeg1Lift\nx : \u211d\nn : \u2115\nm : \u2124\nh : (f ^ n) x = x + \u2191m\nhn : 0 < n\nthis : \u03c4 (f ^ n) = \u2191m\n\u22a2 \u03c4 f = \u2191m / \u2191n", "state_after": "f g : CircleDeg1Lift\nx : \u211d\nn : \u2115\nm : \u2124\nh : (f ^ n) x = x + \u2191m\nhn : 0 < n\nthis : \u03c4 f * \u2191n = \u2191m\n\u22a2 \u2191n \u2260 0"}, {"tactic": "exact Nat.cast_ne_zero.2 (ne_of_gt hn)", "annotated_tactic": ["exact <a>Nat.cast_ne_zero</a>.2 (<a>ne_of_gt</a> hn)", [{"full_name": "Nat.cast_ne_zero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [83, 9], "def_end_pos": [83, 21]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}]], "state_before": "f g : CircleDeg1Lift\nx : \u211d\nn : \u2115\nm : \u2124\nh : (f ^ n) x = x + \u2191m\nhn : 0 < n\nthis : \u03c4 f * \u2191n = \u2191m\n\u22a2 \u2191n \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Subsemiring/Basic.lean", "full_name": "Subsemiring.mem_closure_iff_exists_list", "start": [899, 1], "end": [924, 75], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "R\u271d : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b3 : NonAssocSemiring R\u271d\nM : Submonoid R\u271d\ninst\u271d\u00b2 : NonAssocSemiring S\ninst\u271d\u00b9 : NonAssocSemiring T\nR : Type u_1\ninst\u271d : Semiring R\ns : Set R\nx : R\n\u22a2 x \u2208 closure s \u2194 \u2203 L, (\u2200 t \u2208 L, \u2200 y \u2208 t, y \u2208 s) \u2227 (List.map List.prod L).sum = x", "state_after": "case mp\nR\u271d : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b3 : NonAssocSemiring R\u271d\nM : Submonoid R\u271d\ninst\u271d\u00b2 : NonAssocSemiring S\ninst\u271d\u00b9 : NonAssocSemiring T\nR : Type u_1\ninst\u271d : Semiring R\ns : Set R\nx : R\n\u22a2 x \u2208 closure s \u2192 \u2203 L, (\u2200 t \u2208 L, \u2200 y \u2208 t, y \u2208 s) \u2227 (List.map List.prod L).sum = x\n\ncase mpr\nR\u271d : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b3 : NonAssocSemiring R\u271d\nM : Submonoid R\u271d\ninst\u271d\u00b2 : NonAssocSemiring S\ninst\u271d\u00b9 : NonAssocSemiring T\nR : Type u_1\ninst\u271d : Semiring R\ns : Set R\nx : R\n\u22a2 (\u2203 L, (\u2200 t \u2208 L, \u2200 y \u2208 t, y \u2208 s) \u2227 (List.map List.prod L).sum = x) \u2192 x \u2208 closure s"}, {"tactic": "intro hx", "annotated_tactic": ["intro hx", []], "state_before": "case mp\nR\u271d : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b3 : NonAssocSemiring R\u271d\nM : Submonoid R\u271d\ninst\u271d\u00b2 : NonAssocSemiring S\ninst\u271d\u00b9 : NonAssocSemiring T\nR : Type u_1\ninst\u271d : Semiring R\ns : Set R\nx : R\n\u22a2 x \u2208 closure s \u2192 \u2203 L, (\u2200 t \u2208 L, \u2200 y \u2208 t, y \u2208 s) \u2227 (List.map List.prod L).sum = x", "state_after": "case mp\nR\u271d : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b3 : NonAssocSemiring R\u271d\nM : Submonoid R\u271d\ninst\u271d\u00b2 : NonAssocSemiring S\ninst\u271d\u00b9 : NonAssocSemiring T\nR : Type u_1\ninst\u271d : Semiring R\ns : Set R\nx : R\nhx : x \u2208 closure s\n\u22a2 \u2203 L, (\u2200 t \u2208 L, \u2200 y \u2208 t, y \u2208 s) \u2227 (List.map List.prod L).sum = x"}, {"tactic": "let p : R \u2192 Prop := fun x =>\n  \u2203 (L : List (List R)),\n    (\u2200 (t : List R), t \u2208 L \u2192 \u2200 (y : R), y \u2208 t \u2192 y \u2208 s) \u2227 (List.map List.prod L).sum = x", "annotated_tactic": ["let p : R \u2192 Prop := fun x =>\n      \u2203 (L : <a>List</a> (<a>List</a> R)),\n        (\u2200 (t : <a>List</a> R), t \u2208 L \u2192 \u2200 (y : R), y \u2208 t \u2192 y \u2208 s) \u2227 (<a>List.map</a> <a>List.prod</a> L).<a>sum</a> = x", [{"full_name": "List", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2284, 11], "def_end_pos": [2284, 15]}, {"full_name": "List", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2284, 11], "def_end_pos": [2284, 15]}, {"full_name": "List", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2284, 11], "def_end_pos": [2284, 15]}, {"full_name": "List.map", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [361, 19], "def_end_pos": [361, 22]}, {"full_name": "List.prod", "def_path": "Mathlib/Algebra/BigOperators/Group/List.lean", "def_pos": [39, 5], "def_end_pos": [39, 9]}, {"full_name": "List.sum", "def_path": "Mathlib/Algebra/BigOperators/Group/List.lean", "def_pos": [38, 3], "def_end_pos": [38, 14]}]], "state_before": "case mp\nR\u271d : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b3 : NonAssocSemiring R\u271d\nM : Submonoid R\u271d\ninst\u271d\u00b2 : NonAssocSemiring S\ninst\u271d\u00b9 : NonAssocSemiring T\nR : Type u_1\ninst\u271d : Semiring R\ns : Set R\nx : R\nhx : x \u2208 closure s\n\u22a2 \u2203 L, (\u2200 t \u2208 L, \u2200 y \u2208 t, y \u2208 s) \u2227 (List.map List.prod L).sum = x", "state_after": "case mp\nR\u271d : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b3 : NonAssocSemiring R\u271d\nM : Submonoid R\u271d\ninst\u271d\u00b2 : NonAssocSemiring S\ninst\u271d\u00b9 : NonAssocSemiring T\nR : Type u_1\ninst\u271d : Semiring R\ns : Set R\nx : R\nhx : x \u2208 closure s\np : R \u2192 Prop := fun x => \u2203 L, (\u2200 t \u2208 L, \u2200 y \u2208 t, y \u2208 s) \u2227 (List.map List.prod L).sum = x\n\u22a2 \u2203 L, (\u2200 t \u2208 L, \u2200 y \u2208 t, y \u2208 s) \u2227 (List.map List.prod L).sum = x"}, {"tactic": "exact AddSubmonoid.closure_induction (p := p) (mem_closure_iff.1 hx)\n  (fun x hx =>\n    suffices \u2203 t : List R, (\u2200 y \u2208 t, y \u2208 s) \u2227 t.prod = x from\n      let \u27e8t, ht1, ht2\u27e9 := this\n      \u27e8[t], List.forall_mem_singleton.2 ht1, by\n        rw [List.map_singleton, List.sum_singleton, ht2]\u27e9\n    Submonoid.closure_induction hx\n      (fun x hx => \u27e8[x], List.forall_mem_singleton.2 hx, one_mul x\u27e9)\n      \u27e8[], List.forall_mem_nil _, rfl\u27e9 fun x y \u27e8t, ht1, ht2\u27e9 \u27e8u, hu1, hu2\u27e9 =>\n      \u27e8t ++ u, List.forall_mem_append.2 \u27e8ht1, hu1\u27e9, by rw [List.prod_append, ht2, hu2]\u27e9)\n  \u27e8[], List.forall_mem_nil _, rfl\u27e9 fun x y \u27e8L, HL1, HL2\u27e9 \u27e8M, HM1, HM2\u27e9 =>\n  \u27e8L ++ M, List.forall_mem_append.2 \u27e8HL1, HM1\u27e9, by\n    rw [List.map_append, List.sum_append, HL2, HM2]\u27e9", "annotated_tactic": ["exact <a>AddSubmonoid.closure_induction</a> (p := p) (<a>mem_closure_iff</a>.1 hx)\n      (fun x hx =>\n        suffices \u2203 t : <a>List</a> R, (\u2200 y \u2208 t, y \u2208 s) \u2227 t.prod = x from\n          let \u27e8t, ht1, ht2\u27e9 := this\n          \u27e8[t], <a>List.forall_mem_singleton</a>.2 ht1, by\n            rw [<a>List.map_singleton</a>, <a>List.sum_singleton</a>, ht2]\u27e9\n        <a>Submonoid.closure_induction</a> hx\n          (fun x hx => \u27e8[x], <a>List.forall_mem_singleton</a>.2 hx, <a>one_mul</a> x\u27e9)\n          \u27e8[], <a>List.forall_mem_nil</a> _, <a>rfl</a>\u27e9 fun x y \u27e8t, ht1, ht2\u27e9 \u27e8u, hu1, hu2\u27e9 =>\n          \u27e8t ++ u, <a>List.forall_mem_append</a>.2 \u27e8ht1, hu1\u27e9, by rw [<a>List.prod_append</a>, ht2, hu2]\u27e9)\n      \u27e8[], <a>List.forall_mem_nil</a> _, <a>rfl</a>\u27e9 fun x y \u27e8L, HL1, HL2\u27e9 \u27e8M, HM1, HM2\u27e9 =>\n      \u27e8L ++ M, <a>List.forall_mem_append</a>.2 \u27e8HL1, HM1\u27e9, by\n        rw [<a>List.map_append</a>, <a>List.sum_append</a>, HL2, HM2]\u27e9", [{"full_name": "AddSubmonoid.closure_induction", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [441, 3], "def_end_pos": [441, 14]}, {"full_name": "Subsemiring.mem_closure_iff", "def_path": "Mathlib/Algebra/Ring/Subsemiring/Basic.lean", "def_pos": [841, 9], "def_end_pos": [841, 24]}, {"full_name": "List", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2284, 11], "def_end_pos": [2284, 15]}, {"full_name": "List.forall_mem_singleton", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [317, 9], "def_end_pos": [317, 29]}, {"full_name": "List.map_singleton", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [723, 9], "def_end_pos": [723, 22]}, {"full_name": "List.sum_singleton", "def_path": "Mathlib/Algebra/BigOperators/Group/List.lean", "def_pos": [71, 3], "def_end_pos": [71, 14]}, {"full_name": "Submonoid.closure_induction", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 26]}, {"full_name": "List.forall_mem_singleton", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [317, 9], "def_end_pos": [317, 29]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "List.forall_mem_nil", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [312, 9], "def_end_pos": [312, 23]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "List.forall_mem_append", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [1183, 9], "def_end_pos": [1183, 26]}, {"full_name": "List.prod_append", "def_path": "Mathlib/Algebra/BigOperators/Group/List.lean", "def_pos": [114, 9], "def_end_pos": [114, 20]}, {"full_name": "List.forall_mem_nil", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [312, 9], "def_end_pos": [312, 23]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "List.forall_mem_append", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [1183, 9], "def_end_pos": [1183, 26]}, {"full_name": "List.map_append", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [839, 17], "def_end_pos": [839, 27]}, {"full_name": "List.sum_append", "def_path": "Mathlib/Algebra/BigOperators/Group/List.lean", "def_pos": [113, 3], "def_end_pos": [113, 14]}]], "state_before": "case mp\nR\u271d : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b3 : NonAssocSemiring R\u271d\nM : Submonoid R\u271d\ninst\u271d\u00b2 : NonAssocSemiring S\ninst\u271d\u00b9 : NonAssocSemiring T\nR : Type u_1\ninst\u271d : Semiring R\ns : Set R\nx : R\nhx : x \u2208 closure s\np : R \u2192 Prop := fun x => \u2203 L, (\u2200 t \u2208 L, \u2200 y \u2208 t, y \u2208 s) \u2227 (List.map List.prod L).sum = x\n\u22a2 \u2203 L, (\u2200 t \u2208 L, \u2200 y \u2208 t, y \u2208 s) \u2227 (List.map List.prod L).sum = x", "state_after": "no goals"}, {"tactic": "rw [List.map_singleton, List.sum_singleton, ht2]", "annotated_tactic": ["rw [<a>List.map_singleton</a>, <a>List.sum_singleton</a>, ht2]", [{"full_name": "List.map_singleton", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [723, 9], "def_end_pos": [723, 22]}, {"full_name": "List.sum_singleton", "def_path": "Mathlib/Algebra/BigOperators/Group/List.lean", "def_pos": [71, 3], "def_end_pos": [71, 14]}]], "state_before": "R\u271d : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b3 : NonAssocSemiring R\u271d\nM : Submonoid R\u271d\ninst\u271d\u00b2 : NonAssocSemiring S\ninst\u271d\u00b9 : NonAssocSemiring T\nR : Type u_1\ninst\u271d : Semiring R\ns : Set R\nx\u271d : R\nhx\u271d : x\u271d \u2208 closure s\np : R \u2192 Prop := fun x => \u2203 L, (\u2200 t \u2208 L, \u2200 y \u2208 t, y \u2208 s) \u2227 (List.map List.prod L).sum = x\nx : R\nhx : x \u2208 \u2191(Submonoid.closure s)\nthis : \u2203 t, (\u2200 y \u2208 t, y \u2208 s) \u2227 t.prod = x\nt : List R\nht1 : \u2200 y \u2208 t, y \u2208 s\nht2 : t.prod = x\n\u22a2 (List.map List.prod [t]).sum = x", "state_after": "no goals"}, {"tactic": "rw [List.prod_append, ht2, hu2]", "annotated_tactic": ["rw [<a>List.prod_append</a>, ht2, hu2]", [{"full_name": "List.prod_append", "def_path": "Mathlib/Algebra/BigOperators/Group/List.lean", "def_pos": [114, 9], "def_end_pos": [114, 20]}]], "state_before": "R\u271d : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b3 : NonAssocSemiring R\u271d\nM : Submonoid R\u271d\ninst\u271d\u00b2 : NonAssocSemiring S\ninst\u271d\u00b9 : NonAssocSemiring T\nR : Type u_1\ninst\u271d : Semiring R\ns : Set R\nx\u271d\u00b3 : R\nhx\u271d : x\u271d\u00b3 \u2208 closure s\np : R \u2192 Prop := fun x => \u2203 L, (\u2200 t \u2208 L, \u2200 y \u2208 t, y \u2208 s) \u2227 (List.map List.prod L).sum = x\nx\u271d\u00b2 : R\nhx : x\u271d\u00b2 \u2208 \u2191(Submonoid.closure s)\nx y : R\nx\u271d\u00b9 : \u2203 t, (\u2200 y \u2208 t, y \u2208 s) \u2227 t.prod = x\nx\u271d : \u2203 t, (\u2200 y \u2208 t, y \u2208 s) \u2227 t.prod = y\nt : List R\nht1 : \u2200 y \u2208 t, y \u2208 s\nht2 : t.prod = x\nu : List R\nhu1 : \u2200 y \u2208 u, y \u2208 s\nhu2 : u.prod = y\n\u22a2 (t ++ u).prod = x * y", "state_after": "no goals"}, {"tactic": "rw [List.map_append, List.sum_append, HL2, HM2]", "annotated_tactic": ["rw [<a>List.map_append</a>, <a>List.sum_append</a>, HL2, HM2]", [{"full_name": "List.map_append", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [839, 17], "def_end_pos": [839, 27]}, {"full_name": "List.sum_append", "def_path": "Mathlib/Algebra/BigOperators/Group/List.lean", "def_pos": [113, 3], "def_end_pos": [113, 14]}]], "state_before": "R\u271d : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b3 : NonAssocSemiring R\u271d\nM\u271d : Submonoid R\u271d\ninst\u271d\u00b2 : NonAssocSemiring S\ninst\u271d\u00b9 : NonAssocSemiring T\nR : Type u_1\ninst\u271d : Semiring R\ns : Set R\nx\u271d\u00b2 : R\nhx : x\u271d\u00b2 \u2208 closure s\np : R \u2192 Prop := fun x => \u2203 L, (\u2200 t \u2208 L, \u2200 y \u2208 t, y \u2208 s) \u2227 (List.map List.prod L).sum = x\nx y : R\nx\u271d\u00b9 : p x\nx\u271d : p y\nL : List (List R)\nHL1 : \u2200 t \u2208 L, \u2200 y \u2208 t, y \u2208 s\nHL2 : (List.map List.prod L).sum = x\nM : List (List R)\nHM1 : \u2200 t \u2208 M, \u2200 y \u2208 t, y \u2208 s\nHM2 : (List.map List.prod M).sum = y\n\u22a2 (List.map List.prod (L ++ M)).sum = x + y", "state_after": "no goals"}, {"tactic": "rintro \u27e8L, HL1, HL2\u27e9", "annotated_tactic": ["rintro \u27e8L, HL1, HL2\u27e9", []], "state_before": "case mpr\nR\u271d : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b3 : NonAssocSemiring R\u271d\nM : Submonoid R\u271d\ninst\u271d\u00b2 : NonAssocSemiring S\ninst\u271d\u00b9 : NonAssocSemiring T\nR : Type u_1\ninst\u271d : Semiring R\ns : Set R\nx : R\n\u22a2 (\u2203 L, (\u2200 t \u2208 L, \u2200 y \u2208 t, y \u2208 s) \u2227 (List.map List.prod L).sum = x) \u2192 x \u2208 closure s", "state_after": "case mpr.intro.intro\nR\u271d : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b3 : NonAssocSemiring R\u271d\nM : Submonoid R\u271d\ninst\u271d\u00b2 : NonAssocSemiring S\ninst\u271d\u00b9 : NonAssocSemiring T\nR : Type u_1\ninst\u271d : Semiring R\ns : Set R\nx : R\nL : List (List R)\nHL1 : \u2200 t \u2208 L, \u2200 y \u2208 t, y \u2208 s\nHL2 : (List.map List.prod L).sum = x\n\u22a2 x \u2208 closure s"}, {"tactic": "exact HL2 \u25b8\n  list_sum_mem fun r hr =>\n    let \u27e8t, ht1, ht2\u27e9 := List.mem_map.1 hr\n    ht2 \u25b8 list_prod_mem _ fun y hy => subset_closure <| HL1 t ht1 y hy", "annotated_tactic": ["exact HL2 \u25b8\n      <a>list_sum_mem</a> fun r hr =>\n        let \u27e8t, ht1, ht2\u27e9 := <a>List.mem_map</a>.1 hr\n        ht2 \u25b8 <a>list_prod_mem</a> _ fun y hy => <a>subset_closure</a> <| HL1 t ht1 y hy", [{"full_name": "list_sum_mem", "def_path": "Mathlib/Algebra/Group/Submonoid/Membership.lean", "def_pos": [72, 3], "def_end_pos": [72, 14]}, {"full_name": "List.mem_map", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [750, 17], "def_end_pos": [750, 24]}, {"full_name": "Subsemiring.list_prod_mem", "def_path": "Mathlib/Algebra/Ring/Subsemiring/Basic.lean", "def_pos": [285, 16], "def_end_pos": [285, 29]}, {"full_name": "Subsemiring.subset_closure", "def_path": "Mathlib/Algebra/Ring/Subsemiring/Basic.lean", "def_pos": [749, 9], "def_end_pos": [749, 23]}]], "state_before": "case mpr.intro.intro\nR\u271d : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b3 : NonAssocSemiring R\u271d\nM : Submonoid R\u271d\ninst\u271d\u00b2 : NonAssocSemiring S\ninst\u271d\u00b9 : NonAssocSemiring T\nR : Type u_1\ninst\u271d : Semiring R\ns : Set R\nx : R\nL : List (List R)\nHL1 : \u2200 t \u2208 L, \u2200 y \u2208 t, y \u2208 s\nHL2 : (List.map List.prod L).sum = x\n\u22a2 x \u2208 closure s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/NumberField/Embeddings.lean", "full_name": "NumberField.InfinitePlace.IsReal.comap", "start": [651, 1], "end": [655, 18], "traced_tactics": [{"tactic": "rw [\u2190 mk_embedding w, comap_mk, isReal_mk_iff]", "annotated_tactic": ["rw [\u2190 <a>mk_embedding</a> w, <a>comap_mk</a>, <a>isReal_mk_iff</a>]", [{"full_name": "NumberField.InfinitePlace.mk_embedding", "def_path": "Mathlib/NumberTheory/NumberField/Embeddings.lean", "def_pos": [299, 9], "def_end_pos": [299, 21]}, {"full_name": "NumberField.InfinitePlace.comap_mk", "def_path": "Mathlib/NumberTheory/NumberField/Embeddings.lean", "def_pos": [644, 7], "def_end_pos": [644, 15]}, {"full_name": "NumberField.InfinitePlace.isReal_mk_iff", "def_path": "Mathlib/NumberTheory/NumberField/Embeddings.lean", "def_pos": [442, 7], "def_end_pos": [442, 20]}]], "state_before": "k : Type u_1\ninst\u271d\u00b3 : Field k\nK : Type u_2\ninst\u271d\u00b2 : Field K\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : NumberField K\nf : k \u2192+* K\nw : InfinitePlace K\nh\u03c6 : w.IsReal\n\u22a2 (w.comap f).IsReal", "state_after": "k : Type u_1\ninst\u271d\u00b3 : Field k\nK : Type u_2\ninst\u271d\u00b2 : Field K\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : NumberField K\nf : k \u2192+* K\nw : InfinitePlace K\nh\u03c6 : w.IsReal\n\u22a2 ComplexEmbedding.IsReal (w.embedding.comp f)"}, {"tactic": "rw [\u2190 mk_embedding w, isReal_mk_iff] at h\u03c6", "annotated_tactic": ["rw [\u2190 <a>mk_embedding</a> w, <a>isReal_mk_iff</a>] at h\u03c6", [{"full_name": "NumberField.InfinitePlace.mk_embedding", "def_path": "Mathlib/NumberTheory/NumberField/Embeddings.lean", "def_pos": [299, 9], "def_end_pos": [299, 21]}, {"full_name": "NumberField.InfinitePlace.isReal_mk_iff", "def_path": "Mathlib/NumberTheory/NumberField/Embeddings.lean", "def_pos": [442, 7], "def_end_pos": [442, 20]}]], "state_before": "k : Type u_1\ninst\u271d\u00b3 : Field k\nK : Type u_2\ninst\u271d\u00b2 : Field K\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : NumberField K\nf : k \u2192+* K\nw : InfinitePlace K\nh\u03c6 : w.IsReal\n\u22a2 ComplexEmbedding.IsReal (w.embedding.comp f)", "state_after": "k : Type u_1\ninst\u271d\u00b3 : Field k\nK : Type u_2\ninst\u271d\u00b2 : Field K\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : NumberField K\nf : k \u2192+* K\nw : InfinitePlace K\nh\u03c6 : ComplexEmbedding.IsReal w.embedding\n\u22a2 ComplexEmbedding.IsReal (w.embedding.comp f)"}, {"tactic": "exact h\u03c6.comp f", "annotated_tactic": ["exact h\u03c6.comp f", []], "state_before": "k : Type u_1\ninst\u271d\u00b3 : Field k\nK : Type u_2\ninst\u271d\u00b2 : Field K\nF : Type u_3\ninst\u271d\u00b9 : Field F\ninst\u271d : NumberField K\nf : k \u2192+* K\nw : InfinitePlace K\nh\u03c6 : ComplexEmbedding.IsReal w.embedding\n\u22a2 ComplexEmbedding.IsReal (w.embedding.comp f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/SetSemiring.lean", "full_name": "MeasureTheory.IsSetSemiring.diffFinset_subset", "start": [77, 1], "end": [81, 83], "traced_tactics": [{"tactic": "classical\nsimp only [diffFinset, coe_sdiff, coe_singleton, diff_singleton_subset_iff]\nexact (hC.diff_eq_sUnion' s hs t ht).choose_spec.1.trans (Set.subset_insert _ _)", "annotated_tactic": ["classical\n  simp only [<a>diffFinset</a>, <a>coe_sdiff</a>, <a>coe_singleton</a>, <a>diff_singleton_subset_iff</a>]\n  exact (hC.diff_eq_sUnion' s hs t ht).<a>choose_spec</a>.1.<a>trans</a> (<a>Set.subset_insert</a> _ _)", [{"full_name": "MeasureTheory.IsSetSemiring.diffFinset", "def_path": "Mathlib/MeasureTheory/SetSemiring.lean", "def_pos": [67, 19], "def_end_pos": [67, 29]}, {"full_name": "Finset.coe_sdiff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2177, 9], "def_end_pos": [2177, 18]}, {"full_name": "Finset.coe_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [727, 9], "def_end_pos": [727, 22]}, {"full_name": "Set.diff_singleton_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1903, 9], "def_end_pos": [1903, 34]}, {"full_name": "Exists.choose_spec", "def_path": ".lake/packages/lean4/src/lean/Init/Classical.lean", "def_pos": [177, 9], "def_end_pos": [177, 27]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [655, 7], "def_end_pos": [655, 29]}, {"full_name": "Set.subset_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1078, 9], "def_end_pos": [1078, 22]}]], "state_before": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nhC : IsSetSemiring C\nhs : s \u2208 C\nht : t \u2208 C\n\u22a2 \u2191(hC.diffFinset hs ht) \u2286 C", "state_after": "no goals"}, {"tactic": "simp only [diffFinset, coe_sdiff, coe_singleton, diff_singleton_subset_iff]", "annotated_tactic": ["simp only [<a>diffFinset</a>, <a>coe_sdiff</a>, <a>coe_singleton</a>, <a>diff_singleton_subset_iff</a>]", [{"full_name": "MeasureTheory.IsSetSemiring.diffFinset", "def_path": "Mathlib/MeasureTheory/SetSemiring.lean", "def_pos": [67, 19], "def_end_pos": [67, 29]}, {"full_name": "Finset.coe_sdiff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2177, 9], "def_end_pos": [2177, 18]}, {"full_name": "Finset.coe_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [727, 9], "def_end_pos": [727, 22]}, {"full_name": "Set.diff_singleton_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1903, 9], "def_end_pos": [1903, 34]}]], "state_before": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nhC : IsSetSemiring C\nhs : s \u2208 C\nht : t \u2208 C\n\u22a2 \u2191(hC.diffFinset hs ht) \u2286 C", "state_after": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nhC : IsSetSemiring C\nhs : s \u2208 C\nht : t \u2208 C\n\u22a2 \u2191\u22ef.choose \u2286 insert \u2205 C"}, {"tactic": "exact (hC.diff_eq_sUnion' s hs t ht).choose_spec.1.trans (Set.subset_insert _ _)", "annotated_tactic": ["exact (hC.diff_eq_sUnion' s hs t ht).<a>choose_spec</a>.1.<a>trans</a> (<a>Set.subset_insert</a> _ _)", [{"full_name": "Exists.choose_spec", "def_path": ".lake/packages/lean4/src/lean/Init/Classical.lean", "def_pos": [177, 9], "def_end_pos": [177, 27]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [655, 7], "def_end_pos": [655, 29]}, {"full_name": "Set.subset_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1078, 9], "def_end_pos": [1078, 22]}]], "state_before": "\u03b1 : Type u_1\nC : Set (Set \u03b1)\ns t : Set \u03b1\nhC : IsSetSemiring C\nhs : s \u2208 C\nht : t \u2208 C\n\u22a2 \u2191\u22ef.choose \u2286 insert \u2205 C", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "full_name": "MeasureTheory.integral_pos_iff_support_of_nonneg", "start": [1277, 1], "end": [1279, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "full_name": "equicontinuousOn_iff_range", "start": [482, 1], "end": [484, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/SmoothNumbers.lean", "full_name": "Nat.smoothNumbersUpTo_card_le", "start": [478, 1], "end": [484, 82], "traced_tactics": [{"tactic": "convert (Finset.card_le_card <| smoothNumbersUpTo_subset_image N k).trans <|\n  Finset.card_image_le", "annotated_tactic": ["convert (<a>Finset.card_le_card</a> <| <a>smoothNumbersUpTo_subset_image</a> N k).<a>trans</a> <|\n    <a>Finset.card_image_le</a>", [{"full_name": "Finset.card_le_card", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [64, 9], "def_end_pos": [64, 21]}, {"full_name": "Nat.smoothNumbersUpTo_subset_image", "def_path": "Mathlib/NumberTheory/SmoothNumbers.lean", "def_pos": [460, 7], "def_end_pos": [460, 37]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}, {"full_name": "Finset.card_image_le", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [264, 9], "def_end_pos": [264, 22]}]], "state_before": "N k : \u2115\n\u22a2 (N.smoothNumbersUpTo k).card \u2264 2 ^ k.primesBelow.card * N.sqrt", "state_after": "case h.e'_4\nN k : \u2115\n\u22a2 2 ^ k.primesBelow.card * N.sqrt = (k.primesBelow.powerset \u00d7\u02e2 (Finset.range N.sqrt.succ).erase 0).card"}, {"tactic": "simp only [Finset.card_product, Finset.card_powerset, Finset.mem_range, zero_lt_succ,\n  Finset.card_erase_of_mem, Finset.card_range, succ_sub_succ_eq_sub, tsub_zero]", "annotated_tactic": ["simp only [<a>Finset.card_product</a>, <a>Finset.card_powerset</a>, <a>Finset.mem_range</a>, <a>zero_lt_succ</a>,\n    <a>Finset.card_erase_of_mem</a>, <a>Finset.card_range</a>, <a>succ_sub_succ_eq_sub</a>, <a>tsub_zero</a>]", [{"full_name": "Finset.card_product", "def_path": "Mathlib/Data/Finset/Prod.lean", "def_pos": [143, 9], "def_end_pos": [143, 21]}, {"full_name": "Finset.card_powerset", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [89, 9], "def_end_pos": [89, 22]}, {"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2935, 9], "def_end_pos": [2935, 18]}, {"full_name": "Nat.zero_lt_succ", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1671, 9], "def_end_pos": [1671, 25]}, {"full_name": "Finset.card_erase_of_mem", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [164, 9], "def_end_pos": [164, 26]}, {"full_name": "Finset.card_range", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [206, 9], "def_end_pos": [206, 19]}, {"full_name": "Nat.succ_sub_succ_eq_sub", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [276, 9], "def_end_pos": [276, 29]}, {"full_name": "tsub_zero", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [448, 9], "def_end_pos": [448, 18]}]], "state_before": "case h.e'_4\nN k : \u2115\n\u22a2 2 ^ k.primesBelow.card * N.sqrt = (k.primesBelow.powerset \u00d7\u02e2 (Finset.range N.sqrt.succ).erase 0).card", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/ModelTheory/Skolem.lean", "full_name": "FirstOrder.Language.Substructure.coeSort_elementarySkolem\u2081Reduct", "start": [104, 1], "end": [106, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/Adjoin.lean", "full_name": "IntermediateField.lift_bot", "start": [473, 1], "end": [474, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Cyclotomic/Rat.lean", "full_name": "IsCyclotomicExtension.Rat.cyclotomicRing_isIntegralClosure_of_prime", "start": [144, 1], "end": [147, 54], "traced_tactics": [{"tactic": "rw [\u2190 pow_one p]", "annotated_tactic": ["rw [\u2190 <a>pow_one</a> p]", [{"full_name": "pow_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [663, 7], "def_end_pos": [663, 14]}]], "state_before": "p : \u2115+\nk : \u2115\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\n\u03b6 : K\nhp : Fact (Nat.Prime \u2191p)\n\u22a2 IsIntegralClosure (CyclotomicRing p \u2124 \u211a) \u2124 (CyclotomicField p \u211a)", "state_after": "p : \u2115+\nk : \u2115\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\n\u03b6 : K\nhp : Fact (Nat.Prime \u2191p)\n\u22a2 IsIntegralClosure (CyclotomicRing (p ^ 1) \u2124 \u211a) \u2124 (CyclotomicField (p ^ 1) \u211a)"}, {"tactic": "exact cyclotomicRing_isIntegralClosure_of_prime_pow", "annotated_tactic": ["exact <a>cyclotomicRing_isIntegralClosure_of_prime_pow</a>", [{"full_name": "IsCyclotomicExtension.Rat.cyclotomicRing_isIntegralClosure_of_prime_pow", "def_path": "Mathlib/NumberTheory/Cyclotomic/Rat.lean", "def_pos": [130, 9], "def_end_pos": [130, 54]}]], "state_before": "p : \u2115+\nk : \u2115\nK : Type u\ninst\u271d\u00b9 : Field K\ninst\u271d : CharZero K\n\u03b6 : K\nhp : Fact (Nat.Prime \u2191p)\n\u22a2 IsIntegralClosure (CyclotomicRing (p ^ 1) \u2124 \u211a) \u2124 (CyclotomicField (p ^ 1) \u211a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/MvPowerSeries/Basic.lean", "full_name": "MvPowerSeries.map_id", "start": [570, 1], "end": [571, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/FiberedCategory/BasedCategory.lean", "full_name": "CategoryTheory.BasedFunctor.isHomLift_iff", "start": [124, 1], "end": [125, 67], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Interval/Multiset.lean", "full_name": "Multiset.map_add_right_Icc", "start": [42, 1], "end": [44, 31], "traced_tactics": [{"tactic": "simp_rw [add_comm _ c]", "annotated_tactic": ["simp_rw [<a>add_comm</a> _ c]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map (fun x => x + c) (Icc a b) = Icc (a + c) (b + c)", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map (fun x => c + x) (Icc a b) = Icc (c + a) (c + b)"}, {"tactic": "exact map_add_left_Icc _ _ _", "annotated_tactic": ["exact <a>map_add_left_Icc</a> _ _ _", [{"full_name": "Multiset.map_add_left_Icc", "def_path": "Mathlib/Algebra/Order/Interval/Multiset.lean", "def_pos": [22, 7], "def_end_pos": [22, 23]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1\ninst\u271d\u00b9 : ExistsAddOfLE \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 map (fun x => c + x) (Icc a b) = Icc (c + a) (c + b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "mem_closure_iff_ultrafilter", "start": [1361, 1], "end": [1363, 79], "traced_tactics": [{"tactic": "simp [closure_eq_cluster_pts, ClusterPt, \u2190 exists_ultrafilter_iff, and_comm]", "annotated_tactic": ["simp [<a>closure_eq_cluster_pts</a>, <a>ClusterPt</a>, \u2190 <a>exists_ultrafilter_iff</a>, <a>and_comm</a>]", [{"full_name": "closure_eq_cluster_pts", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1309, 9], "def_end_pos": [1309, 31]}, {"full_name": "ClusterPt", "def_path": "Mathlib/Topology/Defs/Filter.lean", "def_pos": [255, 5], "def_end_pos": [255, 14]}, {"full_name": "Filter.exists_ultrafilter_iff", "def_path": "Mathlib/Order/Filter/Ultrafilter.lean", "def_pos": [470, 9], "def_end_pos": [470, 31]}, {"full_name": "and_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [819, 9], "def_end_pos": [819, 17]}]], "state_before": "X : Type u\nY : Type v\n\u03b9 : Sort w\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nx : X\ns s\u2081 s\u2082 t : Set X\np p\u2081 p\u2082 : X \u2192 Prop\ninst\u271d : TopologicalSpace X\n\u22a2 x \u2208 closure s \u2194 \u2203 u, s \u2208 u \u2227 \u2191u \u2264 \ud835\udcdd x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/QuadraticForm/Basic.lean", "full_name": "QuadraticForm.associated_isOrtho", "start": [994, 1], "end": [997, 36], "traced_tactics": [{"tactic": "simp_rw [isOrtho_def, LinearMap.isOrtho_def, associated_apply, invOf_mul_eq_iff_eq_mul_left,\n  mul_zero, sub_sub, sub_eq_zero]", "annotated_tactic": ["simp_rw [<a>isOrtho_def</a>, <a>LinearMap.isOrtho_def</a>, <a>associated_apply</a>, <a>invOf_mul_eq_iff_eq_mul_left</a>,\n    <a>mul_zero</a>, <a>sub_sub</a>, <a>sub_eq_zero</a>]", [{"full_name": "QuadraticForm.isOrtho_def", "def_path": "Mathlib/LinearAlgebra/QuadraticForm/Basic.lean", "def_pos": [955, 9], "def_end_pos": [955, 20]}, {"full_name": "LinearMap.isOrtho_def", "def_path": "Mathlib/LinearAlgebra/SesquilinearForm.lean", "def_pos": [58, 9], "def_end_pos": [58, 20]}, {"full_name": "QuadraticForm.associated_apply", "def_path": "Mathlib/LinearAlgebra/QuadraticForm/Basic.lean", "def_pos": [831, 9], "def_end_pos": [831, 25]}, {"full_name": "invOf_mul_eq_iff_eq_mul_left", "def_path": "Mathlib/Algebra/Group/Invertible/Defs.lean", "def_pos": [253, 9], "def_end_pos": [253, 37]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}, {"full_name": "sub_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [770, 3], "def_end_pos": [770, 14]}, {"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1070, 3], "def_end_pos": [1070, 14]}]], "state_before": "S : Type u_1\nT : Type u_2\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\nQ : QuadraticForm R M\ninst\u271d : Invertible 2\nx y : M\n\u22a2 LinearMap.IsOrtho (associated Q) x y \u2194 Q.IsOrtho x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Complex/Basic.lean", "full_name": "Complex.ofReal_natCast", "start": [514, 1], "end": [514, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Complex/OperatorNorm.lean", "full_name": "Complex.imCLM_nnnorm", "start": [58, 1], "end": [59, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/AddChar.lean", "full_name": "AddChar.coe_toAddMonoidHom", "start": [157, 1], "end": [157, 99], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Sites/Over.lean", "full_name": "CategoryTheory.Sieve.overEquiv_symm_top", "start": [63, 1], "end": [66, 36], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX : C\nY : Over X\n\u22a2 (overEquiv Y) ((overEquiv Y).symm \u22a4) = (overEquiv Y) \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "full_name": "MeasureTheory.Measure.quasiMeasurePreserving_smul", "start": [368, 1], "end": [372, 34], "traced_tactics": [{"tactic": "refine \u27e8measurable_const_smul r, ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>measurable_const_smul</a> r, ?_\u27e9", [{"full_name": "MeasurableSMul.measurable_const_smul", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [589, 3], "def_end_pos": [589, 24]}]], "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 : \u03bc.IsAddHaarMeasure\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\nhr : r \u2260 0\n\u22a2 QuasiMeasurePreserving (fun x => r \u2022 x) \u03bc \u03bc", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\nhr : r \u2260 0\n\u22a2 map (fun x => r \u2022 x) \u03bc \u226a \u03bc"}, {"tactic": "rw [map_addHaar_smul \u03bc hr]", "annotated_tactic": ["rw [<a>map_addHaar_smul</a> \u03bc hr]", [{"full_name": "MeasureTheory.Measure.map_addHaar_smul", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean", "def_pos": [357, 9], "def_end_pos": [357, 25]}]], "state_before": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\nhr : r \u2260 0\n\u22a2 map (fun x => r \u2022 x) \u03bc \u226a \u03bc", "state_after": "E : Type u_1\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\nhr : r \u2260 0\n\u22a2 ENNReal.ofReal |(r ^ finrank \u211d E)\u207b\u00b9| \u2022 \u03bc \u226a \u03bc"}, {"tactic": "exact smul_absolutelyContinuous", "annotated_tactic": ["exact <a>smul_absolutelyContinuous</a>", [{"full_name": "MeasureTheory.Measure.smul_absolutelyContinuous", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1710, 7], "def_end_pos": [1710, 32]}]], "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 : \u03bc.IsAddHaarMeasure\nF : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup F\ninst\u271d\u00b9 : NormedSpace \u211d F\ninst\u271d : CompleteSpace F\nr : \u211d\nhr : r \u2260 0\n\u22a2 ENNReal.ofReal |(r ^ finrank \u211d E)\u207b\u00b9| \u2022 \u03bc \u226a \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean", "full_name": "CategoryTheory.Subgroupoid.full_objs", "start": [667, 1], "end": [668, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/OrdConnected.lean", "full_name": "OrderEmbedding.image_Ioo", "start": [103, 1], "end": [106, 78], "traced_tactics": [{"tactic": "rw [\u2190 e.preimage_Ioo, image_preimage_eq_inter_range,\n  inter_eq_left.2 <| Ioo_subset_Icc_self.trans <| he.out \u27e8_, rfl\u27e9 \u27e8_, rfl\u27e9]", "annotated_tactic": ["rw [\u2190 e.preimage_Ioo, <a>image_preimage_eq_inter_range</a>,\n    <a>inter_eq_left</a>.2 <| Ioo_subset_Icc_self.trans <| he.out \u27e8_, <a>rfl</a>\u27e9 \u27e8_, <a>rfl</a>\u27e9]", [{"full_name": "Set.image_preimage_eq_inter_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [786, 9], "def_end_pos": [786, 38]}, {"full_name": "Set.inter_eq_left", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [946, 15], "def_end_pos": [946, 28]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\ne : \u03b1 \u21aao \u03b2\nhe : (range \u21d1e).OrdConnected\nx y : \u03b1\n\u22a2 \u21d1e '' Ioo x y = Ioo (e x) (e y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SetFamily/Compression/Down.lean", "full_name": "Finset.image_insert_memberSubfamily", "start": [154, 1], "end": [161, 26], "traced_tactics": [{"tactic": "ext s", "annotated_tactic": ["ext s", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c : Finset (Finset \u03b1)\ns : Finset \u03b1\na\u271d : \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\na : \u03b1\n\u22a2 image (insert a) (memberSubfamily a \ud835\udc9c) = filter (fun x => a \u2208 x) \ud835\udc9c", "state_after": "case a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\na : \u03b1\ns : Finset \u03b1\n\u22a2 s \u2208 image (insert a) (memberSubfamily a \ud835\udc9c) \u2194 s \u2208 filter (fun x => a \u2208 x) \ud835\udc9c"}, {"tactic": "simp only [mem_memberSubfamily, mem_image, mem_filter]", "annotated_tactic": ["simp only [<a>mem_memberSubfamily</a>, <a>mem_image</a>, <a>mem_filter</a>]", [{"full_name": "Finset.mem_memberSubfamily", "def_path": "Mathlib/Combinatorics/SetFamily/Compression/Down.lean", "def_pos": [61, 9], "def_end_pos": [61, 28]}, {"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [357, 9], "def_end_pos": [357, 18]}, {"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2593, 9], "def_end_pos": [2593, 19]}]], "state_before": "case a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\na : \u03b1\ns : Finset \u03b1\n\u22a2 s \u2208 image (insert a) (memberSubfamily a \ud835\udc9c) \u2194 s \u2208 filter (fun x => a \u2208 x) \ud835\udc9c", "state_after": "case a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\na : \u03b1\ns : Finset \u03b1\n\u22a2 (\u2203 a_1, (insert a a_1 \u2208 \ud835\udc9c \u2227 a \u2209 a_1) \u2227 insert a a_1 = s) \u2194 s \u2208 \ud835\udc9c \u2227 a \u2208 s"}, {"tactic": "refine \u27e8?_, fun \u27e8hs, ha\u27e9 \u21a6 \u27e8erase s a, \u27e8?_, not_mem_erase _ _\u27e9, insert_erase ha\u27e9\u27e9", "annotated_tactic": ["refine \u27e8?_, fun \u27e8hs, ha\u27e9 \u21a6 \u27e8<a>erase</a> s a, \u27e8?_, <a>not_mem_erase</a> _ _\u27e9, <a>insert_erase</a> ha\u27e9\u27e9", [{"full_name": "Finset.erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1885, 5], "def_end_pos": [1885, 10]}, {"full_name": "Finset.not_mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1899, 9], "def_end_pos": [1899, 22]}, {"full_name": "Finset.insert_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1974, 17], "def_end_pos": [1974, 29]}]], "state_before": "case a\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\na : \u03b1\ns : Finset \u03b1\n\u22a2 (\u2203 a_1, (insert a a_1 \u2208 \ud835\udc9c \u2227 a \u2209 a_1) \u2227 insert a a_1 = s) \u2194 s \u2208 \ud835\udc9c \u2227 a \u2208 s", "state_after": "case a.refine_1\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\na : \u03b1\ns : Finset \u03b1\n\u22a2 (\u2203 a_1, (insert a a_1 \u2208 \ud835\udc9c \u2227 a \u2209 a_1) \u2227 insert a a_1 = s) \u2192 s \u2208 \ud835\udc9c \u2227 a \u2208 s\n\ncase a.refine_2\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\na : \u03b1\ns : Finset \u03b1\nx\u271d : s \u2208 \ud835\udc9c \u2227 a \u2208 s\nhs : s \u2208 \ud835\udc9c\nha : a \u2208 s\n\u22a2 insert a (s.erase a) \u2208 \ud835\udc9c"}, {"tactic": "rintro \u27e8s, \u27e8hs, -\u27e9, rfl\u27e9", "annotated_tactic": ["rintro \u27e8s, \u27e8hs, -\u27e9, rfl\u27e9", []], "state_before": "case a.refine_1\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\na : \u03b1\ns : Finset \u03b1\n\u22a2 (\u2203 a_1, (insert a a_1 \u2208 \ud835\udc9c \u2227 a \u2209 a_1) \u2227 insert a a_1 = s) \u2192 s \u2208 \ud835\udc9c \u2227 a \u2208 s", "state_after": "case a.refine_1.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\na : \u03b1\ns : Finset \u03b1\nhs : insert a s \u2208 \ud835\udc9c\n\u22a2 insert a s \u2208 \ud835\udc9c \u2227 a \u2208 insert a s"}, {"tactic": "exact \u27e8hs, mem_insert_self _ _\u27e9", "annotated_tactic": ["exact \u27e8hs, <a>mem_insert_self</a> _ _\u27e9", [{"full_name": "Finset.mem_insert_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1117, 9], "def_end_pos": [1117, 24]}]], "state_before": "case a.refine_1.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\na : \u03b1\ns : Finset \u03b1\nhs : insert a s \u2208 \ud835\udc9c\n\u22a2 insert a s \u2208 \ud835\udc9c \u2227 a \u2208 insert a s", "state_after": "no goals"}, {"tactic": "rwa [insert_erase ha]", "annotated_tactic": ["rwa [<a>insert_erase</a> ha]", [{"full_name": "Finset.insert_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1974, 17], "def_end_pos": [1974, 29]}]], "state_before": "case a.refine_2\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\n\ud835\udc9c\u271d \u212c : Finset (Finset \u03b1)\ns\u271d : Finset \u03b1\na\u271d : \u03b1\n\ud835\udc9c : Finset (Finset \u03b1)\na : \u03b1\ns : Finset \u03b1\nx\u271d : s \u2208 \ud835\udc9c \u2227 a \u2208 s\nhs : s \u2208 \ud835\udc9c\nha : a \u2208 s\n\u22a2 insert a (s.erase a) \u2208 \ud835\udc9c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean", "full_name": "CategoryTheory.Limits.coprod.associator_naturality", "start": [1159, 1], "end": [1163, 7], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX Y : C\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : HasBinaryCoproducts C\nX\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C\nf\u2081 : X\u2081 \u27f6 Y\u2081\nf\u2082 : X\u2082 \u27f6 Y\u2082\nf\u2083 : X\u2083 \u27f6 Y\u2083\n\u22a2 map (map f\u2081 f\u2082) f\u2083 \u226b (associator Y\u2081 Y\u2082 Y\u2083).hom = (associator X\u2081 X\u2082 X\u2083).hom \u226b map f\u2081 (map f\u2082 f\u2083)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/DirectLimit.lean", "full_name": "AddCommGroup.DirectLimit.map_apply_of", "start": [481, 1], "end": [485, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Indexes.lean", "full_name": "List.foldrIdxSpec_cons", "start": [233, 1], "end": [235, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean", "full_name": "PrimeSpectrum.union_zeroLocus", "start": [361, 1], "end": [364, 7], "traced_tactics": [{"tactic": "rw [zeroLocus_inf]", "annotated_tactic": ["rw [<a>zeroLocus_inf</a>]", [{"full_name": "PrimeSpectrum.zeroLocus_inf", "def_path": "Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean", "def_pos": [356, 9], "def_end_pos": [356, 22]}]], "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\ns s' : Set R\n\u22a2 zeroLocus s \u222a zeroLocus s' = zeroLocus \u2191(Ideal.span s \u2293 Ideal.span s')", "state_after": "R : Type u\nS : Type v\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\ns s' : Set R\n\u22a2 zeroLocus s \u222a zeroLocus s' = zeroLocus \u2191(Ideal.span s) \u222a zeroLocus \u2191(Ideal.span s')"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "R : Type u\nS : Type v\ninst\u271d\u00b9 : CommSemiring R\ninst\u271d : CommSemiring S\ns s' : Set R\n\u22a2 zeroLocus s \u222a zeroLocus s' = zeroLocus \u2191(Ideal.span s) \u222a zeroLocus \u2191(Ideal.span s')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/RootsOfUnity/Complex.lean", "full_name": "IsPrimitiveRoot.arg_eq_zero_iff", "start": [119, 1], "end": [122, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/WittVector/Basic.lean", "full_name": "WittVector.ghostFun_natCast", "start": [196, 9], "end": [199, 90], "traced_tactics": [{"tactic": "induction i <;>\n  simp [*, Nat.unaryCast, ghostFun_zero, ghostFun_one, ghostFun_add, -Pi.natCast_def]", "annotated_tactic": ["induction i <;>\n      simp [*, <a>Nat.unaryCast</a>, <a>ghostFun_zero</a>, <a>ghostFun_one</a>, <a>ghostFun_add</a>, -<a>Pi.natCast_def</a>]", [{"full_name": "Nat.unaryCast", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [30, 15], "def_end_pos": [30, 28]}, {"full_name": "_private.Mathlib.RingTheory.WittVector.Basic.0.WittVector.ghostFun_zero", "def_path": "Mathlib/RingTheory/WittVector/Basic.lean", "def_pos": [187, 17], "def_end_pos": [187, 30]}, {"full_name": "_private.Mathlib.RingTheory.WittVector.Basic.0.WittVector.ghostFun_one", "def_path": "Mathlib/RingTheory/WittVector/Basic.lean", "def_pos": [190, 17], "def_end_pos": [190, 29]}, {"full_name": "_private.Mathlib.RingTheory.WittVector.Basic.0.WittVector.ghostFun_add", "def_path": "Mathlib/RingTheory/WittVector/Basic.lean", "def_pos": [193, 17], "def_end_pos": [193, 29]}, {"full_name": "Pi.natCast_def", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 20]}]], "state_before": "p : \u2115\nR : Type u_1\nS : Type u_2\nT : Type u_3\nhp : Fact (Nat.Prime p)\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : CommRing S\ninst\u271d : CommRing T\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nx y : \ud835\udd4e R\ni : \u2115\n\u22a2 WittVector.ghostFun i.unaryCast = \u2191i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Distribution/SchwartzSpace.lean", "full_name": "Function.HasTemperateGrowth.const", "start": [649, 1], "end": [651, 99], "traced_tactics": [{"tactic": "simpa using .zero", "annotated_tactic": ["simpa using .zero", []], "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\nV : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nc : F\n\u22a2 Function.HasTemperateGrowth (fderiv \u211d fun x => c)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nD : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\nV : Type u_7\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nc : F\nx : E\n\u22a2 \u2016c\u2016 \u2264 \u2016c\u2016 * (1 + \u2016x\u2016) ^ 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/LineDeriv/Basic.lean", "full_name": "HasLineDerivAt.hasLineDerivWithinAt", "start": [114, 1], "end": [116, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean", "full_name": "MeasureTheory.snorm_le_snorm_fderiv_of_eq_inner", "start": [463, 1], "end": [578, 17], "traced_tactics": [{"tactic": "by_cases hp'0 : p' = 0", "annotated_tactic": ["by_cases hp'0 : p' = 0", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhn : 0 < finrank \u211d E\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191(finrank \u211d E))\u207b\u00b9\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case pos\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhn : 0 < finrank \u211d E\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191(finrank \u211d E))\u207b\u00b9\nhp'0 : p' = 0\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc\n\ncase neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhn : 0 < finrank \u211d E\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191(finrank \u211d E))\u207b\u00b9\nhp'0 : \u00acp' = 0\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "set n := finrank \u211d E", "annotated_tactic": ["set n := <a>finrank</a> \u211d E", [{"full_name": "FiniteDimensional.finrank", "def_path": "Mathlib/LinearAlgebra/Dimension/Finrank.lean", "def_pos": [54, 19], "def_end_pos": [54, 26]}]], "state_before": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhn : 0 < finrank \u211d E\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191(finrank \u211d E))\u207b\u00b9\nhp'0 : \u00acp' = 0\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "let n' := NNReal.conjExponent n", "annotated_tactic": ["let n' := <a>NNReal.conjExponent</a> n", [{"full_name": "NNReal.conjExponent", "def_path": "Mathlib/Data/Real/ConjExponents.lean", "def_pos": [159, 19], "def_end_pos": [159, 31]}]], "state_before": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have h2p : (p : \u211d) < n := by\n  have : 0 < p\u207b\u00b9 - (n : \u211d)\u207b\u00b9 :=\n    NNReal.coe_lt_coe.mpr (pos_iff_ne_zero.mpr (inv_ne_zero hp'0)) |>.trans_eq hp'\n  simp [sub_pos] at this\n  rwa [inv_lt_inv _ (zero_lt_one.trans_le (NNReal.coe_le_coe.mpr hp))] at this\n  exact_mod_cast hn", "annotated_tactic": ["have h2p : (p : \u211d) < n := by\n    have : 0 < p\u207b\u00b9 - (n : \u211d)\u207b\u00b9 :=\n      NNReal.coe_lt_coe.mpr (pos_iff_ne_zero.mpr (<a>inv_ne_zero</a> hp'0)) |>.<a>trans_eq</a> hp'\n    simp [<a>sub_pos</a>] at this\n    rwa [<a>inv_lt_inv</a> _ (zero_lt_one.trans_le (NNReal.coe_le_coe.mpr hp))] at this\n    exact_mod_cast hn", [{"full_name": "inv_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/NeZero.lean", "def_pos": [49, 9], "def_end_pos": [49, 20]}, {"full_name": "LT.lt.trans_eq", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [194, 7], "def_end_pos": [194, 21]}, {"full_name": "sub_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [886, 30], "def_end_pos": [886, 37]}, {"full_name": "inv_lt_inv", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [204, 9], "def_end_pos": [204, 19]}]], "state_before": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have h0n : 2 \u2264 n := Nat.succ_le_of_lt <| Nat.one_lt_cast.mp <| hp.trans_lt h2p", "annotated_tactic": ["have h0n : 2 \u2264 n := <a>Nat.succ_le_of_lt</a> <| Nat.one_lt_cast.mp <| hp.trans_lt h2p", [{"full_name": "Nat.succ_le_of_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [348, 9], "def_end_pos": [348, 22]}]], "state_before": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have hn : NNReal.IsConjExponent n n' := .conjExponent (by norm_cast)", "annotated_tactic": ["have hn : <a>NNReal.IsConjExponent</a> n n' := .conjExponent (by norm_cast)", [{"full_name": "NNReal.IsConjExponent", "def_path": "Mathlib/Data/Real/ConjExponents.lean", "def_pos": [152, 11], "def_end_pos": [152, 25]}]], "state_before": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have h1n : 1 \u2264 (n : \u211d\u22650) := hn.one_le", "annotated_tactic": ["have h1n : 1 \u2264 (n : \u211d\u22650) := hn.one_le", []], "state_before": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have h2n : (0 : \u211d) < n - 1 := by simp_rw [sub_pos]; exact hn.coe.one_lt", "annotated_tactic": ["have h2n : (0 : \u211d) < n - 1 := by simp_rw [<a>sub_pos</a>]; exact hn.coe.one_lt", [{"full_name": "sub_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [886, 30], "def_end_pos": [886, 37]}]], "state_before": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have hnp : (0 : \u211d) < n - p := by simp_rw [sub_pos]; exact h2p", "annotated_tactic": ["have hnp : (0 : \u211d) < n - p := by simp_rw [<a>sub_pos</a>]; exact h2p", [{"full_name": "sub_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [886, 30], "def_end_pos": [886, 37]}]], "state_before": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "rcases hp.eq_or_lt with rfl|hp", "annotated_tactic": ["rcases hp.eq_or_lt with rfl|hp", []], "state_before": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg.inl\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np' : \u211d\u22650\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhp : 1 \u2264 1\nhp' : (\u2191p')\u207b\u00b9 = \u21911\u207b\u00b9 - (\u2191n)\u207b\u00b9\nh2p : \u21911 < \u2191n\nhnp : 0 < \u2191n - \u21911\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u21911) * snorm (fderiv \u211d u) (\u21911) \u03bc\n\ncase neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "let q := Real.conjExponent p", "annotated_tactic": ["let q := <a>Real.conjExponent</a> p", [{"full_name": "Real.conjExponent", "def_path": "Mathlib/Data/Real/ConjExponents.lean", "def_pos": [46, 5], "def_end_pos": [46, 17]}]], "state_before": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have hq : Real.IsConjExponent p q := .conjExponent hp", "annotated_tactic": ["have hq : <a>Real.IsConjExponent</a> p q := .conjExponent hp", [{"full_name": "Real.IsConjExponent", "def_path": "Mathlib/Data/Real/ConjExponents.lean", "def_pos": [40, 11], "def_end_pos": [40, 25]}]], "state_before": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have h0p : p \u2260 0 := zero_lt_one.trans hp |>.ne'", "annotated_tactic": ["have h0p : p \u2260 0 := zero_lt_one.trans hp |>.<a>ne'</a>", [{"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}]], "state_before": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have h1p : (p : \u211d) \u2260 1 := hq.one_lt.ne'", "annotated_tactic": ["have h1p : (p : \u211d) \u2260 1 := hq.one_lt.ne'", []], "state_before": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have h3p : (p : \u211d) - 1 \u2260 0 := sub_ne_zero_of_ne h1p", "annotated_tactic": ["have h3p : (p : \u211d) - 1 \u2260 0 := <a>sub_ne_zero_of_ne</a> h1p", [{"full_name": "sub_ne_zero_of_ne", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [535, 3], "def_end_pos": [535, 14]}]], "state_before": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have h0p' : p' \u2260 0 := by\n  suffices 0 < (p' : \u211d) from (show 0 < p' from this) |>.ne'\n  rw [\u2190 inv_pos, hp', sub_pos]\n  exact inv_lt_inv_of_lt hq.pos h2p", "annotated_tactic": ["have h0p' : p' \u2260 0 := by\n    suffices 0 < (p' : \u211d) from (show 0 < p' from this) |>.<a>ne'</a>\n    rw [\u2190 <a>inv_pos</a>, hp', <a>sub_pos</a>]\n    exact <a>inv_lt_inv_of_lt</a> hq.pos h2p", [{"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "inv_pos", "def_path": "Mathlib/Algebra/Order/Field/Defs.lean", "def_pos": [47, 15], "def_end_pos": [47, 22]}, {"full_name": "sub_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [886, 30], "def_end_pos": [886, 37]}, {"full_name": "inv_lt_inv_of_lt", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 25]}]], "state_before": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have h2q : 1 / n' - 1 / q = 1 / p' := by\n  simp_rw (config := {zeta := false}) [one_div, hp']\n  rw [\u2190 hq.one_sub_inv, \u2190 hn.coe.one_sub_inv, sub_sub_sub_cancel_left]\n  simp only [NNReal.coe_natCast, NNReal.coe_inv]", "annotated_tactic": ["have h2q : 1 / n' - 1 / q = 1 / p' := by\n    simp_rw (config := {zeta := <a>false</a>}) [<a>one_div</a>, hp']\n    rw [\u2190 hq.one_sub_inv, \u2190 hn.coe.one_sub_inv, <a>sub_sub_sub_cancel_left</a>]\n    simp only [<a>NNReal.coe_natCast</a>, <a>NNReal.coe_inv</a>]", [{"full_name": "Bool.false", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [569, 5], "def_end_pos": [569, 10]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 16]}, {"full_name": "sub_sub_sub_cancel_left", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1374, 3], "def_end_pos": [1374, 14]}, {"full_name": "NNReal.coe_natCast", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [352, 19], "def_end_pos": [352, 30]}, {"full_name": "NNReal.coe_inv", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [190, 19], "def_end_pos": [190, 26]}]], "state_before": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "let \u03b3 : \u211d\u22650 := \u27e8p * (n - 1) / (n - p), by positivity\u27e9", "annotated_tactic": ["let \u03b3 : \u211d\u22650 := \u27e8p * (n - 1) / (n - p), by positivity\u27e9", []], "state_before": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have h0\u03b3 : (\u03b3 : \u211d) = p * (n - 1) / (n - p) := rfl", "annotated_tactic": ["have h0\u03b3 : (\u03b3 : \u211d) = p * (n - 1) / (n - p) := <a>rfl</a>", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have h1\u03b3 : 1 < (\u03b3 : \u211d) := by\n  rwa [h0\u03b3, one_lt_div hnp, mul_sub, mul_one, sub_lt_sub_iff_right, lt_mul_iff_one_lt_left]\n  exact hn.coe.pos", "annotated_tactic": ["have h1\u03b3 : 1 < (\u03b3 : \u211d) := by\n    rwa [h0\u03b3, <a>one_lt_div</a> hnp, <a>mul_sub</a>, <a>mul_one</a>, <a>sub_lt_sub_iff_right</a>, <a>lt_mul_iff_one_lt_left</a>]\n    exact hn.coe.pos", [{"full_name": "one_lt_div", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [369, 9], "def_end_pos": [369, 19]}, {"full_name": "mul_sub", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [394, 7], "def_end_pos": [394, 14]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "sub_lt_sub_iff_right", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [874, 3], "def_end_pos": [874, 14]}, {"full_name": "lt_mul_iff_one_lt_left", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [702, 9], "def_end_pos": [702, 31]}]], "state_before": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have h2\u03b3 : \u03b3 * n' = p' := by\n  rw [\u2190 NNReal.coe_inj, \u2190 inv_inj, hp', NNReal.coe_mul, h0\u03b3, hn.coe.conj_eq]\n  field_simp; ring", "annotated_tactic": ["have h2\u03b3 : \u03b3 * n' = p' := by\n    rw [\u2190 <a>NNReal.coe_inj</a>, \u2190 <a>inv_inj</a>, hp', <a>NNReal.coe_mul</a>, h0\u03b3, hn.coe.conj_eq]\n    field_simp; ring", [{"full_name": "NNReal.coe_inj", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [167, 26], "def_end_pos": [167, 33]}, {"full_name": "inv_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [407, 9], "def_end_pos": [407, 16]}, {"full_name": "NNReal.coe_mul", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [185, 19], "def_end_pos": [185, 26]}]], "state_before": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have h3\u03b3 : (\u03b3 - 1) * q = p' := by\n  rw [\u2190 inv_inj, hp', h0\u03b3, hq.conj_eq]\n  have : (p : \u211d) * (n - 1) - (n - p) = n * (p - 1) := by ring\n  field_simp [this]; ring", "annotated_tactic": ["have h3\u03b3 : (\u03b3 - 1) * q = p' := by\n    rw [\u2190 <a>inv_inj</a>, hp', h0\u03b3, hq.conj_eq]\n    have : (p : \u211d) * (n - 1) - (n - p) = n * (p - 1) := by ring\n    field_simp [this]; ring", [{"full_name": "inv_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [407, 9], "def_end_pos": [407, 16]}]], "state_before": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have h4\u03b3 : (\u03b3 : \u211d) \u2260 0 := (zero_lt_one.trans h1\u03b3).ne'", "annotated_tactic": ["have h4\u03b3 : (\u03b3 : \u211d) \u2260 0 := (zero_lt_one.trans h1\u03b3).<a>ne'</a>", [{"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}]], "state_before": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "by_cases h3u : \u222b\u207b x, \u2016u x\u2016\u208a ^ (p' : \u211d) \u2202\u03bc = 0", "annotated_tactic": ["by_cases h3u : \u222b\u207b x, \u2016u x\u2016\u208a ^ (p' : \u211d) \u2202\u03bc = 0", []], "state_before": "case neg.inr\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case pos\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc\n\ncase neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have h4u : \u222b\u207b x, \u2016u x\u2016\u208a ^ (p' : \u211d) \u2202\u03bc \u2260 \u221e := by\n  refine lintegral_rpow_nnnorm_lt_top_of_snorm'_lt_top (pos_iff_ne_zero.mpr h0p') ?_ |>.ne\n  dsimp only\n  rw [NNReal.val_eq_coe, \u2190 snorm_nnreal_eq_snorm' h0p']\n  exact hu.continuous.mem\u2112p_of_hasCompactSupport (\u03bc := \u03bc) h2u |>.snorm_lt_top", "annotated_tactic": ["have h4u : \u222b\u207b x, \u2016u x\u2016\u208a ^ (p' : \u211d) \u2202\u03bc \u2260 \u221e := by\n    refine <a>lintegral_rpow_nnnorm_lt_top_of_snorm'_lt_top</a> (pos_iff_ne_zero.mpr h0p') ?_ |>.<a>ne</a>\n    dsimp only\n    rw [<a>NNReal.val_eq_coe</a>, \u2190 <a>snorm_nnreal_eq_snorm'</a> h0p']\n    exact hu.continuous.mem\u2112p_of_hasCompactSupport (\u03bc := \u03bc) h2u |>.<a>snorm_lt_top</a>", [{"full_name": "MeasureTheory.lintegral_rpow_nnnorm_lt_top_of_snorm'_lt_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [146, 9], "def_end_pos": [146, 54]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [147, 7], "def_end_pos": [147, 15]}, {"full_name": "NNReal.val_eq_coe", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 19]}, {"full_name": "MeasureTheory.snorm_nnreal_eq_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [90, 7], "def_end_pos": [90, 29]}, {"full_name": "MeasureTheory.Mem\u2112p.snorm_lt_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [138, 9], "def_end_pos": [138, 27]}]], "state_before": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have h5u : (\u222b\u207b x, \u2016u x\u2016\u208a ^ (p' : \u211d) \u2202\u03bc) ^ (1 / q) \u2260 0 :=\n  ENNReal.rpow_pos (pos_iff_ne_zero.mpr h3u) h4u |>.ne'", "annotated_tactic": ["have h5u : (\u222b\u207b x, \u2016u x\u2016\u208a ^ (p' : \u211d) \u2202\u03bc) ^ (1 / q) \u2260 0 :=\n    <a>ENNReal.rpow_pos</a> (pos_iff_ne_zero.mpr h3u) h4u |>.<a>ne'</a>", [{"full_name": "ENNReal.rpow_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [808, 9], "def_end_pos": [808, 17]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}]], "state_before": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have h6u : (\u222b\u207b x, \u2016u x\u2016\u208a ^ (p' : \u211d) \u2202\u03bc) ^ (1 / q) \u2260 \u221e :=\n  ENNReal.rpow_ne_top_of_nonneg (div_nonneg zero_le_one hq.symm.nonneg) h4u", "annotated_tactic": ["have h6u : (\u222b\u207b x, \u2016u x\u2016\u208a ^ (p' : \u211d) \u2202\u03bc) ^ (1 / q) \u2260 \u221e :=\n    <a>ENNReal.rpow_ne_top_of_nonneg</a> (<a>div_nonneg</a> <a>zero_le_one</a> hq.symm.nonneg) h4u", [{"full_name": "ENNReal.rpow_ne_top_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [528, 9], "def_end_pos": [528, 30]}, {"full_name": "div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 17]}, {"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\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have h7u := hu.continuous", "annotated_tactic": ["have h7u := hu.continuous", []], "state_before": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have h8u := (hu.fderiv_right (m := 0) le_rfl).continuous", "annotated_tactic": ["have h8u := (hu.fderiv_right (m := 0) <a>le_rfl</a>).<a>continuous</a>", [{"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}, {"full_name": "ContDiff.continuous", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "def_pos": [1519, 9], "def_end_pos": [1519, 28]}]], "state_before": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "let v : E \u2192 \u211d := fun x \u21a6 \u2016u x\u2016 ^ (\u03b3 : \u211d)", "annotated_tactic": ["let v : E \u2192 \u211d := fun x \u21a6 \u2016u x\u2016 ^ (\u03b3 : \u211d)", []], "state_before": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have hv : ContDiff \u211d 1 v := hu.norm_rpow h1\u03b3", "annotated_tactic": ["have hv : <a>ContDiff</a> \u211d 1 v := hu.norm_rpow h1\u03b3", [{"full_name": "ContDiff", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Defs.lean", "def_pos": [1443, 5], "def_end_pos": [1443, 13]}]], "state_before": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "have h2v : HasCompactSupport v := h2u.norm.rpow_const h4\u03b3", "annotated_tactic": ["have h2v : <a>HasCompactSupport</a> v := h2u.norm.rpow_const h4\u03b3", [{"full_name": "HasCompactSupport", "def_path": "Mathlib/Topology/Support.lean", "def_pos": [149, 3], "def_end_pos": [149, 14]}]], "state_before": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "set C := snormLESNormFDerivOneConst \u03bc n'", "annotated_tactic": ["set C := <a>snormLESNormFDerivOneConst</a> \u03bc n'", [{"full_name": "MeasureTheory.snormLESNormFDerivOneConst", "def_path": "Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean", "def_pos": [429, 17], "def_end_pos": [429, 43]}]], "state_before": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc"}, {"tactic": "calc\n  snorm u p' \u03bc = (\u222b\u207b x, \u2016u x\u2016\u208a ^ (p' : \u211d) \u2202\u03bc) ^ (1 / (p' : \u211d)) := snorm_nnreal_eq_lintegral h0p'\n  _ \u2264 C * \u03b3 * (\u222b\u207b x, \u2016fderiv \u211d u x\u2016\u208a ^ (p : \u211d) \u2202\u03bc) ^ (1 / (p : \u211d)) := by\n    rwa [\u2190 h2q, ENNReal.rpow_sub _ _ h3u h4u, ENNReal.div_le_iff h5u h6u]\n  _ = snormLESNormFDerivOfEqInnerConst \u03bc p *  snorm (fderiv \u211d u) (\u2191p) \u03bc := by\n    suffices (C : \u211d) * \u03b3 = snormLESNormFDerivOfEqInnerConst \u03bc p by\n      rw [snorm_nnreal_eq_lintegral h0p]\n      congr\n      norm_cast at this \u22a2\n    simp_rw [snormLESNormFDerivOfEqInnerConst, \u03b3]\n    refold_let n n' C\n    rw [NNReal.coe_mul, NNReal.coe_mk, Real.coe_toNNReal', mul_eq_mul_left_iff, eq_comm,\n      max_eq_left_iff]\n    left\n    positivity", "annotated_tactic": ["calc\n    <a>snorm</a> u p' \u03bc = (\u222b\u207b x, \u2016u x\u2016\u208a ^ (p' : \u211d) \u2202\u03bc) ^ (1 / (p' : \u211d)) := <a>snorm_nnreal_eq_lintegral</a> h0p'\n    _ \u2264 C * \u03b3 * (\u222b\u207b x, \u2016<a>fderiv</a> \u211d u x\u2016\u208a ^ (p : \u211d) \u2202\u03bc) ^ (1 / (p : \u211d)) := by\n      rwa [\u2190 h2q, <a>ENNReal.rpow_sub</a> _ _ h3u h4u, <a>ENNReal.div_le_iff</a> h5u h6u]\n    _ = <a>snormLESNormFDerivOfEqInnerConst</a> \u03bc p *  <a>snorm</a> (<a>fderiv</a> \u211d u) (\u2191p) \u03bc := by\n      suffices (C : \u211d) * \u03b3 = <a>snormLESNormFDerivOfEqInnerConst</a> \u03bc p by\n        rw [<a>snorm_nnreal_eq_lintegral</a> h0p]\n        congr\n        norm_cast at this \u22a2\n      simp_rw [<a>snormLESNormFDerivOfEqInnerConst</a>, \u03b3]\n      refold_let n n' C\n      rw [<a>NNReal.coe_mul</a>, <a>NNReal.coe_mk</a>, <a>Real.coe_toNNReal'</a>, <a>mul_eq_mul_left_iff</a>, <a>eq_comm</a>,\n        <a>max_eq_left_iff</a>]\n      left\n      positivity", [{"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [82, 5], "def_end_pos": [82, 10]}, {"full_name": "MeasureTheory.snorm_nnreal_eq_lintegral", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [98, 7], "def_end_pos": [98, 32]}, {"full_name": "fderiv", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [192, 17], "def_end_pos": [192, 23]}, {"full_name": "ENNReal.rpow_sub", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [564, 9], "def_end_pos": [564, 17]}, {"full_name": "ENNReal.div_le_iff", "def_path": "Mathlib/Data/ENNReal/Inv.lean", "def_pos": [165, 19], "def_end_pos": [165, 29]}, {"full_name": "MeasureTheory.snormLESNormFDerivOfEqInnerConst", "def_path": "Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean", "def_pos": [449, 5], "def_end_pos": [449, 37]}, {"full_name": "MeasureTheory.snorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [82, 5], "def_end_pos": [82, 10]}, {"full_name": "fderiv", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [192, 17], "def_end_pos": [192, 23]}, {"full_name": "MeasureTheory.snormLESNormFDerivOfEqInnerConst", "def_path": "Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean", "def_pos": [449, 5], "def_end_pos": [449, 37]}, {"full_name": "MeasureTheory.snorm_nnreal_eq_lintegral", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [98, 7], "def_end_pos": [98, 32]}, {"full_name": "MeasureTheory.snormLESNormFDerivOfEqInnerConst", "def_path": "Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean", "def_pos": [449, 5], "def_end_pos": [449, 37]}, {"full_name": "NNReal.coe_mul", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [185, 19], "def_end_pos": [185, 26]}, {"full_name": "NNReal.coe_mk", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [139, 28], "def_end_pos": [139, 34]}, {"full_name": "Real.coe_toNNReal'", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [638, 9], "def_end_pos": [638, 22]}, {"full_name": "mul_eq_mul_left_iff", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [231, 9], "def_end_pos": [231, 28]}, {"full_name": "eq_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}, {"full_name": "max_eq_left_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [153, 9], "def_end_pos": [153, 24]}]], "state_before": "case neg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nthis :\n  (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / \u2191n') \u2264\n    \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) * (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q)\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "no goals"}, {"tactic": "simp [hp'0]", "annotated_tactic": ["simp [hp'0]", []], "state_before": "case pos\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhn : 0 < finrank \u211d E\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191(finrank \u211d E))\u207b\u00b9\nhp'0 : p' = 0\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "no goals"}, {"tactic": "have : 0 < p\u207b\u00b9 - (n : \u211d)\u207b\u00b9 :=\n  NNReal.coe_lt_coe.mpr (pos_iff_ne_zero.mpr (inv_ne_zero hp'0)) |>.trans_eq hp'", "annotated_tactic": ["have : 0 < p\u207b\u00b9 - (n : \u211d)\u207b\u00b9 :=\n      NNReal.coe_lt_coe.mpr (pos_iff_ne_zero.mpr (<a>inv_ne_zero</a> hp'0)) |>.<a>trans_eq</a> hp'", [{"full_name": "inv_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/NeZero.lean", "def_pos": [49, 9], "def_end_pos": [49, 20]}, {"full_name": "LT.lt.trans_eq", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [194, 7], "def_end_pos": [194, 21]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\n\u22a2 \u2191p < \u2191n", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nthis : 0 < \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\n\u22a2 \u2191p < \u2191n"}, {"tactic": "simp [sub_pos] at this", "annotated_tactic": ["simp [<a>sub_pos</a>] at this", [{"full_name": "sub_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [886, 30], "def_end_pos": [886, 37]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nthis : 0 < \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\n\u22a2 \u2191p < \u2191n", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nthis : (\u2191n)\u207b\u00b9 < (\u2191p)\u207b\u00b9\n\u22a2 \u2191p < \u2191n"}, {"tactic": "rwa [inv_lt_inv _ (zero_lt_one.trans_le (NNReal.coe_le_coe.mpr hp))] at this", "annotated_tactic": ["rwa [<a>inv_lt_inv</a> _ (zero_lt_one.trans_le (NNReal.coe_le_coe.mpr hp))] at this", [{"full_name": "inv_lt_inv", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [204, 9], "def_end_pos": [204, 19]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nthis : (\u2191n)\u207b\u00b9 < (\u2191p)\u207b\u00b9\n\u22a2 \u2191p < \u2191n", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nthis : (\u2191n)\u207b\u00b9 < (\u2191p)\u207b\u00b9\n\u22a2 0 < \u2191n"}, {"tactic": "exact_mod_cast hn", "annotated_tactic": ["exact_mod_cast hn", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nthis : (\u2191n)\u207b\u00b9 < (\u2191p)\u207b\u00b9\n\u22a2 0 < \u2191n", "state_after": "no goals"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\n\u22a2 1 < \u2191n", "state_after": "no goals"}, {"tactic": "simp_rw [sub_pos]", "annotated_tactic": ["simp_rw [<a>sub_pos</a>]", [{"full_name": "sub_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [886, 30], "def_end_pos": [886, 37]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\n\u22a2 0 < \u2191n - 1", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\n\u22a2 1 < \u2191n"}, {"tactic": "exact hn.coe.one_lt", "annotated_tactic": ["exact hn.coe.one_lt", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\n\u22a2 1 < \u2191n", "state_after": "no goals"}, {"tactic": "simp_rw [sub_pos]", "annotated_tactic": ["simp_rw [<a>sub_pos</a>]", [{"full_name": "sub_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [886, 30], "def_end_pos": [886, 37]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\n\u22a2 0 < \u2191n - \u2191p", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\n\u22a2 \u2191p < \u2191n"}, {"tactic": "exact h2p", "annotated_tactic": ["exact h2p", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\n\u22a2 \u2191p < \u2191n", "state_after": "no goals"}, {"tactic": "convert snorm_le_snorm_fderiv_one \u03bc hu h2u hn using 2", "annotated_tactic": ["convert <a>snorm_le_snorm_fderiv_one</a> \u03bc hu h2u hn using 2", [{"full_name": "MeasureTheory.snorm_le_snorm_fderiv_one", "def_path": "Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean", "def_pos": [436, 9], "def_end_pos": [436, 34]}]], "state_before": "case neg.inl\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np' : \u211d\u22650\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhp : 1 \u2264 1\nhp' : (\u2191p')\u207b\u00b9 = \u21911\u207b\u00b9 - (\u2191n)\u207b\u00b9\nh2p : \u21911 < \u2191n\nhnp : 0 < \u2191n - \u21911\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u21911) * snorm (fderiv \u211d u) (\u21911) \u03bc", "state_after": "case h.e'_3.h.e'_6\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np' : \u211d\u22650\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhp : 1 \u2264 1\nhp' : (\u2191p')\u207b\u00b9 = \u21911\u207b\u00b9 - (\u2191n)\u207b\u00b9\nh2p : \u21911 < \u2191n\nhnp : 0 < \u2191n - \u21911\n\u22a2 \u2191p' = \u2191n'\n\ncase h.e'_4.h.e'_5\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np' : \u211d\u22650\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhp : 1 \u2264 1\nhp' : (\u2191p')\u207b\u00b9 = \u21911\u207b\u00b9 - (\u2191n)\u207b\u00b9\nh2p : \u21911 < \u2191n\nhnp : 0 < \u2191n - \u21911\n\u22a2 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u21911) = \u2191(snormLESNormFDerivOneConst \u03bc \u2191n')"}, {"tactic": "rw [\u2190 inv_inj, hp']", "annotated_tactic": ["rw [\u2190 <a>inv_inj</a>, hp']", [{"full_name": "inv_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [407, 9], "def_end_pos": [407, 16]}]], "state_before": "case h.e'_3.h.e'_6\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np' : \u211d\u22650\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhp : 1 \u2264 1\nhp' : (\u2191p')\u207b\u00b9 = \u21911\u207b\u00b9 - (\u2191n)\u207b\u00b9\nh2p : \u21911 < \u2191n\nhnp : 0 < \u2191n - \u21911\n\u22a2 \u2191p' = \u2191n'", "state_after": "case h.e'_3.h.e'_6\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np' : \u211d\u22650\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhp : 1 \u2264 1\nhp' : (\u2191p')\u207b\u00b9 = \u21911\u207b\u00b9 - (\u2191n)\u207b\u00b9\nh2p : \u21911 < \u2191n\nhnp : 0 < \u2191n - \u21911\n\u22a2 \u21911\u207b\u00b9 - (\u2191n)\u207b\u00b9 = (\u2191n')\u207b\u00b9"}, {"tactic": "field_simp [n', NNReal.conjExponent]", "annotated_tactic": ["field_simp [n', <a>NNReal.conjExponent</a>]", [{"full_name": "NNReal.conjExponent", "def_path": "Mathlib/Data/Real/ConjExponents.lean", "def_pos": [159, 19], "def_end_pos": [159, 31]}]], "state_before": "case h.e'_3.h.e'_6\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np' : \u211d\u22650\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhp : 1 \u2264 1\nhp' : (\u2191p')\u207b\u00b9 = \u21911\u207b\u00b9 - (\u2191n)\u207b\u00b9\nh2p : \u21911 < \u2191n\nhnp : 0 < \u2191n - \u21911\n\u22a2 \u21911\u207b\u00b9 - (\u2191n)\u207b\u00b9 = (\u2191n')\u207b\u00b9", "state_after": "no goals"}, {"tactic": "simpa using this", "annotated_tactic": ["simpa using this", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np' : \u211d\u22650\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhp : 1 \u2264 1\nhp' : (\u2191p')\u207b\u00b9 = \u21911\u207b\u00b9 - (\u2191n)\u207b\u00b9\nh2p : \u21911 < \u2191n\nhnp : 0 < \u2191n - \u21911\nthis : \u2191p' = \u2191n'\n\u22a2 \u2191p' = \u2191n'", "state_after": "no goals"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "case h.e'_4.h.e'_5\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np' : \u211d\u22650\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhp : 1 \u2264 1\nhp' : (\u2191p')\u207b\u00b9 = \u21911\u207b\u00b9 - (\u2191n)\u207b\u00b9\nh2p : \u21911 < \u2191n\nhnp : 0 < \u2191n - \u21911\n\u22a2 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u21911) = \u2191(snormLESNormFDerivOneConst \u03bc \u2191n')", "state_after": "case h.e'_4.h.e'_5\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np' : \u211d\u22650\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhp : 1 \u2264 1\nhp' : (\u2191p')\u207b\u00b9 = \u21911\u207b\u00b9 - (\u2191n)\u207b\u00b9\nh2p : \u21911 < \u2191n\nhnp : 0 < \u2191n - \u21911\n\u22a2 snormLESNormFDerivOfEqInnerConst \u03bc 1 = snormLESNormFDerivOneConst \u03bc \u2191n'"}, {"tactic": "simp_rw [snormLESNormFDerivOfEqInnerConst]", "annotated_tactic": ["simp_rw [<a>snormLESNormFDerivOfEqInnerConst</a>]", [{"full_name": "MeasureTheory.snormLESNormFDerivOfEqInnerConst", "def_path": "Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean", "def_pos": [449, 5], "def_end_pos": [449, 37]}]], "state_before": "case h.e'_4.h.e'_5\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np' : \u211d\u22650\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhp : 1 \u2264 1\nhp' : (\u2191p')\u207b\u00b9 = \u21911\u207b\u00b9 - (\u2191n)\u207b\u00b9\nh2p : \u21911 < \u2191n\nhnp : 0 < \u2191n - \u21911\n\u22a2 snormLESNormFDerivOfEqInnerConst \u03bc 1 = snormLESNormFDerivOneConst \u03bc \u2191n'", "state_after": "case h.e'_4.h.e'_5\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np' : \u211d\u22650\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhp : 1 \u2264 1\nhp' : (\u2191p')\u207b\u00b9 = \u21911\u207b\u00b9 - (\u2191n)\u207b\u00b9\nh2p : \u21911 < \u2191n\nhnp : 0 < \u2191n - \u21911\n\u22a2 snormLESNormFDerivOneConst \u03bc \u2191(\u2191(finrank \u211d E)).conjExponent *\n      (1 * (\u2191(finrank \u211d E) - 1) / (\u2191(finrank \u211d E) - 1)).toNNReal =\n    snormLESNormFDerivOneConst \u03bc \u2191n'"}, {"tactic": "field_simp", "annotated_tactic": ["field_simp", []], "state_before": "case h.e'_4.h.e'_5\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np' : \u211d\u22650\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhp : 1 \u2264 1\nhp' : (\u2191p')\u207b\u00b9 = \u21911\u207b\u00b9 - (\u2191n)\u207b\u00b9\nh2p : \u21911 < \u2191n\nhnp : 0 < \u2191n - \u21911\n\u22a2 snormLESNormFDerivOneConst \u03bc \u2191(\u2191(finrank \u211d E)).conjExponent *\n      (1 * (\u2191(finrank \u211d E) - 1) / (\u2191(finrank \u211d E) - 1)).toNNReal =\n    snormLESNormFDerivOneConst \u03bc \u2191n'", "state_after": "no goals"}, {"tactic": "suffices 0 < (p' : \u211d) from (show 0 < p' from this) |>.ne'", "annotated_tactic": ["suffices 0 < (p' : \u211d) from (show 0 < p' from this) |>.<a>ne'</a>", [{"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\n\u22a2 p' \u2260 0", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\n\u22a2 0 < \u2191p'"}, {"tactic": "rw [\u2190 inv_pos, hp', sub_pos]", "annotated_tactic": ["rw [\u2190 <a>inv_pos</a>, hp', <a>sub_pos</a>]", [{"full_name": "inv_pos", "def_path": "Mathlib/Algebra/Order/Field/Defs.lean", "def_pos": [47, 15], "def_end_pos": [47, 22]}, {"full_name": "sub_pos", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [886, 30], "def_end_pos": [886, 37]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\n\u22a2 0 < \u2191p'", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\n\u22a2 (\u2191n)\u207b\u00b9 < \u2191p\u207b\u00b9"}, {"tactic": "exact inv_lt_inv_of_lt hq.pos h2p", "annotated_tactic": ["exact <a>inv_lt_inv_of_lt</a> hq.pos h2p", [{"full_name": "inv_lt_inv_of_lt", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 25]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\n\u22a2 (\u2191n)\u207b\u00b9 < \u2191p\u207b\u00b9", "state_after": "no goals"}, {"tactic": "simp_rw (config := {zeta := false}) [one_div, hp']", "annotated_tactic": ["simp_rw (config := {zeta := <a>false</a>}) [<a>one_div</a>, hp']", [{"full_name": "Bool.false", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [569, 5], "def_end_pos": [569, 10]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 16]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\n\u22a2 1 / \u2191n' - 1 / q = 1 / \u2191p'", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\n\u22a2 (\u2191n')\u207b\u00b9 - q\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9"}, {"tactic": "rw [\u2190 hq.one_sub_inv, \u2190 hn.coe.one_sub_inv, sub_sub_sub_cancel_left]", "annotated_tactic": ["rw [\u2190 hq.one_sub_inv, \u2190 hn.coe.one_sub_inv, <a>sub_sub_sub_cancel_left</a>]", [{"full_name": "sub_sub_sub_cancel_left", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1374, 3], "def_end_pos": [1374, 14]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\n\u22a2 (\u2191n')\u207b\u00b9 - q\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\n\u22a2 (\u2191p)\u207b\u00b9 - (\u2191\u2191n)\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9"}, {"tactic": "simp only [NNReal.coe_natCast, NNReal.coe_inv]", "annotated_tactic": ["simp only [<a>NNReal.coe_natCast</a>, <a>NNReal.coe_inv</a>]", [{"full_name": "NNReal.coe_natCast", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [352, 19], "def_end_pos": [352, 30]}, {"full_name": "NNReal.coe_inv", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [190, 19], "def_end_pos": [190, 26]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\n\u22a2 (\u2191p)\u207b\u00b9 - (\u2191\u2191n)\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u22a2 0 \u2264 \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)", "state_after": "no goals"}, {"tactic": "rwa [h0\u03b3, one_lt_div hnp, mul_sub, mul_one, sub_lt_sub_iff_right, lt_mul_iff_one_lt_left]", "annotated_tactic": ["rwa [h0\u03b3, <a>one_lt_div</a> hnp, <a>mul_sub</a>, <a>mul_one</a>, <a>sub_lt_sub_iff_right</a>, <a>lt_mul_iff_one_lt_left</a>]", [{"full_name": "one_lt_div", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [369, 9], "def_end_pos": [369, 19]}, {"full_name": "mul_sub", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [394, 7], "def_end_pos": [394, 14]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "sub_lt_sub_iff_right", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [874, 3], "def_end_pos": [874, 14]}, {"full_name": "lt_mul_iff_one_lt_left", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [702, 9], "def_end_pos": [702, 31]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\n\u22a2 1 < \u2191\u03b3", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\n\u22a2 0 < \u2191n"}, {"tactic": "exact hn.coe.pos", "annotated_tactic": ["exact hn.coe.pos", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\n\u22a2 0 < \u2191n", "state_after": "no goals"}, {"tactic": "rw [\u2190 NNReal.coe_inj, \u2190 inv_inj, hp', NNReal.coe_mul, h0\u03b3, hn.coe.conj_eq]", "annotated_tactic": ["rw [\u2190 <a>NNReal.coe_inj</a>, \u2190 <a>inv_inj</a>, hp', <a>NNReal.coe_mul</a>, h0\u03b3, hn.coe.conj_eq]", [{"full_name": "NNReal.coe_inj", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [167, 26], "def_end_pos": [167, 33]}, {"full_name": "inv_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [407, 9], "def_end_pos": [407, 16]}, {"full_name": "NNReal.coe_mul", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [185, 19], "def_end_pos": [185, 26]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\n\u22a2 \u03b3 * n' = p'", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\n\u22a2 (\u2191p * (\u2191n - 1) / (\u2191n - \u2191p) * (\u2191\u2191n / (\u2191\u2191n - 1)))\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9"}, {"tactic": "field_simp", "annotated_tactic": ["field_simp", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\n\u22a2 (\u2191p * (\u2191n - 1) / (\u2191n - \u2191p) * (\u2191\u2191n / (\u2191\u2191n - 1)))\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\n\u22a2 (\u2191n - \u2191p) * (\u2191n - 1) * (\u2191p * \u2191n) = (\u2191n - \u2191p) * (\u2191p * (\u2191n - 1) * \u2191n)"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\n\u22a2 (\u2191n - \u2191p) * (\u2191n - 1) * (\u2191p * \u2191n) = (\u2191n - \u2191p) * (\u2191p * (\u2191n - 1) * \u2191n)", "state_after": "no goals"}, {"tactic": "rw [\u2190 inv_inj, hp', h0\u03b3, hq.conj_eq]", "annotated_tactic": ["rw [\u2190 <a>inv_inj</a>, hp', h0\u03b3, hq.conj_eq]", [{"full_name": "inv_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [407, 9], "def_end_pos": [407, 16]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\n\u22a2 (\u2191\u03b3 - 1) * q = \u2191p'", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\n\u22a2 ((\u2191p * (\u2191n - 1) / (\u2191n - \u2191p) - 1) * (\u2191p / (\u2191p - 1)))\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9"}, {"tactic": "have : (p : \u211d) * (n - 1) - (n - p) = n * (p - 1) := by ring", "annotated_tactic": ["have : (p : \u211d) * (n - 1) - (n - p) = n * (p - 1) := by ring", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\n\u22a2 ((\u2191p * (\u2191n - 1) / (\u2191n - \u2191p) - 1) * (\u2191p / (\u2191p - 1)))\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nthis : \u2191p * (\u2191n - 1) - (\u2191n - \u2191p) = \u2191n * (\u2191p - 1)\n\u22a2 ((\u2191p * (\u2191n - 1) / (\u2191n - \u2191p) - 1) * (\u2191p / (\u2191p - 1)))\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9"}, {"tactic": "field_simp [this]", "annotated_tactic": ["field_simp [this]", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nthis : \u2191p * (\u2191n - 1) - (\u2191n - \u2191p) = \u2191n * (\u2191p - 1)\n\u22a2 ((\u2191p * (\u2191n - 1) / (\u2191n - \u2191p) - 1) * (\u2191p / (\u2191p - 1)))\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nthis : \u2191p * (\u2191n - 1) - (\u2191n - \u2191p) = \u2191n * (\u2191p - 1)\n\u22a2 (\u2191n - \u2191p) * (\u2191p - 1) * (\u2191p * \u2191n) = (\u2191n - \u2191p) * (\u2191n * (\u2191p - 1) * \u2191p)"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nthis : \u2191p * (\u2191n - 1) - (\u2191n - \u2191p) = \u2191n * (\u2191p - 1)\n\u22a2 (\u2191n - \u2191p) * (\u2191p - 1) * (\u2191p * \u2191n) = (\u2191n - \u2191p) * (\u2191n * (\u2191p - 1) * \u2191p)", "state_after": "no goals"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\n\u22a2 \u2191p * (\u2191n - 1) - (\u2191n - \u2191p) = \u2191n * (\u2191p - 1)", "state_after": "no goals"}, {"tactic": "rw [snorm_nnreal_eq_lintegral h0p', h3u, ENNReal.zero_rpow_of_pos] <;> positivity", "annotated_tactic": ["rw [<a>snorm_nnreal_eq_lintegral</a> h0p', h3u, <a>ENNReal.zero_rpow_of_pos</a>] <;> positivity", [{"full_name": "MeasureTheory.snorm_nnreal_eq_lintegral", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [98, 7], "def_end_pos": [98, 32]}, {"full_name": "ENNReal.zero_rpow_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [424, 9], "def_end_pos": [424, 25]}]], "state_before": "case pos\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\n\u22a2 snorm u (\u2191p') \u03bc \u2264 \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "no goals"}, {"tactic": "refine lintegral_rpow_nnnorm_lt_top_of_snorm'_lt_top (pos_iff_ne_zero.mpr h0p') ?_ |>.ne", "annotated_tactic": ["refine <a>lintegral_rpow_nnnorm_lt_top_of_snorm'_lt_top</a> (pos_iff_ne_zero.mpr h0p') ?_ |>.<a>ne</a>", [{"full_name": "MeasureTheory.lintegral_rpow_nnnorm_lt_top_of_snorm'_lt_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [146, 9], "def_end_pos": [146, 54]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [147, 7], "def_end_pos": [147, 15]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\n\u22a2 \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\n\u22a2 snorm' u ((fun a => \u2191a) p') \u03bc < \u22a4"}, {"tactic": "dsimp only", "annotated_tactic": ["dsimp only", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\n\u22a2 snorm' u ((fun a => \u2191a) p') \u03bc < \u22a4", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\n\u22a2 snorm' u (\u2191p') \u03bc < \u22a4"}, {"tactic": "rw [NNReal.val_eq_coe, \u2190 snorm_nnreal_eq_snorm' h0p']", "annotated_tactic": ["rw [<a>NNReal.val_eq_coe</a>, \u2190 <a>snorm_nnreal_eq_snorm'</a> h0p']", [{"full_name": "NNReal.val_eq_coe", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 19]}, {"full_name": "MeasureTheory.snorm_nnreal_eq_snorm'", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [90, 7], "def_end_pos": [90, 29]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\n\u22a2 snorm' u (\u2191p') \u03bc < \u22a4", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\n\u22a2 snorm u (\u2191p') \u03bc < \u22a4"}, {"tactic": "exact hu.continuous.mem\u2112p_of_hasCompactSupport (\u03bc := \u03bc) h2u |>.snorm_lt_top", "annotated_tactic": ["exact hu.continuous.mem\u2112p_of_hasCompactSupport (\u03bc := \u03bc) h2u |>.<a>snorm_lt_top</a>", [{"full_name": "MeasureTheory.Mem\u2112p.snorm_lt_top", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [138, 9], "def_end_pos": [138, 27]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\n\u22a2 snorm u (\u2191p') \u03bc < \u22a4", "state_after": "no goals"}, {"tactic": "rw [\u2190 h2\u03b3, snorm_nnreal_eq_lintegral hn.symm.pos.ne']", "annotated_tactic": ["rw [\u2190 h2\u03b3, <a>snorm_nnreal_eq_lintegral</a> hn.symm.pos.ne']", [{"full_name": "MeasureTheory.snorm_nnreal_eq_lintegral", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [98, 7], "def_end_pos": [98, 32]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / \u2191n') = snorm v (\u2191n') \u03bc", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191(\u03b3 * n') \u2202\u03bc) ^ (1 / \u2191n') = (\u222b\u207b (x : E), \u2191\u2016v x\u2016\u208a ^ \u2191n' \u2202\u03bc) ^ (1 / \u2191n')"}, {"tactic": "simp (discharger := positivity) [v, Real.nnnorm_rpow_of_nonneg, ENNReal.rpow_mul,\n  ENNReal.coe_rpow_of_nonneg]", "annotated_tactic": ["simp (discharger := positivity) [v, <a>Real.nnnorm_rpow_of_nonneg</a>, <a>ENNReal.rpow_mul</a>,\n          <a>ENNReal.coe_rpow_of_nonneg</a>]", [{"full_name": "Real.nnnorm_rpow_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [379, 9], "def_end_pos": [379, 42]}, {"full_name": "ENNReal.rpow_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [571, 9], "def_end_pos": [571, 17]}, {"full_name": "ENNReal.coe_rpow_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [459, 9], "def_end_pos": [459, 27]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191(\u03b3 * n') \u2202\u03bc) ^ (1 / \u2191n') = (\u222b\u207b (x : E), \u2191\u2016v x\u2016\u208a ^ \u2191n' \u2202\u03bc) ^ (1 / \u2191n')", "state_after": "no goals"}, {"tactic": "rw [snorm_one_eq_lintegral_nnnorm]", "annotated_tactic": ["rw [<a>snorm_one_eq_lintegral_nnnorm</a>]", [{"full_name": "MeasureTheory.snorm_one_eq_lintegral_nnnorm", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [102, 9], "def_end_pos": [102, 38]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 \u2191C * snorm (fderiv \u211d v) 1 \u03bc = \u2191C * \u222b\u207b (x : E), \u2191\u2016fderiv \u211d v x\u2016\u208a \u2202\u03bc", "state_after": "no goals"}, {"tactic": "rw [mul_assoc, \u2190 lintegral_const_mul \u03b3]", "annotated_tactic": ["rw [<a>mul_assoc</a>, \u2190 <a>lintegral_const_mul</a> \u03b3]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "MeasureTheory.lintegral_const_mul", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [716, 9], "def_end_pos": [716, 28]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 \u2191C * \u222b\u207b (x : E), \u2191\u2016fderiv \u211d v x\u2016\u208a \u2202\u03bc \u2264 \u2191C * \u2191\u03b3 * \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ (\u2191\u03b3 - 1) * \u2191\u2016fderiv \u211d u x\u2016\u208a \u2202\u03bc", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 \u2191C * \u222b\u207b (x : E), \u2191\u2016fderiv \u211d v x\u2016\u208a \u2202\u03bc \u2264 \u2191C * \u222b\u207b (a : E), \u2191\u03b3 * (\u2191\u2016u a\u2016\u208a ^ (\u2191\u03b3 - 1) * \u2191\u2016fderiv \u211d u a\u2016\u208a) \u2202\u03bc\n\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 Measurable fun x => \u2191\u2016u x\u2016\u208a ^ (\u2191\u03b3 - 1) * \u2191\u2016fderiv \u211d u x\u2016\u208a"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 \u2191C * \u222b\u207b (x : E), \u2191\u2016fderiv \u211d v x\u2016\u208a \u2202\u03bc \u2264 \u2191C * \u222b\u207b (a : E), \u2191\u03b3 * (\u2191\u2016u a\u2016\u208a ^ (\u2191\u03b3 - 1) * \u2191\u2016fderiv \u211d u a\u2016\u208a) \u2202\u03bc\n\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 Measurable fun x => \u2191\u2016u x\u2016\u208a ^ (\u2191\u03b3 - 1) * \u2191\u2016fderiv \u211d u x\u2016\u208a", "state_after": "case bc.hfg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nx\u271d : E\n\u22a2 \u2191\u2016fderiv \u211d v x\u271d\u2016\u208a \u2264 \u2191\u03b3 * (\u2191\u2016u x\u271d\u2016\u208a ^ (\u2191\u03b3 - 1) * \u2191\u2016fderiv \u211d u x\u271d\u2016\u208a)\n\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 Measurable fun x => \u2191\u2016u x\u2016\u208a ^ (\u2191\u03b3 - 1) * \u2191\u2016fderiv \u211d u x\u2016\u208a"}, {"tactic": "simp_rw [\u2190 mul_assoc, ENNReal.coe_rpow_of_nonneg _ (sub_nonneg.mpr h1\u03b3.le)]", "annotated_tactic": ["simp_rw [\u2190 <a>mul_assoc</a>, <a>ENNReal.coe_rpow_of_nonneg</a> _ (sub_nonneg.mpr h1\u03b3.le)]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "ENNReal.coe_rpow_of_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [459, 9], "def_end_pos": [459, 27]}]], "state_before": "case bc.hfg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nx\u271d : E\n\u22a2 \u2191\u2016fderiv \u211d v x\u271d\u2016\u208a \u2264 \u2191\u03b3 * (\u2191\u2016u x\u271d\u2016\u208a ^ (\u2191\u03b3 - 1) * \u2191\u2016fderiv \u211d u x\u271d\u2016\u208a)\n\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 Measurable fun x => \u2191\u2016u x\u2016\u208a ^ (\u2191\u03b3 - 1) * \u2191\u2016fderiv \u211d u x\u2016\u208a", "state_after": "case bc.hfg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nx\u271d : E\n\u22a2 \u2191\u2016fderiv \u211d v x\u271d\u2016\u208a \u2264 \u2191\u03b3 * \u2191(\u2016u x\u271d\u2016\u208a ^ (\u2191\u03b3 - 1)) * \u2191\u2016fderiv \u211d u x\u271d\u2016\u208a\n\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 Measurable fun x => \u2191\u2016u x\u2016\u208a ^ (\u2191\u03b3 - 1) * \u2191\u2016fderiv \u211d u x\u2016\u208a"}, {"tactic": "exact ENNReal.coe_le_coe.mpr <| nnnorm_fderiv_norm_rpow_le (hu.differentiable le_rfl) h1\u03b3", "annotated_tactic": ["exact ENNReal.coe_le_coe.mpr <| <a>nnnorm_fderiv_norm_rpow_le</a> (hu.differentiable <a>le_rfl</a>) h1\u03b3", [{"full_name": "nnnorm_fderiv_norm_rpow_le", "def_path": "Mathlib/Analysis/InnerProductSpace/NormPow.lean", "def_pos": [96, 9], "def_end_pos": [96, 35]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}]], "state_before": "case bc.hfg\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nx\u271d : E\n\u22a2 \u2191\u2016fderiv \u211d v x\u271d\u2016\u208a \u2264 \u2191\u03b3 * \u2191(\u2016u x\u271d\u2016\u208a ^ (\u2191\u03b3 - 1)) * \u2191\u2016fderiv \u211d u x\u271d\u2016\u208a\n\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 Measurable fun x => \u2191\u2016u x\u2016\u208a ^ (\u2191\u03b3 - 1) * \u2191\u2016fderiv \u211d u x\u2016\u208a", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 Measurable fun x => \u2191\u2016u x\u2016\u208a ^ (\u2191\u03b3 - 1) * \u2191\u2016fderiv \u211d u x\u2016\u208a"}, {"tactic": "fun_prop", "annotated_tactic": ["fun_prop", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 Measurable fun x => \u2191\u2016u x\u2016\u208a ^ (\u2191\u03b3 - 1) * \u2191\u2016fderiv \u211d u x\u2016\u208a", "state_after": "no goals"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 \u2191C * \u2191\u03b3 * \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ (\u2191\u03b3 - 1) * \u2191\u2016fderiv \u211d u x\u2016\u208a \u2202\u03bc \u2264\n    \u2191C * \u2191\u03b3 * ((\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p))", "state_after": "case bc\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ (\u2191\u03b3 - 1) * \u2191\u2016fderiv \u211d u x\u2016\u208a \u2202\u03bc \u2264\n    (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p)"}, {"tactic": "convert ENNReal.lintegral_mul_le_Lp_mul_Lq \u03bc\n  (.symm <| .conjExponent <| show 1 < (p : \u211d) from hp) ?_ ?_ using 5", "annotated_tactic": ["convert <a>ENNReal.lintegral_mul_le_Lp_mul_Lq</a> \u03bc\n          (.symm <| .conjExponent <| show 1 < (p : \u211d) from hp) ?_ ?_ using 5", [{"full_name": "ENNReal.lintegral_mul_le_Lp_mul_Lq", "def_path": "Mathlib/MeasureTheory/Integral/MeanInequalities.lean", "def_pos": [162, 9], "def_end_pos": [162, 35]}]], "state_before": "case bc\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ (\u2191\u03b3 - 1) * \u2191\u2016fderiv \u211d u x\u2016\u208a \u2202\u03bc \u2264\n    (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p)", "state_after": "case h.e'_4.h.e'_5.h.e'_5.h.e'_4.h\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nx\u271d : E\n\u22a2 \u2191\u2016u x\u271d\u2016\u208a ^ \u2191p' = (\u2191\u2016u x\u271d\u2016\u208a ^ (\u2191\u03b3 - 1)) ^ (\u2191p).conjExponent\n\ncase bc.convert_3\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 AEMeasurable (fun x => \u2191\u2016u x\u2016\u208a ^ (\u2191\u03b3 - 1)) \u03bc\n\ncase bc.convert_4\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 AEMeasurable (fun x => \u2191\u2016fderiv \u211d u x\u2016\u208a) \u03bc"}, {"tactic": "simp_rw [\u2190 ENNReal.rpow_mul, \u2190 h3\u03b3]", "annotated_tactic": ["simp_rw [\u2190 <a>ENNReal.rpow_mul</a>, \u2190 h3\u03b3]", [{"full_name": "ENNReal.rpow_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [571, 9], "def_end_pos": [571, 17]}]], "state_before": "case h.e'_4.h.e'_5.h.e'_5.h.e'_4.h\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nx\u271d : E\n\u22a2 \u2191\u2016u x\u271d\u2016\u208a ^ \u2191p' = (\u2191\u2016u x\u271d\u2016\u208a ^ (\u2191\u03b3 - 1)) ^ (\u2191p).conjExponent", "state_after": "no goals"}, {"tactic": "borelize F'", "annotated_tactic": ["borelize F'", []], "state_before": "case bc.convert_3\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 AEMeasurable (fun x => \u2191\u2016u x\u2016\u208a ^ (\u2191\u03b3 - 1)) \u03bc", "state_after": "case bc.convert_3\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nthis\u271d\u00b9 : MeasurableSpace F' := borel F'\nthis\u271d : BorelSpace F'\n\u22a2 AEMeasurable (fun x => \u2191\u2016u x\u2016\u208a ^ (\u2191\u03b3 - 1)) \u03bc"}, {"tactic": "fun_prop", "annotated_tactic": ["fun_prop", []], "state_before": "case bc.convert_3\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nthis\u271d\u00b9 : MeasurableSpace F' := borel F'\nthis\u271d : BorelSpace F'\n\u22a2 AEMeasurable (fun x => \u2191\u2016u x\u2016\u208a ^ (\u2191\u03b3 - 1)) \u03bc", "state_after": "no goals"}, {"tactic": "fun_prop", "annotated_tactic": ["fun_prop", []], "state_before": "case bc.convert_4\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 AEMeasurable (fun x => \u2191\u2016fderiv \u211d u x\u2016\u208a) \u03bc", "state_after": "no goals"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\n\u22a2 \u2191C * \u2191\u03b3 * ((\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p)) =\n    \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) * (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q)", "state_after": "no goals"}, {"tactic": "rwa [\u2190 h2q, ENNReal.rpow_sub _ _ h3u h4u, ENNReal.div_le_iff h5u h6u]", "annotated_tactic": ["rwa [\u2190 h2q, <a>ENNReal.rpow_sub</a> _ _ h3u h4u, <a>ENNReal.div_le_iff</a> h5u h6u]", [{"full_name": "ENNReal.rpow_sub", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [564, 9], "def_end_pos": [564, 17]}, {"full_name": "ENNReal.div_le_iff", "def_path": "Mathlib/Data/ENNReal/Inv.lean", "def_pos": [165, 19], "def_end_pos": [165, 29]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nthis :\n  (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / \u2191n') \u2264\n    \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) * (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q)\n\u22a2 (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / \u2191p') \u2264 \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p)", "state_after": "no goals"}, {"tactic": "suffices (C : \u211d) * \u03b3 = snormLESNormFDerivOfEqInnerConst \u03bc p by\n  rw [snorm_nnreal_eq_lintegral h0p]\n  congr\n  norm_cast at this \u22a2", "annotated_tactic": ["suffices (C : \u211d) * \u03b3 = <a>snormLESNormFDerivOfEqInnerConst</a> \u03bc p by\n        rw [<a>snorm_nnreal_eq_lintegral</a> h0p]\n        congr\n        norm_cast at this \u22a2", [{"full_name": "MeasureTheory.snormLESNormFDerivOfEqInnerConst", "def_path": "Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean", "def_pos": [449, 5], "def_end_pos": [449, 37]}, {"full_name": "MeasureTheory.snorm_nnreal_eq_lintegral", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [98, 7], "def_end_pos": [98, 32]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nthis :\n  (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / \u2191n') \u2264\n    \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) * (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q)\n\u22a2 \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) =\n    \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nthis :\n  (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / \u2191n') \u2264\n    \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) * (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q)\n\u22a2 \u2191C * \u2191\u03b3 = \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p)"}, {"tactic": "simp_rw [snormLESNormFDerivOfEqInnerConst, \u03b3]", "annotated_tactic": ["simp_rw [<a>snormLESNormFDerivOfEqInnerConst</a>, \u03b3]", [{"full_name": "MeasureTheory.snormLESNormFDerivOfEqInnerConst", "def_path": "Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean", "def_pos": [449, 5], "def_end_pos": [449, 37]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nthis :\n  (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / \u2191n') \u2264\n    \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) * (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q)\n\u22a2 \u2191C * \u2191\u03b3 = \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p)", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nthis :\n  (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / \u2191n') \u2264\n    \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) * (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q)\n\u22a2 \u2191C * \u2191\u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9 =\n    \u2191(snormLESNormFDerivOneConst \u03bc \u2191(\u2191(finrank \u211d E)).conjExponent *\n        (\u2191p * (\u2191(finrank \u211d E) - 1) / (\u2191(finrank \u211d E) - \u2191p)).toNNReal)"}, {"tactic": "refold_let n n' C", "annotated_tactic": ["refold_let n n' C", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nthis :\n  (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / \u2191n') \u2264\n    \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) * (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q)\n\u22a2 \u2191C * \u2191\u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9 =\n    \u2191(snormLESNormFDerivOneConst \u03bc \u2191(\u2191(finrank \u211d E)).conjExponent *\n        (\u2191p * (\u2191(finrank \u211d E) - 1) / (\u2191(finrank \u211d E) - \u2191p)).toNNReal)", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nthis :\n  (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / \u2191n') \u2264\n    \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) * (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q)\n\u22a2 \u2191C * \u2191\u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9 = \u2191(C * (\u2191p * (\u2191n - 1) / (\u2191n - \u2191p)).toNNReal)"}, {"tactic": "rw [NNReal.coe_mul, NNReal.coe_mk, Real.coe_toNNReal', mul_eq_mul_left_iff, eq_comm,\n  max_eq_left_iff]", "annotated_tactic": ["rw [<a>NNReal.coe_mul</a>, <a>NNReal.coe_mk</a>, <a>Real.coe_toNNReal'</a>, <a>mul_eq_mul_left_iff</a>, <a>eq_comm</a>,\n        <a>max_eq_left_iff</a>]", [{"full_name": "NNReal.coe_mul", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [185, 19], "def_end_pos": [185, 26]}, {"full_name": "NNReal.coe_mk", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [139, 28], "def_end_pos": [139, 34]}, {"full_name": "Real.coe_toNNReal'", "def_path": "Mathlib/Data/NNReal/Basic.lean", "def_pos": [638, 9], "def_end_pos": [638, 22]}, {"full_name": "mul_eq_mul_left_iff", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [231, 9], "def_end_pos": [231, 28]}, {"full_name": "eq_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}, {"full_name": "max_eq_left_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [153, 9], "def_end_pos": [153, 24]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nthis :\n  (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / \u2191n') \u2264\n    \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) * (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q)\n\u22a2 \u2191C * \u2191\u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9 = \u2191(C * (\u2191p * (\u2191n - 1) / (\u2191n - \u2191p)).toNNReal)", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nthis :\n  (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / \u2191n') \u2264\n    \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) * (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q)\n\u22a2 0 \u2264 \u2191p * (\u2191n - 1) / (\u2191n - \u2191p) \u2228 \u2191C = 0"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nthis :\n  (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / \u2191n') \u2264\n    \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) * (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q)\n\u22a2 0 \u2264 \u2191p * (\u2191n - 1) / (\u2191n - \u2191p) \u2228 \u2191C = 0", "state_after": "case h\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nthis :\n  (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / \u2191n') \u2264\n    \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) * (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q)\n\u22a2 0 \u2264 \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "case h\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nthis :\n  (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / \u2191n') \u2264\n    \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) * (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q)\n\u22a2 0 \u2264 \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)", "state_after": "no goals"}, {"tactic": "rw [snorm_nnreal_eq_lintegral h0p]", "annotated_tactic": ["rw [<a>snorm_nnreal_eq_lintegral</a> h0p]", [{"full_name": "MeasureTheory.snorm_nnreal_eq_lintegral", "def_path": "Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean", "def_pos": [98, 7], "def_end_pos": [98, 32]}]], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nthis\u271d :\n  (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / \u2191n') \u2264\n    \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) * (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q)\nthis : \u2191C * \u2191\u03b3 = \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p)\n\u22a2 \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) =\n    \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * snorm (fderiv \u211d u) (\u2191p) \u03bc", "state_after": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nthis\u271d :\n  (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / \u2191n') \u2264\n    \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) * (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q)\nthis : \u2191C * \u2191\u03b3 = \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p)\n\u22a2 \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) =\n    \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p)"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nthis\u271d :\n  (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / \u2191n') \u2264\n    \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) * (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q)\nthis : \u2191C * \u2191\u03b3 = \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p)\n\u22a2 \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) =\n    \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p) * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p)", "state_after": "case e_a\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nthis\u271d :\n  (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / \u2191n') \u2264\n    \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) * (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q)\nthis : \u2191C * \u2191\u03b3 = \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p)\n\u22a2 \u2191C * \u2191\u03b3 = \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p)"}, {"tactic": "norm_cast at this \u22a2", "annotated_tactic": ["norm_cast at this \u22a2", []], "state_before": "case e_a\n\u03b9 : Type u_1\ninst\u271d\u00b9\u2074 : Fintype \u03b9\ninst\u271d\u00b9\u00b3 : DecidableEq \u03b9\nA : \u03b9 \u2192 Type u_2\ninst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 MeasurableSpace (A i)\n\u03bc\u271d : (i : \u03b9) \u2192 Measure (A i)\ninst\u271d\u00b9\u00b9 : \u2200 (i : \u03b9), SigmaFinite (\u03bc\u271d i)\nF : Type u_3\ninst\u271d\u00b9\u2070 : NormedAddCommGroup F\ninst\u271d\u2079 : NormedSpace \u211d F\nE : Type u_4\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : MeasurableSpace E\ninst\u271d\u2075 : BorelSpace E\ninst\u271d\u2074 : FiniteDimensional \u211d E\n\u03bc : Measure E\ninst\u271d\u00b3 : \u03bc.IsAddHaarMeasure\nF' : Type u_5\ninst\u271d\u00b2 : NormedAddCommGroup F'\ninst\u271d\u00b9 : InnerProductSpace \u211d F'\ninst\u271d : CompleteSpace F'\nu : E \u2192 F'\nhu : ContDiff \u211d 1 u\nh2u : HasCompactSupport u\np p' : \u211d\u22650\nhp\u271d : 1 \u2264 p\nhp'0 : \u00acp' = 0\nn : \u2115 := finrank \u211d E\nhn\u271d : 0 < n\nhp' : (\u2191p')\u207b\u00b9 = \u2191p\u207b\u00b9 - (\u2191n)\u207b\u00b9\nn' : \u211d\u22650 := (\u2191n).conjExponent\nh2p : \u2191p < \u2191n\nh0n : 2 \u2264 n\nhn : (\u2191n).IsConjExponent n'\nh1n : 1 \u2264 \u2191n\nh2n : 0 < \u2191n - 1\nhnp : 0 < \u2191n - \u2191p\nhp : 1 < p\nq : \u211d := (\u2191p).conjExponent\nhq : (\u2191p).IsConjExponent q\nh0p : p \u2260 0\nh1p : \u2191p \u2260 1\nh3p : \u2191p - 1 \u2260 0\nh0p' : p' \u2260 0\nh2q : 1 / \u2191n' - 1 / q = 1 / \u2191p'\n\u03b3 : \u211d\u22650 := \u27e8\u2191p * (\u2191n - 1) / (\u2191n - \u2191p), \u22ef\u27e9\nh0\u03b3 : \u2191\u03b3 = \u2191p * (\u2191n - 1) / (\u2191n - \u2191p)\nh1\u03b3 : 1 < \u2191\u03b3\nh2\u03b3 : \u03b3 * n' = p'\nh3\u03b3 : (\u2191\u03b3 - 1) * q = \u2191p'\nh4\u03b3 : \u2191\u03b3 \u2260 0\nh3u : \u00ac\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc = 0\nh4u : \u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc \u2260 \u22a4\nh5u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 0\nh6u : (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q) \u2260 \u22a4\nh7u : Continuous u\nh8u : Continuous (fderiv \u211d u)\nv : E \u2192 \u211d := fun x => \u2016u x\u2016 ^ \u2191\u03b3\nhv : ContDiff \u211d 1 v\nh2v : HasCompactSupport v\nC : \u211d\u22650 := snormLESNormFDerivOneConst \u03bc \u2191n'\nthis\u271d :\n  (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / \u2191n') \u2264\n    \u2191C * \u2191\u03b3 * (\u222b\u207b (x : E), \u2191\u2016fderiv \u211d u x\u2016\u208a ^ \u2191p \u2202\u03bc) ^ (1 / \u2191p) * (\u222b\u207b (x : E), \u2191\u2016u x\u2016\u208a ^ \u2191p' \u2202\u03bc) ^ (1 / q)\nthis : \u2191C * \u2191\u03b3 = \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p)\n\u22a2 \u2191C * \u2191\u03b3 = \u2191(snormLESNormFDerivOfEqInnerConst \u03bc \u2191p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finsupp/Basic.lean", "full_name": "Finsupp.frange_single", "start": [989, 1], "end": [996, 32], "traced_tactics": [{"tactic": "rw [single_apply] at ht2 \u22a2", "annotated_tactic": ["rw [<a>single_apply</a>] at ht2 \u22a2", [{"full_name": "Finsupp.single_apply", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [287, 9], "def_end_pos": [287, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nx : \u03b1\ny r : M\nhr : r \u2208 (single x y).frange\nt : r \u2260 0\nht1 : \u03b1\nht2 : (single x y) ht1 = r\n\u22a2 (single x y) ht1 \u2208 {y}", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nx : \u03b1\ny r : M\nhr : r \u2208 (single x y).frange\nt : r \u2260 0\nht1 : \u03b1\nht2 : (if x = ht1 then y else 0) = r\n\u22a2 (if x = ht1 then y else 0) \u2208 {y}"}, {"tactic": "split_ifs at ht2 \u22a2", "annotated_tactic": ["split_ifs at ht2 \u22a2", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nx : \u03b1\ny r : M\nhr : r \u2208 (single x y).frange\nt : r \u2260 0\nht1 : \u03b1\nht2 : (if x = ht1 then y else 0) = r\n\u22a2 (if x = ht1 then y else 0) \u2208 {y}", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nx : \u03b1\ny r : M\nhr : r \u2208 (single x y).frange\nt : r \u2260 0\nht1 : \u03b1\nh\u271d : x = ht1\nht2 : y = r\n\u22a2 y \u2208 {y}\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nx : \u03b1\ny r : M\nhr : r \u2208 (single x y).frange\nt : r \u2260 0\nht1 : \u03b1\nh\u271d : \u00acx = ht1\nht2 : 0 = r\n\u22a2 0 \u2208 {y}"}, {"tactic": "exact Finset.mem_singleton_self _", "annotated_tactic": ["exact <a>Finset.mem_singleton_self</a> _", [{"full_name": "Finset.mem_singleton_self", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [692, 9], "def_end_pos": [692, 27]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nx : \u03b1\ny r : M\nhr : r \u2208 (single x y).frange\nt : r \u2260 0\nht1 : \u03b1\nh\u271d : x = ht1\nht2 : y = r\n\u22a2 y \u2208 {y}", "state_after": "no goals"}, {"tactic": "exact (t ht2.symm).elim", "annotated_tactic": ["exact (t ht2.symm).<a>elim</a>", [{"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\nM : Type u_5\nM' : Type u_6\nN : Type u_7\nP : Type u_8\nG : Type u_9\nH : Type u_10\nR : Type u_11\nS : Type u_12\ninst\u271d : Zero M\nx : \u03b1\ny r : M\nhr : r \u2208 (single x y).frange\nt : r \u2260 0\nht1 : \u03b1\nh\u271d : \u00acx = ht1\nht2 : 0 = r\n\u22a2 0 \u2208 {y}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/RestrictScalars.lean", "full_name": "RestrictScalars.addEquiv_map_smul", "start": [164, 1], "end": [166, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "full_name": "Ordinal.le_enum_succ", "start": [1164, 1], "end": [1171, 23], "traced_tactics": [{"tactic": "rw [type_lt]", "annotated_tactic": ["rw [<a>type_lt</a>]", [{"full_name": "Ordinal.type_lt", "def_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "def_pos": [222, 9], "def_end_pos": [222, 16]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{?u.104673}\na : (Quotient.out (succ o)).\u03b1\n\u22a2 o < type fun x x_1 => x < x_1", "state_after": "\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{?u.104673}\na : (Quotient.out (succ o)).\u03b1\n\u22a2 o < succ o"}, {"tactic": "exact lt_succ o", "annotated_tactic": ["exact <a>lt_succ</a> o", [{"full_name": "Order.lt_succ", "def_path": "Mathlib/Order/SuccPred/Basic.lean", "def_pos": [351, 9], "def_end_pos": [351, 16]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{?u.104673}\na : (Quotient.out (succ o)).\u03b1\n\u22a2 o < succ o", "state_after": "no goals"}, {"tactic": "apply typein_lt_self", "annotated_tactic": ["apply <a>typein_lt_self</a>", [{"full_name": "Ordinal.typein_lt_self", "def_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "def_pos": [445, 9], "def_end_pos": [445, 23]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\n\u03b3 : Type u_2\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : \u03b2 \u2192 \u03b2 \u2192 Prop\nt : \u03b3 \u2192 \u03b3 \u2192 Prop\no : Ordinal.{u_3}\na : (Quotient.out (succ o)).\u03b1\n\u22a2 typein (fun x x_1 => x < x_1) a < succ o", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/NoncommPiCoprod.lean", "full_name": "MonoidHom.independent_range_of_coprime_order", "start": [228, 1], "end": [259, 19], "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": [443, 9], "def_end_pos": [443, 25]}]], "state_before": "G : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\n\u22a2 CompleteLattice.Independent fun i => (\u03d5 i).range", "state_after": "case intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\n\u22a2 CompleteLattice.Independent fun i => (\u03d5 i).range"}, {"tactic": "letI := Classical.decEq \u03b9", "annotated_tactic": ["letI := <a>Classical.decEq</a> \u03b9", [{"full_name": "Classical.decEq", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1020, 19], "def_end_pos": [1020, 24]}]], "state_before": "case intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\n\u22a2 CompleteLattice.Independent fun i => (\u03d5 i).range", "state_after": "case intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\n\u22a2 CompleteLattice.Independent fun i => (\u03d5 i).range"}, {"tactic": "rintro i", "annotated_tactic": ["rintro i", []], "state_before": "case intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\n\u22a2 CompleteLattice.Independent fun i => (\u03d5 i).range", "state_after": "case intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\n\u22a2 Disjoint ((fun i => (\u03d5 i).range) i) (\u2a06 j, \u2a06 (_ : j \u2260 i), (fun i => (\u03d5 i).range) j)"}, {"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": [131, 9], "def_end_pos": [131, 28]}]], "state_before": "case intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\n\u22a2 Disjoint ((fun i => (\u03d5 i).range) i) (\u2a06 j, \u2a06 (_ : j \u2260 i), (fun i => (\u03d5 i).range) j)", "state_after": "case intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\n\u22a2 (fun i => (\u03d5 i).range) i \u2293 \u2a06 j, \u2a06 (_ : j \u2260 i), (fun i => (\u03d5 i).range) j \u2264 \u22a5"}, {"tactic": "rintro f \u27e8hxi, hxp\u27e9", "annotated_tactic": ["rintro f \u27e8hxi, hxp\u27e9", []], "state_before": "case intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\n\u22a2 (fun i => (\u03d5 i).range) i \u2293 \u2a06 j, \u2a06 (_ : j \u2260 i), (fun i => (\u03d5 i).range) j \u2264 \u22a5", "state_after": "case intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxi : f \u2208 \u2191((fun i => (\u03d5 i).range) i).toSubmonoid\nhxp : f \u2208 \u2191(\u2a06 j, \u2a06 (_ : j \u2260 i), (fun i => (\u03d5 i).range) j).toSubmonoid\n\u22a2 f \u2208 \u22a5"}, {"tactic": "dsimp at hxi hxp", "annotated_tactic": ["dsimp at hxi hxp", []], "state_before": "case intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxi : f \u2208 \u2191((fun i => (\u03d5 i).range) i).toSubmonoid\nhxp : f \u2208 \u2191(\u2a06 j, \u2a06 (_ : j \u2260 i), (fun i => (\u03d5 i).range) j).toSubmonoid\n\u22a2 f \u2208 \u22a5", "state_after": "case intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxi : f \u2208 Set.range \u21d1(\u03d5 i)\nhxp : f \u2208 \u2191(\u2a06 j, \u2a06 (_ : \u00acj = i), (\u03d5 j).range)\n\u22a2 f \u2208 \u22a5"}, {"tactic": "rw [iSup_subtype', \u2190 noncommPiCoprod_range] at hxp", "annotated_tactic": ["rw [<a>iSup_subtype'</a>, \u2190 <a>noncommPiCoprod_range</a>] at hxp", [{"full_name": "iSup_subtype'", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1149, 9], "def_end_pos": [1149, 22]}, {"full_name": "MonoidHom.noncommPiCoprod_range", "def_path": "Mathlib/GroupTheory/NoncommPiCoprod.lean", "def_pos": [193, 9], "def_end_pos": [193, 30]}]], "state_before": "case intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxi : f \u2208 Set.range \u21d1(\u03d5 i)\nhxp : f \u2208 \u2191(\u2a06 j, \u2a06 (_ : \u00acj = i), (\u03d5 j).range)\n\u22a2 f \u2208 \u22a5", "state_after": "case intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxi : f \u2208 Set.range \u21d1(\u03d5 i)\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\nhxp : f \u2208 \u2191(noncommPiCoprod (fun x => \u03d5 \u2191x) ?intro.intro.hcomm).range\n\u22a2 f \u2208 \u22a5\n\ncase intro.intro.hcomm\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxi : f \u2208 Set.range \u21d1(\u03d5 i)\nhxp : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\n\u22a2 Pairwise fun i_1 j => \u2200 (x : H \u2191i_1) (y : H \u2191j), Commute ((\u03d5 \u2191i_1) x) ((\u03d5 \u2191j) y)"}, {"tactic": "rotate_left", "annotated_tactic": ["rotate_left", []], "state_before": "case intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxi : f \u2208 Set.range \u21d1(\u03d5 i)\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\nhxp : f \u2208 \u2191(noncommPiCoprod (fun x => \u03d5 \u2191x) ?intro.intro.hcomm).range\n\u22a2 f \u2208 \u22a5\n\ncase intro.intro.hcomm\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxi : f \u2208 Set.range \u21d1(\u03d5 i)\nhxp : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\n\u22a2 Pairwise fun i_1 j => \u2200 (x : H \u2191i_1) (y : H \u2191j), Commute ((\u03d5 \u2191i_1) x) ((\u03d5 \u2191j) y)", "state_after": "case intro.intro.hcomm\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxi : f \u2208 Set.range \u21d1(\u03d5 i)\nhxp : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\n\u22a2 Pairwise fun i_1 j => \u2200 (x : H \u2191i_1) (y : H \u2191j), Commute ((\u03d5 \u2191i_1) x) ((\u03d5 \u2191j) y)\n\ncase intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxi : f \u2208 Set.range \u21d1(\u03d5 i)\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\nhxp : f \u2208 \u2191(noncommPiCoprod (fun x => \u03d5 \u2191x) ?intro.intro.hcomm).range\n\u22a2 f \u2208 \u22a5"}, {"tactic": "cases' hxp with g hgf", "annotated_tactic": ["cases' hxp with g hgf", []], "state_before": "case intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxi : f \u2208 Set.range \u21d1(\u03d5 i)\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\nhxp : f \u2208 \u2191(noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef).range\n\u22a2 f \u2208 \u22a5", "state_after": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxi : f \u2208 Set.range \u21d1(\u03d5 i)\nhxp : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\n\u22a2 f \u2208 \u22a5"}, {"tactic": "cases' hxi with g' hg'f", "annotated_tactic": ["cases' hxi with g' hg'f", []], "state_before": "case intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxi : f \u2208 Set.range \u21d1(\u03d5 i)\nhxp : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\n\u22a2 f \u2208 \u22a5", "state_after": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\n\u22a2 f \u2208 \u22a5"}, {"tactic": "have hxi : orderOf f \u2223 Fintype.card (H i) := by\n  rw [\u2190 hg'f]\n  exact (orderOf_map_dvd _ _).trans orderOf_dvd_card", "annotated_tactic": ["have hxi : <a>orderOf</a> f \u2223 <a>Fintype.card</a> (H i) := by\n    rw [\u2190 hg'f]\n    exact (<a>orderOf_map_dvd</a> _ _).<a>trans</a> <a>orderOf_dvd_card</a>", [{"full_name": "orderOf", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [155, 19], "def_end_pos": [155, 26]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [63, 5], "def_end_pos": [63, 9]}, {"full_name": "orderOf_map_dvd", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [314, 9], "def_end_pos": [314, 24]}, {"full_name": "Dvd.dvd.trans", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [76, 7], "def_end_pos": [76, 20]}, {"full_name": "orderOf_dvd_card", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [1022, 9], "def_end_pos": [1022, 25]}]], "state_before": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\n\u22a2 f \u2208 \u22a5", "state_after": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\n\u22a2 f \u2208 \u22a5"}, {"tactic": "have hxp : orderOf f \u2223 \u220f j : { j // j \u2260 i }, Fintype.card (H j) := by\n  rw [\u2190 hgf, \u2190 Fintype.card_pi]\n  exact (orderOf_map_dvd _ _).trans orderOf_dvd_card", "annotated_tactic": ["have hxp : <a>orderOf</a> f \u2223 \u220f j : { j // j \u2260 i }, <a>Fintype.card</a> (H j) := by\n    rw [\u2190 hgf, \u2190 <a>Fintype.card_pi</a>]\n    exact (<a>orderOf_map_dvd</a> _ _).<a>trans</a> <a>orderOf_dvd_card</a>", [{"full_name": "orderOf", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [155, 19], "def_end_pos": [155, 26]}, {"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [63, 5], "def_end_pos": [63, 9]}, {"full_name": "Fintype.card_pi", "def_path": "Mathlib/Data/Fintype/BigOperators.lean", "def_pos": [134, 15], "def_end_pos": [134, 22]}, {"full_name": "orderOf_map_dvd", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [314, 9], "def_end_pos": [314, 24]}, {"full_name": "Dvd.dvd.trans", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [76, 7], "def_end_pos": [76, 20]}, {"full_name": "orderOf_dvd_card", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [1022, 9], "def_end_pos": [1022, 25]}]], "state_before": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\n\u22a2 f \u2208 \u22a5", "state_after": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\nhxp : orderOf f \u2223 \u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)\n\u22a2 f \u2208 \u22a5"}, {"tactic": "change f = 1", "annotated_tactic": ["change f = 1", []], "state_before": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\nhxp : orderOf f \u2223 \u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)\n\u22a2 f \u2208 \u22a5", "state_after": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\nhxp : orderOf f \u2223 \u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)\n\u22a2 f = 1"}, {"tactic": "rw [\u2190 pow_one f, \u2190 orderOf_dvd_iff_pow_eq_one]", "annotated_tactic": ["rw [\u2190 <a>pow_one</a> f, \u2190 <a>orderOf_dvd_iff_pow_eq_one</a>]", [{"full_name": "pow_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [663, 7], "def_end_pos": [663, 14]}, {"full_name": "orderOf_dvd_iff_pow_eq_one", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [272, 9], "def_end_pos": [272, 35]}]], "state_before": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\nhxp : orderOf f \u2223 \u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)\n\u22a2 f = 1", "state_after": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\nhxp : orderOf f \u2223 \u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)\n\u22a2 orderOf f \u2223 1"}, {"tactic": "obtain \u27e8c, hc\u27e9 := Nat.dvd_gcd hxp hxi", "annotated_tactic": ["obtain \u27e8c, hc\u27e9 := <a>Nat.dvd_gcd</a> hxp hxi", [{"full_name": "Nat.dvd_gcd", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Gcd.lean", "def_pos": [94, 9], "def_end_pos": [94, 16]}]], "state_before": "case intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\nhxp : orderOf f \u2223 \u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)\n\u22a2 orderOf f \u2223 1", "state_after": "case intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\nhxp : orderOf f \u2223 \u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)\nc : \u2115\nhc : (\u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)).gcd (Fintype.card (H i)) = orderOf f * c\n\u22a2 orderOf f \u2223 1"}, {"tactic": "use c", "annotated_tactic": ["use c", []], "state_before": "case intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\nhxp : orderOf f \u2223 \u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)\nc : \u2115\nhc : (\u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)).gcd (Fintype.card (H i)) = orderOf f * c\n\u22a2 orderOf f \u2223 1", "state_after": "case h\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\nhxp : orderOf f \u2223 \u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)\nc : \u2115\nhc : (\u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)).gcd (Fintype.card (H i)) = orderOf f * c\n\u22a2 1 = orderOf f * c"}, {"tactic": "rw [\u2190 hc]", "annotated_tactic": ["rw [\u2190 hc]", []], "state_before": "case h\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\nhxp : orderOf f \u2223 \u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)\nc : \u2115\nhc : (\u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)).gcd (Fintype.card (H i)) = orderOf f * c\n\u22a2 1 = orderOf f * c", "state_after": "case h\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\nhxp : orderOf f \u2223 \u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)\nc : \u2115\nhc : (\u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)).gcd (Fintype.card (H i)) = orderOf f * c\n\u22a2 1 = (\u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)).gcd (Fintype.card (H i))"}, {"tactic": "symm", "annotated_tactic": ["symm", []], "state_before": "case h\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\nhxp : orderOf f \u2223 \u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)\nc : \u2115\nhc : (\u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)).gcd (Fintype.card (H i)) = orderOf f * c\n\u22a2 1 = (\u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)).gcd (Fintype.card (H i))", "state_after": "case h\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\nhxp : orderOf f \u2223 \u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)\nc : \u2115\nhc : (\u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)).gcd (Fintype.card (H i)) = orderOf f * c\n\u22a2 (\u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)).gcd (Fintype.card (H i)) = 1"}, {"tactic": "rw [\u2190 Nat.coprime_iff_gcd_eq_one, Nat.coprime_fintype_prod_left_iff, Subtype.forall]", "annotated_tactic": ["rw [\u2190 <a>Nat.coprime_iff_gcd_eq_one</a>, <a>Nat.coprime_fintype_prod_left_iff</a>, <a>Subtype.forall</a>]", [{"full_name": "Nat.coprime_iff_gcd_eq_one", "def_path": ".lake/packages/batteries/Batteries/Data/Nat/Gcd.lean", "def_pos": [24, 9], "def_end_pos": [24, 31]}, {"full_name": "Nat.coprime_fintype_prod_left_iff", "def_path": "Mathlib/Data/Nat/GCD/BigOperators.lean", "def_pos": [52, 9], "def_end_pos": [52, 38]}, {"full_name": "Subtype.forall", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [52, 19], "def_end_pos": [52, 27]}]], "state_before": "case h\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\nhxp : orderOf f \u2223 \u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)\nc : \u2115\nhc : (\u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)).gcd (Fintype.card (H i)) = orderOf f * c\n\u22a2 (\u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)).gcd (Fintype.card (H i)) = 1", "state_after": "case h\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\nhxp : orderOf f \u2223 \u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)\nc : \u2115\nhc : (\u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)).gcd (Fintype.card (H i)) = orderOf f * c\n\u22a2 \u2200 (a : \u03b9) (b : a \u2260 i), (Fintype.card (H \u2191\u27e8a, b\u27e9)).Coprime (Fintype.card (H i))"}, {"tactic": "intro j h", "annotated_tactic": ["intro j h", []], "state_before": "case h\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\nhxp : orderOf f \u2223 \u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)\nc : \u2115\nhc : (\u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)).gcd (Fintype.card (H i)) = orderOf f * c\n\u22a2 \u2200 (a : \u03b9) (b : a \u2260 i), (Fintype.card (H \u2191\u27e8a, b\u27e9)).Coprime (Fintype.card (H i))", "state_after": "case h\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\nhxp : orderOf f \u2223 \u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)\nc : \u2115\nhc : (\u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)).gcd (Fintype.card (H i)) = orderOf f * c\nj : \u03b9\nh : j \u2260 i\n\u22a2 (Fintype.card (H \u2191\u27e8j, h\u27e9)).Coprime (Fintype.card (H i))"}, {"tactic": "exact hcoprime h", "annotated_tactic": ["exact hcoprime h", []], "state_before": "case h\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp\u271d : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\nhxp : orderOf f \u2223 \u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)\nc : \u2115\nhc : (\u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)).gcd (Fintype.card (H i)) = orderOf f * c\nj : \u03b9\nh : j \u2260 i\n\u22a2 (Fintype.card (H \u2191\u27e8j, h\u27e9)).Coprime (Fintype.card (H i))", "state_after": "no goals"}, {"tactic": "intro _ _ hj", "annotated_tactic": ["intro _ _ hj", []], "state_before": "case intro.intro.hcomm\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxi : f \u2208 Set.range \u21d1(\u03d5 i)\nhxp : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\n\u22a2 Pairwise fun i_1 j => \u2200 (x : H \u2191i_1) (y : H \u2191j), Commute ((\u03d5 \u2191i_1) x) ((\u03d5 \u2191j) y)", "state_after": "case intro.intro.hcomm\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxi : f \u2208 Set.range \u21d1(\u03d5 i)\nhxp : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ni\u271d j\u271d : { j // \u00acj = i }\nhj : i\u271d \u2260 j\u271d\n\u22a2 \u2200 (x : H \u2191i\u271d) (y : H \u2191j\u271d), Commute ((\u03d5 \u2191i\u271d) x) ((\u03d5 \u2191j\u271d) y)"}, {"tactic": "apply hcomm", "annotated_tactic": ["apply hcomm", []], "state_before": "case intro.intro.hcomm\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxi : f \u2208 Set.range \u21d1(\u03d5 i)\nhxp : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ni\u271d j\u271d : { j // \u00acj = i }\nhj : i\u271d \u2260 j\u271d\n\u22a2 \u2200 (x : H \u2191i\u271d) (y : H \u2191j\u271d), Commute ((\u03d5 \u2191i\u271d) x) ((\u03d5 \u2191j\u271d) y)", "state_after": "case intro.intro.hcomm.a\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxi : f \u2208 Set.range \u21d1(\u03d5 i)\nhxp : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ni\u271d j\u271d : { j // \u00acj = i }\nhj : i\u271d \u2260 j\u271d\n\u22a2 \u2191i\u271d \u2260 \u2191j\u271d"}, {"tactic": "exact hj \u2218 Subtype.ext", "annotated_tactic": ["exact hj \u2218 <a>Subtype.ext</a>", [{"full_name": "Subtype.ext", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [74, 19], "def_end_pos": [74, 22]}]], "state_before": "case intro.intro.hcomm.a\nG : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxi : f \u2208 Set.range \u21d1(\u03d5 i)\nhxp : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ni\u271d j\u271d : { j // \u00acj = i }\nhj : i\u271d \u2260 j\u271d\n\u22a2 \u2191i\u271d \u2260 \u2191j\u271d", "state_after": "no goals"}, {"tactic": "rw [\u2190 hg'f]", "annotated_tactic": ["rw [\u2190 hg'f]", []], "state_before": "G : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\n\u22a2 orderOf f \u2223 Fintype.card (H i)", "state_after": "G : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\n\u22a2 orderOf ((\u03d5 i) g') \u2223 Fintype.card (H i)"}, {"tactic": "exact (orderOf_map_dvd _ _).trans orderOf_dvd_card", "annotated_tactic": ["exact (<a>orderOf_map_dvd</a> _ _).<a>trans</a> <a>orderOf_dvd_card</a>", [{"full_name": "orderOf_map_dvd", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [314, 9], "def_end_pos": [314, 24]}, {"full_name": "Dvd.dvd.trans", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [76, 7], "def_end_pos": [76, 20]}, {"full_name": "orderOf_dvd_card", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [1022, 9], "def_end_pos": [1022, 25]}]], "state_before": "G : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\n\u22a2 orderOf ((\u03d5 i) g') \u2223 Fintype.card (H i)", "state_after": "no goals"}, {"tactic": "rw [\u2190 hgf, \u2190 Fintype.card_pi]", "annotated_tactic": ["rw [\u2190 hgf, \u2190 <a>Fintype.card_pi</a>]", [{"full_name": "Fintype.card_pi", "def_path": "Mathlib/Data/Fintype/BigOperators.lean", "def_pos": [134, 15], "def_end_pos": [134, 22]}]], "state_before": "G : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\n\u22a2 orderOf f \u2223 \u220f j : { j // j \u2260 i }, Fintype.card (H \u2191j)", "state_after": "G : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\n\u22a2 orderOf ((noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g) \u2223 Fintype.card ((i_1 : { j // j \u2260 i }) \u2192 H \u2191i_1)"}, {"tactic": "exact (orderOf_map_dvd _ _).trans orderOf_dvd_card", "annotated_tactic": ["exact (<a>orderOf_map_dvd</a> _ _).<a>trans</a> <a>orderOf_dvd_card</a>", [{"full_name": "orderOf_map_dvd", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [314, 9], "def_end_pos": [314, 24]}, {"full_name": "Dvd.dvd.trans", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [76, 7], "def_end_pos": [76, 20]}, {"full_name": "orderOf_dvd_card", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [1022, 9], "def_end_pos": [1022, 25]}]], "state_before": "G : Type u_1\ninst\u271d\u00b3 : Group G\n\u03b9 : Type u_2\nhdec : DecidableEq \u03b9\nhfin : Fintype \u03b9\nH : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Group (H i)\n\u03d5 : (i : \u03b9) \u2192 H i \u2192* G\nhcomm : Pairwise fun i j => \u2200 (x : H i) (y : H j), Commute ((\u03d5 i) x) ((\u03d5 j) y)\nf\u271d g\u271d : (i : \u03b9) \u2192 H i\ninst\u271d\u00b9 : Finite \u03b9\ninst\u271d : (i : \u03b9) \u2192 Fintype (H i)\nhcoprime : Pairwise fun i j => (Fintype.card (H i)).Coprime (Fintype.card (H j))\nval\u271d : Fintype \u03b9\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\nf : G\nhxp : f \u2208 \u2191(\u2a06 x, (\u03d5 \u2191x).range)\ng : (i_1 : { j // \u00acj = i }) \u2192 H \u2191i_1\nhgf : (noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g = f\ng' : H i\nhg'f : (\u03d5 i) g' = f\nhxi : orderOf f \u2223 Fintype.card (H i)\n\u22a2 orderOf ((noncommPiCoprod (fun x => \u03d5 \u2191x) \u22ef) g) \u2223 Fintype.card ((i_1 : { j // j \u2260 i }) \u2192 H \u2191i_1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Bernoulli.lean", "full_name": "bernoulli'_def", "start": [78, 1], "end": [80, 52], "traced_tactics": [{"tactic": "rw [bernoulli'_def', \u2190 Fin.sum_univ_eq_sum_range]", "annotated_tactic": ["rw [<a>bernoulli'_def'</a>, \u2190 <a>Fin.sum_univ_eq_sum_range</a>]", [{"full_name": "bernoulli'_def'", "def_path": "Mathlib/NumberTheory/Bernoulli.lean", "def_pos": [73, 9], "def_end_pos": [73, 24]}, {"full_name": "Fin.sum_univ_eq_sum_range", "def_path": "Mathlib/Data/Fintype/BigOperators.lean", "def_pos": [190, 3], "def_end_pos": [190, 14]}]], "state_before": "A : Type u_1\ninst\u271d\u00b9 : CommRing A\ninst\u271d : Algebra \u211a A\nn : \u2115\n\u22a2 bernoulli' n = 1 - \u2211 k \u2208 range n, \u2191(n.choose k) / (\u2191n - \u2191k + 1) * bernoulli' k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/SemidirectProduct.lean", "full_name": "SemidirectProduct.rightHom_inl", "start": [194, 1], "end": [194, 85], "traced_tactics": [{"tactic": "simp [rightHom]", "annotated_tactic": ["simp [<a>rightHom</a>]", [{"full_name": "SemidirectProduct.rightHom", "def_path": "Mathlib/GroupTheory/SemidirectProduct.lean", "def_pos": [174, 5], "def_end_pos": [174, 13]}]], "state_before": "N : Type u_1\nG : Type u_2\nH : Type u_3\ninst\u271d\u00b2 : Group N\ninst\u271d\u00b9 : Group G\ninst\u271d : Group H\n\u03c6 : G \u2192* MulAut N\nn : N\n\u22a2 rightHom (inl n) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Periodic.lean", "full_name": "Function.Periodic.div", "start": [71, 11], "end": [72, 38], "traced_tactics": [{"tactic": "simp_all", "annotated_tactic": ["simp_all", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf g : \u03b1 \u2192 \u03b2\nc c\u2081 c\u2082 x : \u03b1\ninst\u271d\u00b9 : Add \u03b1\ninst\u271d : Div \u03b2\nhf : Periodic f c\nhg : Periodic g c\n\u22a2 Periodic (f / g) c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Atoms.lean", "full_name": "GaloisCoinsertion.isCoatom_iff'", "start": [882, 1], "end": [885, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/RieszMarkovKakutani.lean", "full_name": "rieszContentAux_image_nonempty", "start": [51, 1], "end": [56, 67], "traced_tactics": [{"tactic": "rw [image_nonempty]", "annotated_tactic": ["rw [<a>image_nonempty</a>]", [{"full_name": "Set.image_nonempty", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [463, 9], "def_end_pos": [463, 23]}]], "state_before": "X : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK : Compacts X\n\u22a2 (\u21d1\u039b '' {f | \u2200 x \u2208 K, 1 \u2264 f x}).Nonempty", "state_after": "X : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK : Compacts X\n\u22a2 {f | \u2200 x \u2208 K, 1 \u2264 f x}.Nonempty"}, {"tactic": "use (1 : X \u2192\u1d47 \u211d\u22650)", "annotated_tactic": ["use (1 : X \u2192\u1d47 \u211d\u22650)", []], "state_before": "X : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK : Compacts X\n\u22a2 {f | \u2200 x \u2208 K, 1 \u2264 f x}.Nonempty", "state_after": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK : Compacts X\n\u22a2 1 \u2208 {f | \u2200 x \u2208 K, 1 \u2264 f x}"}, {"tactic": "intro x _", "annotated_tactic": ["intro x _", []], "state_before": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK : Compacts X\n\u22a2 1 \u2208 {f | \u2200 x \u2208 K, 1 \u2264 f x}", "state_after": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK : Compacts X\nx : X\na\u271d : x \u2208 K\n\u22a2 1 \u2264 1 x"}, {"tactic": "simp only [BoundedContinuousFunction.coe_one, Pi.one_apply]", "annotated_tactic": ["simp only [<a>BoundedContinuousFunction.coe_one</a>, <a>Pi.one_apply</a>]", [{"full_name": "BoundedContinuousFunction.coe_one", "def_path": "Mathlib/Topology/ContinuousFunction/Bounded.lean", "def_pos": [621, 9], "def_end_pos": [621, 16]}, {"full_name": "Pi.one_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [63, 9], "def_end_pos": [63, 18]}]], "state_before": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK : Compacts X\nx : X\na\u271d : x \u2208 K\n\u22a2 1 \u2264 1 x", "state_after": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK : Compacts X\nx : X\na\u271d : x \u2208 K\n\u22a2 1 \u2264 1"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h\nX : Type u_1\ninst\u271d : TopologicalSpace X\n\u039b : (X \u2192\u1d47 \u211d\u22650) \u2192\u2097[\u211d\u22650] \u211d\u22650\nK : Compacts X\nx : X\na\u271d : x \u2208 K\n\u22a2 1 \u2264 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "full_name": "Submodule.comap_le_comap_iff_of_surjective", "start": [326, 1], "end": [327, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Basic.lean", "full_name": "exists_unique_prop_of_true", "start": [910, 1], "end": [911, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Unique.lean", "full_name": "MeasureTheory.Measure.smul_measure_isMulInvariant_le_of_isCompact_closure", "start": [511, 1], "end": [540, 30], "traced_tactics": [{"tactic": "apply le_of_forall_lt (fun r hr \u21a6 ?_)", "annotated_tactic": ["apply <a>le_of_forall_lt</a> (fun r hr \u21a6 ?_)", [{"full_name": "le_of_forall_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [527, 9], "def_end_pos": [527, 24]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\ninst\u271d\u2074 : LocallyCompactSpace G\n\u03bc' \u03bc : Measure G\ninst\u271d\u00b3 : \u03bc.IsHaarMeasure\ninst\u271d\u00b2 : IsFiniteMeasureOnCompacts \u03bc'\ninst\u271d\u00b9 : \u03bc'.IsMulLeftInvariant\ninst\u271d : \u03bc.InnerRegularCompactLTTop\ns : Set G\nhs : MeasurableSet s\nh's : IsCompact (closure s)\n\u22a2 \u03bc'.haarScalarFactor \u03bc \u2022 \u03bc s \u2264 \u03bc' s", "state_after": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\ninst\u271d\u2074 : LocallyCompactSpace G\n\u03bc' \u03bc : Measure G\ninst\u271d\u00b3 : \u03bc.IsHaarMeasure\ninst\u271d\u00b2 : IsFiniteMeasureOnCompacts \u03bc'\ninst\u271d\u00b9 : \u03bc'.IsMulLeftInvariant\ninst\u271d : \u03bc.InnerRegularCompactLTTop\ns : Set G\nhs : MeasurableSet s\nh's : IsCompact (closure s)\nr : \u211d\u22650\u221e\nhr : r < \u03bc'.haarScalarFactor \u03bc \u2022 \u03bc s\n\u22a2 r < \u03bc' s"}, {"tactic": "let \u03bd := haarScalarFactor \u03bc' \u03bc \u2022 \u03bc", "annotated_tactic": ["let \u03bd := <a>haarScalarFactor</a> \u03bc' \u03bc \u2022 \u03bc", [{"full_name": "MeasureTheory.Measure.haarScalarFactor", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Unique.lean", "def_pos": [294, 19], "def_end_pos": [294, 35]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\ninst\u271d\u2074 : LocallyCompactSpace G\n\u03bc' \u03bc : Measure G\ninst\u271d\u00b3 : \u03bc.IsHaarMeasure\ninst\u271d\u00b2 : IsFiniteMeasureOnCompacts \u03bc'\ninst\u271d\u00b9 : \u03bc'.IsMulLeftInvariant\ninst\u271d : \u03bc.InnerRegularCompactLTTop\ns : Set G\nhs : MeasurableSet s\nh's : IsCompact (closure s)\nr : \u211d\u22650\u221e\nhr : r < \u03bc'.haarScalarFactor \u03bc \u2022 \u03bc s\n\u22a2 r < \u03bc' s", "state_after": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\ninst\u271d\u2074 : LocallyCompactSpace G\n\u03bc' \u03bc : Measure G\ninst\u271d\u00b3 : \u03bc.IsHaarMeasure\ninst\u271d\u00b2 : IsFiniteMeasureOnCompacts \u03bc'\ninst\u271d\u00b9 : \u03bc'.IsMulLeftInvariant\ninst\u271d : \u03bc.InnerRegularCompactLTTop\ns : Set G\nhs : MeasurableSet s\nh's : IsCompact (closure s)\nr : \u211d\u22650\u221e\nhr : r < \u03bc'.haarScalarFactor \u03bc \u2022 \u03bc s\n\u03bd : Measure G := \u03bc'.haarScalarFactor \u03bc \u2022 \u03bc\n\u22a2 r < \u03bc' s"}, {"tactic": "have : \u03bd s \u2260 \u221e := ((measure_mono subset_closure).trans_lt h's.measure_lt_top).ne", "annotated_tactic": ["have : \u03bd s \u2260 \u221e := ((<a>measure_mono</a> <a>subset_closure</a>).<a>trans_lt</a> h's.measure_lt_top).<a>ne</a>", [{"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 21]}, {"full_name": "subset_closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [403, 9], "def_end_pos": [403, 23]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [119, 7], "def_end_pos": [119, 21]}, {"full_name": "LT.lt.ne", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [147, 7], "def_end_pos": [147, 15]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\ninst\u271d\u2074 : LocallyCompactSpace G\n\u03bc' \u03bc : Measure G\ninst\u271d\u00b3 : \u03bc.IsHaarMeasure\ninst\u271d\u00b2 : IsFiniteMeasureOnCompacts \u03bc'\ninst\u271d\u00b9 : \u03bc'.IsMulLeftInvariant\ninst\u271d : \u03bc.InnerRegularCompactLTTop\ns : Set G\nhs : MeasurableSet s\nh's : IsCompact (closure s)\nr : \u211d\u22650\u221e\nhr : r < \u03bc'.haarScalarFactor \u03bc \u2022 \u03bc s\n\u03bd : Measure G := \u03bc'.haarScalarFactor \u03bc \u2022 \u03bc\n\u22a2 r < \u03bc' s", "state_after": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\ninst\u271d\u2074 : LocallyCompactSpace G\n\u03bc' \u03bc : Measure G\ninst\u271d\u00b3 : \u03bc.IsHaarMeasure\ninst\u271d\u00b2 : IsFiniteMeasureOnCompacts \u03bc'\ninst\u271d\u00b9 : \u03bc'.IsMulLeftInvariant\ninst\u271d : \u03bc.InnerRegularCompactLTTop\ns : Set G\nhs : MeasurableSet s\nh's : IsCompact (closure s)\nr : \u211d\u22650\u221e\nhr : r < \u03bc'.haarScalarFactor \u03bc \u2022 \u03bc s\n\u03bd : Measure G := \u03bc'.haarScalarFactor \u03bc \u2022 \u03bc\nthis : \u03bd s \u2260 \u22a4\n\u22a2 r < \u03bc' s"}, {"tactic": "obtain \u27e8-, hf, \u27e8f, f_cont, f_comp, rfl\u27e9, \u03bdf\u27e9 :\n    \u2203 K \u2286 s, (\u2203 f, Continuous f \u2227 HasCompactSupport f \u2227 K = f \u207b\u00b9' {1}) \u2227 r < \u03bd K :=\n  innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_group \u27e8hs, this\u27e9 r\n    (by convert hr)", "annotated_tactic": ["obtain \u27e8-, hf, \u27e8f, f_cont, f_comp, rfl\u27e9, \u03bdf\u27e9 :\n      \u2203 K \u2286 s, (\u2203 f, <a>Continuous</a> f \u2227 <a>HasCompactSupport</a> f \u2227 K = f \u207b\u00b9' {1}) \u2227 r < \u03bd K :=\n    <a>innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_group</a> \u27e8hs, this\u27e9 r\n      (by convert hr)", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Defs/Basic.lean", "def_pos": [137, 11], "def_end_pos": [137, 21]}, {"full_name": "HasCompactSupport", "def_path": "Mathlib/Topology/Support.lean", "def_pos": [149, 3], "def_end_pos": [149, 14]}, {"full_name": "MeasureTheory.Measure.innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_group", "def_path": "Mathlib/MeasureTheory/Measure/EverywherePos.lean", "def_pos": [271, 9], "def_end_pos": [271, 79]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\ninst\u271d\u2074 : LocallyCompactSpace G\n\u03bc' \u03bc : Measure G\ninst\u271d\u00b3 : \u03bc.IsHaarMeasure\ninst\u271d\u00b2 : IsFiniteMeasureOnCompacts \u03bc'\ninst\u271d\u00b9 : \u03bc'.IsMulLeftInvariant\ninst\u271d : \u03bc.InnerRegularCompactLTTop\ns : Set G\nhs : MeasurableSet s\nh's : IsCompact (closure s)\nr : \u211d\u22650\u221e\nhr : r < \u03bc'.haarScalarFactor \u03bc \u2022 \u03bc s\n\u03bd : Measure G := \u03bc'.haarScalarFactor \u03bc \u2022 \u03bc\nthis : \u03bd s \u2260 \u22a4\n\u22a2 r < \u03bc' s", "state_after": "case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\ninst\u271d\u2074 : LocallyCompactSpace G\n\u03bc' \u03bc : Measure G\ninst\u271d\u00b3 : \u03bc.IsHaarMeasure\ninst\u271d\u00b2 : IsFiniteMeasureOnCompacts \u03bc'\ninst\u271d\u00b9 : \u03bc'.IsMulLeftInvariant\ninst\u271d : \u03bc.InnerRegularCompactLTTop\ns : Set G\nhs : MeasurableSet s\nh's : IsCompact (closure s)\nr : \u211d\u22650\u221e\nhr : r < \u03bc'.haarScalarFactor \u03bc \u2022 \u03bc s\n\u03bd : Measure G := \u03bc'.haarScalarFactor \u03bc \u2022 \u03bc\nthis : \u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\nf_cont : Continuous f\nf_comp : HasCompactSupport f\nhf : f \u207b\u00b9' {1} \u2286 s\n\u03bdf : r < \u03bd (f \u207b\u00b9' {1})\n\u22a2 r < \u03bc' s"}, {"tactic": "calc\nr < \u03bd (f \u207b\u00b9' {1}) := \u03bdf\n_ = \u03bc' (f \u207b\u00b9' {1}) :=\n  (measure_preimage_isMulLeftInvariant_eq_smul_of_hasCompactSupport _ _ f_cont f_comp).symm\n_ \u2264 \u03bc' s := measure_mono hf", "annotated_tactic": ["calc\n  r < \u03bd (f \u207b\u00b9' {1}) := \u03bdf\n  _ = \u03bc' (f \u207b\u00b9' {1}) :=\n    (<a>measure_preimage_isMulLeftInvariant_eq_smul_of_hasCompactSupport</a> _ _ f_cont f_comp).<a>symm</a>\n  _ \u2264 \u03bc' s := <a>measure_mono</a> hf", [{"full_name": "MeasureTheory.Measure.measure_preimage_isMulLeftInvariant_eq_smul_of_hasCompactSupport", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Unique.lean", "def_pos": [447, 7], "def_end_pos": [447, 71]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}, {"full_name": "MeasureTheory.measure_mono", "def_path": "Mathlib/MeasureTheory/OuterMeasure/Basic.lean", "def_pos": [52, 9], "def_end_pos": [52, 21]}]], "state_before": "case intro.intro.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\ninst\u271d\u2074 : LocallyCompactSpace G\n\u03bc' \u03bc : Measure G\ninst\u271d\u00b3 : \u03bc.IsHaarMeasure\ninst\u271d\u00b2 : IsFiniteMeasureOnCompacts \u03bc'\ninst\u271d\u00b9 : \u03bc'.IsMulLeftInvariant\ninst\u271d : \u03bc.InnerRegularCompactLTTop\ns : Set G\nhs : MeasurableSet s\nh's : IsCompact (closure s)\nr : \u211d\u22650\u221e\nhr : r < \u03bc'.haarScalarFactor \u03bc \u2022 \u03bc s\n\u03bd : Measure G := \u03bc'.haarScalarFactor \u03bc \u2022 \u03bc\nthis : \u03bd s \u2260 \u22a4\nf : G \u2192 \u211d\nf_cont : Continuous f\nf_comp : HasCompactSupport f\nhf : f \u207b\u00b9' {1} \u2286 s\n\u03bdf : r < \u03bd (f \u207b\u00b9' {1})\n\u22a2 r < \u03bc' s", "state_after": "no goals"}, {"tactic": "convert hr", "annotated_tactic": ["convert hr", []], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\ninst\u271d\u2074 : LocallyCompactSpace G\n\u03bc' \u03bc : Measure G\ninst\u271d\u00b3 : \u03bc.IsHaarMeasure\ninst\u271d\u00b2 : IsFiniteMeasureOnCompacts \u03bc'\ninst\u271d\u00b9 : \u03bc'.IsMulLeftInvariant\ninst\u271d : \u03bc.InnerRegularCompactLTTop\ns : Set G\nhs : MeasurableSet s\nh's : IsCompact (closure s)\nr : \u211d\u22650\u221e\nhr : r < \u03bc'.haarScalarFactor \u03bc \u2022 \u03bc s\n\u03bd : Measure G := \u03bc'.haarScalarFactor \u03bc \u2022 \u03bc\nthis : \u03bd s \u2260 \u22a4\n\u22a2 r < \u03bd s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "full_name": "isComplete_iff_ultrafilter", "start": [366, 1], "end": [371, 72], "traced_tactics": [{"tactic": "refine \u27e8fun h l => h l, fun H => isComplete_iff_clusterPt.2 fun l hl hls => ?_\u27e9", "annotated_tactic": ["refine \u27e8fun h l => h l, fun H => <a>isComplete_iff_clusterPt</a>.2 fun l hl hls => ?_\u27e9", [{"full_name": "isComplete_iff_clusterPt", "def_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "def_pos": [360, 9], "def_end_pos": [360, 33]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\ns : Set \u03b1\n\u22a2 IsComplete s \u2194 \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x \u2208 s, \u2191l \u2264 \ud835\udcdd x", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\ns : Set \u03b1\nH : \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x \u2208 s, \u2191l \u2264 \ud835\udcdd x\nl : Filter \u03b1\nhl : Cauchy l\nhls : l \u2264 \ud835\udcdf s\n\u22a2 \u2203 x \u2208 s, ClusterPt x l"}, {"tactic": "haveI := hl.1", "annotated_tactic": ["haveI := hl.1", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\ns : Set \u03b1\nH : \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x \u2208 s, \u2191l \u2264 \ud835\udcdd x\nl : Filter \u03b1\nhl : Cauchy l\nhls : l \u2264 \ud835\udcdf s\n\u22a2 \u2203 x \u2208 s, ClusterPt x l", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\ns : Set \u03b1\nH : \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x \u2208 s, \u2191l \u2264 \ud835\udcdd x\nl : Filter \u03b1\nhl : Cauchy l\nhls : l \u2264 \ud835\udcdf s\nthis : l.NeBot\n\u22a2 \u2203 x \u2208 s, ClusterPt x l"}, {"tactic": "rcases H (Ultrafilter.of l) hl.ultrafilter_of ((Ultrafilter.of_le l).trans hls) with \u27e8x, hxs, hxl\u27e9", "annotated_tactic": ["rcases H (<a>Ultrafilter.of</a> l) hl.ultrafilter_of ((<a>Ultrafilter.of_le</a> l).<a>trans</a> hls) with \u27e8x, hxs, hxl\u27e9", [{"full_name": "Ultrafilter.of", "def_path": "Mathlib/Order/Filter/Ultrafilter.lean", "def_pos": [381, 19], "def_end_pos": [381, 21]}, {"full_name": "Ultrafilter.of_le", "def_path": "Mathlib/Order/Filter/Ultrafilter.lean", "def_pos": [385, 9], "def_end_pos": [385, 14]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\ns : Set \u03b1\nH : \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x \u2208 s, \u2191l \u2264 \ud835\udcdd x\nl : Filter \u03b1\nhl : Cauchy l\nhls : l \u2264 \ud835\udcdf s\nthis : l.NeBot\n\u22a2 \u2203 x \u2208 s, ClusterPt x l", "state_after": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\ns : Set \u03b1\nH : \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x \u2208 s, \u2191l \u2264 \ud835\udcdd x\nl : Filter \u03b1\nhl : Cauchy l\nhls : l \u2264 \ud835\udcdf s\nthis : l.NeBot\nx : \u03b1\nhxs : x \u2208 s\nhxl : \u2191(Ultrafilter.of l) \u2264 \ud835\udcdd x\n\u22a2 \u2203 x \u2208 s, ClusterPt x l"}, {"tactic": "exact \u27e8x, hxs, (ClusterPt.of_le_nhds hxl).mono (Ultrafilter.of_le l)\u27e9", "annotated_tactic": ["exact \u27e8x, hxs, (<a>ClusterPt.of_le_nhds</a> hxl).<a>mono</a> (<a>Ultrafilter.of_le</a> l)\u27e9", [{"full_name": "ClusterPt.of_le_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1069, 9], "def_end_pos": [1069, 29]}, {"full_name": "ClusterPt.mono", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1082, 9], "def_end_pos": [1082, 23]}, {"full_name": "Ultrafilter.of_le", "def_path": "Mathlib/Order/Filter/Ultrafilter.lean", "def_pos": [385, 9], "def_end_pos": [385, 14]}]], "state_before": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\nuniformSpace : UniformSpace \u03b1\ns : Set \u03b1\nH : \u2200 (l : Ultrafilter \u03b1), Cauchy \u2191l \u2192 \u2191l \u2264 \ud835\udcdf s \u2192 \u2203 x \u2208 s, \u2191l \u2264 \ud835\udcdd x\nl : Filter \u03b1\nhl : Cauchy l\nhls : l \u2264 \ud835\udcdf s\nthis : l.NeBot\nx : \u03b1\nhxs : x \u2208 s\nhxl : \u2191(Ultrafilter.of l) \u2264 \ud835\udcdd x\n\u22a2 \u2203 x \u2208 s, ClusterPt x l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "full_name": "Subgroup.prod_top", "start": [1745, 1], "end": [1746, 53], "traced_tactics": [{"tactic": "simp [mem_prod, MonoidHom.coe_fst]", "annotated_tactic": ["simp [<a>mem_prod</a>, <a>MonoidHom.coe_fst</a>]", [{"full_name": "Subgroup.mem_prod", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [1721, 9], "def_end_pos": [1721, 17]}, {"full_name": "MonoidHom.coe_fst", "def_path": "Mathlib/Algebra/Group/Prod.lean", "def_pos": [508, 9], "def_end_pos": [508, 16]}]], "state_before": "G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst\u271d\u2075 : Group G\ninst\u271d\u2074 : Group G'\ninst\u271d\u00b3 : Group G''\nA : Type u_4\ninst\u271d\u00b2 : AddGroup A\nH K\u271d : Subgroup G\nk : Set G\nN : Type u_5\ninst\u271d\u00b9 : Group N\nP : Type u_6\ninst\u271d : Group P\nK : Subgroup G\nx : G \u00d7 N\n\u22a2 x \u2208 K.prod \u22a4 \u2194 x \u2208 comap (MonoidHom.fst G N) K", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.toList_empty", "start": [3367, 1], "end": [3368, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean", "full_name": "CochainComplex.HomComplex.Cochain.add_comp", "start": [303, 1], "end": [307, 65], "traced_tactics": [{"tactic": "ext p q hpq", "annotated_tactic": ["ext p q hpq", []], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : Linear R C\nF G K L : CochainComplex C \u2124\nn m n\u2081 n\u2082 n\u2081\u2082 : \u2124\nz\u2081 z\u2081' : Cochain F G n\u2081\nz\u2082 : Cochain G K n\u2082\nh : n\u2081 + n\u2082 = n\u2081\u2082\n\u22a2 (z\u2081 + z\u2081').comp z\u2082 h = z\u2081.comp z\u2082 h + z\u2081'.comp z\u2082 h", "state_after": "case h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : Linear R C\nF G K L : CochainComplex C \u2124\nn m n\u2081 n\u2082 n\u2081\u2082 : \u2124\nz\u2081 z\u2081' : Cochain F G n\u2081\nz\u2082 : Cochain G K n\u2082\nh : n\u2081 + n\u2082 = n\u2081\u2082\np q : \u2124\nhpq : p + n\u2081\u2082 = q\n\u22a2 ((z\u2081 + z\u2081').comp z\u2082 h).v p q hpq = (z\u2081.comp z\u2082 h + z\u2081'.comp z\u2082 h).v p q hpq"}, {"tactic": "simp only [comp_v _ _ h p _ q rfl (by omega), add_v, add_comp]", "annotated_tactic": ["simp only [<a>comp_v</a> _ _ h p _ q <a>rfl</a> (by omega), <a>add_v</a>, <a>add_comp</a>]", [{"full_name": "CochainComplex.HomComplex.Cochain.comp_v", "def_path": "Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean", "def_pos": [236, 7], "def_end_pos": [236, 13]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "CochainComplex.HomComplex.Cochain.add_v", "def_path": "Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean", "def_pos": [109, 7], "def_end_pos": [109, 12]}, {"full_name": "CategoryTheory.Preadditive.add_comp", "def_path": "Mathlib/CategoryTheory/Preadditive/Basic.lean", "def_pos": [60, 3], "def_end_pos": [60, 11]}]], "state_before": "case h\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : Linear R C\nF G K L : CochainComplex C \u2124\nn m n\u2081 n\u2082 n\u2081\u2082 : \u2124\nz\u2081 z\u2081' : Cochain F G n\u2081\nz\u2082 : Cochain G K n\u2082\nh : n\u2081 + n\u2082 = n\u2081\u2082\np q : \u2124\nhpq : p + n\u2081\u2082 = q\n\u22a2 ((z\u2081 + z\u2081').comp z\u2082 h).v p q hpq = (z\u2081.comp z\u2082 h + z\u2081'.comp z\u2082 h).v p q hpq", "state_after": "no goals"}, {"tactic": "omega", "annotated_tactic": ["omega", []], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nR : Type u_1\ninst\u271d\u00b9 : Ring R\ninst\u271d : Linear R C\nF G K L : CochainComplex C \u2124\nn m n\u2081 n\u2082 n\u2081\u2082 : \u2124\nz\u2081 z\u2081' : Cochain F G n\u2081\nz\u2082 : Cochain G K n\u2082\nh : n\u2081 + n\u2082 = n\u2081\u2082\np q : \u2124\nhpq : p + n\u2081\u2082 = q\n\u22a2 p + n\u2081 + n\u2082 = q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Finiteness.lean", "full_name": "Subalgebra.fg_bot_toSubmodule", "start": [148, 1], "end": [150, 56], "traced_tactics": [{"tactic": "simp [Algebra.toSubmodule_bot, one_eq_span]", "annotated_tactic": ["simp [<a>Algebra.toSubmodule_bot</a>, <a>one_eq_span</a>]", [{"full_name": "Algebra.toSubmodule_bot", "def_path": "Mathlib/Algebra/Algebra/Subalgebra/Basic.lean", "def_pos": [887, 9], "def_end_pos": [887, 24]}, {"full_name": "Submodule.one_eq_span", "def_path": "Mathlib/Algebra/Algebra/Operations.lean", "def_pos": [106, 9], "def_end_pos": [106, 20]}]], "state_before": "R\u271d : Type u_1\nM : Type u_2\ninst\u271d\u2075 : Semiring R\u271d\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R\u271d M\nR : Type u_3\nA : Type u_4\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : Semiring A\ninst\u271d : Algebra R A\n\u22a2 span R \u2191{1} = Subalgebra.toSubmodule \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/ModularForms/Basic.lean", "full_name": "ModularForm.coe_sub", "start": [231, 1], "end": [232, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Int.ceil_sub_nat", "start": [1281, 1], "end": [1283, 7], "traced_tactics": [{"tactic": "convert ceil_sub_int a n using 1", "annotated_tactic": ["convert <a>ceil_sub_int</a> a n using 1", [{"full_name": "Int.ceil_sub_int", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [1276, 9], "def_end_pos": [1276, 21]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nn : \u2115\n\u22a2 \u2308a - \u2191n\u2309 = \u2308a\u2309 - \u2191n", "state_after": "case h.e'_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nn : \u2115\n\u22a2 \u2308a - \u2191n\u2309 = \u2308a - \u2191\u2191n\u2309"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_2\nF : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedRing \u03b1\ninst\u271d : FloorRing \u03b1\nz : \u2124\na\u271d a : \u03b1\nn : \u2115\n\u22a2 \u2308a - \u2191n\u2309 = \u2308a - \u2191\u2191n\u2309", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean", "full_name": "DirichletCharacter.primitiveCharacter_one", "start": [256, 1], "end": [260, 24], "traced_tactics": [{"tactic": "rw [eq_one_iff_conductor_eq_one <| (@conductor_one R _ _ hn) \u25b8 Nat.one_ne_zero,\n    (isPrimitive_def _).1 (1 : DirichletCharacter R n).primitiveCharacter_isPrimitive,\n    conductor_one hn]", "annotated_tactic": ["rw [<a>eq_one_iff_conductor_eq_one</a> <| (@<a>conductor_one</a> R _ _ hn) \u25b8 <a>Nat.one_ne_zero</a>,\n      (<a>isPrimitive_def</a> _).1 (1 : <a>DirichletCharacter</a> R n).<a>primitiveCharacter_isPrimitive</a>,\n      <a>conductor_one</a> hn]", [{"full_name": "DirichletCharacter.eq_one_iff_conductor_eq_one", "def_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean", "def_pos": [207, 7], "def_end_pos": [207, 34]}, {"full_name": "DirichletCharacter.conductor_one", "def_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean", "def_pos": [198, 7], "def_end_pos": [198, 20]}, {"full_name": "Nat.one_ne_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [702, 19], "def_end_pos": [702, 30]}, {"full_name": "DirichletCharacter.isPrimitive_def", "def_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean", "def_pos": [233, 7], "def_end_pos": [233, 22]}, {"full_name": "DirichletCharacter", "def_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean", "def_pos": [33, 8], "def_end_pos": [33, 26]}, {"full_name": "DirichletCharacter.primitiveCharacter_isPrimitive", "def_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean", "def_pos": [249, 7], "def_end_pos": [249, 37]}, {"full_name": "DirichletCharacter.conductor_one", "def_path": "Mathlib/NumberTheory/DirichletCharacter/Basic.lean", "def_pos": [198, 7], "def_end_pos": [198, 20]}]], "state_before": "R : Type u_1\ninst\u271d : CommMonoidWithZero R\nn : \u2115\n\u03c7 : DirichletCharacter R n\nhn : n \u2260 0\n\u22a2 primitiveCharacter 1 = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/PowerSeries/Basic.lean", "full_name": "PowerSeries.coeff_mul", "start": [330, 1], "end": [335, 6], "traced_tactics": [{"tactic": "refine (MvPowerSeries.coeff_mul _ \u03c6 \u03c8).trans ?_", "annotated_tactic": ["refine (<a>MvPowerSeries.coeff_mul</a> _ \u03c6 \u03c8).<a>trans</a> ?_", [{"full_name": "MvPowerSeries.coeff_mul", "def_path": "Mathlib/RingTheory/MvPowerSeries/Basic.lean", "def_pos": [199, 9], "def_end_pos": [199, 18]}, {"full_name": "Eq.trans", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [335, 9], "def_end_pos": [335, 17]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\nn : \u2115\n\u03c6 \u03c8 : R\u27e6X\u27e7\n\u22a2 (coeff R n) (\u03c6 * \u03c8) = \u2211 p \u2208 antidiagonal n, (coeff R p.1) \u03c6 * (coeff R p.2) \u03c8", "state_after": "R : Type u_1\ninst\u271d : Semiring R\nn : \u2115\n\u03c6 \u03c8 : R\u27e6X\u27e7\n\u22a2 \u2211 p \u2208 antidiagonal (single () n), (MvPowerSeries.coeff R p.1) \u03c6 * (MvPowerSeries.coeff R p.2) \u03c8 =\n    \u2211 p \u2208 antidiagonal n, (coeff R p.1) \u03c6 * (coeff R p.2) \u03c8"}, {"tactic": "rw [Finsupp.antidiagonal_single, Finset.sum_map]", "annotated_tactic": ["rw [<a>Finsupp.antidiagonal_single</a>, <a>Finset.sum_map</a>]", [{"full_name": "Finsupp.antidiagonal_single", "def_path": "Mathlib/Data/Finsupp/Antidiagonal.lean", "def_pos": [61, 9], "def_end_pos": [61, 28]}, {"full_name": "Finset.sum_map", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [407, 3], "def_end_pos": [407, 14]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\nn : \u2115\n\u03c6 \u03c8 : R\u27e6X\u27e7\n\u22a2 \u2211 p \u2208 antidiagonal (single () n), (MvPowerSeries.coeff R p.1) \u03c6 * (MvPowerSeries.coeff R p.2) \u03c8 =\n    \u2211 p \u2208 antidiagonal n, (coeff R p.1) \u03c6 * (coeff R p.2) \u03c8", "state_after": "R : Type u_1\ninst\u271d : Semiring R\nn : \u2115\n\u03c6 \u03c8 : R\u27e6X\u27e7\n\u22a2 \u2211 x \u2208 antidiagonal n,\n      (MvPowerSeries.coeff R (({ toFun := single (), inj' := \u22ef }.prodMap { toFun := single (), inj' := \u22ef }) x).1) \u03c6 *\n        (MvPowerSeries.coeff R (({ toFun := single (), inj' := \u22ef }.prodMap { toFun := single (), inj' := \u22ef }) x).2) \u03c8 =\n    \u2211 p \u2208 antidiagonal n, (coeff R p.1) \u03c6 * (coeff R p.2) \u03c8"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "R : Type u_1\ninst\u271d : Semiring R\nn : \u2115\n\u03c6 \u03c8 : R\u27e6X\u27e7\n\u22a2 \u2211 x \u2208 antidiagonal n,\n      (MvPowerSeries.coeff R (({ toFun := single (), inj' := \u22ef }.prodMap { toFun := single (), inj' := \u22ef }) x).1) \u03c6 *\n        (MvPowerSeries.coeff R (({ toFun := single (), inj' := \u22ef }.prodMap { toFun := single (), inj' := \u22ef }) x).2) \u03c8 =\n    \u2211 p \u2208 antidiagonal n, (coeff R p.1) \u03c6 * (coeff R p.2) \u03c8", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ENNReal/Real.lean", "full_name": "ENNReal.toNNReal_pos", "start": [163, 1], "end": [164, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Prime.lean", "full_name": "Nat.Prime.ne_zero", "start": [61, 1], "end": [62, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/GroupAction/SubMulAction.lean", "full_name": "SubMulAction.image_inclusion", "start": [477, 1], "end": [480, 26], "traced_tactics": [{"tactic": "rw [inclusion.coe_eq]", "annotated_tactic": ["rw [<a>inclusion.coe_eq</a>]", [{"full_name": "SubMulAction.inclusion.coe_eq", "def_path": "Mathlib/GroupTheory/GroupAction/SubMulAction.lean", "def_pos": [474, 9], "def_end_pos": [474, 25]}]], "state_before": "S : Type u'\nT : Type u''\nR : Type u\nM\u271d : Type v\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Monoid M\ninst\u271d : MulAction M \u03b1\ns : SubMulAction M \u03b1\n\u22a2 Set.range \u21d1s.inclusion = s.carrier", "state_after": "S : Type u'\nT : Type u''\nR : Type u\nM\u271d : Type v\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Monoid M\ninst\u271d : MulAction M \u03b1\ns : SubMulAction M \u03b1\n\u22a2 Set.range Subtype.val = s.carrier"}, {"tactic": "exact Subtype.range_coe", "annotated_tactic": ["exact <a>Subtype.range_coe</a>", [{"full_name": "Subtype.range_coe", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1373, 9], "def_end_pos": [1373, 18]}]], "state_before": "S : Type u'\nT : Type u''\nR : Type u\nM\u271d : Type v\nM : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b9 : Monoid M\ninst\u271d : MulAction M \u03b1\ns : SubMulAction M \u03b1\n\u22a2 Set.range Subtype.val = s.carrier", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/Exact.lean", "full_name": "CategoryTheory.exact_kernel_\u03b9", "start": [230, 1], "end": [232, 36], "traced_tactics": [{"tactic": "rw [\u2190 kernelSubobject_arrow', exact_iso_comp]", "annotated_tactic": ["rw [\u2190 <a>kernelSubobject_arrow'</a>, <a>exact_iso_comp</a>]", [{"full_name": "CategoryTheory.Limits.kernelSubobject_arrow'", "def_path": "Mathlib/CategoryTheory/Subobject/Limits.lean", "def_pos": [104, 9], "def_end_pos": [104, 31]}, {"full_name": "CategoryTheory.exact_iso_comp", "def_path": "Mathlib/Algebra/Homology/Exact.lean", "def_pos": [197, 9], "def_end_pos": [197, 23]}]], "state_before": "V : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasImages V\nA B C D : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\ninst\u271d\u00b9 : HasZeroMorphisms V\ninst\u271d : HasEqualizers V\n\u22a2 Exact (kernel.\u03b9 f) f", "state_after": "V : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasImages V\nA B C D : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\ninst\u271d\u00b9 : HasZeroMorphisms V\ninst\u271d : HasEqualizers V\n\u22a2 Exact (kernelSubobject f).arrow f"}, {"tactic": "exact exact_kernelSubobject_arrow", "annotated_tactic": ["exact <a>exact_kernelSubobject_arrow</a>", [{"full_name": "CategoryTheory.exact_kernelSubobject_arrow", "def_path": "Mathlib/Algebra/Homology/Exact.lean", "def_pos": [224, 9], "def_end_pos": [224, 36]}]], "state_before": "V : Type u\ninst\u271d\u00b3 : Category.{v, u} V\ninst\u271d\u00b2 : HasImages V\nA B C D : V\nf : A \u27f6 B\ng : B \u27f6 C\nh : C \u27f6 D\ninst\u271d\u00b9 : HasZeroMorphisms V\ninst\u271d : HasEqualizers V\n\u22a2 Exact (kernelSubobject f).arrow f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/PNat/Defs.lean", "full_name": "PNat.not_lt_one", "start": [183, 1], "end": [184, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.filter_congr_decidable", "start": [2620, 1], "end": [2621, 65], "traced_tactics": [{"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d q : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\u271d\ninst\u271d\u00b9 : DecidablePred q\ns\u271d s : Finset \u03b1\np : \u03b1 \u2192 Prop\nh inst\u271d : DecidablePred p\n\u22a2 filter p s = filter p s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Nat.floor_zero", "start": [175, 1], "end": [175, 78], "traced_tactics": [{"tactic": "rw [\u2190 Nat.cast_zero, floor_natCast]", "annotated_tactic": ["rw [\u2190 <a>Nat.cast_zero</a>, <a>floor_natCast</a>]", [{"full_name": "Nat.cast_zero", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}, {"full_name": "Nat.floor_natCast", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [166, 9], "def_end_pos": [166, 22]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : LinearOrderedSemiring \u03b1\ninst\u271d : FloorSemiring \u03b1\na : \u03b1\nn : \u2115\n\u22a2 \u230a0\u230b\u208a = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Infsep.lean", "full_name": "Set.infsep_le_of_mem_of_edist_le", "start": [418, 1], "end": [420, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Separation.lean", "full_name": "lim_eq", "start": [1590, 1], "end": [1591, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean", "full_name": "ContinuousMultilinearMap.uncurryRight_apply", "start": [248, 1], "end": [251, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "full_name": "Cardinal.lift_mk_le_lift_mk_of_surjective", "start": [1982, 1], "end": [1984, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Coprime/Lemmas.lean", "full_name": "exists_sum_eq_one_iff_pairwise_coprime'", "start": [178, 1], "end": [181, 97], "traced_tactics": [{"tactic": "convert exists_sum_eq_one_iff_pairwise_coprime Finset.univ_nonempty (s := s) using 1", "annotated_tactic": ["convert <a>exists_sum_eq_one_iff_pairwise_coprime</a> <a>Finset.univ_nonempty</a> (s := s) using 1", [{"full_name": "exists_sum_eq_one_iff_pairwise_coprime", "def_path": "Mathlib/RingTheory/Coprime/Lemmas.lean", "def_pos": [120, 9], "def_end_pos": [120, 47]}, {"full_name": "Finset.univ_nonempty", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [109, 9], "def_end_pos": [109, 22]}]], "state_before": "R : Type u\nI : Type v\ninst\u271d\u00b3 : CommSemiring R\nx y z : R\ns : I \u2192 R\nt : Finset I\ninst\u271d\u00b2 : Fintype I\ninst\u271d\u00b9 : Nonempty I\ninst\u271d : DecidableEq I\n\u22a2 (\u2203 \u03bc, \u2211 i : I, \u03bc i * \u220f j \u2208 {i}\u1d9c, s j = 1) \u2194 Pairwise (IsCoprime on s)", "state_after": "case h.e'_2.a\nR : Type u\nI : Type v\ninst\u271d\u00b3 : CommSemiring R\nx y z : R\ns : I \u2192 R\nt : Finset I\ninst\u271d\u00b2 : Fintype I\ninst\u271d\u00b9 : Nonempty I\ninst\u271d : DecidableEq I\n\u22a2 Pairwise (IsCoprime on s) \u2194 Pairwise (IsCoprime on fun i => s \u2191i)"}, {"tactic": "simp only [Function.onFun, pairwise_subtype_iff_pairwise_finset', coe_univ, Set.pairwise_univ]", "annotated_tactic": ["simp only [<a>Function.onFun</a>, <a>pairwise_subtype_iff_pairwise_finset'</a>, <a>coe_univ</a>, <a>Set.pairwise_univ</a>]", [{"full_name": "Function.onFun", "def_path": "Mathlib/Init/Function.lean", "def_pos": [59, 8], "def_end_pos": [59, 13]}, {"full_name": "Finset.pairwise_subtype_iff_pairwise_finset'", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3456, 9], "def_end_pos": [3456, 46]}, {"full_name": "Finset.coe_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [92, 9], "def_end_pos": [92, 17]}, {"full_name": "Set.pairwise_univ", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [200, 9], "def_end_pos": [200, 22]}]], "state_before": "case h.e'_2.a\nR : Type u\nI : Type v\ninst\u271d\u00b3 : CommSemiring R\nx y z : R\ns : I \u2192 R\nt : Finset I\ninst\u271d\u00b2 : Fintype I\ninst\u271d\u00b9 : Nonempty I\ninst\u271d : DecidableEq I\n\u22a2 Pairwise (IsCoprime on s) \u2194 Pairwise (IsCoprime on fun i => s \u2191i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/OpenSubgroup.lean", "full_name": "OpenSubgroup.isClosed", "start": [168, 1], "end": [176, 19], "traced_tactics": [{"tactic": "apply isOpen_compl_iff.1", "annotated_tactic": ["apply <a>isOpen_compl_iff</a>.1", [{"full_name": "isOpen_compl_iff", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [160, 17], "def_end_pos": [160, 33]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\n\u22a2 IsClosed \u2191U", "state_after": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\n\u22a2 IsOpen (\u2191U)\u1d9c"}, {"tactic": "refine isOpen_iff_forall_mem_open.2 fun x hx => \u27e8(fun y => y * x\u207b\u00b9) \u207b\u00b9' U, ?_, ?_, ?_\u27e9", "annotated_tactic": ["refine <a>isOpen_iff_forall_mem_open</a>.2 fun x hx => \u27e8(fun y => y * x\u207b\u00b9) \u207b\u00b9' U, ?_, ?_, ?_\u27e9", [{"full_name": "isOpen_iff_forall_mem_open", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 35]}]], "state_before": "G : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\n\u22a2 IsOpen (\u2191U)\u1d9c", "state_after": "case refine_1\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\nx : G\nhx : x \u2208 (\u2191U)\u1d9c\n\u22a2 (fun y => y * x\u207b\u00b9) \u207b\u00b9' \u2191U \u2286 (\u2191U)\u1d9c\n\ncase refine_2\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\nx : G\nhx : x \u2208 (\u2191U)\u1d9c\n\u22a2 IsOpen ((fun y => y * x\u207b\u00b9) \u207b\u00b9' \u2191U)\n\ncase refine_3\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\nx : G\nhx : x \u2208 (\u2191U)\u1d9c\n\u22a2 x \u2208 (fun y => y * x\u207b\u00b9) \u207b\u00b9' \u2191U"}, {"tactic": "refine fun u hux hu => hx ?_", "annotated_tactic": ["refine fun u hux hu => hx ?_", []], "state_before": "case refine_1\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\nx : G\nhx : x \u2208 (\u2191U)\u1d9c\n\u22a2 (fun y => y * x\u207b\u00b9) \u207b\u00b9' \u2191U \u2286 (\u2191U)\u1d9c", "state_after": "case refine_1\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\nx : G\nhx : x \u2208 (\u2191U)\u1d9c\nu : G\nhux : u \u2208 (fun y => y * x\u207b\u00b9) \u207b\u00b9' \u2191U\nhu : u \u2208 \u2191U\n\u22a2 x \u2208 \u2191U"}, {"tactic": "simp only [Set.mem_preimage, SetLike.mem_coe] at hux hu \u22a2", "annotated_tactic": ["simp only [<a>Set.mem_preimage</a>, <a>SetLike.mem_coe</a>] at hux hu \u22a2", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [120, 9], "def_end_pos": [120, 21]}, {"full_name": "SetLike.mem_coe", "def_path": "Mathlib/Data/SetLike/Basic.lean", "def_pos": [180, 9], "def_end_pos": [180, 16]}]], "state_before": "case refine_1\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\nx : G\nhx : x \u2208 (\u2191U)\u1d9c\nu : G\nhux : u \u2208 (fun y => y * x\u207b\u00b9) \u207b\u00b9' \u2191U\nhu : u \u2208 \u2191U\n\u22a2 x \u2208 \u2191U", "state_after": "case refine_1\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\nx : G\nhx : x \u2208 (\u2191U)\u1d9c\nu : G\nhux : u * x\u207b\u00b9 \u2208 U\nhu : u \u2208 U\n\u22a2 x \u2208 U"}, {"tactic": "convert U.mul_mem (U.inv_mem hux) hu", "annotated_tactic": ["convert U.mul_mem (U.inv_mem hux) hu", []], "state_before": "case refine_1\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\nx : G\nhx : x \u2208 (\u2191U)\u1d9c\nu : G\nhux : u * x\u207b\u00b9 \u2208 U\nhu : u \u2208 U\n\u22a2 x \u2208 U", "state_after": "case a\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\nx : G\nhx : x \u2208 (\u2191U)\u1d9c\nu : G\nhux : u * x\u207b\u00b9 \u2208 U\nhu : u \u2208 U\n\u22a2 x \u2208 U \u2194 (u * x\u207b\u00b9)\u207b\u00b9 * u \u2208 \u2191U"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case a\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\nx : G\nhx : x \u2208 (\u2191U)\u1d9c\nu : G\nhux : u * x\u207b\u00b9 \u2208 U\nhu : u \u2208 U\n\u22a2 x \u2208 U \u2194 (u * x\u207b\u00b9)\u207b\u00b9 * u \u2208 \u2191U", "state_after": "no goals"}, {"tactic": "exact U.isOpen.preimage (continuous_mul_right _)", "annotated_tactic": ["exact U.isOpen.preimage (<a>continuous_mul_right</a> _)", [{"full_name": "continuous_mul_right", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [110, 9], "def_end_pos": [110, 29]}]], "state_before": "case refine_2\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\nx : G\nhx : x \u2208 (\u2191U)\u1d9c\n\u22a2 IsOpen ((fun y => y * x\u207b\u00b9) \u207b\u00b9' \u2191U)", "state_after": "no goals"}, {"tactic": "simp [one_mem]", "annotated_tactic": ["simp [<a>one_mem</a>]", [{"full_name": "OneMemClass.one_mem", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [72, 3], "def_end_pos": [72, 10]}]], "state_before": "case refine_3\nG : Type u_1\ninst\u271d\u00b2 : Group G\ninst\u271d\u00b9 : TopologicalSpace G\nU\u271d V : OpenSubgroup G\ng : G\ninst\u271d : ContinuousMul G\nU : OpenSubgroup G\nx : G\nhx : x \u2208 (\u2191U)\u1d9c\n\u22a2 x \u2208 (fun y => y * x\u207b\u00b9) \u207b\u00b9' \u2191U", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Units.lean", "full_name": "inv_eq_one_divp", "start": [570, 1], "end": [570, 69], "traced_tactics": [{"tactic": "rw [one_divp]", "annotated_tactic": ["rw [<a>one_divp</a>]", [{"full_name": "one_divp", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [564, 9], "def_end_pos": [564, 17]}]], "state_before": "\u03b1 : Type u\ninst\u271d : Monoid \u03b1\na b c : \u03b1\nu : \u03b1\u02e3\n\u22a2 \u2191u\u207b\u00b9 = 1 /\u209a u", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Subring/Basic.lean", "full_name": "AddSubgroup.int_mul_mem", "start": [1374, 1], "end": [1377, 7], "traced_tactics": [{"tactic": "convert AddSubgroup.zsmul_mem G h k using 1", "annotated_tactic": ["convert <a>AddSubgroup.zsmul_mem</a> G h k using 1", [{"full_name": "AddSubgroup.zsmul_mem", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [623, 3], "def_end_pos": [623, 14]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : Ring T\nG : AddSubgroup R\nk : \u2124\ng : R\nh : g \u2208 G\n\u22a2 \u2191k * g \u2208 G", "state_after": "case h.e'_4\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : Ring T\nG : AddSubgroup R\nk : \u2124\ng : R\nh : g \u2208 G\n\u22a2 \u2191k * g = k \u2022 g"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_4\nR : Type u\nS : Type v\nT : Type w\ninst\u271d\u00b2 : Ring R\ninst\u271d\u00b9 : Ring S\ninst\u271d : Ring T\nG : AddSubgroup R\nk : \u2124\ng : R\nh : g \u2208 G\n\u22a2 \u2191k * g = k \u2022 g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Abelian/RightDerived.lean", "full_name": "CategoryTheory.NatTrans.rightDerivedToHomotopyCategory_comp", "start": [203, 1], "end": [208, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/EuclideanDomain/Basic.lean", "full_name": "EuclideanDomain.zero_mod", "start": [73, 1], "end": [74, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/ModularForms/SlashActions.lean", "full_name": "ModularForm.smul_slash", "start": [135, 9], "end": [141, 7], "traced_tactics": [{"tactic": "simp_rw [\u2190 smul_one_smul \u2102 c f, \u2190 smul_one_smul \u2102 c (f \u2223[k]A)]", "annotated_tactic": ["simp_rw [\u2190 <a>smul_one_smul</a> \u2102 c f, \u2190 <a>smul_one_smul</a> \u2102 c (f \u2223[k]A)]", [{"full_name": "smul_one_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [638, 7], "def_end_pos": [638, 20]}, {"full_name": "smul_one_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [638, 7], "def_end_pos": [638, 20]}]], "state_before": "\u0393 : Subgroup SL(2, \u2124)\nk\u271d : \u2124\nf\u271d : \u210d \u2192 \u2102\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SMul \u03b1 \u2102\ninst\u271d : IsScalarTower \u03b1 \u2102 \u2102\nk : \u2124\nA : \u21a5GL(2, \u211d)\u207a\nf : \u210d \u2192 \u2102\nc : \u03b1\n\u22a2 (c \u2022 f) \u2223[k]A = c \u2022 f \u2223[k]A", "state_after": "\u0393 : Subgroup SL(2, \u2124)\nk\u271d : \u2124\nf\u271d : \u210d \u2192 \u2102\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SMul \u03b1 \u2102\ninst\u271d : IsScalarTower \u03b1 \u2102 \u2102\nk : \u2124\nA : \u21a5GL(2, \u211d)\u207a\nf : \u210d \u2192 \u2102\nc : \u03b1\n\u22a2 ((c \u2022 1) \u2022 f) \u2223[k]A = (c \u2022 1) \u2022 f \u2223[k]A"}, {"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "\u0393 : Subgroup SL(2, \u2124)\nk\u271d : \u2124\nf\u271d : \u210d \u2192 \u2102\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SMul \u03b1 \u2102\ninst\u271d : IsScalarTower \u03b1 \u2102 \u2102\nk : \u2124\nA : \u21a5GL(2, \u211d)\u207a\nf : \u210d \u2192 \u2102\nc : \u03b1\n\u22a2 ((c \u2022 1) \u2022 f) \u2223[k]A = (c \u2022 1) \u2022 f \u2223[k]A", "state_after": "case h\n\u0393 : Subgroup SL(2, \u2124)\nk\u271d : \u2124\nf\u271d : \u210d \u2192 \u2102\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SMul \u03b1 \u2102\ninst\u271d : IsScalarTower \u03b1 \u2102 \u2102\nk : \u2124\nA : \u21a5GL(2, \u211d)\u207a\nf : \u210d \u2192 \u2102\nc : \u03b1\nx\u271d : \u210d\n\u22a2 (((c \u2022 1) \u2022 f) \u2223[k]A) x\u271d = ((c \u2022 1) \u2022 f \u2223[k]A) x\u271d"}, {"tactic": "simp_rw [slash]", "annotated_tactic": ["simp_rw [<a>slash</a>]", [{"full_name": "ModularForm.slash", "def_path": "Mathlib/NumberTheory/ModularForms/SlashActions.lean", "def_pos": [92, 5], "def_end_pos": [92, 10]}]], "state_before": "case h\n\u0393 : Subgroup SL(2, \u2124)\nk\u271d : \u2124\nf\u271d : \u210d \u2192 \u2102\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SMul \u03b1 \u2102\ninst\u271d : IsScalarTower \u03b1 \u2102 \u2102\nk : \u2124\nA : \u21a5GL(2, \u211d)\u207a\nf : \u210d \u2192 \u2102\nc : \u03b1\nx\u271d : \u210d\n\u22a2 (((c \u2022 1) \u2022 f) \u2223[k]A) x\u271d = ((c \u2022 1) \u2022 f \u2223[k]A) x\u271d", "state_after": "case h\n\u0393 : Subgroup SL(2, \u2124)\nk\u271d : \u2124\nf\u271d : \u210d \u2192 \u2102\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SMul \u03b1 \u2102\ninst\u271d : IsScalarTower \u03b1 \u2102 \u2102\nk : \u2124\nA : \u21a5GL(2, \u211d)\u207a\nf : \u210d \u2192 \u2102\nc : \u03b1\nx\u271d : \u210d\n\u22a2 ((c \u2022 1) \u2022 f) (A \u2022 x\u271d) * \u2191(\u2191\u2191A).det ^ (k - 1) * denom A x\u271d ^ (-k) = ((c \u2022 1) \u2022 f \u2223[k]A) x\u271d"}, {"tactic": "simp only [slash, Algebra.id.smul_eq_mul, Matrix.GeneralLinearGroup.val_det_apply, Pi.smul_apply]", "annotated_tactic": ["simp only [<a>slash</a>, <a>Algebra.id.smul_eq_mul</a>, <a>Matrix.GeneralLinearGroup.val_det_apply</a>, <a>Pi.smul_apply</a>]", [{"full_name": "ModularForm.slash", "def_path": "Mathlib/NumberTheory/ModularForms/SlashActions.lean", "def_pos": [92, 5], "def_end_pos": [92, 10]}, {"full_name": "Algebra.id.smul_eq_mul", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [449, 9], "def_end_pos": [449, 20]}, {"full_name": "Matrix.GeneralLinearGroup.val_det_apply", "def_path": "Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup.lean", "def_pos": [66, 3], "def_end_pos": [66, 8]}, {"full_name": "Pi.smul_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [129, 60], "def_end_pos": [129, 70]}]], "state_before": "case h\n\u0393 : Subgroup SL(2, \u2124)\nk\u271d : \u2124\nf\u271d : \u210d \u2192 \u2102\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SMul \u03b1 \u2102\ninst\u271d : IsScalarTower \u03b1 \u2102 \u2102\nk : \u2124\nA : \u21a5GL(2, \u211d)\u207a\nf : \u210d \u2192 \u2102\nc : \u03b1\nx\u271d : \u210d\n\u22a2 ((c \u2022 1) \u2022 f) (A \u2022 x\u271d) * \u2191(\u2191\u2191A).det ^ (k - 1) * denom A x\u271d ^ (-k) = ((c \u2022 1) \u2022 f \u2223[k]A) x\u271d", "state_after": "case h\n\u0393 : Subgroup SL(2, \u2124)\nk\u271d : \u2124\nf\u271d : \u210d \u2192 \u2102\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SMul \u03b1 \u2102\ninst\u271d : IsScalarTower \u03b1 \u2102 \u2102\nk : \u2124\nA : \u21a5GL(2, \u211d)\u207a\nf : \u210d \u2192 \u2102\nc : \u03b1\nx\u271d : \u210d\n\u22a2 c \u2022 1 * f (A \u2022 x\u271d) * \u2191(\u2191\u2191A).det ^ (k - 1) * denom A x\u271d ^ (-k) =\n    c \u2022 1 * (f (A \u2022 x\u271d) * \u2191(\u2191\u2191A).det ^ (k - 1) * denom A x\u271d ^ (-k))"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case h\n\u0393 : Subgroup SL(2, \u2124)\nk\u271d : \u2124\nf\u271d : \u210d \u2192 \u2102\n\u03b1 : Type u_1\ninst\u271d\u00b9 : SMul \u03b1 \u2102\ninst\u271d : IsScalarTower \u03b1 \u2102 \u2102\nk : \u2124\nA : \u21a5GL(2, \u211d)\u207a\nf : \u210d \u2192 \u2102\nc : \u03b1\nx\u271d : \u210d\n\u22a2 c \u2022 1 * f (A \u2022 x\u271d) * \u2191(\u2191\u2191A).det ^ (k - 1) * denom A x\u271d ^ (-k) =\n    c \u2022 1 * (f (A \u2022 x\u271d) * \u2191(\u2191\u2191A).det ^ (k - 1) * denom A x\u271d ^ (-k))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Exponential.lean", "full_name": "Ordinal.log_one_left", "start": [295, 1], "end": [296, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matroid/Restrict.lean", "full_name": "Matroid.restrict_eq_self_iff", "start": [205, 9], "end": [206, 50], "traced_tactics": [{"tactic": "rw [\u2190 h]", "annotated_tactic": ["rw [\u2190 h]", []], "state_before": "\u03b1 : Type u_1\nM : Matroid \u03b1\nR I J X Y : Set \u03b1\nh : M \u21be R = M\n\u22a2 R = M.E", "state_after": "\u03b1 : Type u_1\nM : Matroid \u03b1\nR I J X Y : Set \u03b1\nh : M \u21be R = M\n\u22a2 R = (M \u21be R).E"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\u03b1 : Type u_1\nM : Matroid \u03b1\nR I J X Y : Set \u03b1\nh : M \u21be R = M\n\u22a2 R = (M \u21be R).E", "state_after": "no goals"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "\u03b1 : Type u_1\nM : Matroid \u03b1\nR I J X Y : Set \u03b1\nh : R = M.E\n\u22a2 M \u21be R = M", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Derivative.lean", "full_name": "Polynomial.iterate_derivative_map", "start": [325, 1], "end": [329, 90], "traced_tactics": [{"tactic": "induction' k with k ih generalizing p", "annotated_tactic": ["induction' k with k ih generalizing p", []], "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\np : R[X]\nf : R \u2192+* S\nk : \u2115\n\u22a2 (\u21d1derivative)^[k] (map f p) = map f ((\u21d1derivative)^[k] p)", "state_after": "case zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nf : R \u2192+* S\np : R[X]\n\u22a2 (\u21d1derivative)^[0] (map f p) = map f ((\u21d1derivative)^[0] p)\n\ncase succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nf : R \u2192+* S\nk : \u2115\nih : \u2200 (p : R[X]), (\u21d1derivative)^[k] (map f p) = map f ((\u21d1derivative)^[k] p)\np : R[X]\n\u22a2 (\u21d1derivative)^[k + 1] (map f p) = map f ((\u21d1derivative)^[k + 1] p)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case zero\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nf : R \u2192+* S\np : R[X]\n\u22a2 (\u21d1derivative)^[0] (map f p) = map f ((\u21d1derivative)^[0] p)", "state_after": "no goals"}, {"tactic": "simp only [ih, Function.iterate_succ, Polynomial.derivative_map, Function.comp_apply]", "annotated_tactic": ["simp only [ih, <a>Function.iterate_succ</a>, <a>Polynomial.derivative_map</a>, <a>Function.comp_apply</a>]", [{"full_name": "Function.iterate_succ", "def_path": "Mathlib/Logic/Function/Iterate.lean", "def_pos": [62, 9], "def_end_pos": [62, 21]}, {"full_name": "Polynomial.derivative_map", "def_path": "Mathlib/Algebra/Polynomial/Derivative.lean", "def_pos": [313, 9], "def_end_pos": [313, 23]}, {"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}]], "state_before": "case succ\nR : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b9 : Semiring R\ninst\u271d : Semiring S\nf : R \u2192+* S\nk : \u2115\nih : \u2200 (p : R[X]), (\u21d1derivative)^[k] (map f p) = map f ((\u21d1derivative)^[k] p)\np : R[X]\n\u22a2 (\u21d1derivative)^[k + 1] (map f p) = map f ((\u21d1derivative)^[k + 1] p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "full_name": "Polynomial.natDegree_X", "start": [527, 1], "end": [528, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.mem_singleton", "start": [680, 1], "end": [681, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Tactic/Abel.lean", "full_name": "Mathlib.Tactic.Abel.subst_into_negg", "start": [291, 1], "end": [293, 18], "traced_tactics": [{"tactic": "simp [pra, prt]", "annotated_tactic": ["simp [pra, prt]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : AddCommGroup \u03b1\na ta t : \u03b1\npra : a = ta\nprt : -ta = t\n\u22a2 -a = t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Tactic/Positivity/Basic.lean", "full_name": "Mathlib.Meta.Positivity.ite_ne_zero_of_ne_zero_of_pos", "start": [48, 1], "end": [50, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "full_name": "MeasureTheory.integrableOn_congr_fun", "start": [151, 1], "end": [153, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Comma.lean", "full_name": "CategoryTheory.CostructuredArrow.epi_iff_epi_left", "start": [294, 1], "end": [296, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Extr.lean", "full_name": "IsMaxOn.bicomp_mono", "start": [387, 1], "end": [390, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Acyclic.lean", "full_name": "SimpleGraph.isAcyclic_iff_forall_edge_isBridge", "start": [83, 1], "end": [85, 56], "traced_tactics": [{"tactic": "simp [isAcyclic_iff_forall_adj_isBridge, Sym2.forall]", "annotated_tactic": ["simp [<a>isAcyclic_iff_forall_adj_isBridge</a>, <a>Sym2.forall</a>]", [{"full_name": "SimpleGraph.isAcyclic_iff_forall_adj_isBridge", "def_path": "Mathlib/Combinatorics/SimpleGraph/Acyclic.lean", "def_pos": [68, 9], "def_end_pos": [68, 42]}, {"full_name": "Sym2.forall", "def_path": "Mathlib/Data/Sym/Sym2.lean", "def_pos": [166, 19], "def_end_pos": [166, 27]}]], "state_before": "V : Type u\nG : SimpleGraph V\n\u22a2 G.IsAcyclic \u2194 \u2200 \u2983e : Sym2 V\u2984, e \u2208 G.edgeSet \u2192 G.IsBridge e", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/SpecificGroups/Alternating.lean", "full_name": "Equiv.Perm.isThreeCycle_sq_of_three_mem_cycleType_five", "start": [197, 1], "end": [210, 38], "traced_tactics": [{"tactic": "obtain \u27e8c, g', rfl, hd, _, h3\u27e9 := mem_cycleType_iff.1 h", "annotated_tactic": ["obtain \u27e8c, g', rfl, hd, _, h3\u27e9 := <a>mem_cycleType_iff</a>.1 h", [{"full_name": "Equiv.Perm.mem_cycleType_iff", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Type.lean", "def_pos": [290, 9], "def_end_pos": [290, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\ng : Perm (Fin 5)\nh : 3 \u2208 g.cycleType\n\u22a2 (g * g).IsThreeCycle", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\n\u22a2 (c * g' * (c * g')).IsThreeCycle"}, {"tactic": "simp only [mul_assoc]", "annotated_tactic": ["simp only [<a>mul_assoc</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\n\u22a2 (c * g' * (c * g')).IsThreeCycle", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\n\u22a2 (c * (g' * (c * g'))).IsThreeCycle"}, {"tactic": "rw [hd.commute.eq, \u2190 mul_assoc g']", "annotated_tactic": ["rw [hd.commute.eq, \u2190 <a>mul_assoc</a> g']", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\n\u22a2 (c * (g' * (c * g'))).IsThreeCycle", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\n\u22a2 (c * (g' * g' * c)).IsThreeCycle"}, {"tactic": "suffices hg' : orderOf g' \u2223 2 by\n  rw [\u2190 pow_two, orderOf_dvd_iff_pow_eq_one.1 hg', one_mul]\n  exact (card_support_eq_three_iff.1 h3).isThreeCycle_sq", "annotated_tactic": ["suffices hg' : <a>orderOf</a> g' \u2223 2 by\n    rw [\u2190 <a>pow_two</a>, <a>orderOf_dvd_iff_pow_eq_one</a>.1 hg', <a>one_mul</a>]\n    exact (<a>card_support_eq_three_iff</a>.1 h3).<a>isThreeCycle_sq</a>", [{"full_name": "orderOf", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [155, 19], "def_end_pos": [155, 26]}, {"full_name": "pow_two", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [679, 32], "def_end_pos": [679, 39]}, {"full_name": "orderOf_dvd_iff_pow_eq_one", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [272, 9], "def_end_pos": [272, 35]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "card_support_eq_three_iff", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Type.lean", "def_pos": [593, 9], "def_end_pos": [593, 41]}, {"full_name": "Equiv.Perm.IsThreeCycle.isThreeCycle_sq", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Type.lean", "def_pos": [632, 9], "def_end_pos": [632, 24]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\n\u22a2 (c * (g' * g' * c)).IsThreeCycle", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\n\u22a2 orderOf g' \u2223 2"}, {"tactic": "rw [\u2190 lcm_cycleType, Multiset.lcm_dvd]", "annotated_tactic": ["rw [\u2190 <a>lcm_cycleType</a>, <a>Multiset.lcm_dvd</a>]", [{"full_name": "Equiv.Perm.lcm_cycleType", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Type.lean", "def_pos": [172, 9], "def_end_pos": [172, 22]}, {"full_name": "Multiset.lcm_dvd", "def_path": "Mathlib/Algebra/GCDMonoid/Multiset.lean", "def_pos": [63, 9], "def_end_pos": [63, 16]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\n\u22a2 orderOf g' \u2223 2", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\n\u22a2 \u2200 b \u2208 g'.cycleType, b \u2223 2"}, {"tactic": "intro n hn", "annotated_tactic": ["intro n hn", []], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\n\u22a2 \u2200 b \u2208 g'.cycleType, b \u2223 2", "state_after": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\nn : \u2115\nhn : n \u2208 g'.cycleType\n\u22a2 n \u2223 2"}, {"tactic": "rw [le_antisymm (two_le_of_mem_cycleType hn) (le_trans (le_card_support_of_mem_cycleType hn) _)]", "annotated_tactic": ["rw [<a>le_antisymm</a> (<a>two_le_of_mem_cycleType</a> hn) (<a>le_trans</a> (<a>le_card_support_of_mem_cycleType</a> hn) _)]", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "Equiv.Perm.two_le_of_mem_cycleType", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Type.lean", "def_pos": [94, 9], "def_end_pos": [94, 32]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "Equiv.Perm.le_card_support_of_mem_cycleType", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Type.lean", "def_pos": [305, 9], "def_end_pos": [305, 41]}]], "state_before": "case intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\nn : \u2115\nhn : n \u2208 g'.cycleType\n\u22a2 n \u2223 2", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\nn : \u2115\nhn : n \u2208 g'.cycleType\n\u22a2 g'.support.card \u2264 2"}, {"tactic": "apply le_of_add_le_add_left", "annotated_tactic": ["apply <a>le_of_add_le_add_left</a>", [{"full_name": "le_of_add_le_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [56, 15], "def_end_pos": [56, 36]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\nn : \u2115\nhn : n \u2208 g'.cycleType\n\u22a2 g'.support.card \u2264 2", "state_after": "case bc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\nn : \u2115\nhn : n \u2208 g'.cycleType\n\u22a2 ?a + g'.support.card \u2264 ?a + 2\n\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\nn : \u2115\nhn : n \u2208 g'.cycleType\n\u22a2 \u2115"}, {"tactic": "rw [\u2190 hd.card_support_mul, h3]", "annotated_tactic": ["rw [\u2190 hd.card_support_mul, h3]", []], "state_before": "case bc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\nn : \u2115\nhn : n \u2208 g'.cycleType\n\u22a2 ?a + g'.support.card \u2264 ?a + 2\n\ncase a\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\nn : \u2115\nhn : n \u2208 g'.cycleType\n\u22a2 \u2115", "state_after": "case bc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\nn : \u2115\nhn : n \u2208 g'.cycleType\n\u22a2 (c * g').support.card \u2264 3 + 2"}, {"tactic": "exact (c * g').support.card_le_univ", "annotated_tactic": ["exact (c * g').support.card_le_univ", []], "state_before": "case bc\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\nn : \u2115\nhn : n \u2208 g'.cycleType\n\u22a2 (c * g').support.card \u2264 3 + 2", "state_after": "no goals"}, {"tactic": "rw [\u2190 pow_two, orderOf_dvd_iff_pow_eq_one.1 hg', one_mul]", "annotated_tactic": ["rw [\u2190 <a>pow_two</a>, <a>orderOf_dvd_iff_pow_eq_one</a>.1 hg', <a>one_mul</a>]", [{"full_name": "pow_two", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [679, 32], "def_end_pos": [679, 39]}, {"full_name": "orderOf_dvd_iff_pow_eq_one", "def_path": "Mathlib/GroupTheory/OrderOfElement.lean", "def_pos": [272, 9], "def_end_pos": [272, 35]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\nhg' : orderOf g' \u2223 2\n\u22a2 (c * (g' * g' * c)).IsThreeCycle", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\nhg' : orderOf g' \u2223 2\n\u22a2 (c * c).IsThreeCycle"}, {"tactic": "exact (card_support_eq_three_iff.1 h3).isThreeCycle_sq", "annotated_tactic": ["exact (<a>card_support_eq_three_iff</a>.1 h3).<a>isThreeCycle_sq</a>", [{"full_name": "card_support_eq_three_iff", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Type.lean", "def_pos": [593, 9], "def_end_pos": [593, 41]}, {"full_name": "Equiv.Perm.IsThreeCycle.isThreeCycle_sq", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Type.lean", "def_pos": [632, 9], "def_end_pos": [632, 24]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : DecidableEq \u03b1\nc g' : Perm (Fin 5)\nh : 3 \u2208 (c * g').cycleType\nhd : c.Disjoint g'\nleft\u271d : c.IsCycle\nh3 : c.support.card = 3\nhg' : orderOf g' \u2223 2\n\u22a2 (c * c).IsThreeCycle", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Defs.lean", "full_name": "map_natCast_smul", "start": [434, 1], "end": [437, 70], "traced_tactics": [{"tactic": "simp only [\u2190 nsmul_eq_smul_cast, AddMonoidHom.map_nsmul, map_nsmul]", "annotated_tactic": ["simp only [\u2190 <a>nsmul_eq_smul_cast</a>, <a>AddMonoidHom.map_nsmul</a>, <a>map_nsmul</a>]", [{"full_name": "nsmul_eq_smul_cast", "def_path": "Mathlib/Algebra/Module/Defs.lean", "def_pos": [365, 9], "def_end_pos": [365, 27]}, {"full_name": "AddMonoidHom.map_nsmul", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [1019, 3], "def_end_pos": [1019, 14]}, {"full_name": "map_nsmul", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [472, 3], "def_end_pos": [472, 14]}]], "state_before": "\u03b1 : Type u_1\nR\u271d : Type u_2\nk : Type u_3\nS\u271d : Type u_4\nM : Type u_5\nM\u2082 : Type u_6\nM\u2083 : Type u_7\n\u03b9 : Type u_8\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\nF : Type u_9\ninst\u271d\u2075 : FunLike F M M\u2082\ninst\u271d\u2074 : AddMonoidHomClass F M M\u2082\nf : F\nR : Type u_10\nS : Type u_11\ninst\u271d\u00b3 : Semiring R\ninst\u271d\u00b2 : Semiring S\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module S M\u2082\nx : \u2115\na : M\n\u22a2 f (\u2191x \u2022 a) = \u2191x \u2022 f a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/WithDensityFinite.lean", "full_name": "MeasureTheory.densityToFinite_def", "start": [152, 1], "end": [153, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/Closure.lean", "full_name": "Equiv.Perm.closure_cycle_adjacent_swap", "start": [46, 1], "end": [93, 18], "traced_tactics": [{"tactic": "let H := closure ({\u03c3, swap x (\u03c3 x)} : Set (Perm \u03b1))", "annotated_tactic": ["let H := <a>closure</a> ({\u03c3, <a>swap</a> x (\u03c3 x)} : <a>Set</a> (<a>Perm</a> \u03b1))", [{"full_name": "Subgroup.closure", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [1073, 5], "def_end_pos": [1073, 12]}, {"full_name": "Equiv.swap", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1623, 5], "def_end_pos": [1623, 9]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "Equiv.Perm", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [87, 8], "def_end_pos": [87, 18]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\n\u22a2 closure {\u03c3, swap x (\u03c3 x)} = \u22a4", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\n\u22a2 closure {\u03c3, swap x (\u03c3 x)} = \u22a4"}, {"tactic": "have h3 : \u03c3 \u2208 H := subset_closure (Set.mem_insert \u03c3 _)", "annotated_tactic": ["have h3 : \u03c3 \u2208 H := <a>subset_closure</a> (<a>Set.mem_insert</a> \u03c3 _)", [{"full_name": "Subgroup.subset_closure", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [1089, 9], "def_end_pos": [1089, 23]}, {"full_name": "Set.mem_insert", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1081, 9], "def_end_pos": [1081, 19]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\n\u22a2 closure {\u03c3, swap x (\u03c3 x)} = \u22a4", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\n\u22a2 closure {\u03c3, swap x (\u03c3 x)} = \u22a4"}, {"tactic": "have h4 : swap x (\u03c3 x) \u2208 H := subset_closure (Set.mem_insert_of_mem _ (Set.mem_singleton _))", "annotated_tactic": ["have h4 : <a>swap</a> x (\u03c3 x) \u2208 H := <a>subset_closure</a> (<a>Set.mem_insert_of_mem</a> _ (<a>Set.mem_singleton</a> _))", [{"full_name": "Equiv.swap", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1623, 5], "def_end_pos": [1623, 9]}, {"full_name": "Subgroup.subset_closure", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [1089, 9], "def_end_pos": [1089, 23]}, {"full_name": "Set.mem_insert_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1085, 9], "def_end_pos": [1085, 26]}, {"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1246, 9], "def_end_pos": [1246, 22]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\n\u22a2 closure {\u03c3, swap x (\u03c3 x)} = \u22a4", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\n\u22a2 closure {\u03c3, swap x (\u03c3 x)} = \u22a4"}, {"tactic": "have step3 : \u2200 y : \u03b1, swap x y \u2208 H := by\n  intro y\n  have hx : x \u2208 (\u22a4 : Finset \u03b1) := Finset.mem_univ x\n  rw [\u2190 h2, mem_support] at hx\n  have hy : y \u2208 (\u22a4 : Finset \u03b1) := Finset.mem_univ y\n  rw [\u2190 h2, mem_support] at hy\n  cases' IsCycle.exists_pow_eq h1 hx hy with n hn\n  rw [\u2190 hn]\n  exact step2 n", "annotated_tactic": ["have step3 : \u2200 y : \u03b1, <a>swap</a> x y \u2208 H := by\n    intro y\n    have hx : x \u2208 (\u22a4 : <a>Finset</a> \u03b1) := <a>Finset.mem_univ</a> x\n    rw [\u2190 h2, <a>mem_support</a>] at hx\n    have hy : y \u2208 (\u22a4 : <a>Finset</a> \u03b1) := <a>Finset.mem_univ</a> y\n    rw [\u2190 h2, <a>mem_support</a>] at hy\n    cases' <a>IsCycle.exists_pow_eq</a> h1 hx hy with n hn\n    rw [\u2190 hn]\n    exact step2 n", [{"full_name": "Equiv.swap", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1623, 5], "def_end_pos": [1623, 9]}, {"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 17]}, {"full_name": "Equiv.Perm.mem_support", "def_path": "Mathlib/GroupTheory/Perm/Support.lean", "def_pos": [297, 9], "def_end_pos": [297, 20]}, {"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 17]}, {"full_name": "Equiv.Perm.mem_support", "def_path": "Mathlib/GroupTheory/Perm/Support.lean", "def_pos": [297, 9], "def_end_pos": [297, 20]}, {"full_name": "Equiv.Perm.IsCycle.exists_pow_eq", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Basic.lean", "def_pos": [347, 9], "def_end_pos": [347, 30]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\n\u22a2 closure {\u03c3, swap x (\u03c3 x)} = \u22a4", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\n\u22a2 closure {\u03c3, swap x (\u03c3 x)} = \u22a4"}, {"tactic": "rw [eq_top_iff, \u2190 closure_isSwap, closure_le]", "annotated_tactic": ["rw [<a>eq_top_iff</a>, \u2190 <a>closure_isSwap</a>, <a>closure_le</a>]", [{"full_name": "eq_top_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [133, 9], "def_end_pos": [133, 19]}, {"full_name": "Equiv.Perm.closure_isSwap", "def_path": "Mathlib/GroupTheory/Perm/Sign.lean", "def_pos": [121, 9], "def_end_pos": [121, 23]}, {"full_name": "Subgroup.closure_le", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [1104, 9], "def_end_pos": [1104, 19]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\nstep4 : \u2200 (y z : \u03b1), swap y z \u2208 H\n\u22a2 closure {\u03c3, swap x (\u03c3 x)} = \u22a4", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\nstep4 : \u2200 (y z : \u03b1), swap y z \u2208 H\n\u22a2 {\u03c3 | \u03c3.IsSwap} \u2286 \u2191(closure {\u03c3, swap x (\u03c3 x)})"}, {"tactic": "rintro \u03c4 \u27e8y, z, _, h6\u27e9", "annotated_tactic": ["rintro \u03c4 \u27e8y, z, _, h6\u27e9", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\nstep4 : \u2200 (y z : \u03b1), swap y z \u2208 H\n\u22a2 {\u03c3 | \u03c3.IsSwap} \u2286 \u2191(closure {\u03c3, swap x (\u03c3 x)})", "state_after": "case intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\nstep4 : \u2200 (y z : \u03b1), swap y z \u2208 H\n\u03c4 : Perm \u03b1\ny z : \u03b1\nleft\u271d : y \u2260 z\nh6 : \u03c4 = swap y z\n\u22a2 \u03c4 \u2208 \u2191(closure {\u03c3, swap x (\u03c3 x)})"}, {"tactic": "rw [h6]", "annotated_tactic": ["rw [h6]", []], "state_before": "case intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\nstep4 : \u2200 (y z : \u03b1), swap y z \u2208 H\n\u03c4 : Perm \u03b1\ny z : \u03b1\nleft\u271d : y \u2260 z\nh6 : \u03c4 = swap y z\n\u22a2 \u03c4 \u2208 \u2191(closure {\u03c3, swap x (\u03c3 x)})", "state_after": "case intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\nstep4 : \u2200 (y z : \u03b1), swap y z \u2208 H\n\u03c4 : Perm \u03b1\ny z : \u03b1\nleft\u271d : y \u2260 z\nh6 : \u03c4 = swap y z\n\u22a2 swap y z \u2208 \u2191(closure {\u03c3, swap x (\u03c3 x)})"}, {"tactic": "exact step4 y z", "annotated_tactic": ["exact step4 y z", []], "state_before": "case intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\nstep4 : \u2200 (y z : \u03b1), swap y z \u2208 H\n\u03c4 : Perm \u03b1\ny z : \u03b1\nleft\u271d : y \u2260 z\nh6 : \u03c4 = swap y z\n\u22a2 swap y z \u2208 \u2191(closure {\u03c3, swap x (\u03c3 x)})", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\n\u22a2 \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nn : \u2115\n\u22a2 swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H"}, {"tactic": "induction' n with n ih", "annotated_tactic": ["induction' n with n ih", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nn : \u2115\n\u22a2 swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H", "state_after": "case zero\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\n\u22a2 swap ((\u03c3 ^ 0) x) ((\u03c3 ^ (0 + 1)) x) \u2208 H\n\ncase succ\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nn : \u2115\nih : swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\n\u22a2 swap ((\u03c3 ^ (n + 1)) x) ((\u03c3 ^ (n + 1 + 1)) x) \u2208 H"}, {"tactic": "exact subset_closure (Set.mem_insert_of_mem _ (Set.mem_singleton _))", "annotated_tactic": ["exact <a>subset_closure</a> (<a>Set.mem_insert_of_mem</a> _ (<a>Set.mem_singleton</a> _))", [{"full_name": "Subgroup.subset_closure", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [1089, 9], "def_end_pos": [1089, 23]}, {"full_name": "Set.mem_insert_of_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1085, 9], "def_end_pos": [1085, 26]}, {"full_name": "Set.mem_singleton", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1246, 9], "def_end_pos": [1246, 22]}]], "state_before": "case zero\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\n\u22a2 swap ((\u03c3 ^ 0) x) ((\u03c3 ^ (0 + 1)) x) \u2208 H", "state_after": "no goals"}, {"tactic": "convert H.mul_mem (H.mul_mem h3 ih) (H.inv_mem h3)", "annotated_tactic": ["convert H.mul_mem (H.mul_mem h3 ih) (H.inv_mem h3)", []], "state_before": "case succ\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nn : \u2115\nih : swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\n\u22a2 swap ((\u03c3 ^ (n + 1)) x) ((\u03c3 ^ (n + 1 + 1)) x) \u2208 H", "state_after": "case h.e'_4\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nn : \u2115\nih : swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\n\u22a2 swap ((\u03c3 ^ (n + 1)) x) ((\u03c3 ^ (n + 1 + 1)) x) = \u03c3 * swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) * \u03c3\u207b\u00b9"}, {"tactic": "simp_rw [mul_swap_eq_swap_mul, mul_inv_cancel_right, pow_succ']", "annotated_tactic": ["simp_rw [<a>mul_swap_eq_swap_mul</a>, <a>mul_inv_cancel_right</a>, <a>pow_succ'</a>]", [{"full_name": "Equiv.mul_swap_eq_swap_mul", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [537, 9], "def_end_pos": [537, 29]}, {"full_name": "mul_inv_cancel_right", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1262, 9], "def_end_pos": [1262, 29]}, {"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [667, 34], "def_end_pos": [667, 43]}]], "state_before": "case h.e'_4\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nn : \u2115\nih : swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\n\u22a2 swap ((\u03c3 ^ (n + 1)) x) ((\u03c3 ^ (n + 1 + 1)) x) = \u03c3 * swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) * \u03c3\u207b\u00b9", "state_after": "case h.e'_4\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nn : \u2115\nih : swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\n\u22a2 swap ((\u03c3 * \u03c3 ^ n) x) ((\u03c3 * (\u03c3 * \u03c3 ^ n)) x) = swap (\u03c3 ((\u03c3 ^ n) x)) (\u03c3 ((\u03c3 * \u03c3 ^ n) x))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.e'_4\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nn : \u2115\nih : swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\n\u22a2 swap ((\u03c3 * \u03c3 ^ n) x) ((\u03c3 * (\u03c3 * \u03c3 ^ n)) x) = swap (\u03c3 ((\u03c3 ^ n) x)) (\u03c3 ((\u03c3 * \u03c3 ^ n) x))", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\n\u22a2 \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nn : \u2115\n\u22a2 swap x ((\u03c3 ^ n) x) \u2208 H"}, {"tactic": "induction' n with n ih", "annotated_tactic": ["induction' n with n ih", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nn : \u2115\n\u22a2 swap x ((\u03c3 ^ n) x) \u2208 H", "state_after": "case zero\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\n\u22a2 swap x ((\u03c3 ^ 0) x) \u2208 H\n\ncase succ\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nn : \u2115\nih : swap x ((\u03c3 ^ n) x) \u2208 H\n\u22a2 swap x ((\u03c3 ^ (n + 1)) x) \u2208 H"}, {"tactic": "simp only [Nat.zero_eq, pow_zero, coe_one, id_eq, swap_self, Set.mem_singleton_iff]", "annotated_tactic": ["simp only [<a>Nat.zero_eq</a>, <a>pow_zero</a>, <a>coe_one</a>, <a>id_eq</a>, <a>swap_self</a>, <a>Set.mem_singleton_iff</a>]", [{"full_name": "Nat.zero_eq", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [106, 17], "def_end_pos": [106, 24]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [651, 9], "def_end_pos": [651, 17]}, {"full_name": "Equiv.Perm.coe_one", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [105, 26], "def_end_pos": [105, 33]}, {"full_name": "id_eq", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [297, 17], "def_end_pos": [297, 22]}, {"full_name": "Equiv.swap_self", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1629, 9], "def_end_pos": [1629, 18]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 26]}]], "state_before": "case zero\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\n\u22a2 swap x ((\u03c3 ^ 0) x) \u2208 H", "state_after": "case zero\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\n\u22a2 Equiv.refl \u03b1 \u2208 H"}, {"tactic": "convert H.one_mem", "annotated_tactic": ["convert H.one_mem", []], "state_before": "case zero\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\n\u22a2 Equiv.refl \u03b1 \u2208 H", "state_after": "no goals"}, {"tactic": "by_cases h5 : x = (\u03c3 ^ n) x", "annotated_tactic": ["by_cases h5 : x = (\u03c3 ^ n) x", []], "state_before": "case succ\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nn : \u2115\nih : swap x ((\u03c3 ^ n) x) \u2208 H\n\u22a2 swap x ((\u03c3 ^ (n + 1)) x) \u2208 H", "state_after": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nn : \u2115\nih : swap x ((\u03c3 ^ n) x) \u2208 H\nh5 : x = (\u03c3 ^ n) x\n\u22a2 swap x ((\u03c3 ^ (n + 1)) x) \u2208 H\n\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nn : \u2115\nih : swap x ((\u03c3 ^ n) x) \u2208 H\nh5 : \u00acx = (\u03c3 ^ n) x\n\u22a2 swap x ((\u03c3 ^ (n + 1)) x) \u2208 H"}, {"tactic": "by_cases h6 : x = (\u03c3 ^ (n + 1) : Perm \u03b1) x", "annotated_tactic": ["by_cases h6 : x = (\u03c3 ^ (n + 1) : <a>Perm</a> \u03b1) x", [{"full_name": "Equiv.Perm", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [87, 8], "def_end_pos": [87, 18]}]], "state_before": "case neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nn : \u2115\nih : swap x ((\u03c3 ^ n) x) \u2208 H\nh5 : \u00acx = (\u03c3 ^ n) x\n\u22a2 swap x ((\u03c3 ^ (n + 1)) x) \u2208 H", "state_after": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nn : \u2115\nih : swap x ((\u03c3 ^ n) x) \u2208 H\nh5 : \u00acx = (\u03c3 ^ n) x\nh6 : x = (\u03c3 ^ (n + 1)) x\n\u22a2 swap x ((\u03c3 ^ (n + 1)) x) \u2208 H\n\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nn : \u2115\nih : swap x ((\u03c3 ^ n) x) \u2208 H\nh5 : \u00acx = (\u03c3 ^ n) x\nh6 : \u00acx = (\u03c3 ^ (n + 1)) x\n\u22a2 swap x ((\u03c3 ^ (n + 1)) x) \u2208 H"}, {"tactic": "rw [swap_comm, \u2190 swap_mul_swap_mul_swap h5 h6]", "annotated_tactic": ["rw [<a>swap_comm</a>, \u2190 <a>swap_mul_swap_mul_swap</a> h5 h6]", [{"full_name": "Equiv.swap_comm", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1633, 9], "def_end_pos": [1633, 18]}, {"full_name": "Equiv.swap_mul_swap_mul_swap", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [595, 9], "def_end_pos": [595, 31]}]], "state_before": "case neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nn : \u2115\nih : swap x ((\u03c3 ^ n) x) \u2208 H\nh5 : \u00acx = (\u03c3 ^ n) x\nh6 : \u00acx = (\u03c3 ^ (n + 1)) x\n\u22a2 swap x ((\u03c3 ^ (n + 1)) x) \u2208 H", "state_after": "case neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nn : \u2115\nih : swap x ((\u03c3 ^ n) x) \u2208 H\nh5 : \u00acx = (\u03c3 ^ n) x\nh6 : \u00acx = (\u03c3 ^ (n + 1)) x\n\u22a2 swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) * swap x ((\u03c3 ^ n) x) * swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H"}, {"tactic": "exact H.mul_mem (H.mul_mem (step1 n) ih) (step1 n)", "annotated_tactic": ["exact H.mul_mem (H.mul_mem (step1 n) ih) (step1 n)", []], "state_before": "case neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nn : \u2115\nih : swap x ((\u03c3 ^ n) x) \u2208 H\nh5 : \u00acx = (\u03c3 ^ n) x\nh6 : \u00acx = (\u03c3 ^ (n + 1)) x\n\u22a2 swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) * swap x ((\u03c3 ^ n) x) * swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H", "state_after": "no goals"}, {"tactic": "rw [pow_succ', mul_apply, \u2190 h5]", "annotated_tactic": ["rw [<a>pow_succ'</a>, <a>mul_apply</a>, \u2190 h5]", [{"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [667, 34], "def_end_pos": [667, 43]}, {"full_name": "Equiv.Perm.mul_apply", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 18]}]], "state_before": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nn : \u2115\nih : swap x ((\u03c3 ^ n) x) \u2208 H\nh5 : x = (\u03c3 ^ n) x\n\u22a2 swap x ((\u03c3 ^ (n + 1)) x) \u2208 H", "state_after": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nn : \u2115\nih : swap x ((\u03c3 ^ n) x) \u2208 H\nh5 : x = (\u03c3 ^ n) x\n\u22a2 swap x (\u03c3 x) \u2208 H"}, {"tactic": "exact h4", "annotated_tactic": ["exact h4", []], "state_before": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nn : \u2115\nih : swap x ((\u03c3 ^ n) x) \u2208 H\nh5 : x = (\u03c3 ^ n) x\n\u22a2 swap x (\u03c3 x) \u2208 H", "state_after": "no goals"}, {"tactic": "rw [\u2190 h6, swap_self]", "annotated_tactic": ["rw [\u2190 h6, <a>swap_self</a>]", [{"full_name": "Equiv.swap_self", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1629, 9], "def_end_pos": [1629, 18]}]], "state_before": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nn : \u2115\nih : swap x ((\u03c3 ^ n) x) \u2208 H\nh5 : \u00acx = (\u03c3 ^ n) x\nh6 : x = (\u03c3 ^ (n + 1)) x\n\u22a2 swap x ((\u03c3 ^ (n + 1)) x) \u2208 H", "state_after": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nn : \u2115\nih : swap x ((\u03c3 ^ n) x) \u2208 H\nh5 : \u00acx = (\u03c3 ^ n) x\nh6 : x = (\u03c3 ^ (n + 1)) x\n\u22a2 Equiv.refl \u03b1 \u2208 H"}, {"tactic": "exact H.one_mem", "annotated_tactic": ["exact H.one_mem", []], "state_before": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nn : \u2115\nih : swap x ((\u03c3 ^ n) x) \u2208 H\nh5 : \u00acx = (\u03c3 ^ n) x\nh6 : x = (\u03c3 ^ (n + 1)) x\n\u22a2 Equiv.refl \u03b1 \u2208 H", "state_after": "no goals"}, {"tactic": "intro y", "annotated_tactic": ["intro y", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\n\u22a2 \u2200 (y : \u03b1), swap x y \u2208 H", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\ny : \u03b1\n\u22a2 swap x y \u2208 H"}, {"tactic": "have hx : x \u2208 (\u22a4 : Finset \u03b1) := Finset.mem_univ x", "annotated_tactic": ["have hx : x \u2208 (\u22a4 : <a>Finset</a> \u03b1) := <a>Finset.mem_univ</a> x", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 17]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\ny : \u03b1\n\u22a2 swap x y \u2208 H", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\ny : \u03b1\nhx : x \u2208 \u22a4\n\u22a2 swap x y \u2208 H"}, {"tactic": "rw [\u2190 h2, mem_support] at hx", "annotated_tactic": ["rw [\u2190 h2, <a>mem_support</a>] at hx", [{"full_name": "Equiv.Perm.mem_support", "def_path": "Mathlib/GroupTheory/Perm/Support.lean", "def_pos": [297, 9], "def_end_pos": [297, 20]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\ny : \u03b1\nhx : x \u2208 \u22a4\n\u22a2 swap x y \u2208 H", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\ny : \u03b1\nhx : \u03c3 x \u2260 x\n\u22a2 swap x y \u2208 H"}, {"tactic": "have hy : y \u2208 (\u22a4 : Finset \u03b1) := Finset.mem_univ y", "annotated_tactic": ["have hy : y \u2208 (\u22a4 : <a>Finset</a> \u03b1) := <a>Finset.mem_univ</a> y", [{"full_name": "Finset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [135, 11], "def_end_pos": [135, 17]}, {"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 17]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\ny : \u03b1\nhx : \u03c3 x \u2260 x\n\u22a2 swap x y \u2208 H", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\ny : \u03b1\nhx : \u03c3 x \u2260 x\nhy : y \u2208 \u22a4\n\u22a2 swap x y \u2208 H"}, {"tactic": "rw [\u2190 h2, mem_support] at hy", "annotated_tactic": ["rw [\u2190 h2, <a>mem_support</a>] at hy", [{"full_name": "Equiv.Perm.mem_support", "def_path": "Mathlib/GroupTheory/Perm/Support.lean", "def_pos": [297, 9], "def_end_pos": [297, 20]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\ny : \u03b1\nhx : \u03c3 x \u2260 x\nhy : y \u2208 \u22a4\n\u22a2 swap x y \u2208 H", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\ny : \u03b1\nhx : \u03c3 x \u2260 x\nhy : \u03c3 y \u2260 y\n\u22a2 swap x y \u2208 H"}, {"tactic": "cases' IsCycle.exists_pow_eq h1 hx hy with n hn", "annotated_tactic": ["cases' <a>IsCycle.exists_pow_eq</a> h1 hx hy with n hn", [{"full_name": "Equiv.Perm.IsCycle.exists_pow_eq", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Basic.lean", "def_pos": [347, 9], "def_end_pos": [347, 30]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\ny : \u03b1\nhx : \u03c3 x \u2260 x\nhy : \u03c3 y \u2260 y\n\u22a2 swap x y \u2208 H", "state_after": "case intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\ny : \u03b1\nhx : \u03c3 x \u2260 x\nhy : \u03c3 y \u2260 y\nn : \u2115\nhn : (\u03c3 ^ n) x = y\n\u22a2 swap x y \u2208 H"}, {"tactic": "rw [\u2190 hn]", "annotated_tactic": ["rw [\u2190 hn]", []], "state_before": "case intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\ny : \u03b1\nhx : \u03c3 x \u2260 x\nhy : \u03c3 y \u2260 y\nn : \u2115\nhn : (\u03c3 ^ n) x = y\n\u22a2 swap x y \u2208 H", "state_after": "case intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\ny : \u03b1\nhx : \u03c3 x \u2260 x\nhy : \u03c3 y \u2260 y\nn : \u2115\nhn : (\u03c3 ^ n) x = y\n\u22a2 swap x ((\u03c3 ^ n) x) \u2208 H"}, {"tactic": "exact step2 n", "annotated_tactic": ["exact step2 n", []], "state_before": "case intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\ny : \u03b1\nhx : \u03c3 x \u2260 x\nhy : \u03c3 y \u2260 y\nn : \u2115\nhn : (\u03c3 ^ n) x = y\n\u22a2 swap x ((\u03c3 ^ n) x) \u2208 H", "state_after": "no goals"}, {"tactic": "intro y z", "annotated_tactic": ["intro y z", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\n\u22a2 \u2200 (y z : \u03b1), swap y z \u2208 H", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\ny z : \u03b1\n\u22a2 swap y z \u2208 H"}, {"tactic": "by_cases h5 : z = x", "annotated_tactic": ["by_cases h5 : z = x", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\ny z : \u03b1\n\u22a2 swap y z \u2208 H", "state_after": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\ny z : \u03b1\nh5 : z = x\n\u22a2 swap y z \u2208 H\n\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\ny z : \u03b1\nh5 : \u00acz = x\n\u22a2 swap y z \u2208 H"}, {"tactic": "by_cases h6 : z = y", "annotated_tactic": ["by_cases h6 : z = y", []], "state_before": "case neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\ny z : \u03b1\nh5 : \u00acz = x\n\u22a2 swap y z \u2208 H", "state_after": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\ny z : \u03b1\nh5 : \u00acz = x\nh6 : z = y\n\u22a2 swap y z \u2208 H\n\ncase neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\ny z : \u03b1\nh5 : \u00acz = x\nh6 : \u00acz = y\n\u22a2 swap y z \u2208 H"}, {"tactic": "rw [\u2190 swap_mul_swap_mul_swap h5 h6, swap_comm z x]", "annotated_tactic": ["rw [\u2190 <a>swap_mul_swap_mul_swap</a> h5 h6, <a>swap_comm</a> z x]", [{"full_name": "Equiv.swap_mul_swap_mul_swap", "def_path": "Mathlib/GroupTheory/Perm/Basic.lean", "def_pos": [595, 9], "def_end_pos": [595, 31]}, {"full_name": "Equiv.swap_comm", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1633, 9], "def_end_pos": [1633, 18]}]], "state_before": "case neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\ny z : \u03b1\nh5 : \u00acz = x\nh6 : \u00acz = y\n\u22a2 swap y z \u2208 H", "state_after": "case neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\ny z : \u03b1\nh5 : \u00acz = x\nh6 : \u00acz = y\n\u22a2 swap x y * swap x z * swap x y \u2208 H"}, {"tactic": "exact H.mul_mem (H.mul_mem (step3 y) (step3 z)) (step3 y)", "annotated_tactic": ["exact H.mul_mem (H.mul_mem (step3 y) (step3 z)) (step3 y)", []], "state_before": "case neg\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\ny z : \u03b1\nh5 : \u00acz = x\nh6 : \u00acz = y\n\u22a2 swap x y * swap x z * swap x y \u2208 H", "state_after": "no goals"}, {"tactic": "rw [h5, swap_comm]", "annotated_tactic": ["rw [h5, <a>swap_comm</a>]", [{"full_name": "Equiv.swap_comm", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1633, 9], "def_end_pos": [1633, 18]}]], "state_before": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\ny z : \u03b1\nh5 : z = x\n\u22a2 swap y z \u2208 H", "state_after": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\ny z : \u03b1\nh5 : z = x\n\u22a2 swap x y \u2208 H"}, {"tactic": "exact step3 y", "annotated_tactic": ["exact step3 y", []], "state_before": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\ny z : \u03b1\nh5 : z = x\n\u22a2 swap x y \u2208 H", "state_after": "no goals"}, {"tactic": "rw [h6, swap_self]", "annotated_tactic": ["rw [h6, <a>swap_self</a>]", [{"full_name": "Equiv.swap_self", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1629, 9], "def_end_pos": [1629, 18]}]], "state_before": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\ny z : \u03b1\nh5 : \u00acz = x\nh6 : z = y\n\u22a2 swap y z \u2208 H", "state_after": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\ny z : \u03b1\nh5 : \u00acz = x\nh6 : z = y\n\u22a2 Equiv.refl \u03b1 \u2208 H"}, {"tactic": "exact H.one_mem", "annotated_tactic": ["exact H.one_mem", []], "state_before": "case pos\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b2 : Finite \u03b2\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\n\u03c3 : Perm \u03b1\nh1 : \u03c3.IsCycle\nh2 : \u03c3.support = \u22a4\nx : \u03b1\nH : Subgroup (Perm \u03b1) := closure {\u03c3, swap x (\u03c3 x)}\nh3 : \u03c3 \u2208 H\nh4 : swap x (\u03c3 x) \u2208 H\nstep1 : \u2200 (n : \u2115), swap ((\u03c3 ^ n) x) ((\u03c3 ^ (n + 1)) x) \u2208 H\nstep2 : \u2200 (n : \u2115), swap x ((\u03c3 ^ n) x) \u2208 H\nstep3 : \u2200 (y : \u03b1), swap x y \u2208 H\ny z : \u03b1\nh5 : \u00acz = x\nh6 : z = y\n\u22a2 Equiv.refl \u03b1 \u2208 H", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.condexpL1_of_aestronglyMeasurable'", "start": [595, 1], "end": [600, 73], "traced_tactics": [{"tactic": "rw [condexpL1_eq hfi]", "annotated_tactic": ["rw [<a>condexpL1_eq</a> hfi]", [{"full_name": "MeasureTheory.condexpL1_eq", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [530, 9], "def_end_pos": [530, 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 : RCLike \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 (\u03bc.trim hm)\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhfm : AEStronglyMeasurable' m f \u03bc\nhfi : Integrable f \u03bc\n\u22a2 \u2191\u2191(condexpL1 hm \u03bc f) =\u1da0[ae \u03bc] f", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : RCLike \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 (\u03bc.trim hm)\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhfm : AEStronglyMeasurable' m f \u03bc\nhfi : Integrable f \u03bc\n\u22a2 \u2191\u2191((condexpL1CLM F' hm \u03bc) (Integrable.toL1 f hfi)) =\u1da0[ae \u03bc] f"}, {"tactic": "refine EventuallyEq.trans ?_ (Integrable.coeFn_toL1 hfi)", "annotated_tactic": ["refine <a>EventuallyEq.trans</a> ?_ (<a>Integrable.coeFn_toL1</a> hfi)", [{"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1535, 9], "def_end_pos": [1535, 27]}, {"full_name": "MeasureTheory.Integrable.coeFn_toL1", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1494, 9], "def_end_pos": [1494, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : RCLike \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 (\u03bc.trim hm)\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhfm : AEStronglyMeasurable' m f \u03bc\nhfi : Integrable f \u03bc\n\u22a2 \u2191\u2191((condexpL1CLM F' hm \u03bc) (Integrable.toL1 f hfi)) =\u1da0[ae \u03bc] f", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : RCLike \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 (\u03bc.trim hm)\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhfm : AEStronglyMeasurable' m f \u03bc\nhfi : Integrable f \u03bc\n\u22a2 \u2191\u2191((condexpL1CLM F' hm \u03bc) (Integrable.toL1 f hfi)) =\u1da0[ae \u03bc] \u2191\u2191(Integrable.toL1 f hfi)"}, {"tactic": "rw [condexpL1CLM_of_aestronglyMeasurable']", "annotated_tactic": ["rw [<a>condexpL1CLM_of_aestronglyMeasurable'</a>]", [{"full_name": "MeasureTheory.condexpL1CLM_of_aestronglyMeasurable'", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [515, 9], "def_end_pos": [515, 46]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : RCLike \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 (\u03bc.trim hm)\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhfm : AEStronglyMeasurable' m f \u03bc\nhfi : Integrable f \u03bc\n\u22a2 \u2191\u2191((condexpL1CLM F' hm \u03bc) (Integrable.toL1 f hfi)) =\u1da0[ae \u03bc] \u2191\u2191(Integrable.toL1 f hfi)", "state_after": "case hfm\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : RCLike \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 (\u03bc.trim hm)\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhfm : AEStronglyMeasurable' m f \u03bc\nhfi : Integrable f \u03bc\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(Integrable.toL1 f hfi)) \u03bc"}, {"tactic": "exact AEStronglyMeasurable'.congr hfm (Integrable.coeFn_toL1 hfi).symm", "annotated_tactic": ["exact <a>AEStronglyMeasurable'.congr</a> hfm (<a>Integrable.coeFn_toL1</a> hfi).<a>symm</a>", [{"full_name": "MeasureTheory.AEStronglyMeasurable'.congr", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean", "def_pos": [62, 9], "def_end_pos": [62, 14]}, {"full_name": "MeasureTheory.Integrable.coeFn_toL1", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [1494, 9], "def_end_pos": [1494, 19]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1530, 9], "def_end_pos": [1530, 26]}]], "state_before": "case hfm\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : RCLike \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 (\u03bc.trim hm)\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhfm : AEStronglyMeasurable' m f \u03bc\nhfi : Integrable f \u03bc\n\u22a2 AEStronglyMeasurable' m (\u2191\u2191(Integrable.toL1 f hfi)) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Prime.lean", "full_name": "Nat.Prime.five_le_of_ne_two_of_ne_three", "start": [178, 1], "end": [189, 47], "traced_tactics": [{"tactic": "by_contra! h", "annotated_tactic": ["by_contra! h", []], "state_before": "n p : \u2115\nhp : Prime p\nh_two : p \u2260 2\nh_three : p \u2260 3\n\u22a2 5 \u2264 p", "state_after": "n p : \u2115\nhp : Prime p\nh_two : p \u2260 2\nh_three : p \u2260 3\nh : p < 5\n\u22a2 False"}, {"tactic": "revert h_two h_three hp", "annotated_tactic": ["revert h_two h_three hp", []], "state_before": "n p : \u2115\nhp : Prime p\nh_two : p \u2260 2\nh_three : p \u2260 3\nh : p < 5\n\u22a2 False", "state_after": "n p : \u2115\nh : p < 5\n\u22a2 Prime p \u2192 p \u2260 2 \u2192 p \u2260 3 \u2192 False"}, {"tactic": "match p with\n| 0 => decide\n| 1 => decide\n| 2 => decide\n| 3 => decide\n| 4 => decide\n| n + 5 => exact (h.not_le le_add_self).elim", "annotated_tactic": ["match p with\n  | 0 => decide\n  | 1 => decide\n  | 2 => decide\n  | 3 => decide\n  | 4 => decide\n  | n + 5 => exact (h.not_le <a>le_add_self</a>).<a>elim</a>", [{"full_name": "le_add_self", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [147, 3], "def_end_pos": [147, 14]}, {"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "n p : \u2115\nh : p < 5\n\u22a2 Prime p \u2192 p \u2260 2 \u2192 p \u2260 3 \u2192 False", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "n p : \u2115\nh : 0 < 5\n\u22a2 Prime 0 \u2192 0 \u2260 2 \u2192 0 \u2260 3 \u2192 False", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "n p : \u2115\nh : 1 < 5\n\u22a2 Prime 1 \u2192 1 \u2260 2 \u2192 1 \u2260 3 \u2192 False", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "n p : \u2115\nh : 2 < 5\n\u22a2 Prime 2 \u2192 2 \u2260 2 \u2192 2 \u2260 3 \u2192 False", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "n p : \u2115\nh : 3 < 5\n\u22a2 Prime 3 \u2192 3 \u2260 2 \u2192 3 \u2260 3 \u2192 False", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "n p : \u2115\nh : 4 < 5\n\u22a2 Prime 4 \u2192 4 \u2260 2 \u2192 4 \u2260 3 \u2192 False", "state_after": "no goals"}, {"tactic": "exact (h.not_le le_add_self).elim", "annotated_tactic": ["exact (h.not_le <a>le_add_self</a>).<a>elim</a>", [{"full_name": "le_add_self", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [147, 3], "def_end_pos": [147, 14]}, {"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "n\u271d p n : \u2115\nh : n + 5 < 5\n\u22a2 Prime (n + 5) \u2192 n + 5 \u2260 2 \u2192 n + 5 \u2260 3 \u2192 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean", "full_name": "GromovHausdorff.candidatesBOfCandidates_mem", "start": [212, 1], "end": [214, 5], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/RingHom/FiniteType.lean", "full_name": "RingHom.finiteType_holdsForLocalizationAway", "start": [30, 1], "end": [36, 70], "traced_tactics": [{"tactic": "introv R _", "annotated_tactic": ["introv R _", []], "state_before": "\u22a2 HoldsForLocalizationAway @FiniteType", "state_after": "R S : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nr : R\ninst\u271d : IsLocalization.Away r S\n\u22a2 (algebraMap R S).FiniteType"}, {"tactic": "suffices Algebra.FiniteType R S by\n  rw [RingHom.FiniteType]\n  convert this; ext;\n  rw [Algebra.smul_def]; rfl", "annotated_tactic": ["suffices <a>Algebra.FiniteType</a> R S by\n    rw [<a>RingHom.FiniteType</a>]\n    convert this; ext;\n    rw [<a>Algebra.smul_def</a>]; rfl", [{"full_name": "Algebra.FiniteType", "def_path": "Mathlib/RingTheory/FiniteType.lean", "def_pos": [37, 7], "def_end_pos": [37, 25]}, {"full_name": "RingHom.FiniteType", "def_path": "Mathlib/RingTheory/FiniteType.lean", "def_pos": [229, 5], "def_end_pos": [229, 15]}, {"full_name": "Algebra.smul_def", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [326, 9], "def_end_pos": [326, 17]}]], "state_before": "R S : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nr : R\ninst\u271d : IsLocalization.Away r S\n\u22a2 (algebraMap R S).FiniteType", "state_after": "R S : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nr : R\ninst\u271d : IsLocalization.Away r S\n\u22a2 Algebra.FiniteType R S"}, {"tactic": "exact IsLocalization.finiteType_of_monoid_fg (Submonoid.powers r) S", "annotated_tactic": ["exact <a>IsLocalization.finiteType_of_monoid_fg</a> (<a>Submonoid.powers</a> r) S", [{"full_name": "IsLocalization.finiteType_of_monoid_fg", "def_path": "Mathlib/RingTheory/Localization/InvSubmonoid.lean", "def_pos": [115, 9], "def_end_pos": [115, 32]}, {"full_name": "Submonoid.powers", "def_path": "Mathlib/Algebra/Group/Submonoid/Membership.lean", "def_pos": [464, 5], "def_end_pos": [464, 11]}]], "state_before": "R S : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nr : R\ninst\u271d : IsLocalization.Away r S\n\u22a2 Algebra.FiniteType R S", "state_after": "no goals"}, {"tactic": "rw [RingHom.FiniteType]", "annotated_tactic": ["rw [<a>RingHom.FiniteType</a>]", [{"full_name": "RingHom.FiniteType", "def_path": "Mathlib/RingTheory/FiniteType.lean", "def_pos": [229, 5], "def_end_pos": [229, 15]}]], "state_before": "R S : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nr : R\ninst\u271d : IsLocalization.Away r S\nthis : Algebra.FiniteType R S\n\u22a2 (algebraMap R S).FiniteType", "state_after": "R S : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nr : R\ninst\u271d : IsLocalization.Away r S\nthis : Algebra.FiniteType R S\n\u22a2 Algebra.FiniteType R S"}, {"tactic": "convert this", "annotated_tactic": ["convert this", []], "state_before": "R S : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nr : R\ninst\u271d : IsLocalization.Away r S\nthis : Algebra.FiniteType R S\n\u22a2 Algebra.FiniteType R S", "state_after": "case h.e'_5\nR S : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nr : R\ninst\u271d : IsLocalization.Away r S\nthis : Algebra.FiniteType R S\n\u22a2 (algebraMap R S).toAlgebra = inst\u271d\u00b9"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "case h.e'_5\nR S : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nr : R\ninst\u271d : IsLocalization.Away r S\nthis : Algebra.FiniteType R S\n\u22a2 (algebraMap R S).toAlgebra = inst\u271d\u00b9", "state_after": "case h.e'_5.h\nR S : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nr : R\ninst\u271d : IsLocalization.Away r S\nthis : Algebra.FiniteType R S\nr\u271d : R\nx\u271d : S\n\u22a2 (let_fun I := (algebraMap R S).toAlgebra;\n    r\u271d \u2022 x\u271d) =\n    r\u271d \u2022 x\u271d"}, {"tactic": "rw [Algebra.smul_def]", "annotated_tactic": ["rw [<a>Algebra.smul_def</a>]", [{"full_name": "Algebra.smul_def", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [326, 9], "def_end_pos": [326, 17]}]], "state_before": "case h.e'_5.h\nR S : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nr : R\ninst\u271d : IsLocalization.Away r S\nthis : Algebra.FiniteType R S\nr\u271d : R\nx\u271d : S\n\u22a2 (let_fun I := (algebraMap R S).toAlgebra;\n    r\u271d \u2022 x\u271d) =\n    r\u271d \u2022 x\u271d", "state_after": "case h.e'_5.h\nR S : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nr : R\ninst\u271d : IsLocalization.Away r S\nthis : Algebra.FiniteType R S\nr\u271d : R\nx\u271d : S\n\u22a2 (let_fun I := (algebraMap R S).toAlgebra;\n    r\u271d \u2022 x\u271d) =\n    (algebraMap R S) r\u271d * x\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.e'_5.h\nR S : Type u_1\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing S\ninst\u271d\u00b9 : Algebra R S\nr : R\ninst\u271d : IsLocalization.Away r S\nthis : Algebra.FiniteType R S\nr\u271d : R\nx\u271d : S\n\u22a2 (let_fun I := (algebraMap R S).toAlgebra;\n    r\u271d \u2022 x\u271d) =\n    (algebraMap R S) r\u271d * x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Semicontinuous.lean", "full_name": "UpperSemicontinuousWithinAt.mono", "start": [760, 1], "end": [762, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Submodule/Range.lean", "full_name": "LinearMap.ker_rangeRestrict", "start": [440, 9], "end": [440, 99], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Topology.lean", "full_name": "Convex.combo_interior_self_subset_interior", "start": [138, 1], "end": [143, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/List/Count.lean", "full_name": "List.count_le_count_map", "start": [222, 1], "end": [225, 66], "traced_tactics": [{"tactic": "rw [count, count, countP_map]", "annotated_tactic": ["rw [<a>count</a>, <a>count</a>, <a>countP_map</a>]", [{"full_name": "List.count", "def_path": ".lake/packages/batteries/Batteries/Data/List/Basic.lean", "def_pos": [448, 15], "def_end_pos": [448, 20]}, {"full_name": "List.count", "def_path": ".lake/packages/batteries/Batteries/Data/List/Basic.lean", "def_pos": [448, 15], "def_end_pos": [448, 20]}, {"full_name": "List.countP_map", "def_path": ".lake/packages/batteries/Batteries/Data/List/Count.lean", "def_pos": [100, 17], "def_end_pos": [100, 27]}]], "state_before": "\u03b1 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\n\u03b2 : Type u_1\ninst\u271d : DecidableEq \u03b2\nl : List \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 count x l \u2264 count (f x) (map f l)", "state_after": "\u03b1 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\n\u03b2 : Type u_1\ninst\u271d : DecidableEq \u03b2\nl : List \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 countP (fun x_1 => x_1 == x) l \u2264 countP ((fun x_1 => x_1 == f x) \u2218 f) l"}, {"tactic": "apply countP_mono_left", "annotated_tactic": ["apply <a>countP_mono_left</a>", [{"full_name": "List.countP_mono_left", "def_path": ".lake/packages/batteries/Batteries/Data/List/Count.lean", "def_pos": [107, 9], "def_end_pos": [107, 25]}]], "state_before": "\u03b1 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\n\u03b2 : Type u_1\ninst\u271d : DecidableEq \u03b2\nl : List \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 countP (fun x_1 => x_1 == x) l \u2264 countP ((fun x_1 => x_1 == f x) \u2218 f) l", "state_after": "case h\n\u03b1 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\n\u03b2 : Type u_1\ninst\u271d : DecidableEq \u03b2\nl : List \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 \u2200 (x_1 : \u03b1), x_1 \u2208 l \u2192 (x_1 == x) = true \u2192 ((fun x_2 => x_2 == f x) \u2218 f) x_1 = true"}, {"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": [571, 5], "def_end_pos": [571, 9]}]], "state_before": "case h\n\u03b1 : Type u_2\ninst\u271d\u00b9 : DecidableEq \u03b1\n\u03b2 : Type u_1\ninst\u271d : DecidableEq \u03b2\nl : List \u03b1\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 \u2200 (x_1 : \u03b1), x_1 \u2208 l \u2192 (x_1 == x) = true \u2192 ((fun x_2 => x_2 == f x) \u2218 f) x_1 = true", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Complex/Basic.lean", "full_name": "Complex.nnnorm_nat", "start": [203, 1], "end": [204, 25], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "z : \u2102\nE : Type u_1\ninst\u271d\u00b9 : SeminormedAddCommGroup E\ninst\u271d : NormedSpace \u2102 E\nn : \u2115\n\u22a2 \u2191\u2016\u2191n\u2016\u208a = \u2191\u2191n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Analytic/Basic.lean", "full_name": "HasFPowerSeriesOnBall.uniform_geometric_approx", "start": [704, 1], "end": [715, 79], "traced_tactics": [{"tactic": "obtain \u27e8a, ha, C, hC, hp\u27e9 : \u2203 a \u2208 Ioo (0 : \u211d) 1, \u2203 C > 0, \u2200 y \u2208 Metric.ball (0 : E) r', \u2200 n,\n    \u2016f (x + y) - p.partialSum n y\u2016 \u2264 C * (a * (\u2016y\u2016 / r')) ^ n :=\n  hf.uniform_geometric_approx' h", "annotated_tactic": ["obtain \u27e8a, ha, C, hC, hp\u27e9 : \u2203 a \u2208 <a>Ioo</a> (0 : \u211d) 1, \u2203 C > 0, \u2200 y \u2208 <a>Metric.ball</a> (0 : E) r', \u2200 n,\n      \u2016f (x + y) - p.partialSum n y\u2016 \u2264 C * (a * (\u2016y\u2016 / r')) ^ n :=\n    hf.uniform_geometric_approx' h", [{"full_name": "Set.Ioo", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 8]}, {"full_name": "Metric.ball", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [399, 5], "def_end_pos": [399, 9]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx : E\nr r'\u271d : \u211d\u22650\u221e\nr' : \u211d\u22650\nhf : HasFPowerSeriesOnBall f p x r\nh : \u2191r' < r\n\u22a2 \u2203 a \u2208 Ioo 0 1, \u2203 C > 0, \u2200 y \u2208 Metric.ball 0 \u2191r', \u2200 (n : \u2115), \u2016f (x + y) - p.partialSum n y\u2016 \u2264 C * a ^ n", "state_after": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx : E\nr r'\u271d : \u211d\u22650\u221e\nr' : \u211d\u22650\nhf : HasFPowerSeriesOnBall f p x r\nh : \u2191r' < r\na : \u211d\nha : a \u2208 Ioo 0 1\nC : \u211d\nhC : C > 0\nhp : \u2200 y \u2208 Metric.ball 0 \u2191r', \u2200 (n : \u2115), \u2016f (x + y) - p.partialSum n y\u2016 \u2264 C * (a * (\u2016y\u2016 / \u2191r')) ^ n\n\u22a2 \u2203 a \u2208 Ioo 0 1, \u2203 C > 0, \u2200 y \u2208 Metric.ball 0 \u2191r', \u2200 (n : \u2115), \u2016f (x + y) - p.partialSum n y\u2016 \u2264 C * a ^ n"}, {"tactic": "refine \u27e8a, ha, C, hC, fun y hy n => (hp y hy n).trans ?_\u27e9", "annotated_tactic": ["refine \u27e8a, ha, C, hC, fun y hy n => (hp y hy n).<a>trans</a> ?_\u27e9", [{"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [115, 7], "def_end_pos": [115, 18]}]], "state_before": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx : E\nr r'\u271d : \u211d\u22650\u221e\nr' : \u211d\u22650\nhf : HasFPowerSeriesOnBall f p x r\nh : \u2191r' < r\na : \u211d\nha : a \u2208 Ioo 0 1\nC : \u211d\nhC : C > 0\nhp : \u2200 y \u2208 Metric.ball 0 \u2191r', \u2200 (n : \u2115), \u2016f (x + y) - p.partialSum n y\u2016 \u2264 C * (a * (\u2016y\u2016 / \u2191r')) ^ n\n\u22a2 \u2203 a \u2208 Ioo 0 1, \u2203 C > 0, \u2200 y \u2208 Metric.ball 0 \u2191r', \u2200 (n : \u2115), \u2016f (x + y) - p.partialSum n y\u2016 \u2264 C * a ^ n", "state_after": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx : E\nr r'\u271d : \u211d\u22650\u221e\nr' : \u211d\u22650\nhf : HasFPowerSeriesOnBall f p x r\nh : \u2191r' < r\na : \u211d\nha : a \u2208 Ioo 0 1\nC : \u211d\nhC : C > 0\nhp : \u2200 y \u2208 Metric.ball 0 \u2191r', \u2200 (n : \u2115), \u2016f (x + y) - p.partialSum n y\u2016 \u2264 C * (a * (\u2016y\u2016 / \u2191r')) ^ n\ny : E\nhy : y \u2208 Metric.ball 0 \u2191r'\nn : \u2115\n\u22a2 C * (a * (\u2016y\u2016 / \u2191r')) ^ n \u2264 C * a ^ n"}, {"tactic": "have yr' : \u2016y\u2016 < r' := by rwa [ball_zero_eq] at hy", "annotated_tactic": ["have yr' : \u2016y\u2016 < r' := by rwa [<a>ball_zero_eq</a>] at hy", [{"full_name": "ball_zero_eq", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [604, 3], "def_end_pos": [604, 14]}]], "state_before": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx : E\nr r'\u271d : \u211d\u22650\u221e\nr' : \u211d\u22650\nhf : HasFPowerSeriesOnBall f p x r\nh : \u2191r' < r\na : \u211d\nha : a \u2208 Ioo 0 1\nC : \u211d\nhC : C > 0\nhp : \u2200 y \u2208 Metric.ball 0 \u2191r', \u2200 (n : \u2115), \u2016f (x + y) - p.partialSum n y\u2016 \u2264 C * (a * (\u2016y\u2016 / \u2191r')) ^ n\ny : E\nhy : y \u2208 Metric.ball 0 \u2191r'\nn : \u2115\n\u22a2 C * (a * (\u2016y\u2016 / \u2191r')) ^ n \u2264 C * a ^ n", "state_after": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx : E\nr r'\u271d : \u211d\u22650\u221e\nr' : \u211d\u22650\nhf : HasFPowerSeriesOnBall f p x r\nh : \u2191r' < r\na : \u211d\nha : a \u2208 Ioo 0 1\nC : \u211d\nhC : C > 0\nhp : \u2200 y \u2208 Metric.ball 0 \u2191r', \u2200 (n : \u2115), \u2016f (x + y) - p.partialSum n y\u2016 \u2264 C * (a * (\u2016y\u2016 / \u2191r')) ^ n\ny : E\nhy : y \u2208 Metric.ball 0 \u2191r'\nn : \u2115\nyr' : \u2016y\u2016 < \u2191r'\n\u22a2 C * (a * (\u2016y\u2016 / \u2191r')) ^ n \u2264 C * a ^ n"}, {"tactic": "have := ha.1.le", "annotated_tactic": ["have := ha.1.<a>le</a>", [{"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}]], "state_before": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx : E\nr r'\u271d : \u211d\u22650\u221e\nr' : \u211d\u22650\nhf : HasFPowerSeriesOnBall f p x r\nh : \u2191r' < r\na : \u211d\nha : a \u2208 Ioo 0 1\nC : \u211d\nhC : C > 0\nhp : \u2200 y \u2208 Metric.ball 0 \u2191r', \u2200 (n : \u2115), \u2016f (x + y) - p.partialSum n y\u2016 \u2264 C * (a * (\u2016y\u2016 / \u2191r')) ^ n\ny : E\nhy : y \u2208 Metric.ball 0 \u2191r'\nn : \u2115\nyr' : \u2016y\u2016 < \u2191r'\n\u22a2 C * (a * (\u2016y\u2016 / \u2191r')) ^ n \u2264 C * a ^ n", "state_after": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx : E\nr r'\u271d : \u211d\u22650\u221e\nr' : \u211d\u22650\nhf : HasFPowerSeriesOnBall f p x r\nh : \u2191r' < r\na : \u211d\nha : a \u2208 Ioo 0 1\nC : \u211d\nhC : C > 0\nhp : \u2200 y \u2208 Metric.ball 0 \u2191r', \u2200 (n : \u2115), \u2016f (x + y) - p.partialSum n y\u2016 \u2264 C * (a * (\u2016y\u2016 / \u2191r')) ^ n\ny : E\nhy : y \u2208 Metric.ball 0 \u2191r'\nn : \u2115\nyr' : \u2016y\u2016 < \u2191r'\nthis : 0 \u2264 a\n\u22a2 C * (a * (\u2016y\u2016 / \u2191r')) ^ n \u2264 C * a ^ n"}, {"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "case intro.intro.intro.intro\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx : E\nr r'\u271d : \u211d\u22650\u221e\nr' : \u211d\u22650\nhf : HasFPowerSeriesOnBall f p x r\nh : \u2191r' < r\na : \u211d\nha : a \u2208 Ioo 0 1\nC : \u211d\nhC : C > 0\nhp : \u2200 y \u2208 Metric.ball 0 \u2191r', \u2200 (n : \u2115), \u2016f (x + y) - p.partialSum n y\u2016 \u2264 C * (a * (\u2016y\u2016 / \u2191r')) ^ n\ny : E\nhy : y \u2208 Metric.ball 0 \u2191r'\nn : \u2115\nyr' : \u2016y\u2016 < \u2191r'\nthis : 0 \u2264 a\n\u22a2 C * (a * (\u2016y\u2016 / \u2191r')) ^ n \u2264 C * a ^ n", "state_after": "case intro.intro.intro.intro.h.hab\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx : E\nr r'\u271d : \u211d\u22650\u221e\nr' : \u211d\u22650\nhf : HasFPowerSeriesOnBall f p x r\nh : \u2191r' < r\na : \u211d\nha : a \u2208 Ioo 0 1\nC : \u211d\nhC : C > 0\nhp : \u2200 y \u2208 Metric.ball 0 \u2191r', \u2200 (n : \u2115), \u2016f (x + y) - p.partialSum n y\u2016 \u2264 C * (a * (\u2016y\u2016 / \u2191r')) ^ n\ny : E\nhy : y \u2208 Metric.ball 0 \u2191r'\nn : \u2115\nyr' : \u2016y\u2016 < \u2191r'\nthis : 0 \u2264 a\n\u22a2 a * (\u2016y\u2016 / \u2191r') \u2264 a"}, {"tactic": "exact mul_le_of_le_one_right ha.1.le (div_le_one_of_le yr'.le r'.coe_nonneg)", "annotated_tactic": ["exact <a>mul_le_of_le_one_right</a> ha.1.<a>le</a> (<a>div_le_one_of_le</a> yr'.le r'.coe_nonneg)", [{"full_name": "mul_le_of_le_one_right", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [732, 9], "def_end_pos": [732, 31]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}, {"full_name": "div_le_one_of_le", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [165, 9], "def_end_pos": [165, 25]}]], "state_before": "case intro.intro.intro.intro.h.hab\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx : E\nr r'\u271d : \u211d\u22650\u221e\nr' : \u211d\u22650\nhf : HasFPowerSeriesOnBall f p x r\nh : \u2191r' < r\na : \u211d\nha : a \u2208 Ioo 0 1\nC : \u211d\nhC : C > 0\nhp : \u2200 y \u2208 Metric.ball 0 \u2191r', \u2200 (n : \u2115), \u2016f (x + y) - p.partialSum n y\u2016 \u2264 C * (a * (\u2016y\u2016 / \u2191r')) ^ n\ny : E\nhy : y \u2208 Metric.ball 0 \u2191r'\nn : \u2115\nyr' : \u2016y\u2016 < \u2191r'\nthis : 0 \u2264 a\n\u22a2 a * (\u2016y\u2016 / \u2191r') \u2264 a", "state_after": "no goals"}, {"tactic": "rwa [ball_zero_eq] at hy", "annotated_tactic": ["rwa [<a>ball_zero_eq</a>] at hy", [{"full_name": "ball_zero_eq", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [604, 3], "def_end_pos": [604, 14]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2075 : NormedAddCommGroup E\ninst\u271d\u2074 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b3 : NormedAddCommGroup F\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c F\ninst\u271d\u00b9 : NormedAddCommGroup G\ninst\u271d : NormedSpace \ud835\udd5c G\nf g : E \u2192 F\np pf pg : FormalMultilinearSeries \ud835\udd5c E F\nx : E\nr r'\u271d : \u211d\u22650\u221e\nr' : \u211d\u22650\nhf : HasFPowerSeriesOnBall f p x r\nh : \u2191r' < r\na : \u211d\nha : a \u2208 Ioo 0 1\nC : \u211d\nhC : C > 0\nhp : \u2200 y \u2208 Metric.ball 0 \u2191r', \u2200 (n : \u2115), \u2016f (x + y) - p.partialSum n y\u2016 \u2264 C * (a * (\u2016y\u2016 / \u2191r')) ^ n\ny : E\nhy : y \u2208 Metric.ball 0 \u2191r'\nn : \u2115\n\u22a2 \u2016y\u2016 < \u2191r'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/RatFunc/Basic.lean", "full_name": "RatFunc.div_smul", "start": [612, 1], "end": [616, 39], "traced_tactics": [{"tactic": "rw [\u2190 mk_eq_div, mk_smul, mk_eq_div]", "annotated_tactic": ["rw [\u2190 <a>mk_eq_div</a>, <a>mk_smul</a>, <a>mk_eq_div</a>]", [{"full_name": "RatFunc.mk_eq_div", "def_path": "Mathlib/FieldTheory/RatFunc/Basic.lean", "def_pos": [607, 9], "def_end_pos": [607, 18]}, {"full_name": "RatFunc.mk_smul", "def_path": "Mathlib/FieldTheory/RatFunc/Basic.lean", "def_pos": [235, 9], "def_end_pos": [235, 16]}, {"full_name": "RatFunc.mk_eq_div", "def_path": "Mathlib/FieldTheory/RatFunc/Basic.lean", "def_pos": [607, 9], "def_end_pos": [607, 18]}]], "state_before": "K : Type u\ninst\u271d\u2074 : CommRing K\ninst\u271d\u00b3 : IsDomain K\nR : Type u_1\ninst\u271d\u00b2 : Monoid R\ninst\u271d\u00b9 : DistribMulAction R K[X]\ninst\u271d : IsScalarTower R K[X] K[X]\nc : R\np q : K[X]\n\u22a2 (algebraMap K[X] (RatFunc K)) (c \u2022 p) / (algebraMap K[X] (RatFunc K)) q =\n    c \u2022 ((algebraMap K[X] (RatFunc K)) p / (algebraMap K[X] (RatFunc K)) q)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/GroupAction/DomAct/Basic.lean", "full_name": "DomMulAct.mk_one", "start": [124, 1], "end": [125, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/CauSeq/BigOperators.lean", "full_name": "IsCauSeq.geo_series_const", "start": [203, 1], "end": [205, 69], "traced_tactics": [{"tactic": "simpa [mul_sum, Pi.mul_def] using (const a).mul (geo_series x hx1)", "annotated_tactic": ["simpa [<a>mul_sum</a>, <a>Pi.mul_def</a>] using (<a>const</a> a).<a>mul</a> (<a>geo_series</a> x hx1)", [{"full_name": "Finset.mul_sum", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [61, 7], "def_end_pos": [61, 14]}, {"full_name": "Pi.mul_def", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 16]}, {"full_name": "IsCauSeq.const", "def_path": "Mathlib/Algebra/Order/CauSeq/Basic.lean", "def_pos": [133, 7], "def_end_pos": [133, 12]}, {"full_name": "IsCauSeq.mul", "def_path": "Mathlib/Algebra/Order/CauSeq/Basic.lean", "def_pos": [144, 7], "def_end_pos": [144, 10]}, {"full_name": "IsCauSeq.geo_series", "def_path": "Mathlib/Algebra/Order/CauSeq/BigOperators.lean", "def_pos": [184, 7], "def_end_pos": [184, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : LinearOrderedField \u03b1\ninst\u271d\u00b2 : Ring \u03b2\nabv : \u03b2 \u2192 \u03b1\ninst\u271d\u00b9 : IsAbsoluteValue abv\nf g : \u2115 \u2192 \u03b2\na\u271d : \u2115 \u2192 \u03b1\ninst\u271d : Archimedean \u03b1\na x : \u03b1\nhx1 : |x| < 1\n\u22a2 IsCauSeq abs fun m => \u2211 n \u2208 range m, a * x ^ n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean", "full_name": "MvPolynomial.isWeightedHomogeneous_monomial", "start": [196, 1], "end": [204, 18], "traced_tactics": [{"tactic": "intro c hc", "annotated_tactic": ["intro c hc", []], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nd : \u03c3 \u2192\u2080 \u2115\nr : R\nm : M\nhm : (weightedDegree w) d = m\n\u22a2 IsWeightedHomogeneous w ((monomial d) r) m", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nd : \u03c3 \u2192\u2080 \u2115\nr : R\nm : M\nhm : (weightedDegree w) d = m\nc : \u03c3 \u2192\u2080 \u2115\nhc : coeff c ((monomial d) r) \u2260 0\n\u22a2 (weightedDegree w) c = m"}, {"tactic": "rw [coeff_monomial] at hc", "annotated_tactic": ["rw [<a>coeff_monomial</a>] at hc", [{"full_name": "MvPolynomial.coeff_monomial", "def_path": "Mathlib/Algebra/MvPolynomial/Basic.lean", "def_pos": [665, 9], "def_end_pos": [665, 23]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nd : \u03c3 \u2192\u2080 \u2115\nr : R\nm : M\nhm : (weightedDegree w) d = m\nc : \u03c3 \u2192\u2080 \u2115\nhc : coeff c ((monomial d) r) \u2260 0\n\u22a2 (weightedDegree w) c = m", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nd : \u03c3 \u2192\u2080 \u2115\nr : R\nm : M\nhm : (weightedDegree w) d = m\nc : \u03c3 \u2192\u2080 \u2115\nhc : (if d = c then r else 0) \u2260 0\n\u22a2 (weightedDegree w) c = m"}, {"tactic": "split_ifs at hc with h", "annotated_tactic": ["split_ifs at hc with h", []], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nd : \u03c3 \u2192\u2080 \u2115\nr : R\nm : M\nhm : (weightedDegree w) d = m\nc : \u03c3 \u2192\u2080 \u2115\nhc : (if d = c then r else 0) \u2260 0\n\u22a2 (weightedDegree w) c = m", "state_after": "case pos\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nd : \u03c3 \u2192\u2080 \u2115\nr : R\nm : M\nhm : (weightedDegree w) d = m\nc : \u03c3 \u2192\u2080 \u2115\nh : d = c\nhc : r \u2260 0\n\u22a2 (weightedDegree w) c = m\n\ncase neg\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nd : \u03c3 \u2192\u2080 \u2115\nr : R\nm : M\nhm : (weightedDegree w) d = m\nc : \u03c3 \u2192\u2080 \u2115\nh : \u00acd = c\nhc : 0 \u2260 0\n\u22a2 (weightedDegree w) c = m"}, {"tactic": "subst c", "annotated_tactic": ["subst c", []], "state_before": "case pos\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nd : \u03c3 \u2192\u2080 \u2115\nr : R\nm : M\nhm : (weightedDegree w) d = m\nc : \u03c3 \u2192\u2080 \u2115\nh : d = c\nhc : r \u2260 0\n\u22a2 (weightedDegree w) c = m", "state_after": "case pos\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nd : \u03c3 \u2192\u2080 \u2115\nr : R\nm : M\nhm : (weightedDegree w) d = m\nhc : r \u2260 0\n\u22a2 (weightedDegree w) d = m"}, {"tactic": "exact hm", "annotated_tactic": ["exact hm", []], "state_before": "case pos\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nd : \u03c3 \u2192\u2080 \u2115\nr : R\nm : M\nhm : (weightedDegree w) d = m\nhc : r \u2260 0\n\u22a2 (weightedDegree w) d = m", "state_after": "no goals"}, {"tactic": "contradiction", "annotated_tactic": ["contradiction", []], "state_before": "case neg\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b9 : CommSemiring R\n\u03c3 : Type u_3\ninst\u271d : AddCommMonoid M\nw : \u03c3 \u2192 M\nd : \u03c3 \u2192\u2080 \u2115\nr : R\nm : M\nhm : (weightedDegree w) d = m\nc : \u03c3 \u2192\u2080 \u2115\nh : \u00acd = c\nhc : 0 \u2260 0\n\u22a2 (weightedDegree w) c = m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Symmetrized.lean", "full_name": "SymAlg.unsym_inj", "start": [121, 1], "end": [122, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/SplittingField/IsSplittingField.lean", "full_name": "Polynomial.IsSplittingField.splits", "start": [57, 1], "end": [58, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean", "full_name": "NonUnitalAlgebra.adjoin_univ", "start": [655, 1], "end": [656, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean", "full_name": "HurwitzZeta.hurwitzZetaEven_one_sub_two_mul_nat", "start": [126, 1], "end": [146, 22], "traced_tactics": [{"tactic": "have h1 (n : \u2115) : (2 * k : \u2102) \u2260 -n := by\n  rw [\u2190 Int.cast_ofNat, \u2190 Int.cast_natCast, \u2190 Int.cast_mul, \u2190 Int.cast_natCast n, \u2190 Int.cast_neg,\n    Ne, Int.cast_inj, \u2190 Ne]\n  refine ne_of_gt ((neg_nonpos_of_nonneg n.cast_nonneg).trans_lt (mul_pos two_pos ?_))\n  exact Nat.cast_pos.mpr (Nat.pos_of_ne_zero hk)", "annotated_tactic": ["have h1 (n : \u2115) : (2 * k : \u2102) \u2260 -n := by\n    rw [\u2190 <a>Int.cast_ofNat</a>, \u2190 <a>Int.cast_natCast</a>, \u2190 <a>Int.cast_mul</a>, \u2190 <a>Int.cast_natCast</a> n, \u2190 <a>Int.cast_neg</a>,\n      <a>Ne</a>, <a>Int.cast_inj</a>, \u2190 <a>Ne</a>]\n    refine <a>ne_of_gt</a> ((<a>neg_nonpos_of_nonneg</a> n.cast_nonneg).<a>trans_lt</a> (<a>mul_pos</a> <a>two_pos</a> ?_))\n    exact Nat.cast_pos.mpr (<a>Nat.pos_of_ne_zero</a> hk)", [{"full_name": "Int.cast_ofNat", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [74, 9], "def_end_pos": [74, 19]}, {"full_name": "Int.cast_natCast", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [66, 9], "def_end_pos": [66, 21]}, {"full_name": "Int.cast_mul", "def_path": "Mathlib/Algebra/Ring/Int.lean", "def_pos": [58, 7], "def_end_pos": [58, 15]}, {"full_name": "Int.cast_natCast", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [66, 9], "def_end_pos": [66, 21]}, {"full_name": "Int.cast_neg", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [85, 9], "def_end_pos": [85, 17]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "Int.cast_inj", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [76, 7], "def_end_pos": [76, 15]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}, {"full_name": "neg_nonpos_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [1218, 3], "def_end_pos": [1218, 14]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [119, 7], "def_end_pos": [119, 21]}, {"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [406, 7], "def_end_pos": [406, 14]}, {"full_name": "two_pos", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [113, 7], "def_end_pos": [113, 14]}, {"full_name": "Nat.pos_of_ne_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [354, 19], "def_end_pos": [354, 33]}]], "state_before": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\n\u22a2 hurwitzZetaEven (\u2191x) (1 - 2 * \u2191k) =\n    -1 / (2 * \u2191k) * Polynomial.eval (\u2191x) (Polynomial.map (algebraMap \u211a \u2102) (Polynomial.bernoulli (2 * k)))", "state_after": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\nh1 : \u2200 (n : \u2115), 2 * \u2191k \u2260 -\u2191n\n\u22a2 hurwitzZetaEven (\u2191x) (1 - 2 * \u2191k) =\n    -1 / (2 * \u2191k) * Polynomial.eval (\u2191x) (Polynomial.map (algebraMap \u211a \u2102) (Polynomial.bernoulli (2 * k)))"}, {"tactic": "have h2 : (2 * k : \u2102) \u2260 1 := by norm_cast; simp only [mul_eq_one, OfNat.ofNat_ne_one,\n  false_and, not_false_eq_true]", "annotated_tactic": ["have h2 : (2 * k : \u2102) \u2260 1 := by norm_cast; simp only [<a>mul_eq_one</a>, <a>OfNat.ofNat_ne_one</a>,\n    <a>false_and</a>, <a>not_false_eq_true</a>]", [{"full_name": "mul_eq_one", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [628, 9], "def_end_pos": [628, 19]}, {"full_name": "OfNat.ofNat_ne_one", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [115, 15], "def_end_pos": [115, 27]}, {"full_name": "false_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [110, 17], "def_end_pos": [110, 26]}, {"full_name": "not_false_eq_true", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [134, 17], "def_end_pos": [134, 34]}]], "state_before": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\nh1 : \u2200 (n : \u2115), 2 * \u2191k \u2260 -\u2191n\n\u22a2 hurwitzZetaEven (\u2191x) (1 - 2 * \u2191k) =\n    -1 / (2 * \u2191k) * Polynomial.eval (\u2191x) (Polynomial.map (algebraMap \u211a \u2102) (Polynomial.bernoulli (2 * k)))", "state_after": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\nh1 : \u2200 (n : \u2115), 2 * \u2191k \u2260 -\u2191n\nh2 : 2 * \u2191k \u2260 1\n\u22a2 hurwitzZetaEven (\u2191x) (1 - 2 * \u2191k) =\n    -1 / (2 * \u2191k) * Polynomial.eval (\u2191x) (Polynomial.map (algebraMap \u211a \u2102) (Polynomial.bernoulli (2 * k)))"}, {"tactic": "have h3 : Gamma\u2102 (2 * k) \u2260 0 := by\n  refine mul_ne_zero (mul_ne_zero two_ne_zero ?_) (Gamma_ne_zero h1)\n  simp only [ne_eq, cpow_eq_zero_iff, mul_eq_zero, OfNat.ofNat_ne_zero, ofReal_eq_zero,\n    pi_ne_zero, Nat.cast_eq_zero, false_or, false_and, not_false_eq_true]", "annotated_tactic": ["have h3 : <a>Gamma\u2102</a> (2 * k) \u2260 0 := by\n    refine <a>mul_ne_zero</a> (<a>mul_ne_zero</a> <a>two_ne_zero</a> ?_) (<a>Gamma_ne_zero</a> h1)\n    simp only [<a>ne_eq</a>, <a>cpow_eq_zero_iff</a>, <a>mul_eq_zero</a>, <a>OfNat.ofNat_ne_zero</a>, <a>ofReal_eq_zero</a>,\n      <a>pi_ne_zero</a>, <a>Nat.cast_eq_zero</a>, <a>false_or</a>, <a>false_and</a>, <a>not_false_eq_true</a>]", [{"full_name": "Complex.Gamma\u2102", "def_path": "Mathlib/Analysis/SpecialFunctions/Gamma/Deligne.lean", "def_pos": [49, 19], "def_end_pos": [49, 25]}, {"full_name": "mul_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 20]}, {"full_name": "mul_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 20]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [65, 7], "def_end_pos": [65, 18]}, {"full_name": "Complex.Gamma_ne_zero", "def_path": "Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean", "def_pos": [450, 9], "def_end_pos": [450, 22]}, {"full_name": "ne_eq", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [89, 17], "def_end_pos": [89, 22]}, {"full_name": "Complex.cpow_eq_zero_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean", "def_pos": [49, 9], "def_end_pos": [49, 25]}, {"full_name": "mul_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [282, 9], "def_end_pos": [282, 20]}, {"full_name": "OfNat.ofNat_ne_zero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [109, 15], "def_end_pos": [109, 28]}, {"full_name": "Complex.ofReal_eq_zero", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [158, 9], "def_end_pos": [158, 23]}, {"full_name": "Real.pi_ne_zero", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "def_pos": [176, 9], "def_end_pos": [176, 19]}, {"full_name": "Nat.cast_eq_zero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 21]}, {"full_name": "false_or", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [122, 17], "def_end_pos": [122, 25]}, {"full_name": "false_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [110, 17], "def_end_pos": [110, 26]}, {"full_name": "not_false_eq_true", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [134, 17], "def_end_pos": [134, 34]}]], "state_before": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\nh1 : \u2200 (n : \u2115), 2 * \u2191k \u2260 -\u2191n\nh2 : 2 * \u2191k \u2260 1\n\u22a2 hurwitzZetaEven (\u2191x) (1 - 2 * \u2191k) =\n    -1 / (2 * \u2191k) * Polynomial.eval (\u2191x) (Polynomial.map (algebraMap \u211a \u2102) (Polynomial.bernoulli (2 * k)))", "state_after": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\nh1 : \u2200 (n : \u2115), 2 * \u2191k \u2260 -\u2191n\nh2 : 2 * \u2191k \u2260 1\nh3 : (2 * \u2191k).Gamma\u2102 \u2260 0\n\u22a2 hurwitzZetaEven (\u2191x) (1 - 2 * \u2191k) =\n    -1 / (2 * \u2191k) * Polynomial.eval (\u2191x) (Polynomial.map (algebraMap \u211a \u2102) (Polynomial.bernoulli (2 * k)))"}, {"tactic": "rw [hurwitzZetaEven_one_sub _ h1 (Or.inr h2), \u2190 Gamma\u2102, cosZeta_two_mul_nat' hk hx, \u2190 mul_assoc,\n  \u2190 mul_div_assoc, mul_assoc, mul_div_cancel_left\u2080 _ h3, \u2190 mul_div_assoc]", "annotated_tactic": ["rw [<a>hurwitzZetaEven_one_sub</a> _ h1 (<a>Or.inr</a> h2), \u2190 <a>Gamma\u2102</a>, <a>cosZeta_two_mul_nat'</a> hk hx, \u2190 <a>mul_assoc</a>,\n    \u2190 <a>mul_div_assoc</a>, <a>mul_assoc</a>, <a>mul_div_cancel_left\u2080</a> _ h3, \u2190 <a>mul_div_assoc</a>]", [{"full_name": "HurwitzZeta.hurwitzZetaEven_one_sub", "def_path": "Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean", "def_pos": [784, 7], "def_end_pos": [784, 30]}, {"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}, {"full_name": "Complex.Gamma\u2102", "def_path": "Mathlib/Analysis/SpecialFunctions/Gamma/Deligne.lean", "def_pos": [49, 19], "def_end_pos": [49, 25]}, {"full_name": "HurwitzZeta.cosZeta_two_mul_nat'", "def_path": "Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean", "def_pos": [100, 9], "def_end_pos": [100, 29]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "mul_div_assoc", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [456, 9], "def_end_pos": [456, 22]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "mul_div_cancel_left\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [198, 15], "def_end_pos": [198, 35]}, {"full_name": "mul_div_assoc", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [456, 9], "def_end_pos": [456, 22]}]], "state_before": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\nh1 : \u2200 (n : \u2115), 2 * \u2191k \u2260 -\u2191n\nh2 : 2 * \u2191k \u2260 1\nh3 : (2 * \u2191k).Gamma\u2102 \u2260 0\n\u22a2 hurwitzZetaEven (\u2191x) (1 - 2 * \u2191k) =\n    -1 / (2 * \u2191k) * Polynomial.eval (\u2191x) (Polynomial.map (algebraMap \u211a \u2102) (Polynomial.bernoulli (2 * k)))", "state_after": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\nh1 : \u2200 (n : \u2115), 2 * \u2191k \u2260 -\u2191n\nh2 : 2 * \u2191k \u2260 1\nh3 : (2 * \u2191k).Gamma\u2102 \u2260 0\n\u22a2 Complex.cos (\u2191\u03c0 * (2 * \u2191k) / 2) * (-1) ^ (k + 1) / (2 * \u2191k) *\n      Polynomial.eval (\u2191x) (Polynomial.map (algebraMap \u211a \u2102) (Polynomial.bernoulli (2 * k))) =\n    -1 / (2 * \u2191k) * Polynomial.eval (\u2191x) (Polynomial.map (algebraMap \u211a \u2102) (Polynomial.bernoulli (2 * k)))"}, {"tactic": "congr 2", "annotated_tactic": ["congr 2", []], "state_before": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\nh1 : \u2200 (n : \u2115), 2 * \u2191k \u2260 -\u2191n\nh2 : 2 * \u2191k \u2260 1\nh3 : (2 * \u2191k).Gamma\u2102 \u2260 0\n\u22a2 Complex.cos (\u2191\u03c0 * (2 * \u2191k) / 2) * (-1) ^ (k + 1) / (2 * \u2191k) *\n      Polynomial.eval (\u2191x) (Polynomial.map (algebraMap \u211a \u2102) (Polynomial.bernoulli (2 * k))) =\n    -1 / (2 * \u2191k) * Polynomial.eval (\u2191x) (Polynomial.map (algebraMap \u211a \u2102) (Polynomial.bernoulli (2 * k)))", "state_after": "case e_a.e_a\nk : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\nh1 : \u2200 (n : \u2115), 2 * \u2191k \u2260 -\u2191n\nh2 : 2 * \u2191k \u2260 1\nh3 : (2 * \u2191k).Gamma\u2102 \u2260 0\n\u22a2 Complex.cos (\u2191\u03c0 * (2 * \u2191k) / 2) * (-1) ^ (k + 1) = -1"}, {"tactic": "rw [mul_div_assoc, mul_div_cancel_left\u2080 _ two_ne_zero, \u2190 ofReal_natCast, \u2190 ofReal_mul,\n  \u2190 ofReal_cos, mul_comm \u03c0, \u2190 sub_zero (k * \u03c0), cos_nat_mul_pi_sub, Real.cos_zero, mul_one,\n  ofReal_pow, ofReal_neg, ofReal_one, pow_succ, mul_neg_one, mul_neg, \u2190 mul_pow, neg_one_mul,\n  neg_neg, one_pow]", "annotated_tactic": ["rw [<a>mul_div_assoc</a>, <a>mul_div_cancel_left\u2080</a> _ <a>two_ne_zero</a>, \u2190 <a>ofReal_natCast</a>, \u2190 <a>ofReal_mul</a>,\n    \u2190 <a>ofReal_cos</a>, <a>mul_comm</a> \u03c0, \u2190 <a>sub_zero</a> (k * \u03c0), <a>cos_nat_mul_pi_sub</a>, <a>Real.cos_zero</a>, <a>mul_one</a>,\n    <a>ofReal_pow</a>, <a>ofReal_neg</a>, <a>ofReal_one</a>, <a>pow_succ</a>, <a>mul_neg_one</a>, <a>mul_neg</a>, \u2190 <a>mul_pow</a>, <a>neg_one_mul</a>,\n    <a>neg_neg</a>, <a>one_pow</a>]", [{"full_name": "mul_div_assoc", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [456, 9], "def_end_pos": [456, 22]}, {"full_name": "mul_div_cancel_left\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [198, 15], "def_end_pos": [198, 35]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [65, 7], "def_end_pos": [65, 18]}, {"full_name": "Complex.ofReal_natCast", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [514, 26], "def_end_pos": [514, 40]}, {"full_name": "Complex.ofReal_mul", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 19]}, {"full_name": "Complex.ofReal_cos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [652, 9], "def_end_pos": [652, 19]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [489, 3], "def_end_pos": [489, 14]}, {"full_name": "Real.cos_nat_mul_pi_sub", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "def_pos": [438, 9], "def_end_pos": [438, 27]}, {"full_name": "Real.cos_zero", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [883, 9], "def_end_pos": [883, 17]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "Complex.ofReal_pow", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [794, 9], "def_end_pos": [794, 19]}, {"full_name": "Complex.ofReal_neg", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [236, 9], "def_end_pos": [236, 19]}, {"full_name": "Complex.ofReal_one", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [180, 9], "def_end_pos": [180, 19]}, {"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [657, 9], "def_end_pos": [657, 17]}, {"full_name": "mul_neg_one", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [352, 9], "def_end_pos": [352, 20]}, {"full_name": "mul_neg", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}, {"full_name": "mul_pow", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [310, 32], "def_end_pos": [310, 39]}, {"full_name": "neg_one_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [357, 9], "def_end_pos": [357, 20]}, {"full_name": "neg_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [869, 3], "def_end_pos": [869, 14]}, {"full_name": "one_pow", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [696, 39], "def_end_pos": [696, 46]}]], "state_before": "case e_a.e_a\nk : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\nh1 : \u2200 (n : \u2115), 2 * \u2191k \u2260 -\u2191n\nh2 : 2 * \u2191k \u2260 1\nh3 : (2 * \u2191k).Gamma\u2102 \u2260 0\n\u22a2 Complex.cos (\u2191\u03c0 * (2 * \u2191k) / 2) * (-1) ^ (k + 1) = -1", "state_after": "no goals"}, {"tactic": "refine ne_of_gt ((neg_nonpos_of_nonneg n.cast_nonneg).trans_lt (mul_pos two_pos ?_))", "annotated_tactic": ["refine <a>ne_of_gt</a> ((<a>neg_nonpos_of_nonneg</a> n.cast_nonneg).<a>trans_lt</a> (<a>mul_pos</a> <a>two_pos</a> ?_))", [{"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}, {"full_name": "neg_nonpos_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [1218, 3], "def_end_pos": [1218, 14]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [119, 7], "def_end_pos": [119, 21]}, {"full_name": "mul_pos", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [406, 7], "def_end_pos": [406, 14]}, {"full_name": "two_pos", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [113, 7], "def_end_pos": [113, 14]}]], "state_before": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\nn : \u2115\n\u22a2 2 * \u2191k \u2260 -\u2191n", "state_after": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\nn : \u2115\n\u22a2 0 < \u2191k"}, {"tactic": "exact Nat.cast_pos.mpr (Nat.pos_of_ne_zero hk)", "annotated_tactic": ["exact Nat.cast_pos.mpr (<a>Nat.pos_of_ne_zero</a> hk)", [{"full_name": "Nat.pos_of_ne_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [354, 19], "def_end_pos": [354, 33]}]], "state_before": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\nn : \u2115\n\u22a2 0 < \u2191k", "state_after": "no goals"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\nh1 : \u2200 (n : \u2115), 2 * \u2191k \u2260 -\u2191n\n\u22a2 2 * \u2191k \u2260 1", "state_after": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\nh1 : \u2200 (n : \u2115), 2 * \u2191k \u2260 -\u2191n\n\u22a2 \u00ac2 * k = 1"}, {"tactic": "simp only [mul_eq_one, OfNat.ofNat_ne_one,\nfalse_and, not_false_eq_true]", "annotated_tactic": ["simp only [<a>mul_eq_one</a>, <a>OfNat.ofNat_ne_one</a>,\n    <a>false_and</a>, <a>not_false_eq_true</a>]", [{"full_name": "mul_eq_one", "def_path": "Mathlib/Algebra/Group/Units.lean", "def_pos": [628, 9], "def_end_pos": [628, 19]}, {"full_name": "OfNat.ofNat_ne_one", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [115, 15], "def_end_pos": [115, 27]}, {"full_name": "false_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [110, 17], "def_end_pos": [110, 26]}, {"full_name": "not_false_eq_true", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [134, 17], "def_end_pos": [134, 34]}]], "state_before": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\nh1 : \u2200 (n : \u2115), 2 * \u2191k \u2260 -\u2191n\n\u22a2 \u00ac2 * k = 1", "state_after": "no goals"}, {"tactic": "refine mul_ne_zero (mul_ne_zero two_ne_zero ?_) (Gamma_ne_zero h1)", "annotated_tactic": ["refine <a>mul_ne_zero</a> (<a>mul_ne_zero</a> <a>two_ne_zero</a> ?_) (<a>Gamma_ne_zero</a> h1)", [{"full_name": "mul_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 20]}, {"full_name": "mul_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 20]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [65, 7], "def_end_pos": [65, 18]}, {"full_name": "Complex.Gamma_ne_zero", "def_path": "Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean", "def_pos": [450, 9], "def_end_pos": [450, 22]}]], "state_before": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\nh1 : \u2200 (n : \u2115), 2 * \u2191k \u2260 -\u2191n\nh2 : 2 * \u2191k \u2260 1\n\u22a2 (2 * \u2191k).Gamma\u2102 \u2260 0", "state_after": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\nh1 : \u2200 (n : \u2115), 2 * \u2191k \u2260 -\u2191n\nh2 : 2 * \u2191k \u2260 1\n\u22a2 (2 * \u2191\u03c0) ^ (-(2 * \u2191k)) \u2260 0"}, {"tactic": "simp only [ne_eq, cpow_eq_zero_iff, mul_eq_zero, OfNat.ofNat_ne_zero, ofReal_eq_zero,\n  pi_ne_zero, Nat.cast_eq_zero, false_or, false_and, not_false_eq_true]", "annotated_tactic": ["simp only [<a>ne_eq</a>, <a>cpow_eq_zero_iff</a>, <a>mul_eq_zero</a>, <a>OfNat.ofNat_ne_zero</a>, <a>ofReal_eq_zero</a>,\n      <a>pi_ne_zero</a>, <a>Nat.cast_eq_zero</a>, <a>false_or</a>, <a>false_and</a>, <a>not_false_eq_true</a>]", [{"full_name": "ne_eq", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [89, 17], "def_end_pos": [89, 22]}, {"full_name": "Complex.cpow_eq_zero_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean", "def_pos": [49, 9], "def_end_pos": [49, 25]}, {"full_name": "mul_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [282, 9], "def_end_pos": [282, 20]}, {"full_name": "OfNat.ofNat_ne_zero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [109, 15], "def_end_pos": [109, 28]}, {"full_name": "Complex.ofReal_eq_zero", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [158, 9], "def_end_pos": [158, 23]}, {"full_name": "Real.pi_ne_zero", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "def_pos": [176, 9], "def_end_pos": [176, 19]}, {"full_name": "Nat.cast_eq_zero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 21]}, {"full_name": "false_or", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [122, 17], "def_end_pos": [122, 25]}, {"full_name": "false_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [110, 17], "def_end_pos": [110, 26]}, {"full_name": "not_false_eq_true", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [134, 17], "def_end_pos": [134, 34]}]], "state_before": "k : \u2115\nx : \u211d\nhk : k \u2260 0\nhx : x \u2208 Icc 0 1\nh1 : \u2200 (n : \u2115), 2 * \u2191k \u2260 -\u2191n\nh2 : 2 * \u2191k \u2260 1\n\u22a2 (2 * \u2191\u03c0) ^ (-(2 * \u2191k)) \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Eval.lean", "full_name": "Polynomial.coe_evalRingHom", "start": [1116, 1], "end": [1117, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Real.cos_pos_of_le_one", "start": [1573, 1], "end": [1584, 67], "traced_tactics": [{"tactic": "gcongr", "annotated_tactic": ["gcongr", []], "state_before": "x : \u211d\nhx : |x| \u2264 1\n\u22a2 |x| ^ 4 * (5 / 96) + x ^ 2 / 2 \u2264 1 * (5 / 96) + 1 / 2", "state_after": "case h\u2081.h\nx : \u211d\nhx : |x| \u2264 1\n\u22a2 |x| ^ 4 \u2264 1\n\ncase h\u2082.hab\nx : \u211d\nhx : |x| \u2264 1\n\u22a2 x ^ 2 \u2264 1"}, {"tactic": "exact pow_le_one _ (abs_nonneg _) hx", "annotated_tactic": ["exact <a>pow_le_one</a> _ (<a>abs_nonneg</a> _) hx", [{"full_name": "pow_le_one", "def_path": "Mathlib/Algebra/Order/Ring/Basic.lean", "def_pos": [70, 9], "def_end_pos": [70, 19]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [122, 30], "def_end_pos": [122, 40]}]], "state_before": "case h\u2081.h\nx : \u211d\nhx : |x| \u2264 1\n\u22a2 |x| ^ 4 \u2264 1", "state_after": "no goals"}, {"tactic": "rw [sq, \u2190 abs_mul_self, abs_mul]", "annotated_tactic": ["rw [<a>sq</a>, \u2190 <a>abs_mul_self</a>, <a>abs_mul</a>]", [{"full_name": "sq", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [684, 41], "def_end_pos": [684, 43]}, {"full_name": "abs_mul_self", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [87, 7], "def_end_pos": [87, 19]}, {"full_name": "abs_mul", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [48, 7], "def_end_pos": [48, 14]}]], "state_before": "case h\u2082.hab\nx : \u211d\nhx : |x| \u2264 1\n\u22a2 x ^ 2 \u2264 1", "state_after": "case h\u2082.hab\nx : \u211d\nhx : |x| \u2264 1\n\u22a2 |x| * |x| \u2264 1"}, {"tactic": "exact mul_le_one hx (abs_nonneg _) hx", "annotated_tactic": ["exact <a>mul_le_one</a> hx (<a>abs_nonneg</a> _) hx", [{"full_name": "mul_le_one", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [313, 9], "def_end_pos": [313, 19]}, {"full_name": "abs_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [122, 30], "def_end_pos": [122, 40]}]], "state_before": "case h\u2082.hab\nx : \u211d\nhx : |x| \u2264 1\n\u22a2 |x| * |x| \u2264 1", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "x : \u211d\nhx : |x| \u2264 1\n\u22a2 1 * (5 / 96) + 1 / 2 < 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Projectivization/Independence.lean", "full_name": "Projectivization.independent_iff", "start": [48, 1], "end": [58, 24], "traced_tactics": [{"tactic": "refine \u27e8?_, fun h => ?_\u27e9", "annotated_tactic": ["refine \u27e8?_, fun h => ?_\u27e9", []], "state_before": "\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\n\u22a2 Independent f \u2194 LinearIndependent K (Projectivization.rep \u2218 f)", "state_after": "case refine_1\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\n\u22a2 Independent f \u2192 LinearIndependent K (Projectivization.rep \u2218 f)\n\ncase refine_2\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nh : LinearIndependent K (Projectivization.rep \u2218 f)\n\u22a2 Independent f"}, {"tactic": "rintro \u27e8ff, hff, hh\u27e9", "annotated_tactic": ["rintro \u27e8ff, hff, hh\u27e9", []], "state_before": "case refine_1\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\n\u22a2 Independent f \u2192 LinearIndependent K (Projectivization.rep \u2218 f)", "state_after": "case refine_1.mk\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nff : \u03b9 \u2192 V\nhff : \u2200 (i : \u03b9), ff i \u2260 0\nhh : LinearIndependent K ff\n\u22a2 LinearIndependent K (Projectivization.rep \u2218 fun i => mk K (ff i) \u22ef)"}, {"tactic": "choose a ha using fun i : \u03b9 => exists_smul_eq_mk_rep K (ff i) (hff i)", "annotated_tactic": ["choose a ha using fun i : \u03b9 => <a>exists_smul_eq_mk_rep</a> K (ff i) (hff i)", [{"full_name": "Projectivization.exists_smul_eq_mk_rep", "def_path": "Mathlib/LinearAlgebra/Projectivization/Basic.lean", "def_pos": [119, 9], "def_end_pos": [119, 30]}]], "state_before": "case refine_1.mk\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nff : \u03b9 \u2192 V\nhff : \u2200 (i : \u03b9), ff i \u2260 0\nhh : LinearIndependent K ff\n\u22a2 LinearIndependent K (Projectivization.rep \u2218 fun i => mk K (ff i) \u22ef)", "state_after": "case refine_1.mk\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nff : \u03b9 \u2192 V\nhff : \u2200 (i : \u03b9), ff i \u2260 0\nhh : LinearIndependent K ff\na : \u03b9 \u2192 K\u02e3\nha : \u2200 (i : \u03b9), a i \u2022 ff i = (mk K (ff i) \u22ef).rep\n\u22a2 LinearIndependent K (Projectivization.rep \u2218 fun i => mk K (ff i) \u22ef)"}, {"tactic": "convert hh.units_smul a", "annotated_tactic": ["convert hh.units_smul a", []], "state_before": "case refine_1.mk\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nff : \u03b9 \u2192 V\nhff : \u2200 (i : \u03b9), ff i \u2260 0\nhh : LinearIndependent K ff\na : \u03b9 \u2192 K\u02e3\nha : \u2200 (i : \u03b9), a i \u2022 ff i = (mk K (ff i) \u22ef).rep\n\u22a2 LinearIndependent K (Projectivization.rep \u2218 fun i => mk K (ff i) \u22ef)", "state_after": "case h.e'_4\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nff : \u03b9 \u2192 V\nhff : \u2200 (i : \u03b9), ff i \u2260 0\nhh : LinearIndependent K ff\na : \u03b9 \u2192 K\u02e3\nha : \u2200 (i : \u03b9), a i \u2022 ff i = (mk K (ff i) \u22ef).rep\n\u22a2 (Projectivization.rep \u2218 fun i => mk K (ff i) \u22ef) = a \u2022 ff"}, {"tactic": "ext i", "annotated_tactic": ["ext i", []], "state_before": "case h.e'_4\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nff : \u03b9 \u2192 V\nhff : \u2200 (i : \u03b9), ff i \u2260 0\nhh : LinearIndependent K ff\na : \u03b9 \u2192 K\u02e3\nha : \u2200 (i : \u03b9), a i \u2022 ff i = (mk K (ff i) \u22ef).rep\n\u22a2 (Projectivization.rep \u2218 fun i => mk K (ff i) \u22ef) = a \u2022 ff", "state_after": "case h.e'_4.h\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nff : \u03b9 \u2192 V\nhff : \u2200 (i : \u03b9), ff i \u2260 0\nhh : LinearIndependent K ff\na : \u03b9 \u2192 K\u02e3\nha : \u2200 (i : \u03b9), a i \u2022 ff i = (mk K (ff i) \u22ef).rep\ni : \u03b9\n\u22a2 (Projectivization.rep \u2218 fun i => mk K (ff i) \u22ef) i = (a \u2022 ff) i"}, {"tactic": "exact (ha i).symm", "annotated_tactic": ["exact (ha i).<a>symm</a>", [{"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case h.e'_4.h\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nff : \u03b9 \u2192 V\nhff : \u2200 (i : \u03b9), ff i \u2260 0\nhh : LinearIndependent K ff\na : \u03b9 \u2192 K\u02e3\nha : \u2200 (i : \u03b9), a i \u2022 ff i = (mk K (ff i) \u22ef).rep\ni : \u03b9\n\u22a2 (Projectivization.rep \u2218 fun i => mk K (ff i) \u22ef) i = (a \u2022 ff) i", "state_after": "no goals"}, {"tactic": "convert Independent.mk _ _ h", "annotated_tactic": ["convert <a>Independent.mk</a> _ _ h", [{"full_name": "Projectivization.Independent.mk", "def_path": "Mathlib/LinearAlgebra/Projectivization/Independence.lean", "def_pos": [42, 5], "def_end_pos": [42, 7]}]], "state_before": "case refine_2\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nh : LinearIndependent K (Projectivization.rep \u2218 f)\n\u22a2 Independent f", "state_after": "case h.e'_7.h\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nh : LinearIndependent K (Projectivization.rep \u2218 f)\nx\u271d : \u03b9\n\u22a2 f x\u271d = mk K ((Projectivization.rep \u2218 f) x\u271d) \u22ef\n\ncase refine_2\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nh : LinearIndependent K (Projectivization.rep \u2218 f)\n\u22a2 \u2200 (i : \u03b9), (Projectivization.rep \u2218 f) i \u2260 0"}, {"tactic": "simp only [mk_rep, Function.comp_apply]", "annotated_tactic": ["simp only [<a>mk_rep</a>, <a>Function.comp_apply</a>]", [{"full_name": "Projectivization.mk_rep", "def_path": "Mathlib/LinearAlgebra/Projectivization/Basic.lean", "def_pos": [87, 9], "def_end_pos": [87, 15]}, {"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}]], "state_before": "case h.e'_7.h\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nh : LinearIndependent K (Projectivization.rep \u2218 f)\nx\u271d : \u03b9\n\u22a2 f x\u271d = mk K ((Projectivization.rep \u2218 f) x\u271d) \u22ef", "state_after": "no goals"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "case refine_2\n\u03b9 : Type u_1\nK : Type u_2\nV : Type u_3\ninst\u271d\u00b2 : DivisionRing K\ninst\u271d\u00b9 : AddCommGroup V\ninst\u271d : Module K V\nf : \u03b9 \u2192 \u2119 K V\nh : LinearIndependent K 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"annotated_tactic": ["dsimp [<a>finsuppTotal</a>]", [{"full_name": "Ideal.finsuppTotal", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [1342, 19], "def_end_pos": [1342, 31]}]], "state_before": "\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Module R M\nI : Ideal R\nv : \u03b9 \u2192 M\nhv : Submodule.span R (Set.range v) = \u22a4\nf : \u03b9 \u2192\u2080 \u21a5I\n\u22a2 (finsuppTotal \u03b9 M I v) f = f.sum fun i x => \u2191x \u2022 v i", "state_after": "\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Module R M\nI : Ideal R\nv : \u03b9 \u2192 M\nhv : Submodule.span R (Set.range v) = \u22a4\nf : \u03b9 \u2192\u2080 \u21a5I\n\u22a2 (Finsupp.total \u03b9 M R v) (Finsupp.mapRange Subtype.val \u22ef f) = f.sum fun i x => \u2191x \u2022 v i"}, {"tactic": "rw [Finsupp.total_apply, Finsupp.sum_mapRange_index]", "annotated_tactic": ["rw [<a>Finsupp.total_apply</a>, <a>Finsupp.sum_mapRange_index</a>]", [{"full_name": "Finsupp.total_apply", "def_path": "Mathlib/LinearAlgebra/Finsupp.lean", "def_pos": [664, 9], "def_end_pos": [664, 20]}, {"full_name": "Finsupp.sum_mapRange_index", "def_path": "Mathlib/Algebra/BigOperators/Finsupp.lean", "def_pos": [79, 3], "def_end_pos": [79, 14]}]], "state_before": "\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Module R M\nI : Ideal R\nv : \u03b9 \u2192 M\nhv : Submodule.span R (Set.range v) = \u22a4\nf : \u03b9 \u2192\u2080 \u21a5I\n\u22a2 (Finsupp.total \u03b9 M R v) (Finsupp.mapRange Subtype.val \u22ef f) = f.sum fun i x => \u2191x \u2022 v i", "state_after": "\u03b9 : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\nR : Type u_3\ninst\u271d\u00b9 : CommRing R\ninst\u271d : Module R M\nI : Ideal R\nv : \u03b9 \u2192 M\nhv : Submodule.span R (Set.range v) = \u22a4\nf : 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"29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Martingale/Basic.lean", "full_name": "MeasureTheory.martingale_of_condexp_sub_eq_zero_nat", "start": [485, 1], "end": [493, 56], "traced_tactics": [{"tactic": "refine martingale_iff.2 \u27e8supermartingale_of_condexp_sub_nonneg_nat hadp hint fun i => ?_,\n  submartingale_of_condexp_sub_nonneg_nat hadp hint fun i => (hf i).symm.le\u27e9", "annotated_tactic": ["refine <a>martingale_iff</a>.2 \u27e8<a>supermartingale_of_condexp_sub_nonneg_nat</a> hadp hint fun i => ?_,\n    <a>submartingale_of_condexp_sub_nonneg_nat</a> hadp hint fun i => (hf i).symm.le\u27e9", [{"full_name": "MeasureTheory.martingale_iff", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [148, 9], "def_end_pos": [148, 23]}, {"full_name": "MeasureTheory.supermartingale_of_condexp_sub_nonneg_nat", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [477, 9], "def_end_pos": [477, 50]}, {"full_name": "MeasureTheory.submartingale_of_condexp_sub_nonneg_nat", "def_path": "Mathlib/Probability/Martingale/Basic.lean", "def_pos": [469, 9], "def_end_pos": [469, 48]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i) \u03bc\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1da0[ae \u03bc] 0\n\u22a2 Martingale f \ud835\udca2 \u03bc", "state_after": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i) \u03bc\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1da0[ae \u03bc] 0\ni : \u2115\n\u22a2 0 \u2264\u1da0[ae \u03bc] \u03bc[f i - f (i + 1)|\u2191\ud835\udca2 i]"}, {"tactic": "rw [\u2190 neg_sub]", "annotated_tactic": ["rw [\u2190 <a>neg_sub</a>]", [{"full_name": "neg_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [553, 3], "def_end_pos": [553, 14]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i) \u03bc\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1da0[ae \u03bc] 0\ni : \u2115\n\u22a2 0 \u2264\u1da0[ae \u03bc] \u03bc[f i - f (i + 1)|\u2191\ud835\udca2 i]", "state_after": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i) \u03bc\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1da0[ae \u03bc] 0\ni : \u2115\n\u22a2 0 \u2264\u1da0[ae \u03bc] \u03bc[-(f (i + 1) - f i)|\u2191\ud835\udca2 i]"}, {"tactic": "refine (EventuallyEq.trans ?_ (condexp_neg _).symm).le", "annotated_tactic": ["refine (<a>EventuallyEq.trans</a> ?_ (<a>condexp_neg</a> _).<a>symm</a>).<a>le</a>", [{"full_name": "Filter.EventuallyEq.trans", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1535, 9], "def_end_pos": [1535, 27]}, {"full_name": "MeasureTheory.condexp_neg", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean", "def_pos": [327, 9], "def_end_pos": [327, 20]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1530, 9], "def_end_pos": [1530, 26]}, {"full_name": "Filter.EventuallyEq.le", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1697, 9], "def_end_pos": [1697, 24]}]], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i) \u03bc\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1da0[ae \u03bc] 0\ni : \u2115\n\u22a2 0 \u2264\u1da0[ae \u03bc] \u03bc[-(f (i + 1) - f i)|\u2191\ud835\udca2 i]", "state_after": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i) \u03bc\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1da0[ae \u03bc] 0\ni : \u2115\n\u22a2 0 =\u1da0[ae \u03bc] -\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]"}, {"tactic": "filter_upwards [hf i] with x hx", "annotated_tactic": ["filter_upwards [hf i] with x hx", []], "state_before": "\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i) \u03bc\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1da0[ae \u03bc] 0\ni : \u2115\n\u22a2 0 =\u1da0[ae \u03bc] -\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]", "state_after": "case h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i) \u03bc\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1da0[ae \u03bc] 0\ni : \u2115\nx : \u03a9\nhx : (\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]) x = 0 x\n\u22a2 0 x = (-\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]) x"}, {"tactic": "simpa only [Pi.zero_apply, Pi.neg_apply, zero_eq_neg]", "annotated_tactic": ["simpa only [<a>Pi.zero_apply</a>, <a>Pi.neg_apply</a>, <a>zero_eq_neg</a>]", [{"full_name": "Pi.zero_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [62, 3], "def_end_pos": [62, 14]}, {"full_name": "Pi.neg_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [166, 3], "def_end_pos": [166, 14]}, {"full_name": "zero_eq_neg", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [640, 3], "def_end_pos": [640, 14]}]], "state_before": "case h\n\u03a9 : Type u_1\nE : Type u_2\n\u03b9 : Type u_3\ninst\u271d\u2074 : Preorder \u03b9\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \u211d E\ninst\u271d\u00b9 : CompleteSpace E\nf\u271d g : \u03b9 \u2192 \u03a9 \u2192 E\n\u2131 : Filtration \u03b9 m0\n\ud835\udca2 : Filtration \u2115 m0\ninst\u271d : IsFiniteMeasure \u03bc\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nhadp : Adapted \ud835\udca2 f\nhint : \u2200 (i : \u2115), Integrable (f i) \u03bc\nhf : \u2200 (i : \u2115), \u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i] =\u1da0[ae \u03bc] 0\ni : \u2115\nx : \u03a9\nhx : (\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]) x = 0 x\n\u22a2 0 x = (-\u03bc[f (i + 1) - f i|\u2191\ud835\udca2 i]) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/String/Basic.lean", "full_name": "String.toList_empty", "start": [131, 1], "end": [132, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/BigOperators/Group/Finset.lean", "full_name": "Finset.mul_card_image_le_card", "start": [304, 1], "end": [306, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "full_name": "ContinuousLinearEquiv.symm_toLinearEquiv", "start": [2054, 1], "end": [2056, 6], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "R\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u00b2\u2074 : Semiring R\u2081\ninst\u271d\u00b2\u00b3 : Semiring R\u2082\ninst\u271d\u00b2\u00b2 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\ninst\u271d\u00b2\u00b9 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2\u2070 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\ninst\u271d\u00b9\u2079 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b9\u2078 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u00b9\u2077 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\ninst\u271d\u00b9\u2076 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ninst\u271d\u00b9\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b9\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\u2081\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\nM'\u2081 : Type u_5\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\u2081\nM\u2082 : Type u_6\ninst\u271d\u2079 : TopologicalSpace M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2082\nM\u2083 : Type u_7\ninst\u271d\u2077 : TopologicalSpace M\u2083\ninst\u271d\u2076 : AddCommMonoid M\u2083\nM\u2084 : Type u_8\ninst\u271d\u2075 : TopologicalSpace M\u2084\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R\u2081 M\u2081\ninst\u271d\u00b2 : Module R\u2081 M'\u2081\ninst\u271d\u00b9 : Module R\u2082 M\u2082\ninst\u271d : Module R\u2083 M\u2083\ne : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082\n\u22a2 e.symm.toLinearEquiv = e.symm", "state_after": "case h\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u00b2\u2074 : Semiring R\u2081\ninst\u271d\u00b2\u00b3 : Semiring R\u2082\ninst\u271d\u00b2\u00b2 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\ninst\u271d\u00b2\u00b9 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2\u2070 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\ninst\u271d\u00b9\u2079 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b9\u2078 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u00b9\u2077 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\ninst\u271d\u00b9\u2076 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ninst\u271d\u00b9\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b9\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\u2081\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\nM'\u2081 : Type u_5\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\u2081\nM\u2082 : Type u_6\ninst\u271d\u2079 : TopologicalSpace M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2082\nM\u2083 : Type u_7\ninst\u271d\u2077 : TopologicalSpace M\u2083\ninst\u271d\u2076 : AddCommMonoid M\u2083\nM\u2084 : Type u_8\ninst\u271d\u2075 : TopologicalSpace M\u2084\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R\u2081 M\u2081\ninst\u271d\u00b2 : Module R\u2081 M'\u2081\ninst\u271d\u00b9 : Module R\u2082 M\u2082\ninst\u271d : Module R\u2083 M\u2083\ne : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082\nx\u271d : M\u2082\n\u22a2 e.symm.toLinearEquiv x\u271d = e.symm x\u271d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h\nR\u2081 : Type u_1\nR\u2082 : Type u_2\nR\u2083 : Type u_3\ninst\u271d\u00b2\u2074 : Semiring R\u2081\ninst\u271d\u00b2\u00b3 : Semiring R\u2082\ninst\u271d\u00b2\u00b2 : Semiring R\u2083\n\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082\n\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081\ninst\u271d\u00b2\u00b9 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081\ninst\u271d\u00b2\u2070 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082\n\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083\n\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082\ninst\u271d\u00b9\u2079 : RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082\ninst\u271d\u00b9\u2078 : RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083\n\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083\n\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081\ninst\u271d\u00b9\u2077 : RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081\ninst\u271d\u00b9\u2076 : RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083\ninst\u271d\u00b9\u2075 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083\ninst\u271d\u00b9\u2074 : RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081\nM\u2081 : Type u_4\ninst\u271d\u00b9\u00b3 : TopologicalSpace M\u2081\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2081\nM'\u2081 : Type u_5\ninst\u271d\u00b9\u00b9 : TopologicalSpace M'\u2081\ninst\u271d\u00b9\u2070 : AddCommMonoid M'\u2081\nM\u2082 : Type u_6\ninst\u271d\u2079 : TopologicalSpace M\u2082\ninst\u271d\u2078 : AddCommMonoid M\u2082\nM\u2083 : Type u_7\ninst\u271d\u2077 : TopologicalSpace M\u2083\ninst\u271d\u2076 : AddCommMonoid M\u2083\nM\u2084 : Type u_8\ninst\u271d\u2075 : TopologicalSpace M\u2084\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R\u2081 M\u2081\ninst\u271d\u00b2 : Module R\u2081 M'\u2081\ninst\u271d\u00b9 : Module R\u2082 M\u2082\ninst\u271d : Module R\u2083 M\u2083\ne : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082\nx\u271d : M\u2082\n\u22a2 e.symm.toLinearEquiv x\u271d = e.symm x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean", "full_name": "Fin.list_succ", "start": [43, 1], "end": [44, 54], "traced_tactics": [{"tactic": "apply List.ext_get", "annotated_tactic": ["apply <a>List.ext_get</a>", [{"full_name": "List.ext_get", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [267, 9], "def_end_pos": [267, 16]}]], "state_before": "n : Nat\n\u22a2 list (n + 1) = 0 :: List.map succ (list n)", "state_after": "case hl\nn : Nat\n\u22a2 (list (n + 1)).length = (0 :: List.map succ (list n)).length\n\ncase h\nn : Nat\n\u22a2 \u2200 (n_1 : Nat) (h\u2081 : n_1 < (list (n + 1)).length) (h\u2082 : n_1 < (0 :: List.map succ (list n)).length),\n    (list (n + 1)).get \u27e8n_1, h\u2081\u27e9 = (0 :: List.map succ (list n)).get \u27e8n_1, h\u2082\u27e9"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case hl\nn : Nat\n\u22a2 (list (n + 1)).length = (0 :: List.map succ (list n)).length\n\ncase h\nn : Nat\n\u22a2 \u2200 (n_1 : Nat) (h\u2081 : n_1 < (list (n + 1)).length) (h\u2082 : n_1 < (0 :: List.map succ (list n)).length),\n    (list (n + 1)).get \u27e8n_1, h\u2081\u27e9 = (0 :: List.map succ (list n)).get \u27e8n_1, h\u2082\u27e9", "state_after": "case h\nn : Nat\n\u22a2 \u2200 (n_1 : Nat) (h\u2081 : n_1 < (list (n + 1)).length) (h\u2082 : n_1 < (0 :: List.map succ (list n)).length),\n    (list (n + 1)).get \u27e8n_1, h\u2081\u27e9 = (0 :: List.map succ (list n)).get \u27e8n_1, h\u2082\u27e9"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "case h\nn : Nat\n\u22a2 \u2200 (n_1 : Nat) (h\u2081 : n_1 < (list (n + 1)).length) (h\u2082 : n_1 < (0 :: List.map succ (list n)).length),\n    (list (n + 1)).get \u27e8n_1, h\u2081\u27e9 = (0 :: List.map succ (list n)).get \u27e8n_1, h\u2082\u27e9", "state_after": "case h\nn i : Nat\n\u22a2 \u2200 (h\u2081 : i < (list (n + 1)).length) (h\u2082 : i < (0 :: List.map succ (list n)).length),\n    (list (n + 1)).get \u27e8i, h\u2081\u27e9 = (0 :: List.map succ (list n)).get \u27e8i, h\u2082\u27e9"}, {"tactic": "cases i <;> simp", "annotated_tactic": ["cases i <;> simp", []], "state_before": "case h\nn i : Nat\n\u22a2 \u2200 (h\u2081 : i < (list (n + 1)).length) (h\u2082 : i < (0 :: List.map succ (list n)).length),\n    (list (n + 1)).get \u27e8i, h\u2081\u27e9 = (0 :: List.map succ (list n)).get \u27e8i, h\u2082\u27e9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/WittVector/Truncated.lean", "full_name": "WittVector.truncateFun_neg", "start": [264, 1], "end": [265, 23], "traced_tactics": [{"tactic": "witt_truncateFun_tac", "annotated_tactic": ["witt_truncateFun_tac", []], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx : \ud835\udd4e R\n\u22a2 truncateFun n (-x) = -truncateFun n x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "full_name": "inner_smul_left", "start": [464, 1], "end": [465, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Valuation/RankOne.lean", "full_name": "Valuation.RankOne.zero_of_hom_zero", "start": [51, 1], "end": [55, 17], "traced_tactics": [{"tactic": "refine (eq_of_le_of_not_lt (zero_le' (a := x)) fun h_lt \u21a6 ?_).symm", "annotated_tactic": ["refine (<a>eq_of_le_of_not_lt</a> (<a>zero_le'</a> (a := x)) fun h_lt \u21a6 ?_).<a>symm</a>", [{"full_name": "eq_of_le_of_not_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [404, 9], "def_end_pos": [404, 27]}, {"full_name": "zero_le'", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Canonical.lean", "def_pos": [78, 15], "def_end_pos": [78, 23]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : Ring R\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\ninst\u271d : v.RankOne\nx : \u0393\u2080\nhx : (hom v) x = 0\n\u22a2 x = 0", "state_after": "R : Type u_1\ninst\u271d\u00b2 : Ring R\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\ninst\u271d : v.RankOne\nx : \u0393\u2080\nhx : (hom v) x = 0\nh_lt : 0 < x\n\u22a2 False"}, {"tactic": "have hs := strictMono v h_lt", "annotated_tactic": ["have hs := <a>strictMono</a> v h_lt", [{"full_name": "Valuation.RankOne.strictMono", "def_path": "Mathlib/RingTheory/Valuation/RankOne.lean", "def_pos": [45, 7], "def_end_pos": [45, 17]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : Ring R\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\ninst\u271d : v.RankOne\nx : \u0393\u2080\nhx : (hom v) x = 0\nh_lt : 0 < x\n\u22a2 False", "state_after": "R : Type u_1\ninst\u271d\u00b2 : Ring R\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\ninst\u271d : v.RankOne\nx : \u0393\u2080\nhx : (hom v) x = 0\nh_lt : 0 < x\nhs : (hom v) 0 < (hom v) x\n\u22a2 False"}, {"tactic": "rw [_root_.map_zero, hx] at hs", "annotated_tactic": ["rw [<a>_root_.map_zero</a>, hx] at hs", [{"full_name": "map_zero", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [203, 3], "def_end_pos": [203, 14]}]], "state_before": "R : Type u_1\ninst\u271d\u00b2 : Ring R\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\ninst\u271d : v.RankOne\nx : \u0393\u2080\nhx : (hom v) x = 0\nh_lt : 0 < x\nhs : (hom v) 0 < (hom v) x\n\u22a2 False", "state_after": "R : Type u_1\ninst\u271d\u00b2 : Ring R\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\ninst\u271d : v.RankOne\nx : \u0393\u2080\nhx : (hom v) x = 0\nh_lt : 0 < x\nhs : 0 < 0\n\u22a2 False"}, {"tactic": "exact hs.false", "annotated_tactic": ["exact hs.false", []], "state_before": "R : Type u_1\ninst\u271d\u00b2 : Ring R\n\u0393\u2080 : Type u_2\ninst\u271d\u00b9 : LinearOrderedCommGroupWithZero \u0393\u2080\nv : Valuation R \u0393\u2080\ninst\u271d : v.RankOne\nx : \u0393\u2080\nhx : (hom v) x = 0\nh_lt : 0 < x\nhs : 0 < 0\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Lemmas.lean", "full_name": "List.foldl_range_subset_of_range_subset", "start": [55, 1], "end": [66, 49], "traced_tactics": [{"tactic": "change (Set.range fun l => _) \u2286 Set.range fun l => _", "annotated_tactic": ["change (<a>Set.range</a> fun l => _) \u2286 <a>Set.range</a> fun l => _", [{"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [157, 5], "def_end_pos": [157, 10]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [157, 5], "def_end_pos": [157, 10]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b1\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b1\nhfg : (Set.range fun a c => f c a) \u2286 Set.range fun b c => g c b\na : \u03b1\n\u22a2 Set.range (foldl f a) \u2286 Set.range (foldl g a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b1\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b1\nhfg : (Set.range fun a c => f c a) \u2286 Set.range fun b c => g c b\na : \u03b1\n\u22a2 (Set.range fun l => foldl f a l) \u2286 Set.range fun l => foldl g a l"}, {"tactic": "simp_rw [\u2190 foldr_reverse _ (fun z w => g w z), \u2190 foldr_reverse _ (fun z w => f w z)]", "annotated_tactic": ["simp_rw [\u2190 <a>foldr_reverse</a> _ (fun z w => g w z), \u2190 <a>foldr_reverse</a> _ (fun z w => f w z)]", [{"full_name": "List.foldr_reverse", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [1508, 17], "def_end_pos": [1508, 30]}, {"full_name": "List.foldr_reverse", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [1508, 17], "def_end_pos": [1508, 30]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b1\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b1\nhfg : (Set.range fun a c => f c a) \u2286 Set.range fun b c => g c b\na : \u03b1\n\u22a2 (Set.range fun l => foldl f a l) \u2286 Set.range fun l => foldl g a l", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b1\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b1\nhfg : (Set.range fun a c => f c a) \u2286 Set.range fun b c => g c b\na : \u03b1\n\u22a2 (Set.range fun l => foldr (fun z w => f w z) a l.reverse) \u2286 Set.range fun l => foldr (fun z w => g w z) a l.reverse"}, {"tactic": "change (Set.range (foldr (fun z w => f w z) a \u2218 reverse)) \u2286\n  Set.range (foldr (fun z w => g w z) a \u2218 reverse)", "annotated_tactic": ["change (<a>Set.range</a> (<a>foldr</a> (fun z w => f w z) a \u2218 <a>reverse</a>)) \u2286\n    <a>Set.range</a> (<a>foldr</a> (fun z w => g w z) a \u2218 <a>reverse</a>)", [{"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [157, 5], "def_end_pos": [157, 10]}, {"full_name": "List.foldr", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [416, 19], "def_end_pos": [416, 24]}, {"full_name": "List.reverse", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [441, 5], "def_end_pos": [441, 12]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [157, 5], "def_end_pos": [157, 10]}, {"full_name": "List.foldr", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [416, 19], "def_end_pos": [416, 24]}, {"full_name": "List.reverse", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [441, 5], "def_end_pos": [441, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b1\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b1\nhfg : (Set.range fun a c => f c a) \u2286 Set.range fun b c => g c b\na : \u03b1\n\u22a2 (Set.range fun l => foldr (fun z w => f w z) a l.reverse) \u2286 Set.range fun l => foldr (fun z w => g w z) a l.reverse", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b1\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b1\nhfg : (Set.range fun a c => f c a) \u2286 Set.range fun b c => g c b\na : \u03b1\n\u22a2 Set.range (foldr (fun z w => f w z) a \u2218 reverse) \u2286 Set.range (foldr (fun z w => g w z) a \u2218 reverse)"}, {"tactic": "simp_rw [Set.range_comp _ reverse, reverse_involutive.bijective.surjective.range_eq,\n  Set.image_univ]", "annotated_tactic": ["simp_rw [<a>Set.range_comp</a> _ <a>reverse</a>, reverse_involutive.bijective.surjective.range_eq,\n    <a>Set.image_univ</a>]", [{"full_name": "Set.range_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [731, 9], "def_end_pos": [731, 19]}, {"full_name": "List.reverse", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [441, 5], "def_end_pos": [441, 12]}, {"full_name": "Set.image_univ", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [701, 9], "def_end_pos": [701, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b1\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b1\nhfg : (Set.range fun a c => f c a) \u2286 Set.range fun b c => g c b\na : \u03b1\n\u22a2 Set.range (foldr (fun z w => f w z) a \u2218 reverse) \u2286 Set.range (foldr (fun z w => g w z) a \u2218 reverse)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b1\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b1\nhfg : (Set.range fun a c => f c a) \u2286 Set.range fun b c => g c b\na : \u03b1\n\u22a2 Set.range (foldr (fun z w => f w z) a) \u2286 Set.range (foldr (fun z w => g w z) a)"}, {"tactic": "exact foldr_range_subset_of_range_subset hfg a", "annotated_tactic": ["exact <a>foldr_range_subset_of_range_subset</a> hfg a", [{"full_name": "List.foldr_range_subset_of_range_subset", "def_path": "Mathlib/Data/List/Lemmas.lean", "def_pos": [44, 9], "def_end_pos": [44, 43]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b1\ng : \u03b1 \u2192 \u03b3 \u2192 \u03b1\nhfg : (Set.range fun a c => f c a) \u2286 Set.range fun b c => g c b\na : \u03b1\n\u22a2 Set.range (foldr (fun z w => f w z) a) \u2286 Set.range (foldr (fun z w => g w z) a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Smeval.lean", "full_name": "Polynomial.smeval_mul_X", "start": [202, 1], "end": [208, 53], "traced_tactics": [{"tactic": "induction p using Polynomial.induction_on' with\n| h_add p q ph qh =>\nsimp only [add_mul, smeval_add, ph, qh]\n| h_monomial n a =>\nsimp only [\u2190 monomial_one_one_eq_X, monomial_mul_monomial, smeval_monomial, mul_one, pow_succ',\n  mul_assoc, npow_add, smul_mul_assoc, npow_one]", "annotated_tactic": ["induction p using <a>Polynomial.induction_on'</a> with\n  | h_add p q ph qh =>\n    simp only [<a>add_mul</a>, <a>smeval_add</a>, ph, qh]\n  | h_monomial n a =>\n    simp only [\u2190 <a>monomial_one_one_eq_X</a>, <a>monomial_mul_monomial</a>, <a>smeval_monomial</a>, <a>mul_one</a>, <a>pow_succ'</a>,\n      <a>mul_assoc</a>, <a>npow_add</a>, <a>smul_mul_assoc</a>, <a>npow_one</a>]", [{"full_name": "Polynomial.induction_on'", "def_path": "Mathlib/Algebra/Polynomial/Induction.lean", "def_pos": [63, 19], "def_end_pos": [63, 32]}, {"full_name": "add_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [91, 7], "def_end_pos": [91, 14]}, {"full_name": "Polynomial.smeval_add", "def_path": "Mathlib/Algebra/Polynomial/Smeval.lean", "def_pos": [105, 9], "def_end_pos": [105, 19]}, {"full_name": "Polynomial.monomial_one_one_eq_X", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [568, 9], "def_end_pos": [568, 30]}, {"full_name": "Polynomial.monomial_mul_monomial", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [467, 9], "def_end_pos": [467, 30]}, {"full_name": "Polynomial.smeval_monomial", "def_path": "Mathlib/Algebra/Polynomial/Smeval.lean", "def_pos": [61, 9], "def_end_pos": [61, 24]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [667, 34], "def_end_pos": [667, 43]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "npow_add", "def_path": "Mathlib/Algebra/Group/NatPowAssoc.lean", "def_pos": [54, 9], "def_end_pos": [54, 17]}, {"full_name": "smul_mul_assoc", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [408, 7], "def_end_pos": [408, 21]}, {"full_name": "npow_one", "def_path": "Mathlib/Algebra/Group/NatPowAssoc.lean", "def_pos": [62, 9], "def_end_pos": [62, 17]}]], "state_before": "R : Type u_1\ninst\u271d\u2076 : Semiring R\np\u271d : R[X]\nr : R\np q : R[X]\nS : Type u_2\ninst\u271d\u2075 : NonAssocSemiring S\ninst\u271d\u2074 : Module R S\ninst\u271d\u00b3 : IsScalarTower R S S\ninst\u271d\u00b2 : SMulCommClass R S S\ninst\u271d\u00b9 : Pow S \u2115\ninst\u271d : NatPowAssoc S\nx : S\n\u22a2 (p * X).smeval x = p.smeval x * x", "state_after": "no goals"}, {"tactic": "simp only [add_mul, smeval_add, ph, qh]", "annotated_tactic": ["simp only [<a>add_mul</a>, <a>smeval_add</a>, ph, qh]", [{"full_name": "add_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [91, 7], "def_end_pos": [91, 14]}, {"full_name": "Polynomial.smeval_add", "def_path": "Mathlib/Algebra/Polynomial/Smeval.lean", "def_pos": [105, 9], "def_end_pos": [105, 19]}]], "state_before": "case h_add\nR : Type u_1\ninst\u271d\u2076 : Semiring R\np\u271d\u00b9 : R[X]\nr : R\np\u271d q\u271d : R[X]\nS : Type u_2\ninst\u271d\u2075 : NonAssocSemiring S\ninst\u271d\u2074 : Module R S\ninst\u271d\u00b3 : IsScalarTower R S S\ninst\u271d\u00b2 : SMulCommClass R S S\ninst\u271d\u00b9 : Pow S \u2115\ninst\u271d : NatPowAssoc S\nx : S\np q : R[X]\nph : (p * X).smeval x = p.smeval x * x\nqh : (q * X).smeval x = q.smeval x * x\n\u22a2 ((p + q) * X).smeval x = (p + q).smeval x * x", "state_after": "no goals"}, {"tactic": "simp only [\u2190 monomial_one_one_eq_X, monomial_mul_monomial, smeval_monomial, mul_one, pow_succ',\n  mul_assoc, npow_add, smul_mul_assoc, npow_one]", "annotated_tactic": ["simp only [\u2190 <a>monomial_one_one_eq_X</a>, <a>monomial_mul_monomial</a>, <a>smeval_monomial</a>, <a>mul_one</a>, <a>pow_succ'</a>,\n      <a>mul_assoc</a>, <a>npow_add</a>, <a>smul_mul_assoc</a>, <a>npow_one</a>]", [{"full_name": "Polynomial.monomial_one_one_eq_X", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [568, 9], "def_end_pos": [568, 30]}, {"full_name": "Polynomial.monomial_mul_monomial", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [467, 9], "def_end_pos": [467, 30]}, {"full_name": "Polynomial.smeval_monomial", "def_path": "Mathlib/Algebra/Polynomial/Smeval.lean", "def_pos": [61, 9], "def_end_pos": [61, 24]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [667, 34], "def_end_pos": [667, 43]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "npow_add", "def_path": "Mathlib/Algebra/Group/NatPowAssoc.lean", "def_pos": [54, 9], "def_end_pos": [54, 17]}, {"full_name": "smul_mul_assoc", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [408, 7], "def_end_pos": [408, 21]}, {"full_name": "npow_one", "def_path": "Mathlib/Algebra/Group/NatPowAssoc.lean", "def_pos": [62, 9], "def_end_pos": [62, 17]}]], "state_before": "case h_monomial\nR : Type u_1\ninst\u271d\u2076 : Semiring R\np\u271d : R[X]\nr : R\np q : R[X]\nS : Type u_2\ninst\u271d\u2075 : NonAssocSemiring S\ninst\u271d\u2074 : Module R S\ninst\u271d\u00b3 : IsScalarTower R S S\ninst\u271d\u00b2 : SMulCommClass R S S\ninst\u271d\u00b9 : Pow S \u2115\ninst\u271d : NatPowAssoc S\nx : S\nn : \u2115\na : R\n\u22a2 ((monomial n) a * X).smeval x = ((monomial n) a).smeval x * x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/Cycle/Basic.lean", "full_name": "Equiv.Perm.sameCycle_pow_right", "start": [162, 1], "end": [163, 44], "traced_tactics": [{"tactic": "rw [\u2190 zpow_natCast, sameCycle_zpow_right]", "annotated_tactic": ["rw [\u2190 <a>zpow_natCast</a>, <a>sameCycle_zpow_right</a>]", [{"full_name": "zpow_natCast", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1021, 9], "def_end_pos": [1021, 21]}, {"full_name": "Equiv.Perm.sameCycle_zpow_right", "def_path": "Mathlib/GroupTheory/Perm/Cycle/Basic.lean", "def_pos": [152, 9], "def_end_pos": [152, 29]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nf g : Perm \u03b1\np : \u03b1 \u2192 Prop\nx y z : \u03b1\nn : \u2115\n\u22a2 f.SameCycle x ((f ^ n) y) \u2194 f.SameCycle x y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "full_name": "InnerProductSpace.Core.inner_smul_left", "start": [236, 1], "end": [237, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Opposite.lean", "full_name": "MulOpposite.commute_op", "start": [304, 1], "end": [305, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Regular.lean", "full_name": "MeasurableSet.exists_lt_isCompact_of_ne_top", "start": [742, 1], "end": [745, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Haar/Unique.lean", "full_name": "MeasureTheory.Measure.integral_isMulLeftInvariant_isMulRightInvariant_combo", "start": [143, 1], "end": [236, 69], "traced_tactics": [{"tactic": "rcases h'f.eq_zero_or_locallyCompactSpace_of_group hf with Hf|Hf", "annotated_tactic": ["rcases h'f.eq_zero_or_locallyCompactSpace_of_group hf with Hf|Hf", []], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\n\u22a2 \u222b (x : G), f x \u2202\u03bc = (\u222b (y : G), f y * (\u222b (z : G), g (z\u207b\u00b9 * y) \u2202\u03bd)\u207b\u00b9 \u2202\u03bd) * \u222b (x : G), g x \u2202\u03bc", "state_after": "case inl\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : f = 0\n\u22a2 \u222b (x : G), f x \u2202\u03bc = (\u222b (y : G), f y * (\u222b (z : G), g (z\u207b\u00b9 * y) \u2202\u03bd)\u207b\u00b9 \u2202\u03bd) * \u222b (x : G), g x \u2202\u03bc\n\ncase inr\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\n\u22a2 \u222b (x : G), f x \u2202\u03bc = (\u222b (y : G), f y * (\u222b (z : G), g (z\u207b\u00b9 * y) \u2202\u03bd)\u207b\u00b9 \u2202\u03bd) * \u222b (x : G), g x \u2202\u03bc"}, {"tactic": "let D : G \u2192 \u211d := fun (x : G) \u21a6 \u222b y, g (y\u207b\u00b9 * x) \u2202\u03bd", "annotated_tactic": ["let D : G \u2192 \u211d := fun (x : G) \u21a6 \u222b y, g (y\u207b\u00b9 * x) \u2202\u03bd", []], "state_before": "case inr\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\n\u22a2 \u222b (x : G), f x \u2202\u03bc = (\u222b (y : G), f y * (\u222b (z : G), g (z\u207b\u00b9 * y) \u2202\u03bd)\u207b\u00b9 \u2202\u03bd) * \u222b (x : G), g x \u2202\u03bc", "state_after": "case inr\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\n\u22a2 \u222b (x : G), f x \u2202\u03bc = (\u222b (y : G), f y * (\u222b (z : G), g (z\u207b\u00b9 * y) \u2202\u03bd)\u207b\u00b9 \u2202\u03bd) * \u222b (x : G), g x \u2202\u03bc"}, {"tactic": "have D_cont : Continuous D := continuous_integral_apply_inv_mul hg h'g", "annotated_tactic": ["have D_cont : <a>Continuous</a> D := <a>continuous_integral_apply_inv_mul</a> hg h'g", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Defs/Basic.lean", "def_pos": [137, 11], "def_end_pos": [137, 21]}, {"full_name": "MeasureTheory.continuous_integral_apply_inv_mul", "def_path": "Mathlib/MeasureTheory/Measure/Haar/Unique.lean", "def_pos": [102, 7], "def_end_pos": [102, 40]}]], "state_before": "case inr\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\n\u22a2 \u222b (x : G), f x \u2202\u03bc = (\u222b (y : G), f y * (\u222b (z : G), g (z\u207b\u00b9 * y) \u2202\u03bd)\u207b\u00b9 \u2202\u03bd) * \u222b (x : G), g x \u2202\u03bc", "state_after": "case inr\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\n\u22a2 \u222b (x : G), f x \u2202\u03bc = (\u222b (y : G), f y * (\u222b (z : G), g (z\u207b\u00b9 * y) \u2202\u03bd)\u207b\u00b9 \u2202\u03bd) * \u222b (x : G), g x \u2202\u03bc"}, {"tactic": "simp [Hf]", "annotated_tactic": ["simp [Hf]", []], "state_before": "case inl\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : f = 0\n\u22a2 \u222b (x : G), f x \u2202\u03bc = (\u222b (y : G), f y * (\u222b (z : G), g (z\u207b\u00b9 * y) \u2202\u03bd)\u207b\u00b9 \u2202\u03bd) * \u222b (x : G), g x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "intro x", "annotated_tactic": ["intro x", []], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\n\u22a2 \u2200 (x : G), 0 < D x", "state_after": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nx : G\n\u22a2 0 < D x"}, {"tactic": "have C : Continuous (fun y \u21a6 g (y\u207b\u00b9 * x)) := hg.comp (continuous_inv.mul continuous_const)", "annotated_tactic": ["have C : <a>Continuous</a> (fun y \u21a6 g (y\u207b\u00b9 * x)) := hg.comp (continuous_inv.mul <a>continuous_const</a>)", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Defs/Basic.lean", "def_pos": [137, 11], "def_end_pos": [137, 21]}, {"full_name": "continuous_const", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1675, 9], "def_end_pos": [1675, 25]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nx : G\n\u22a2 0 < D x", "state_after": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nx : G\nC : Continuous fun y => g (y\u207b\u00b9 * x)\n\u22a2 0 < D x"}, {"tactic": "apply (integral_pos_iff_support_of_nonneg _ _).2", "annotated_tactic": ["apply (<a>integral_pos_iff_support_of_nonneg</a> _ _).2", [{"full_name": "MeasureTheory.integral_pos_iff_support_of_nonneg", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [1277, 9], "def_end_pos": [1277, 43]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nx : G\nC : Continuous fun y => g (y\u207b\u00b9 * x)\n\u22a2 0 < D x", "state_after": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nx : G\nC : Continuous fun y => g (y\u207b\u00b9 * x)\n\u22a2 0 < \u03bd (support fun x_1 => g (x_1\u207b\u00b9 * x))\n\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nx : G\nC : Continuous fun y => g (y\u207b\u00b9 * x)\n\u22a2 0 \u2264 fun x_1 => g (x_1\u207b\u00b9 * x)\n\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nx : G\nC : Continuous fun y => g (y\u207b\u00b9 * x)\n\u22a2 Integrable (fun x_1 => g (x_1\u207b\u00b9 * x)) \u03bd"}, {"tactic": "apply C.isOpen_support.measure_pos \u03bd", "annotated_tactic": ["apply C.isOpen_support.measure_pos \u03bd", []], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nx : G\nC : Continuous fun y => g (y\u207b\u00b9 * x)\n\u22a2 0 < \u03bd (support fun x_1 => g (x_1\u207b\u00b9 * x))", "state_after": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nx : G\nC : Continuous fun y => g (y\u207b\u00b9 * x)\n\u22a2 (support fun y => g (y\u207b\u00b9 * x)).Nonempty"}, {"tactic": "exact \u27e8x * x\u2080\u207b\u00b9, by simpa using g_pos\u27e9", "annotated_tactic": ["exact \u27e8x * x\u2080\u207b\u00b9, by simpa using g_pos\u27e9", []], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nx : G\nC : Continuous fun y => g (y\u207b\u00b9 * x)\n\u22a2 (support fun y => g (y\u207b\u00b9 * x)).Nonempty", "state_after": "no goals"}, {"tactic": "simpa using g_pos", "annotated_tactic": ["simpa using g_pos", []], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nx : G\nC : Continuous fun y => g (y\u207b\u00b9 * x)\n\u22a2 x * x\u2080\u207b\u00b9 \u2208 support fun y => g (y\u207b\u00b9 * x)", "state_after": "no goals"}, {"tactic": "exact fun y \u21a6 g_nonneg (y\u207b\u00b9 * x)", "annotated_tactic": ["exact fun y \u21a6 g_nonneg (y\u207b\u00b9 * x)", []], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nx : G\nC : Continuous fun y => g (y\u207b\u00b9 * x)\n\u22a2 0 \u2264 fun x_1 => g (x_1\u207b\u00b9 * x)", "state_after": "no goals"}, {"tactic": "apply C.integrable_of_hasCompactSupport", "annotated_tactic": ["apply C.integrable_of_hasCompactSupport", []], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nx : G\nC : Continuous fun y => g (y\u207b\u00b9 * x)\n\u22a2 Integrable (fun x_1 => g (x_1\u207b\u00b9 * x)) \u03bd", "state_after": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nx : G\nC : Continuous fun y => g (y\u207b\u00b9 * x)\n\u22a2 HasCompactSupport fun y => g (y\u207b\u00b9 * x)"}, {"tactic": "exact h'g.comp_homeomorph ((Homeomorph.inv G).trans (Homeomorph.mulRight x))", "annotated_tactic": ["exact h'g.comp_homeomorph ((<a>Homeomorph.inv</a> G).<a>trans</a> (<a>Homeomorph.mulRight</a> x))", [{"full_name": "Homeomorph.inv", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [354, 15], "def_end_pos": [354, 29]}, {"full_name": "Homeomorph.trans", "def_path": "Mathlib/Topology/Homeomorph.lean", "def_pos": [123, 15], "def_end_pos": [123, 20]}, {"full_name": "Homeomorph.mulRight", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [101, 15], "def_end_pos": [101, 34]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nx : G\nC : Continuous fun y => g (y\u207b\u00b9 * x)\n\u22a2 HasCompactSupport fun y => g (y\u207b\u00b9 * x)", "state_after": "no goals"}, {"tactic": "congr with x", "annotated_tactic": ["congr with x", []], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 \u222b (x : G), f x \u2202\u03bc = \u222b (x : G), f x * (D x)\u207b\u00b9 * D x \u2202\u03bc", "state_after": "case e_f.h\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nx : G\n\u22a2 f x = f x * (D x)\u207b\u00b9 * D x"}, {"tactic": "rw [mul_assoc, inv_mul_cancel (D_pos x).ne', mul_one]", "annotated_tactic": ["rw [<a>mul_assoc</a>, <a>inv_mul_cancel</a> (D_pos x).<a>ne'</a>, <a>mul_one</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "inv_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/NeZero.lean", "def_pos": [55, 9], "def_end_pos": [55, 23]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "case e_f.h\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nx : G\n\u22a2 f x = f x * (D x)\u207b\u00b9 * D x", "state_after": "no goals"}, {"tactic": "simp_rw [integral_mul_left]", "annotated_tactic": ["simp_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": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 \u222b (x : G), f x * (D x)\u207b\u00b9 * D x \u2202\u03bc = \u222b (x : G), \u222b (y : G), f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x) \u2202\u03bd \u2202\u03bc", "state_after": "no goals"}, {"tactic": "apply integral_integral_swap_of_hasCompactSupport", "annotated_tactic": ["apply <a>integral_integral_swap_of_hasCompactSupport</a>", [{"full_name": "MeasureTheory.integral_integral_swap_of_hasCompactSupport", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "def_pos": [553, 7], "def_end_pos": [553, 50]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 \u222b (x : G), \u222b (y : G), f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x) \u2202\u03bd \u2202\u03bc = \u222b (y : G), \u222b (x : G), f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x) \u2202\u03bc \u2202\u03bd", "state_after": "case hf\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 Continuous (uncurry fun x y => f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x))\n\ncase h'f\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 HasCompactSupport (uncurry fun x y => f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x))"}, {"tactic": "apply Continuous.mul", "annotated_tactic": ["apply <a>Continuous.mul</a>", [{"full_name": "Continuous.mul", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [97, 9], "def_end_pos": [97, 23]}]], "state_before": "case hf\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 Continuous (uncurry fun x y => f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x))", "state_after": "case hf.hf\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 Continuous fun x => f x.1 * (D x.1)\u207b\u00b9\n\ncase hf.hg\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 Continuous fun x => g (x.2\u207b\u00b9 * x.1)"}, {"tactic": "exact (hf.comp continuous_fst).mul\n  ((D_cont.comp continuous_fst).inv\u2080 (fun x \u21a6 (D_pos _).ne'))", "annotated_tactic": ["exact (hf.comp <a>continuous_fst</a>).<a>mul</a>\n            ((D_cont.comp <a>continuous_fst</a>).<a>inv\u2080</a> (fun x \u21a6 (D_pos _).<a>ne'</a>))", [{"full_name": "continuous_fst", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [339, 9], "def_end_pos": [339, 23]}, {"full_name": "Continuous.mul", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [97, 9], "def_end_pos": [97, 23]}, {"full_name": "continuous_fst", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [339, 9], "def_end_pos": [339, 23]}, {"full_name": "Continuous.inv\u2080", "def_path": "Mathlib/Topology/Algebra/GroupWithZero.lean", "def_pos": [133, 9], "def_end_pos": [133, 24]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}]], "state_before": "case hf.hf\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 Continuous fun x => f x.1 * (D x.1)\u207b\u00b9", "state_after": "no goals"}, {"tactic": "exact hg.comp (continuous_snd.inv.mul continuous_fst)", "annotated_tactic": ["exact hg.comp (continuous_snd.inv.mul <a>continuous_fst</a>)", [{"full_name": "continuous_fst", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [339, 9], "def_end_pos": [339, 23]}]], "state_before": "case hf.hg\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 Continuous fun x => g (x.2\u207b\u00b9 * x.1)", "state_after": "no goals"}, {"tactic": "let K := tsupport f", "annotated_tactic": ["let K := <a>tsupport</a> f", [{"full_name": "tsupport", "def_path": "Mathlib/Topology/Support.lean", "def_pos": [44, 3], "def_end_pos": [44, 14]}]], "state_before": "case h'f\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 HasCompactSupport (uncurry fun x y => f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x))", "state_after": "case h'f\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\n\u22a2 HasCompactSupport (uncurry fun x y => f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x))"}, {"tactic": "have K_comp : IsCompact K := h'f", "annotated_tactic": ["have K_comp : <a>IsCompact</a> K := h'f", [{"full_name": "IsCompact", "def_path": "Mathlib/Topology/Defs/Filter.lean", "def_pos": [273, 5], "def_end_pos": [273, 14]}]], "state_before": "case h'f\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\n\u22a2 HasCompactSupport (uncurry fun x y => f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x))", "state_after": "case h'f\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\n\u22a2 HasCompactSupport (uncurry fun x y => f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x))"}, {"tactic": "let L := tsupport g", "annotated_tactic": ["let L := <a>tsupport</a> g", [{"full_name": "tsupport", "def_path": "Mathlib/Topology/Support.lean", "def_pos": [44, 3], "def_end_pos": [44, 14]}]], "state_before": "case h'f\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\n\u22a2 HasCompactSupport (uncurry fun x y => f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x))", "state_after": "case h'f\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\n\u22a2 HasCompactSupport (uncurry fun x y => f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x))"}, {"tactic": "have L_comp : IsCompact L := h'g", "annotated_tactic": ["have L_comp : <a>IsCompact</a> L := h'g", [{"full_name": "IsCompact", "def_path": "Mathlib/Topology/Defs/Filter.lean", "def_pos": [273, 5], "def_end_pos": [273, 14]}]], "state_before": "case h'f\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\n\u22a2 HasCompactSupport (uncurry fun x y => f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x))", "state_after": "case h'f\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\n\u22a2 HasCompactSupport (uncurry fun x y => f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x))"}, {"tactic": "let M := (fun (p : G \u00d7 G) \u21a6 p.1 * p.2\u207b\u00b9) '' (K \u00d7\u02e2 L)", "annotated_tactic": ["let M := (fun (p : G \u00d7 G) \u21a6 p.1 * p.2\u207b\u00b9) '' (K \u00d7\u02e2 L)", []], "state_before": "case h'f\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\n\u22a2 HasCompactSupport (uncurry fun x y => f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x))", "state_after": "case h'f\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\n\u22a2 HasCompactSupport (uncurry fun x y => f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x))"}, {"tactic": "have M_comp : IsCompact M :=\n  (K_comp.prod L_comp).image (continuous_fst.mul continuous_snd.inv)", "annotated_tactic": ["have M_comp : <a>IsCompact</a> M :=\n          (K_comp.prod L_comp).<a>image</a> (continuous_fst.mul continuous_snd.inv)", [{"full_name": "IsCompact", "def_path": "Mathlib/Topology/Defs/Filter.lean", "def_pos": [273, 5], "def_end_pos": [273, 14]}, {"full_name": "IsCompact.image", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [119, 9], "def_end_pos": [119, 24]}]], "state_before": "case h'f\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\n\u22a2 HasCompactSupport (uncurry fun x y => f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x))", "state_after": "case h'f\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\n\u22a2 HasCompactSupport (uncurry fun x y => f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x))"}, {"tactic": "have M'_comp : IsCompact (closure M) := M_comp.closure", "annotated_tactic": ["have M'_comp : <a>IsCompact</a> (<a>closure</a> M) := M_comp.closure", [{"full_name": "IsCompact", "def_path": "Mathlib/Topology/Defs/Filter.lean", "def_pos": [273, 5], "def_end_pos": [273, 14]}, {"full_name": "closure", "def_path": "Mathlib/Topology/Defs/Basic.lean", "def_pos": [116, 5], "def_end_pos": [116, 12]}]], "state_before": "case h'f\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\n\u22a2 HasCompactSupport (uncurry fun x y => f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x))", "state_after": "case h'f\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\n\u22a2 HasCompactSupport (uncurry fun x y => f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x))"}, {"tactic": "apply HasCompactSupport.intro' (K_comp.prod M'_comp) ?_ this", "annotated_tactic": ["apply <a>HasCompactSupport.intro'</a> (K_comp.prod M'_comp) ?_ this", [{"full_name": "HasCompactSupport.intro'", "def_path": "Mathlib/Topology/Support.lean", "def_pos": [179, 3], "def_end_pos": [179, 14]}]], "state_before": "case h'f\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nthis : \u2200 p \u2209 K \u00d7\u02e2 closure M, f p.1 * (D p.1)\u207b\u00b9 * g (p.2\u207b\u00b9 * p.1) = 0\n\u22a2 HasCompactSupport (uncurry fun x y => f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x))", "state_after": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nthis : \u2200 p \u2209 K \u00d7\u02e2 closure M, f p.1 * (D p.1)\u207b\u00b9 * g (p.2\u207b\u00b9 * p.1) = 0\n\u22a2 IsClosed (K \u00d7\u02e2 closure M)"}, {"tactic": "exact (isClosed_tsupport f).prod isClosed_closure", "annotated_tactic": ["exact (<a>isClosed_tsupport</a> f).<a>prod</a> <a>isClosed_closure</a>", [{"full_name": "isClosed_tsupport", "def_path": "Mathlib/Topology/Support.lean", "def_pos": [56, 3], "def_end_pos": [56, 14]}, {"full_name": "IsClosed.prod", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [858, 9], "def_end_pos": [858, 22]}, {"full_name": "isClosed_closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [399, 9], "def_end_pos": [399, 25]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nthis : \u2200 p \u2209 K \u00d7\u02e2 closure M, f p.1 * (D p.1)\u207b\u00b9 * g (p.2\u207b\u00b9 * p.1) = 0\n\u22a2 IsClosed (K \u00d7\u02e2 closure M)", "state_after": "no goals"}, {"tactic": "rintro \u27e8x, y\u27e9 hxy", "annotated_tactic": ["rintro \u27e8x, y\u27e9 hxy", []], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\n\u22a2 \u2200 p \u2209 K \u00d7\u02e2 closure M, f p.1 * (D p.1)\u207b\u00b9 * g (p.2\u207b\u00b9 * p.1) = 0", "state_after": "case mk\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 K \u00d7\u02e2 closure M\n\u22a2 f (x, y).1 * (D (x, y).1)\u207b\u00b9 * g ((x, y).2\u207b\u00b9 * (x, y).1) = 0"}, {"tactic": "by_cases H : x \u2208 K", "annotated_tactic": ["by_cases H : x \u2208 K", []], "state_before": "case mk\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 K \u00d7\u02e2 closure M\n\u22a2 f (x, y).1 * (D (x, y).1)\u207b\u00b9 * g ((x, y).2\u207b\u00b9 * (x, y).1) = 0", "state_after": "case pos\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 K \u00d7\u02e2 closure M\nH : x \u2208 K\n\u22a2 f (x, y).1 * (D (x, y).1)\u207b\u00b9 * g ((x, y).2\u207b\u00b9 * (x, y).1) = 0\n\ncase neg\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 K \u00d7\u02e2 closure M\nH : x \u2209 K\n\u22a2 f (x, y).1 * (D (x, y).1)\u207b\u00b9 * g ((x, y).2\u207b\u00b9 * (x, y).1) = 0"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 K \u00d7\u02e2 closure M\nH : x \u2208 K\n\u22a2 f (x, y).1 * (D (x, y).1)\u207b\u00b9 * g ((x, y).2\u207b\u00b9 * (x, y).1) = 0\n\ncase neg\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 K \u00d7\u02e2 closure M\nH : x \u2209 K\n\u22a2 f (x, y).1 * (D (x, y).1)\u207b\u00b9 * g ((x, y).2\u207b\u00b9 * (x, y).1) = 0", "state_after": "case neg\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 K \u00d7\u02e2 closure M\nH : x \u2209 K\n\u22a2 f (x, y).1 * (D (x, y).1)\u207b\u00b9 * g ((x, y).2\u207b\u00b9 * (x, y).1) = 0\n\ncase pos\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 K \u00d7\u02e2 closure M\nH : x \u2208 K\n\u22a2 f (x, y).1 * (D (x, y).1)\u207b\u00b9 * g ((x, y).2\u207b\u00b9 * (x, y).1) = 0"}, {"tactic": "have : g (y\u207b\u00b9 * x) = 0 := by\n  apply image_eq_zero_of_nmem_tsupport\n  contrapose! hxy\n  simp only [mem_prod, H, true_and]\n  apply subset_closure\n  simp only [M, mem_image, mem_prod, Prod.exists]\n  exact \u27e8x, y\u207b\u00b9 * x, \u27e8H, hxy\u27e9, by group\u27e9", "annotated_tactic": ["have : g (y\u207b\u00b9 * x) = 0 := by\n            apply <a>image_eq_zero_of_nmem_tsupport</a>\n            contrapose! hxy\n            simp only [<a>mem_prod</a>, H, <a>true_and</a>]\n            apply <a>subset_closure</a>\n            simp only [M, <a>mem_image</a>, <a>mem_prod</a>, <a>Prod.exists</a>]\n            exact \u27e8x, y\u207b\u00b9 * x, \u27e8H, hxy\u27e9, by group\u27e9", [{"full_name": "image_eq_zero_of_nmem_tsupport", "def_path": "Mathlib/Topology/Support.lean", "def_pos": [68, 3], "def_end_pos": [68, 14]}, {"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [207, 9], "def_end_pos": [207, 17]}, {"full_name": "true_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [105, 17], "def_end_pos": [105, 25]}, {"full_name": "subset_closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [403, 9], "def_end_pos": [403, 23]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}, {"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [207, 9], "def_end_pos": [207, 17]}, {"full_name": "Prod.exists", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [37, 9], "def_end_pos": [37, 17]}]], "state_before": "case pos\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 K \u00d7\u02e2 closure M\nH : x \u2208 K\n\u22a2 f (x, y).1 * (D (x, y).1)\u207b\u00b9 * g ((x, y).2\u207b\u00b9 * (x, y).1) = 0", "state_after": "case pos\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 K \u00d7\u02e2 closure M\nH : x \u2208 K\nthis : g (y\u207b\u00b9 * x) = 0\n\u22a2 f (x, y).1 * (D (x, y).1)\u207b\u00b9 * g ((x, y).2\u207b\u00b9 * (x, y).1) = 0"}, {"tactic": "simp [this]", "annotated_tactic": ["simp [this]", []], "state_before": "case pos\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 K \u00d7\u02e2 closure M\nH : x \u2208 K\nthis : g (y\u207b\u00b9 * x) = 0\n\u22a2 f (x, y).1 * (D (x, y).1)\u207b\u00b9 * g ((x, y).2\u207b\u00b9 * (x, y).1) = 0", "state_after": "no goals"}, {"tactic": "simp [image_eq_zero_of_nmem_tsupport H]", "annotated_tactic": ["simp [<a>image_eq_zero_of_nmem_tsupport</a> H]", [{"full_name": "image_eq_zero_of_nmem_tsupport", "def_path": "Mathlib/Topology/Support.lean", "def_pos": [68, 3], "def_end_pos": [68, 14]}]], "state_before": "case neg\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 K \u00d7\u02e2 closure M\nH : x \u2209 K\n\u22a2 f (x, y).1 * (D (x, y).1)\u207b\u00b9 * g ((x, y).2\u207b\u00b9 * (x, y).1) = 0", "state_after": "no goals"}, {"tactic": "apply image_eq_zero_of_nmem_tsupport", "annotated_tactic": ["apply <a>image_eq_zero_of_nmem_tsupport</a>", [{"full_name": "image_eq_zero_of_nmem_tsupport", "def_path": "Mathlib/Topology/Support.lean", "def_pos": [68, 3], "def_end_pos": [68, 14]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 K \u00d7\u02e2 closure M\nH : x \u2208 K\n\u22a2 g (y\u207b\u00b9 * x) = 0", "state_after": "case hx\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 K \u00d7\u02e2 closure M\nH : x \u2208 K\n\u22a2 y\u207b\u00b9 * x \u2209 tsupport g"}, {"tactic": "contrapose! hxy", "annotated_tactic": ["contrapose! hxy", []], "state_before": "case hx\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 K \u00d7\u02e2 closure M\nH : x \u2208 K\n\u22a2 y\u207b\u00b9 * x \u2209 tsupport g", "state_after": "case hx\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nH : x \u2208 K\nhxy : y\u207b\u00b9 * x \u2208 tsupport g\n\u22a2 (x, y) \u2208 K \u00d7\u02e2 closure M"}, {"tactic": "simp only [mem_prod, H, true_and]", "annotated_tactic": ["simp only [<a>mem_prod</a>, H, <a>true_and</a>]", [{"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [207, 9], "def_end_pos": [207, 17]}, {"full_name": "true_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [105, 17], "def_end_pos": [105, 25]}]], "state_before": "case hx\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nH : x \u2208 K\nhxy : y\u207b\u00b9 * x \u2208 tsupport g\n\u22a2 (x, y) \u2208 K \u00d7\u02e2 closure M", "state_after": "case hx\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nH : x \u2208 K\nhxy : y\u207b\u00b9 * x \u2208 tsupport g\n\u22a2 y \u2208 closure M"}, {"tactic": "apply subset_closure", "annotated_tactic": ["apply <a>subset_closure</a>", [{"full_name": "subset_closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [403, 9], "def_end_pos": [403, 23]}]], "state_before": "case hx\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nH : x \u2208 K\nhxy : y\u207b\u00b9 * x \u2208 tsupport g\n\u22a2 y \u2208 closure M", "state_after": "case hx.a\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nH : x \u2208 K\nhxy : y\u207b\u00b9 * x \u2208 tsupport g\n\u22a2 y \u2208 M"}, {"tactic": "simp only [M, mem_image, mem_prod, Prod.exists]", "annotated_tactic": ["simp only [M, <a>mem_image</a>, <a>mem_prod</a>, <a>Prod.exists</a>]", [{"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}, {"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [207, 9], "def_end_pos": [207, 17]}, {"full_name": "Prod.exists", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [37, 9], "def_end_pos": [37, 17]}]], "state_before": "case hx.a\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nH : x \u2208 K\nhxy : y\u207b\u00b9 * x \u2208 tsupport g\n\u22a2 y \u2208 M", "state_after": "case hx.a\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nH : x \u2208 K\nhxy : y\u207b\u00b9 * x \u2208 tsupport g\n\u22a2 \u2203 a b, (a \u2208 K \u2227 b \u2208 L) \u2227 a * b\u207b\u00b9 = y"}, {"tactic": "exact \u27e8x, y\u207b\u00b9 * x, \u27e8H, hxy\u27e9, by group\u27e9", "annotated_tactic": ["exact \u27e8x, y\u207b\u00b9 * x, \u27e8H, hxy\u27e9, by group\u27e9", []], "state_before": "case hx.a\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nH : x \u2208 K\nhxy : y\u207b\u00b9 * x \u2208 tsupport g\n\u22a2 \u2203 a b, (a \u2208 K \u2227 b \u2208 L) \u2227 a * b\u207b\u00b9 = y", "state_after": "no goals"}, {"tactic": "group", "annotated_tactic": ["group", []], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nH : x \u2208 K\nhxy : y\u207b\u00b9 * x \u2208 tsupport g\n\u22a2 x * (y\u207b\u00b9 * x)\u207b\u00b9 = y", "state_after": "no goals"}, {"tactic": "congr with y", "annotated_tactic": ["congr with y", []], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 \u222b (y : G), \u222b (x : G), f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x) \u2202\u03bc \u2202\u03bd = \u222b (y : G), \u222b (x : G), f (y * x) * (D (y * x))\u207b\u00b9 * g x \u2202\u03bc \u2202\u03bd", "state_after": "case e_f.h\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\ny : G\n\u22a2 \u222b (x : G), f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x) \u2202\u03bc = \u222b (x : G), f (y * x) * (D (y * x))\u207b\u00b9 * g x \u2202\u03bc"}, {"tactic": "rw [\u2190 integral_mul_left_eq_self _ y]", "annotated_tactic": ["rw [\u2190 <a>integral_mul_left_eq_self</a> _ y]", [{"full_name": "MeasureTheory.integral_mul_left_eq_self", "def_path": "Mathlib/MeasureTheory/Group/Integral.lean", "def_pos": [58, 9], "def_end_pos": [58, 34]}]], "state_before": "case e_f.h\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\ny : G\n\u22a2 \u222b (x : G), f x * (D x)\u207b\u00b9 * g (y\u207b\u00b9 * x) \u2202\u03bc = \u222b (x : G), f (y * x) * (D (y * x))\u207b\u00b9 * g x \u2202\u03bc", "state_after": "case e_f.h\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\ny : G\n\u22a2 \u222b (x : G), f (y * x) * (D (y * x))\u207b\u00b9 * g (y\u207b\u00b9 * (y * x)) \u2202\u03bc = \u222b (x : G), f (y * x) * (D (y * x))\u207b\u00b9 * g x \u2202\u03bc"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case e_f.h\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\ny : G\n\u22a2 \u222b (x : G), f (y * x) * (D (y * x))\u207b\u00b9 * g (y\u207b\u00b9 * (y * x)) \u2202\u03bc = \u222b (x : G), f (y * x) * (D (y * x))\u207b\u00b9 * g x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "apply (integral_integral_swap_of_hasCompactSupport _ _).symm", "annotated_tactic": ["apply (<a>integral_integral_swap_of_hasCompactSupport</a> _ _).<a>symm</a>", [{"full_name": "MeasureTheory.integral_integral_swap_of_hasCompactSupport", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Integral.lean", "def_pos": [553, 7], "def_end_pos": [553, 50]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 \u222b (y : G), \u222b (x : G), f (y * x) * (D (y * x))\u207b\u00b9 * g x \u2202\u03bc \u2202\u03bd =\n    \u222b (x : G), \u222b (y : G), f (y * x) * (D (y * x))\u207b\u00b9 * g x \u2202\u03bd \u2202\u03bc", "state_after": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 Continuous (uncurry fun x y => f (y * x) * (D (y * x))\u207b\u00b9 * g x)\n\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 HasCompactSupport (uncurry fun x y => f (y * x) * (D (y * x))\u207b\u00b9 * g x)"}, {"tactic": "apply Continuous.mul ?_ (hg.comp continuous_fst)", "annotated_tactic": ["apply <a>Continuous.mul</a> ?_ (hg.comp <a>continuous_fst</a>)", [{"full_name": "Continuous.mul", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [97, 9], "def_end_pos": [97, 23]}, {"full_name": "continuous_fst", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [339, 9], "def_end_pos": [339, 23]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 Continuous (uncurry fun x y => f (y * x) * (D (y * x))\u207b\u00b9 * g x)", "state_after": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 Continuous fun x => f (x.2 * x.1) * (D (x.2 * x.1))\u207b\u00b9"}, {"tactic": "exact (hf.comp (continuous_snd.mul continuous_fst)).mul\n  ((D_cont.comp (continuous_snd.mul continuous_fst)).inv\u2080 (fun x \u21a6 (D_pos _).ne'))", "annotated_tactic": ["exact (hf.comp (continuous_snd.mul <a>continuous_fst</a>)).<a>mul</a>\n          ((D_cont.comp (continuous_snd.mul <a>continuous_fst</a>)).<a>inv\u2080</a> (fun x \u21a6 (D_pos _).<a>ne'</a>))", [{"full_name": "continuous_fst", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [339, 9], "def_end_pos": [339, 23]}, {"full_name": "Continuous.mul", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [97, 9], "def_end_pos": [97, 23]}, {"full_name": "continuous_fst", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [339, 9], "def_end_pos": [339, 23]}, {"full_name": "Continuous.inv\u2080", "def_path": "Mathlib/Topology/Algebra/GroupWithZero.lean", "def_pos": [133, 9], "def_end_pos": [133, 24]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 Continuous fun x => f (x.2 * x.1) * (D (x.2 * x.1))\u207b\u00b9", "state_after": "no goals"}, {"tactic": "let K := tsupport f", "annotated_tactic": ["let K := <a>tsupport</a> f", [{"full_name": "tsupport", "def_path": "Mathlib/Topology/Support.lean", "def_pos": [44, 3], "def_end_pos": [44, 14]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 HasCompactSupport (uncurry fun x y => f (y * x) * (D (y * x))\u207b\u00b9 * g x)", "state_after": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\n\u22a2 HasCompactSupport (uncurry fun x y => f (y * x) * (D (y * x))\u207b\u00b9 * g x)"}, {"tactic": "have K_comp : IsCompact K := h'f", "annotated_tactic": ["have K_comp : <a>IsCompact</a> K := h'f", [{"full_name": "IsCompact", "def_path": "Mathlib/Topology/Defs/Filter.lean", "def_pos": [273, 5], "def_end_pos": [273, 14]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\n\u22a2 HasCompactSupport (uncurry fun x y => f (y * x) * (D (y * x))\u207b\u00b9 * g x)", "state_after": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\n\u22a2 HasCompactSupport (uncurry fun x y => f (y * x) * (D (y * x))\u207b\u00b9 * g x)"}, {"tactic": "let L := tsupport g", "annotated_tactic": ["let L := <a>tsupport</a> g", [{"full_name": "tsupport", "def_path": "Mathlib/Topology/Support.lean", "def_pos": [44, 3], "def_end_pos": [44, 14]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\n\u22a2 HasCompactSupport (uncurry fun x y => f (y * x) * (D (y * x))\u207b\u00b9 * g x)", "state_after": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\n\u22a2 HasCompactSupport (uncurry fun x y => f (y * x) * (D (y * x))\u207b\u00b9 * g x)"}, {"tactic": "have L_comp : IsCompact L := h'g", "annotated_tactic": ["have L_comp : <a>IsCompact</a> L := h'g", [{"full_name": "IsCompact", "def_path": "Mathlib/Topology/Defs/Filter.lean", "def_pos": [273, 5], "def_end_pos": [273, 14]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\n\u22a2 HasCompactSupport (uncurry fun x y => f (y * x) * (D (y * x))\u207b\u00b9 * g x)", "state_after": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\n\u22a2 HasCompactSupport (uncurry fun x y => f (y * x) * (D (y * x))\u207b\u00b9 * g x)"}, {"tactic": "let M := (fun (p : G \u00d7 G) \u21a6 p.1 * p.2\u207b\u00b9) '' (K \u00d7\u02e2 L)", "annotated_tactic": ["let M := (fun (p : G \u00d7 G) \u21a6 p.1 * p.2\u207b\u00b9) '' (K \u00d7\u02e2 L)", []], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\n\u22a2 HasCompactSupport (uncurry fun x y => f (y * x) * (D (y * x))\u207b\u00b9 * g x)", "state_after": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\n\u22a2 HasCompactSupport (uncurry fun x y => f (y * x) * (D (y * x))\u207b\u00b9 * g x)"}, {"tactic": "have M_comp : IsCompact M :=\n  (K_comp.prod L_comp).image (continuous_fst.mul continuous_snd.inv)", "annotated_tactic": ["have M_comp : <a>IsCompact</a> M :=\n          (K_comp.prod L_comp).<a>image</a> (continuous_fst.mul continuous_snd.inv)", [{"full_name": "IsCompact", "def_path": "Mathlib/Topology/Defs/Filter.lean", "def_pos": [273, 5], "def_end_pos": [273, 14]}, {"full_name": "IsCompact.image", "def_path": "Mathlib/Topology/Compactness/Compact.lean", "def_pos": [119, 9], "def_end_pos": [119, 24]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\n\u22a2 HasCompactSupport (uncurry fun x y => f (y * x) * (D (y * x))\u207b\u00b9 * g x)", "state_after": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\n\u22a2 HasCompactSupport (uncurry fun x y => f (y * x) * (D (y * x))\u207b\u00b9 * g x)"}, {"tactic": "have M'_comp : IsCompact (closure M) := M_comp.closure", "annotated_tactic": ["have M'_comp : <a>IsCompact</a> (<a>closure</a> M) := M_comp.closure", [{"full_name": "IsCompact", "def_path": "Mathlib/Topology/Defs/Filter.lean", "def_pos": [273, 5], "def_end_pos": [273, 14]}, {"full_name": "closure", "def_path": "Mathlib/Topology/Defs/Basic.lean", "def_pos": [116, 5], "def_end_pos": [116, 12]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\n\u22a2 HasCompactSupport (uncurry fun x y => f (y * x) * (D (y * x))\u207b\u00b9 * g x)", "state_after": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\n\u22a2 HasCompactSupport (uncurry fun x y => f (y * x) * (D (y * x))\u207b\u00b9 * g x)"}, {"tactic": "apply HasCompactSupport.intro' (L_comp.prod M'_comp) ?_ this", "annotated_tactic": ["apply <a>HasCompactSupport.intro'</a> (L_comp.prod M'_comp) ?_ this", [{"full_name": "HasCompactSupport.intro'", "def_path": "Mathlib/Topology/Support.lean", "def_pos": [179, 3], "def_end_pos": [179, 14]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nthis : \u2200 p \u2209 L \u00d7\u02e2 closure M, f (p.2 * p.1) * (D (p.2 * p.1))\u207b\u00b9 * g p.1 = 0\n\u22a2 HasCompactSupport (uncurry fun x y => f (y * x) * (D (y * x))\u207b\u00b9 * g x)", "state_after": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nthis : \u2200 p \u2209 L \u00d7\u02e2 closure M, f (p.2 * p.1) * (D (p.2 * p.1))\u207b\u00b9 * g p.1 = 0\n\u22a2 IsClosed (L \u00d7\u02e2 closure M)"}, {"tactic": "exact (isClosed_tsupport g).prod isClosed_closure", "annotated_tactic": ["exact (<a>isClosed_tsupport</a> g).<a>prod</a> <a>isClosed_closure</a>", [{"full_name": "isClosed_tsupport", "def_path": "Mathlib/Topology/Support.lean", "def_pos": [56, 3], "def_end_pos": [56, 14]}, {"full_name": "IsClosed.prod", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [858, 9], "def_end_pos": [858, 22]}, {"full_name": "isClosed_closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [399, 9], "def_end_pos": [399, 25]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nthis : \u2200 p \u2209 L \u00d7\u02e2 closure M, f (p.2 * p.1) * (D (p.2 * p.1))\u207b\u00b9 * g p.1 = 0\n\u22a2 IsClosed (L \u00d7\u02e2 closure M)", "state_after": "no goals"}, {"tactic": "rintro \u27e8x, y\u27e9 hxy", "annotated_tactic": ["rintro \u27e8x, y\u27e9 hxy", []], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\n\u22a2 \u2200 p \u2209 L \u00d7\u02e2 closure M, f (p.2 * p.1) * (D (p.2 * p.1))\u207b\u00b9 * g p.1 = 0", "state_after": "case mk\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 L \u00d7\u02e2 closure M\n\u22a2 f ((x, y).2 * (x, y).1) * (D ((x, y).2 * (x, y).1))\u207b\u00b9 * g (x, y).1 = 0"}, {"tactic": "by_cases H : x \u2208 L", "annotated_tactic": ["by_cases H : x \u2208 L", []], "state_before": "case mk\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 L \u00d7\u02e2 closure M\n\u22a2 f ((x, y).2 * (x, y).1) * (D ((x, y).2 * (x, y).1))\u207b\u00b9 * g (x, y).1 = 0", "state_after": "case pos\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 L \u00d7\u02e2 closure M\nH : x \u2208 L\n\u22a2 f ((x, y).2 * (x, y).1) * (D ((x, y).2 * (x, y).1))\u207b\u00b9 * g (x, y).1 = 0\n\ncase neg\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 L \u00d7\u02e2 closure M\nH : x \u2209 L\n\u22a2 f ((x, y).2 * (x, y).1) * (D ((x, y).2 * (x, y).1))\u207b\u00b9 * g (x, y).1 = 0"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "case pos\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 L \u00d7\u02e2 closure M\nH : x \u2208 L\n\u22a2 f ((x, y).2 * (x, y).1) * (D ((x, y).2 * (x, y).1))\u207b\u00b9 * g (x, y).1 = 0\n\ncase neg\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 L \u00d7\u02e2 closure M\nH : x \u2209 L\n\u22a2 f ((x, y).2 * (x, y).1) * (D ((x, y).2 * (x, y).1))\u207b\u00b9 * g (x, y).1 = 0", "state_after": "case neg\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 L \u00d7\u02e2 closure M\nH : x \u2209 L\n\u22a2 f ((x, y).2 * (x, y).1) * (D ((x, y).2 * (x, y).1))\u207b\u00b9 * g (x, y).1 = 0\n\ncase pos\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 L \u00d7\u02e2 closure M\nH : x \u2208 L\n\u22a2 f ((x, y).2 * (x, y).1) * (D ((x, y).2 * (x, y).1))\u207b\u00b9 * g (x, y).1 = 0"}, {"tactic": "have : f (y * x) = 0 := by\n  apply image_eq_zero_of_nmem_tsupport\n  contrapose! hxy\n  simp only [mem_prod, H, true_and]\n  apply subset_closure\n  simp only [M, mem_image, mem_prod, Prod.exists]\n  exact \u27e8y * x, x, \u27e8hxy, H\u27e9, by group\u27e9", "annotated_tactic": ["have : f (y * x) = 0 := by\n            apply <a>image_eq_zero_of_nmem_tsupport</a>\n            contrapose! hxy\n            simp only [<a>mem_prod</a>, H, <a>true_and</a>]\n            apply <a>subset_closure</a>\n            simp only [M, <a>mem_image</a>, <a>mem_prod</a>, <a>Prod.exists</a>]\n            exact \u27e8y * x, x, \u27e8hxy, H\u27e9, by group\u27e9", [{"full_name": "image_eq_zero_of_nmem_tsupport", "def_path": "Mathlib/Topology/Support.lean", "def_pos": [68, 3], "def_end_pos": [68, 14]}, {"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [207, 9], "def_end_pos": [207, 17]}, {"full_name": "true_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [105, 17], "def_end_pos": [105, 25]}, {"full_name": "subset_closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [403, 9], "def_end_pos": [403, 23]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}, {"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [207, 9], "def_end_pos": [207, 17]}, {"full_name": "Prod.exists", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [37, 9], "def_end_pos": [37, 17]}]], "state_before": "case pos\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 L \u00d7\u02e2 closure M\nH : x \u2208 L\n\u22a2 f ((x, y).2 * (x, y).1) * (D ((x, y).2 * (x, y).1))\u207b\u00b9 * g (x, y).1 = 0", "state_after": "case pos\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 L \u00d7\u02e2 closure M\nH : x \u2208 L\nthis : f (y * x) = 0\n\u22a2 f ((x, y).2 * (x, y).1) * (D ((x, y).2 * (x, y).1))\u207b\u00b9 * g (x, y).1 = 0"}, {"tactic": "simp [this]", "annotated_tactic": ["simp [this]", []], "state_before": "case pos\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 L \u00d7\u02e2 closure M\nH : x \u2208 L\nthis : f (y * x) = 0\n\u22a2 f ((x, y).2 * (x, y).1) * (D ((x, y).2 * (x, y).1))\u207b\u00b9 * g (x, y).1 = 0", "state_after": "no goals"}, {"tactic": "simp [image_eq_zero_of_nmem_tsupport H]", "annotated_tactic": ["simp [<a>image_eq_zero_of_nmem_tsupport</a> H]", [{"full_name": "image_eq_zero_of_nmem_tsupport", "def_path": "Mathlib/Topology/Support.lean", "def_pos": [68, 3], "def_end_pos": [68, 14]}]], "state_before": "case neg\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 L \u00d7\u02e2 closure M\nH : x \u2209 L\n\u22a2 f ((x, y).2 * (x, y).1) * (D ((x, y).2 * (x, y).1))\u207b\u00b9 * g (x, y).1 = 0", "state_after": "no goals"}, {"tactic": "apply image_eq_zero_of_nmem_tsupport", "annotated_tactic": ["apply <a>image_eq_zero_of_nmem_tsupport</a>", [{"full_name": "image_eq_zero_of_nmem_tsupport", "def_path": "Mathlib/Topology/Support.lean", "def_pos": [68, 3], "def_end_pos": [68, 14]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 L \u00d7\u02e2 closure M\nH : x \u2208 L\n\u22a2 f (y * x) = 0", "state_after": "case hx\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 L \u00d7\u02e2 closure M\nH : x \u2208 L\n\u22a2 y * x \u2209 tsupport f"}, {"tactic": "contrapose! hxy", "annotated_tactic": ["contrapose! hxy", []], "state_before": "case hx\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nhxy : (x, y) \u2209 L \u00d7\u02e2 closure M\nH : x \u2208 L\n\u22a2 y * x \u2209 tsupport f", "state_after": "case hx\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nH : x \u2208 L\nhxy : y * x \u2208 tsupport f\n\u22a2 (x, y) \u2208 L \u00d7\u02e2 closure M"}, {"tactic": "simp only [mem_prod, H, true_and]", "annotated_tactic": ["simp only [<a>mem_prod</a>, H, <a>true_and</a>]", [{"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [207, 9], "def_end_pos": [207, 17]}, {"full_name": "true_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [105, 17], "def_end_pos": [105, 25]}]], "state_before": "case hx\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nH : x \u2208 L\nhxy : y * x \u2208 tsupport f\n\u22a2 (x, y) \u2208 L \u00d7\u02e2 closure M", "state_after": "case hx\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nH : x \u2208 L\nhxy : y * x \u2208 tsupport f\n\u22a2 y \u2208 closure M"}, {"tactic": "apply subset_closure", "annotated_tactic": ["apply <a>subset_closure</a>", [{"full_name": "subset_closure", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [403, 9], "def_end_pos": [403, 23]}]], "state_before": "case hx\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nH : x \u2208 L\nhxy : y * x \u2208 tsupport f\n\u22a2 y \u2208 closure M", "state_after": "case hx.a\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nH : x \u2208 L\nhxy : y * x \u2208 tsupport f\n\u22a2 y \u2208 M"}, {"tactic": "simp only [M, mem_image, mem_prod, Prod.exists]", "annotated_tactic": ["simp only [M, <a>mem_image</a>, <a>mem_prod</a>, <a>Prod.exists</a>]", [{"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}, {"full_name": "Set.mem_prod", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [207, 9], "def_end_pos": [207, 17]}, {"full_name": "Prod.exists", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [37, 9], "def_end_pos": [37, 17]}]], "state_before": "case hx.a\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nH : x \u2208 L\nhxy : y * x \u2208 tsupport f\n\u22a2 y \u2208 M", "state_after": "case hx.a\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nH : x \u2208 L\nhxy : y * x \u2208 tsupport f\n\u22a2 \u2203 a b, (a \u2208 K \u2227 b \u2208 L) \u2227 a * b\u207b\u00b9 = y"}, {"tactic": "exact \u27e8y * x, x, \u27e8hxy, H\u27e9, by group\u27e9", "annotated_tactic": ["exact \u27e8y * x, x, \u27e8hxy, H\u27e9, by group\u27e9", []], "state_before": "case hx.a\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nH : x \u2208 L\nhxy : y * x \u2208 tsupport f\n\u22a2 \u2203 a b, (a \u2208 K \u2227 b \u2208 L) \u2227 a * b\u207b\u00b9 = y", "state_after": "no goals"}, {"tactic": "group", "annotated_tactic": ["group", []], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nK : Set G := tsupport f\nK_comp : IsCompact K\nL : Set G := tsupport g\nL_comp : IsCompact L\nM : Set G := (fun p => p.1 * p.2\u207b\u00b9) '' K \u00d7\u02e2 L\nM_comp : IsCompact M\nM'_comp : IsCompact (closure M)\nx y : G\nH : x \u2208 L\nhxy : y * x \u2208 tsupport f\n\u22a2 y * x * x\u207b\u00b9 = y", "state_after": "no goals"}, {"tactic": "simp_rw [integral_mul_right]", "annotated_tactic": ["simp_rw [<a>integral_mul_right</a>]", [{"full_name": "MeasureTheory.integral_mul_right", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [928, 9], "def_end_pos": [928, 27]}]], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 \u222b (x : G), \u222b (y : G), f (y * x) * (D (y * x))\u207b\u00b9 * g x \u2202\u03bd \u2202\u03bc = \u222b (x : G), (\u222b (y : G), f y * (D y)\u207b\u00b9 \u2202\u03bd) * g x \u2202\u03bc", "state_after": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 \u222b (x : G), (\u222b (a : G), f (a * x) * (D (a * x))\u207b\u00b9 \u2202\u03bd) * g x \u2202\u03bc = \u222b (x : G), (\u222b (y : G), f y * (D y)\u207b\u00b9 \u2202\u03bd) * g x \u2202\u03bc"}, {"tactic": "congr with x", "annotated_tactic": ["congr with x", []], "state_before": "G : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\n\u22a2 \u222b (x : G), (\u222b (a : G), f (a * x) * (D (a * x))\u207b\u00b9 \u2202\u03bd) * g x \u2202\u03bc = \u222b (x : G), (\u222b (y : G), f y * (D y)\u207b\u00b9 \u2202\u03bd) * g x \u2202\u03bc", "state_after": "case e_f.h\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nx : G\n\u22a2 (\u222b (a : G), f (a * x) * (D (a * x))\u207b\u00b9 \u2202\u03bd) * g x = (\u222b (y : G), f y * (D y)\u207b\u00b9 \u2202\u03bd) * g x"}, {"tactic": "conv_rhs => rw [\u2190 integral_mul_right_eq_self _ x]", "annotated_tactic": ["conv_rhs => rw [\u2190 <a>integral_mul_right_eq_self</a> _ x]", [{"full_name": "MeasureTheory.integral_mul_right_eq_self", "def_path": "Mathlib/MeasureTheory/Group/Integral.lean", "def_pos": [70, 9], "def_end_pos": [70, 35]}]], "state_before": "case e_f.h\nG : Type u_1\ninst\u271d\u2079 : TopologicalSpace G\ninst\u271d\u2078 : Group G\ninst\u271d\u2077 : TopologicalGroup G\ninst\u271d\u2076 : MeasurableSpace G\ninst\u271d\u2075 : BorelSpace G\n\u03bc \u03bd : Measure G\ninst\u271d\u2074 : IsFiniteMeasureOnCompacts \u03bc\ninst\u271d\u00b3 : IsFiniteMeasureOnCompacts \u03bd\ninst\u271d\u00b2 : \u03bc.IsMulLeftInvariant\ninst\u271d\u00b9 : \u03bd.IsMulRightInvariant\ninst\u271d : \u03bd.IsOpenPosMeasure\nf g : G \u2192 \u211d\nhf : Continuous f\nh'f : HasCompactSupport f\nhg : Continuous g\nh'g : HasCompactSupport g\ng_nonneg : 0 \u2264 g\nx\u2080 : G\ng_pos : g x\u2080 \u2260 0\nHf : LocallyCompactSpace G\nD : G \u2192 \u211d := fun x => \u222b (y : G), g (y\u207b\u00b9 * x) \u2202\u03bd\nD_cont : Continuous D\nD_pos : \u2200 (x : G), 0 < D x\nx : G\n\u22a2 (\u222b (a : G), f (a * x) * (D (a * x))\u207b\u00b9 \u2202\u03bd) * g x = (\u222b (y : G), f y * (D y)\u207b\u00b9 \u2202\u03bd) * g x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/ContinuousFunction/UniqueCFC.lean", "full_name": "ContinuousMap.toNNReal_neg_algebraMap", "start": [83, 1], "end": [85, 12], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "X : Type u_1\ninst\u271d : TopologicalSpace X\nr : \u211d\u22650\n\u22a2 (-(algebraMap \u211d C(X, \u211d)) \u2191r).toNNReal = 0", "state_after": "case h.a\nX : Type u_1\ninst\u271d : TopologicalSpace X\nr : \u211d\u22650\na\u271d : X\n\u22a2 \u2191((-(algebraMap \u211d C(X, \u211d)) \u2191r).toNNReal a\u271d) = \u2191(0 a\u271d)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.a\nX : Type u_1\ninst\u271d : TopologicalSpace X\nr : \u211d\u22650\na\u271d : X\n\u22a2 \u2191((-(algebraMap \u211d C(X, \u211d)) \u2191r).toNNReal a\u271d) = \u2191(0 a\u271d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Basic.lean", "full_name": "Polynomial.smul_C", "start": [534, 1], "end": [535, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Module/Algebra.lean", "full_name": "algebraMap_le_algebraMap", "start": [44, 1], "end": [45, 40], "traced_tactics": [{"tactic": "simp [Algebra.algebraMap_eq_smul_one]", "annotated_tactic": ["simp [<a>Algebra.algebraMap_eq_smul_one</a>]", [{"full_name": "Algebra.algebraMap_eq_smul_one", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [330, 9], "def_end_pos": [330, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2074 : OrderedCommSemiring \u03b1\ninst\u271d\u00b3 : StrictOrderedSemiring \u03b2\ninst\u271d\u00b2 : Algebra \u03b1 \u03b2\ninst\u271d\u00b9 : SMulPosMono \u03b1 \u03b2\ninst\u271d : SMulPosReflectLE \u03b1 \u03b2\na\u2081 a\u2082 : \u03b1\n\u22a2 (algebraMap \u03b1 \u03b2) a\u2081 \u2264 (algebraMap \u03b1 \u03b2) a\u2082 \u2194 a\u2081 \u2264 a\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Function/OfArity.lean", "full_name": "Function.ofArity_zero", "start": [33, 1], "end": [33, 78], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Part.lean", "full_name": "Part.div_def", "start": [703, 1], "end": [703, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Polynomial/Basic.lean", "full_name": "Polynomial.degree_restriction", "start": [382, 1], "end": [382, 94], "traced_tactics": [{"tactic": "simp [degree]", "annotated_tactic": ["simp [<a>degree</a>]", [{"full_name": "Polynomial.degree", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [52, 5], "def_end_pos": [52, 11]}]], "state_before": "R : Type u\nS : Type u_1\ninst\u271d : Ring R\np : R[X]\n\u22a2 p.restriction.degree = p.degree", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean", "full_name": "AlgebraicGeometry.quasiSeparated_eq_affineProperty_diagonal", "start": [127, 1], "end": [131, 94], "traced_tactics": [{"tactic": "rw [quasiSeparated_eq_diagonal_is_quasiCompact, quasiCompact_eq_affineProperty]", "annotated_tactic": ["rw [<a>quasiSeparated_eq_diagonal_is_quasiCompact</a>, <a>quasiCompact_eq_affineProperty</a>]", [{"full_name": "AlgebraicGeometry.quasiSeparated_eq_diagonal_is_quasiCompact", "def_path": "Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean", "def_pos": [118, 9], "def_end_pos": [118, 51]}, {"full_name": "AlgebraicGeometry.quasiCompact_eq_affineProperty", "def_path": "Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean", "def_pos": [116, 9], "def_end_pos": [116, 39]}]], "state_before": "X Y : Scheme\nf : X \u27f6 Y\n\u22a2 @QuasiSeparated = targetAffineLocally QuasiCompact.affineProperty.diagonal", "state_after": "X Y : Scheme\nf : X \u27f6 Y\n\u22a2 (targetAffineLocally QuasiCompact.affineProperty).diagonal = targetAffineLocally QuasiCompact.affineProperty.diagonal"}, {"tactic": "exact\n  diagonal_targetAffineLocally_eq_targetAffineLocally _ QuasiCompact.affineProperty_isLocal", "annotated_tactic": ["exact\n    <a>diagonal_targetAffineLocally_eq_targetAffineLocally</a> _ <a>QuasiCompact.affineProperty_isLocal</a>", [{"full_name": "AlgebraicGeometry.diagonal_targetAffineLocally_eq_targetAffineLocally", "def_path": "Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean", "def_pos": [140, 9], "def_end_pos": [140, 60]}, {"full_name": "AlgebraicGeometry.QuasiCompact.affineProperty_isLocal", "def_path": "Mathlib/AlgebraicGeometry/Morphisms/QuasiCompact.lean", "def_pos": [154, 9], "def_end_pos": [154, 44]}]], "state_before": "X Y : Scheme\nf : X \u27f6 Y\n\u22a2 (targetAffineLocally QuasiCompact.affineProperty).diagonal = targetAffineLocally QuasiCompact.affineProperty.diagonal", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "full_name": "Subgroup.top_prod_top", "start": 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DistribMulAction S M\nsR : Set R\ns : Set S\nN : Submodule R M\nx\u271d : M\n\u22a2 x\u271d \u2208 \u2205 \u2022 N \u2194 x\u271d \u2208 \u22a5"}, {"tactic": "fconstructor", "annotated_tactic": ["fconstructor", []], "state_before": "case h\n\u03b1 : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nS : Type u_4\ninst\u271d\u00b3 : Monoid S\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : DistribMulAction S M\nsR : Set R\ns : Set S\nN : Submodule R M\nx\u271d : M\n\u22a2 x\u271d \u2208 \u2205 \u2022 N \u2194 x\u271d \u2208 \u22a5", "state_after": "case h.mp\n\u03b1 : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nS : Type u_4\ninst\u271d\u00b3 : Monoid S\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : DistribMulAction S M\nsR : Set R\ns : Set S\nN : Submodule R M\nx\u271d : M\n\u22a2 x\u271d \u2208 \u2205 \u2022 N \u2192 x\u271d \u2208 \u22a5\n\ncase h.mpr\n\u03b1 : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nS : Type u_4\ninst\u271d\u00b3 : Monoid S\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : DistribMulAction S M\nsR : Set R\ns : Set S\nN : Submodule R M\nx\u271d : M\n\u22a2 x\u271d \u2208 \u22a5 \u2192 x\u271d \u2208 \u2205 \u2022 N"}, {"tactic": "intro hx", "annotated_tactic": ["intro hx", []], "state_before": "case h.mp\n\u03b1 : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nS : Type u_4\ninst\u271d\u00b3 : Monoid S\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : DistribMulAction S M\nsR : Set R\ns : Set S\nN : Submodule R M\nx\u271d : M\n\u22a2 x\u271d \u2208 \u2205 \u2022 N \u2192 x\u271d \u2208 \u22a5", "state_after": "case h.mp\n\u03b1 : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nS : Type u_4\ninst\u271d\u00b3 : Monoid S\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : DistribMulAction S M\nsR : Set R\ns : Set S\nN : Submodule R M\nx\u271d : M\nhx : x\u271d \u2208 \u2205 \u2022 N\n\u22a2 x\u271d \u2208 \u22a5"}, {"tactic": "rw [mem_set_smul_def, Submodule.mem_sInf] at hx", "annotated_tactic": ["rw [<a>mem_set_smul_def</a>, <a>Submodule.mem_sInf</a>] at hx", [{"full_name": "Submodule.mem_set_smul_def", "def_path": "Mathlib/Algebra/Module/Submodule/Pointwise.lean", "def_pos": [350, 7], "def_end_pos": [350, 23]}, {"full_name": "Submodule.mem_sInf", "def_path": "Mathlib/Algebra/Module/Submodule/Lattice.lean", "def_pos": [257, 9], "def_end_pos": [257, 17]}]], "state_before": "case h.mp\n\u03b1 : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nS : Type u_4\ninst\u271d\u00b3 : Monoid S\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : DistribMulAction S M\nsR : Set R\ns : Set S\nN : Submodule R M\nx\u271d : M\nhx : x\u271d \u2208 \u2205 \u2022 N\n\u22a2 x\u271d \u2208 \u22a5", "state_after": "case h.mp\n\u03b1 : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nS : Type u_4\ninst\u271d\u00b3 : Monoid S\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : DistribMulAction S M\nsR : Set R\ns : Set S\nN : Submodule R M\nx\u271d : M\nhx : \u2200 p \u2208 {p | \u2200 \u2983r : S\u2984 {n : M}, r \u2208 \u2205 \u2192 n \u2208 N \u2192 r \u2022 n \u2208 p}, x\u271d \u2208 p\n\u22a2 x\u271d \u2208 \u22a5"}, {"tactic": "exact hx \u22a5 (fun r _ hr \u21a6 hr.elim)", "annotated_tactic": ["exact hx \u22a5 (fun r _ hr \u21a6 hr.elim)", []], "state_before": "case h.mp\n\u03b1 : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nS : Type u_4\ninst\u271d\u00b3 : Monoid S\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : DistribMulAction S M\nsR : Set R\ns : Set S\nN : Submodule R M\nx\u271d : M\nhx : \u2200 p \u2208 {p | \u2200 \u2983r : S\u2984 {n : M}, r \u2208 \u2205 \u2192 n \u2208 N \u2192 r \u2022 n \u2208 p}, x\u271d \u2208 p\n\u22a2 x\u271d \u2208 \u22a5", "state_after": "no goals"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "case h.mpr\n\u03b1 : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nS : Type u_4\ninst\u271d\u00b3 : Monoid S\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : DistribMulAction S M\nsR : Set R\ns : Set S\nN : Submodule R M\nx\u271d : M\n\u22a2 x\u271d \u2208 \u22a5 \u2192 x\u271d \u2208 \u2205 \u2022 N", "state_after": "case h.mpr\n\u03b1 : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nS : Type u_4\ninst\u271d\u00b3 : Monoid S\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : DistribMulAction S M\nsR : Set R\ns : Set S\nN : Submodule R M\n\u22a2 0 \u2208 \u2205 \u2022 N"}, {"tactic": "exact Submodule.zero_mem _", "annotated_tactic": ["exact <a>Submodule.zero_mem</a> _", [{"full_name": "Submodule.zero_mem", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [216, 19], "def_end_pos": [216, 27]}]], "state_before": "case h.mpr\n\u03b1 : Type u_1\nR : Type u_2\nM : Type u_3\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\nS : Type u_4\ninst\u271d\u00b3 : Monoid S\ninst\u271d\u00b2 : AddCommMonoid M\ninst\u271d\u00b9 : Module R M\ninst\u271d : DistribMulAction S M\nsR : Set R\ns : Set S\nN : Submodule R M\n\u22a2 0 \u2208 \u2205 \u2022 N", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Defs.lean", "full_name": "Nat.dvd_iff_le_div_mul", "start": [1403, 1], "end": [1404, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "full_name": "Pairwise.range_pairwise", "start": [227, 1], "end": [228, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean", "full_name": "CochainComplex.HomComplex.\u03b4_zero", "start": [457, 1], "end": [457, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/RootsOfUnity/Basic.lean", "full_name": "card_rootsOfUnity", "start": [240, 1], "end": [246, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.Nonempty.of_smul_right", "start": [1461, 1], "end": [1462, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Shift/CommShift.lean", "full_name": "CategoryTheory.Functor.map_shiftFunctorComm_hom_app", "start": [193, 1], "end": [210, 66], "traced_tactics": [{"tactic": "have eq := NatTrans.congr_app (congr_arg Iso.hom (F.commShiftIso_add a b)) X", "annotated_tactic": ["have eq := <a>NatTrans.congr_app</a> (<a>congr_arg</a> <a>Iso.hom</a> (F.commShiftIso_add a b)) X", [{"full_name": "CategoryTheory.NatTrans.congr_app", "def_path": "Mathlib/CategoryTheory/Functor/Category.lean", "def_pos": [68, 9], "def_end_pos": [68, 18]}, {"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "CategoryTheory.Iso.hom", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [53, 3], "def_end_pos": [53, 6]}]], "state_before": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b9\u2070 : Category.{u_6, u_1} C\ninst\u271d\u2079 : Category.{u_7, u_2} D\ninst\u271d\u2078 : Category.{?u.73897, u_3} E\nF : C \u2964 D\nG : D \u2964 E\nA : Type u_4\nB : Type u_5\ninst\u271d\u2077 : AddMonoid A\ninst\u271d\u2076 : AddCommMonoid B\ninst\u271d\u2075 : HasShift C A\ninst\u271d\u2074 : HasShift D A\ninst\u271d\u00b3 : HasShift E A\ninst\u271d\u00b2 : HasShift C B\ninst\u271d\u00b9 : HasShift D B\ninst\u271d : F.CommShift B\nX : C\na b : B\n\u22a2 F.map ((shiftFunctorComm C a b).hom.app X) =\n    (F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) \u226b\n      (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) \u226b\n        (shiftFunctorComm D a b).hom.app (F.obj X) \u226b\n          (shiftFunctor D a).map ((F.commShiftIso b).inv.app X) \u226b (F.commShiftIso a).inv.app ((shiftFunctor C b).obj X)", "state_after": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b9\u2070 : Category.{u_6, u_1} C\ninst\u271d\u2079 : Category.{u_7, u_2} D\ninst\u271d\u2078 : Category.{?u.73897, u_3} E\nF : C \u2964 D\nG : D \u2964 E\nA : Type u_4\nB : Type u_5\ninst\u271d\u2077 : AddMonoid A\ninst\u271d\u2076 : AddCommMonoid B\ninst\u271d\u2075 : HasShift C A\ninst\u271d\u2074 : HasShift D A\ninst\u271d\u00b3 : HasShift E A\ninst\u271d\u00b2 : HasShift C B\ninst\u271d\u00b9 : HasShift D B\ninst\u271d : F.CommShift B\nX : C\na b : B\neq : (F.commShiftIso (a + b)).hom.app X = (CommShift.isoAdd (F.commShiftIso a) (F.commShiftIso b)).hom.app X\n\u22a2 F.map ((shiftFunctorComm C a b).hom.app X) =\n    (F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) \u226b\n      (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) \u226b\n        (shiftFunctorComm D a b).hom.app (F.obj X) \u226b\n          (shiftFunctor D a).map ((F.commShiftIso b).inv.app X) \u226b (F.commShiftIso a).inv.app ((shiftFunctor C b).obj X)"}, {"tactic": "simp only [comp_obj, CommShift.isoAdd_hom_app,\n  \u2190 cancel_epi (F.map ((shiftFunctorAdd C a b).inv.app X)), Category.assoc,\n  \u2190 F.map_comp_assoc, Iso.inv_hom_id_app, F.map_id, Category.id_comp, F.map_comp] at eq", "annotated_tactic": ["simp only [<a>comp_obj</a>, <a>CommShift.isoAdd_hom_app</a>,\n    \u2190 <a>cancel_epi</a> (F.map ((<a>shiftFunctorAdd</a> C a b).inv.app X)), <a>Category.assoc</a>,\n    \u2190 F.map_comp_assoc, <a>Iso.inv_hom_id_app</a>, F.map_id, <a>Category.id_comp</a>, F.map_comp] at eq", [{"full_name": "CategoryTheory.Functor.comp_obj", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [110, 9], "def_end_pos": [110, 12]}, {"full_name": "CategoryTheory.Functor.CommShift.isoAdd_hom_app", "def_path": "Mathlib/CategoryTheory/Shift/CommShift.lean", "def_pos": [73, 7], "def_end_pos": [73, 21]}, {"full_name": "CategoryTheory.cancel_epi", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 19]}, {"full_name": "CategoryTheory.shiftFunctorAdd", "def_path": "Mathlib/CategoryTheory/Shift/Basic.lean", "def_pos": [173, 5], "def_end_pos": [173, 20]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.Iso.inv_hom_id_app", "def_path": "Mathlib/CategoryTheory/NatIso.lean", "def_pos": [71, 9], "def_end_pos": [71, 23]}, {"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [177, 3], "def_end_pos": [177, 10]}]], "state_before": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b9\u2070 : Category.{u_6, u_1} C\ninst\u271d\u2079 : Category.{u_7, u_2} D\ninst\u271d\u2078 : Category.{?u.73897, u_3} E\nF : C \u2964 D\nG : D \u2964 E\nA : Type u_4\nB : Type u_5\ninst\u271d\u2077 : AddMonoid A\ninst\u271d\u2076 : AddCommMonoid B\ninst\u271d\u2075 : HasShift C A\ninst\u271d\u2074 : HasShift D A\ninst\u271d\u00b3 : HasShift E A\ninst\u271d\u00b2 : HasShift C B\ninst\u271d\u00b9 : HasShift D B\ninst\u271d : F.CommShift B\nX : C\na b : B\neq : (F.commShiftIso (a + b)).hom.app X = (CommShift.isoAdd (F.commShiftIso a) (F.commShiftIso b)).hom.app X\n\u22a2 F.map ((shiftFunctorComm C a b).hom.app X) =\n    (F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) \u226b\n      (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) \u226b\n        (shiftFunctorComm D a b).hom.app (F.obj X) \u226b\n          (shiftFunctor D a).map ((F.commShiftIso b).inv.app X) \u226b (F.commShiftIso a).inv.app ((shiftFunctor C b).obj X)", "state_after": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b9\u2070 : Category.{u_6, u_1} C\ninst\u271d\u2079 : Category.{u_7, u_2} D\ninst\u271d\u2078 : Category.{?u.73897, u_3} E\nF : C \u2964 D\nG : D \u2964 E\nA : Type u_4\nB : Type u_5\ninst\u271d\u2077 : AddMonoid A\ninst\u271d\u2076 : AddCommMonoid B\ninst\u271d\u2075 : HasShift C A\ninst\u271d\u2074 : HasShift D A\ninst\u271d\u00b3 : HasShift E A\ninst\u271d\u00b2 : HasShift C B\ninst\u271d\u00b9 : HasShift D B\ninst\u271d : F.CommShift B\nX : C\na b : B\neq :\n  F.map ((shiftFunctorAdd C a b).inv.app X) \u226b (F.commShiftIso (a + b)).hom.app X =\n    (F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) \u226b\n      (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) \u226b (shiftFunctorAdd D a b).inv.app (F.obj X)\n\u22a2 F.map ((shiftFunctorComm C a b).hom.app X) =\n    (F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) \u226b\n      (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) \u226b\n        (shiftFunctorComm D a b).hom.app (F.obj X) \u226b\n          (shiftFunctor D a).map ((F.commShiftIso b).inv.app X) \u226b (F.commShiftIso a).inv.app ((shiftFunctor C b).obj X)"}, {"tactic": "simp only [shiftFunctorComm_eq D a b _ rfl]", "annotated_tactic": ["simp only [<a>shiftFunctorComm_eq</a> D a b _ <a>rfl</a>]", [{"full_name": "CategoryTheory.shiftFunctorComm_eq", "def_path": "Mathlib/CategoryTheory/Shift/Basic.lean", "def_pos": [562, 7], "def_end_pos": [562, 26]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b9\u2070 : Category.{u_6, u_1} C\ninst\u271d\u2079 : Category.{u_7, u_2} D\ninst\u271d\u2078 : Category.{?u.73897, u_3} E\nF : C \u2964 D\nG : D \u2964 E\nA : Type u_4\nB : Type u_5\ninst\u271d\u2077 : AddMonoid A\ninst\u271d\u2076 : AddCommMonoid B\ninst\u271d\u2075 : HasShift C A\ninst\u271d\u2074 : HasShift D A\ninst\u271d\u00b3 : HasShift E A\ninst\u271d\u00b2 : HasShift C B\ninst\u271d\u00b9 : HasShift D B\ninst\u271d : F.CommShift B\nX : C\na b : B\neq :\n  F.map ((shiftFunctorAdd C a b).inv.app X) \u226b (F.commShiftIso (a + b)).hom.app X =\n    (F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) \u226b\n      (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) \u226b (shiftFunctorAdd D a b).inv.app (F.obj X)\n\u22a2 F.map ((shiftFunctorComm C a b).hom.app X) =\n    (F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) \u226b\n      (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) \u226b\n        (shiftFunctorComm D a b).hom.app (F.obj X) \u226b\n          (shiftFunctor D a).map ((F.commShiftIso b).inv.app X) \u226b (F.commShiftIso a).inv.app ((shiftFunctor C b).obj X)", "state_after": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b9\u2070 : Category.{u_6, u_1} C\ninst\u271d\u2079 : Category.{u_7, u_2} D\ninst\u271d\u2078 : Category.{?u.73897, u_3} E\nF : C \u2964 D\nG : D \u2964 E\nA : Type u_4\nB : Type u_5\ninst\u271d\u2077 : AddMonoid A\ninst\u271d\u2076 : AddCommMonoid B\ninst\u271d\u2075 : HasShift C A\ninst\u271d\u2074 : HasShift D A\ninst\u271d\u00b3 : HasShift E A\ninst\u271d\u00b2 : HasShift C B\ninst\u271d\u00b9 : HasShift D B\ninst\u271d : F.CommShift B\nX : C\na b : B\neq :\n  F.map ((shiftFunctorAdd C a b).inv.app X) \u226b (F.commShiftIso (a + b)).hom.app X =\n    (F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) \u226b\n      (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) \u226b (shiftFunctorAdd D a b).inv.app (F.obj X)\n\u22a2 F.map ((shiftFunctorComm C a b).hom.app X) =\n    (F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) \u226b\n      (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) \u226b\n        ((shiftFunctorAdd' D a b (a + b) \u22ef).symm \u226a\u226b shiftFunctorAdd' D b a (a + b) \u22ef).hom.app (F.obj X) \u226b\n          (shiftFunctor D a).map ((F.commShiftIso b).inv.app X) \u226b (F.commShiftIso a).inv.app ((shiftFunctor C b).obj X)"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b9\u2070 : Category.{u_6, u_1} C\ninst\u271d\u2079 : Category.{u_7, u_2} D\ninst\u271d\u2078 : Category.{?u.73897, u_3} E\nF : C \u2964 D\nG : D \u2964 E\nA : Type u_4\nB : Type u_5\ninst\u271d\u2077 : AddMonoid A\ninst\u271d\u2076 : AddCommMonoid B\ninst\u271d\u2075 : HasShift C A\ninst\u271d\u2074 : HasShift D A\ninst\u271d\u00b3 : HasShift E A\ninst\u271d\u00b2 : HasShift C B\ninst\u271d\u00b9 : HasShift D B\ninst\u271d : F.CommShift B\nX : C\na b : B\neq :\n  F.map ((shiftFunctorAdd C a b).inv.app X) \u226b (F.commShiftIso (a + b)).hom.app X =\n    (F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) \u226b\n      (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) \u226b (shiftFunctorAdd D a b).inv.app (F.obj X)\n\u22a2 F.map ((shiftFunctorComm C a b).hom.app X) =\n    (F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) \u226b\n      (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) \u226b\n        ((shiftFunctorAdd' D a b (a + b) \u22ef).symm \u226a\u226b shiftFunctorAdd' D b a (a + b) \u22ef).hom.app (F.obj X) \u226b\n          (shiftFunctor D a).map ((F.commShiftIso b).inv.app X) \u226b (F.commShiftIso a).inv.app ((shiftFunctor C b).obj X)", "state_after": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b9\u2070 : Category.{u_6, u_1} C\ninst\u271d\u2079 : Category.{u_7, u_2} D\ninst\u271d\u2078 : Category.{?u.73897, u_3} E\nF : C \u2964 D\nG : D \u2964 E\nA : Type u_4\nB : Type u_5\ninst\u271d\u2077 : AddMonoid A\ninst\u271d\u2076 : AddCommMonoid B\ninst\u271d\u2075 : HasShift C A\ninst\u271d\u2074 : HasShift D A\ninst\u271d\u00b3 : HasShift E A\ninst\u271d\u00b2 : HasShift C B\ninst\u271d\u00b9 : HasShift D B\ninst\u271d : F.CommShift B\nX : C\na b : B\neq :\n  F.map ((shiftFunctorAdd C a b).inv.app X) \u226b (F.commShiftIso (a + b)).hom.app X =\n    (F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) \u226b\n      (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) \u226b (shiftFunctorAdd D a b).inv.app (F.obj X)\n\u22a2 F.map ((shiftFunctorComm C a b).hom.app X) =\n    (F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) \u226b\n      (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) \u226b\n        ((shiftFunctorAdd' D a b (a + b) \u22ef).inv.app (F.obj X) \u226b (shiftFunctorAdd' D b a (a + b) \u22ef).hom.app (F.obj X)) \u226b\n          (shiftFunctor D a).map ((F.commShiftIso b).inv.app X) \u226b (F.commShiftIso a).inv.app ((shiftFunctor C b).obj X)"}, {"tactic": "simp only [Functor.map_comp, shiftFunctorAdd'_eq_shiftFunctorAdd, Category.assoc,\n  \u2190 reassoc_of% eq, shiftFunctorComm_eq C a b _ rfl]", "annotated_tactic": ["simp only [<a>Functor.map_comp</a>, <a>shiftFunctorAdd'_eq_shiftFunctorAdd</a>, <a>Category.assoc</a>,\n    \u2190 reassoc_of% eq, <a>shiftFunctorComm_eq</a> C a b _ <a>rfl</a>]", [{"full_name": "CategoryTheory.Functor.map_comp", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [43, 3], "def_end_pos": [43, 11]}, {"full_name": "CategoryTheory.shiftFunctorAdd'_eq_shiftFunctorAdd", "def_path": "Mathlib/CategoryTheory/Shift/Basic.lean", "def_pos": [183, 7], "def_end_pos": [183, 42]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.shiftFunctorComm_eq", "def_path": "Mathlib/CategoryTheory/Shift/Basic.lean", "def_pos": [562, 7], "def_end_pos": [562, 26]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b9\u2070 : Category.{u_6, u_1} C\ninst\u271d\u2079 : Category.{u_7, u_2} D\ninst\u271d\u2078 : Category.{?u.73897, u_3} E\nF : C \u2964 D\nG : D \u2964 E\nA : Type u_4\nB : Type u_5\ninst\u271d\u2077 : AddMonoid A\ninst\u271d\u2076 : AddCommMonoid B\ninst\u271d\u2075 : HasShift C A\ninst\u271d\u2074 : HasShift D A\ninst\u271d\u00b3 : HasShift E A\ninst\u271d\u00b2 : HasShift C B\ninst\u271d\u00b9 : HasShift D B\ninst\u271d : F.CommShift B\nX : C\na b : B\neq :\n  F.map ((shiftFunctorAdd C a b).inv.app X) \u226b (F.commShiftIso (a + b)).hom.app X =\n    (F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) \u226b\n      (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) \u226b (shiftFunctorAdd D a b).inv.app (F.obj X)\n\u22a2 F.map ((shiftFunctorComm C a b).hom.app X) =\n    (F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) \u226b\n      (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) \u226b\n        ((shiftFunctorAdd' D a b (a + b) \u22ef).inv.app (F.obj X) \u226b (shiftFunctorAdd' D b a (a + b) \u22ef).hom.app (F.obj X)) \u226b\n          (shiftFunctor D a).map ((F.commShiftIso b).inv.app X) \u226b (F.commShiftIso a).inv.app ((shiftFunctor C b).obj X)", "state_after": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b9\u2070 : Category.{u_6, u_1} C\ninst\u271d\u2079 : Category.{u_7, u_2} D\ninst\u271d\u2078 : Category.{?u.73897, u_3} E\nF : C \u2964 D\nG : D \u2964 E\nA : Type u_4\nB : Type u_5\ninst\u271d\u2077 : AddMonoid A\ninst\u271d\u2076 : AddCommMonoid B\ninst\u271d\u2075 : HasShift C A\ninst\u271d\u2074 : HasShift D A\ninst\u271d\u00b3 : HasShift E A\ninst\u271d\u00b2 : HasShift C B\ninst\u271d\u00b9 : HasShift D B\ninst\u271d : F.CommShift B\nX : C\na b : B\neq :\n  F.map ((shiftFunctorAdd C a b).inv.app X) \u226b (F.commShiftIso (a + b)).hom.app X =\n    (F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) \u226b\n      (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) \u226b (shiftFunctorAdd D a b).inv.app (F.obj X)\n\u22a2 F.map (((shiftFunctorAdd C a b).symm \u226a\u226b shiftFunctorAdd' C b a (a + b) \u22ef).hom.app X) =\n    F.map ((shiftFunctorAdd C a b).inv.app X) \u226b\n      (F.commShiftIso (a + b)).hom.app X \u226b\n        (shiftFunctorAdd' D b a (a + b) \u22ef).hom.app (F.obj X) \u226b\n          (shiftFunctor D a).map ((F.commShiftIso b).inv.app X) \u226b (F.commShiftIso a).inv.app ((shiftFunctor C b).obj X)"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b9\u2070 : Category.{u_6, u_1} C\ninst\u271d\u2079 : Category.{u_7, u_2} D\ninst\u271d\u2078 : Category.{?u.73897, u_3} E\nF : C \u2964 D\nG : D \u2964 E\nA : Type u_4\nB : Type u_5\ninst\u271d\u2077 : AddMonoid A\ninst\u271d\u2076 : AddCommMonoid B\ninst\u271d\u2075 : HasShift C A\ninst\u271d\u2074 : HasShift D A\ninst\u271d\u00b3 : HasShift E A\ninst\u271d\u00b2 : HasShift C B\ninst\u271d\u00b9 : HasShift D B\ninst\u271d : F.CommShift B\nX : C\na b : B\neq :\n  F.map ((shiftFunctorAdd C a b).inv.app X) \u226b (F.commShiftIso (a + b)).hom.app X =\n    (F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) \u226b\n      (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) \u226b (shiftFunctorAdd D a b).inv.app (F.obj X)\n\u22a2 F.map (((shiftFunctorAdd C a b).symm \u226a\u226b shiftFunctorAdd' C b a (a + b) \u22ef).hom.app X) =\n    F.map ((shiftFunctorAdd C a b).inv.app X) \u226b\n      (F.commShiftIso (a + b)).hom.app X \u226b\n        (shiftFunctorAdd' D b a (a + b) \u22ef).hom.app (F.obj X) \u226b\n          (shiftFunctor D a).map ((F.commShiftIso b).inv.app X) \u226b (F.commShiftIso a).inv.app ((shiftFunctor C b).obj X)", "state_after": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b9\u2070 : Category.{u_6, u_1} C\ninst\u271d\u2079 : Category.{u_7, u_2} D\ninst\u271d\u2078 : Category.{?u.73897, u_3} E\nF : C \u2964 D\nG : D \u2964 E\nA : Type u_4\nB : Type u_5\ninst\u271d\u2077 : AddMonoid A\ninst\u271d\u2076 : AddCommMonoid B\ninst\u271d\u2075 : HasShift C A\ninst\u271d\u2074 : HasShift D A\ninst\u271d\u00b3 : HasShift E A\ninst\u271d\u00b2 : HasShift C B\ninst\u271d\u00b9 : HasShift D B\ninst\u271d : F.CommShift B\nX : C\na b : B\neq :\n  F.map ((shiftFunctorAdd C a b).inv.app X) \u226b (F.commShiftIso (a + b)).hom.app X =\n    (F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) \u226b\n      (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) \u226b (shiftFunctorAdd D a b).inv.app (F.obj X)\n\u22a2 F.map ((shiftFunctorAdd C a b).inv.app X \u226b (shiftFunctorAdd' C b a (a + b) \u22ef).hom.app X) =\n    F.map ((shiftFunctorAdd C a b).inv.app X) \u226b\n      (F.commShiftIso (a + b)).hom.app X \u226b\n        (shiftFunctorAdd' D b a (a + b) \u22ef).hom.app (F.obj X) \u226b\n          (shiftFunctor D a).map ((F.commShiftIso b).inv.app X) \u226b (F.commShiftIso a).inv.app ((shiftFunctor C b).obj X)"}, {"tactic": "rw [Functor.map_comp]", "annotated_tactic": ["rw [<a>Functor.map_comp</a>]", [{"full_name": "CategoryTheory.Functor.map_comp", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [43, 3], "def_end_pos": [43, 11]}]], "state_before": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b9\u2070 : Category.{u_6, u_1} C\ninst\u271d\u2079 : Category.{u_7, u_2} D\ninst\u271d\u2078 : Category.{?u.73897, u_3} E\nF : C \u2964 D\nG : D \u2964 E\nA : Type u_4\nB : Type u_5\ninst\u271d\u2077 : AddMonoid A\ninst\u271d\u2076 : AddCommMonoid B\ninst\u271d\u2075 : HasShift C A\ninst\u271d\u2074 : HasShift D A\ninst\u271d\u00b3 : HasShift E A\ninst\u271d\u00b2 : HasShift C B\ninst\u271d\u00b9 : HasShift D B\ninst\u271d : F.CommShift B\nX : C\na b : B\neq :\n  F.map ((shiftFunctorAdd C a b).inv.app X) \u226b (F.commShiftIso (a + b)).hom.app X =\n    (F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) \u226b\n      (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) \u226b (shiftFunctorAdd D a b).inv.app (F.obj X)\n\u22a2 F.map ((shiftFunctorAdd C a b).inv.app X \u226b (shiftFunctorAdd' C b a (a + b) \u22ef).hom.app X) =\n    F.map ((shiftFunctorAdd C a b).inv.app X) \u226b\n      (F.commShiftIso (a + b)).hom.app X \u226b\n        (shiftFunctorAdd' D b a (a + b) \u22ef).hom.app (F.obj X) \u226b\n          (shiftFunctor D a).map ((F.commShiftIso b).inv.app X) \u226b (F.commShiftIso a).inv.app ((shiftFunctor C b).obj X)", "state_after": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b9\u2070 : Category.{u_6, u_1} C\ninst\u271d\u2079 : Category.{u_7, u_2} D\ninst\u271d\u2078 : Category.{?u.73897, u_3} E\nF : C \u2964 D\nG : D \u2964 E\nA : Type u_4\nB : Type u_5\ninst\u271d\u2077 : AddMonoid A\ninst\u271d\u2076 : AddCommMonoid B\ninst\u271d\u2075 : HasShift C A\ninst\u271d\u2074 : HasShift D A\ninst\u271d\u00b3 : HasShift E A\ninst\u271d\u00b2 : HasShift C B\ninst\u271d\u00b9 : HasShift D B\ninst\u271d : F.CommShift B\nX : C\na b : B\neq :\n  F.map ((shiftFunctorAdd C a b).inv.app X) \u226b (F.commShiftIso (a + b)).hom.app X =\n    (F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) \u226b\n      (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) \u226b (shiftFunctorAdd D a b).inv.app (F.obj X)\n\u22a2 F.map ((shiftFunctorAdd C a b).inv.app X) \u226b F.map ((shiftFunctorAdd' C b a (a + b) \u22ef).hom.app X) =\n    F.map ((shiftFunctorAdd C a b).inv.app X) \u226b\n      (F.commShiftIso (a + b)).hom.app X \u226b\n        (shiftFunctorAdd' D b a (a + b) \u22ef).hom.app (F.obj X) \u226b\n          (shiftFunctor D a).map ((F.commShiftIso b).inv.app X) \u226b (F.commShiftIso a).inv.app ((shiftFunctor C b).obj X)"}, {"tactic": "simp only [NatTrans.congr_app (congr_arg Iso.hom (F.commShiftIso_add' (add_comm b a))) X,\n  CommShift.isoAdd'_hom_app, Category.assoc, Iso.inv_hom_id_app_assoc,\n  \u2190 Functor.map_comp_assoc, Iso.hom_inv_id_app,\n  Functor.map_id, Category.id_comp, comp_obj, Category.comp_id]", "annotated_tactic": ["simp only [<a>NatTrans.congr_app</a> (<a>congr_arg</a> <a>Iso.hom</a> (F.commShiftIso_add' (<a>add_comm</a> b a))) X,\n    <a>CommShift.isoAdd'_hom_app</a>, <a>Category.assoc</a>, <a>Iso.inv_hom_id_app_assoc</a>,\n    \u2190 <a>Functor.map_comp_assoc</a>, <a>Iso.hom_inv_id_app</a>,\n    <a>Functor.map_id</a>, <a>Category.id_comp</a>, <a>comp_obj</a>, <a>Category.comp_id</a>]", [{"full_name": "CategoryTheory.NatTrans.congr_app", "def_path": "Mathlib/CategoryTheory/Functor/Category.lean", "def_pos": [68, 9], "def_end_pos": [68, 18]}, {"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "CategoryTheory.Iso.hom", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [53, 3], "def_end_pos": [53, 6]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "CategoryTheory.Functor.CommShift.isoAdd'_hom_app", "def_path": "Mathlib/CategoryTheory/Shift/CommShift.lean", "def_pos": [55, 3], "def_end_pos": [55, 9]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.Iso.inv_hom_id_app_assoc", "def_path": "Mathlib/CategoryTheory/NatIso.lean", "def_pos": [70, 3], "def_end_pos": [70, 25]}, {"full_name": "CategoryTheory.Functor.map_comp_assoc", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [64, 7], "def_end_pos": [64, 29]}, {"full_name": "CategoryTheory.Iso.hom_inv_id_app", "def_path": "Mathlib/CategoryTheory/NatIso.lean", "def_pos": [64, 9], "def_end_pos": [64, 23]}, {"full_name": "CategoryTheory.Functor.map_id", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [41, 3], "def_end_pos": [41, 9]}, {"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [177, 3], "def_end_pos": [177, 10]}, {"full_name": "CategoryTheory.Functor.comp_obj", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [110, 9], "def_end_pos": [110, 12]}, {"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [179, 3], "def_end_pos": [179, 10]}]], "state_before": "C : Type u_1\nD : Type u_2\nE : Type u_3\ninst\u271d\u00b9\u2070 : Category.{u_6, u_1} C\ninst\u271d\u2079 : Category.{u_7, u_2} D\ninst\u271d\u2078 : Category.{?u.73897, u_3} E\nF : C \u2964 D\nG : D \u2964 E\nA : Type u_4\nB : Type u_5\ninst\u271d\u2077 : AddMonoid A\ninst\u271d\u2076 : AddCommMonoid B\ninst\u271d\u2075 : HasShift C A\ninst\u271d\u2074 : HasShift D A\ninst\u271d\u00b3 : HasShift E A\ninst\u271d\u00b2 : HasShift C B\ninst\u271d\u00b9 : HasShift D B\ninst\u271d : F.CommShift B\nX : C\na b : B\neq :\n  F.map ((shiftFunctorAdd C a b).inv.app X) \u226b (F.commShiftIso (a + b)).hom.app X =\n    (F.commShiftIso b).hom.app ((shiftFunctor C a).obj X) \u226b\n      (shiftFunctor D b).map ((F.commShiftIso a).hom.app X) \u226b (shiftFunctorAdd D a b).inv.app (F.obj X)\n\u22a2 F.map ((shiftFunctorAdd C a b).inv.app X) \u226b F.map ((shiftFunctorAdd' C b a (a + b) \u22ef).hom.app X) =\n    F.map ((shiftFunctorAdd C a b).inv.app X) \u226b\n      (F.commShiftIso (a + b)).hom.app X \u226b\n        (shiftFunctorAdd' D b a (a + b) \u22ef).hom.app (F.obj X) \u226b\n          (shiftFunctor D a).map ((F.commShiftIso b).inv.app X) \u226b (F.commShiftIso a).inv.app ((shiftFunctor C b).obj X)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/FiberBundle/Constructions.lean", "full_name": "Trivialization.Prod.continuous_to_fun", "start": [148, 1], "end": [166, 12], "traced_tactics": [{"tactic": "let f\u2081 : TotalSpace (F\u2081 \u00d7 F\u2082) (E\u2081 \u00d7\u1d47 E\u2082) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p \u21a6 ((\u27e8p.1, p.2.1\u27e9 : TotalSpace F\u2081 E\u2081), (\u27e8p.1, p.2.2\u27e9 : TotalSpace F\u2082 E\u2082))", "annotated_tactic": ["let f\u2081 : <a>TotalSpace</a> (F\u2081 \u00d7 F\u2082) (E\u2081 \u00d7\u1d47 E\u2082) \u2192 <a>TotalSpace</a> F\u2081 E\u2081 \u00d7 <a>TotalSpace</a> F\u2082 E\u2082 :=\n    fun p \u21a6 ((\u27e8p.1, p.2.1\u27e9 : <a>TotalSpace</a> F\u2081 E\u2081), (\u27e8p.1, p.2.2\u27e9 : <a>TotalSpace</a> F\u2082 E\u2082))", [{"full_name": "Bundle.TotalSpace", "def_path": "Mathlib/Data/Bundle.lean", "def_pos": [52, 11], "def_end_pos": [52, 21]}, {"full_name": "Bundle.TotalSpace", "def_path": "Mathlib/Data/Bundle.lean", "def_pos": [52, 11], "def_end_pos": [52, 21]}, {"full_name": "Bundle.TotalSpace", "def_path": "Mathlib/Data/Bundle.lean", "def_pos": [52, 11], "def_end_pos": [52, 21]}, {"full_name": "Bundle.TotalSpace", "def_path": "Mathlib/Data/Bundle.lean", "def_pos": [52, 11], "def_end_pos": [52, 21]}, {"full_name": "Bundle.TotalSpace", "def_path": "Mathlib/Data/Bundle.lean", "def_pos": [52, 11], "def_end_pos": [52, 21]}]], "state_before": "B : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\n\u22a2 ContinuousOn (toFun' e\u2081 e\u2082) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))", "state_after": "B : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\n\u22a2 ContinuousOn (toFun' e\u2081 e\u2082) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))"}, {"tactic": "let f\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p \u21a6 \u27e8e\u2081 p.1, e\u2082 p.2\u27e9", "annotated_tactic": ["let f\u2082 : <a>TotalSpace</a> F\u2081 E\u2081 \u00d7 <a>TotalSpace</a> F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p \u21a6 \u27e8e\u2081 p.1, e\u2082 p.2\u27e9", [{"full_name": "Bundle.TotalSpace", "def_path": "Mathlib/Data/Bundle.lean", "def_pos": [52, 11], "def_end_pos": [52, 21]}, {"full_name": "Bundle.TotalSpace", "def_path": "Mathlib/Data/Bundle.lean", "def_pos": [52, 11], "def_end_pos": [52, 21]}]], "state_before": "B : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\n\u22a2 ContinuousOn (toFun' e\u2081 e\u2082) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))", "state_after": "B : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\n\u22a2 ContinuousOn (toFun' e\u2081 e\u2082) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))"}, {"tactic": "let f\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p \u21a6 \u27e8p.1.1, p.1.2, p.2.2\u27e9", "annotated_tactic": ["let f\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p \u21a6 \u27e8p.1.1, p.1.2, p.2.2\u27e9", []], "state_before": "B : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\n\u22a2 ContinuousOn (toFun' e\u2081 e\u2082) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))", "state_after": "B : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\n\u22a2 ContinuousOn (toFun' e\u2081 e\u2082) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))"}, {"tactic": "have hf\u2081 : Continuous f\u2081 := (Prod.inducing_diag F\u2081 E\u2081 F\u2082 E\u2082).continuous", "annotated_tactic": ["have hf\u2081 : <a>Continuous</a> f\u2081 := (<a>Prod.inducing_diag</a> F\u2081 E\u2081 F\u2082 E\u2082).<a>continuous</a>", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Defs/Basic.lean", "def_pos": [137, 11], "def_end_pos": [137, 21]}, {"full_name": "FiberBundle.Prod.inducing_diag", "def_path": "Mathlib/Topology/FiberBundle/Constructions.lean", "def_pos": [120, 9], "def_end_pos": [120, 39]}, {"full_name": "Inducing.continuous", "def_path": "Mathlib/Topology/Maps.lean", "def_pos": [141, 19], "def_end_pos": [141, 29]}]], "state_before": "B : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\n\u22a2 ContinuousOn (toFun' e\u2081 e\u2082) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))", "state_after": "B : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\nhf\u2081 : Continuous f\u2081\n\u22a2 ContinuousOn (toFun' e\u2081 e\u2082) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))"}, {"tactic": "have hf\u2082 : ContinuousOn f\u2082 (e\u2081.source \u00d7\u02e2 e\u2082.source) :=\n  e\u2081.toPartialHomeomorph.continuousOn.prod_map e\u2082.toPartialHomeomorph.continuousOn", "annotated_tactic": ["have hf\u2082 : <a>ContinuousOn</a> f\u2082 (e\u2081.source \u00d7\u02e2 e\u2082.source) :=\n    e\u2081.toPartialHomeomorph.continuousOn.prod_map e\u2082.toPartialHomeomorph.continuousOn", [{"full_name": "ContinuousOn", "def_path": "Mathlib/Topology/Defs/Filter.lean", "def_pos": [170, 5], "def_end_pos": [170, 17]}]], "state_before": "B : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\nhf\u2081 : Continuous f\u2081\n\u22a2 ContinuousOn (toFun' e\u2081 e\u2082) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))", "state_after": "B : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\nhf\u2081 : Continuous f\u2081\nhf\u2082 : ContinuousOn f\u2082 (e\u2081.source \u00d7\u02e2 e\u2082.source)\n\u22a2 ContinuousOn (toFun' e\u2081 e\u2082) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))"}, {"tactic": "have hf\u2083 : Continuous f\u2083 :=\n  (continuous_fst.comp continuous_fst).prod_mk (continuous_snd.prod_map continuous_snd)", "annotated_tactic": ["have hf\u2083 : <a>Continuous</a> f\u2083 :=\n    (continuous_fst.comp <a>continuous_fst</a>).<a>prod_mk</a> (continuous_snd.prod_map <a>continuous_snd</a>)", [{"full_name": "Continuous", "def_path": "Mathlib/Topology/Defs/Basic.lean", "def_pos": [137, 11], "def_end_pos": [137, 21]}, {"full_name": "continuous_fst", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [339, 9], "def_end_pos": [339, 23]}, {"full_name": "Continuous.prod_mk", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [425, 9], "def_end_pos": [425, 27]}, {"full_name": "continuous_snd", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [382, 9], "def_end_pos": [382, 23]}]], "state_before": "B : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\nhf\u2081 : Continuous f\u2081\nhf\u2082 : ContinuousOn f\u2082 (e\u2081.source \u00d7\u02e2 e\u2082.source)\n\u22a2 ContinuousOn (toFun' e\u2081 e\u2082) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))", "state_after": "B : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\nhf\u2081 : Continuous f\u2081\nhf\u2082 : ContinuousOn f\u2082 (e\u2081.source \u00d7\u02e2 e\u2082.source)\nhf\u2083 : Continuous f\u2083\n\u22a2 ContinuousOn (toFun' e\u2081 e\u2082) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))"}, {"tactic": "refine ((hf\u2083.comp_continuousOn hf\u2082).comp hf\u2081.continuousOn ?_).congr ?_", "annotated_tactic": ["refine ((hf\u2083.comp_continuousOn hf\u2082).<a>comp</a> hf\u2081.continuousOn ?_).<a>congr</a> ?_", [{"full_name": "ContinuousOn.comp", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [943, 9], "def_end_pos": [943, 26]}, {"full_name": "ContinuousOn.congr", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [887, 9], "def_end_pos": [887, 27]}]], "state_before": "B : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\nhf\u2081 : Continuous f\u2081\nhf\u2082 : ContinuousOn f\u2082 (e\u2081.source \u00d7\u02e2 e\u2082.source)\nhf\u2083 : Continuous f\u2083\n\u22a2 ContinuousOn (toFun' e\u2081 e\u2082) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))", "state_after": "case refine_1\nB : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\nhf\u2081 : Continuous f\u2081\nhf\u2082 : ContinuousOn f\u2082 (e\u2081.source \u00d7\u02e2 e\u2082.source)\nhf\u2083 : Continuous f\u2083\n\u22a2 MapsTo f\u2081 (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet)) (e\u2081.source \u00d7\u02e2 e\u2082.source)\n\ncase refine_2\nB : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\nhf\u2081 : Continuous f\u2081\nhf\u2082 : ContinuousOn f\u2082 (e\u2081.source \u00d7\u02e2 e\u2082.source)\nhf\u2083 : Continuous f\u2083\n\u22a2 EqOn (toFun' e\u2081 e\u2082) ((f\u2083 \u2218 f\u2082) \u2218 f\u2081) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))"}, {"tactic": "rintro \u27e8b, v\u2081, v\u2082\u27e9 \u27e8hb\u2081, _\u27e9", "annotated_tactic": ["rintro \u27e8b, v\u2081, v\u2082\u27e9 \u27e8hb\u2081, _\u27e9", []], "state_before": "case refine_2\nB : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\nhf\u2081 : Continuous f\u2081\nhf\u2082 : ContinuousOn f\u2082 (e\u2081.source \u00d7\u02e2 e\u2082.source)\nhf\u2083 : Continuous f\u2083\n\u22a2 EqOn (toFun' e\u2081 e\u2082) ((f\u2083 \u2218 f\u2082) \u2218 f\u2081) (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))", "state_after": "case refine_2.mk.mk.intro\nB : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\nhf\u2081 : Continuous f\u2081\nhf\u2082 : ContinuousOn f\u2082 (e\u2081.source \u00d7\u02e2 e\u2082.source)\nhf\u2083 : Continuous f\u2083\nb : B\nv\u2081 : E\u2081 b\nv\u2082 : E\u2082 b\nhb\u2081 : { proj := b, snd := (v\u2081, v\u2082) }.proj \u2208 e\u2081.baseSet\nright\u271d : { proj := b, snd := (v\u2081, v\u2082) }.proj \u2208 e\u2082.baseSet\n\u22a2 toFun' e\u2081 e\u2082 { proj := b, snd := (v\u2081, v\u2082) } = ((f\u2083 \u2218 f\u2082) \u2218 f\u2081) { proj := b, snd := (v\u2081, v\u2082) }"}, {"tactic": "simp only [f\u2083, Prod.toFun', Prod.mk.inj_iff, Function.comp_apply, and_true_iff]", "annotated_tactic": ["simp only [f\u2083, <a>Prod.toFun'</a>, <a>Prod.mk.inj_iff</a>, <a>Function.comp_apply</a>, <a>and_true_iff</a>]", [{"full_name": "Trivialization.Prod.toFun'", "def_path": "Mathlib/Topology/FiberBundle/Constructions.lean", "def_pos": [142, 5], "def_end_pos": [142, 16]}, {"full_name": "Prod.mk.inj_iff", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [103, 9], "def_end_pos": [103, 19]}, {"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}, {"full_name": "and_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [116, 9], "def_end_pos": [116, 21]}]], "state_before": "case refine_2.mk.mk.intro\nB : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\nhf\u2081 : Continuous f\u2081\nhf\u2082 : ContinuousOn f\u2082 (e\u2081.source \u00d7\u02e2 e\u2082.source)\nhf\u2083 : Continuous f\u2083\nb : B\nv\u2081 : E\u2081 b\nv\u2082 : E\u2082 b\nhb\u2081 : { proj := b, snd := (v\u2081, v\u2082) }.proj \u2208 e\u2081.baseSet\nright\u271d : { proj := b, snd := (v\u2081, v\u2082) }.proj \u2208 e\u2082.baseSet\n\u22a2 toFun' e\u2081 e\u2082 { proj := b, snd := (v\u2081, v\u2082) } = ((f\u2083 \u2218 f\u2082) \u2218 f\u2081) { proj := b, snd := (v\u2081, v\u2082) }", "state_after": "case refine_2.mk.mk.intro\nB : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\nhf\u2081 : Continuous f\u2081\nhf\u2082 : ContinuousOn f\u2082 (e\u2081.source \u00d7\u02e2 e\u2082.source)\nhf\u2083 : Continuous f\u2083\nb : B\nv\u2081 : E\u2081 b\nv\u2082 : E\u2082 b\nhb\u2081 : { proj := b, snd := (v\u2081, v\u2082) }.proj \u2208 e\u2081.baseSet\nright\u271d : { proj := b, snd := (v\u2081, v\u2082) }.proj \u2208 e\u2082.baseSet\n\u22a2 b = (\u2191e\u2081 { proj := b, snd := v\u2081 }).1"}, {"tactic": "rw [e\u2081.coe_fst]", "annotated_tactic": ["rw [e\u2081.coe_fst]", []], "state_before": "case refine_2.mk.mk.intro\nB : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\nhf\u2081 : Continuous f\u2081\nhf\u2082 : ContinuousOn f\u2082 (e\u2081.source \u00d7\u02e2 e\u2082.source)\nhf\u2083 : Continuous f\u2083\nb : B\nv\u2081 : E\u2081 b\nv\u2082 : E\u2082 b\nhb\u2081 : { proj := b, snd := (v\u2081, v\u2082) }.proj \u2208 e\u2081.baseSet\nright\u271d : { proj := b, snd := (v\u2081, v\u2082) }.proj \u2208 e\u2082.baseSet\n\u22a2 b = (\u2191e\u2081 { proj := b, snd := v\u2081 }).1", "state_after": "case refine_2.mk.mk.intro\nB : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\nhf\u2081 : Continuous f\u2081\nhf\u2082 : ContinuousOn f\u2082 (e\u2081.source \u00d7\u02e2 e\u2082.source)\nhf\u2083 : Continuous f\u2083\nb : B\nv\u2081 : E\u2081 b\nv\u2082 : E\u2082 b\nhb\u2081 : { proj := b, snd := (v\u2081, v\u2082) }.proj \u2208 e\u2081.baseSet\nright\u271d : { proj := b, snd := (v\u2081, v\u2082) }.proj \u2208 e\u2082.baseSet\n\u22a2 { proj := b, snd := v\u2081 } \u2208 e\u2081.source"}, {"tactic": "rw [e\u2081.source_eq, mem_preimage]", "annotated_tactic": ["rw [e\u2081.source_eq, <a>mem_preimage</a>]", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [120, 9], "def_end_pos": [120, 21]}]], "state_before": "case refine_2.mk.mk.intro\nB : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\nhf\u2081 : Continuous f\u2081\nhf\u2082 : ContinuousOn f\u2082 (e\u2081.source \u00d7\u02e2 e\u2082.source)\nhf\u2083 : Continuous f\u2083\nb : B\nv\u2081 : E\u2081 b\nv\u2082 : E\u2082 b\nhb\u2081 : { proj := b, snd := (v\u2081, v\u2082) }.proj \u2208 e\u2081.baseSet\nright\u271d : { proj := b, snd := (v\u2081, v\u2082) }.proj \u2208 e\u2082.baseSet\n\u22a2 { proj := b, snd := v\u2081 } \u2208 e\u2081.source", "state_after": "case refine_2.mk.mk.intro\nB : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\nhf\u2081 : Continuous f\u2081\nhf\u2082 : ContinuousOn f\u2082 (e\u2081.source \u00d7\u02e2 e\u2082.source)\nhf\u2083 : Continuous f\u2083\nb : B\nv\u2081 : E\u2081 b\nv\u2082 : E\u2082 b\nhb\u2081 : { proj := b, snd := (v\u2081, v\u2082) }.proj \u2208 e\u2081.baseSet\nright\u271d : { proj := b, snd := (v\u2081, v\u2082) }.proj \u2208 e\u2082.baseSet\n\u22a2 { proj := b, snd := v\u2081 }.proj \u2208 e\u2081.baseSet"}, {"tactic": "exact hb\u2081", "annotated_tactic": ["exact hb\u2081", []], "state_before": "case refine_2.mk.mk.intro\nB : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\nhf\u2081 : Continuous f\u2081\nhf\u2082 : ContinuousOn f\u2082 (e\u2081.source \u00d7\u02e2 e\u2082.source)\nhf\u2083 : Continuous f\u2083\nb : B\nv\u2081 : E\u2081 b\nv\u2082 : E\u2082 b\nhb\u2081 : { proj := b, snd := (v\u2081, v\u2082) }.proj \u2208 e\u2081.baseSet\nright\u271d : { proj := b, snd := (v\u2081, v\u2082) }.proj \u2208 e\u2082.baseSet\n\u22a2 { proj := b, snd := v\u2081 }.proj \u2208 e\u2081.baseSet", "state_after": "no goals"}, {"tactic": "rw [e\u2081.source_eq, e\u2082.source_eq]", "annotated_tactic": ["rw [e\u2081.source_eq, e\u2082.source_eq]", []], "state_before": "case refine_1\nB : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\nhf\u2081 : Continuous f\u2081\nhf\u2082 : ContinuousOn f\u2082 (e\u2081.source \u00d7\u02e2 e\u2082.source)\nhf\u2083 : Continuous f\u2083\n\u22a2 MapsTo f\u2081 (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet)) (e\u2081.source \u00d7\u02e2 e\u2082.source)", "state_after": "case refine_1\nB : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\nhf\u2081 : Continuous f\u2081\nhf\u2082 : ContinuousOn f\u2082 (e\u2081.source \u00d7\u02e2 e\u2082.source)\nhf\u2083 : Continuous f\u2083\n\u22a2 MapsTo f\u2081 (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))\n    ((TotalSpace.proj \u207b\u00b9' e\u2081.baseSet) \u00d7\u02e2 (TotalSpace.proj \u207b\u00b9' e\u2082.baseSet))"}, {"tactic": "exact mapsTo_preimage _ _", "annotated_tactic": ["exact <a>mapsTo_preimage</a> _ _", [{"full_name": "Set.mapsTo_preimage", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [273, 9], "def_end_pos": [273, 24]}]], "state_before": "case refine_1\nB : Type u_1\ninst\u271d\u2074 : TopologicalSpace B\nF\u2081 : Type u_2\ninst\u271d\u00b3 : TopologicalSpace F\u2081\nE\u2081 : B \u2192 Type u_3\ninst\u271d\u00b2 : TopologicalSpace (TotalSpace F\u2081 E\u2081)\nF\u2082 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace F\u2082\nE\u2082 : B \u2192 Type u_5\ninst\u271d : TopologicalSpace (TotalSpace F\u2082 E\u2082)\ne\u2081 : Trivialization F\u2081 TotalSpace.proj\ne\u2082 : Trivialization F\u2082 TotalSpace.proj\nf\u2081 : (TotalSpace (F\u2081 \u00d7 F\u2082) fun x => E\u2081 x \u00d7 E\u2082 x) \u2192 TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 :=\n  fun p => ({ proj := p.proj, snd := p.snd.1 }, { proj := p.proj, snd := p.snd.2 })\nf\u2082 : TotalSpace F\u2081 E\u2081 \u00d7 TotalSpace F\u2082 E\u2082 \u2192 (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 := fun p => (\u2191e\u2081 p.1, \u2191e\u2082 p.2)\nf\u2083 : (B \u00d7 F\u2081) \u00d7 B \u00d7 F\u2082 \u2192 B \u00d7 F\u2081 \u00d7 F\u2082 := fun p => (p.1.1, p.1.2, p.2.2)\nhf\u2081 : Continuous f\u2081\nhf\u2082 : ContinuousOn f\u2082 (e\u2081.source \u00d7\u02e2 e\u2082.source)\nhf\u2083 : Continuous f\u2083\n\u22a2 MapsTo f\u2081 (TotalSpace.proj \u207b\u00b9' (e\u2081.baseSet \u2229 e\u2082.baseSet))\n    ((TotalSpace.proj \u207b\u00b9' e\u2081.baseSet) \u00d7\u02e2 (TotalSpace.proj \u207b\u00b9' e\u2082.baseSet))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "MeasureTheory.Measure.map_prod_map", "start": [817, 1], "end": [826, 38], "traced_tactics": [{"tactic": "simp_rw [\u2190 sum_sFiniteSeq \u03bca, \u2190 sum_sFiniteSeq \u03bcc, map_sum hf.aemeasurable,\n  map_sum hg.aemeasurable, prod_sum, map_sum (hf.prod_map hg).aemeasurable]", "annotated_tactic": ["simp_rw [\u2190 <a>sum_sFiniteSeq</a> \u03bca, \u2190 <a>sum_sFiniteSeq</a> \u03bcc, <a>map_sum</a> hf.aemeasurable,\n    <a>map_sum</a> hg.aemeasurable, <a>prod_sum</a>, <a>map_sum</a> (hf.prod_map hg).<a>aemeasurable</a>]", [{"full_name": "MeasureTheory.sum_sFiniteSeq", "def_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "def_pos": [568, 7], "def_end_pos": [568, 21]}, {"full_name": "MeasureTheory.sum_sFiniteSeq", "def_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "def_pos": [568, 7], "def_end_pos": [568, 21]}, {"full_name": "MeasureTheory.Measure.map_sum", "def_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "def_pos": [449, 7], "def_end_pos": [449, 14]}, {"full_name": "MeasureTheory.Measure.map_sum", "def_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "def_pos": [449, 7], "def_end_pos": [449, 14]}, {"full_name": "MeasureTheory.Measure.prod_sum", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [599, 7], "def_end_pos": [599, 15]}, {"full_name": "MeasureTheory.Measure.map_sum", "def_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "def_pos": [449, 7], "def_end_pos": [449, 14]}, {"full_name": "Measurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [423, 9], "def_end_pos": [423, 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\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 : SFinite \u03bd\ninst\u271d\u00b3 : SFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SFinite \u03bca\ninst\u271d : SFinite \u03bcc\nhf : Measurable f\nhg : Measurable g\n\u22a2 (map f \u03bca).prod (map g \u03bcc) = map (Prod.map f g) (\u03bca.prod \u03bcc)", "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 : SFinite \u03bd\ninst\u271d\u00b3 : SFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SFinite \u03bca\ninst\u271d : SFinite \u03bcc\nhf : Measurable f\nhg : Measurable g\n\u22a2 (sum fun p => (map f (sFiniteSeq \u03bca p.1)).prod (map g (sFiniteSeq \u03bcc p.2))) =\n    sum fun i => map (Prod.map f g) ((sFiniteSeq \u03bca i.1).prod (sFiniteSeq \u03bcc i.2))"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "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 : SFinite \u03bd\ninst\u271d\u00b3 : SFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SFinite \u03bca\ninst\u271d : SFinite \u03bcc\nhf : Measurable f\nhg : Measurable g\n\u22a2 (sum fun p => (map f (sFiniteSeq \u03bca p.1)).prod (map g (sFiniteSeq \u03bcc p.2))) =\n    sum fun i => map (Prod.map f g) ((sFiniteSeq \u03bca i.1).prod (sFiniteSeq \u03bcc i.2))", "state_after": "case e_f\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 : SFinite \u03bd\ninst\u271d\u00b3 : SFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SFinite \u03bca\ninst\u271d : SFinite \u03bcc\nhf : Measurable f\nhg : Measurable g\n\u22a2 (fun p => (map f (sFiniteSeq \u03bca p.1)).prod (map g (sFiniteSeq \u03bcc p.2))) = fun i =>\n    map (Prod.map f g) ((sFiniteSeq \u03bca i.1).prod (sFiniteSeq \u03bcc i.2))"}, {"tactic": "ext1 i", "annotated_tactic": ["ext1 i", []], "state_before": "case e_f\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 : SFinite \u03bd\ninst\u271d\u00b3 : SFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SFinite \u03bca\ninst\u271d : SFinite \u03bcc\nhf : Measurable f\nhg : Measurable g\n\u22a2 (fun p => (map f (sFiniteSeq \u03bca p.1)).prod (map g (sFiniteSeq \u03bcc p.2))) = fun i =>\n    map (Prod.map f g) ((sFiniteSeq \u03bca i.1).prod (sFiniteSeq \u03bcc i.2))", "state_after": "case e_f.h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\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 : SFinite \u03bd\ninst\u271d\u00b3 : SFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SFinite \u03bca\ninst\u271d : SFinite \u03bcc\nhf : Measurable f\nhg : Measurable g\ni : \u2115 \u00d7 \u2115\n\u22a2 (map f (sFiniteSeq \u03bca i.1)).prod (map g (sFiniteSeq \u03bcc i.2)) =\n    map (Prod.map f g) ((sFiniteSeq \u03bca i.1).prod (sFiniteSeq \u03bcc i.2))"}, {"tactic": "refine prod_eq fun s t hs ht => ?_", "annotated_tactic": ["refine <a>prod_eq</a> fun s t hs ht => ?_", [{"full_name": "MeasureTheory.Measure.prod_eq", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [653, 9], "def_end_pos": [653, 16]}]], "state_before": "case e_f.h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\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 : SFinite \u03bd\ninst\u271d\u00b3 : SFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SFinite \u03bca\ninst\u271d : SFinite \u03bcc\nhf : Measurable f\nhg : Measurable g\ni : \u2115 \u00d7 \u2115\n\u22a2 (map f (sFiniteSeq \u03bca i.1)).prod (map g (sFiniteSeq \u03bcc i.2)) =\n    map (Prod.map f g) ((sFiniteSeq \u03bca i.1).prod (sFiniteSeq \u03bcc i.2))", "state_after": "case e_f.h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\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 : SFinite \u03bd\ninst\u271d\u00b3 : SFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SFinite \u03bca\ninst\u271d : SFinite \u03bcc\nhf : Measurable f\nhg : Measurable g\ni : \u2115 \u00d7 \u2115\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 (map (Prod.map f g) ((sFiniteSeq \u03bca i.1).prod (sFiniteSeq \u03bcc i.2))) (s \u00d7\u02e2 t) =\n    (map f (sFiniteSeq \u03bca i.1)) s * (map g (sFiniteSeq \u03bcc i.2)) t"}, {"tactic": "rw [map_apply (hf.prod_map hg) (hs.prod ht), map_apply hf hs, map_apply hg ht]", "annotated_tactic": ["rw [<a>map_apply</a> (hf.prod_map hg) (hs.prod ht), <a>map_apply</a> hf hs, <a>map_apply</a> hg ht]", [{"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1291, 9], "def_end_pos": [1291, 18]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1291, 9], "def_end_pos": [1291, 18]}, {"full_name": "MeasureTheory.Measure.map_apply", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1291, 9], "def_end_pos": [1291, 18]}]], "state_before": "case e_f.h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\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 : SFinite \u03bd\ninst\u271d\u00b3 : SFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SFinite \u03bca\ninst\u271d : SFinite \u03bcc\nhf : Measurable f\nhg : Measurable g\ni : \u2115 \u00d7 \u2115\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 (map (Prod.map f g) ((sFiniteSeq \u03bca i.1).prod (sFiniteSeq \u03bcc i.2))) (s \u00d7\u02e2 t) =\n    (map f (sFiniteSeq \u03bca i.1)) s * (map g (sFiniteSeq \u03bcc i.2)) t", "state_after": "case e_f.h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\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 : SFinite \u03bd\ninst\u271d\u00b3 : SFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SFinite \u03bca\ninst\u271d : SFinite \u03bcc\nhf : Measurable f\nhg : Measurable g\ni : \u2115 \u00d7 \u2115\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 ((sFiniteSeq \u03bca i.1).prod (sFiniteSeq \u03bcc i.2)) (Prod.map f g \u207b\u00b9' s \u00d7\u02e2 t) =\n    (sFiniteSeq \u03bca i.1) (f \u207b\u00b9' s) * (sFiniteSeq \u03bcc i.2) (g \u207b\u00b9' t)"}, {"tactic": "exact prod_prod (f \u207b\u00b9' s) (g \u207b\u00b9' t)", "annotated_tactic": ["exact <a>prod_prod</a> (f \u207b\u00b9' s) (g \u207b\u00b9' t)", [{"full_name": "MeasureTheory.Measure.prod_prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [350, 9], "def_end_pos": [350, 18]}]], "state_before": "case e_f.h\n\u03b1 : Type u_1\n\u03b1' : Type u_2\n\u03b2 : Type u_3\n\u03b2' : Type u_4\n\u03b3 : Type u_5\nE : Type u_6\ninst\u271d\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 : SFinite \u03bd\ninst\u271d\u00b3 : SFinite \u03bc\n\u03b4 : Type u_7\ninst\u271d\u00b2 : MeasurableSpace \u03b4\nf : \u03b1 \u2192 \u03b2\ng : \u03b3 \u2192 \u03b4\n\u03bca : Measure \u03b1\n\u03bcc : Measure \u03b3\ninst\u271d\u00b9 : SFinite \u03bca\ninst\u271d : SFinite \u03bcc\nhf : Measurable f\nhg : Measurable g\ni : \u2115 \u00d7 \u2115\ns : Set \u03b2\nt : Set \u03b4\nhs : MeasurableSet s\nht : MeasurableSet t\n\u22a2 ((sFiniteSeq \u03bca i.1).prod (sFiniteSeq \u03bcc i.2)) (Prod.map f g \u207b\u00b9' s \u00d7\u02e2 t) =\n    (sFiniteSeq \u03bca i.1) (f \u207b\u00b9' s) * (sFiniteSeq \u03bcc i.2) (g \u207b\u00b9' t)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.encard_eq_encard_iff_encard_diff_eq_encard_diff", "start": [174, 1], "end": [177, 53], "traced_tactics": [{"tactic": "rw [\u2190 encard_diff_add_encard_inter s t, \u2190 encard_diff_add_encard_inter t s, inter_comm t s,\n  WithTop.add_right_cancel_iff h.encard_lt_top.ne]", "annotated_tactic": ["rw [\u2190 <a>encard_diff_add_encard_inter</a> s t, \u2190 <a>encard_diff_add_encard_inter</a> t s, <a>inter_comm</a> t s,\n    <a>WithTop.add_right_cancel_iff</a> h.encard_lt_top.ne]", [{"full_name": "Set.encard_diff_add_encard_inter", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [164, 9], "def_end_pos": [164, 37]}, {"full_name": "Set.encard_diff_add_encard_inter", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [164, 9], "def_end_pos": [164, 37]}, {"full_name": "Set.inter_comm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [905, 9], "def_end_pos": [905, 19]}, {"full_name": "WithTop.add_right_cancel_iff", "def_path": "Mathlib/Algebra/Order/Monoid/WithTop.lean", "def_pos": [164, 9], "def_end_pos": [164, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\nh : (s \u2229 t).Finite\n\u22a2 s.encard = t.encard \u2194 (s \\ t).encard = (t \\ s).encard", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/SupClosed.lean", "full_name": "LinearOrder.isSublattice", "start": [248, 1], "end": [249, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/PushoutI.lean", "full_name": "Monoid.PushoutI.NormalWord.ext", "start": [277, 1], "end": [281, 11], "traced_tactics": [{"tactic": "rcases w\u2081 with \u27e8\u27e8_, _, _\u27e9, _, _\u27e9", "annotated_tactic": ["rcases w\u2081 with \u27e8\u27e8_, _, _\u27e9, _, _\u27e9", []], "state_before": "\u03b9 : Type u_1\nG : \u03b9 \u2192 Type u_2\nH : Type u_3\nK : Type u_4\ninst\u271d\u2074 : Monoid K\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Group (G i)\ninst\u271d\u00b2 : Group H\n\u03c6 : (i : \u03b9) \u2192 H \u2192* G i\nd : Transversal \u03c6\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (G i)\nw\u2081 w\u2082 : NormalWord d\nhhead : w\u2081.head = w\u2082.head\nhlist : w\u2081.toList = w\u2082.toList\n\u22a2 w\u2081 = w\u2082", "state_after": "case mk.mk\n\u03b9 : Type u_1\nG : \u03b9 \u2192 Type u_2\nH : Type u_3\nK : Type u_4\ninst\u271d\u2074 : Monoid K\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Group (G i)\ninst\u271d\u00b2 : Group H\n\u03c6 : (i : \u03b9) \u2192 H \u2192* G i\nd : Transversal \u03c6\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (G i)\nw\u2082 : NormalWord d\nhead\u271d : H\ntoList\u271d : List ((i : \u03b9) \u00d7 G i)\nne_one\u271d : \u2200 l \u2208 toList\u271d, l.snd \u2260 1\nchain_ne\u271d : Chain' (fun l l' => l.fst \u2260 l'.fst) toList\u271d\nnormalized\u271d :\n  \u2200 (i : \u03b9) (g : G i), \u27e8i, g\u27e9 \u2208 { toList := toList\u271d, ne_one := ne_one\u271d, chain_ne := chain_ne\u271d }.toList \u2192 g \u2208 d.set i\nhhead :\n  { toList := toList\u271d, ne_one := ne_one\u271d, chain_ne := chain_ne\u271d, head := head\u271d, normalized := normalized\u271d }.head =\n    w\u2082.head\nhlist :\n  { toList := toList\u271d, ne_one := ne_one\u271d, chain_ne := chain_ne\u271d, head := head\u271d, normalized := normalized\u271d }.toList =\n    w\u2082.toList\n\u22a2 { toList := toList\u271d, ne_one := ne_one\u271d, chain_ne := chain_ne\u271d, head := head\u271d, normalized := normalized\u271d } = w\u2082"}, {"tactic": "rcases w\u2082 with \u27e8\u27e8_, _, _\u27e9, _, _\u27e9", "annotated_tactic": ["rcases w\u2082 with \u27e8\u27e8_, _, _\u27e9, _, _\u27e9", []], "state_before": "case mk.mk\n\u03b9 : Type u_1\nG : \u03b9 \u2192 Type u_2\nH : Type u_3\nK : Type u_4\ninst\u271d\u2074 : Monoid K\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Group (G i)\ninst\u271d\u00b2 : Group H\n\u03c6 : (i : \u03b9) \u2192 H \u2192* G i\nd : Transversal \u03c6\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (G i)\nw\u2082 : NormalWord d\nhead\u271d : H\ntoList\u271d : List ((i : \u03b9) \u00d7 G i)\nne_one\u271d : \u2200 l \u2208 toList\u271d, l.snd \u2260 1\nchain_ne\u271d : Chain' (fun l l' => l.fst \u2260 l'.fst) toList\u271d\nnormalized\u271d :\n  \u2200 (i : \u03b9) (g : G i), \u27e8i, g\u27e9 \u2208 { toList := toList\u271d, ne_one := ne_one\u271d, chain_ne := chain_ne\u271d }.toList \u2192 g \u2208 d.set i\nhhead :\n  { toList := toList\u271d, ne_one := ne_one\u271d, chain_ne := chain_ne\u271d, head := head\u271d, normalized := normalized\u271d }.head =\n    w\u2082.head\nhlist :\n  { toList := toList\u271d, ne_one := ne_one\u271d, chain_ne := chain_ne\u271d, head := head\u271d, normalized := normalized\u271d }.toList =\n    w\u2082.toList\n\u22a2 { toList := toList\u271d, ne_one := ne_one\u271d, chain_ne := chain_ne\u271d, head := head\u271d, normalized := normalized\u271d } = w\u2082", "state_after": "case mk.mk.mk.mk\n\u03b9 : Type u_1\nG : \u03b9 \u2192 Type u_2\nH : Type u_3\nK : Type u_4\ninst\u271d\u2074 : Monoid K\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Group (G i)\ninst\u271d\u00b2 : Group H\n\u03c6 : (i : \u03b9) \u2192 H \u2192* G i\nd : Transversal \u03c6\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (G i)\nhead\u271d\u00b9 : H\ntoList\u271d\u00b9 : List ((i : \u03b9) \u00d7 G i)\nne_one\u271d\u00b9 : \u2200 l \u2208 toList\u271d\u00b9, l.snd \u2260 1\nchain_ne\u271d\u00b9 : Chain' (fun l l' => l.fst \u2260 l'.fst) toList\u271d\u00b9\nnormalized\u271d\u00b9 :\n  \u2200 (i : \u03b9) (g : G i), \u27e8i, g\u27e9 \u2208 { toList := toList\u271d\u00b9, ne_one := ne_one\u271d\u00b9, chain_ne := chain_ne\u271d\u00b9 }.toList \u2192 g \u2208 d.set i\nhead\u271d : H\ntoList\u271d : List ((i : \u03b9) \u00d7 G i)\nne_one\u271d : \u2200 l \u2208 toList\u271d, l.snd \u2260 1\nchain_ne\u271d : Chain' (fun l l' => l.fst \u2260 l'.fst) toList\u271d\nnormalized\u271d :\n  \u2200 (i : \u03b9) (g : G i), \u27e8i, g\u27e9 \u2208 { toList := toList\u271d, ne_one := ne_one\u271d, chain_ne := chain_ne\u271d }.toList \u2192 g \u2208 d.set i\nhhead :\n  { toList := toList\u271d\u00b9, ne_one := ne_one\u271d\u00b9, chain_ne := chain_ne\u271d\u00b9, head := head\u271d\u00b9, normalized := normalized\u271d\u00b9 }.head =\n    { toList := toList\u271d, ne_one := ne_one\u271d, chain_ne := chain_ne\u271d, head := head\u271d, normalized := normalized\u271d }.head\nhlist :\n  { toList := toList\u271d\u00b9, ne_one := ne_one\u271d\u00b9, chain_ne := chain_ne\u271d\u00b9, head := head\u271d\u00b9,\n        normalized := normalized\u271d\u00b9 }.toList =\n    { toList := toList\u271d, ne_one := ne_one\u271d, chain_ne := chain_ne\u271d, head := head\u271d, normalized := normalized\u271d }.toList\n\u22a2 { toList := toList\u271d\u00b9, ne_one := ne_one\u271d\u00b9, chain_ne := chain_ne\u271d\u00b9, head := head\u271d\u00b9, normalized := normalized\u271d\u00b9 } =\n    { toList := toList\u271d, ne_one := ne_one\u271d, chain_ne := chain_ne\u271d, head := head\u271d, normalized := normalized\u271d }"}, {"tactic": "simp_all", "annotated_tactic": ["simp_all", []], "state_before": "case mk.mk.mk.mk\n\u03b9 : Type u_1\nG : \u03b9 \u2192 Type u_2\nH : Type u_3\nK : Type u_4\ninst\u271d\u2074 : Monoid K\ninst\u271d\u00b3 : (i : \u03b9) \u2192 Group (G i)\ninst\u271d\u00b2 : Group H\n\u03c6 : (i : \u03b9) \u2192 H \u2192* G i\nd : Transversal \u03c6\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : (i : \u03b9) \u2192 DecidableEq (G i)\nhead\u271d\u00b9 : H\ntoList\u271d\u00b9 : List ((i : \u03b9) \u00d7 G i)\nne_one\u271d\u00b9 : \u2200 l \u2208 toList\u271d\u00b9, l.snd \u2260 1\nchain_ne\u271d\u00b9 : Chain' (fun l l' => l.fst \u2260 l'.fst) toList\u271d\u00b9\nnormalized\u271d\u00b9 :\n  \u2200 (i : \u03b9) (g : G i), \u27e8i, g\u27e9 \u2208 { toList := toList\u271d\u00b9, ne_one := ne_one\u271d\u00b9, chain_ne := chain_ne\u271d\u00b9 }.toList \u2192 g \u2208 d.set i\nhead\u271d : H\ntoList\u271d : List ((i : \u03b9) \u00d7 G i)\nne_one\u271d : \u2200 l \u2208 toList\u271d, l.snd \u2260 1\nchain_ne\u271d : Chain' (fun l l' => l.fst \u2260 l'.fst) toList\u271d\nnormalized\u271d :\n  \u2200 (i : \u03b9) (g : G i), \u27e8i, g\u27e9 \u2208 { toList := toList\u271d, ne_one := ne_one\u271d, chain_ne := chain_ne\u271d }.toList \u2192 g \u2208 d.set i\nhhead :\n  { toList := toList\u271d\u00b9, ne_one := ne_one\u271d\u00b9, chain_ne := chain_ne\u271d\u00b9, head := head\u271d\u00b9, normalized := normalized\u271d\u00b9 }.head =\n    { toList := toList\u271d, ne_one := ne_one\u271d, chain_ne := chain_ne\u271d, head := head\u271d, normalized := normalized\u271d }.head\nhlist :\n  { toList := toList\u271d\u00b9, ne_one := ne_one\u271d\u00b9, chain_ne := chain_ne\u271d\u00b9, head := head\u271d\u00b9,\n        normalized := normalized\u271d\u00b9 }.toList =\n    { toList := toList\u271d, ne_one := ne_one\u271d, chain_ne := chain_ne\u271d, head := head\u271d, normalized := normalized\u271d }.toList\n\u22a2 { toList := toList\u271d\u00b9, ne_one := ne_one\u271d\u00b9, chain_ne := chain_ne\u271d\u00b9, head := head\u271d\u00b9, normalized := normalized\u271d\u00b9 } =\n    { toList := toList\u271d, ne_one := ne_one\u271d, chain_ne := chain_ne\u271d, head := head\u271d, normalized := normalized\u271d }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Independence/Basic.lean", "full_name": "ProbabilityTheory.indepSet_iff_indepSets_singleton", "start": [526, 1], "end": [529, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/ContinuousFunction/Bounded.lean", "full_name": "BoundedContinuousFunction.restrict_apply", "start": [401, 1], "end": [401, 86], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/RelClasses.lean", "full_name": "WellFoundedLT.fix_eq", "start": [420, 1], "end": [422, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Num/Lemmas.lean", "full_name": "Num.cast_sub'", "start": [1204, 1], "end": [1208, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/Pointwise.lean", "full_name": "Real.smul_iSup_of_nonpos", "start": [87, 1], "end": [88, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean", "full_name": "AffineIndependent.vectorSpan_eq_of_le_of_card_eq_finrank_add_one", "start": [286, 1], "end": [290, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Kaehler/Basic.lean", "full_name": "KaehlerDifferential.span_range_derivation", "start": [245, 1], "end": [264, 34], "traced_tactics": [{"tactic": "rw [_root_.eq_top_iff]", "annotated_tactic": ["rw [<a>_root_.eq_top_iff</a>]", [{"full_name": "eq_top_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [133, 9], "def_end_pos": [133, 19]}]], "state_before": "R : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\n\u22a2 Submodule.span S (Set.range \u21d1(D R S)) = \u22a4", "state_after": "R : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\n\u22a2 \u22a4 \u2264 Submodule.span S (Set.range \u21d1(D R S))"}, {"tactic": "rintro x -", "annotated_tactic": ["rintro x -", []], "state_before": "R : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\n\u22a2 \u22a4 \u2264 Submodule.span S (Set.range \u21d1(D R S))", "state_after": "R : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx : \u03a9[S\u2044R]\n\u22a2 x \u2208 Submodule.span S (Set.range \u21d1(D R S))"}, {"tactic": "obtain \u27e8\u27e8x, hx\u27e9, rfl\u27e9 := Ideal.toCotangent_surjective _ x", "annotated_tactic": ["obtain \u27e8\u27e8x, hx\u27e9, rfl\u27e9 := <a>Ideal.toCotangent_surjective</a> _ x", [{"full_name": "Ideal.toCotangent_surjective", "def_path": "Mathlib/RingTheory/Ideal/Cotangent.lean", "def_pos": [82, 9], "def_end_pos": [82, 31]}]], "state_before": "R : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx : \u03a9[S\u2044R]\n\u22a2 x \u2208 Submodule.span S (Set.range \u21d1(D R S))", "state_after": "case intro.mk\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx : S \u2297[R] S\nhx : x \u2208 ideal R S\n\u22a2 (ideal R S).toCotangent \u27e8x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))"}, {"tactic": "have : x \u2208 (KaehlerDifferential.ideal R S).restrictScalars S := hx", "annotated_tactic": ["have : x \u2208 (<a>KaehlerDifferential.ideal</a> R S).<a>restrictScalars</a> S := hx", [{"full_name": "KaehlerDifferential.ideal", "def_path": "Mathlib/RingTheory/Kaehler/Basic.lean", "def_pos": [60, 8], "def_end_pos": [60, 33]}, {"full_name": "Submodule.restrictScalars", "def_path": "Mathlib/Algebra/Module/Submodule/RestrictScalars.lean", "def_pos": [31, 5], "def_end_pos": [31, 20]}]], "state_before": "case intro.mk\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx : S \u2297[R] S\nhx : x \u2208 ideal R S\n\u22a2 (ideal R S).toCotangent \u27e8x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))", "state_after": "case intro.mk\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx : S \u2297[R] S\nhx : x \u2208 ideal R S\nthis : x \u2208 Submodule.restrictScalars S (ideal R S)\n\u22a2 (ideal R S).toCotangent \u27e8x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))"}, {"tactic": "rw [\u2190 KaehlerDifferential.submodule_span_range_eq_ideal] at this", "annotated_tactic": ["rw [\u2190 <a>KaehlerDifferential.submodule_span_range_eq_ideal</a>] at this", [{"full_name": "KaehlerDifferential.submodule_span_range_eq_ideal", "def_path": "Mathlib/RingTheory/Kaehler/Basic.lean", "def_pos": [109, 9], "def_end_pos": [109, 58]}]], "state_before": "case intro.mk\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx : S \u2297[R] S\nhx : x \u2208 ideal R S\nthis : x \u2208 Submodule.restrictScalars S (ideal R S)\n\u22a2 (ideal R S).toCotangent \u27e8x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))", "state_after": "case intro.mk\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx : S \u2297[R] S\nhx : x \u2208 ideal R S\nthis : x \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\n\u22a2 (ideal R S).toCotangent \u27e8x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))"}, {"tactic": "suffices \u2203 hx, (KaehlerDifferential.ideal R S).toCotangent \u27e8x, hx\u27e9 \u2208\n    Submodule.span S (Set.range <| KaehlerDifferential.D R S) by\n  exact this.choose_spec", "annotated_tactic": ["suffices \u2203 hx, (<a>KaehlerDifferential.ideal</a> R S).<a>toCotangent</a> \u27e8x, hx\u27e9 \u2208\n      <a>Submodule.span</a> S (<a>Set.range</a> <| <a>KaehlerDifferential.D</a> R S) by\n    exact this.choose_spec", [{"full_name": "KaehlerDifferential.ideal", "def_path": "Mathlib/RingTheory/Kaehler/Basic.lean", "def_pos": [60, 8], "def_end_pos": [60, 33]}, {"full_name": "Ideal.toCotangent", "def_path": "Mathlib/RingTheory/Ideal/Cotangent.lean", "def_pos": [60, 5], "def_end_pos": [60, 16]}, {"full_name": "Submodule.span", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [51, 5], "def_end_pos": [51, 9]}, {"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [157, 5], "def_end_pos": [157, 10]}, {"full_name": "KaehlerDifferential.D", "def_path": "Mathlib/RingTheory/Kaehler/Basic.lean", "def_pos": [217, 5], "def_end_pos": [217, 26]}]], "state_before": "case intro.mk\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx : S \u2297[R] S\nhx : x \u2208 ideal R S\nthis : x \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\n\u22a2 (ideal R S).toCotangent \u27e8x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))", "state_after": "case intro.mk\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx : S \u2297[R] S\nhx : x \u2208 ideal R S\nthis : x \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\n\u22a2 \u2203 (hx : x \u2208 ideal R S), (ideal R S).toCotangent \u27e8x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))"}, {"tactic": "refine Submodule.span_induction this ?_ ?_ ?_ ?_", "annotated_tactic": ["refine <a>Submodule.span_induction</a> this ?_ ?_ ?_ ?_", [{"full_name": "Submodule.span_induction", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [182, 9], "def_end_pos": [182, 23]}]], "state_before": "case intro.mk\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx : S \u2297[R] S\nhx : x \u2208 ideal R S\nthis : x \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\n\u22a2 \u2203 (hx : x \u2208 ideal R S), (ideal R S).toCotangent \u27e8x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))", "state_after": "case intro.mk.refine_1\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx : S \u2297[R] S\nhx : x \u2208 ideal R S\nthis : x \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\n\u22a2 \u2200 x \u2208 Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1,\n    \u2203 (hx : x \u2208 ideal R S), (ideal R S).toCotangent \u27e8x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))\n\ncase intro.mk.refine_2\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx : S \u2297[R] S\nhx : x \u2208 ideal R S\nthis : x \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\n\u22a2 \u2203 (hx : 0 \u2208 ideal R S), (ideal R S).toCotangent \u27e80, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))\n\ncase intro.mk.refine_3\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx : S \u2297[R] S\nhx : x \u2208 ideal R S\nthis : x \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\n\u22a2 \u2200 (x y : S \u2297[R] S),\n    (\u2203 (hx : x \u2208 ideal R S), (ideal R S).toCotangent \u27e8x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))) \u2192\n      (\u2203 (hx : y \u2208 ideal R S), (ideal R S).toCotangent \u27e8y, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))) \u2192\n        \u2203 (hx : x + y \u2208 ideal R S), (ideal R S).toCotangent \u27e8x + y, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))\n\ncase intro.mk.refine_4\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx : S \u2297[R] S\nhx : x \u2208 ideal R S\nthis : x \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\n\u22a2 \u2200 (a : S) (x : S \u2297[R] S),\n    (\u2203 (hx : x \u2208 ideal R S), (ideal R S).toCotangent \u27e8x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))) \u2192\n      \u2203 (hx : a \u2022 x \u2208 ideal R S), (ideal R S).toCotangent \u27e8a \u2022 x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))"}, {"tactic": "exact this.choose_spec", "annotated_tactic": ["exact this.choose_spec", []], "state_before": "R : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx : S \u2297[R] S\nhx : x \u2208 ideal R S\nthis\u271d : x \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\nthis : \u2203 (hx : x \u2208 ideal R S), (ideal R S).toCotangent \u27e8x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))\n\u22a2 (ideal R S).toCotangent \u27e8x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))", "state_after": "no goals"}, {"tactic": "rintro _ \u27e8x, rfl\u27e9", "annotated_tactic": ["rintro _ \u27e8x, rfl\u27e9", []], "state_before": "case intro.mk.refine_1\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx : S \u2297[R] S\nhx : x \u2208 ideal R S\nthis : x \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\n\u22a2 \u2200 x \u2208 Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1,\n    \u2203 (hx : x \u2208 ideal R S), (ideal R S).toCotangent \u27e8x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))", "state_after": "case intro.mk.refine_1.intro\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx\u271d : S \u2297[R] S\nhx : x\u271d \u2208 ideal R S\nthis : x\u271d \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\nx : S\n\u22a2 \u2203 (hx : (fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1) x \u2208 ideal R S),\n    (ideal R S).toCotangent \u27e8(fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1) x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))"}, {"tactic": "refine \u27e8KaehlerDifferential.one_smul_sub_smul_one_mem_ideal R x, ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>KaehlerDifferential.one_smul_sub_smul_one_mem_ideal</a> R x, ?_\u27e9", [{"full_name": "KaehlerDifferential.one_smul_sub_smul_one_mem_ideal", "def_path": "Mathlib/RingTheory/Kaehler/Basic.lean", "def_pos": [66, 9], "def_end_pos": [66, 60]}]], "state_before": "case intro.mk.refine_1.intro\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx\u271d : S \u2297[R] S\nhx : x\u271d \u2208 ideal R S\nthis : x\u271d \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\nx : S\n\u22a2 \u2203 (hx : (fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1) x \u2208 ideal R S),\n    (ideal R S).toCotangent \u27e8(fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1) x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))", "state_after": "case intro.mk.refine_1.intro\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx\u271d : S \u2297[R] S\nhx : x\u271d \u2208 ideal R S\nthis : x\u271d \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\nx : S\n\u22a2 (ideal R S).toCotangent \u27e8(fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1) x, \u22ef\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))"}, {"tactic": "apply Submodule.subset_span", "annotated_tactic": ["apply <a>Submodule.subset_span</a>", [{"full_name": "Submodule.subset_span", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [78, 9], "def_end_pos": [78, 20]}]], "state_before": "case intro.mk.refine_1.intro\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx\u271d : S \u2297[R] S\nhx : x\u271d \u2208 ideal R S\nthis : x\u271d \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\nx : S\n\u22a2 (ideal R S).toCotangent \u27e8(fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1) x, \u22ef\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))", "state_after": "case intro.mk.refine_1.intro.a\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx\u271d : S \u2297[R] S\nhx : x\u271d \u2208 ideal R S\nthis : x\u271d \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\nx : S\n\u22a2 (ideal R S).toCotangent \u27e8(fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1) x, \u22ef\u27e9 \u2208 Set.range \u21d1(D R S)"}, {"tactic": "exact \u27e8x, KaehlerDifferential.DLinearMap_apply R S x\u27e9", "annotated_tactic": ["exact \u27e8x, <a>KaehlerDifferential.DLinearMap_apply</a> R S x\u27e9", [{"full_name": "KaehlerDifferential.DLinearMap_apply", "def_path": "Mathlib/RingTheory/Kaehler/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 45]}]], "state_before": "case intro.mk.refine_1.intro.a\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx\u271d : S \u2297[R] S\nhx : x\u271d \u2208 ideal R S\nthis : x\u271d \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\nx : S\n\u22a2 (ideal R S).toCotangent \u27e8(fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1) x, \u22ef\u27e9 \u2208 Set.range \u21d1(D R S)", "state_after": "no goals"}, {"tactic": "exact \u27e8zero_mem _, Submodule.zero_mem _\u27e9", "annotated_tactic": ["exact \u27e8<a>zero_mem</a> _, <a>Submodule.zero_mem</a> _\u27e9", [{"full_name": "ZeroMemClass.zero_mem", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [80, 3], "def_end_pos": [80, 11]}, {"full_name": "Submodule.zero_mem", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [216, 19], "def_end_pos": [216, 27]}]], "state_before": "case intro.mk.refine_2\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx : S \u2297[R] S\nhx : x \u2208 ideal R S\nthis : x \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\n\u22a2 \u2203 (hx : 0 \u2208 ideal R S), (ideal R S).toCotangent \u27e80, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))", "state_after": "no goals"}, {"tactic": "rintro x y \u27e8hx\u2081, hx\u2082\u27e9 \u27e8hy\u2081, hy\u2082\u27e9", "annotated_tactic": ["rintro x y \u27e8hx\u2081, hx\u2082\u27e9 \u27e8hy\u2081, hy\u2082\u27e9", []], "state_before": "case intro.mk.refine_3\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx : S \u2297[R] S\nhx : x \u2208 ideal R S\nthis : x \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\n\u22a2 \u2200 (x y : S \u2297[R] S),\n    (\u2203 (hx : x \u2208 ideal R S), (ideal R S).toCotangent \u27e8x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))) \u2192\n      (\u2203 (hx : y \u2208 ideal R S), (ideal R S).toCotangent \u27e8y, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))) \u2192\n        \u2203 (hx : x + y \u2208 ideal R S), (ideal R S).toCotangent \u27e8x + y, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))", "state_after": "case intro.mk.refine_3.intro.intro\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx\u271d : S \u2297[R] S\nhx : x\u271d \u2208 ideal R S\nthis : x\u271d \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\nx y : S \u2297[R] S\nhx\u2081 : x \u2208 ideal R S\nhx\u2082 : (ideal R S).toCotangent \u27e8x, hx\u2081\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))\nhy\u2081 : y \u2208 ideal R S\nhy\u2082 : (ideal R S).toCotangent \u27e8y, hy\u2081\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))\n\u22a2 \u2203 (hx : x + y \u2208 ideal R S), (ideal R S).toCotangent \u27e8x + y, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))"}, {"tactic": "exact \u27e8add_mem hx\u2081 hy\u2081, Submodule.add_mem _ hx\u2082 hy\u2082\u27e9", "annotated_tactic": ["exact \u27e8<a>add_mem</a> hx\u2081 hy\u2081, <a>Submodule.add_mem</a> _ hx\u2082 hy\u2082\u27e9", [{"full_name": "AddMemClass.add_mem", "def_path": "Mathlib/Algebra/Group/Subsemigroup/Basic.lean", "def_pos": [72, 3], "def_end_pos": [72, 10]}, {"full_name": "Submodule.add_mem", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [220, 19], "def_end_pos": [220, 26]}]], "state_before": "case intro.mk.refine_3.intro.intro\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx\u271d : S \u2297[R] S\nhx : x\u271d \u2208 ideal R S\nthis : x\u271d \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\nx y : S \u2297[R] S\nhx\u2081 : x \u2208 ideal R S\nhx\u2082 : (ideal R S).toCotangent \u27e8x, hx\u2081\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))\nhy\u2081 : y \u2208 ideal R S\nhy\u2082 : (ideal R S).toCotangent \u27e8y, hy\u2081\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))\n\u22a2 \u2203 (hx : x + y \u2208 ideal R S), (ideal R S).toCotangent \u27e8x + y, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))", "state_after": "no goals"}, {"tactic": "rintro r x \u27e8hx\u2081, hx\u2082\u27e9", "annotated_tactic": ["rintro r x \u27e8hx\u2081, hx\u2082\u27e9", []], "state_before": "case intro.mk.refine_4\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx : S \u2297[R] S\nhx : x \u2208 ideal R S\nthis : x \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\n\u22a2 \u2200 (a : S) (x : S \u2297[R] S),\n    (\u2203 (hx : x \u2208 ideal R S), (ideal R S).toCotangent \u27e8x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))) \u2192\n      \u2203 (hx : a \u2022 x \u2208 ideal R S), (ideal R S).toCotangent \u27e8a \u2022 x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))", "state_after": "case intro.mk.refine_4.intro\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx\u271d : S \u2297[R] S\nhx : x\u271d \u2208 ideal R S\nthis : x\u271d \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\nr : S\nx : S \u2297[R] S\nhx\u2081 : x \u2208 ideal R S\nhx\u2082 : (ideal R S).toCotangent \u27e8x, hx\u2081\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))\n\u22a2 \u2203 (hx : r \u2022 x \u2208 ideal R S), (ideal R S).toCotangent \u27e8r \u2022 x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))"}, {"tactic": "exact \u27e8((KaehlerDifferential.ideal R S).restrictScalars S).smul_mem r hx\u2081,\n  Submodule.smul_mem _ r hx\u2082\u27e9", "annotated_tactic": ["exact \u27e8((<a>KaehlerDifferential.ideal</a> R S).<a>restrictScalars</a> S).<a>smul_mem</a> r hx\u2081,\n      <a>Submodule.smul_mem</a> _ r hx\u2082\u27e9", [{"full_name": "KaehlerDifferential.ideal", "def_path": "Mathlib/RingTheory/Kaehler/Basic.lean", "def_pos": [60, 8], "def_end_pos": [60, 33]}, {"full_name": "Submodule.restrictScalars", "def_path": "Mathlib/Algebra/Module/Submodule/RestrictScalars.lean", "def_pos": [31, 5], "def_end_pos": [31, 20]}, {"full_name": "Submodule.smul_mem", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [224, 9], "def_end_pos": [224, 17]}, {"full_name": "Submodule.smul_mem", "def_path": "Mathlib/Algebra/Module/Submodule/Basic.lean", "def_pos": [224, 9], "def_end_pos": [224, 17]}]], "state_before": "case intro.mk.refine_4.intro\nR : Type u\nS : Type v\ninst\u271d\u2076 : CommRing R\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : Algebra R S\nM : Type u_1\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : Module S M\ninst\u271d : IsScalarTower R S M\nx\u271d : S \u2297[R] S\nhx : x\u271d \u2208 ideal R S\nthis : x\u271d \u2208 Submodule.span S (Set.range fun s => 1 \u2297\u209c[R] s - s \u2297\u209c[R] 1)\nr : S\nx : S \u2297[R] S\nhx\u2081 : x \u2208 ideal R S\nhx\u2082 : (ideal R S).toCotangent \u27e8x, hx\u2081\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))\n\u22a2 \u2203 (hx : r \u2022 x \u2208 ideal R S), (ideal R S).toCotangent \u27e8r \u2022 x, hx\u27e9 \u2208 Submodule.span S (Set.range \u21d1(D R S))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/Adjoin.lean", "full_name": "IntermediateField.adjoin_eq_range_algebraMap_adjoin", "start": [341, 1], "end": [343, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean", "full_name": "Complex.cos_eq_iff_quadratic", "start": [187, 1], "end": [192, 7], "traced_tactics": [{"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": [1070, 3], "def_end_pos": [1070, 14]}]], "state_before": "z w : \u2102\n\u22a2 cos z = w \u2194 cexp (z * I) ^ 2 - 2 * w * cexp (z * I) + 1 = 0", "state_after": "z w : \u2102\n\u22a2 cos z - w = 0 \u2194 cexp (z * I) ^ 2 - 2 * w * cexp (z * I) + 1 = 0"}, {"tactic": "field_simp [cos, exp_neg, exp_ne_zero]", "annotated_tactic": ["field_simp [<a>cos</a>, <a>exp_neg</a>, <a>exp_ne_zero</a>]", [{"full_name": "Complex.cos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [67, 5], "def_end_pos": [67, 8]}, {"full_name": "Complex.exp_neg", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [239, 9], "def_end_pos": [239, 16]}, {"full_name": "Complex.exp_ne_zero", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [235, 9], "def_end_pos": [235, 20]}]], "state_before": "z w : \u2102\n\u22a2 cos z - w = 0 \u2194 cexp (z * I) ^ 2 - 2 * w * cexp (z * I) + 1 = 0", "state_after": "z w : \u2102\n\u22a2 cexp (z * I) * cexp (z * I) + 1 - cexp (z * I) * 2 * w = 0 \u2194 cexp (z * I) ^ 2 - 2 * w * cexp (z * I) + 1 = 0"}, {"tactic": "refine Eq.congr ?_ rfl", "annotated_tactic": ["refine <a>Eq.congr</a> ?_ <a>rfl</a>", [{"full_name": "Eq.congr", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [69, 19], "def_end_pos": [69, 27]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "z w : \u2102\n\u22a2 cexp (z * I) * cexp (z * I) + 1 - cexp (z * I) * 2 * w = 0 \u2194 cexp (z * I) ^ 2 - 2 * w * cexp (z * I) + 1 = 0", "state_after": "z w : \u2102\n\u22a2 cexp (z * I) * cexp (z * I) + 1 - cexp (z * I) * 2 * w = cexp (z * I) ^ 2 - 2 * w * cexp (z * I) + 1"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "z w : \u2102\n\u22a2 cexp (z * I) * cexp (z * I) + 1 - cexp (z * I) * 2 * w = cexp (z * I) ^ 2 - 2 * w * cexp (z * I) + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean", "full_name": "continuousMultilinearCurryRightEquiv_symm_apply", "start": [353, 1], "end": [356, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "full_name": "norm_div_eq_zero_iff", "start": [1549, 1], "end": [1549, 89], "traced_tactics": [{"tactic": "rw [norm_eq_zero'', div_eq_one]", "annotated_tactic": ["rw [<a>norm_eq_zero''</a>, <a>div_eq_one</a>]", [{"full_name": "norm_eq_zero''", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1525, 9], "def_end_pos": [1525, 23]}, {"full_name": "div_eq_one", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1071, 9], "def_end_pos": [1071, 19]}]], "state_before": "\ud835\udcd5 : Type u_1\n\ud835\udd5c : Type u_2\n\u03b1 : Type u_3\n\u03b9 : Type u_4\n\u03ba : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\ninst\u271d\u00b9 : NormedGroup E\ninst\u271d : NormedGroup F\na b : E\n\u22a2 \u2016a / b\u2016 = 0 \u2194 a = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Weights/RootSystem.lean", "full_name": "LieAlgebra.IsKilling.chainLength_nsmul", "start": [66, 1], "end": [76, 94], "traced_tactics": [{"tactic": "by_cases h\u03b1 : \u03b1.IsZero", "annotated_tactic": ["by_cases h\u03b1 : \u03b1.IsZero", []], "state_before": "K : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1 : \u03b1.IsNonZero\nx : L\nhx : x \u2208 rootSpace H \u21d1(chainTop (\u21d1\u03b1) \u03b2)\n\u22a2 chainLength \u03b1 \u03b2 \u2022 x = \u2045coroot \u03b1, x\u2046", "state_after": "case pos\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nx : L\nhx : x \u2208 rootSpace H \u21d1(chainTop (\u21d1\u03b1) \u03b2)\nh\u03b1 : \u03b1.IsZero\n\u22a2 chainLength \u03b1 \u03b2 \u2022 x = \u2045coroot \u03b1, x\u2046\n\ncase neg\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nx : L\nhx : x \u2208 rootSpace H \u21d1(chainTop (\u21d1\u03b1) \u03b2)\nh\u03b1 : \u00ac\u03b1.IsZero\n\u22a2 chainLength \u03b1 \u03b2 \u2022 x = \u2045coroot \u03b1, x\u2046"}, {"tactic": "let x' := (chainTop \u03b1 \u03b2).exists_ne_zero.choose", "annotated_tactic": ["let x' := (<a>chainTop</a> \u03b1 \u03b2).exists_ne_zero.choose", [{"full_name": "LieModule.chainTop", "def_path": "Mathlib/Algebra/Lie/Weights/Chain.lean", "def_pos": [300, 5], "def_end_pos": [300, 13]}]], "state_before": "case neg\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nx : L\nhx : x \u2208 rootSpace H \u21d1(chainTop (\u21d1\u03b1) \u03b2)\nh\u03b1 : \u00ac\u03b1.IsZero\n\u22a2 chainLength \u03b1 \u03b2 \u2022 x = \u2045coroot \u03b1, x\u2046", "state_after": "case neg\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nx : L\nhx : x \u2208 rootSpace H \u21d1(chainTop (\u21d1\u03b1) \u03b2)\nh\u03b1 : \u00ac\u03b1.IsZero\nx' : L := \u22ef.choose\n\u22a2 chainLength \u03b1 \u03b2 \u2022 x = \u2045coroot \u03b1, x\u2046"}, {"tactic": "have h : x' \u2208 rootSpace H (chainTop \u03b1 \u03b2) \u2227 x' \u2260 0 :=\n  (chainTop \u03b1 \u03b2).exists_ne_zero.choose_spec", "annotated_tactic": ["have h : x' \u2208 <a>rootSpace</a> H (<a>chainTop</a> \u03b1 \u03b2) \u2227 x' \u2260 0 :=\n    (<a>chainTop</a> \u03b1 \u03b2).exists_ne_zero.choose_spec", [{"full_name": "LieAlgebra.rootSpace", "def_path": "Mathlib/Algebra/Lie/Weights/Cartan.lean", "def_pos": [44, 8], "def_end_pos": [44, 17]}, {"full_name": "LieModule.chainTop", "def_path": "Mathlib/Algebra/Lie/Weights/Chain.lean", "def_pos": [300, 5], "def_end_pos": [300, 13]}, {"full_name": "LieModule.chainTop", "def_path": "Mathlib/Algebra/Lie/Weights/Chain.lean", "def_pos": [300, 5], "def_end_pos": [300, 13]}]], "state_before": "case neg\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nx : L\nhx : x \u2208 rootSpace H \u21d1(chainTop (\u21d1\u03b1) \u03b2)\nh\u03b1 : \u00ac\u03b1.IsZero\nx' : L := \u22ef.choose\n\u22a2 chainLength \u03b1 \u03b2 \u2022 x = \u2045coroot \u03b1, x\u2046", "state_after": "case neg\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nx : L\nhx : x \u2208 rootSpace H \u21d1(chainTop (\u21d1\u03b1) \u03b2)\nh\u03b1 : \u00ac\u03b1.IsZero\nx' : L := \u22ef.choose\nh : x' \u2208 rootSpace H \u21d1(chainTop (\u21d1\u03b1) \u03b2) \u2227 x' \u2260 0\n\u22a2 chainLength \u03b1 \u03b2 \u2022 x = \u2045coroot \u03b1, x\u2046"}, {"tactic": "obtain \u27e8k, rfl\u27e9 : \u2203 k : K, k \u2022 x' = x := by\n  simpa using (finrank_eq_one_iff_of_nonzero' \u27e8x', h.1\u27e9 (by simpa using h.2)).mp\n    (finrank_rootSpace_eq_one _ (chainTop_isNonZero \u03b1 \u03b2 h\u03b1)) \u27e8_, hx\u27e9", "annotated_tactic": ["obtain \u27e8k, rfl\u27e9 : \u2203 k : K, k \u2022 x' = x := by\n    simpa using (<a>finrank_eq_one_iff_of_nonzero'</a> \u27e8x', h.1\u27e9 (by simpa using h.2)).<a>mp</a>\n      (<a>finrank_rootSpace_eq_one</a> _ (<a>chainTop_isNonZero</a> \u03b1 \u03b2 h\u03b1)) \u27e8_, hx\u27e9", [{"full_name": "finrank_eq_one_iff_of_nonzero'", "def_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 39]}, {"full_name": "Iff.mp", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [118, 3], "def_end_pos": [118, 5]}, {"full_name": "LieAlgebra.IsKilling.finrank_rootSpace_eq_one", "def_path": "Mathlib/Algebra/Lie/Weights/Killing.lean", "def_pos": [522, 7], "def_end_pos": [522, 31]}, {"full_name": "LieModule.chainTop_isNonZero", "def_path": "Mathlib/Algebra/Lie/Weights/Chain.lean", "def_pos": [334, 7], "def_end_pos": [334, 25]}]], "state_before": "case neg\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nx : L\nhx : x \u2208 rootSpace H \u21d1(chainTop (\u21d1\u03b1) \u03b2)\nh\u03b1 : \u00ac\u03b1.IsZero\nx' : L := \u22ef.choose\nh : x' \u2208 rootSpace H \u21d1(chainTop (\u21d1\u03b1) \u03b2) \u2227 x' \u2260 0\n\u22a2 chainLength \u03b1 \u03b2 \u2022 x = \u2045coroot \u03b1, x\u2046", "state_after": "case neg.intro\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b1 : \u00ac\u03b1.IsZero\nx' : L := \u22ef.choose\nh : x' \u2208 rootSpace H \u21d1(chainTop (\u21d1\u03b1) \u03b2) \u2227 x' \u2260 0\nk : K\nhx : k \u2022 x' \u2208 rootSpace H \u21d1(chainTop (\u21d1\u03b1) \u03b2)\n\u22a2 chainLength \u03b1 \u03b2 \u2022 k \u2022 x' = \u2045coroot \u03b1, k \u2022 x'\u2046"}, {"tactic": "rw [lie_smul, smul_comm, chainLength, dif_neg h\u03b1, (chainLength_aux \u03b1 \u03b2 h\u03b1 h.1).choose_spec]", "annotated_tactic": ["rw [<a>lie_smul</a>, <a>smul_comm</a>, <a>chainLength</a>, <a>dif_neg</a> h\u03b1, (<a>chainLength_aux</a> \u03b1 \u03b2 h\u03b1 h.1).<a>choose_spec</a>]", [{"full_name": "lie_smul", "def_path": "Mathlib/Algebra/Lie/Basic.lean", "def_pos": [124, 9], "def_end_pos": [124, 17]}, {"full_name": "SMulCommClass.smul_comm", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [183, 3], "def_end_pos": [183, 12]}, {"full_name": "LieAlgebra.IsKilling.chainLength", "def_path": "Mathlib/Algebra/Lie/Weights/RootSystem.lean", "def_pos": [59, 5], "def_end_pos": [59, 16]}, {"full_name": "dif_neg", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [954, 9], "def_end_pos": [954, 16]}, {"full_name": "_private.Mathlib.Algebra.Lie.Weights.RootSystem.0.LieAlgebra.IsKilling.chainLength_aux", "def_path": "Mathlib/Algebra/Lie/Weights/RootSystem.lean", "def_pos": [45, 15], "def_end_pos": [45, 30]}, {"full_name": "Exists.choose_spec", "def_path": ".lake/packages/lean4/src/lean/Init/Classical.lean", "def_pos": [177, 9], "def_end_pos": [177, 27]}]], "state_before": "case neg.intro\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nh\u03b1 : \u00ac\u03b1.IsZero\nx' : L := \u22ef.choose\nh : x' \u2208 rootSpace H \u21d1(chainTop (\u21d1\u03b1) \u03b2) \u2227 x' \u2260 0\nk : K\nhx : k \u2022 x' \u2208 rootSpace H \u21d1(chainTop (\u21d1\u03b1) \u03b2)\n\u22a2 chainLength \u03b1 \u03b2 \u2022 k \u2022 x' = \u2045coroot \u03b1, k \u2022 x'\u2046", "state_after": "no goals"}, {"tactic": "rw [coroot_eq_zero_iff.mpr h\u03b1, chainLength_of_isZero _ _ h\u03b1, zero_smul, zero_lie]", "annotated_tactic": ["rw [coroot_eq_zero_iff.mpr h\u03b1, <a>chainLength_of_isZero</a> _ _ h\u03b1, <a>zero_smul</a>, <a>zero_lie</a>]", [{"full_name": "LieAlgebra.IsKilling.chainLength_of_isZero", "def_path": "Mathlib/Algebra/Lie/Weights/RootSystem.lean", "def_pos": [64, 7], "def_end_pos": [64, 28]}, {"full_name": "zero_smul", "def_path": "Mathlib/Algebra/SMulWithZero.lean", "def_pos": [70, 9], "def_end_pos": [70, 18]}, {"full_name": "zero_lie", "def_path": "Mathlib/Algebra/Lie/Basic.lean", "def_pos": [138, 9], "def_end_pos": [138, 17]}]], "state_before": "case pos\nK : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nx : L\nhx : x \u2208 rootSpace H \u21d1(chainTop (\u21d1\u03b1) \u03b2)\nh\u03b1 : \u03b1.IsZero\n\u22a2 chainLength \u03b1 \u03b2 \u2022 x = \u2045coroot \u03b1, x\u2046", "state_after": "no goals"}, {"tactic": "simpa using (finrank_eq_one_iff_of_nonzero' \u27e8x', h.1\u27e9 (by simpa using h.2)).mp\n  (finrank_rootSpace_eq_one _ (chainTop_isNonZero \u03b1 \u03b2 h\u03b1)) \u27e8_, hx\u27e9", "annotated_tactic": ["simpa using (<a>finrank_eq_one_iff_of_nonzero'</a> \u27e8x', h.1\u27e9 (by simpa using h.2)).<a>mp</a>\n      (<a>finrank_rootSpace_eq_one</a> _ (<a>chainTop_isNonZero</a> \u03b1 \u03b2 h\u03b1)) \u27e8_, hx\u27e9", [{"full_name": "finrank_eq_one_iff_of_nonzero'", "def_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "def_pos": [1049, 9], "def_end_pos": [1049, 39]}, {"full_name": "Iff.mp", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [118, 3], "def_end_pos": [118, 5]}, {"full_name": "LieAlgebra.IsKilling.finrank_rootSpace_eq_one", "def_path": "Mathlib/Algebra/Lie/Weights/Killing.lean", "def_pos": [522, 7], "def_end_pos": [522, 31]}, {"full_name": "LieModule.chainTop_isNonZero", "def_path": "Mathlib/Algebra/Lie/Weights/Chain.lean", "def_pos": [334, 7], "def_end_pos": [334, 25]}]], "state_before": "K : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nx : L\nhx : x \u2208 rootSpace H \u21d1(chainTop (\u21d1\u03b1) \u03b2)\nh\u03b1 : \u00ac\u03b1.IsZero\nx' : L := \u22ef.choose\nh : x' \u2208 rootSpace H \u21d1(chainTop (\u21d1\u03b1) \u03b2) \u2227 x' \u2260 0\n\u22a2 \u2203 k, k \u2022 x' = x", "state_after": "no goals"}, {"tactic": "simpa using h.2", "annotated_tactic": ["simpa using h.2", []], "state_before": "K : Type u_1\nL : Type u_2\ninst\u271d\u2077 : Field K\ninst\u271d\u2076 : CharZero K\ninst\u271d\u2075 : LieRing L\ninst\u271d\u2074 : LieAlgebra K L\ninst\u271d\u00b3 : IsKilling K L\ninst\u271d\u00b2 : FiniteDimensional K L\nH : LieSubalgebra K L\ninst\u271d\u00b9 : H.IsCartanSubalgebra\ninst\u271d : IsTriangularizable K (\u21a5H) L\n\u03b1 \u03b2 : Weight K (\u21a5H) L\nh\u03b1\u271d : \u03b1.IsNonZero\nx : L\nhx : x \u2208 rootSpace H \u21d1(chainTop (\u21d1\u03b1) \u03b2)\nh\u03b1 : \u00ac\u03b1.IsZero\nx' : L := \u22ef.choose\nh : x' \u2208 rootSpace H \u21d1(chainTop (\u21d1\u03b1) \u03b2) \u2227 x' \u2260 0\n\u22a2 \u27e8x', \u22ef\u27e9 \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Option/Basic.lean", "full_name": "Option.bind_pmap", "start": [224, 1], "end": [226, 74], "traced_tactics": [{"tactic": "cases x <;> simp only [pmap, bind_eq_bind, none_bind, some_bind, pbind]", "annotated_tactic": ["cases x <;> simp only [<a>pmap</a>, <a>bind_eq_bind</a>, <a>none_bind</a>, <a>some_bind</a>, <a>pbind</a>]", [{"full_name": "Option.pmap", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Option/Instances.lean", "def_pos": [68, 21], "def_end_pos": [68, 25]}, {"full_name": "Option.bind_eq_bind", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Option/Lemmas.lean", "def_pos": [97, 17], "def_end_pos": [97, 29]}, {"full_name": "Option.none_bind", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Option/Basic.lean", "def_pos": [117, 17], "def_end_pos": [117, 26]}, {"full_name": "Option.some_bind", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Option/Basic.lean", "def_pos": [118, 17], "def_end_pos": [118, 26]}, {"full_name": "Option.pbind", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Option/Instances.lean", "def_pos": [59, 5], "def_end_pos": [59, 10]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3\u271d : Type u_3\n\u03b4 : Type u_4\np\u271d : \u03b1\u271d \u2192 Prop\nf\u271d : (a : \u03b1\u271d) \u2192 p\u271d a \u2192 \u03b2\u271d\nx\u271d : Option \u03b1\u271d\n\u03b1 : Type u_5\n\u03b2 \u03b3 : Type u_6\np : \u03b1 \u2192 Prop\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nx : Option \u03b1\ng : \u03b2 \u2192 Option \u03b3\nH : \u2200 (a : \u03b1), a \u2208 x \u2192 p a\n\u22a2 pmap f x H >>= g = x.pbind fun a h => g (f a \u22ef)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/PerfectPairing.lean", "full_name": "PerfectPairing.apply_apply_toDualRight_symm", "start": [85, 1], "end": [89, 45], "traced_tactics": [{"tactic": "have h := LinearEquiv.apply_symm_apply p.toDualRight f", "annotated_tactic": ["have h := <a>LinearEquiv.apply_symm_apply</a> p.toDualRight f", [{"full_name": "LinearEquiv.apply_symm_apply", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [376, 9], "def_end_pos": [376, 25]}]], "state_before": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : PerfectPairing R M N\nx : M\nf : Dual R M\n\u22a2 (p x) (p.toDualRight.symm f) = f x", "state_after": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : PerfectPairing R M N\nx : M\nf : Dual R M\nh : p.toDualRight (p.toDualRight.symm f) = f\n\u22a2 (p x) (p.toDualRight.symm f) = f x"}, {"tactic": "rw [toDualRight_apply] at h", "annotated_tactic": ["rw [<a>toDualRight_apply</a>] at h", [{"full_name": "PerfectPairing.toDualRight_apply", "def_path": "Mathlib/LinearAlgebra/PerfectPairing.lean", "def_pos": [81, 9], "def_end_pos": [81, 26]}]], "state_before": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : PerfectPairing R M N\nx : M\nf : Dual R M\nh : p.toDualRight (p.toDualRight.symm f) = f\n\u22a2 (p x) (p.toDualRight.symm f) = f x", "state_after": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : PerfectPairing R M N\nx : M\nf : Dual R M\nh : p.flip (p.toDualRight.symm f) = f\n\u22a2 (p x) (p.toDualRight.symm f) = f x"}, {"tactic": "exact congrFun (congrArg DFunLike.coe h) x", "annotated_tactic": ["exact <a>congrFun</a> (<a>congrArg</a> <a>DFunLike.coe</a> h) x", [{"full_name": "congrFun", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [376, 9], "def_end_pos": [376, 17]}, {"full_name": "congrArg", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [362, 9], "def_end_pos": [362, 17]}, {"full_name": "DFunLike.coe", "def_path": "Mathlib/Data/FunLike/Basic.lean", "def_pos": [147, 3], "def_end_pos": [147, 6]}]], "state_before": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : AddCommGroup M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommGroup N\ninst\u271d : Module R N\np : PerfectPairing R M N\nx : M\nf : Dual R M\nh : p.flip (p.toDualRight.symm f) = f\n\u22a2 (p x) (p.toDualRight.symm f) = f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/SupIndep.lean", "full_name": "Finset.supIndep_iff_disjoint_erase", "start": [100, 1], "end": [103, 98], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/SpecificFunctions/Pow.lean", "full_name": "NNReal.strictConcaveOn_rpow", "start": [34, 1], "end": [45, 58], "traced_tactics": [{"tactic": "have hp\u2080' : 0 < 1 / p := div_pos zero_lt_one hp\u2080", "annotated_tactic": ["have hp\u2080' : 0 < 1 / p := <a>div_pos</a> <a>zero_lt_one</a> hp\u2080", [{"full_name": "div_pos", "def_path": "Mathlib/Algebra/Order/Field/Defs.lean", "def_pos": [79, 7], "def_end_pos": [79, 14]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}]], "state_before": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\n\u22a2 StrictConcaveOn \u211d\u22650 univ fun x => x ^ p", "state_after": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\n\u22a2 StrictConcaveOn \u211d\u22650 univ fun x => x ^ p"}, {"tactic": "have hp\u2081' : 1 < 1 / p := by rw [one_lt_div hp\u2080]; exact hp\u2081", "annotated_tactic": ["have hp\u2081' : 1 < 1 / p := by rw [<a>one_lt_div</a> hp\u2080]; exact hp\u2081", [{"full_name": "one_lt_div", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [369, 9], "def_end_pos": [369, 19]}]], "state_before": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\n\u22a2 StrictConcaveOn \u211d\u22650 univ fun x => x ^ p", "state_after": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\nhp\u2081' : 1 < 1 / p\n\u22a2 StrictConcaveOn \u211d\u22650 univ fun x => x ^ p"}, {"tactic": "let f := NNReal.orderIsoRpow (1 / p) hp\u2080'", "annotated_tactic": ["let f := <a>NNReal.orderIsoRpow</a> (1 / p) hp\u2080'", [{"full_name": "NNReal.orderIsoRpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [369, 5], "def_end_pos": [369, 17]}]], "state_before": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\nhp\u2081' : 1 < 1 / p\n\u22a2 StrictConcaveOn \u211d\u22650 univ fun x => x ^ p", "state_after": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\nhp\u2081' : 1 < 1 / p\nf : \u211d\u22650 \u2243o \u211d\u22650 := orderIsoRpow (1 / p) hp\u2080'\n\u22a2 StrictConcaveOn \u211d\u22650 univ fun x => x ^ p"}, {"tactic": "have h\u2081 : StrictConvexOn \u211d\u22650 univ f := by\n  refine \u27e8convex_univ, fun x _ y _ hxy a b ha hb hab => ?_\u27e9\n  exact (strictConvexOn_rpow hp\u2081').2 x.2 y.2 (by simp [hxy]) ha hb (by simp; norm_cast)", "annotated_tactic": ["have h\u2081 : <a>StrictConvexOn</a> \u211d\u22650 <a>univ</a> f := by\n    refine \u27e8<a>convex_univ</a>, fun x _ y _ hxy a b ha hb hab => ?_\u27e9\n    exact (<a>strictConvexOn_rpow</a> hp\u2081').2 x.2 y.2 (by simp [hxy]) ha hb (by simp; norm_cast)", [{"full_name": "StrictConvexOn", "def_path": "Mathlib/Analysis/Convex/Function.lean", "def_pos": [66, 5], "def_end_pos": [66, 19]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [153, 5], "def_end_pos": [153, 9]}, {"full_name": "convex_univ", "def_path": "Mathlib/Analysis/Convex/Basic.lean", "def_pos": [91, 9], "def_end_pos": [91, 20]}, {"full_name": "strictConvexOn_rpow", "def_path": "Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean", "def_pos": [178, 9], "def_end_pos": [178, 28]}]], "state_before": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\nhp\u2081' : 1 < 1 / p\nf : \u211d\u22650 \u2243o \u211d\u22650 := orderIsoRpow (1 / p) hp\u2080'\n\u22a2 StrictConcaveOn \u211d\u22650 univ fun x => x ^ p", "state_after": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\nhp\u2081' : 1 < 1 / p\nf : \u211d\u22650 \u2243o \u211d\u22650 := orderIsoRpow (1 / p) hp\u2080'\nh\u2081 : StrictConvexOn \u211d\u22650 univ \u21d1f\n\u22a2 StrictConcaveOn \u211d\u22650 univ fun x => x ^ p"}, {"tactic": "have h\u2082 : \u2200 x, f.symm x = x ^ p := by simp [f, NNReal.orderIsoRpow_symm_eq]", "annotated_tactic": ["have h\u2082 : \u2200 x, f.symm x = x ^ p := by simp [f, <a>NNReal.orderIsoRpow_symm_eq</a>]", [{"full_name": "NNReal.orderIsoRpow_symm_eq", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [375, 9], "def_end_pos": [375, 29]}]], "state_before": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\nhp\u2081' : 1 < 1 / p\nf : \u211d\u22650 \u2243o \u211d\u22650 := orderIsoRpow (1 / p) hp\u2080'\nh\u2081 : StrictConvexOn \u211d\u22650 univ \u21d1f\n\u22a2 StrictConcaveOn \u211d\u22650 univ fun x => x ^ p", "state_after": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\nhp\u2081' : 1 < 1 / p\nf : \u211d\u22650 \u2243o \u211d\u22650 := orderIsoRpow (1 / p) hp\u2080'\nh\u2081 : StrictConvexOn \u211d\u22650 univ \u21d1f\nh\u2082 : \u2200 (x : \u211d\u22650), f.symm x = x ^ p\n\u22a2 StrictConcaveOn \u211d\u22650 univ fun x => x ^ p"}, {"tactic": "refine \u27e8convex_univ, fun x mx y my hxy a b ha hb hab => ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>convex_univ</a>, fun x mx y my hxy a b ha hb hab => ?_\u27e9", [{"full_name": "convex_univ", "def_path": "Mathlib/Analysis/Convex/Basic.lean", "def_pos": [91, 9], "def_end_pos": [91, 20]}]], "state_before": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\nhp\u2081' : 1 < 1 / p\nf : \u211d\u22650 \u2243o \u211d\u22650 := orderIsoRpow (1 / p) hp\u2080'\nh\u2081 : StrictConvexOn \u211d\u22650 univ \u21d1f\nh\u2082 : \u2200 (x : \u211d\u22650), f.symm x = x ^ p\n\u22a2 StrictConcaveOn \u211d\u22650 univ fun x => x ^ p", "state_after": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\nhp\u2081' : 1 < 1 / p\nf : \u211d\u22650 \u2243o \u211d\u22650 := orderIsoRpow (1 / p) hp\u2080'\nh\u2081 : StrictConvexOn \u211d\u22650 univ \u21d1f\nh\u2082 : \u2200 (x : \u211d\u22650), f.symm x = x ^ p\nx : \u211d\u22650\nmx : x \u2208 univ\ny : \u211d\u22650\nmy : y \u2208 univ\nhxy : x \u2260 y\na b : \u211d\u22650\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 (fun x => x ^ p) x + b \u2022 (fun x => x ^ p) y < (fun x => x ^ p) (a \u2022 x + b \u2022 y)"}, {"tactic": "simp only [\u2190 h\u2082]", "annotated_tactic": ["simp only [\u2190 h\u2082]", []], "state_before": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\nhp\u2081' : 1 < 1 / p\nf : \u211d\u22650 \u2243o \u211d\u22650 := orderIsoRpow (1 / p) hp\u2080'\nh\u2081 : StrictConvexOn \u211d\u22650 univ \u21d1f\nh\u2082 : \u2200 (x : \u211d\u22650), f.symm x = x ^ p\nx : \u211d\u22650\nmx : x \u2208 univ\ny : \u211d\u22650\nmy : y \u2208 univ\nhxy : x \u2260 y\na b : \u211d\u22650\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 (fun x => x ^ p) x + b \u2022 (fun x => x ^ p) y < (fun x => x ^ p) (a \u2022 x + b \u2022 y)", "state_after": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\nhp\u2081' : 1 < 1 / p\nf : \u211d\u22650 \u2243o \u211d\u22650 := orderIsoRpow (1 / p) hp\u2080'\nh\u2081 : StrictConvexOn \u211d\u22650 univ \u21d1f\nh\u2082 : \u2200 (x : \u211d\u22650), f.symm x = x ^ p\nx : \u211d\u22650\nmx : x \u2208 univ\ny : \u211d\u22650\nmy : y \u2208 univ\nhxy : x \u2260 y\na b : \u211d\u22650\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 f.symm x + b \u2022 f.symm y < f.symm (a \u2022 x + b \u2022 y)"}, {"tactic": "exact (f.strictConcaveOn_symm h\u2081).2 mx my hxy ha hb hab", "annotated_tactic": ["exact (f.strictConcaveOn_symm h\u2081).2 mx my hxy ha hb hab", []], "state_before": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\nhp\u2081' : 1 < 1 / p\nf : \u211d\u22650 \u2243o \u211d\u22650 := orderIsoRpow (1 / p) hp\u2080'\nh\u2081 : StrictConvexOn \u211d\u22650 univ \u21d1f\nh\u2082 : \u2200 (x : \u211d\u22650), f.symm x = x ^ p\nx : \u211d\u22650\nmx : x \u2208 univ\ny : \u211d\u22650\nmy : y \u2208 univ\nhxy : x \u2260 y\na b : \u211d\u22650\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 a \u2022 f.symm x + b \u2022 f.symm y < f.symm (a \u2022 x + b \u2022 y)", "state_after": "no goals"}, {"tactic": "rw [one_lt_div hp\u2080]", "annotated_tactic": ["rw [<a>one_lt_div</a> hp\u2080]", [{"full_name": "one_lt_div", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [369, 9], "def_end_pos": [369, 19]}]], "state_before": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\n\u22a2 1 < 1 / p", "state_after": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\n\u22a2 p < 1"}, {"tactic": "exact hp\u2081", "annotated_tactic": ["exact hp\u2081", []], "state_before": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\n\u22a2 p < 1", "state_after": "no goals"}, {"tactic": "refine \u27e8convex_univ, fun x _ y _ hxy a b ha hb hab => ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>convex_univ</a>, fun x _ y _ hxy a b ha hb hab => ?_\u27e9", [{"full_name": "convex_univ", "def_path": "Mathlib/Analysis/Convex/Basic.lean", "def_pos": [91, 9], "def_end_pos": [91, 20]}]], "state_before": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\nhp\u2081' : 1 < 1 / p\nf : \u211d\u22650 \u2243o \u211d\u22650 := orderIsoRpow (1 / p) hp\u2080'\n\u22a2 StrictConvexOn \u211d\u22650 univ \u21d1f", "state_after": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\nhp\u2081' : 1 < 1 / p\nf : \u211d\u22650 \u2243o \u211d\u22650 := orderIsoRpow (1 / p) hp\u2080'\nx : \u211d\u22650\nx\u271d\u00b9 : x \u2208 univ\ny : \u211d\u22650\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \u211d\u22650\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y"}, {"tactic": "exact (strictConvexOn_rpow hp\u2081').2 x.2 y.2 (by simp [hxy]) ha hb (by simp; norm_cast)", "annotated_tactic": ["exact (<a>strictConvexOn_rpow</a> hp\u2081').2 x.2 y.2 (by simp [hxy]) ha hb (by simp; norm_cast)", [{"full_name": "strictConvexOn_rpow", "def_path": "Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean", "def_pos": [178, 9], "def_end_pos": [178, 28]}]], "state_before": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\nhp\u2081' : 1 < 1 / p\nf : \u211d\u22650 \u2243o \u211d\u22650 := orderIsoRpow (1 / p) hp\u2080'\nx : \u211d\u22650\nx\u271d\u00b9 : x \u2208 univ\ny : \u211d\u22650\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \u211d\u22650\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 f (a \u2022 x + b \u2022 y) < a \u2022 f x + b \u2022 f y", "state_after": "no goals"}, {"tactic": "simp [hxy]", "annotated_tactic": ["simp [hxy]", []], "state_before": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\nhp\u2081' : 1 < 1 / p\nf : \u211d\u22650 \u2243o \u211d\u22650 := orderIsoRpow (1 / p) hp\u2080'\nx : \u211d\u22650\nx\u271d\u00b9 : x \u2208 univ\ny : \u211d\u22650\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \u211d\u22650\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 \u2191x \u2260 \u2191y", "state_after": "no goals"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\nhp\u2081' : 1 < 1 / p\nf : \u211d\u22650 \u2243o \u211d\u22650 := orderIsoRpow (1 / p) hp\u2080'\nx : \u211d\u22650\nx\u271d\u00b9 : x \u2208 univ\ny : \u211d\u22650\nx\u271d : y \u2208 univ\nhxy : x \u2260 y\na b : \u211d\u22650\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n\u22a2 \u2191a + \u2191b = 1", "state_after": "no goals"}, {"tactic": "simp [f, NNReal.orderIsoRpow_symm_eq]", "annotated_tactic": ["simp [f, <a>NNReal.orderIsoRpow_symm_eq</a>]", [{"full_name": "NNReal.orderIsoRpow_symm_eq", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [375, 9], "def_end_pos": [375, 29]}]], "state_before": "p : \u211d\nhp\u2080 : 0 < p\nhp\u2081 : p < 1\nhp\u2080' : 0 < 1 / p\nhp\u2081' : 1 < 1 / p\nf : \u211d\u22650 \u2243o \u211d\u22650 := orderIsoRpow (1 / p) hp\u2080'\nh\u2081 : StrictConvexOn \u211d\u22650 univ \u21d1f\n\u22a2 \u2200 (x : \u211d\u22650), f.symm x = x ^ p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "Submodule.smul_inf_le", "start": [254, 11], "end": [255, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "full_name": "MeasureTheory.iUnion_spanningSets", "start": [659, 1], "end": [660, 79], "traced_tactics": [{"tactic": "simp_rw [spanningSets, iUnion_accumulate, \u03bc.toFiniteSpanningSetsIn.spanning]", "annotated_tactic": ["simp_rw [<a>spanningSets</a>, <a>iUnion_accumulate</a>, \u03bc.toFiniteSpanningSetsIn.spanning]", [{"full_name": "MeasureTheory.spanningSets", "def_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "def_pos": [641, 5], "def_end_pos": [641, 17]}, {"full_name": "Set.iUnion_accumulate", "def_path": "Mathlib/Data/Set/Accumulate.lean", "def_pos": [56, 9], "def_end_pos": [56, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b4 : Type u_3\n\u03b9 : Type u_4\nm0 : MeasurableSpace \u03b1\ninst\u271d\u00b9 : MeasurableSpace \u03b2\n\u03bc\u271d \u03bd \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\ns t : Set \u03b1\n\u03bc : Measure \u03b1\ninst\u271d : SigmaFinite \u03bc\n\u22a2 \u22c3 i, spanningSets \u03bc i = univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Expand.lean", "full_name": "Polynomial.rootMultiplicity_expand", "start": [297, 1], "end": [299, 48], "traced_tactics": [{"tactic": "rw [\u2190 pow_one p, rootMultiplicity_expand_pow]", "annotated_tactic": ["rw [\u2190 <a>pow_one</a> p, <a>rootMultiplicity_expand_pow</a>]", [{"full_name": "pow_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [663, 7], "def_end_pos": [663, 14]}, {"full_name": "Polynomial.rootMultiplicity_expand_pow", "def_path": "Mathlib/Algebra/Polynomial/Expand.lean", "def_pos": [288, 9], "def_end_pos": [288, 36]}]], "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\np n : \u2115\ninst\u271d : ExpChar R p\nf : R[X]\nr : R\n\u22a2 rootMultiplicity r ((expand R p) f) = p * rootMultiplicity (r ^ p) f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Types.lean", "full_name": "CategoryTheory.FunctorToTypes.map_hom_map_inv_apply", "start": [186, 1], "end": [187, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Surreal/Multiplication.lean", "full_name": "Surreal.Multiplication.P24_of_ih", "start": [234, 1], "end": [235, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.span_eq_take_drop", "start": [2976, 1], "end": [2977, 44], "traced_tactics": [{"tactic": "simpa using span.loop_eq_take_drop p l []", "annotated_tactic": ["simpa using <a>span.loop_eq_take_drop</a> p l []", [{"full_name": "List.span.loop_eq_take_drop", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [2969, 9], "def_end_pos": [2969, 31]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\np : \u03b1 \u2192 Bool\nl : List \u03b1\n\u22a2 span p l = (takeWhile p l, dropWhile p l)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean", "full_name": "LinearMap.BilinForm.isOrtho_def", "start": [53, 1], "end": [54, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Function.lean", "full_name": "Set.EqOn.mono", "start": [232, 1], "end": [232, 96], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Triangle/Removal.lean", "full_name": "SimpleGraph.triangleRemovalBound_pos", "start": [35, 1], "end": [38, 13], "traced_tactics": [{"tactic": "have : 0 < 1 - \u03b5 / 4 := by linarith", "annotated_tactic": ["have : 0 < 1 - \u03b5 / 4 := by linarith", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\nG : SimpleGraph \u03b1\ninst\u271d : DecidableRel G.Adj\ns t : Finset \u03b1\nP : Finpartition univ\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b5\u2081 : \u03b5 \u2264 1\n\u22a2 0 < triangleRemovalBound \u03b5", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\nG : SimpleGraph \u03b1\ninst\u271d : DecidableRel G.Adj\ns t : Finset \u03b1\nP : Finpartition univ\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : 0 < 1 - \u03b5 / 4\n\u22a2 0 < triangleRemovalBound \u03b5"}, {"tactic": "unfold triangleRemovalBound", "annotated_tactic": ["unfold <a>triangleRemovalBound</a>", [{"full_name": "SimpleGraph.triangleRemovalBound", "def_path": "Mathlib/Combinatorics/SimpleGraph/Triangle/Removal.lean", "def_pos": [32, 19], "def_end_pos": [32, 39]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\nG : SimpleGraph \u03b1\ninst\u271d : DecidableRel G.Adj\ns t : Finset \u03b1\nP : Finpartition univ\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : 0 < 1 - \u03b5 / 4\n\u22a2 0 < triangleRemovalBound \u03b5", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\nG : SimpleGraph \u03b1\ninst\u271d : DecidableRel G.Adj\ns t : Finset \u03b1\nP : Finpartition univ\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : 0 < 1 - \u03b5 / 4\n\u22a2 0 < min (2 * \u2191\u23084 / \u03b5\u2309\u208a ^ 3)\u207b\u00b9 ((1 - \u03b5 / 4) * (\u03b5 / (16 * \u2191(bound (\u03b5 / 8) \u23084 / \u03b5\u2309\u208a))) ^ 3)"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\nG : SimpleGraph \u03b1\ninst\u271d : DecidableRel G.Adj\ns t : Finset \u03b1\nP : Finpartition univ\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b5\u2081 : \u03b5 \u2264 1\nthis : 0 < 1 - \u03b5 / 4\n\u22a2 0 < min (2 * \u2191\u23084 / \u03b5\u2309\u208a ^ 3)\u207b\u00b9 ((1 - \u03b5 / 4) * (\u03b5 / (16 * \u2191(bound (\u03b5 / 8) \u23084 / \u03b5\u2309\u208a))) ^ 3)", "state_after": "no goals"}, {"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : DecidableEq \u03b1\ninst\u271d\u00b9 : Fintype \u03b1\nG : SimpleGraph \u03b1\ninst\u271d : DecidableRel G.Adj\ns t : Finset \u03b1\nP : Finpartition univ\n\u03b5 : \u211d\nh\u03b5 : 0 < \u03b5\nh\u03b5\u2081 : \u03b5 \u2264 1\n\u22a2 0 < 1 - \u03b5 / 4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Finset/Fin.lean", "full_name": "Fin.card_fintype_uIcc", "start": [152, 1], "end": [153, 42], "traced_tactics": [{"tactic": "rw [\u2190 card_uIcc, Fintype.card_ofFinset]", "annotated_tactic": ["rw [\u2190 <a>card_uIcc</a>, <a>Fintype.card_ofFinset</a>]", [{"full_name": "Fin.card_uIcc", "def_path": "Mathlib/Order/Interval/Finset/Fin.lean", "def_pos": [124, 9], "def_end_pos": [124, 18]}, {"full_name": "Fintype.card_ofFinset", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [134, 9], "def_end_pos": [134, 22]}]], "state_before": "n : \u2115\na b : Fin n\n\u22a2 Fintype.card \u2191(Set.uIcc a b) = (\u2191\u2191b - \u2191\u2191a).natAbs + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Functor/KanExtension/Pointwise.lean", "full_name": "CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtension.isIso_hom", "start": [129, 1], "end": [132, 34], "traced_tactics": [{"tactic": "have := fun X => (h (L.obj X)).isIso_hom_app", "annotated_tactic": ["have := fun X => (h (L.obj X)).<a>isIso_hom_app</a>", [{"full_name": "CategoryTheory.Functor.LeftExtension.IsPointwiseLeftKanExtensionAt.isIso_hom_app", "def_path": "Mathlib/CategoryTheory/Functor/KanExtension/Pointwise.lean", "def_pos": [79, 7], "def_end_pos": [79, 50]}]], "state_before": "C : Type u_1\nD : Type u_2\nH : Type u_3\ninst\u271d\u2074 : Category.{u_4, u_1} C\ninst\u271d\u00b3 : Category.{u_5, u_2} D\ninst\u271d\u00b2 : Category.{u_6, u_3} H\nL : C \u2964 D\nF : C \u2964 H\nE : L.LeftExtension F\nh : E.IsPointwiseLeftKanExtension\ninst\u271d\u00b9 : L.Full\ninst\u271d : L.Faithful\n\u22a2 IsIso E.hom", "state_after": "C : Type u_1\nD : Type u_2\nH : Type u_3\ninst\u271d\u2074 : Category.{u_4, u_1} C\ninst\u271d\u00b3 : Category.{u_5, u_2} D\ninst\u271d\u00b2 : Category.{u_6, u_3} H\nL : C \u2964 D\nF : C \u2964 H\nE : L.LeftExtension F\nh : E.IsPointwiseLeftKanExtension\ninst\u271d\u00b9 : L.Full\ninst\u271d : L.Faithful\nthis : \u2200 (X : C), IsIso (E.hom.app X)\n\u22a2 IsIso E.hom"}, {"tactic": "apply NatIso.isIso_of_isIso_app", "annotated_tactic": ["apply <a>NatIso.isIso_of_isIso_app</a>", [{"full_name": "CategoryTheory.NatIso.isIso_of_isIso_app", "def_path": "Mathlib/CategoryTheory/NatIso.lean", "def_pos": [238, 9], "def_end_pos": [238, 27]}]], "state_before": "C : Type u_1\nD : Type u_2\nH : Type u_3\ninst\u271d\u2074 : Category.{u_4, u_1} C\ninst\u271d\u00b3 : Category.{u_5, u_2} D\ninst\u271d\u00b2 : Category.{u_6, u_3} H\nL : C \u2964 D\nF : C \u2964 H\nE : L.LeftExtension F\nh : E.IsPointwiseLeftKanExtension\ninst\u271d\u00b9 : L.Full\ninst\u271d : L.Faithful\nthis : \u2200 (X : C), IsIso (E.hom.app X)\n\u22a2 IsIso E.hom", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/WittVector/Defs.lean", "full_name": "WittVector.mul_coeff_zero", "start": [389, 1], "end": [390, 26], "traced_tactics": [{"tactic": "simp [mul_coeff, peval]", "annotated_tactic": ["simp [<a>mul_coeff</a>, <a>peval</a>]", [{"full_name": "WittVector.mul_coeff", "def_path": "Mathlib/RingTheory/WittVector/Defs.lean", "def_pos": [361, 9], "def_end_pos": [361, 18]}, {"full_name": "WittVector.peval", "def_path": "Mathlib/RingTheory/WittVector/Defs.lean", "def_pos": [161, 5], "def_end_pos": [161, 10]}]], "state_before": "p : \u2115\nR : Type u_1\nhp : Fact (Nat.Prime p)\ninst\u271d : CommRing R\nx y : \ud835\udd4e R\n\u22a2 (x * y).coeff 0 = x.coeff 0 * y.coeff 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Projectivization/Basic.lean", "full_name": "Projectivization.rep_nonzero", "start": [82, 1], "end": [83, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Instances/ENNReal.lean", "full_name": "ENNReal.nhdsWithin_Ioi_ofNat_nebot", "start": [238, 1], "end": [239, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean", "full_name": "Batteries.RBNode.Ordered.toList_sorted", "start": [238, 1], "end": [239, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/VonMangoldt.lean", "full_name": "ArithmeticFunction.vonMangoldt_apply_one", "start": [79, 1], "end": [79, 71], "traced_tactics": [{"tactic": "simp [vonMangoldt_apply]", "annotated_tactic": ["simp [<a>vonMangoldt_apply</a>]", [{"full_name": "ArithmeticFunction.vonMangoldt_apply", "def_path": "Mathlib/NumberTheory/VonMangoldt.lean", "def_pos": [74, 9], "def_end_pos": [74, 26]}]], "state_before": "\u22a2 \u039b 1 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.StronglyMeasurable.inf", "start": [575, 11], "end": [578, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.get?_zero_scanl", "start": [2037, 1], "end": [2038, 29], "traced_tactics": [{"tactic": "simp [getElem?_scanl_zero]", "annotated_tactic": ["simp [<a>getElem?_scanl_zero</a>]", [{"full_name": "List.getElem?_scanl_zero", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [2030, 9], "def_end_pos": [2030, 28]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nf : \u03b2 \u2192 \u03b1 \u2192 \u03b2\nb : \u03b2\na : \u03b1\nl : List \u03b1\n\u22a2 (scanl f b l).get? 0 = some b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/ConcreteCategory.lean", "full_name": "CategoryTheory.Limits.Concrete.to_product_injective_of_isLimit", "start": [41, 1], "end": [54, 20], "traced_tactics": [{"tactic": "let E := (forget C).mapCone D", "annotated_tactic": ["let E := (<a>forget</a> C).<a>mapCone</a> D", [{"full_name": "CategoryTheory.forget", "def_path": "Mathlib/CategoryTheory/ConcreteCategory/Basic.lean", "def_pos": [66, 8], "def_end_pos": [66, 14]}, {"full_name": "CategoryTheory.Functor.mapCone", "def_path": "Mathlib/CategoryTheory/Limits/Cones.lean", "def_pos": [727, 5], "def_end_pos": [727, 12]}]], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\n\u22a2 Function.Injective fun x j => (D.\u03c0.app j) x", "state_after": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\n\u22a2 Function.Injective fun x j => (D.\u03c0.app j) x"}, {"tactic": "let hE : IsLimit E := isLimitOfPreserves _ hD", "annotated_tactic": ["let hE : <a>IsLimit</a> E := <a>isLimitOfPreserves</a> _ hD", [{"full_name": "CategoryTheory.Limits.IsLimit", "def_path": "Mathlib/CategoryTheory/Limits/IsLimit.lean", "def_pos": [55, 11], "def_end_pos": [55, 18]}, {"full_name": "CategoryTheory.Limits.isLimitOfPreserves", "def_path": "Mathlib/CategoryTheory/Limits/Preserves/Basic.lean", "def_pos": [121, 5], "def_end_pos": [121, 23]}]], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\n\u22a2 Function.Injective fun x j => (D.\u03c0.app j) x", "state_after": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\nhE : IsLimit E := isLimitOfPreserves (forget C) hD\n\u22a2 Function.Injective fun x j => (D.\u03c0.app j) x"}, {"tactic": "let G := Types.limitCone.{w, v} (F \u22d9 forget C)", "annotated_tactic": ["let G := <a>Types.limitCone</a>.{w, v} (F \u22d9 <a>forget</a> C)", [{"full_name": "CategoryTheory.Limits.Types.limitCone", "def_path": "Mathlib/CategoryTheory/Limits/Types.lean", "def_pos": [160, 19], "def_end_pos": [160, 28]}, {"full_name": "CategoryTheory.forget", "def_path": "Mathlib/CategoryTheory/ConcreteCategory/Basic.lean", "def_pos": [66, 8], "def_end_pos": [66, 14]}]], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\nhE : IsLimit E := isLimitOfPreserves (forget C) hD\n\u22a2 Function.Injective fun x j => (D.\u03c0.app j) x", "state_after": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\nhE : IsLimit E := isLimitOfPreserves (forget C) hD\nG : Cone (F \u22d9 forget C) := Types.limitCone (F \u22d9 forget C)\n\u22a2 Function.Injective fun x j => (D.\u03c0.app j) x"}, {"tactic": "let hG := Types.limitConeIsLimit.{w, v} (F \u22d9 forget C)", "annotated_tactic": ["let hG := <a>Types.limitConeIsLimit</a>.{w, v} (F \u22d9 <a>forget</a> C)", [{"full_name": "CategoryTheory.Limits.Types.limitConeIsLimit", "def_path": "Mathlib/CategoryTheory/Limits/Types.lean", "def_pos": [171, 19], "def_end_pos": [171, 35]}, {"full_name": "CategoryTheory.forget", "def_path": "Mathlib/CategoryTheory/ConcreteCategory/Basic.lean", "def_pos": [66, 8], "def_end_pos": [66, 14]}]], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\nhE : IsLimit E := isLimitOfPreserves (forget C) hD\nG : Cone (F \u22d9 forget C) := Types.limitCone (F \u22d9 forget C)\n\u22a2 Function.Injective fun x j => (D.\u03c0.app j) x", "state_after": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\nhE : IsLimit E := isLimitOfPreserves (forget C) hD\nG : Cone (F \u22d9 forget C) := Types.limitCone (F \u22d9 forget C)\nhG : IsLimit (Types.limitCone (F \u22d9 forget C)) := Types.limitConeIsLimit (F \u22d9 forget C)\n\u22a2 Function.Injective fun x j => (D.\u03c0.app j) x"}, {"tactic": "let T : E.pt \u2245 G.pt := hE.conePointUniqueUpToIso hG", "annotated_tactic": ["let T : E.pt \u2245 G.pt := hE.conePointUniqueUpToIso hG", []], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\nhE : IsLimit E := isLimitOfPreserves (forget C) hD\nG : Cone (F \u22d9 forget C) := Types.limitCone (F \u22d9 forget C)\nhG : IsLimit (Types.limitCone (F \u22d9 forget C)) := Types.limitConeIsLimit (F \u22d9 forget C)\n\u22a2 Function.Injective fun x j => (D.\u03c0.app j) x", "state_after": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\nhE : IsLimit E := isLimitOfPreserves (forget C) hD\nG : Cone (F \u22d9 forget C) := Types.limitCone (F \u22d9 forget C)\nhG : IsLimit (Types.limitCone (F \u22d9 forget C)) := Types.limitConeIsLimit (F \u22d9 forget C)\nT : E.pt \u2245 G.pt := hE.conePointUniqueUpToIso hG\n\u22a2 Function.Injective fun x j => (D.\u03c0.app j) x"}, {"tactic": "change Function.Injective (T.hom \u226b fun x j => G.\u03c0.app j x)", "annotated_tactic": ["change <a>Function.Injective</a> (T.hom \u226b fun x j => G.\u03c0.app j x)", [{"full_name": "Function.Injective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [123, 5], "def_end_pos": [123, 14]}]], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\nhE : IsLimit E := isLimitOfPreserves (forget C) hD\nG : Cone (F \u22d9 forget C) := Types.limitCone (F \u22d9 forget C)\nhG : IsLimit (Types.limitCone (F \u22d9 forget C)) := Types.limitConeIsLimit (F \u22d9 forget C)\nT : E.pt \u2245 G.pt := hE.conePointUniqueUpToIso hG\n\u22a2 Function.Injective fun x j => (D.\u03c0.app j) x", "state_after": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\nhE : IsLimit E := isLimitOfPreserves (forget C) hD\nG : Cone (F \u22d9 forget C) := Types.limitCone (F \u22d9 forget C)\nhG : IsLimit (Types.limitCone (F \u22d9 forget C)) := Types.limitConeIsLimit (F \u22d9 forget C)\nT : E.pt \u2245 G.pt := hE.conePointUniqueUpToIso hG\n\u22a2 Function.Injective (T.hom \u226b fun x j => G.\u03c0.app j x)"}, {"tactic": "have h : Function.Injective T.hom := by\n  intro a b h\n  suffices T.inv (T.hom a) = T.inv (T.hom b) by simpa\n  rw [h]", "annotated_tactic": ["have h : <a>Function.Injective</a> T.hom := by\n    intro a b h\n    suffices T.inv (T.hom a) = T.inv (T.hom b) by simpa\n    rw [h]", [{"full_name": "Function.Injective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [123, 5], "def_end_pos": [123, 14]}]], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\nhE : IsLimit E := isLimitOfPreserves (forget C) hD\nG : Cone (F \u22d9 forget C) := Types.limitCone (F \u22d9 forget C)\nhG : IsLimit (Types.limitCone (F \u22d9 forget C)) := Types.limitConeIsLimit (F \u22d9 forget C)\nT : E.pt \u2245 G.pt := hE.conePointUniqueUpToIso hG\n\u22a2 Function.Injective (T.hom \u226b fun x j => G.\u03c0.app j x)", "state_after": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\nhE : IsLimit E := isLimitOfPreserves (forget C) hD\nG : Cone (F \u22d9 forget C) := Types.limitCone (F \u22d9 forget C)\nhG : IsLimit (Types.limitCone (F \u22d9 forget C)) := Types.limitConeIsLimit (F \u22d9 forget C)\nT : E.pt \u2245 G.pt := hE.conePointUniqueUpToIso hG\nh : Function.Injective T.hom\n\u22a2 Function.Injective (T.hom \u226b fun x j => G.\u03c0.app j x)"}, {"tactic": "suffices Function.Injective fun (x : G.pt) j => G.\u03c0.app j x by exact this.comp h", "annotated_tactic": ["suffices <a>Function.Injective</a> fun (x : G.pt) j => G.\u03c0.app j x by exact this.comp h", [{"full_name": "Function.Injective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [123, 5], "def_end_pos": [123, 14]}]], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\nhE : IsLimit E := isLimitOfPreserves (forget C) hD\nG : Cone (F \u22d9 forget C) := Types.limitCone (F \u22d9 forget C)\nhG : IsLimit (Types.limitCone (F \u22d9 forget C)) := Types.limitConeIsLimit (F \u22d9 forget C)\nT : E.pt \u2245 G.pt := hE.conePointUniqueUpToIso hG\nh : Function.Injective T.hom\n\u22a2 Function.Injective (T.hom \u226b fun x j => G.\u03c0.app j x)", "state_after": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\nhE : IsLimit E := isLimitOfPreserves (forget C) hD\nG : Cone (F \u22d9 forget C) := Types.limitCone (F \u22d9 forget C)\nhG : IsLimit (Types.limitCone (F \u22d9 forget C)) := Types.limitConeIsLimit (F \u22d9 forget C)\nT : E.pt \u2245 G.pt := hE.conePointUniqueUpToIso hG\nh : Function.Injective T.hom\n\u22a2 Function.Injective fun x j => G.\u03c0.app j x"}, {"tactic": "apply Subtype.ext", "annotated_tactic": ["apply <a>Subtype.ext</a>", [{"full_name": "Subtype.ext", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [74, 19], "def_end_pos": [74, 22]}]], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\nhE : IsLimit E := isLimitOfPreserves (forget C) hD\nG : Cone (F \u22d9 forget C) := Types.limitCone (F \u22d9 forget C)\nhG : IsLimit (Types.limitCone (F \u22d9 forget C)) := Types.limitConeIsLimit (F \u22d9 forget C)\nT : E.pt \u2245 G.pt := hE.conePointUniqueUpToIso hG\nh : Function.Injective T.hom\n\u22a2 Function.Injective fun x j => G.\u03c0.app j x", "state_after": "no goals"}, {"tactic": "intro a b h", "annotated_tactic": ["intro a b h", []], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\nhE : IsLimit E := isLimitOfPreserves (forget C) hD\nG : Cone (F \u22d9 forget C) := Types.limitCone (F \u22d9 forget C)\nhG : IsLimit (Types.limitCone (F \u22d9 forget C)) := Types.limitConeIsLimit (F \u22d9 forget C)\nT : E.pt \u2245 G.pt := hE.conePointUniqueUpToIso hG\n\u22a2 Function.Injective T.hom", "state_after": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\nhE : IsLimit E := isLimitOfPreserves (forget C) hD\nG : Cone (F \u22d9 forget C) := Types.limitCone (F \u22d9 forget C)\nhG : IsLimit (Types.limitCone (F \u22d9 forget C)) := Types.limitConeIsLimit (F \u22d9 forget C)\nT : E.pt \u2245 G.pt := hE.conePointUniqueUpToIso hG\na b : E.pt\nh : T.hom a = T.hom b\n\u22a2 a = b"}, {"tactic": "suffices T.inv (T.hom a) = T.inv (T.hom b) by simpa", "annotated_tactic": ["suffices T.inv (T.hom a) = T.inv (T.hom b) by simpa", []], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\nhE : IsLimit E := isLimitOfPreserves (forget C) hD\nG : Cone (F \u22d9 forget C) := Types.limitCone (F \u22d9 forget C)\nhG : IsLimit (Types.limitCone (F \u22d9 forget C)) := Types.limitConeIsLimit (F \u22d9 forget C)\nT : E.pt \u2245 G.pt := hE.conePointUniqueUpToIso hG\na b : E.pt\nh : T.hom a = T.hom b\n\u22a2 a = b", "state_after": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\nhE : IsLimit E := isLimitOfPreserves (forget C) hD\nG : Cone (F \u22d9 forget C) := Types.limitCone (F \u22d9 forget C)\nhG : IsLimit (Types.limitCone (F \u22d9 forget C)) := Types.limitConeIsLimit (F \u22d9 forget C)\nT : E.pt \u2245 G.pt := hE.conePointUniqueUpToIso hG\na b : E.pt\nh : T.hom a = T.hom b\n\u22a2 T.inv (T.hom a) = T.inv (T.hom b)"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\nhE : IsLimit E := isLimitOfPreserves (forget C) hD\nG : Cone (F \u22d9 forget C) := Types.limitCone (F \u22d9 forget C)\nhG : IsLimit (Types.limitCone (F \u22d9 forget C)) := Types.limitConeIsLimit (F \u22d9 forget C)\nT : E.pt \u2245 G.pt := hE.conePointUniqueUpToIso hG\na b : E.pt\nh : T.hom a = T.hom b\n\u22a2 T.inv (T.hom a) = T.inv (T.hom b)", "state_after": "no goals"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\nhE : IsLimit E := isLimitOfPreserves (forget C) hD\nG : Cone (F \u22d9 forget C) := Types.limitCone (F \u22d9 forget C)\nhG : IsLimit (Types.limitCone (F \u22d9 forget C)) := Types.limitConeIsLimit (F \u22d9 forget C)\nT : E.pt \u2245 G.pt := hE.conePointUniqueUpToIso hG\na b : E.pt\nh : T.hom a = T.hom b\nthis : T.inv (T.hom a) = T.inv (T.hom b)\n\u22a2 a = b", "state_after": "no goals"}, {"tactic": "exact this.comp h", "annotated_tactic": ["exact this.comp h", []], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : ConcreteCategory C\nJ : Type w\ninst\u271d\u00b9 : Category.{t, w} J\nF : J \u2964 C\ninst\u271d : PreservesLimit F (forget C)\nD : Cone F\nhD : IsLimit D\nE : Cone (F \u22d9 forget C) := (forget C).mapCone D\nhE : IsLimit E := isLimitOfPreserves (forget C) hD\nG : Cone (F \u22d9 forget C) := Types.limitCone (F \u22d9 forget C)\nhG : IsLimit (Types.limitCone (F \u22d9 forget C)) := Types.limitConeIsLimit (F \u22d9 forget C)\nT : E.pt \u2245 G.pt := hE.conePointUniqueUpToIso hG\nh : Function.Injective T.hom\nthis : Function.Injective fun x j => G.\u03c0.app j x\n\u22a2 Function.Injective (T.hom \u226b fun x j => G.\u03c0.app j x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/EssSup.lean", "full_name": "essSup_map_measure", "start": [245, 1], "end": [251, 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": [236, 9], "def_end_pos": [236, 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 (Measure.map f \u03bc)\nhf : AEMeasurable f \u03bc\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 (Measure.map f \u03bc)\nhf : AEMeasurable f \u03bc\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 (Measure.map f \u03bc)\nhf : AEMeasurable f \u03bc\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 (Measure.map f \u03bc)\nhf : AEMeasurable f \u03bc\n\u22a2 AEMeasurable.mk g hg \u2218 f =\u1da0[ae \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": [2027, 9], "def_end_pos": [2027, 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 (Measure.map f \u03bc)\nhf : AEMeasurable f \u03bc\n\u22a2 AEMeasurable.mk g hg \u2218 f =\u1da0[ae \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 (Measure.map f \u03bc)\nhf : AEMeasurable f \u03bc\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 =\u1da0[ae \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": [1470, 5], "def_end_pos": [1470, 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 (Measure.map f \u03bc)\nhf : AEMeasurable f \u03bc\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 =\u1da0[ae \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 (Measure.map f \u03bc)\nhf : AEMeasurable f \u03bc\nh_eq : (fun x => g (f x)) =\u1da0[ae \u03bc] fun x => AEMeasurable.mk g hg (f x)\n\u22a2 AEMeasurable.mk g hg \u2218 f =\u1da0[ae \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 (Measure.map f \u03bc)\nhf : AEMeasurable f \u03bc\nh_eq : (fun x => g (f x)) =\u1da0[ae \u03bc] fun x => AEMeasurable.mk g hg (f x)\n\u22a2 AEMeasurable.mk g hg \u2218 f =\u1da0[ae \u03bc] g \u2218 f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/PFun.lean", "full_name": "PFun.mem_dom", "start": [80, 1], "end": [80, 99], "traced_tactics": [{"tactic": "simp [Dom, Part.dom_iff_mem]", "annotated_tactic": ["simp [<a>Dom</a>, <a>Part.dom_iff_mem</a>]", [{"full_name": "PFun.Dom", "def_path": "Mathlib/Data/PFun.lean", "def_pos": [75, 5], "def_end_pos": [75, 8]}, {"full_name": "Part.dom_iff_mem", "def_path": "Mathlib/Data/Part.lean", "def_pos": [103, 9], "def_end_pos": [103, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b5 : Type u_5\n\u03b9 : Type u_6\nf : \u03b1 \u2192. \u03b2\nx : \u03b1\n\u22a2 x \u2208 f.Dom \u2194 \u2203 y, y \u2208 f x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean", "full_name": "tfae_of_isNoetherianRing_of_localRing_of_isDomain", "start": [166, 1], "end": [205, 14], "traced_tactics": [{"tactic": "tfae_have 1 \u2192 2", "annotated_tactic": ["tfae_have 1 \u2192 2", []], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u22a2 [IsPrincipalIdealRing R, ValuationRing R, IsDedekindDomain R,\n      IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R,\n      Submodule.IsPrincipal (maximalIdeal R), finrank (ResidueField R) (CotangentSpace R) \u2264 1,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n].TFAE", "state_after": "case tfae_1_to_2\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u22a2 IsPrincipalIdealRing R \u2192 ValuationRing R\n\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\n\u22a2 [IsPrincipalIdealRing R, ValuationRing R, IsDedekindDomain R,\n      IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R,\n      Submodule.IsPrincipal (maximalIdeal R), finrank (ResidueField R) (CotangentSpace R) \u2264 1,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n].TFAE"}, {"tactic": "tfae_have 2 \u2192 1", "annotated_tactic": ["tfae_have 2 \u2192 1", []], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\n\u22a2 [IsPrincipalIdealRing R, ValuationRing R, IsDedekindDomain R,\n      IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R,\n      Submodule.IsPrincipal (maximalIdeal R), finrank (ResidueField R) (CotangentSpace R) \u2264 1,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n].TFAE", "state_after": "case tfae_2_to_1\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\n\u22a2 ValuationRing R \u2192 IsPrincipalIdealRing R\n\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\n\u22a2 [IsPrincipalIdealRing R, ValuationRing R, IsDedekindDomain R,\n      IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R,\n      Submodule.IsPrincipal (maximalIdeal R), finrank (ResidueField R) (CotangentSpace R) \u2264 1,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n].TFAE"}, {"tactic": "tfae_have 1 \u2192 4", "annotated_tactic": ["tfae_have 1 \u2192 4", []], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\n\u22a2 [IsPrincipalIdealRing R, ValuationRing R, IsDedekindDomain R,\n      IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R,\n      Submodule.IsPrincipal (maximalIdeal R), finrank (ResidueField R) (CotangentSpace R) \u2264 1,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n].TFAE", "state_after": "case tfae_1_to_4\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\n\u22a2 IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\n\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\n\u22a2 [IsPrincipalIdealRing R, ValuationRing R, IsDedekindDomain R,\n      IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R,\n      Submodule.IsPrincipal (maximalIdeal R), finrank (ResidueField R) (CotangentSpace R) \u2264 1,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n].TFAE"}, {"tactic": "tfae_have 4 \u2192 3", "annotated_tactic": ["tfae_have 4 \u2192 3", []], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\n\u22a2 [IsPrincipalIdealRing R, ValuationRing R, IsDedekindDomain R,\n      IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R,\n      Submodule.IsPrincipal (maximalIdeal R), finrank (ResidueField R) (CotangentSpace R) \u2264 1,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n].TFAE", "state_after": "case tfae_4_to_3\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\n\u22a2 (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\n\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\n\u22a2 [IsPrincipalIdealRing R, ValuationRing R, IsDedekindDomain R,\n      IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R,\n      Submodule.IsPrincipal (maximalIdeal R), finrank (ResidueField R) (CotangentSpace R) \u2264 1,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n].TFAE"}, {"tactic": "tfae_have 3 \u2192 5", "annotated_tactic": ["tfae_have 3 \u2192 5", []], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\n\u22a2 [IsPrincipalIdealRing R, ValuationRing R, IsDedekindDomain R,\n      IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R,\n      Submodule.IsPrincipal (maximalIdeal R), finrank (ResidueField R) (CotangentSpace R) \u2264 1,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n].TFAE", "state_after": "case tfae_3_to_5\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\n\u22a2 IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\n\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\n\u22a2 [IsPrincipalIdealRing R, ValuationRing R, IsDedekindDomain R,\n      IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R,\n      Submodule.IsPrincipal (maximalIdeal R), finrank (ResidueField R) (CotangentSpace R) \u2264 1,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n].TFAE"}, {"tactic": "tfae_have 6 \u2194 5", "annotated_tactic": ["tfae_have 6 \u2194 5", []], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\n\u22a2 [IsPrincipalIdealRing R, ValuationRing R, IsDedekindDomain R,\n      IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R,\n      Submodule.IsPrincipal (maximalIdeal R), finrank (ResidueField R) (CotangentSpace R) \u2264 1,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n].TFAE", "state_after": "case tfae_6_iff_5\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\n\u22a2 finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\n\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\n\u22a2 [IsPrincipalIdealRing R, ValuationRing R, IsDedekindDomain R,\n      IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R,\n      Submodule.IsPrincipal (maximalIdeal R), finrank (ResidueField R) (CotangentSpace R) \u2264 1,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n].TFAE"}, {"tactic": "tfae_have 5 \u2192 7", "annotated_tactic": ["tfae_have 5 \u2192 7", []], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\n\u22a2 [IsPrincipalIdealRing R, ValuationRing R, IsDedekindDomain R,\n      IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R,\n      Submodule.IsPrincipal (maximalIdeal R), finrank (ResidueField R) (CotangentSpace R) \u2264 1,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n].TFAE", "state_after": "case tfae_5_to_7\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\n\u22a2 Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\n\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\n\u22a2 [IsPrincipalIdealRing R, ValuationRing R, IsDedekindDomain R,\n      IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R,\n      Submodule.IsPrincipal (maximalIdeal R), finrank (ResidueField R) (CotangentSpace R) \u2264 1,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n].TFAE"}, {"tactic": "tfae_have 7 \u2192 2", "annotated_tactic": ["tfae_have 7 \u2192 2", []], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\n\u22a2 [IsPrincipalIdealRing R, ValuationRing R, IsDedekindDomain R,\n      IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R,\n      Submodule.IsPrincipal (maximalIdeal R), finrank (ResidueField R) (CotangentSpace R) \u2264 1,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n].TFAE", "state_after": "case tfae_7_to_2\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\n\u22a2 (\u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n) \u2192 ValuationRing R\n\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\ntfae_7_to_2 : (\u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n) \u2192 ValuationRing R\n\u22a2 [IsPrincipalIdealRing R, ValuationRing R, IsDedekindDomain R,\n      IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R,\n      Submodule.IsPrincipal (maximalIdeal R), finrank (ResidueField R) (CotangentSpace R) \u2264 1,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n].TFAE"}, {"tactic": "tfae_finish", "annotated_tactic": ["tfae_finish", []], "state_before": "R : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\ntfae_7_to_2 : (\u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n) \u2192 ValuationRing R\n\u22a2 [IsPrincipalIdealRing R, ValuationRing R, IsDedekindDomain R,\n      IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R,\n      Submodule.IsPrincipal (maximalIdeal R), finrank (ResidueField R) (CotangentSpace R) \u2264 1,\n      \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n].TFAE", "state_after": "no goals"}, {"tactic": "exact fun _ \u21a6 inferInstance", "annotated_tactic": ["exact fun _ \u21a6 <a>inferInstance</a>", [{"full_name": "inferInstance", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [99, 8], "def_end_pos": [99, 21]}]], "state_before": "case tfae_1_to_2\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\n\u22a2 IsPrincipalIdealRing R \u2192 ValuationRing R", "state_after": "no goals"}, {"tactic": "exact fun _ \u21a6 ((IsBezout.TFAE (R := R)).out 0 1).mp \u2039_\u203a", "annotated_tactic": ["exact fun _ \u21a6 ((<a>IsBezout.TFAE</a> (R := R)).<a>out</a> 0 1).<a>mp</a> \u2039_\u203a", [{"full_name": "IsBezout.TFAE", "def_path": "Mathlib/RingTheory/Bezout.lean", "def_pos": [53, 9], "def_end_pos": [53, 13]}, {"full_name": "List.TFAE.out", "def_path": "Mathlib/Data/List/TFAE.lean", "def_pos": [74, 9], "def_end_pos": [74, 17]}, {"full_name": "Iff.mp", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [118, 3], "def_end_pos": [118, 5]}]], "state_before": "case tfae_2_to_1\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\n\u22a2 ValuationRing R \u2192 IsPrincipalIdealRing R", "state_after": "no goals"}, {"tactic": "intro H", "annotated_tactic": ["intro H", []], "state_before": "case tfae_1_to_4\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\n\u22a2 IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R", "state_after": "case tfae_1_to_4\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\nH : IsPrincipalIdealRing R\n\u22a2 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R"}, {"tactic": "exact \u27e8inferInstance, fun P hP hP' \u21a6 eq_maximalIdeal (hP'.isMaximal hP)\u27e9", "annotated_tactic": ["exact \u27e8<a>inferInstance</a>, fun P hP hP' \u21a6 <a>eq_maximalIdeal</a> (hP'.isMaximal hP)\u27e9", [{"full_name": "inferInstance", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [99, 8], "def_end_pos": [99, 21]}, {"full_name": "LocalRing.eq_maximalIdeal", "def_path": "Mathlib/RingTheory/Ideal/LocalRing.lean", "def_pos": [132, 9], "def_end_pos": [132, 24]}]], "state_before": "case tfae_1_to_4\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\nH : IsPrincipalIdealRing R\n\u22a2 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R", "state_after": "no goals"}, {"tactic": "exact fun h \u21a6 maximalIdeal_isPrincipal_of_isDedekindDomain R", "annotated_tactic": ["exact fun h \u21a6 <a>maximalIdeal_isPrincipal_of_isDedekindDomain</a> R", [{"full_name": "maximalIdeal_isPrincipal_of_isDedekindDomain", "def_path": "Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean", "def_pos": [92, 9], "def_end_pos": [92, 53]}]], "state_before": "case tfae_3_to_5\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\n\u22a2 IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)", "state_after": "no goals"}, {"tactic": "exact finrank_cotangentSpace_le_one_iff", "annotated_tactic": ["exact <a>finrank_cotangentSpace_le_one_iff</a>", [{"full_name": "LocalRing.finrank_cotangentSpace_le_one_iff", "def_path": "Mathlib/RingTheory/Ideal/Cotangent.lean", "def_pos": [272, 9], "def_end_pos": [272, 42]}]], "state_before": "case tfae_6_iff_5\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\n\u22a2 finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)", "state_after": "no goals"}, {"tactic": "exact exists_maximalIdeal_pow_eq_of_principal R", "annotated_tactic": ["exact <a>exists_maximalIdeal_pow_eq_of_principal</a> R", [{"full_name": "exists_maximalIdeal_pow_eq_of_principal", "def_path": "Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean", "def_pos": [37, 9], "def_end_pos": [37, 48]}]], "state_before": "case tfae_5_to_7\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\n\u22a2 Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n", "state_after": "no goals"}, {"tactic": "rw [ValuationRing.iff_ideal_total]", "annotated_tactic": ["rw [<a>ValuationRing.iff_ideal_total</a>]", [{"full_name": "ValuationRing.iff_ideal_total", "def_path": "Mathlib/RingTheory/Valuation/ValuationRing.lean", "def_pos": [289, 9], "def_end_pos": [289, 24]}]], "state_before": "case tfae_7_to_2\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\n\u22a2 (\u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n) \u2192 ValuationRing R", "state_after": "case tfae_7_to_2\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\n\u22a2 (\u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n) \u2192 IsTotal (Ideal R) fun x x_1 => x \u2264 x_1"}, {"tactic": "intro H", "annotated_tactic": ["intro H", []], "state_before": "case tfae_7_to_2\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\n\u22a2 (\u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n) \u2192 IsTotal (Ideal R) fun x x_1 => x \u2264 x_1", "state_after": "case tfae_7_to_2\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\n\u22a2 IsTotal (Ideal R) fun x x_1 => x \u2264 x_1"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case tfae_7_to_2\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\n\u22a2 IsTotal (Ideal R) fun x x_1 => x \u2264 x_1", "state_after": "case tfae_7_to_2.total\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\n\u22a2 \u2200 (a b : Ideal R), a \u2264 b \u2228 b \u2264 a"}, {"tactic": "intro I J", "annotated_tactic": ["intro I J", []], "state_before": "case tfae_7_to_2.total\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\n\u22a2 \u2200 (a b : Ideal R), a \u2264 b \u2228 b \u2264 a", "state_after": "case tfae_7_to_2.total\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI J : Ideal R\n\u22a2 I \u2264 J \u2228 J \u2264 I"}, {"tactic": "let _ := Classical.decEq (Ideal R)", "annotated_tactic": ["let _ := <a>Classical.decEq</a> (<a>Ideal</a> R)", [{"full_name": "Classical.decEq", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1020, 19], "def_end_pos": [1020, 24]}, {"full_name": "Ideal", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [41, 8], "def_end_pos": [41, 13]}]], "state_before": "case tfae_7_to_2.total\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI J : Ideal R\n\u22a2 I \u2264 J \u2228 J \u2264 I", "state_after": "case tfae_7_to_2.total\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI J : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\n\u22a2 I \u2264 J \u2228 J \u2264 I"}, {"tactic": "by_cases hI : I = \u22a5", "annotated_tactic": ["by_cases hI : I = \u22a5", []], "state_before": "case tfae_7_to_2.total\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI J : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\n\u22a2 I \u2264 J \u2228 J \u2264 I", "state_after": "case pos\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI J : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nhI : I = \u22a5\n\u22a2 I \u2264 J \u2228 J \u2264 I\n\ncase neg\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI J : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nhI : \u00acI = \u22a5\n\u22a2 I \u2264 J \u2228 J \u2264 I"}, {"tactic": "by_cases hJ : J = \u22a5", "annotated_tactic": ["by_cases hJ : J = \u22a5", []], "state_before": "case neg\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI J : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nhI : \u00acI = \u22a5\n\u22a2 I \u2264 J \u2228 J \u2264 I", "state_after": "case pos\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI J : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nhI : \u00acI = \u22a5\nhJ : J = \u22a5\n\u22a2 I \u2264 J \u2228 J \u2264 I\n\ncase neg\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI J : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nhI : \u00acI = \u22a5\nhJ : \u00acJ = \u22a5\n\u22a2 I \u2264 J \u2228 J \u2264 I"}, {"tactic": "obtain \u27e8n, rfl\u27e9 := H I hI", "annotated_tactic": ["obtain \u27e8n, rfl\u27e9 := H I hI", []], "state_before": "case neg\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI J : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nhI : \u00acI = \u22a5\nhJ : \u00acJ = \u22a5\n\u22a2 I \u2264 J \u2228 J \u2264 I", "state_after": "case neg.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nJ : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nhJ : \u00acJ = \u22a5\nn : \u2115\nhI : \u00acmaximalIdeal R ^ n = \u22a5\n\u22a2 maximalIdeal R ^ n \u2264 J \u2228 J \u2264 maximalIdeal R ^ n"}, {"tactic": "obtain \u27e8m, rfl\u27e9 := H J hJ", "annotated_tactic": ["obtain \u27e8m, rfl\u27e9 := H J hJ", []], "state_before": "case neg.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nJ : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nhJ : \u00acJ = \u22a5\nn : \u2115\nhI : \u00acmaximalIdeal R ^ n = \u22a5\n\u22a2 maximalIdeal R ^ n \u2264 J \u2228 J \u2264 maximalIdeal R ^ n", "state_after": "case neg.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nn : \u2115\nhI : \u00acmaximalIdeal R ^ n = \u22a5\nm : \u2115\nhJ : \u00acmaximalIdeal R ^ m = \u22a5\n\u22a2 maximalIdeal R ^ n \u2264 maximalIdeal R ^ m \u2228 maximalIdeal R ^ m \u2264 maximalIdeal R ^ n"}, {"tactic": "exact (le_total m n).imp Ideal.pow_le_pow_right Ideal.pow_le_pow_right", "annotated_tactic": ["exact (<a>le_total</a> m n).<a>imp</a> <a>Ideal.pow_le_pow_right</a> <a>Ideal.pow_le_pow_right</a>", [{"full_name": "le_total", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [297, 9], "def_end_pos": [297, 17]}, {"full_name": "Or.imp", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [172, 9], "def_end_pos": [172, 15]}, {"full_name": "Ideal.pow_le_pow_right", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [778, 9], "def_end_pos": [778, 25]}, {"full_name": "Ideal.pow_le_pow_right", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [778, 9], "def_end_pos": [778, 25]}]], "state_before": "case neg.intro.intro\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nn : \u2115\nhI : \u00acmaximalIdeal R ^ n = \u22a5\nm : \u2115\nhJ : \u00acmaximalIdeal R ^ m = \u22a5\n\u22a2 maximalIdeal R ^ n \u2264 maximalIdeal R ^ m \u2228 maximalIdeal R ^ m \u2264 maximalIdeal R ^ n", "state_after": "no goals"}, {"tactic": "subst hI", "annotated_tactic": ["subst hI", []], "state_before": "case pos\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI J : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nhI : I = \u22a5\n\u22a2 I \u2264 J \u2228 J \u2264 I", "state_after": "case pos\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nJ : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\n\u22a2 \u22a5 \u2264 J \u2228 J \u2264 \u22a5"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "case pos\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nJ : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\n\u22a2 \u22a5 \u2264 J \u2228 J \u2264 \u22a5", "state_after": "case pos.h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nJ : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\n\u22a2 \u22a5 \u2264 J"}, {"tactic": "exact bot_le", "annotated_tactic": ["exact <a>bot_le</a>", [{"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [224, 9], "def_end_pos": [224, 15]}]], "state_before": "case pos.h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nJ : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\n\u22a2 \u22a5 \u2264 J", "state_after": "no goals"}, {"tactic": "subst hJ", "annotated_tactic": ["subst hJ", []], "state_before": "case pos\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI J : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nhI : \u00acI = \u22a5\nhJ : J = \u22a5\n\u22a2 I \u2264 J \u2228 J \u2264 I", "state_after": "case pos\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nhI : \u00acI = \u22a5\n\u22a2 I \u2264 \u22a5 \u2228 \u22a5 \u2264 I"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "case pos\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nhI : \u00acI = \u22a5\n\u22a2 I \u2264 \u22a5 \u2228 \u22a5 \u2264 I", "state_after": "case pos.h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nhI : \u00acI = \u22a5\n\u22a2 \u22a5 \u2264 I"}, {"tactic": "exact bot_le", "annotated_tactic": ["exact <a>bot_le</a>", [{"full_name": "bot_le", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [224, 9], "def_end_pos": [224, 15]}]], "state_before": "case pos.h\nR : Type u_1\ninst\u271d\u2076 : CommRing R\nK : Type u_2\ninst\u271d\u2075 : Field K\ninst\u271d\u2074 : Algebra R K\ninst\u271d\u00b3 : IsFractionRing R K\ninst\u271d\u00b2 : IsNoetherianRing R\ninst\u271d\u00b9 : LocalRing R\ninst\u271d : IsDomain R\ntfae_1_to_2 : IsPrincipalIdealRing R \u2192 ValuationRing R\ntfae_2_to_1 : ValuationRing R \u2192 IsPrincipalIdealRing R\ntfae_1_to_4 : IsPrincipalIdealRing R \u2192 IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R\ntfae_4_to_3 : (IsIntegrallyClosed R \u2227 \u2200 (P : Ideal R), P \u2260 \u22a5 \u2192 P.IsPrime \u2192 P = maximalIdeal R) \u2192 IsDedekindDomain R\ntfae_3_to_5 : IsDedekindDomain R \u2192 Submodule.IsPrincipal (maximalIdeal R)\ntfae_6_iff_5 : finrank (ResidueField R) (CotangentSpace R) \u2264 1 \u2194 Submodule.IsPrincipal (maximalIdeal R)\ntfae_5_to_7 : Submodule.IsPrincipal (maximalIdeal R) \u2192 \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nH : \u2200 (I : Ideal R), I \u2260 \u22a5 \u2192 \u2203 n, I = maximalIdeal R ^ n\nI : Ideal R\nx\u271d : DecidableEq (Ideal R) := Classical.decEq (Ideal R)\nhI : \u00acI = \u22a5\n\u22a2 \u22a5 \u2264 I", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Padics/PadicIntegers.lean", "full_name": "PadicInt.pow_p_dvd_int_iff", "start": [584, 1], "end": [586, 44], "traced_tactics": [{"tactic": "rw [\u2190 Nat.cast_pow, \u2190 norm_int_le_pow_iff_dvd, norm_le_pow_iff_mem_span_pow,\n  Ideal.mem_span_singleton, Nat.cast_pow]", "annotated_tactic": ["rw [\u2190 <a>Nat.cast_pow</a>, \u2190 <a>norm_int_le_pow_iff_dvd</a>, <a>norm_le_pow_iff_mem_span_pow</a>,\n    <a>Ideal.mem_span_singleton</a>, <a>Nat.cast_pow</a>]", [{"full_name": "Nat.cast_pow", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [93, 7], "def_end_pos": [93, 15]}, {"full_name": "PadicInt.norm_int_le_pow_iff_dvd", "def_path": "Mathlib/NumberTheory/Padics/PadicIntegers.lean", "def_pos": [369, 9], "def_end_pos": [369, 32]}, {"full_name": "PadicInt.norm_le_pow_iff_mem_span_pow", "def_path": "Mathlib/NumberTheory/Padics/PadicIntegers.lean", "def_pos": [557, 9], "def_end_pos": [557, 37]}, {"full_name": "Ideal.mem_span_singleton", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [503, 9], "def_end_pos": [503, 27]}, {"full_name": "Nat.cast_pow", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [93, 7], "def_end_pos": [93, 15]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\na : \u2124\n\u22a2 \u2191p ^ n \u2223 \u2191a \u2194 \u2191p ^ n \u2223 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Sets/Closeds.lean", "full_name": "TopologicalSpace.Closeds.compl_bijective", "start": [222, 1], "end": [223, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Group/MeasurableEquiv.lean", "full_name": "MeasurableEquiv.toEquiv_mulLeft", "start": [115, 1], "end": [116, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Abelian/NonPreadditive.lean", "full_name": "CategoryTheory.NonPreadditiveAbelian.diag_\u03c3", "start": [285, 1], "end": [285, 87], "traced_tactics": [{"tactic": "rw [cokernel.condition_assoc, zero_comp]", "annotated_tactic": ["rw [<a>cokernel.condition_assoc</a>, <a>zero_comp</a>]", [{"full_name": "CategoryTheory.Limits.cokernel.condition_assoc", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean", "def_pos": [760, 3], "def_end_pos": [760, 25]}, {"full_name": "CategoryTheory.Limits.zero_comp", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean", "def_pos": [72, 9], "def_end_pos": [72, 18]}]], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : NonPreadditiveAbelian C\nX : C\n\u22a2 diag X \u226b \u03c3 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": 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P\ns : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 s\n\u22a2 (affineSpan k (insert p2 \u2191s)).direction = Submodule.span k {p2 -\u1d65 p1} \u2294 s.direction", "state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\ns : AffineSubspace k P\np1 p2 : P\nhp1 : p1 \u2208 s\n\u22a2 (affineSpan k (\u2191s \u222a \u2191(affineSpan k {p2}))).direction = s.direction \u2294 Submodule.span k {p2 -\u1d65 p1}"}, {"tactic": "change (s \u2294 affineSpan k {p2}).direction = _", "annotated_tactic": ["change (s \u2294 <a>affineSpan</a> k {p2}).<a>direction</a> = _", [{"full_name": "affineSpan", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [539, 5], "def_end_pos": [539, 15]}, {"full_name": "AffineSubspace.direction", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [194, 5], "def_end_pos": [194, 14]}]], 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"annotated_tactic": ["use x", []], "state_before": "case inl\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nx dx y dy : \u03b1\nh : dy < dx\nhx : 0 < dx\nh' : x < y\n\u22a2 Nonempty \u2191(Ico x (x + dx) \\ Ico y (y + dy))", "state_after": "case property\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nx dx y dy : \u03b1\nh : dy < dx\nhx : 0 < dx\nh' : x < y\n\u22a2 x \u2208 Ico x (x + dx) \\ Ico y (y + dy)"}, {"tactic": "simp [*, not_le.2 h']", "annotated_tactic": ["simp [*, <a>not_le</a>.2 h']", [{"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [375, 9], "def_end_pos": [375, 15]}]], "state_before": "case property\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nx dx y dy : \u03b1\nh : dy < dx\nhx : 0 < dx\nh' : x < y\n\u22a2 x \u2208 Ico x (x + dx) \\ Ico y (y + dy)", "state_after": "no goals"}, {"tactic": "use max x (x + dy)", "annotated_tactic": ["use <a>max</a> x (x + dy)", [{"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}]], "state_before": "case inr\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nx dx y dy : \u03b1\nh : dy < dx\nhx : 0 < dx\nh' : y \u2264 x\n\u22a2 Nonempty \u2191(Ico x (x + dx) \\ Ico y (y + dy))", "state_after": "case property\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nx dx y dy : \u03b1\nh : dy < dx\nhx : 0 < dx\nh' : y \u2264 x\n\u22a2 max x (x + dy) \u2208 Ico x (x + dx) \\ Ico y (y + dy)"}, {"tactic": "simp [*, le_refl]", "annotated_tactic": ["simp [*, <a>le_refl</a>]", [{"full_name": "le_refl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [45, 9], "def_end_pos": [45, 16]}]], "state_before": "case property\n\u03b1 : Type u_1\ninst\u271d : LinearOrderedAddCommGroup \u03b1\nx dx y dy : \u03b1\nh : dy < dx\nhx : 0 < dx\nh' : y \u2264 x\n\u22a2 max x (x + dy) \u2208 Ico x (x + dx) \\ Ico y (y + dy)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Basic.lean", "full_name": "Ordinal.le_zero", "start": [400, 11], "end": [401, 13], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "full_name": "Besicovitch.card_le_multiplicity", "start": [160, 1], "end": [167, 28], "traced_tactics": [{"tactic": "apply le_csSup", "annotated_tactic": ["apply <a>le_csSup</a>", [{"full_name": "le_csSup", "def_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "def_pos": [459, 9], "def_end_pos": [459, 17]}]], "state_before": "E : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Finset E\nhs : \u2200 c \u2208 s, \u2016c\u2016 \u2264 2\nh's : \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016\n\u22a2 s.card \u2264 multiplicity E", "state_after": "case h\u2081\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Finset E\nhs : \u2200 c \u2208 s, \u2016c\u2016 \u2264 2\nh's : \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016\n\u22a2 BddAbove {N | \u2203 s, s.card = N \u2227 (\u2200 c \u2208 s, \u2016c\u2016 \u2264 2) \u2227 \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016}\n\ncase h\u2082\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Finset E\nhs : \u2200 c \u2208 s, \u2016c\u2016 \u2264 2\nh's : \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016\n\u22a2 s.card \u2208 {N | \u2203 s, s.card = N \u2227 (\u2200 c \u2208 s, \u2016c\u2016 \u2264 2) \u2227 \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016}"}, {"tactic": "refine \u27e85 ^ finrank \u211d E, ?_\u27e9", "annotated_tactic": ["refine \u27e85 ^ <a>finrank</a> \u211d E, ?_\u27e9", [{"full_name": "FiniteDimensional.finrank", "def_path": "Mathlib/LinearAlgebra/Dimension/Finrank.lean", "def_pos": [54, 19], "def_end_pos": [54, 26]}]], "state_before": "case h\u2081\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Finset E\nhs : \u2200 c \u2208 s, \u2016c\u2016 \u2264 2\nh's : \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016\n\u22a2 BddAbove {N | \u2203 s, s.card = N \u2227 (\u2200 c \u2208 s, \u2016c\u2016 \u2264 2) \u2227 \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016}", "state_after": "case h\u2081\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Finset E\nhs : \u2200 c \u2208 s, \u2016c\u2016 \u2264 2\nh's : \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016\n\u22a2 5 ^ finrank \u211d E \u2208 upperBounds {N | \u2203 s, s.card = N \u2227 (\u2200 c \u2208 s, \u2016c\u2016 \u2264 2) \u2227 \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016}"}, {"tactic": "rintro _ \u27e8s, \u27e8rfl, h\u27e9\u27e9", "annotated_tactic": ["rintro _ \u27e8s, \u27e8rfl, h\u27e9\u27e9", []], "state_before": "case h\u2081\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Finset E\nhs : \u2200 c \u2208 s, \u2016c\u2016 \u2264 2\nh's : \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016\n\u22a2 5 ^ finrank \u211d E \u2208 upperBounds {N | \u2203 s, s.card = N \u2227 (\u2200 c \u2208 s, \u2016c\u2016 \u2264 2) \u2227 \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016}", "state_after": "case h\u2081.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns\u271d : Finset E\nhs : \u2200 c \u2208 s\u271d, \u2016c\u2016 \u2264 2\nh's : \u2200 c \u2208 s\u271d, \u2200 d \u2208 s\u271d, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016\ns : Finset E\nh : (\u2200 c \u2208 s, \u2016c\u2016 \u2264 2) \u2227 \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016\n\u22a2 s.card \u2264 5 ^ finrank \u211d E"}, {"tactic": "exact Besicovitch.card_le_of_separated s h.1 h.2", "annotated_tactic": ["exact <a>Besicovitch.card_le_of_separated</a> s h.1 h.2", [{"full_name": "Besicovitch.card_le_of_separated", "def_path": "Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean", "def_pos": [110, 9], "def_end_pos": [110, 29]}]], "state_before": "case h\u2081.intro.intro\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns\u271d : Finset E\nhs : \u2200 c \u2208 s\u271d, \u2016c\u2016 \u2264 2\nh's : \u2200 c \u2208 s\u271d, \u2200 d \u2208 s\u271d, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016\ns : Finset E\nh : (\u2200 c \u2208 s, \u2016c\u2016 \u2264 2) \u2227 \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016\n\u22a2 s.card \u2264 5 ^ finrank \u211d E", "state_after": "no goals"}, {"tactic": "simp only [mem_setOf_eq, Ne]", "annotated_tactic": ["simp only [<a>mem_setOf_eq</a>, <a>Ne</a>]", [{"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [79, 29], "def_end_pos": [79, 41]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}]], "state_before": "case h\u2082\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Finset E\nhs : \u2200 c \u2208 s, \u2016c\u2016 \u2264 2\nh's : \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016\n\u22a2 s.card \u2208 {N | \u2203 s, s.card = N \u2227 (\u2200 c \u2208 s, \u2016c\u2016 \u2264 2) \u2227 \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016}", "state_after": "case h\u2082\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Finset E\nhs : \u2200 c \u2208 s, \u2016c\u2016 \u2264 2\nh's : \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016\n\u22a2 \u2203 s_1, s_1.card = s.card \u2227 (\u2200 c \u2208 s_1, \u2016c\u2016 \u2264 2) \u2227 \u2200 c \u2208 s_1, \u2200 d \u2208 s_1, \u00acc = d \u2192 1 \u2264 \u2016c - d\u2016"}, {"tactic": "exact \u27e8s, rfl, hs, h's\u27e9", "annotated_tactic": ["exact \u27e8s, <a>rfl</a>, hs, h's\u27e9", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case h\u2082\nE : Type u_1\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : FiniteDimensional \u211d E\ns : Finset E\nhs : \u2200 c \u2208 s, \u2016c\u2016 \u2264 2\nh's : \u2200 c \u2208 s, \u2200 d \u2208 s, c \u2260 d \u2192 1 \u2264 \u2016c - d\u2016\n\u22a2 \u2203 s_1, s_1.card = s.card \u2227 (\u2200 c \u2208 s_1, \u2016c\u2016 \u2264 2) \u2227 \u2200 c \u2208 s_1, \u2200 d \u2208 s_1, \u00acc = d \u2192 1 \u2264 \u2016c - d\u2016", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/FieldDivision.lean", "full_name": "Polynomial.derivative_rootMultiplicity_of_root", "start": [143, 1], "end": [148, 94], "traced_tactics": [{"tactic": "by_cases h : p = 0", "annotated_tactic": ["by_cases h : p = 0", []], "state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : p.IsRoot t\n\u22a2 rootMultiplicity t (derivative p) = rootMultiplicity t p - 1", "state_after": "case pos\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : p.IsRoot t\nh : p = 0\n\u22a2 rootMultiplicity t (derivative p) = rootMultiplicity t p - 1\n\ncase neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : p.IsRoot t\nh : \u00acp = 0\n\u22a2 rootMultiplicity t (derivative p) = rootMultiplicity t p - 1"}, {"tactic": "exact derivative_rootMultiplicity_of_root_of_mem_nonZeroDivisors hpt <|\n  mem_nonZeroDivisors_of_ne_zero <| Nat.cast_ne_zero.2 ((rootMultiplicity_pos h).2 hpt).ne'", "annotated_tactic": ["exact <a>derivative_rootMultiplicity_of_root_of_mem_nonZeroDivisors</a> hpt <|\n    <a>mem_nonZeroDivisors_of_ne_zero</a> <| <a>Nat.cast_ne_zero</a>.2 ((<a>rootMultiplicity_pos</a> h).2 hpt).<a>ne'</a>", [{"full_name": "Polynomial.derivative_rootMultiplicity_of_root_of_mem_nonZeroDivisors", "def_path": "Mathlib/Algebra/Polynomial/FieldDivision.lean", "def_pos": [40, 9], "def_end_pos": [40, 67]}, {"full_name": "mem_nonZeroDivisors_of_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean", "def_pos": [213, 9], "def_end_pos": [213, 39]}, {"full_name": "Nat.cast_ne_zero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [83, 9], "def_end_pos": [83, 21]}, {"full_name": "Polynomial.rootMultiplicity_pos", "def_path": "Mathlib/Algebra/Polynomial/Div.lean", "def_pos": [693, 9], "def_end_pos": [693, 29]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 12]}]], "state_before": "case neg\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : p.IsRoot t\nh : \u00acp = 0\n\u22a2 rootMultiplicity t (derivative p) = rootMultiplicity t p - 1", "state_after": "no goals"}, {"tactic": "rw [h, map_zero, rootMultiplicity_zero]", "annotated_tactic": ["rw [h, <a>map_zero</a>, <a>rootMultiplicity_zero</a>]", [{"full_name": "map_zero", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [203, 3], "def_end_pos": [203, 14]}, {"full_name": "Polynomial.rootMultiplicity_zero", "def_path": "Mathlib/Algebra/Polynomial/Div.lean", "def_pos": [552, 9], "def_end_pos": [552, 30]}]], "state_before": "case pos\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : \u2115\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : CharZero R\np : R[X]\nt : R\nhpt : p.IsRoot t\nh : p = 0\n\u22a2 rootMultiplicity t (derivative p) = rootMultiplicity t p - 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Star/Pointwise.lean", "full_name": "Set.compl_star", "start": [94, 1], "end": [94, 58], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/ZetaValues.lean", "full_name": "hasSum_one_div_nat_pow_mul_cos", "start": [251, 1], "end": [285, 9], "traced_tactics": [{"tactic": "have ofReal_two : ((2 : \u211d) : \u2102) = 2 := by norm_cast", "annotated_tactic": ["have ofReal_two : ((2 : \u211d) : \u2102) = 2 := by norm_cast", []], "state_before": "k : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\n\u22a2 HasSum (fun n => 1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * x))\n    ((-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))", "state_after": "k : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 HasSum (fun n => 1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * x))\n    ((-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))"}, {"tactic": "convert ((hasSum_iff _ _).mp (this.div_const 2)).1 with n", "annotated_tactic": ["convert ((<a>hasSum_iff</a> _ _).<a>mp</a> (this.div_const 2)).1 with n", [{"full_name": "Complex.hasSum_iff", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [694, 9], "def_end_pos": [694, 19]}, {"full_name": "Iff.mp", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [118, 3], "def_end_pos": [118, 5]}]], "state_before": "k : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 HasSum (fun n => 1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * x))\n    ((-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))", "state_after": "case h.e'_5.h\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * x) = (1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x) / 2).re\n\ncase h.e'_6\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))) =\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x) / 2).re"}, {"tactic": "convert\n  hasSum_one_div_nat_pow_mul_fourier (by omega : 2 \u2264 2 * k)\n    hx using 3", "annotated_tactic": ["convert\n      <a>hasSum_one_div_nat_pow_mul_fourier</a> (by omega : 2 \u2264 2 * k)\n        hx using 3", [{"full_name": "hasSum_one_div_nat_pow_mul_fourier", "def_path": "Mathlib/NumberTheory/ZetaValues.lean", "def_pos": [234, 9], "def_end_pos": [234, 43]}]], "state_before": "k : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))", "state_after": "case h.e'_5.h.h.e'_6\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nx\u271d : \u2115\n\u22a2 (fourier \u2191x\u271d) \u2191x + (fourier (-\u2191x\u271d)) \u2191x = (fourier \u2191x\u271d) \u2191x + (-1) ^ (2 * k) * (fourier (-\u2191x\u271d)) \u2191x\n\ncase h.e'_6.h.e'_5.h.e'_5\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) = -(2 * \u2191\u03c0 * I) ^ (2 * k)"}, {"tactic": "omega", "annotated_tactic": ["omega", []], "state_before": "k : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 2 \u2264 2 * k", "state_after": "no goals"}, {"tactic": "rw [pow_mul (-1 : \u2102), neg_one_sq, one_pow, one_mul]", "annotated_tactic": ["rw [<a>pow_mul</a> (-1 : \u2102), <a>neg_one_sq</a>, <a>one_pow</a>, <a>one_mul</a>]", [{"full_name": "pow_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [713, 32], "def_end_pos": [713, 39]}, {"full_name": "neg_one_sq", "def_path": "Mathlib/Algebra/Ring/Commute.lean", "def_pos": [200, 7], "def_end_pos": [200, 17]}, {"full_name": "one_pow", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [696, 39], "def_end_pos": [696, 46]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "case h.e'_5.h.h.e'_6\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nx\u271d : \u2115\n\u22a2 (fourier \u2191x\u271d) \u2191x + (fourier (-\u2191x\u271d)) \u2191x = (fourier \u2191x\u271d) \u2191x + (-1) ^ (2 * k) * (fourier (-\u2191x\u271d)) \u2191x", "state_after": "no goals"}, {"tactic": "rw [pow_add, pow_one]", "annotated_tactic": ["rw [<a>pow_add</a>, <a>pow_one</a>]", [{"full_name": "pow_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [703, 7], "def_end_pos": [703, 14]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [663, 7], "def_end_pos": [663, 14]}]], "state_before": "case h.e'_6.h.e'_5.h.e'_5\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) = -(2 * \u2191\u03c0 * I) ^ (2 * k)", "state_after": "case h.e'_6.h.e'_5.h.e'_5\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 (-1) ^ k * -1 * (2 * \u2191\u03c0) ^ (2 * k) = -(2 * \u2191\u03c0 * I) ^ (2 * k)"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case h.e'_6.h.e'_5.h.e'_5\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\n\u22a2 (-1) ^ k * -1 * (2 * \u2191\u03c0) ^ (2 * k) = -((2 * \u2191\u03c0) ^ (2 * k) * (-1) ^ k)", "state_after": "no goals"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "k : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\n\u22a2 \u21912 = 2", "state_after": "no goals"}, {"tactic": "convert (ofReal_re _).symm", "annotated_tactic": ["convert (<a>ofReal_re</a> _).<a>symm</a>", [{"full_name": "Complex.ofReal_re", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [96, 9], "def_end_pos": [96, 18]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case h.e'_5.h\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * x) = (1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x) / 2).re", "state_after": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x) / 2 = \u2191(1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * x))"}, {"tactic": "rw [ofReal_mul]", "annotated_tactic": ["rw [<a>ofReal_mul</a>]", [{"full_name": "Complex.ofReal_mul", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 19]}]], "state_before": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x) / 2 = \u2191(1 / \u2191n ^ (2 * k) * Real.cos (2 * \u03c0 * \u2191n * x))", "state_after": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x) / 2 = \u2191(1 / \u2191n ^ (2 * k)) * \u2191(Real.cos (2 * \u03c0 * \u2191n * x))"}, {"tactic": "rw [\u2190 mul_div]", "annotated_tactic": ["rw [\u2190 <a>mul_div</a>]", [{"full_name": "mul_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [474, 9], "def_end_pos": [474, 16]}]], "state_before": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x) / 2 = \u2191(1 / \u2191n ^ (2 * k)) * \u2191(Real.cos (2 * \u03c0 * \u2191n * x))", "state_after": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) * (((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x) / 2) = \u2191(1 / \u2191n ^ (2 * k)) * \u2191(Real.cos (2 * \u03c0 * \u2191n * x))"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) * (((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x) / 2) = \u2191(1 / \u2191n ^ (2 * k)) * \u2191(Real.cos (2 * \u03c0 * \u2191n * x))", "state_after": "case h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) = \u2191(1 / \u2191n ^ (2 * k))\n\ncase h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x) / 2 = \u2191(Real.cos (2 * \u03c0 * \u2191n * x))"}, {"tactic": "rw [ofReal_div, ofReal_one, ofReal_pow]", "annotated_tactic": ["rw [<a>ofReal_div</a>, <a>ofReal_one</a>, <a>ofReal_pow</a>]", [{"full_name": "Complex.ofReal_div", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [874, 9], "def_end_pos": [874, 19]}, {"full_name": "Complex.ofReal_one", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [180, 9], "def_end_pos": [180, 19]}, {"full_name": "Complex.ofReal_pow", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [794, 9], "def_end_pos": [794, 19]}]], "state_before": "case h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) = \u2191(1 / \u2191n ^ (2 * k))", "state_after": "case h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) = 1 / \u2191\u2191n ^ (2 * k)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 1 / \u2191n ^ (2 * k) = 1 / \u2191\u2191n ^ (2 * k)", "state_after": "no goals"}, {"tactic": "rw [ofReal_cos, ofReal_mul, fourier_coe_apply, fourier_coe_apply, cos, ofReal_one, div_one,\n  div_one, ofReal_mul, ofReal_mul, ofReal_two, Int.cast_neg, Int.cast_natCast,\n  ofReal_natCast]", "annotated_tactic": ["rw [<a>ofReal_cos</a>, <a>ofReal_mul</a>, <a>fourier_coe_apply</a>, <a>fourier_coe_apply</a>, <a>cos</a>, <a>ofReal_one</a>, <a>div_one</a>,\n        <a>div_one</a>, <a>ofReal_mul</a>, <a>ofReal_mul</a>, ofReal_two, <a>Int.cast_neg</a>, <a>Int.cast_natCast</a>,\n        <a>ofReal_natCast</a>]", [{"full_name": "Complex.ofReal_cos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [652, 9], "def_end_pos": [652, 19]}, {"full_name": "Complex.ofReal_mul", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 19]}, {"full_name": "fourier_coe_apply", "def_path": "Mathlib/Analysis/Fourier/AddCircle.lean", "def_pos": [117, 9], "def_end_pos": [117, 26]}, {"full_name": "fourier_coe_apply", "def_path": "Mathlib/Analysis/Fourier/AddCircle.lean", "def_pos": [117, 9], "def_end_pos": [117, 26]}, {"full_name": "Complex.cos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [67, 5], "def_end_pos": [67, 8]}, {"full_name": "Complex.ofReal_one", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [180, 9], "def_end_pos": [180, 19]}, {"full_name": "div_one", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [490, 9], "def_end_pos": [490, 16]}, {"full_name": "div_one", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [490, 9], "def_end_pos": [490, 16]}, {"full_name": "Complex.ofReal_mul", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 19]}, {"full_name": "Complex.ofReal_mul", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 19]}, {"full_name": "Int.cast_neg", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [85, 9], "def_end_pos": [85, 17]}, {"full_name": "Int.cast_natCast", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [66, 9], "def_end_pos": [66, 21]}, {"full_name": "Complex.ofReal_natCast", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [514, 26], "def_end_pos": [514, 40]}]], "state_before": "case h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x) / 2 = \u2191(Real.cos (2 * \u03c0 * \u2191n * x))", "state_after": "case h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 (cexp (2 * \u2191\u03c0 * I * \u2191n * \u2191x) + cexp (2 * \u2191\u03c0 * I * -\u2191n * \u2191x)) / 2 =\n    (cexp (2 * \u2191\u03c0 * \u2191n * \u2191x * I) + cexp (-(2 * \u2191\u03c0 * \u2191n * \u2191x) * I)) / 2"}, {"tactic": "congr 3", "annotated_tactic": ["congr 3", []], "state_before": "case h.e'_3.h.e'_1.e_a\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 (cexp (2 * \u2191\u03c0 * I * \u2191n * \u2191x) + cexp (2 * \u2191\u03c0 * I * -\u2191n * \u2191x)) / 2 =\n    (cexp (2 * \u2191\u03c0 * \u2191n * \u2191x * I) + cexp (-(2 * \u2191\u03c0 * \u2191n * \u2191x) * I)) / 2", "state_after": "case h.e'_3.h.e'_1.e_a.e_a.e_a.e_z\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 2 * \u2191\u03c0 * I * \u2191n * \u2191x = 2 * \u2191\u03c0 * \u2191n * \u2191x * I\n\ncase h.e'_3.h.e'_1.e_a.e_a.e_a.e_z\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 2 * \u2191\u03c0 * I * -\u2191n * \u2191x = -(2 * \u2191\u03c0 * \u2191n * \u2191x) * I"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case h.e'_3.h.e'_1.e_a.e_a.e_a.e_z\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 2 * \u2191\u03c0 * I * \u2191n * \u2191x = 2 * \u2191\u03c0 * \u2191n * \u2191x * I", "state_after": "no goals"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case h.e'_3.h.e'_1.e_a.e_a.e_a.e_z\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\nn : \u2115\n\u22a2 2 * \u2191\u03c0 * I * -\u2191n * \u2191x = -(2 * \u2191\u03c0 * \u2191n * \u2191x) * I", "state_after": "no goals"}, {"tactic": "convert (ofReal_re _).symm", "annotated_tactic": ["convert (<a>ofReal_re</a> _).<a>symm</a>", [{"full_name": "Complex.ofReal_re", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [96, 9], "def_end_pos": [96, 18]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case h.e'_6\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))) =\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x) / 2).re", "state_after": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x) / 2 =\n    \u2191((-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n        Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))"}, {"tactic": "rw [ofReal_mul, ofReal_div, ofReal_div, ofReal_mul, ofReal_pow, ofReal_pow, ofReal_neg,\n  ofReal_natCast, ofReal_mul, ofReal_two, ofReal_one]", "annotated_tactic": ["rw [<a>ofReal_mul</a>, <a>ofReal_div</a>, <a>ofReal_div</a>, <a>ofReal_mul</a>, <a>ofReal_pow</a>, <a>ofReal_pow</a>, <a>ofReal_neg</a>,\n      <a>ofReal_natCast</a>, <a>ofReal_mul</a>, ofReal_two, <a>ofReal_one</a>]", [{"full_name": "Complex.ofReal_mul", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 19]}, {"full_name": "Complex.ofReal_div", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [874, 9], "def_end_pos": [874, 19]}, {"full_name": "Complex.ofReal_div", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [874, 9], "def_end_pos": [874, 19]}, {"full_name": "Complex.ofReal_mul", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 19]}, {"full_name": "Complex.ofReal_pow", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [794, 9], "def_end_pos": [794, 19]}, {"full_name": "Complex.ofReal_pow", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [794, 9], "def_end_pos": [794, 19]}, {"full_name": "Complex.ofReal_neg", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [236, 9], "def_end_pos": [236, 19]}, {"full_name": "Complex.ofReal_natCast", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [514, 26], "def_end_pos": [514, 40]}, {"full_name": "Complex.ofReal_mul", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [257, 9], "def_end_pos": [257, 19]}, {"full_name": "Complex.ofReal_one", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [180, 9], "def_end_pos": [180, 19]}]], "state_before": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x) / 2 =\n    \u2191((-1) ^ (k + 1) * (2 * \u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n        Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))", "state_after": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x) / 2 =\n    (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))"}, {"tactic": "rw [bernoulliFun]", "annotated_tactic": ["rw [<a>bernoulliFun</a>]", [{"full_name": "bernoulliFun", "def_path": "Mathlib/NumberTheory/ZetaValues.lean", "def_pos": [45, 5], "def_end_pos": [45, 17]}]], "state_before": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x) / 2 =\n    (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))", "state_after": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! *\n        \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k)))) /\n      2 =\n    (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case h.e'_3.h.e'_1\nk : \u2115\nhk : k \u2260 0\nx : \u211d\nhx : x \u2208 Icc 0 1\nthis :\n  HasSum (fun n => 1 / \u2191n ^ (2 * k) * ((fourier \u2191n) \u2191x + (fourier (-\u2191n)) \u2191x))\n    ((-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! * \u2191(bernoulliFun (2 * k) x))\nofReal_two : \u21912 = 2\n\u22a2 (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / \u2191(2 * k)! *\n        \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k)))) /\n      2 =\n    (-1) ^ (k + 1) * (2 * \u2191\u03c0) ^ (2 * k) / 2 / \u2191(2 * k)! *\n      \u2191(Polynomial.eval x (Polynomial.map (algebraMap \u211a \u211d) (Polynomial.bernoulli (2 * k))))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Bounds/Basic.lean", "full_name": "Antitone.map_bddBelow", "start": [1350, 1], "end": [1351, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/ConditionalProbability.lean", "full_name": "ProbabilityTheory.cond_empty", "start": [130, 1], "end": [130, 49], "traced_tactics": [{"tactic": "simp [cond]", "annotated_tactic": ["simp [<a>cond</a>]", [{"full_name": "ProbabilityTheory.cond", "def_path": "Mathlib/Probability/ConditionalProbability.lean", "def_pos": [73, 5], "def_end_pos": [73, 9]}]], "state_before": "\u03a9 : Type u_1\n\u03a9' : Type u_2\n\u03b1 : Type u_3\nm : MeasurableSpace \u03a9\nm' : MeasurableSpace \u03a9'\n\u03bc : Measure \u03a9\ns t : Set \u03a9\n\u22a2 \u03bc[|\u2205] = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Add.lean", "full_name": "HasFDerivAtFilter.const_add", "start": [264, 1], "end": [266, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/ByteArray.lean", "full_name": "ByteArray.get_append_left", "start": [79, 1], "end": [82, 66], "traced_tactics": [{"tactic": "simp [getElem_eq_data_getElem]", "annotated_tactic": ["simp [<a>getElem_eq_data_getElem</a>]", [{"full_name": "ByteArray.getElem_eq_data_getElem", "def_path": ".lake/packages/batteries/Batteries/Data/ByteArray.lean", "def_pos": [13, 9], "def_end_pos": [13, 32]}]], "state_before": "i : Nat\na b : ByteArray\nhlt : i < a.size\nh : optParam (i < (a ++ b).size) \u22ef\n\u22a2 (a ++ b)[i] = a[i]", "state_after": "i : Nat\na b : ByteArray\nhlt : i < a.size\nh : optParam (i < (a ++ b).size) \u22ef\n\u22a2 (a.data ++ b.data)[i] = a.data[i]"}, {"tactic": "exact Array.get_append_left hlt", "annotated_tactic": ["exact <a>Array.get_append_left</a> hlt", [{"full_name": "Array.get_append_left", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Array/Lemmas.lean", "def_pos": [776, 9], "def_end_pos": [776, 24]}]], "state_before": "i : Nat\na b : ByteArray\nhlt : i < a.size\nh : optParam (i < (a ++ b).size) \u22ef\n\u22a2 (a.data ++ b.data)[i] = a.data[i]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Normed/Order/UpperLower.lean", "full_name": "IsClopen.upperClosure_pi", "start": [246, 1], "end": [247, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Classes/SatisfiesM.lean", "full_name": "SatisfiesM_Option_eq", "start": [153, 9], "end": [155, 85], "traced_tactics": [{"tactic": "revert x", "annotated_tactic": ["revert x", []], "state_before": "\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx : Option \u03b1\u271d\n\u22a2 SatisfiesM p x \u2192 \u2200 (a : \u03b1\u271d), x = some a \u2192 p a", "state_after": "\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\n\u22a2 \u2200 {x : Option \u03b1\u271d}, SatisfiesM p x \u2192 \u2200 (a : \u03b1\u271d), x = some a \u2192 p a"}, {"tactic": "intro | some _, \u27e8some \u27e8_, h\u27e9, rfl\u27e9, _, rfl => exact h", "annotated_tactic": ["intro | <a>some</a> _, \u27e8<a>some</a> \u27e8_, h\u27e9, <a>rfl</a>\u27e9, _, <a>rfl</a> => exact h", [{"full_name": "Option.some", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2244, 5], "def_end_pos": [2244, 9]}, {"full_name": "Option.some", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2244, 5], "def_end_pos": [2244, 9]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\n\u22a2 \u2200 {x : Option \u03b1\u271d}, SatisfiesM p x \u2192 \u2200 (a : \u03b1\u271d), x = some a \u2192 p a", "state_after": "no goals"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "\u03b1\u271d : Type u_1\np : \u03b1\u271d \u2192 Prop\nx\u271d\u00b3 : Option \u03b1\u271d\nx\u271d\u00b2 : SatisfiesM p x\u271d\u00b3\nx\u271d\u00b9 : \u03b1\u271d\nx\u271d : x\u271d\u00b3 = some x\u271d\u00b9\nval\u271d : \u03b1\u271d\nh : p val\u271d\n\u22a2 p \u27e8val\u271d, h\u27e9.val", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "full_name": "MeasureTheory.AEStronglyMeasurable.comp_quasiMeasurePreserving", "start": [1637, 1], "end": [1640, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/Basic.lean", "full_name": "Set.mem_Ici", "start": [146, 1], "end": [147, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Determinant.lean", "full_name": "Basis.det_apply", "start": [526, 1], "end": [527, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/DirectSum/Module.lean", "full_name": "DirectSum.isInternal_ne_bot_iff", "start": [449, 1], "end": [452, 78], "traced_tactics": [{"tactic": "simp only [isInternal_submodule_iff_independent_and_iSup_eq_top]", "annotated_tactic": ["simp only [<a>isInternal_submodule_iff_independent_and_iSup_eq_top</a>]", [{"full_name": "DirectSum.isInternal_submodule_iff_independent_and_iSup_eq_top", "def_path": "Mathlib/Algebra/DirectSum/Module.lean", "def_pos": [432, 9], "def_end_pos": [432, 61]}]], "state_before": "R : Type u\ninst\u271d\u00b2 : Ring R\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\nM : Type u_1\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nA : \u03b9 \u2192 Submodule R M\n\u22a2 (IsInternal fun i => A \u2191i) \u2194 IsInternal A", "state_after": "R : Type u\ninst\u271d\u00b2 : Ring R\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\nM : Type u_1\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nA : \u03b9 \u2192 Submodule R M\n\u22a2 (CompleteLattice.Independent fun i => A \u2191i) \u2227 \u2a06 i, A \u2191i = \u22a4 \u2194 CompleteLattice.Independent A \u2227 iSup A = \u22a4"}, {"tactic": "exact Iff.and CompleteLattice.independent_ne_bot_iff_independent <| by simp", "annotated_tactic": ["exact <a>Iff.and</a> <a>CompleteLattice.independent_ne_bot_iff_independent</a> <| by simp", [{"full_name": "Iff.and", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [331, 7], "def_end_pos": [331, 14]}, {"full_name": "CompleteLattice.independent_ne_bot_iff_independent", "def_path": "Mathlib/Order/SupIndep.lean", "def_pos": [415, 9], "def_end_pos": [415, 43]}]], "state_before": "R : Type u\ninst\u271d\u00b2 : Ring R\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\nM : Type u_1\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nA : \u03b9 \u2192 Submodule R M\n\u22a2 (CompleteLattice.Independent fun i => A \u2191i) \u2227 \u2a06 i, A \u2191i = \u22a4 \u2194 CompleteLattice.Independent A \u2227 iSup A = \u22a4", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "R : Type u\ninst\u271d\u00b2 : Ring R\n\u03b9 : Type v\ndec_\u03b9 : DecidableEq \u03b9\nM : Type u_1\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nA : \u03b9 \u2192 Submodule R M\n\u22a2 \u2a06 i, A \u2191i = \u22a4 \u2194 iSup A = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/DedekindDomain/Ideal.lean", "full_name": "FractionalIdeal.mul_right_strictMono", "start": [562, 1], "end": [564, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Monoid/Defs.lean", "full_name": "one_lt_mul_self_iff", "start": [172, 1], "end": [173, 86], "traced_tactics": [{"tactic": "push_neg at h \u22a2", "annotated_tactic": ["push_neg at h \u22a2", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrderedCommMonoid \u03b1\na : \u03b1\nh : \u00ac1 < a\n\u22a2 \u00ac1 < a * a", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrderedCommMonoid \u03b1\na : \u03b1\nh : a \u2264 1\n\u22a2 a * a \u2264 1"}, {"tactic": "exact mul_le_one' h h", "annotated_tactic": ["exact <a>mul_le_one'</a> h h", [{"full_name": "mul_le_one'", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [1045, 7], "def_end_pos": [1045, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : LinearOrderedCommMonoid \u03b1\na : \u03b1\nh : a \u2264 1\n\u22a2 a * a \u2264 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/LinearPMap.lean", "full_name": "LinearPMap.domain_mkSpanSingleton", "start": [146, 1], "end": [148, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.filter_congr", "start": [1992, 1], "end": [1994, 87], "traced_tactics": [{"tactic": "simpa using h", "annotated_tactic": ["simpa using h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\np\u271d : \u03b1 \u2192 Prop\ninst\u271d\u00b2 : DecidablePred p\u271d\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\ns : Multiset \u03b1\n_l : List \u03b1\nh : \u2200 x \u2208 Quot.mk Setoid.r _l, p x \u2194 q x\n\u22a2 \u2200 x \u2208 _l, decide (p x) = decide (q x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.tendsto_iff_forall_eventually_mem", "start": [3005, 1], "end": [3007, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "full_name": "MeasureTheory.condexpL1CLM_indicatorConst", "start": [399, 1], "end": [401, 86], "traced_tactics": [{"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": [688, 9], "def_end_pos": [688, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : RCLike \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 (\u03bc.trim hm)\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u03bc s \u2260 \u22a4\nx : F'\n\u22a2 (condexpL1CLM F' hm \u03bc) \u2191(simpleFunc.indicatorConst 1 hs h\u03bcs x) = (condexpInd F' hm \u03bc s) x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\nG' : Type u_6\n\ud835\udd5c : Type u_7\np : \u211d\u22650\u221e\ninst\u271d\u00b9\u00b9 : RCLike \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 (\u03bc.trim hm)\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u03bc s \u2260 \u22a4\nx : F'\n\u22a2 (condexpL1CLM F' hm \u03bc) (indicatorConstLp 1 hs h\u03bcs x) = (condexpInd F' hm \u03bc s) x"}, {"tactic": "exact condexpL1CLM_indicatorConstLp hs h\u03bcs x", "annotated_tactic": ["exact <a>condexpL1CLM_indicatorConstLp</a> hs h\u03bcs x", [{"full_name": "MeasureTheory.condexpL1CLM_indicatorConstLp", "def_path": "Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean", "def_pos": [394, 9], "def_end_pos": [394, 38]}]], "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 : RCLike \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 (\u03bc.trim hm)\nf g : \u03b1 \u2192 F'\ns : Set \u03b1\nhs : MeasurableSet s\nh\u03bcs : \u03bc s \u2260 \u22a4\nx : F'\n\u22a2 (condexpL1CLM F' hm \u03bc) (indicatorConstLp 1 hs h\u03bcs x) = (condexpInd F' hm \u03bc s) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "full_name": "MeasureTheory.Measure.toOuterMeasure_apply", "start": [156, 1], "end": [158, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean", "full_name": "LinearMap.toMatrix\u2082_mul_basis_toMatrix", "start": [456, 1], "end": [461, 62], "traced_tactics": [{"tactic": "simp_rw [\u2190 LinearMap.toMatrix_id_eq_basis_toMatrix]", "annotated_tactic": ["simp_rw [\u2190 <a>LinearMap.toMatrix_id_eq_basis_toMatrix</a>]", [{"full_name": "LinearMap.toMatrix_id_eq_basis_toMatrix", "def_path": "Mathlib/LinearAlgebra/Matrix/Basis.lean", "def_pos": [188, 9], "def_end_pos": [188, 48]}]], "state_before": "R : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nc\u2081 : Basis n' R M\u2081\nc\u2082 : Basis m' R M\u2082\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\n\u22a2 (b\u2081.toMatrix \u21d1c\u2081)\u1d40 * (toMatrix\u2082 b\u2081 b\u2082) B * b\u2082.toMatrix \u21d1c\u2082 = (toMatrix\u2082 c\u2081 c\u2082) B", "state_after": "R : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nc\u2081 : Basis n' R M\u2081\nc\u2082 : Basis m' R M\u2082\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\n\u22a2 ((toMatrix c\u2081 b\u2081) id)\u1d40 * (toMatrix\u2082 b\u2081 b\u2082) B * (toMatrix c\u2082 b\u2082) id = (toMatrix\u2082 c\u2081 c\u2082) B"}, {"tactic": "rw [\u2190 LinearMap.toMatrix\u2082_compl\u2081\u2082, LinearMap.compl\u2081\u2082_id_id]", "annotated_tactic": ["rw [\u2190 <a>LinearMap.toMatrix\u2082_compl\u2081\u2082</a>, <a>LinearMap.compl\u2081\u2082_id_id</a>]", [{"full_name": "LinearMap.toMatrix\u2082_compl\u2081\u2082", "def_path": "Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean", "def_pos": [420, 9], "def_end_pos": [420, 36]}, {"full_name": "LinearMap.compl\u2081\u2082_id_id", "def_path": "Mathlib/LinearAlgebra/BilinearMap.lean", "def_pos": [363, 9], "def_end_pos": [363, 22]}]], "state_before": "R : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nM : Type u_4\nM\u2081 : Type u_5\nM\u2082 : Type u_6\nM\u2081' : Type u_7\nM\u2082' : Type u_8\nn : Type u_9\nm : Type u_10\nn' : Type u_11\nm' : Type u_12\n\u03b9 : Type u_13\ninst\u271d\u00b9\u2076 : CommSemiring R\ninst\u271d\u00b9\u2075 : AddCommMonoid M\u2081\ninst\u271d\u00b9\u2074 : Module R M\u2081\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b2 : Module R M\u2082\ninst\u271d\u00b9\u00b9 : DecidableEq n\ninst\u271d\u00b9\u2070 : Fintype n\ninst\u271d\u2079 : DecidableEq m\ninst\u271d\u2078 : Fintype m\nb\u2081 : Basis n R M\u2081\nb\u2082 : Basis m R M\u2082\ninst\u271d\u2077 : AddCommMonoid M\u2081'\ninst\u271d\u2076 : Module R M\u2081'\ninst\u271d\u2075 : AddCommMonoid M\u2082'\ninst\u271d\u2074 : Module R M\u2082'\nb\u2081' : Basis n' R M\u2081'\nb\u2082' : Basis m' R M\u2082'\ninst\u271d\u00b3 : Fintype n'\ninst\u271d\u00b2 : Fintype m'\ninst\u271d\u00b9 : DecidableEq n'\ninst\u271d : DecidableEq m'\nc\u2081 : Basis n' R M\u2081\nc\u2082 : Basis m' R M\u2082\nB : M\u2081 \u2192\u2097[R] M\u2082 \u2192\u2097[R] R\n\u22a2 ((toMatrix c\u2081 b\u2081) id)\u1d40 * (toMatrix\u2082 b\u2081 b\u2082) B * (toMatrix c\u2082 b\u2082) id = (toMatrix\u2082 c\u2081 c\u2082) B", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.val_lt_iff", "start": [416, 1], "end": [417, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Infix.lean", "full_name": "List.prefix_concat_iff", "start": [73, 1], "end": [76, 58], "traced_tactics": [{"tactic": "simpa only [\u2190 reverse_concat', reverse_inj, reverse_suffix] using\n  suffix_cons_iff (l\u2081 := l\u2081.reverse) (l\u2082 := l\u2082.reverse)", "annotated_tactic": ["simpa only [\u2190 <a>reverse_concat'</a>, <a>reverse_inj</a>, <a>reverse_suffix</a>] using\n    <a>suffix_cons_iff</a> (l\u2081 := l\u2081.reverse) (l\u2082 := l\u2082.reverse)", [{"full_name": "List.reverse_concat'", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [521, 9], "def_end_pos": [521, 24]}, {"full_name": "List.reverse_inj", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [553, 9], "def_end_pos": [553, 20]}, {"full_name": "List.reverse_suffix", "def_path": ".lake/packages/batteries/Batteries/Data/List/Lemmas.lean", "def_pos": [1113, 17], "def_end_pos": [1113, 31]}, {"full_name": "List.suffix_cons_iff", "def_path": ".lake/packages/batteries/Batteries/Data/List/Lemmas.lean", "def_pos": [1175, 9], "def_end_pos": [1175, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl l\u2081\u271d l\u2082\u271d l\u2083 : List \u03b1\na\u271d b : \u03b1\nm n : \u2115\nl\u2081 l\u2082 : List \u03b1\na : \u03b1\n\u22a2 l\u2081 <+: l\u2082 ++ [a] \u2194 l\u2081 = l\u2082 ++ [a] \u2228 l\u2081 <+: l\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "full_name": "LinearIsometryEquiv.toLinearEquiv_injective", "start": [549, 1], "end": [550, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/Archimedean.lean", "full_name": "Real.isCauSeq_iff_lift", "start": [33, 1], "end": [38, 90], "traced_tactics": [{"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "f : \u2115 \u2192 \u211a\nH : IsCauSeq abs f\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b4 : \u211a\n\u03b40 : 0 < \u03b4\n\u03b4\u03b5 : \u2191\u03b4 < \u03b5\ni : \u2115\nhi : \u2200 j \u2265 i, |f j - f i| < \u03b4\nj : \u2115\nij : j \u2265 i\n\u22a2 |(fun i => \u2191(f i)) j - (fun i => \u2191(f i)) i| < \u03b5", "state_after": "f : \u2115 \u2192 \u211a\nH : IsCauSeq abs f\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b4 : \u211a\n\u03b40 : 0 < \u03b4\n\u03b4\u03b5 : \u2191\u03b4 < \u03b5\ni : \u2115\nhi : \u2200 j \u2265 i, |f j - f i| < \u03b4\nj : \u2115\nij : j \u2265 i\n\u22a2 |\u2191(f j) - \u2191(f i)| < \u03b5"}, {"tactic": "exact lt_trans (mod_cast hi _ ij) \u03b4\u03b5", "annotated_tactic": ["exact <a>lt_trans</a> (mod_cast hi _ ij) \u03b4\u03b5", [{"full_name": "lt_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [90, 9], "def_end_pos": [90, 17]}]], "state_before": "f : \u2115 \u2192 \u211a\nH : IsCauSeq abs f\n\u03b5 : \u211d\n\u03b50 : \u03b5 > 0\n\u03b4 : \u211a\n\u03b40 : 0 < \u03b4\n\u03b4\u03b5 : \u2191\u03b4 < \u03b5\ni : \u2115\nhi : \u2200 j \u2265 i, |f j - f i| < \u03b4\nj : \u2115\nij : j \u2265 i\n\u22a2 |\u2191(f j) - \u2191(f i)| < \u03b5", "state_after": "no goals"}, {"tactic": "dsimp at hi", "annotated_tactic": ["dsimp at hi", []], "state_before": "f : \u2115 \u2192 \u211a\nH : IsCauSeq abs fun i => \u2191(f i)\n\u03b5 : \u211a\n\u03b50 : \u03b5 > 0\ni : \u2115\nhi : \u2200 j \u2265 i, |(fun i => \u2191(f i)) j - (fun i => \u2191(f i)) i| < \u2191\u03b5\nj : \u2115\nij : j \u2265 i\n\u22a2 |f j - f i| < \u03b5", "state_after": "f : \u2115 \u2192 \u211a\nH : IsCauSeq abs fun i => \u2191(f i)\n\u03b5 : \u211a\n\u03b50 : \u03b5 > 0\ni : \u2115\nhi : \u2200 j \u2265 i, |\u2191(f j) - \u2191(f i)| < \u2191\u03b5\nj : \u2115\nij : j \u2265 i\n\u22a2 |f j - f i| < \u03b5"}, {"tactic": "exact mod_cast hi _ ij", "annotated_tactic": ["exact mod_cast hi _ ij", []], "state_before": "f : \u2115 \u2192 \u211a\nH : IsCauSeq abs fun i => \u2191(f i)\n\u03b5 : \u211a\n\u03b50 : \u03b5 > 0\ni : \u2115\nhi : \u2200 j \u2265 i, |\u2191(f j) - \u2191(f i)| < \u2191\u03b5\nj : \u2115\nij : j \u2265 i\n\u22a2 |f j - f i| < \u03b5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean", "full_name": "Orientation.eq_iff_norm_eq_of_oangle_eq_zero", "start": [501, 1], "end": [503, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/NNRat/Defs.lean", "full_name": "Rat.coe_nnabs", "start": [347, 1], "end": [347, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finsupp/Defs.lean", "full_name": "Finsupp.support_mapRange", "start": [821, 1], "end": [823, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Instances/Irrational.lean", "full_name": "Irrational.eventually_forall_le_dist_cast_div_of_denom_le", "start": [92, 1], "end": [94, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Subobject/Lattice.lean", "full_name": "CategoryTheory.Subobject.symm_apply_mem_iff_mem_image", "start": [694, 1], "end": [698, 19], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : WellPowered C\ninst\u271d\u00b9 : HasCoproducts C\ninst\u271d : HasImages C\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ne : \u03b1 \u2243 \u03b2\ns : Set \u03b1\nx : \u03b2\nh : e.symm x \u2208 s\n\u22a2 e (e.symm x) = x", "state_after": "no goals"}, {"tactic": "rintro \u27e8a, m, rfl\u27e9", "annotated_tactic": ["rintro \u27e8a, m, rfl\u27e9", []], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : WellPowered C\ninst\u271d\u00b9 : HasCoproducts C\ninst\u271d : HasImages C\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ne : \u03b1 \u2243 \u03b2\ns : Set \u03b1\nx : \u03b2\n\u22a2 x \u2208 \u21d1e '' s \u2192 e.symm x \u2208 s", "state_after": "case intro.intro\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : WellPowered C\ninst\u271d\u00b9 : HasCoproducts C\ninst\u271d : HasImages C\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ne : \u03b1 \u2243 \u03b2\ns : Set \u03b1\na : \u03b1\nm : a \u2208 s\n\u22a2 e.symm (e a) \u2208 s"}, {"tactic": "simpa using m", "annotated_tactic": ["simpa using m", []], "state_before": "case intro.intro\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nX Y Z : C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\ninst\u271d\u00b2 : WellPowered C\ninst\u271d\u00b9 : HasCoproducts C\ninst\u271d : HasImages C\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ne : \u03b1 \u2243 \u03b2\ns : Set \u03b1\na : \u03b1\nm : a \u2208 s\n\u22a2 e.symm (e a) \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Sum/Order.lean", "full_name": "OrderIso.sumAssoc_symm_apply_inr_inl", "start": [590, 1], "end": [591, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "full_name": "Even.mod_even_iff", "start": [256, 1], "end": [259, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Basic.lean", "full_name": "forall\u2083_true_iff", "start": [725, 1], "end": [726, 54], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Sort u_1\n\u03b2\u271d : Sort u_2\np q : \u03b1 \u2192 Prop\n\u03b2 : \u03b1 \u2192 Sort u_3\n\u03b3 : (a : \u03b1) \u2192 \u03b2 a \u2192 Sort u_4\n\u22a2 (\u2200 (a : \u03b1) (b : \u03b2 a), \u03b3 a b \u2192 True) \u2194 True", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Quaternion.lean", "full_name": "QuaternionAlgebra.imK_star", "start": [690, 1], "end": [691, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean", "full_name": "Summable.star", "start": [310, 1], "end": [311, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Complex/Exponential.lean", "full_name": "Complex.sin_ofReal_im", "start": [635, 1], "end": [635, 92], "traced_tactics": [{"tactic": "rw [\u2190 ofReal_sin_ofReal_re, ofReal_im]", "annotated_tactic": ["rw [\u2190 <a>ofReal_sin_ofReal_re</a>, <a>ofReal_im</a>]", [{"full_name": "Complex.ofReal_sin_ofReal_re", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [625, 9], "def_end_pos": [625, 29]}, {"full_name": "Complex.ofReal_im", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [101, 9], "def_end_pos": [101, 18]}]], "state_before": "x\u271d y : \u2102\nx : \u211d\n\u22a2 (sin \u2191x).im = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/DFA.lean", "full_name": "DFA.pumping_lemma", "start": [150, 1], "end": [165, 67], "traced_tactics": [{"tactic": "obtain \u27e8_, a, b, c, hx, hlen, hnil, rfl, hb, hc\u27e9 := M.evalFrom_split (s := M.start) hlen rfl", "annotated_tactic": ["obtain \u27e8_, a, b, c, hx, hlen, hnil, rfl, hb, hc\u27e9 := M.evalFrom_split (s := M.start) hlen <a>rfl</a>", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx : x \u2208 M.accepts\nhlen : Fintype.card \u03c3 \u2264 x.length\n\u22a2 \u2203 a b c, x = a ++ b ++ c \u2227 a.length + b.length \u2264 Fintype.card \u03c3 \u2227 b \u2260 [] \u2227 {a} * {b}\u2217 * {c} \u2264 M.accepts", "state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 M.accepts\nhlen\u271d : Fintype.card \u03c3 \u2264 x.length\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : a.length + b.length \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : M.evalFrom (M.evalFrom M.start a) b = M.evalFrom M.start a\nhc : M.evalFrom (M.evalFrom M.start a) c = M.evalFrom M.start x\n\u22a2 \u2203 a b c, x = a ++ b ++ c \u2227 a.length + b.length \u2264 Fintype.card \u03c3 \u2227 b \u2260 [] \u2227 {a} * {b}\u2217 * {c} \u2264 M.accepts"}, {"tactic": "use a, b, c, hx, hlen, hnil", "annotated_tactic": ["use a, b, c, hx, hlen, hnil", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 M.accepts\nhlen\u271d : Fintype.card \u03c3 \u2264 x.length\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : a.length + b.length \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : M.evalFrom (M.evalFrom M.start a) b = M.evalFrom M.start a\nhc : M.evalFrom (M.evalFrom M.start a) c = M.evalFrom M.start x\n\u22a2 \u2203 a b c, x = a ++ b ++ c \u2227 a.length + b.length \u2264 Fintype.card \u03c3 \u2227 b \u2260 [] \u2227 {a} * {b}\u2217 * {c} \u2264 M.accepts", "state_after": "case right\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 M.accepts\nhlen\u271d : Fintype.card \u03c3 \u2264 x.length\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : a.length + b.length \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : M.evalFrom (M.evalFrom M.start a) b = M.evalFrom M.start a\nhc : M.evalFrom (M.evalFrom M.start a) c = M.evalFrom M.start x\n\u22a2 {a} * {b}\u2217 * {c} \u2264 M.accepts"}, {"tactic": "intro y hy", "annotated_tactic": ["intro y hy", []], "state_before": "case right\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 M.accepts\nhlen\u271d : Fintype.card \u03c3 \u2264 x.length\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : a.length + b.length \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : M.evalFrom (M.evalFrom M.start a) b = M.evalFrom M.start a\nhc : M.evalFrom (M.evalFrom M.start a) c = M.evalFrom M.start x\n\u22a2 {a} * {b}\u2217 * {c} \u2264 M.accepts", "state_after": "case right\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 M.accepts\nhlen\u271d : Fintype.card \u03c3 \u2264 x.length\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : a.length + b.length \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : M.evalFrom (M.evalFrom M.start a) b = M.evalFrom M.start a\nhc : M.evalFrom (M.evalFrom M.start a) c = M.evalFrom M.start x\ny : List \u03b1\nhy : y \u2208 {a} * {b}\u2217 * {c}\n\u22a2 y \u2208 M.accepts"}, {"tactic": "rw [Language.mem_mul] at hy", "annotated_tactic": ["rw [<a>Language.mem_mul</a>] at hy", [{"full_name": "Language.mem_mul", "def_path": "Mathlib/Computability/Language.lean", "def_pos": [115, 9], "def_end_pos": [115, 16]}]], "state_before": "case right\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 M.accepts\nhlen\u271d : Fintype.card \u03c3 \u2264 x.length\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : a.length + b.length \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : M.evalFrom (M.evalFrom M.start a) b = M.evalFrom M.start a\nhc : M.evalFrom (M.evalFrom M.start a) c = M.evalFrom M.start x\ny : List \u03b1\nhy : y \u2208 {a} * {b}\u2217 * {c}\n\u22a2 y \u2208 M.accepts", "state_after": "case right\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 M.accepts\nhlen\u271d : Fintype.card \u03c3 \u2264 x.length\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : a.length + b.length \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : M.evalFrom (M.evalFrom M.start a) b = M.evalFrom M.start a\nhc : M.evalFrom (M.evalFrom M.start a) c = M.evalFrom M.start x\ny : List \u03b1\nhy : \u2203 a_1 \u2208 {a} * {b}\u2217, \u2203 b \u2208 {c}, a_1 ++ b = y\n\u22a2 y \u2208 M.accepts"}, {"tactic": "rcases hy with \u27e8ab, hab, c', hc', rfl\u27e9", "annotated_tactic": ["rcases hy with \u27e8ab, hab, c', hc', rfl\u27e9", []], "state_before": "case right\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 M.accepts\nhlen\u271d : Fintype.card \u03c3 \u2264 x.length\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : a.length + b.length \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : M.evalFrom (M.evalFrom M.start a) b = M.evalFrom M.start a\nhc : M.evalFrom (M.evalFrom M.start a) c = M.evalFrom M.start x\ny : List \u03b1\nhy : \u2203 a_1 \u2208 {a} * {b}\u2217, \u2203 b \u2208 {c}, a_1 ++ b = y\n\u22a2 y \u2208 M.accepts", "state_after": "case right.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 M.accepts\nhlen\u271d : Fintype.card \u03c3 \u2264 x.length\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : a.length + b.length \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : M.evalFrom (M.evalFrom M.start a) b = M.evalFrom M.start a\nhc : M.evalFrom (M.evalFrom M.start a) c = M.evalFrom M.start x\nab : List \u03b1\nhab : ab \u2208 {a} * {b}\u2217\nc' : List \u03b1\nhc' : c' \u2208 {c}\n\u22a2 ab ++ c' \u2208 M.accepts"}, {"tactic": "rw [Language.mem_mul] at hab", "annotated_tactic": ["rw [<a>Language.mem_mul</a>] at hab", [{"full_name": "Language.mem_mul", "def_path": "Mathlib/Computability/Language.lean", "def_pos": [115, 9], "def_end_pos": [115, 16]}]], "state_before": "case right.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 M.accepts\nhlen\u271d : Fintype.card \u03c3 \u2264 x.length\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : a.length + b.length \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : M.evalFrom (M.evalFrom M.start a) b = M.evalFrom M.start a\nhc : M.evalFrom (M.evalFrom M.start a) c = M.evalFrom M.start x\nab : List \u03b1\nhab : ab \u2208 {a} * {b}\u2217\nc' : List \u03b1\nhc' : c' \u2208 {c}\n\u22a2 ab ++ c' \u2208 M.accepts", "state_after": "case right.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 M.accepts\nhlen\u271d : Fintype.card \u03c3 \u2264 x.length\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : a.length + b.length \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : M.evalFrom (M.evalFrom M.start a) b = M.evalFrom M.start a\nhc : M.evalFrom (M.evalFrom M.start a) c = M.evalFrom M.start x\nab : List \u03b1\nhab : \u2203 a_1 \u2208 {a}, \u2203 b_1 \u2208 {b}\u2217, a_1 ++ b_1 = ab\nc' : List \u03b1\nhc' : c' \u2208 {c}\n\u22a2 ab ++ c' \u2208 M.accepts"}, {"tactic": "rcases hab with \u27e8a', ha', b', hb', rfl\u27e9", "annotated_tactic": ["rcases hab with \u27e8a', ha', b', hb', rfl\u27e9", []], "state_before": "case right.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 M.accepts\nhlen\u271d : Fintype.card \u03c3 \u2264 x.length\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : a.length + b.length \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : M.evalFrom (M.evalFrom M.start a) b = M.evalFrom M.start a\nhc : M.evalFrom (M.evalFrom M.start a) c = M.evalFrom M.start x\nab : List \u03b1\nhab : \u2203 a_1 \u2208 {a}, \u2203 b_1 \u2208 {b}\u2217, a_1 ++ b_1 = ab\nc' : List \u03b1\nhc' : c' \u2208 {c}\n\u22a2 ab ++ c' \u2208 M.accepts", "state_after": "case right.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 M.accepts\nhlen\u271d : Fintype.card \u03c3 \u2264 x.length\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : a.length + b.length \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : M.evalFrom (M.evalFrom M.start a) b = M.evalFrom M.start a\nhc : M.evalFrom (M.evalFrom M.start a) c = M.evalFrom M.start x\nc' : List \u03b1\nhc' : c' \u2208 {c}\na' : List \u03b1\nha' : a' \u2208 {a}\nb' : List \u03b1\nhb' : b' \u2208 {b}\u2217\n\u22a2 a' ++ b' ++ c' \u2208 M.accepts"}, {"tactic": "rw [Set.mem_singleton_iff] at ha' hc'", "annotated_tactic": ["rw [<a>Set.mem_singleton_iff</a>] at ha' hc'", [{"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 26]}]], "state_before": "case right.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 M.accepts\nhlen\u271d : Fintype.card \u03c3 \u2264 x.length\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : a.length + b.length \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : M.evalFrom (M.evalFrom M.start a) b = M.evalFrom M.start a\nhc : M.evalFrom (M.evalFrom M.start a) c = M.evalFrom M.start x\nc' : List \u03b1\nhc' : c' \u2208 {c}\na' : List \u03b1\nha' : a' \u2208 {a}\nb' : List \u03b1\nhb' : b' \u2208 {b}\u2217\n\u22a2 a' ++ b' ++ c' \u2208 M.accepts", "state_after": "case right.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 M.accepts\nhlen\u271d : Fintype.card \u03c3 \u2264 x.length\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : a.length + b.length \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : M.evalFrom (M.evalFrom M.start a) b = M.evalFrom M.start a\nhc : M.evalFrom (M.evalFrom M.start a) c = M.evalFrom M.start x\nc' : List \u03b1\nhc' : c' = c\na' : List \u03b1\nha' : a' = a\nb' : List \u03b1\nhb' : b' \u2208 {b}\u2217\n\u22a2 a' ++ b' ++ c' \u2208 M.accepts"}, {"tactic": "substs ha' hc'", "annotated_tactic": ["substs ha' hc'", []], "state_before": "case right.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 M.accepts\nhlen\u271d : Fintype.card \u03c3 \u2264 x.length\na b c : List \u03b1\nhx : x = a ++ b ++ c\nhlen : a.length + b.length \u2264 Fintype.card \u03c3\nhnil : b \u2260 []\nhb : M.evalFrom (M.evalFrom M.start a) b = M.evalFrom M.start a\nhc : M.evalFrom (M.evalFrom M.start a) c = M.evalFrom M.start x\nc' : List \u03b1\nhc' : c' = c\na' : List \u03b1\nha' : a' = a\nb' : List \u03b1\nhb' : b' \u2208 {b}\u2217\n\u22a2 a' ++ b' ++ c' \u2208 M.accepts", "state_after": "case right.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 M.accepts\nhlen\u271d : Fintype.card \u03c3 \u2264 x.length\nb : List \u03b1\nhnil : b \u2260 []\nc' a' b' : List \u03b1\nhb' : b' \u2208 {b}\u2217\nhlen : a'.length + b.length \u2264 Fintype.card \u03c3\nhb : M.evalFrom (M.evalFrom M.start a') b = M.evalFrom M.start a'\nhx : x = a' ++ b ++ c'\nhc : M.evalFrom (M.evalFrom M.start a') c' = M.evalFrom M.start x\n\u22a2 a' ++ b' ++ c' \u2208 M.accepts"}, {"tactic": "have h := M.evalFrom_of_pow hb hb'", "annotated_tactic": ["have h := M.evalFrom_of_pow hb hb'", []], "state_before": "case right.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 M.accepts\nhlen\u271d : Fintype.card \u03c3 \u2264 x.length\nb : List \u03b1\nhnil : b \u2260 []\nc' a' b' : List \u03b1\nhb' : b' \u2208 {b}\u2217\nhlen : a'.length + b.length \u2264 Fintype.card \u03c3\nhb : M.evalFrom (M.evalFrom M.start a') b = M.evalFrom M.start a'\nhx : x = a' ++ b ++ c'\nhc : M.evalFrom (M.evalFrom M.start a') c' = M.evalFrom M.start x\n\u22a2 a' ++ b' ++ c' \u2208 M.accepts", "state_after": "case right.intro.intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u\n\u03c3 : Type v\nM : DFA \u03b1 \u03c3\ninst\u271d : Fintype \u03c3\nx : List \u03b1\nhx\u271d : x \u2208 M.accepts\nhlen\u271d : Fintype.card \u03c3 \u2264 x.length\nb : List \u03b1\nhnil : b \u2260 []\nc' a' b' : List \u03b1\nhb' : b' \u2208 {b}\u2217\nhlen : a'.length + b.length \u2264 Fintype.card 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g.Derives v w\np : List (Symbol T g.NT)\n\u22a2 g.Derives (v ++ p) (w ++ p)", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case refl\nT : Type uT\ng : ContextFreeGrammar T\nv w p : List (Symbol T g.NT)\n\u22a2 g.Derives (v ++ p) (v ++ p)", "state_after": "no goals"}, {"tactic": "exact ih.trans_produces <| last.append_right p", "annotated_tactic": ["exact ih.trans_produces <| last.append_right p", []], "state_before": "case tail\nT : Type uT\ng : ContextFreeGrammar T\nv w p b\u271d c\u271d : List (Symbol T g.NT)\na\u271d : Relation.ReflTransGen g.Produces v b\u271d\nlast : g.Produces b\u271d c\u271d\nih : g.Derives (v ++ p) (b\u271d ++ p)\n\u22a2 g.Derives (v ++ p) (c\u271d ++ p)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/NFA.lean", "full_name": "NFA.evalFrom_nil", "start": [67, 1], "end": [68, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Monoid.lean", "full_name": "continuous_finprod", "start": [841, 1], "end": [846, 61], "traced_tactics": [{"tactic": "refine continuous_iff_continuousAt.2 fun x => ?_", "annotated_tactic": ["refine <a>continuous_iff_continuousAt</a>.2 fun x => ?_", [{"full_name": "continuous_iff_continuousAt", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1664, 9], "def_end_pos": [1664, 36]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\nhc : \u2200 (i : \u03b9), Continuous (f i)\nhf : LocallyFinite fun i => mulSupport (f i)\n\u22a2 Continuous fun x => \u220f\u1da0 (i : \u03b9), f i x", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\nhc : \u2200 (i : \u03b9), Continuous (f i)\nhf : LocallyFinite fun i => mulSupport (f i)\nx : X\n\u22a2 ContinuousAt (fun x => \u220f\u1da0 (i : \u03b9), f i x) x"}, {"tactic": "rcases finprod_eventually_eq_prod hf x with \u27e8s, hs\u27e9", "annotated_tactic": ["rcases <a>finprod_eventually_eq_prod</a> hf x with \u27e8s, hs\u27e9", [{"full_name": "finprod_eventually_eq_prod", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [832, 9], "def_end_pos": [832, 35]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\nhc : \u2200 (i : \u03b9), Continuous (f i)\nhf : LocallyFinite fun i => mulSupport (f i)\nx : X\n\u22a2 ContinuousAt (fun x => \u220f\u1da0 (i : \u03b9), f i x) x", "state_after": "case intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\nhc : \u2200 (i : \u03b9), Continuous (f i)\nhf : LocallyFinite fun i => mulSupport (f i)\nx : X\ns : Finset \u03b9\nhs : \u2200\u1da0 (y : X) in \ud835\udcdd x, \u220f\u1da0 (i : \u03b9), f i y = \u220f i \u2208 s, f i y\n\u22a2 ContinuousAt (fun x => \u220f\u1da0 (i : \u03b9), f i x) x"}, {"tactic": "refine ContinuousAt.congr ?_ (EventuallyEq.symm hs)", "annotated_tactic": ["refine <a>ContinuousAt.congr</a> ?_ (<a>EventuallyEq.symm</a> hs)", [{"full_name": "ContinuousAt.congr", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1573, 9], "def_end_pos": [1573, 27]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1530, 9], "def_end_pos": [1530, 26]}]], "state_before": "case intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\nhc : \u2200 (i : \u03b9), Continuous (f i)\nhf : LocallyFinite fun i => mulSupport (f i)\nx : X\ns : Finset \u03b9\nhs : \u2200\u1da0 (y : X) in \ud835\udcdd x, \u220f\u1da0 (i : \u03b9), f i y = \u220f i \u2208 s, f i y\n\u22a2 ContinuousAt (fun x => \u220f\u1da0 (i : \u03b9), f i x) x", "state_after": "case intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\nhc : \u2200 (i : \u03b9), Continuous (f i)\nhf : LocallyFinite fun i => mulSupport (f i)\nx : X\ns : Finset \u03b9\nhs : \u2200\u1da0 (y : X) in \ud835\udcdd x, \u220f\u1da0 (i : \u03b9), f i y = \u220f i \u2208 s, f i y\n\u22a2 ContinuousAt (fun x => \u220f i \u2208 s, f i x) x"}, {"tactic": "exact tendsto_finset_prod _ fun i _ => (hc i).continuousAt", "annotated_tactic": ["exact <a>tendsto_finset_prod</a> _ fun i _ => (hc i).<a>continuousAt</a>", [{"full_name": "tendsto_finset_prod", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [772, 9], "def_end_pos": [772, 28]}, {"full_name": "Continuous.continuousAt", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1660, 9], "def_end_pos": [1660, 32]}]], "state_before": "case intro\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nX : Type u_5\ninst\u271d\u00b3 : TopologicalSpace X\ninst\u271d\u00b2 : TopologicalSpace M\ninst\u271d\u00b9 : CommMonoid M\ninst\u271d : ContinuousMul M\nf : \u03b9 \u2192 X \u2192 M\nhc : \u2200 (i : \u03b9), Continuous (f i)\nhf : LocallyFinite fun i => mulSupport (f i)\nx : X\ns : Finset \u03b9\nhs : \u2200\u1da0 (y : X) in \ud835\udcdd x, \u220f\u1da0 (i : \u03b9), f i y = \u220f i \u2208 s, f i y\n\u22a2 ContinuousAt (fun x => \u220f i \u2208 s, f i x) x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/ClosureSwap.lean", "full_name": "surjective_of_isSwap_of_isPretransitive", "start": [132, 1], "end": [138, 74], "traced_tactics": [{"tactic": "rw [\u2190 MonoidHom.range_top_iff_surjective]", "annotated_tactic": ["rw [\u2190 <a>MonoidHom.range_top_iff_surjective</a>]", [{"full_name": "MonoidHom.range_top_iff_surjective", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [2566, 9], "def_end_pos": [2566, 33]}]], "state_before": "G : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Finite \u03b1\nS : Set G\nhS1 : \u2200 \u03c3 \u2208 S, ((toPermHom G \u03b1) \u03c3).IsSwap\nhS2 : closure S = \u22a4\nh : IsPretransitive G \u03b1\n\u22a2 Function.Surjective \u21d1(toPermHom G \u03b1)", "state_after": "G : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Finite \u03b1\nS : Set G\nhS1 : \u2200 \u03c3 \u2208 S, ((toPermHom G \u03b1) \u03c3).IsSwap\nhS2 : closure S = \u22a4\nh : IsPretransitive G \u03b1\n\u22a2 (toPermHom G \u03b1).range = \u22a4"}, {"tactic": "have := MulAction.IsPretransitive.of_compHom (\u03b1 := \u03b1) (MulAction.toPermHom G \u03b1).rangeRestrict", "annotated_tactic": ["have := <a>MulAction.IsPretransitive.of_compHom</a> (\u03b1 := \u03b1) (<a>MulAction.toPermHom</a> G \u03b1).<a>rangeRestrict</a>", [{"full_name": "MulAction.IsPretransitive.of_compHom", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [628, 7], "def_end_pos": [628, 33]}, {"full_name": "MulAction.toPermHom", "def_path": "Mathlib/GroupTheory/GroupAction/Group.lean", "def_pos": [66, 5], "def_end_pos": [66, 24]}, {"full_name": "MonoidHom.rangeRestrict", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [2525, 5], "def_end_pos": [2525, 18]}]], "state_before": "G : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Finite \u03b1\nS : Set G\nhS1 : \u2200 \u03c3 \u2208 S, ((toPermHom G \u03b1) \u03c3).IsSwap\nhS2 : closure S = \u22a4\nh : IsPretransitive G \u03b1\n\u22a2 (toPermHom G \u03b1).range = \u22a4", "state_after": "G : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Finite \u03b1\nS : Set G\nhS1 : \u2200 \u03c3 \u2208 S, ((toPermHom G \u03b1) \u03c3).IsSwap\nhS2 : closure S = \u22a4\nh : IsPretransitive G \u03b1\nthis : IsPretransitive (\u21a5(toPermHom G \u03b1).range) \u03b1\n\u22a2 (toPermHom G \u03b1).range = \u22a4"}, {"tactic": "rw [MonoidHom.range_eq_map, \u2190 hS2, MonoidHom.map_closure] at this \u22a2", "annotated_tactic": ["rw [<a>MonoidHom.range_eq_map</a>, \u2190 hS2, <a>MonoidHom.map_closure</a>] at this \u22a2", [{"full_name": "MonoidHom.range_eq_map", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [2509, 9], "def_end_pos": [2509, 21]}, {"full_name": "MonoidHom.map_closure", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [2882, 9], "def_end_pos": [2882, 20]}]], "state_before": "G : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Finite \u03b1\nS : Set G\nhS1 : \u2200 \u03c3 \u2208 S, ((toPermHom G \u03b1) \u03c3).IsSwap\nhS2 : closure S = \u22a4\nh : IsPretransitive G \u03b1\nthis : IsPretransitive (\u21a5(toPermHom G \u03b1).range) \u03b1\n\u22a2 (toPermHom G \u03b1).range = \u22a4", "state_after": "G : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Finite \u03b1\nS : Set G\nhS1 : \u2200 \u03c3 \u2208 S, ((toPermHom G \u03b1) \u03c3).IsSwap\nhS2 : closure S = \u22a4\nh : IsPretransitive G \u03b1\nthis : IsPretransitive (\u21a5(closure (\u21d1(toPermHom G \u03b1) '' S))) \u03b1\n\u22a2 closure (\u21d1(toPermHom G \u03b1) '' S) = \u22a4"}, {"tactic": "exact closure_of_isSwap_of_isPretransitive (Set.forall_mem_image.2 hS1)", "annotated_tactic": ["exact <a>closure_of_isSwap_of_isPretransitive</a> (<a>Set.forall_mem_image</a>.2 hS1)", [{"full_name": "closure_of_isSwap_of_isPretransitive", "def_path": "Mathlib/GroupTheory/Perm/ClosureSwap.lean", "def_pos": [127, 9], "def_end_pos": [127, 45]}, {"full_name": "Set.forall_mem_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [223, 9], "def_end_pos": [223, 25]}]], "state_before": "G : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b3 : Group G\ninst\u271d\u00b2 : MulAction G \u03b1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Finite \u03b1\nS : Set G\nhS1 : \u2200 \u03c3 \u2208 S, ((toPermHom G \u03b1) \u03c3).IsSwap\nhS2 : closure S = \u22a4\nh : IsPretransitive G \u03b1\nthis : IsPretransitive (\u21a5(closure (\u21d1(toPermHom G \u03b1) '' S))) \u03b1\n\u22a2 closure (\u21d1(toPermHom G \u03b1) '' S) = \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Operations.lean", "full_name": "Submodule.mul_iSup", "start": [366, 1], "end": [367, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Prime.lean", "full_name": "Nat.Prime.coprime_pow_of_not_dvd", "start": [688, 1], "end": [689, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "full_name": "StrictAnti.const_mul", "start": [624, 1], "end": [625, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Antidiag/Pi.lean", "full_name": "Finset.mem_finAntidiagonal", "start": [74, 1], "end": [75, 79], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/Charpoly/FiniteField.lean", "full_name": "ZMod.charpoly_pow_card", "start": [47, 1], "end": [50, 23], "traced_tactics": [{"tactic": "have h := FiniteField.Matrix.charpoly_pow_card M", "annotated_tactic": ["have h := <a>FiniteField.Matrix.charpoly_pow_card</a> M", [{"full_name": "FiniteField.Matrix.charpoly_pow_card", "def_path": "Mathlib/LinearAlgebra/Matrix/Charpoly/FiniteField.lean", "def_pos": [26, 9], "def_end_pos": [26, 45]}]], "state_before": "n : Type u_1\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nM : Matrix n n (ZMod p)\n\u22a2 (M ^ p).charpoly = M.charpoly", "state_after": "n : Type u_1\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nM : Matrix n n (ZMod p)\nh : (M ^ Fintype.card (ZMod p)).charpoly = M.charpoly\n\u22a2 (M ^ p).charpoly = M.charpoly"}, {"tactic": "rwa [ZMod.card] at h", "annotated_tactic": ["rwa [<a>ZMod.card</a>] at h", [{"full_name": "ZMod.card", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [124, 9], "def_end_pos": [124, 13]}]], "state_before": "n : Type u_1\ninst\u271d\u00b2 : DecidableEq n\ninst\u271d\u00b9 : Fintype n\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nM : Matrix n n (ZMod p)\nh : (M ^ Fintype.card (ZMod p)).charpoly = M.charpoly\n\u22a2 (M ^ p).charpoly = M.charpoly", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Shift/Pullback.lean", "full_name": "CategoryTheory.pullbackShiftFunctorAdd'_inv_app", "start": [77, 1], "end": [91, 20], "traced_tactics": [{"tactic": "rw [h\u2081, h\u2082, h\u2083, \u2190 h, \u03c6.map_add]", "annotated_tactic": ["rw [h\u2081, h\u2082, h\u2083, \u2190 h, \u03c6.map_add]", []], "state_before": "C : Type u_1\ninst\u271d\u00b3 : Category.{?u.14320, u_1} C\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\n\u03c6 : A \u2192+ B\ninst\u271d : HasShift C B\nX : PullbackShift C \u03c6\na\u2081 a\u2082 a\u2083 : A\nh : a\u2081 + a\u2082 = a\u2083\nb\u2081 b\u2082 b\u2083 : B\nh\u2081 : b\u2081 = \u03c6 a\u2081\nh\u2082 : b\u2082 = \u03c6 a\u2082\nh\u2083 : b\u2083 = \u03c6 a\u2083\n\u22a2 b\u2081 + b\u2082 = b\u2083", "state_after": "no goals"}, {"tactic": "subst h\u2081 h\u2082 h", "annotated_tactic": ["subst h\u2081 h\u2082 h", []], "state_before": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_4, u_1} C\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\n\u03c6 : A \u2192+ B\ninst\u271d : HasShift C B\nX : PullbackShift C \u03c6\na\u2081 a\u2082 a\u2083 : A\nh : a\u2081 + a\u2082 = a\u2083\nb\u2081 b\u2082 b\u2083 : B\nh\u2081 : b\u2081 = \u03c6 a\u2081\nh\u2082 : b\u2082 = \u03c6 a\u2082\nh\u2083 : b\u2083 = \u03c6 a\u2083\n\u22a2 (shiftFunctorAdd' (PullbackShift C \u03c6) a\u2081 a\u2082 a\u2083 h).inv.app X =\n    (shiftFunctor (PullbackShift C \u03c6) a\u2082).map ((pullbackShiftIso C \u03c6 a\u2081 b\u2081 h\u2081).hom.app X) \u226b\n      (pullbackShiftIso C \u03c6 a\u2082 b\u2082 h\u2082).hom.app ((shiftFunctor C b\u2081).obj X) \u226b\n        (shiftFunctorAdd' C b\u2081 b\u2082 b\u2083 \u22ef).inv.app X \u226b (pullbackShiftIso C \u03c6 a\u2083 b\u2083 h\u2083).inv.app X", "state_after": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_4, u_1} C\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\n\u03c6 : A \u2192+ B\ninst\u271d : HasShift C B\nX : PullbackShift C \u03c6\na\u2081 a\u2082 : A\nb\u2083 : B\nh\u2083 : b\u2083 = \u03c6 (a\u2081 + a\u2082)\n\u22a2 (shiftFunctorAdd' (PullbackShift C \u03c6) a\u2081 a\u2082 (a\u2081 + a\u2082) \u22ef).inv.app X =\n    (shiftFunctor (PullbackShift C \u03c6) a\u2082).map ((pullbackShiftIso C \u03c6 a\u2081 (\u03c6 a\u2081) \u22ef).hom.app X) \u226b\n      (pullbackShiftIso C \u03c6 a\u2082 (\u03c6 a\u2082) \u22ef).hom.app ((shiftFunctor C (\u03c6 a\u2081)).obj X) \u226b\n        (shiftFunctorAdd' C (\u03c6 a\u2081) (\u03c6 a\u2082) b\u2083 \u22ef).inv.app X \u226b (pullbackShiftIso C \u03c6 (a\u2081 + a\u2082) b\u2083 h\u2083).inv.app X"}, {"tactic": "obtain rfl : b\u2083 = \u03c6 a\u2081 + \u03c6 a\u2082 := by rw [h\u2083, \u03c6.map_add]", "annotated_tactic": ["obtain rfl : b\u2083 = \u03c6 a\u2081 + \u03c6 a\u2082 := by rw [h\u2083, \u03c6.map_add]", []], "state_before": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_4, u_1} C\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\n\u03c6 : A \u2192+ B\ninst\u271d : HasShift C B\nX : PullbackShift C \u03c6\na\u2081 a\u2082 : A\nb\u2083 : B\nh\u2083 : b\u2083 = \u03c6 (a\u2081 + a\u2082)\n\u22a2 (shiftFunctorAdd' (PullbackShift C \u03c6) a\u2081 a\u2082 (a\u2081 + a\u2082) \u22ef).inv.app X =\n    (shiftFunctor (PullbackShift C \u03c6) a\u2082).map ((pullbackShiftIso C \u03c6 a\u2081 (\u03c6 a\u2081) \u22ef).hom.app X) \u226b\n      (pullbackShiftIso C \u03c6 a\u2082 (\u03c6 a\u2082) \u22ef).hom.app ((shiftFunctor C (\u03c6 a\u2081)).obj X) \u226b\n        (shiftFunctorAdd' C (\u03c6 a\u2081) (\u03c6 a\u2082) b\u2083 \u22ef).inv.app X \u226b (pullbackShiftIso C \u03c6 (a\u2081 + a\u2082) b\u2083 h\u2083).inv.app X", "state_after": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_4, u_1} C\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\n\u03c6 : A \u2192+ B\ninst\u271d : HasShift C B\nX : PullbackShift C \u03c6\na\u2081 a\u2082 : A\nh\u2083 : \u03c6 a\u2081 + \u03c6 a\u2082 = \u03c6 (a\u2081 + a\u2082)\n\u22a2 (shiftFunctorAdd' (PullbackShift C \u03c6) a\u2081 a\u2082 (a\u2081 + a\u2082) \u22ef).inv.app X =\n    (shiftFunctor (PullbackShift C \u03c6) a\u2082).map ((pullbackShiftIso C \u03c6 a\u2081 (\u03c6 a\u2081) \u22ef).hom.app X) \u226b\n      (pullbackShiftIso C \u03c6 a\u2082 (\u03c6 a\u2082) \u22ef).hom.app ((shiftFunctor C (\u03c6 a\u2081)).obj X) \u226b\n        (shiftFunctorAdd' C (\u03c6 a\u2081) (\u03c6 a\u2082) (\u03c6 a\u2081 + \u03c6 a\u2082) \u22ef).inv.app X \u226b\n          (pullbackShiftIso C \u03c6 (a\u2081 + a\u2082) (\u03c6 a\u2081 + \u03c6 a\u2082) h\u2083).inv.app X"}, {"tactic": "erw [Functor.map_id, id_comp, id_comp, shiftFunctorAdd'_eq_shiftFunctorAdd,\n  shiftFunctorAdd'_eq_shiftFunctorAdd]", "annotated_tactic": ["erw [<a>Functor.map_id</a>, <a>id_comp</a>, <a>id_comp</a>, <a>shiftFunctorAdd'_eq_shiftFunctorAdd</a>,\n    <a>shiftFunctorAdd'_eq_shiftFunctorAdd</a>]", [{"full_name": "CategoryTheory.Functor.map_id", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [41, 3], "def_end_pos": [41, 9]}, {"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [177, 3], "def_end_pos": [177, 10]}, {"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [177, 3], "def_end_pos": [177, 10]}, {"full_name": "CategoryTheory.shiftFunctorAdd'_eq_shiftFunctorAdd", "def_path": "Mathlib/CategoryTheory/Shift/Basic.lean", "def_pos": [183, 7], "def_end_pos": [183, 42]}, {"full_name": "CategoryTheory.shiftFunctorAdd'_eq_shiftFunctorAdd", "def_path": "Mathlib/CategoryTheory/Shift/Basic.lean", "def_pos": [183, 7], "def_end_pos": [183, 42]}]], "state_before": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_4, u_1} C\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\n\u03c6 : A \u2192+ B\ninst\u271d : HasShift C B\nX : PullbackShift C \u03c6\na\u2081 a\u2082 : A\nh\u2083 : \u03c6 a\u2081 + \u03c6 a\u2082 = \u03c6 (a\u2081 + a\u2082)\n\u22a2 (shiftFunctorAdd' (PullbackShift C \u03c6) a\u2081 a\u2082 (a\u2081 + a\u2082) \u22ef).inv.app X =\n    (shiftFunctor (PullbackShift C \u03c6) a\u2082).map ((pullbackShiftIso C \u03c6 a\u2081 (\u03c6 a\u2081) \u22ef).hom.app X) \u226b\n      (pullbackShiftIso C \u03c6 a\u2082 (\u03c6 a\u2082) \u22ef).hom.app ((shiftFunctor C (\u03c6 a\u2081)).obj X) \u226b\n        (shiftFunctorAdd' C (\u03c6 a\u2081) (\u03c6 a\u2082) (\u03c6 a\u2081 + \u03c6 a\u2082) \u22ef).inv.app X \u226b\n          (pullbackShiftIso C \u03c6 (a\u2081 + a\u2082) (\u03c6 a\u2081 + \u03c6 a\u2082) h\u2083).inv.app X", "state_after": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_4, u_1} C\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\n\u03c6 : A \u2192+ B\ninst\u271d : HasShift C B\nX : PullbackShift C \u03c6\na\u2081 a\u2082 : A\nh\u2083 : \u03c6 a\u2081 + \u03c6 a\u2082 = \u03c6 (a\u2081 + a\u2082)\n\u22a2 (shiftFunctorAdd (PullbackShift C \u03c6) a\u2081 a\u2082).inv.app X =\n    (shiftFunctorAdd C (\u03c6 a\u2081) (\u03c6 a\u2082)).inv.app X \u226b (pullbackShiftIso C \u03c6 (a\u2081 + a\u2082) (\u03c6 a\u2081 + \u03c6 a\u2082) h\u2083).inv.app X"}, {"tactic": "change _ \u226b _ = _", "annotated_tactic": ["change _ \u226b _ = _", []], "state_before": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_4, u_1} C\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\n\u03c6 : A \u2192+ B\ninst\u271d : HasShift C B\nX : PullbackShift C \u03c6\na\u2081 a\u2082 : A\nh\u2083 : \u03c6 a\u2081 + \u03c6 a\u2082 = \u03c6 (a\u2081 + a\u2082)\n\u22a2 (shiftFunctorAdd (PullbackShift C \u03c6) a\u2081 a\u2082).inv.app X =\n    (shiftFunctorAdd C (\u03c6 a\u2081) (\u03c6 a\u2082)).inv.app X \u226b (pullbackShiftIso C \u03c6 (a\u2081 + a\u2082) (\u03c6 a\u2081 + \u03c6 a\u2082) h\u2083).inv.app X", "state_after": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_4, u_1} C\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\n\u03c6 : A \u2192+ B\ninst\u271d : HasShift C B\nX : PullbackShift C \u03c6\na\u2081 a\u2082 : A\nh\u2083 : \u03c6 a\u2081 + \u03c6 a\u2082 = \u03c6 (a\u2081 + a\u2082)\n\u22a2 (HasShift.shift.\u03bc ((Discrete.addMonoidalFunctor \u03c6).obj { as := a\u2081 })\n            ((Discrete.addMonoidalFunctor \u03c6).obj { as := a\u2082 })).app\n        X \u226b\n      (HasShift.shift.map ((Discrete.addMonoidalFunctor \u03c6).\u03bc { as := a\u2081 } { as := a\u2082 })).app X =\n    (shiftFunctorAdd C (\u03c6 a\u2081) (\u03c6 a\u2082)).inv.app X \u226b (pullbackShiftIso C \u03c6 (a\u2081 + a\u2082) (\u03c6 a\u2081 + \u03c6 a\u2082) h\u2083).inv.app X"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_4, u_1} C\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\n\u03c6 : A \u2192+ B\ninst\u271d : HasShift C B\nX : PullbackShift C \u03c6\na\u2081 a\u2082 : A\nh\u2083 : \u03c6 a\u2081 + \u03c6 a\u2082 = \u03c6 (a\u2081 + a\u2082)\n\u22a2 (HasShift.shift.\u03bc ((Discrete.addMonoidalFunctor \u03c6).obj { as := a\u2081 })\n            ((Discrete.addMonoidalFunctor \u03c6).obj { as := a\u2082 })).app\n        X \u226b\n      (HasShift.shift.map ((Discrete.addMonoidalFunctor \u03c6).\u03bc { as := a\u2081 } { as := a\u2082 })).app X =\n    (shiftFunctorAdd C (\u03c6 a\u2081) (\u03c6 a\u2082)).inv.app X \u226b (pullbackShiftIso C \u03c6 (a\u2081 + a\u2082) (\u03c6 a\u2081 + \u03c6 a\u2082) h\u2083).inv.app X", "state_after": "case e_a\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_4, u_1} C\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\n\u03c6 : A \u2192+ B\ninst\u271d : HasShift C B\nX : PullbackShift C \u03c6\na\u2081 a\u2082 : A\nh\u2083 : \u03c6 a\u2081 + \u03c6 a\u2082 = \u03c6 (a\u2081 + a\u2082)\n\u22a2 (HasShift.shift.map ((Discrete.addMonoidalFunctor \u03c6).\u03bc { as := a\u2081 } { as := a\u2082 })).app X =\n    (pullbackShiftIso C \u03c6 (a\u2081 + a\u2082) (\u03c6 a\u2081 + \u03c6 a\u2082) h\u2083).inv.app X"}, {"tactic": "dsimp [Discrete.eqToHom]", "annotated_tactic": ["dsimp [<a>Discrete.eqToHom</a>]", [{"full_name": "CategoryTheory.Discrete.eqToHom", "def_path": "Mathlib/CategoryTheory/DiscreteCategory.lean", "def_pos": [138, 18], "def_end_pos": [138, 25]}]], "state_before": "case e_a\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_4, u_1} C\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\n\u03c6 : A \u2192+ B\ninst\u271d : HasShift C B\nX : PullbackShift C \u03c6\na\u2081 a\u2082 : A\nh\u2083 : \u03c6 a\u2081 + \u03c6 a\u2082 = \u03c6 (a\u2081 + a\u2082)\n\u22a2 (HasShift.shift.map ((Discrete.addMonoidalFunctor \u03c6).\u03bc { as := a\u2081 } { as := a\u2082 })).app X =\n    (pullbackShiftIso C \u03c6 (a\u2081 + a\u2082) (\u03c6 a\u2081 + \u03c6 a\u2082) h\u2083).inv.app X", "state_after": "case e_a\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_4, u_1} C\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\n\u03c6 : A \u2192+ B\ninst\u271d : HasShift C B\nX : PullbackShift C \u03c6\na\u2081 a\u2082 : A\nh\u2083 : \u03c6 a\u2081 + \u03c6 a\u2082 = \u03c6 (a\u2081 + a\u2082)\n\u22a2 (HasShift.shift.map (eqToHom \u22ef)).app X = (pullbackShiftIso C \u03c6 (a\u2081 + a\u2082) (\u03c6 a\u2081 + \u03c6 a\u2082) h\u2083).inv.app X"}, {"tactic": "congr 2", "annotated_tactic": ["congr 2", []], "state_before": "case e_a\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_4, u_1} C\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\n\u03c6 : A \u2192+ B\ninst\u271d : HasShift C B\nX : PullbackShift C \u03c6\na\u2081 a\u2082 : A\nh\u2083 : \u03c6 a\u2081 + \u03c6 a\u2082 = \u03c6 (a\u2081 + a\u2082)\n\u22a2 (HasShift.shift.map (eqToHom \u22ef)).app X = (pullbackShiftIso C \u03c6 (a\u2081 + a\u2082) (\u03c6 a\u2081 + \u03c6 a\u2082) h\u2083).inv.app X", "state_after": "case e_a.e_self\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_4, u_1} C\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\n\u03c6 : A \u2192+ B\ninst\u271d : HasShift C B\nX : PullbackShift C \u03c6\na\u2081 a\u2082 : A\nh\u2083 : \u03c6 a\u2081 + \u03c6 a\u2082 = \u03c6 (a\u2081 + a\u2082)\n\u22a2 HasShift.shift.map (eqToHom \u22ef) = (pullbackShiftIso C \u03c6 (a\u2081 + a\u2082) (\u03c6 a\u2081 + \u03c6 a\u2082) h\u2083).inv"}, {"tactic": "apply eqToHom_map", "annotated_tactic": ["apply <a>eqToHom_map</a>", [{"full_name": "CategoryTheory.eqToHom_map", "def_path": "Mathlib/CategoryTheory/EqToHom.lean", "def_pos": [301, 9], "def_end_pos": [301, 20]}]], "state_before": "case e_a.e_self\nC : Type u_1\ninst\u271d\u00b3 : Category.{u_4, u_1} C\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\n\u03c6 : A \u2192+ B\ninst\u271d : HasShift C B\nX : PullbackShift C \u03c6\na\u2081 a\u2082 : A\nh\u2083 : \u03c6 a\u2081 + \u03c6 a\u2082 = \u03c6 (a\u2081 + a\u2082)\n\u22a2 HasShift.shift.map (eqToHom \u22ef) = (pullbackShiftIso C \u03c6 (a\u2081 + a\u2082) (\u03c6 a\u2081 + \u03c6 a\u2082) h\u2083).inv", "state_after": "no goals"}, {"tactic": "rw [h\u2083, \u03c6.map_add]", "annotated_tactic": ["rw [h\u2083, \u03c6.map_add]", []], "state_before": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_4, u_1} C\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\n\u03c6 : A \u2192+ B\ninst\u271d : HasShift C B\nX : PullbackShift C \u03c6\na\u2081 a\u2082 : A\nb\u2083 : B\nh\u2083 : b\u2083 = \u03c6 (a\u2081 + a\u2082)\n\u22a2 b\u2083 = \u03c6 a\u2081 + \u03c6 a\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Cone/Basic.lean", "full_name": "ConvexCone.map_id", "start": [242, 1], "end": [243, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/VonMangoldt.lean", "full_name": "ArithmeticFunction.sum_moebius_mul_log_eq", "start": [144, 1], "end": [160, 51], "traced_tactics": [{"tactic": "simp only [\u2190 log_mul_moebius_eq_vonMangoldt, mul_comm log, mul_apply, log_apply, intCoe_apply, \u2190\n  Finset.sum_neg_distrib, neg_mul_eq_mul_neg]", "annotated_tactic": ["simp only [\u2190 <a>log_mul_moebius_eq_vonMangoldt</a>, <a>mul_comm</a> <a>log</a>, <a>mul_apply</a>, <a>log_apply</a>, <a>intCoe_apply</a>, \u2190\n    <a>Finset.sum_neg_distrib</a>, <a>neg_mul_eq_mul_neg</a>]", [{"full_name": "ArithmeticFunction.log_mul_moebius_eq_vonMangoldt", "def_path": "Mathlib/NumberTheory/VonMangoldt.lean", "def_pos": [135, 9], "def_end_pos": [135, 39]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "ArithmeticFunction.log", "def_path": "Mathlib/NumberTheory/VonMangoldt.lean", "def_pos": [46, 19], "def_end_pos": [46, 22]}, {"full_name": "ArithmeticFunction.mul_apply", "def_path": "Mathlib/NumberTheory/ArithmeticFunction.lean", "def_pos": [291, 9], "def_end_pos": [291, 18]}, {"full_name": "ArithmeticFunction.log_apply", "def_path": "Mathlib/NumberTheory/VonMangoldt.lean", "def_pos": [51, 9], "def_end_pos": [51, 18]}, {"full_name": "ArithmeticFunction.intCoe_apply", "def_path": "Mathlib/NumberTheory/ArithmeticFunction.lean", "def_pos": [197, 9], "def_end_pos": [197, 21]}, {"full_name": "Finset.sum_neg_distrib", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [2116, 3], "def_end_pos": [2116, 14]}, {"full_name": "neg_mul_eq_mul_neg", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [334, 9], "def_end_pos": [334, 27]}]], "state_before": "n : \u2115\n\u22a2 \u2211 d \u2208 n.divisors, \u2191(\u03bc d) * log d = -\u039b n", "state_after": "n : \u2115\n\u22a2 \u2211 x \u2208 n.divisors, \u2191(\u03bc x) * Real.log \u2191x = \u2211 x \u2208 n.divisorsAntidiagonal, \u2191(\u03bc x.1) * -Real.log \u2191x.2"}, {"tactic": "rw [sum_divisorsAntidiagonal fun i j => (\u03bc i : \u211d) * -Real.log j]", "annotated_tactic": ["rw [<a>sum_divisorsAntidiagonal</a> fun i j => (\u03bc i : \u211d) * -<a>Real.log</a> j]", [{"full_name": "Nat.sum_divisorsAntidiagonal", "def_path": "Mathlib/NumberTheory/Divisors.lean", "def_pos": [521, 3], "def_end_pos": [521, 14]}, {"full_name": "Real.log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [41, 19], "def_end_pos": [41, 22]}]], "state_before": "n : \u2115\n\u22a2 \u2211 x \u2208 n.divisors, \u2191(\u03bc x) * Real.log \u2191x = \u2211 x \u2208 n.divisorsAntidiagonal, \u2191(\u03bc x.1) * -Real.log \u2191x.2", "state_after": "n : \u2115\n\u22a2 \u2211 x \u2208 n.divisors, \u2191(\u03bc x) * Real.log \u2191x = \u2211 i \u2208 n.divisors, \u2191(\u03bc i) * -Real.log \u2191(n / i)"}, {"tactic": "have : (\u2211 i \u2208 n.divisors, (\u03bc i : \u211d) * -Real.log (n / i : \u2115)) =\n    \u2211 i \u2208 n.divisors, ((\u03bc i : \u211d) * Real.log i - \u03bc i * Real.log n) := by\n  apply sum_congr rfl\n  simp only [and_imp, Int.cast_eq_zero, mul_eq_mul_left_iff, Ne, neg_inj, mem_divisors]\n  intro m mn hn\n  have : (m : \u211d) \u2260 0 := by\n    rw [cast_ne_zero]\n    rintro rfl\n    exact hn (by simpa using mn)\n  rw [Nat.cast_div mn this, Real.log_div (cast_ne_zero.2 hn) this, neg_sub, mul_sub]", "annotated_tactic": ["have : (\u2211 i \u2208 n.divisors, (\u03bc i : \u211d) * -<a>Real.log</a> (n / i : \u2115)) =\n      \u2211 i \u2208 n.divisors, ((\u03bc i : \u211d) * <a>Real.log</a> i - \u03bc i * <a>Real.log</a> n) := by\n    apply <a>sum_congr</a> <a>rfl</a>\n    simp only [<a>and_imp</a>, <a>Int.cast_eq_zero</a>, <a>mul_eq_mul_left_iff</a>, <a>Ne</a>, <a>neg_inj</a>, <a>mem_divisors</a>]\n    intro m mn hn\n    have : (m : \u211d) \u2260 0 := by\n      rw [<a>cast_ne_zero</a>]\n      rintro rfl\n      exact hn (by simpa using mn)\n    rw [<a>Nat.cast_div</a> mn this, <a>Real.log_div</a> (<a>cast_ne_zero</a>.2 hn) this, <a>neg_sub</a>, <a>mul_sub</a>]", [{"full_name": "Real.log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [41, 19], "def_end_pos": [41, 22]}, {"full_name": "Real.log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [41, 19], "def_end_pos": [41, 22]}, {"full_name": "Real.log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [41, 19], "def_end_pos": [41, 22]}, {"full_name": "Finset.sum_congr", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [420, 3], "def_end_pos": [420, 14]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "and_imp", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [115, 17], "def_end_pos": [115, 24]}, {"full_name": "Int.cast_eq_zero", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [65, 15], "def_end_pos": [65, 27]}, {"full_name": "mul_eq_mul_left_iff", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [231, 9], "def_end_pos": [231, 28]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "neg_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [406, 3], "def_end_pos": [406, 14]}, {"full_name": "Nat.mem_divisors", "def_path": "Mathlib/NumberTheory/Divisors.lean", "def_pos": [95, 9], "def_end_pos": [95, 21]}, {"full_name": "Nat.cast_ne_zero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [83, 9], "def_end_pos": [83, 21]}, {"full_name": "Nat.cast_div", "def_path": "Mathlib/Data/Nat/Cast/Field.lean", "def_pos": [29, 9], "def_end_pos": [29, 17]}, {"full_name": "Real.log_div", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}, {"full_name": "Nat.cast_ne_zero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [83, 9], "def_end_pos": [83, 21]}, {"full_name": "neg_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [553, 3], "def_end_pos": [553, 14]}, {"full_name": "mul_sub", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [394, 7], "def_end_pos": [394, 14]}]], "state_before": "n : \u2115\n\u22a2 \u2211 x \u2208 n.divisors, \u2191(\u03bc x) * Real.log \u2191x = \u2211 i \u2208 n.divisors, \u2191(\u03bc i) * -Real.log \u2191(n / i)", "state_after": "n : \u2115\nthis : \u2211 i \u2208 n.divisors, \u2191(\u03bc i) * -Real.log \u2191(n / i) = \u2211 i \u2208 n.divisors, (\u2191(\u03bc i) * Real.log \u2191i - \u2191(\u03bc i) * Real.log \u2191n)\n\u22a2 \u2211 x \u2208 n.divisors, \u2191(\u03bc x) * Real.log \u2191x = \u2211 i \u2208 n.divisors, \u2191(\u03bc i) * -Real.log \u2191(n / i)"}, {"tactic": "rw [this, sum_sub_distrib, \u2190 sum_mul, \u2190 Int.cast_sum, \u2190 coe_mul_zeta_apply, eq_comm, sub_eq_self,\n  moebius_mul_coe_zeta]", "annotated_tactic": ["rw [this, <a>sum_sub_distrib</a>, \u2190 <a>sum_mul</a>, \u2190 <a>Int.cast_sum</a>, \u2190 <a>coe_mul_zeta_apply</a>, <a>eq_comm</a>, <a>sub_eq_self</a>,\n    <a>moebius_mul_coe_zeta</a>]", [{"full_name": "Finset.sum_sub_distrib", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [2122, 3], "def_end_pos": [2122, 14]}, {"full_name": "Finset.sum_mul", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [57, 7], "def_end_pos": [57, 14]}, {"full_name": "Int.cast_sum", "def_path": "Mathlib/Algebra/BigOperators/Ring.lean", "def_pos": [396, 7], "def_end_pos": [396, 15]}, {"full_name": "ArithmeticFunction.coe_mul_zeta_apply", "def_path": "Mathlib/NumberTheory/ArithmeticFunction.lean", "def_pos": [468, 9], "def_end_pos": [468, 27]}, {"full_name": "eq_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}, {"full_name": "sub_eq_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1088, 3], "def_end_pos": [1088, 14]}, {"full_name": "ArithmeticFunction.moebius_mul_coe_zeta", "def_path": "Mathlib/NumberTheory/ArithmeticFunction.lean", "def_pos": [1170, 9], "def_end_pos": [1170, 29]}]], "state_before": "n : \u2115\nthis : \u2211 i \u2208 n.divisors, \u2191(\u03bc i) * -Real.log \u2191(n / i) = \u2211 i \u2208 n.divisors, (\u2191(\u03bc i) * Real.log \u2191i - \u2191(\u03bc i) * Real.log \u2191n)\n\u22a2 \u2211 x \u2208 n.divisors, \u2191(\u03bc x) * Real.log \u2191x = \u2211 i \u2208 n.divisors, \u2191(\u03bc i) * -Real.log \u2191(n / i)", "state_after": "n : \u2115\nthis : \u2211 i \u2208 n.divisors, \u2191(\u03bc i) * -Real.log \u2191(n / i) = \u2211 i \u2208 n.divisors, (\u2191(\u03bc i) * Real.log \u2191i - \u2191(\u03bc i) * Real.log \u2191n)\n\u22a2 \u2191(1 n) * Real.log \u2191n = 0"}, {"tactic": "rcases eq_or_ne n 1 with (hn | hn) <;> simp [hn]", "annotated_tactic": ["rcases <a>eq_or_ne</a> n 1 with (hn | hn) <;> simp [hn]", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 17]}]], "state_before": "n : \u2115\nthis : \u2211 i \u2208 n.divisors, \u2191(\u03bc i) * -Real.log \u2191(n / i) = \u2211 i \u2208 n.divisors, (\u2191(\u03bc i) * Real.log \u2191i - \u2191(\u03bc i) * Real.log \u2191n)\n\u22a2 \u2191(1 n) * Real.log \u2191n = 0", "state_after": "no goals"}, {"tactic": "apply sum_congr rfl", "annotated_tactic": ["apply <a>sum_congr</a> <a>rfl</a>", [{"full_name": "Finset.sum_congr", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [420, 3], "def_end_pos": [420, 14]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "n : \u2115\n\u22a2 \u2211 i \u2208 n.divisors, \u2191(\u03bc i) * -Real.log \u2191(n / i) = \u2211 i \u2208 n.divisors, (\u2191(\u03bc i) * Real.log \u2191i - \u2191(\u03bc i) * Real.log \u2191n)", "state_after": "n : \u2115\n\u22a2 \u2200 x \u2208 n.divisors, \u2191(\u03bc x) * -Real.log \u2191(n / x) = \u2191(\u03bc x) * Real.log \u2191x - \u2191(\u03bc x) * Real.log \u2191n"}, {"tactic": "simp only [and_imp, Int.cast_eq_zero, mul_eq_mul_left_iff, Ne, neg_inj, mem_divisors]", "annotated_tactic": ["simp only [<a>and_imp</a>, <a>Int.cast_eq_zero</a>, <a>mul_eq_mul_left_iff</a>, <a>Ne</a>, <a>neg_inj</a>, <a>mem_divisors</a>]", [{"full_name": "and_imp", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [115, 17], "def_end_pos": [115, 24]}, {"full_name": "Int.cast_eq_zero", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [65, 15], "def_end_pos": [65, 27]}, {"full_name": "mul_eq_mul_left_iff", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [231, 9], "def_end_pos": [231, 28]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "neg_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [406, 3], "def_end_pos": [406, 14]}, {"full_name": "Nat.mem_divisors", "def_path": "Mathlib/NumberTheory/Divisors.lean", "def_pos": [95, 9], "def_end_pos": [95, 21]}]], "state_before": "n : \u2115\n\u22a2 \u2200 x \u2208 n.divisors, \u2191(\u03bc x) * -Real.log \u2191(n / x) = \u2191(\u03bc x) * Real.log \u2191x - \u2191(\u03bc x) * Real.log \u2191n", "state_after": "n : \u2115\n\u22a2 \u2200 (x : \u2115), x \u2223 n \u2192 \u00acn = 0 \u2192 \u2191(\u03bc x) * -Real.log \u2191(n / x) = \u2191(\u03bc x) * Real.log \u2191x - \u2191(\u03bc x) * Real.log \u2191n"}, {"tactic": "intro m mn hn", "annotated_tactic": ["intro m mn hn", []], "state_before": "n : \u2115\n\u22a2 \u2200 (x : \u2115), x \u2223 n \u2192 \u00acn = 0 \u2192 \u2191(\u03bc x) * -Real.log \u2191(n / x) = \u2191(\u03bc x) * Real.log \u2191x - \u2191(\u03bc x) * Real.log \u2191n", "state_after": "n m : \u2115\nmn : m \u2223 n\nhn : \u00acn = 0\n\u22a2 \u2191(\u03bc m) * -Real.log \u2191(n / m) = \u2191(\u03bc m) * Real.log \u2191m - \u2191(\u03bc m) * Real.log \u2191n"}, {"tactic": "have : (m : \u211d) \u2260 0 := by\n  rw [cast_ne_zero]\n  rintro rfl\n  exact hn (by simpa using mn)", "annotated_tactic": ["have : (m : \u211d) \u2260 0 := by\n      rw [<a>cast_ne_zero</a>]\n      rintro rfl\n      exact hn (by simpa using mn)", [{"full_name": "Nat.cast_ne_zero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [83, 9], "def_end_pos": [83, 21]}]], "state_before": "n m : \u2115\nmn : m \u2223 n\nhn : \u00acn = 0\n\u22a2 \u2191(\u03bc m) * -Real.log \u2191(n / m) = \u2191(\u03bc m) * Real.log \u2191m - \u2191(\u03bc m) * Real.log \u2191n", "state_after": "n m : \u2115\nmn : m \u2223 n\nhn : \u00acn = 0\nthis : \u2191m \u2260 0\n\u22a2 \u2191(\u03bc m) * -Real.log \u2191(n / m) = \u2191(\u03bc m) * Real.log \u2191m - \u2191(\u03bc m) * Real.log \u2191n"}, {"tactic": "rw [Nat.cast_div mn this, Real.log_div (cast_ne_zero.2 hn) this, neg_sub, mul_sub]", "annotated_tactic": ["rw [<a>Nat.cast_div</a> mn this, <a>Real.log_div</a> (<a>cast_ne_zero</a>.2 hn) this, <a>neg_sub</a>, <a>mul_sub</a>]", [{"full_name": "Nat.cast_div", "def_path": "Mathlib/Data/Nat/Cast/Field.lean", "def_pos": [29, 9], "def_end_pos": [29, 17]}, {"full_name": "Real.log_div", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [131, 9], "def_end_pos": [131, 16]}, {"full_name": "Nat.cast_ne_zero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [83, 9], "def_end_pos": [83, 21]}, {"full_name": "neg_sub", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [553, 3], "def_end_pos": [553, 14]}, {"full_name": "mul_sub", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [394, 7], "def_end_pos": [394, 14]}]], "state_before": "n m : \u2115\nmn : m \u2223 n\nhn : \u00acn = 0\nthis : \u2191m \u2260 0\n\u22a2 \u2191(\u03bc m) * -Real.log \u2191(n / m) = \u2191(\u03bc m) * Real.log \u2191m - \u2191(\u03bc m) * Real.log \u2191n", "state_after": "no goals"}, {"tactic": "rw [cast_ne_zero]", "annotated_tactic": ["rw [<a>cast_ne_zero</a>]", [{"full_name": "Nat.cast_ne_zero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [83, 9], "def_end_pos": [83, 21]}]], "state_before": "n m : \u2115\nmn : m \u2223 n\nhn : \u00acn = 0\n\u22a2 \u2191m \u2260 0", "state_after": "n m : \u2115\nmn : m \u2223 n\nhn : \u00acn = 0\n\u22a2 m \u2260 0"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "n m : \u2115\nmn : m \u2223 n\nhn : \u00acn = 0\n\u22a2 m \u2260 0", "state_after": "n : \u2115\nhn : \u00acn = 0\nmn : 0 \u2223 n\n\u22a2 False"}, {"tactic": "exact hn (by simpa using mn)", "annotated_tactic": ["exact hn (by simpa using mn)", []], "state_before": "n : \u2115\nhn : \u00acn = 0\nmn : 0 \u2223 n\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simpa using mn", "annotated_tactic": ["simpa using mn", []], "state_before": "n : \u2115\nhn : \u00acn = 0\nmn : 0 \u2223 n\n\u22a2 n = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean", "full_name": "fourierIntegral_gaussian_innerProductSpace", "start": [382, 1], "end": [385, 65], "traced_tactics": [{"tactic": "simpa using fourierIntegral_gaussian_innerProductSpace' hb 0 w", "annotated_tactic": ["simpa using <a>fourierIntegral_gaussian_innerProductSpace'</a> hb 0 w", [{"full_name": "fourierIntegral_gaussian_innerProductSpace'", "def_path": "Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean", "def_pos": [369, 9], "def_end_pos": [369, 59]}]], "state_before": "b : \u2102\nV : Type u_1\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : FiniteDimensional \u211d V\ninst\u271d\u00b9 : MeasurableSpace V\ninst\u271d : BorelSpace V\nhb : 0 < b.re\nw : V\n\u22a2 \ud835\udcd5 (fun v => cexp (-b * \u2191\u2016v\u2016 ^ 2)) w =\n    (\u2191\u03c0 / b) ^ (\u2191(FiniteDimensional.finrank \u211d V) / 2) * cexp (-\u2191\u03c0 ^ 2 * \u2191\u2016w\u2016 ^ 2 / b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Pairwise.lean", "full_name": "Set.PairwiseDisjoint.image_finset_of_le", "start": [44, 1], "end": [48, 26], "traced_tactics": [{"tactic": "rw [coe_image]", "annotated_tactic": ["rw [<a>coe_image</a>]", [{"full_name": "Finset.coe_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [419, 9], "def_end_pos": [419, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b9' : Type u_3\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : (\u2191s).PairwiseDisjoint f\ng : \u03b9 \u2192 \u03b9\nhf : \u2200 (a : \u03b9), f (g a) \u2264 f a\n\u22a2 (\u2191(Finset.image g s)).PairwiseDisjoint f", "state_after": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b9' : Type u_3\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : (\u2191s).PairwiseDisjoint f\ng : \u03b9 \u2192 \u03b9\nhf : \u2200 (a : \u03b9), f (g a) \u2264 f a\n\u22a2 (g '' \u2191s).PairwiseDisjoint f"}, {"tactic": "exact hs.image_of_le hf", "annotated_tactic": ["exact hs.image_of_le hf", []], "state_before": "\u03b1 : Type u_1\n\u03b9 : Type u_2\n\u03b9' : Type u_3\ninst\u271d\u00b2 : SemilatticeInf \u03b1\ninst\u271d\u00b9 : OrderBot \u03b1\ns\u271d : Finset \u03b9\nf\u271d : \u03b9 \u2192 \u03b1\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nf : \u03b9 \u2192 \u03b1\nhs : (\u2191s).PairwiseDisjoint f\ng : \u03b9 \u2192 \u03b9\nhf : \u2200 (a : \u03b9), f (g a) \u2264 f a\n\u22a2 (g '' \u2191s).PairwiseDisjoint f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Finset.inter_mul_union_subset_union", "start": [517, 1], "end": [518, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/FiberBundle/Trivialization.lean", "full_name": "Pretrivialization.mem_source", "start": [118, 1], "end": [118, 92], "traced_tactics": [{"tactic": "rw [e.source_eq, mem_preimage]", "annotated_tactic": ["rw [e.source_eq, <a>mem_preimage</a>]", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [120, 9], "def_end_pos": [120, 21]}]], "state_before": "\u03b9 : Type u_1\nB : Type u_2\nF : Type u_3\nE : B \u2192 Type u_4\nZ : Type u_5\ninst\u271d\u00b9 : TopologicalSpace B\ninst\u271d : TopologicalSpace F\nproj : Z \u2192 B\ne : Pretrivialization F proj\nx : Z\n\u22a2 x \u2208 e.source \u2194 proj x \u2208 e.baseSet", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Iso.lean", "full_name": "CategoryTheory.Iso.cancel_iso_hom_left", "start": [554, 1], "end": [556, 25], "traced_tactics": [{"tactic": "simp only [cancel_epi]", "annotated_tactic": ["simp only [<a>cancel_epi</a>]", [{"full_name": "CategoryTheory.cancel_epi", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 19]}]], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX\u271d Y\u271d Z\u271d X Y Z : C\nf : X \u2245 Y\ng g' : Y \u27f6 Z\n\u22a2 f.hom \u226b g = f.hom \u226b g' \u2194 g = g'", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/ModelTheory/Substructures.lean", "full_name": "FirstOrder.Language.Substructure.closure_empty", "start": [381, 1], "end": [382, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Card.lean", "full_name": "Finset.card_insert_of_not_mem", "start": [108, 1], "end": [109, 41], "traced_tactics": [{"tactic": "rw [\u2190 cons_eq_insert _ _ h, card_cons]", "annotated_tactic": ["rw [\u2190 <a>cons_eq_insert</a> _ _ h, <a>card_cons</a>]", [{"full_name": "Finset.cons_eq_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1137, 9], "def_end_pos": [1137, 23]}, {"full_name": "Finset.card_cons", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [99, 9], "def_end_pos": [99, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t : Finset \u03b1\na b : \u03b1\ninst\u271d : DecidableEq \u03b1\nh : a \u2209 s\n\u22a2 (insert a s).card = s.card + 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/MFDeriv/Basic.lean", "full_name": "mfderivWithin_of_isOpen", "start": [361, 1], "end": [363, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "full_name": "Set.Subsingleton.differentiableOn", "start": [1232, 1], "end": [1233, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/Support.lean", "full_name": "Equiv.Perm.card_support_swap", "start": [606, 1], "end": [608, 68], "traced_tactics": [{"tactic": "simp [hxy]", "annotated_tactic": ["simp [hxy]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nhxy : x \u2260 y\n\u22a2 (x ::\u2098 y ::\u2098 0).Nodup", "state_after": "no goals"}, {"tactic": "simp [support_swap hxy, *, Finset.ext_iff]", "annotated_tactic": ["simp [<a>support_swap</a> hxy, *, <a>Finset.ext_iff</a>]", [{"full_name": "Equiv.Perm.support_swap", "def_path": "Mathlib/GroupTheory/Perm/Support.lean", "def_pos": [435, 9], "def_end_pos": [435, 21]}, {"full_name": "Finset.ext_iff", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [234, 9], "def_end_pos": [234, 16]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nhxy : x \u2260 y\n\u22a2 (swap x y).support = { val := x ::\u2098 y ::\u2098 0, nodup := \u22ef }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/Star/Spectrum.lean", "full_name": "IsSelfAdjoint.spectralRadius_eq_nnnorm", "start": [60, 1], "end": [69, 7], "traced_tactics": [{"tactic": "have hconst : Tendsto (fun _n : \u2115 => (\u2016a\u2016\u208a : \u211d\u22650\u221e)) atTop _ := tendsto_const_nhds", "annotated_tactic": ["have hconst : <a>Tendsto</a> (fun _n : \u2115 => (\u2016a\u2016\u208a : \u211d\u22650\u221e)) <a>atTop</a> _ := <a>tendsto_const_nhds</a>", [{"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2991, 5], "def_end_pos": [2991, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [991, 9], "def_end_pos": [991, 27]}]], "state_before": "A : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : CompleteSpace A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : CstarRing A\na : A\nha : IsSelfAdjoint a\n\u22a2 spectralRadius \u2102 a = \u2191\u2016a\u2016\u208a", "state_after": "A : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : CompleteSpace A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : CstarRing A\na : A\nha : IsSelfAdjoint a\nhconst : Tendsto (fun _n => \u2191\u2016a\u2016\u208a) atTop (\ud835\udcdd \u2191\u2016a\u2016\u208a)\n\u22a2 spectralRadius \u2102 a = \u2191\u2016a\u2016\u208a"}, {"tactic": "refine tendsto_nhds_unique ?_ hconst", "annotated_tactic": ["refine <a>tendsto_nhds_unique</a> ?_ hconst", [{"full_name": "tendsto_nhds_unique", "def_path": "Mathlib/Topology/Separation.lean", "def_pos": [1508, 9], "def_end_pos": [1508, 28]}]], "state_before": "A : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : CompleteSpace A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : CstarRing A\na : A\nha : IsSelfAdjoint a\nhconst : Tendsto (fun _n => \u2191\u2016a\u2016\u208a) atTop (\ud835\udcdd \u2191\u2016a\u2016\u208a)\n\u22a2 spectralRadius \u2102 a = \u2191\u2016a\u2016\u208a", "state_after": "A : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : CompleteSpace A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : CstarRing A\na : A\nha : IsSelfAdjoint a\nhconst : Tendsto (fun _n => \u2191\u2016a\u2016\u208a) atTop (\ud835\udcdd \u2191\u2016a\u2016\u208a)\n\u22a2 Tendsto (fun _n => \u2191\u2016a\u2016\u208a) atTop (\ud835\udcdd (spectralRadius \u2102 a))"}, {"tactic": "convert\n  (spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius (a : A)).comp\n    (Nat.tendsto_pow_atTop_atTop_of_one_lt one_lt_two) using 1", "annotated_tactic": ["convert\n    (<a>spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius</a> (a : A)).<a>comp</a>\n      (<a>Nat.tendsto_pow_atTop_atTop_of_one_lt</a> <a>one_lt_two</a>) using 1", [{"full_name": "spectrum.pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius", "def_path": "Mathlib/Analysis/NormedSpace/Spectrum.lean", "def_pos": [387, 9], "def_end_pos": [387, 59]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3098, 9], "def_end_pos": [3098, 21]}, {"full_name": "Nat.tendsto_pow_atTop_atTop_of_one_lt", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [130, 9], "def_end_pos": [130, 46]}, {"full_name": "one_lt_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [106, 7], "def_end_pos": [106, 17]}]], "state_before": "A : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : CompleteSpace A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : CstarRing A\na : A\nha : IsSelfAdjoint a\nhconst : Tendsto (fun _n => \u2191\u2016a\u2016\u208a) atTop (\ud835\udcdd \u2191\u2016a\u2016\u208a)\n\u22a2 Tendsto (fun _n => \u2191\u2016a\u2016\u208a) atTop (\ud835\udcdd (spectralRadius \u2102 a))", "state_after": "case h.e'_3\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : CompleteSpace A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : CstarRing A\na : A\nha : IsSelfAdjoint a\nhconst : Tendsto (fun _n => \u2191\u2016a\u2016\u208a) atTop (\ud835\udcdd \u2191\u2016a\u2016\u208a)\n\u22a2 (fun _n => \u2191\u2016a\u2016\u208a) = (fun n => \u2191\u2016a ^ n\u2016\u208a ^ (1 / \u2191n)) \u2218 fun n => 2 ^ n"}, {"tactic": "refine funext fun n => ?_", "annotated_tactic": ["refine <a>funext</a> fun n => ?_", [{"full_name": "funext", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1817, 9], "def_end_pos": [1817, 15]}]], "state_before": "case h.e'_3\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : CompleteSpace A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : CstarRing A\na : A\nha : IsSelfAdjoint a\nhconst : Tendsto (fun _n => \u2191\u2016a\u2016\u208a) atTop (\ud835\udcdd \u2191\u2016a\u2016\u208a)\n\u22a2 (fun _n => \u2191\u2016a\u2016\u208a) = (fun n => \u2191\u2016a ^ n\u2016\u208a ^ (1 / \u2191n)) \u2218 fun n => 2 ^ n", "state_after": "case h.e'_3\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : CompleteSpace A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : CstarRing A\na : A\nha : IsSelfAdjoint a\nhconst : Tendsto (fun _n => \u2191\u2016a\u2016\u208a) atTop (\ud835\udcdd \u2191\u2016a\u2016\u208a)\nn : \u2115\n\u22a2 \u2191\u2016a\u2016\u208a = ((fun n => \u2191\u2016a ^ n\u2016\u208a ^ (1 / \u2191n)) \u2218 fun n => 2 ^ n) n"}, {"tactic": "rw [Function.comp_apply, ha.nnnorm_pow_two_pow, ENNReal.coe_pow, \u2190 rpow_natCast, \u2190 rpow_mul]", "annotated_tactic": ["rw [<a>Function.comp_apply</a>, ha.nnnorm_pow_two_pow, <a>ENNReal.coe_pow</a>, \u2190 <a>rpow_natCast</a>, \u2190 <a>rpow_mul</a>]", [{"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}, {"full_name": "ENNReal.coe_pow", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [418, 26], "def_end_pos": [418, 33]}, {"full_name": "ENNReal.rpow_natCast", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [587, 9], "def_end_pos": [587, 21]}, {"full_name": "ENNReal.rpow_mul", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean", "def_pos": [571, 9], "def_end_pos": [571, 17]}]], "state_before": "case h.e'_3\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : CompleteSpace A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : CstarRing A\na : A\nha : IsSelfAdjoint a\nhconst : Tendsto (fun _n => \u2191\u2016a\u2016\u208a) atTop (\ud835\udcdd \u2191\u2016a\u2016\u208a)\nn : \u2115\n\u22a2 \u2191\u2016a\u2016\u208a = ((fun n => \u2191\u2016a ^ n\u2016\u208a ^ (1 / \u2191n)) \u2218 fun n => 2 ^ n) n", "state_after": "case h.e'_3\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : CompleteSpace A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : CstarRing A\na : A\nha : IsSelfAdjoint a\nhconst : Tendsto (fun _n => \u2191\u2016a\u2016\u208a) atTop (\ud835\udcdd \u2191\u2016a\u2016\u208a)\nn : \u2115\n\u22a2 \u2191\u2016a\u2016\u208a = \u2191\u2016a\u2016\u208a ^ (\u2191(2 ^ n) * (1 / \u2191(2 ^ n)))"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_3\nA : Type u_1\ninst\u271d\u2074 : NormedRing A\ninst\u271d\u00b3 : NormedAlgebra \u2102 A\ninst\u271d\u00b2 : CompleteSpace A\ninst\u271d\u00b9 : StarRing A\ninst\u271d : CstarRing A\na : A\nha : IsSelfAdjoint a\nhconst : Tendsto (fun _n => \u2191\u2016a\u2016\u208a) atTop (\ud835\udcdd \u2191\u2016a\u2016\u208a)\nn : \u2115\n\u22a2 \u2191\u2016a\u2016\u208a = \u2191\u2016a\u2016\u208a ^ (\u2191(2 ^ n) * (1 / \u2191(2 ^ n)))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean", "full_name": "CategoryTheory.ShortComplex.liftCycles_i", "start": [938, 1], "end": [940, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean", "full_name": "Complex.abs_arg_lt_pi_div_two_iff", "start": [427, 1], "end": [431, 65], "traced_tactics": [{"tactic": "rw [abs_lt, arg_lt_pi_div_two_iff, neg_pi_div_two_lt_arg_iff, \u2190 or_and_left]", "annotated_tactic": ["rw [<a>abs_lt</a>, <a>arg_lt_pi_div_two_iff</a>, <a>neg_pi_div_two_lt_arg_iff</a>, \u2190 <a>or_and_left</a>]", [{"full_name": "abs_lt", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [311, 3], "def_end_pos": [311, 14]}, {"full_name": "Complex.arg_lt_pi_div_two_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean", "def_pos": [411, 7], "def_end_pos": [411, 28]}, {"full_name": "Complex.neg_pi_div_two_lt_arg_iff", "def_path": "Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean", "def_pos": [404, 7], "def_end_pos": [404, 32]}, {"full_name": "or_and_left", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [113, 9], "def_end_pos": [113, 20]}]], "state_before": "a x z\u271d z : \u2102\n\u22a2 |z.arg| < \u03c0 / 2 \u2194 0 < z.re \u2228 z = 0", "state_after": "a x z\u271d z : \u2102\n\u22a2 0 < z.re \u2228 0 \u2264 z.im \u2227 (z.im < 0 \u2228 z = 0) \u2194 0 < z.re \u2228 z = 0"}, {"tactic": "rcases eq_or_ne z 0 with hz | hz", "annotated_tactic": ["rcases <a>eq_or_ne</a> z 0 with hz | hz", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 17]}]], "state_before": "a x z\u271d z : \u2102\n\u22a2 0 < z.re \u2228 0 \u2264 z.im \u2227 (z.im < 0 \u2228 z = 0) \u2194 0 < z.re \u2228 z = 0", "state_after": "case inl\na x z\u271d z : \u2102\nhz : z = 0\n\u22a2 0 < z.re \u2228 0 \u2264 z.im \u2227 (z.im < 0 \u2228 z = 0) \u2194 0 < z.re \u2228 z = 0\n\ncase inr\na x z\u271d z : \u2102\nhz : z \u2260 0\n\u22a2 0 < z.re \u2228 0 \u2264 z.im \u2227 (z.im < 0 \u2228 z = 0) \u2194 0 < z.re \u2228 z = 0"}, {"tactic": "simp [hz]", "annotated_tactic": ["simp [hz]", []], "state_before": "case inl\na x z\u271d z : \u2102\nhz : z = 0\n\u22a2 0 < z.re \u2228 0 \u2264 z.im \u2227 (z.im < 0 \u2228 z = 0) \u2194 0 < z.re \u2228 z = 0", "state_after": "no goals"}, {"tactic": "simp_rw [hz, or_false, \u2190 not_lt, not_and_self_iff, or_false]", "annotated_tactic": ["simp_rw [hz, <a>or_false</a>, \u2190 <a>not_lt</a>, <a>not_and_self_iff</a>, <a>or_false</a>]", [{"full_name": "or_false", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [121, 17], "def_end_pos": [121, 25]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [370, 9], "def_end_pos": [370, 15]}, {"full_name": "not_and_self_iff", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [37, 9], "def_end_pos": [37, 25]}, {"full_name": "or_false", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [121, 17], "def_end_pos": [121, 25]}]], "state_before": "case inr\na x z\u271d z : \u2102\nhz : z \u2260 0\n\u22a2 0 < z.re \u2228 0 \u2264 z.im \u2227 (z.im < 0 \u2228 z = 0) \u2194 0 < z.re \u2228 z = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/ZFC/Basic.lean", "full_name": "ZFSet.toSet_equiv_aux", "start": [1783, 1], "end": [1791, 29], "traced_tactics": [{"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "x : ZFSet\nh : \u2205 \u2209 x\ns : Set ZFSet\nhs : Small.{u, u + 1} \u2191s\n\u22a2 (mk (PSet.mk (Shrink.{u, u + 1} \u2191s) fun x => Quotient.out \u2191((equivShrink \u2191s).symm x))).toSet = s", "state_after": "case h\nx\u271d : ZFSet\nh : \u2205 \u2209 x\u271d\ns : Set ZFSet\nhs : Small.{u, u + 1} \u2191s\nx : ZFSet\n\u22a2 x \u2208 (mk (PSet.mk (Shrink.{u, u + 1} \u2191s) fun x => Quotient.out \u2191((equivShrink \u2191s).symm x))).toSet \u2194 x \u2208 s"}, {"tactic": "rw [mem_toSet, \u2190 mk_out x, mk_mem_iff, mk_out]", "annotated_tactic": ["rw [<a>mem_toSet</a>, \u2190 <a>mk_out</a> x, <a>mk_mem_iff</a>, <a>mk_out</a>]", [{"full_name": "ZFSet.mem_toSet", "def_path": "Mathlib/SetTheory/ZFC/Basic.lean", "def_pos": [690, 9], "def_end_pos": [690, 18]}, {"full_name": "ZFSet.mk_out", "def_path": "Mathlib/SetTheory/ZFC/Basic.lean", "def_pos": [648, 9], "def_end_pos": [648, 15]}, {"full_name": "ZFSet.mk_mem_iff", "def_path": "Mathlib/SetTheory/ZFC/Basic.lean", "def_pos": [680, 9], "def_end_pos": [680, 19]}, {"full_name": "ZFSet.mk_out", "def_path": "Mathlib/SetTheory/ZFC/Basic.lean", "def_pos": [648, 9], "def_end_pos": [648, 15]}]], "state_before": "case h\nx\u271d : ZFSet\nh : \u2205 \u2209 x\u271d\ns : Set ZFSet\nhs : Small.{u, u + 1} \u2191s\nx : ZFSet\n\u22a2 x \u2208 (mk (PSet.mk (Shrink.{u, u + 1} \u2191s) fun x => Quotient.out \u2191((equivShrink \u2191s).symm x))).toSet \u2194 x \u2208 s", "state_after": "case h\nx\u271d : ZFSet\nh : \u2205 \u2209 x\u271d\ns : Set ZFSet\nhs : Small.{u, u + 1} \u2191s\nx : ZFSet\n\u22a2 (Quotient.out x \u2208 PSet.mk (Shrink.{u, u + 1} \u2191s) fun x => Quotient.out \u2191((equivShrink \u2191s).symm x)) \u2194 x \u2208 s"}, {"tactic": "refine \u27e8?_, fun xs \u21a6 \u27e8equivShrink s (Subtype.mk x xs), ?_\u27e9\u27e9", "annotated_tactic": ["refine \u27e8?_, fun xs \u21a6 \u27e8<a>equivShrink</a> s (<a>Subtype.mk</a> x xs), ?_\u27e9\u27e9", [{"full_name": "equivShrink", "def_path": "Mathlib/Logic/Small/Defs.lean", "def_pos": [51, 19], "def_end_pos": [51, 30]}, {"full_name": "Subtype.mk", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [583, 11], "def_end_pos": [583, 18]}]], "state_before": "case h\nx\u271d : ZFSet\nh : \u2205 \u2209 x\u271d\ns : Set ZFSet\nhs : Small.{u, u + 1} \u2191s\nx : ZFSet\n\u22a2 (Quotient.out x \u2208 PSet.mk (Shrink.{u, u + 1} \u2191s) fun x => Quotient.out \u2191((equivShrink \u2191s).symm x)) \u2194 x \u2208 s", "state_after": "case h.refine_1\nx\u271d : ZFSet\nh : \u2205 \u2209 x\u271d\ns : Set ZFSet\nhs : Small.{u, u + 1} \u2191s\nx : ZFSet\n\u22a2 (Quotient.out x \u2208 PSet.mk (Shrink.{u, u + 1} \u2191s) fun x => Quotient.out \u2191((equivShrink \u2191s).symm x)) \u2192 x \u2208 s\n\ncase h.refine_2\nx\u271d : ZFSet\nh : \u2205 \u2209 x\u271d\ns : Set ZFSet\nhs : Small.{u, u + 1} \u2191s\nx : ZFSet\nxs : x \u2208 s\n\u22a2 (Quotient.out x).Equiv\n    ((PSet.mk (Shrink.{u, u + 1} \u2191s) fun x => Quotient.out \u2191((equivShrink \u2191s).symm x)).Func ((equivShrink \u2191s) \u27e8x, xs\u27e9))"}, {"tactic": "rintro \u27e8b, h2\u27e9", "annotated_tactic": ["rintro \u27e8b, h2\u27e9", []], "state_before": "case h.refine_1\nx\u271d : ZFSet\nh : \u2205 \u2209 x\u271d\ns : Set ZFSet\nhs : Small.{u, u + 1} \u2191s\nx : ZFSet\n\u22a2 (Quotient.out x \u2208 PSet.mk (Shrink.{u, u + 1} \u2191s) fun x => Quotient.out \u2191((equivShrink \u2191s).symm x)) \u2192 x \u2208 s", "state_after": "case h.refine_1.intro\nx\u271d : ZFSet\nh : \u2205 \u2209 x\u271d\ns : Set ZFSet\nhs : Small.{u, u + 1} \u2191s\nx : ZFSet\nb : (PSet.mk (Shrink.{u, u + 1} \u2191s) fun x => Quotient.out \u2191((equivShrink \u2191s).symm x)).Type\nh2 : (Quotient.out x).Equiv ((PSet.mk (Shrink.{u, u + 1} \u2191s) fun x => Quotient.out \u2191((equivShrink \u2191s).symm x)).Func b)\n\u22a2 x \u2208 s"}, {"tactic": "rw [\u2190 ZFSet.eq, ZFSet.mk_out] at h2", "annotated_tactic": ["rw [\u2190 <a>ZFSet.eq</a>, <a>ZFSet.mk_out</a>] at h2", [{"full_name": "ZFSet.eq", "def_path": "Mathlib/SetTheory/ZFC/Basic.lean", "def_pos": [652, 9], "def_end_pos": [652, 11]}, {"full_name": "ZFSet.mk_out", "def_path": "Mathlib/SetTheory/ZFC/Basic.lean", "def_pos": [648, 9], "def_end_pos": [648, 15]}]], "state_before": "case h.refine_1.intro\nx\u271d : ZFSet\nh : \u2205 \u2209 x\u271d\ns : Set ZFSet\nhs : Small.{u, u + 1} \u2191s\nx : ZFSet\nb : (PSet.mk (Shrink.{u, u + 1} \u2191s) fun x => Quotient.out \u2191((equivShrink \u2191s).symm x)).Type\nh2 : (Quotient.out x).Equiv ((PSet.mk (Shrink.{u, u + 1} \u2191s) fun x => Quotient.out \u2191((equivShrink \u2191s).symm x)).Func b)\n\u22a2 x \u2208 s", "state_after": "case h.refine_1.intro\nx\u271d : ZFSet\nh : \u2205 \u2209 x\u271d\ns : Set ZFSet\nhs : Small.{u, u + 1} \u2191s\nx : ZFSet\nb : (PSet.mk (Shrink.{u, u + 1} \u2191s) fun x => Quotient.out \u2191((equivShrink \u2191s).symm x)).Type\nh2 : x = mk ((PSet.mk (Shrink.{u, u + 1} \u2191s) fun x => Quotient.out \u2191((equivShrink \u2191s).symm x)).Func b)\n\u22a2 x \u2208 s"}, {"tactic": "simp [h2]", "annotated_tactic": ["simp [h2]", []], "state_before": "case h.refine_1.intro\nx\u271d : ZFSet\nh : \u2205 \u2209 x\u271d\ns : Set ZFSet\nhs : Small.{u, u + 1} \u2191s\nx : ZFSet\nb : (PSet.mk (Shrink.{u, u + 1} \u2191s) fun x => Quotient.out \u2191((equivShrink \u2191s).symm x)).Type\nh2 : x = mk ((PSet.mk (Shrink.{u, u + 1} \u2191s) fun x => Quotient.out \u2191((equivShrink \u2191s).symm x)).Func b)\n\u22a2 x \u2208 s", "state_after": "no goals"}, {"tactic": "simp [PSet.Equiv.refl]", "annotated_tactic": ["simp [<a>PSet.Equiv.refl</a>]", [{"full_name": "PSet.Equiv.refl", "def_path": "Mathlib/SetTheory/ZFC/Basic.lean", "def_pos": [124, 19], "def_end_pos": [124, 29]}]], "state_before": "case h.refine_2\nx\u271d : ZFSet\nh : \u2205 \u2209 x\u271d\ns : Set ZFSet\nhs : Small.{u, u + 1} \u2191s\nx : ZFSet\nxs : x \u2208 s\n\u22a2 (Quotient.out x).Equiv\n    ((PSet.mk (Shrink.{u, u + 1} \u2191s) fun x => Quotient.out \u2191((equivShrink \u2191s).symm x)).Func ((equivShrink \u2191s) \u27e8x, xs\u27e9))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Pointwise.lean", "full_name": "Set.Finite.toFinset_mul", "start": [2462, 1], "end": [2464, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean", "full_name": "mdifferentiableAt_const", "start": [195, 1], "end": [196, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Submodule/Lattice.lean", "full_name": "Submodule.finset_inf_coe", "start": [242, 1], "end": [248, 9], "traced_tactics": [{"tactic": "letI := Classical.decEq \u03b9", "annotated_tactic": ["letI := <a>Classical.decEq</a> \u03b9", [{"full_name": "Classical.decEq", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1020, 19], "def_end_pos": [1020, 24]}]], "state_before": "R : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module S M\ninst\u271d\u00b9 : SMul S R\ninst\u271d : IsScalarTower S R M\np\u271d q : Submodule R M\n\u03b9 : Type u_4\ns : Finset \u03b9\np : \u03b9 \u2192 Submodule R M\n\u22a2 \u2191(s.inf p) = \u22c2 i \u2208 s, \u2191(p i)", "state_after": "R : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module S M\ninst\u271d\u00b9 : SMul S R\ninst\u271d : IsScalarTower S R M\np\u271d q : Submodule R M\n\u03b9 : Type u_4\ns : Finset \u03b9\np : \u03b9 \u2192 Submodule R M\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\n\u22a2 \u2191(s.inf p) = \u22c2 i \u2208 s, \u2191(p i)"}, {"tactic": "refine s.induction_on ?_ fun i s _ ih \u21a6 ?_", "annotated_tactic": ["refine s.induction_on ?_ fun i s _ ih \u21a6 ?_", []], "state_before": "R : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module S M\ninst\u271d\u00b9 : SMul S R\ninst\u271d : IsScalarTower S R M\np\u271d q : Submodule R M\n\u03b9 : Type u_4\ns : Finset \u03b9\np : \u03b9 \u2192 Submodule R M\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\n\u22a2 \u2191(s.inf p) = \u22c2 i \u2208 s, \u2191(p i)", "state_after": "case refine_1\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module S M\ninst\u271d\u00b9 : SMul S R\ninst\u271d : IsScalarTower S R M\np\u271d q : Submodule R M\n\u03b9 : Type u_4\ns : Finset \u03b9\np : \u03b9 \u2192 Submodule R M\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\n\u22a2 \u2191(\u2205.inf p) = \u22c2 i \u2208 \u2205, \u2191(p i)\n\ncase refine_2\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module S M\ninst\u271d\u00b9 : SMul S R\ninst\u271d : IsScalarTower S R M\np\u271d q : Submodule R M\n\u03b9 : Type u_4\ns\u271d : Finset \u03b9\np : \u03b9 \u2192 Submodule R M\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\ns : Finset \u03b9\nx\u271d : i \u2209 s\nih : \u2191(s.inf p) = \u22c2 i \u2208 s, \u2191(p i)\n\u22a2 \u2191((insert i s).inf p) = \u22c2 i_1 \u2208 insert i s, \u2191(p i_1)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case refine_1\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module S M\ninst\u271d\u00b9 : SMul S R\ninst\u271d : IsScalarTower S R M\np\u271d q : Submodule R M\n\u03b9 : Type u_4\ns : Finset \u03b9\np : \u03b9 \u2192 Submodule R M\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\n\u22a2 \u2191(\u2205.inf p) = \u22c2 i \u2208 \u2205, \u2191(p i)", "state_after": "no goals"}, {"tactic": "rw [Finset.inf_insert, inf_coe, ih]", "annotated_tactic": ["rw [<a>Finset.inf_insert</a>, <a>inf_coe</a>, ih]", [{"full_name": "Finset.inf_insert", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [347, 9], "def_end_pos": [347, 19]}, {"full_name": "Submodule.inf_coe", "def_path": "Mathlib/Algebra/Module/Submodule/Lattice.lean", "def_pos": [227, 9], "def_end_pos": [227, 16]}]], "state_before": "case refine_2\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module S M\ninst\u271d\u00b9 : SMul S R\ninst\u271d : IsScalarTower S R M\np\u271d q : Submodule R M\n\u03b9 : Type u_4\ns\u271d : Finset \u03b9\np : \u03b9 \u2192 Submodule R M\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\ns : Finset \u03b9\nx\u271d : i \u2209 s\nih : \u2191(s.inf p) = \u22c2 i \u2208 s, \u2191(p i)\n\u22a2 \u2191((insert i s).inf p) = \u22c2 i_1 \u2208 insert i s, \u2191(p i_1)", "state_after": "case refine_2\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module S M\ninst\u271d\u00b9 : SMul S R\ninst\u271d : IsScalarTower S R M\np\u271d q : Submodule R M\n\u03b9 : Type u_4\ns\u271d : Finset \u03b9\np : \u03b9 \u2192 Submodule R M\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\ns : Finset \u03b9\nx\u271d : i \u2209 s\nih : \u2191(s.inf p) = \u22c2 i \u2208 s, \u2191(p i)\n\u22a2 \u2191(p i) \u2229 \u22c2 i \u2208 s, \u2191(p i) = \u22c2 i_1 \u2208 insert i s, \u2191(p i_1)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case refine_2\nR : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d\u2076 : Semiring R\ninst\u271d\u2075 : Semiring S\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module S M\ninst\u271d\u00b9 : SMul S R\ninst\u271d : IsScalarTower S R M\np\u271d q : Submodule R M\n\u03b9 : Type u_4\ns\u271d : Finset \u03b9\np : \u03b9 \u2192 Submodule R M\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\ni : \u03b9\ns : Finset \u03b9\nx\u271d : i \u2209 s\nih : \u2191(s.inf p) = \u22c2 i \u2208 s, \u2191(p i)\n\u22a2 \u2191(p i) \u2229 \u22c2 i \u2208 s, \u2191(p i) = \u22c2 i_1 \u2208 insert i s, \u2191(p i_1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/AnnihilatingPolynomial.lean", "full_name": "Polynomial.annIdealGenerator_aeval_eq_zero", "start": [134, 1], "end": [135, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "full_name": "PartialHomeomorph.extend_symm_preimage_inter_range_eventuallyEq", "start": [1048, 1], "end": [1052, 56], "traced_tactics": [{"tactic": "rw [\u2190 nhdsWithin_eq_iff_eventuallyEq, \u2190 map_extend_nhdsWithin _ _ hx,\n  map_extend_nhdsWithin_eq_image_of_subset _ _ hx hs]", "annotated_tactic": ["rw [\u2190 <a>nhdsWithin_eq_iff_eventuallyEq</a>, \u2190 <a>map_extend_nhdsWithin</a> _ _ hx,\n    <a>map_extend_nhdsWithin_eq_image_of_subset</a> _ _ hx hs]", [{"full_name": "nhdsWithin_eq_iff_eventuallyEq", "def_path": "Mathlib/Topology/ContinuousOn.lean", "def_pos": [126, 9], "def_end_pos": [126, 39]}, {"full_name": "PartialHomeomorph.map_extend_nhdsWithin", "def_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "def_pos": [958, 9], "def_end_pos": [958, 30]}, {"full_name": "PartialHomeomorph.map_extend_nhdsWithin_eq_image_of_subset", "def_path": "Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean", "def_pos": [953, 9], "def_end_pos": [953, 49]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\ninst\u271d\u2075 : TopologicalSpace H\ninst\u271d\u2074 : TopologicalSpace M\nf f' : PartialHomeomorph M H\nI : ModelWithCorners \ud835\udd5c E H\ninst\u271d\u00b3 : NormedAddCommGroup E'\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E'\ninst\u271d\u00b9 : TopologicalSpace H'\ninst\u271d : TopologicalSpace M'\nI' : ModelWithCorners \ud835\udd5c E' H'\ns\u271d t s : Set M\nx : M\nhs : s \u2286 f.source\nhx : x \u2208 f.source\n\u22a2 \u2191(f.extend I).symm \u207b\u00b9' s \u2229 range \u2191I =\u1da0[\ud835\udcdd (\u2191(f.extend I) x)] \u2191(f.extend I) '' s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/l2Space.lean", "full_name": "OrthogonalFamily.summable_of_lp", "start": [187, 11], "end": [191, 13], "traced_tactics": [{"tactic": "rw [hV.summable_iff_norm_sq_summable]", "annotated_tactic": ["rw [hV.summable_iff_norm_sq_summable]", []], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nf : \u21a5(lp G 2)\n\u22a2 Summable fun i => (V i) (\u2191f i)", "state_after": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nf : \u21a5(lp G 2)\n\u22a2 Summable fun i => \u2016\u2191f i\u2016 ^ 2"}, {"tactic": "convert (lp.mem\u2113p f).summable _", "annotated_tactic": ["convert (<a>lp.mem\u2113p</a> f).<a>summable</a> _", [{"full_name": "lp.mem\u2113p", "def_path": "Mathlib/Analysis/NormedSpace/lpSpace.lean", "def_pos": [353, 19], "def_end_pos": [353, 24]}, {"full_name": "Mem\u2113p.summable", "def_path": "Mathlib/Analysis/NormedSpace/lpSpace.lean", "def_pos": [154, 9], "def_end_pos": [154, 17]}]], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nf : \u21a5(lp G 2)\n\u22a2 Summable fun i => \u2016\u2191f i\u2016 ^ 2", "state_after": "case h.e'_5.h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nf : \u21a5(lp G 2)\nx\u271d : \u03b9\n\u22a2 \u2016\u2191f x\u271d\u2016 ^ 2 = \u2016\u2191f x\u271d\u2016 ^ ENNReal.toReal 2\n\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nf : \u21a5(lp G 2)\n\u22a2 0 < ENNReal.toReal 2"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "case h.e'_5.h\n\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nf : \u21a5(lp G 2)\nx\u271d : \u03b9\n\u22a2 \u2016\u2191f x\u271d\u2016 ^ 2 = \u2016\u2191f x\u271d\u2016 ^ ENNReal.toReal 2", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "\u03b9 : Type u_1\n\ud835\udd5c : Type u_2\ninst\u271d\u2074 : RCLike \ud835\udd5c\nE : Type u_3\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E\ncplt : CompleteSpace E\nG : \u03b9 \u2192 Type u_4\ninst\u271d\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (G i)\ninst\u271d : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i)\nV : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E\nhV : OrthogonalFamily \ud835\udd5c G V\nf : \u21a5(lp G 2)\n\u22a2 0 < ENNReal.toReal 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Nth.lean", "full_name": "Nat.nth_of_card_le", "start": [62, 1], "end": [63, 91], "traced_tactics": [{"tactic": "rw [nth, dif_pos hf, List.getD_eq_default]", "annotated_tactic": ["rw [<a>nth</a>, <a>dif_pos</a> hf, <a>List.getD_eq_default</a>]", [{"full_name": "Nat.nth", "def_path": "Mathlib/Data/Nat/Nth.lean", "def_pos": [49, 19], "def_end_pos": [49, 22]}, {"full_name": "dif_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [949, 9], "def_end_pos": [949, 16]}, {"full_name": "List.getD_eq_default", "def_path": "Mathlib/Data/List/GetD.lean", "def_pos": [50, 9], "def_end_pos": [50, 24]}]], "state_before": "p : \u2115 \u2192 Prop\nhf : (setOf p).Finite\nn : \u2115\nhn : hf.toFinset.card \u2264 n\n\u22a2 nth p n = 0", "state_after": "case hn\np : \u2115 \u2192 Prop\nhf : (setOf p).Finite\nn : \u2115\nhn : hf.toFinset.card \u2264 n\n\u22a2 (sort (fun x x_1 => x \u2264 x_1) hf.toFinset).length \u2264 n"}, {"tactic": "rwa [Finset.length_sort]", "annotated_tactic": ["rwa [<a>Finset.length_sort</a>]", [{"full_name": "Finset.length_sort", "def_path": "Mathlib/Data/Finset/Sort.lean", "def_pos": [64, 9], "def_end_pos": [64, 20]}]], "state_before": "case hn\np : \u2115 \u2192 Prop\nhf : (setOf p).Finite\nn : \u2115\nhn : hf.toFinset.card \u2264 n\n\u22a2 (sort (fun x x_1 => x \u2264 x_1) hf.toFinset).length \u2264 n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/SpecificGroups/KleinFour.lean", "full_name": "IsKleinFour.not_isCyclic", "start": [92, 1], "end": [94, 77], "traced_tactics": [{"tactic": "let _inst := Fintype.ofFinite G", "annotated_tactic": ["let _inst := <a>Fintype.ofFinite</a> G", [{"full_name": "Fintype.ofFinite", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [449, 19], "def_end_pos": [449, 35]}]], "state_before": "G : Type u_1\ninst\u271d\u00b9 : Group G\ninst\u271d : IsKleinFour G\nh : IsCyclic G\n\u22a2 False", "state_after": "G : Type u_1\ninst\u271d\u00b9 : Group G\ninst\u271d : IsKleinFour G\nh : IsCyclic G\n_inst : Fintype G := Fintype.ofFinite G\n\u22a2 False"}, {"tactic": "simpa using h.exponent_eq_card", "annotated_tactic": ["simpa using h.exponent_eq_card", []], "state_before": "G : Type u_1\ninst\u271d\u00b9 : Group G\ninst\u271d : IsKleinFour G\nh : IsCyclic G\n_inst : Fintype G := Fintype.ofFinite G\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean", "full_name": "Real.two_mul_arctan_inv_3_add_arctan_inv_7", "start": [296, 1], "end": [297, 47], "traced_tactics": [{"tactic": "rw [two_mul_arctan, arctan_add] <;> norm_num", "annotated_tactic": ["rw [<a>two_mul_arctan</a>, <a>arctan_add</a>] <;> norm_num", [{"full_name": "Real.two_mul_arctan", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean", "def_pos": [278, 9], "def_end_pos": [278, 23]}, {"full_name": "Real.arctan_add", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean", "def_pos": [247, 9], "def_end_pos": [247, 19]}]], "state_before": "\u22a2 2 * arctan 3\u207b\u00b9 + arctan 7\u207b\u00b9 = \u03c0 / 4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "full_name": "MeasureTheory.ae_restrict_neBot", "start": [739, 1], "end": [740, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/QuotientGroup.lean", "full_name": "QuotientGroup.quotientQuotientEquivQuotientAux_mk_mk", "start": [682, 1], "end": [684, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Complex.lean", "full_name": "convex_halfspace_im_ge", "start": [47, 1], "end": [48, 72], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Coeff.lean", "full_name": "Polynomial.support_smul", "start": [57, 1], "end": [62, 12], "traced_tactics": [{"tactic": "intro i hi", "annotated_tactic": ["intro i hi", []], "state_before": "R : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r\u271d : R[X]\ninst\u271d : SMulZeroClass S R\nr : S\np : R[X]\n\u22a2 (r \u2022 p).support \u2286 p.support", "state_after": "R : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r\u271d : R[X]\ninst\u271d : SMulZeroClass S R\nr : S\np : R[X]\ni : \u2115\nhi : i \u2208 (r \u2022 p).support\n\u22a2 i \u2208 p.support"}, {"tactic": "simp? [mem_support_iff] at hi \u22a2 says simp only [mem_support_iff, coeff_smul, ne_eq] at hi \u22a2", "annotated_tactic": ["simp? [mem_support_iff] at hi \u22a2 says simp only [<a>mem_support_iff</a>, <a>coeff_smul</a>, <a>ne_eq</a>] at hi \u22a2", [{"full_name": "Polynomial.mem_support_iff", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [726, 9], "def_end_pos": [726, 24]}, {"full_name": "Polynomial.coeff_smul", "def_path": "Mathlib/Algebra/Polynomial/Coeff.lean", "def_pos": [50, 9], "def_end_pos": [50, 19]}, {"full_name": "ne_eq", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [89, 17], "def_end_pos": [89, 22]}]], "state_before": "R : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r\u271d : R[X]\ninst\u271d : SMulZeroClass S R\nr : S\np : R[X]\ni : \u2115\nhi : i \u2208 (r \u2022 p).support\n\u22a2 i \u2208 p.support", "state_after": "R : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r\u271d : R[X]\ninst\u271d : SMulZeroClass S R\nr : S\np : R[X]\ni : \u2115\nhi : \u00acr \u2022 p.coeff i = 0\n\u22a2 \u00acp.coeff i = 0"}, {"tactic": "contrapose! hi", "annotated_tactic": ["contrapose! hi", []], "state_before": "R : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r\u271d : R[X]\ninst\u271d : SMulZeroClass S R\nr : S\np : R[X]\ni : \u2115\nhi : \u00acr \u2022 p.coeff i = 0\n\u22a2 \u00acp.coeff i = 0", "state_after": "R : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r\u271d : R[X]\ninst\u271d : SMulZeroClass S R\nr : S\np : R[X]\ni : \u2115\nhi : p.coeff i = 0\n\u22a2 r \u2022 p.coeff i = 0"}, {"tactic": "simp [hi]", "annotated_tactic": ["simp [hi]", []], "state_before": "R : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r\u271d : R[X]\ninst\u271d : SMulZeroClass S R\nr : S\np : R[X]\ni : \u2115\nhi : p.coeff i = 0\n\u22a2 r \u2022 p.coeff i = 0", "state_after": "no goals"}, {"tactic": "simp only [mem_support_iff, coeff_smul, ne_eq] at hi \u22a2", "annotated_tactic": ["simp only [<a>mem_support_iff</a>, <a>coeff_smul</a>, <a>ne_eq</a>] at hi \u22a2", [{"full_name": "Polynomial.mem_support_iff", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [726, 9], "def_end_pos": [726, 24]}, {"full_name": "Polynomial.coeff_smul", "def_path": "Mathlib/Algebra/Polynomial/Coeff.lean", "def_pos": [50, 9], "def_end_pos": [50, 19]}, {"full_name": "ne_eq", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [89, 17], "def_end_pos": [89, 22]}]], "state_before": "R : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r\u271d : R[X]\ninst\u271d : SMulZeroClass S R\nr : S\np : R[X]\ni : \u2115\nhi : i \u2208 (r \u2022 p).support\n\u22a2 i \u2208 p.support", "state_after": "R : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d\u00b9 : Semiring R\np\u271d q r\u271d : R[X]\ninst\u271d : SMulZeroClass S R\nr : S\np : R[X]\ni : \u2115\nhi : \u00acr \u2022 p.coeff i = 0\n\u22a2 \u00acp.coeff i = 0"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Euclidean/Angle/Sphere.lean", "full_name": "EuclideanGeometry.Sphere.two_zsmul_oangle_eq", "start": [93, 1], "end": [100, 38], "traced_tactics": [{"tactic": "rw [mem_sphere, @dist_eq_norm_vsub V] at hp\u2081 hp\u2082 hp\u2083 hp\u2084", "annotated_tactic": ["rw [<a>mem_sphere</a>, @<a>dist_eq_norm_vsub</a> V] at hp\u2081 hp\u2082 hp\u2083 hp\u2084", [{"full_name": "EuclideanGeometry.mem_sphere", "def_path": "Mathlib/Geometry/Euclidean/Sphere/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 19]}, {"full_name": "dist_eq_norm_vsub", "def_path": "Mathlib/Analysis/Normed/Group/AddTorsor.lean", "def_pos": [75, 9], "def_end_pos": [75, 26]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : p\u2081 \u2208 s\nhp\u2082 : p\u2082 \u2208 s\nhp\u2083 : p\u2083 \u2208 s\nhp\u2084 : p\u2084 \u2208 s\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2084 : p\u2082 \u2260 p\u2084\nhp\u2083p\u2081 : p\u2083 \u2260 p\u2081\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : \u2016p\u2081 -\u1d65 s.center\u2016 = s.radius\nhp\u2082 : \u2016p\u2082 -\u1d65 s.center\u2016 = s.radius\nhp\u2083 : \u2016p\u2083 -\u1d65 s.center\u2016 = s.radius\nhp\u2084 : \u2016p\u2084 -\u1d65 s.center\u2016 = s.radius\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2084 : p\u2082 \u2260 p\u2084\nhp\u2083p\u2081 : p\u2083 \u2260 p\u2081\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084"}, {"tactic": "rw [oangle, oangle, \u2190 vsub_sub_vsub_cancel_right p\u2081 p\u2082 s.center, \u2190\n    vsub_sub_vsub_cancel_right p\u2084 p\u2082 s.center,\n    o.two_zsmul_oangle_sub_eq_two_zsmul_oangle_sub_of_norm_eq _ _ _ _ hp\u2082 hp\u2083 hp\u2081 hp\u2084] <;>\n  simp [hp\u2082p\u2081, hp\u2082p\u2084, hp\u2083p\u2081, hp\u2083p\u2084]", "annotated_tactic": ["rw [<a>oangle</a>, <a>oangle</a>, \u2190 <a>vsub_sub_vsub_cancel_right</a> p\u2081 p\u2082 s.center, \u2190\n      <a>vsub_sub_vsub_cancel_right</a> p\u2084 p\u2082 s.center,\n      o.two_zsmul_oangle_sub_eq_two_zsmul_oangle_sub_of_norm_eq _ _ _ _ hp\u2082 hp\u2083 hp\u2081 hp\u2084] <;>\n    simp [hp\u2082p\u2081, hp\u2082p\u2084, hp\u2083p\u2081, hp\u2083p\u2084]", [{"full_name": "EuclideanGeometry.oangle", "def_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean", "def_pos": [42, 5], "def_end_pos": [42, 11]}, {"full_name": "EuclideanGeometry.oangle", "def_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean", "def_pos": [42, 5], "def_end_pos": [42, 11]}, {"full_name": "vsub_sub_vsub_cancel_right", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [172, 9], "def_end_pos": [172, 35]}, {"full_name": "vsub_sub_vsub_cancel_right", "def_path": "Mathlib/Algebra/AddTorsor.lean", "def_pos": [172, 9], "def_end_pos": [172, 35]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\ns : Sphere P\np\u2081 p\u2082 p\u2083 p\u2084 : P\nhp\u2081 : \u2016p\u2081 -\u1d65 s.center\u2016 = s.radius\nhp\u2082 : \u2016p\u2082 -\u1d65 s.center\u2016 = s.radius\nhp\u2083 : \u2016p\u2083 -\u1d65 s.center\u2016 = s.radius\nhp\u2084 : \u2016p\u2084 -\u1d65 s.center\u2016 = s.radius\nhp\u2082p\u2081 : p\u2082 \u2260 p\u2081\nhp\u2082p\u2084 : p\u2082 \u2260 p\u2084\nhp\u2083p\u2081 : p\u2083 \u2260 p\u2081\nhp\u2083p\u2084 : p\u2083 \u2260 p\u2084\n\u22a2 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Compactness/Lindelof.lean", "full_name": "IsLindelof.compl_mem_sets", "start": [52, 1], "end": [56, 24], "traced_tactics": [{"tactic": "contrapose! hf", "annotated_tactic": ["contrapose! hf", []], "state_before": "X : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ns t : Set X\nhs : IsLindelof s\nf : Filter X\ninst\u271d : CountableInterFilter f\nhf : \u2200 x \u2208 s, s\u1d9c \u2208 \ud835\udcdd x \u2293 f\n\u22a2 s\u1d9c \u2208 f", "state_after": "X : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ns t : Set X\nhs : IsLindelof s\nf : Filter X\ninst\u271d : CountableInterFilter f\nhf : s\u1d9c \u2209 f\n\u22a2 \u2203 x \u2208 s, s\u1d9c \u2209 \ud835\udcdd x \u2293 f"}, {"tactic": "simp only [not_mem_iff_inf_principal_compl, compl_compl, inf_assoc] at hf \u22a2", "annotated_tactic": ["simp only [<a>not_mem_iff_inf_principal_compl</a>, <a>compl_compl</a>, <a>inf_assoc</a>] at hf \u22a2", [{"full_name": "Filter.not_mem_iff_inf_principal_compl", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [700, 9], "def_end_pos": [700, 40]}, {"full_name": "compl_compl", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [655, 9], "def_end_pos": [655, 20]}, {"full_name": "inf_assoc", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [466, 9], "def_end_pos": [466, 18]}]], "state_before": "X : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ns t : Set X\nhs : IsLindelof s\nf : Filter X\ninst\u271d : CountableInterFilter f\nhf : s\u1d9c \u2209 f\n\u22a2 \u2203 x \u2208 s, s\u1d9c \u2209 \ud835\udcdd x \u2293 f", "state_after": "X : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ns t : Set X\nhs : IsLindelof s\nf : Filter X\ninst\u271d : CountableInterFilter f\nhf : (f \u2293 \ud835\udcdf s).NeBot\n\u22a2 \u2203 x \u2208 s, (\ud835\udcdd x \u2293 (f \u2293 \ud835\udcdf s)).NeBot"}, {"tactic": "exact hs inf_le_right", "annotated_tactic": ["exact hs <a>inf_le_right</a>", [{"full_name": "inf_le_right", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [366, 9], "def_end_pos": [366, 21]}]], "state_before": "X : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ns t : Set X\nhs : IsLindelof s\nf : Filter X\ninst\u271d : CountableInterFilter f\nhf : (f \u2293 \ud835\udcdf s).NeBot\n\u22a2 \u2203 x \u2208 s, (\ud835\udcdd x \u2293 (f \u2293 \ud835\udcdf s)).NeBot", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Category/ModuleCat/Presheaf.lean", "full_name": "PresheafOfModules.unit_map_one", "start": [443, 1], "end": [444, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/UpperLower/Basic.lean", "full_name": "upperClosure_anti", "start": [1520, 1], "end": [1521, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Euclidean/MongePoint.lean", "full_name": "Affine.Triangle.orthocenter_eq_of_range_eq", "start": [464, 1], "end": [466, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Lattice.lean", "full_name": "ofDual_sup", "start": [910, 1], "end": [911, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Control/Fold.lean", "full_name": "Traversable.foldr_map", "start": [384, 1], "end": [385, 96], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/List/Perm.lean", "full_name": "List.subperm_cons_erase", "start": [463, 1], "end": [467, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Intervals.lean", "full_name": "Finset.prod_range_mul_prod_Ico", "start": [155, 1], "end": [157, 85], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean", "full_name": "CategoryTheory.ShortComplex.LeftHomologyMapData.quasiIso_iff", "start": [97, 1], "end": [107, 19], "traced_tactics": [{"tactic": "rw [ShortComplex.quasiIso_iff, \u03b3.homologyMap_eq]", "annotated_tactic": ["rw [<a>ShortComplex.quasiIso_iff</a>, \u03b3.homologyMap_eq]", [{"full_name": "CategoryTheory.ShortComplex.quasiIso_iff", "def_path": "Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean", "def_pos": [36, 7], "def_end_pos": [36, 19]}]], "state_before": "C : Type u_2\ninst\u271d\u2075 : Category.{u_1, u_2} C\ninst\u271d\u2074 : HasZeroMorphisms C\nS\u2081 S\u2082 S\u2083 S\u2084 : ShortComplex C\ninst\u271d\u00b3 : S\u2081.HasHomology\ninst\u271d\u00b2 : S\u2082.HasHomology\ninst\u271d\u00b9 : S\u2083.HasHomology\ninst\u271d : S\u2084.HasHomology\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.LeftHomologyData\nh\u2082 : S\u2082.LeftHomologyData\n\u03b3 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\n\u22a2 QuasiIso \u03c6 \u2194 IsIso \u03b3.\u03c6H", "state_after": "C : Type u_2\ninst\u271d\u2075 : Category.{u_1, u_2} C\ninst\u271d\u2074 : HasZeroMorphisms C\nS\u2081 S\u2082 S\u2083 S\u2084 : ShortComplex C\ninst\u271d\u00b3 : S\u2081.HasHomology\ninst\u271d\u00b2 : S\u2082.HasHomology\ninst\u271d\u00b9 : S\u2083.HasHomology\ninst\u271d : S\u2084.HasHomology\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.LeftHomologyData\nh\u2082 : S\u2082.LeftHomologyData\n\u03b3 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\n\u22a2 IsIso (h\u2081.homologyIso.hom \u226b \u03b3.\u03c6H \u226b h\u2082.homologyIso.inv) \u2194 IsIso \u03b3.\u03c6H"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "C : Type u_2\ninst\u271d\u2075 : Category.{u_1, u_2} C\ninst\u271d\u2074 : HasZeroMorphisms C\nS\u2081 S\u2082 S\u2083 S\u2084 : ShortComplex C\ninst\u271d\u00b3 : S\u2081.HasHomology\ninst\u271d\u00b2 : S\u2082.HasHomology\ninst\u271d\u00b9 : S\u2083.HasHomology\ninst\u271d : S\u2084.HasHomology\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.LeftHomologyData\nh\u2082 : S\u2082.LeftHomologyData\n\u03b3 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\n\u22a2 IsIso (h\u2081.homologyIso.hom \u226b \u03b3.\u03c6H \u226b h\u2082.homologyIso.inv) \u2194 IsIso \u03b3.\u03c6H", "state_after": "case mp\nC : Type u_2\ninst\u271d\u2075 : Category.{u_1, u_2} C\ninst\u271d\u2074 : HasZeroMorphisms C\nS\u2081 S\u2082 S\u2083 S\u2084 : ShortComplex C\ninst\u271d\u00b3 : S\u2081.HasHomology\ninst\u271d\u00b2 : S\u2082.HasHomology\ninst\u271d\u00b9 : S\u2083.HasHomology\ninst\u271d : S\u2084.HasHomology\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.LeftHomologyData\nh\u2082 : S\u2082.LeftHomologyData\n\u03b3 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\n\u22a2 IsIso (h\u2081.homologyIso.hom \u226b \u03b3.\u03c6H \u226b h\u2082.homologyIso.inv) \u2192 IsIso \u03b3.\u03c6H\n\ncase mpr\nC : Type u_2\ninst\u271d\u2075 : Category.{u_1, u_2} C\ninst\u271d\u2074 : HasZeroMorphisms C\nS\u2081 S\u2082 S\u2083 S\u2084 : ShortComplex C\ninst\u271d\u00b3 : S\u2081.HasHomology\ninst\u271d\u00b2 : S\u2082.HasHomology\ninst\u271d\u00b9 : S\u2083.HasHomology\ninst\u271d : S\u2084.HasHomology\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.LeftHomologyData\nh\u2082 : S\u2082.LeftHomologyData\n\u03b3 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\n\u22a2 IsIso \u03b3.\u03c6H \u2192 IsIso (h\u2081.homologyIso.hom \u226b \u03b3.\u03c6H \u226b h\u2082.homologyIso.inv)"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mp\nC : Type u_2\ninst\u271d\u2075 : Category.{u_1, u_2} C\ninst\u271d\u2074 : HasZeroMorphisms C\nS\u2081 S\u2082 S\u2083 S\u2084 : ShortComplex C\ninst\u271d\u00b3 : S\u2081.HasHomology\ninst\u271d\u00b2 : S\u2082.HasHomology\ninst\u271d\u00b9 : S\u2083.HasHomology\ninst\u271d : S\u2084.HasHomology\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.LeftHomologyData\nh\u2082 : S\u2082.LeftHomologyData\n\u03b3 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\n\u22a2 IsIso (h\u2081.homologyIso.hom \u226b \u03b3.\u03c6H \u226b h\u2082.homologyIso.inv) \u2192 IsIso \u03b3.\u03c6H", "state_after": "case mp\nC : Type u_2\ninst\u271d\u2075 : Category.{u_1, u_2} C\ninst\u271d\u2074 : HasZeroMorphisms C\nS\u2081 S\u2082 S\u2083 S\u2084 : ShortComplex C\ninst\u271d\u00b3 : S\u2081.HasHomology\ninst\u271d\u00b2 : S\u2082.HasHomology\ninst\u271d\u00b9 : S\u2083.HasHomology\ninst\u271d : S\u2084.HasHomology\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.LeftHomologyData\nh\u2082 : S\u2082.LeftHomologyData\n\u03b3 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\nh : IsIso (h\u2081.homologyIso.hom \u226b \u03b3.\u03c6H \u226b h\u2082.homologyIso.inv)\n\u22a2 IsIso \u03b3.\u03c6H"}, {"tactic": "haveI : IsIso (\u03b3.\u03c6H \u226b (LeftHomologyData.homologyIso h\u2082).inv) :=\n  IsIso.of_isIso_comp_left (LeftHomologyData.homologyIso h\u2081).hom _", "annotated_tactic": ["haveI : <a>IsIso</a> (\u03b3.\u03c6H \u226b (<a>LeftHomologyData.homologyIso</a> h\u2082).<a>inv</a>) :=\n      <a>IsIso.of_isIso_comp_left</a> (<a>LeftHomologyData.homologyIso</a> h\u2081).<a>hom</a> _", [{"full_name": "CategoryTheory.IsIso", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [272, 7], "def_end_pos": [272, 12]}, {"full_name": "CategoryTheory.ShortComplex.LeftHomologyData.homologyIso", "def_path": "Mathlib/Algebra/Homology/ShortComplex/Homology.lean", "def_pos": [387, 19], "def_end_pos": [387, 47]}, {"full_name": "CategoryTheory.Iso.inv", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [55, 3], "def_end_pos": [55, 6]}, {"full_name": "CategoryTheory.IsIso.of_isIso_comp_left", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [460, 9], "def_end_pos": [460, 27]}, {"full_name": "CategoryTheory.ShortComplex.LeftHomologyData.homologyIso", "def_path": "Mathlib/Algebra/Homology/ShortComplex/Homology.lean", "def_pos": [387, 19], "def_end_pos": [387, 47]}, {"full_name": "CategoryTheory.Iso.hom", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [53, 3], "def_end_pos": [53, 6]}]], "state_before": "case mp\nC : Type u_2\ninst\u271d\u2075 : Category.{u_1, u_2} C\ninst\u271d\u2074 : HasZeroMorphisms C\nS\u2081 S\u2082 S\u2083 S\u2084 : ShortComplex C\ninst\u271d\u00b3 : S\u2081.HasHomology\ninst\u271d\u00b2 : S\u2082.HasHomology\ninst\u271d\u00b9 : S\u2083.HasHomology\ninst\u271d : S\u2084.HasHomology\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.LeftHomologyData\nh\u2082 : S\u2082.LeftHomologyData\n\u03b3 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\nh : IsIso (h\u2081.homologyIso.hom \u226b \u03b3.\u03c6H \u226b h\u2082.homologyIso.inv)\n\u22a2 IsIso \u03b3.\u03c6H", "state_after": "case mp\nC : Type u_2\ninst\u271d\u2075 : Category.{u_1, u_2} C\ninst\u271d\u2074 : HasZeroMorphisms C\nS\u2081 S\u2082 S\u2083 S\u2084 : ShortComplex C\ninst\u271d\u00b3 : S\u2081.HasHomology\ninst\u271d\u00b2 : S\u2082.HasHomology\ninst\u271d\u00b9 : S\u2083.HasHomology\ninst\u271d : S\u2084.HasHomology\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.LeftHomologyData\nh\u2082 : S\u2082.LeftHomologyData\n\u03b3 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\nh : IsIso (h\u2081.homologyIso.hom \u226b \u03b3.\u03c6H \u226b h\u2082.homologyIso.inv)\nthis : IsIso (\u03b3.\u03c6H \u226b h\u2082.homologyIso.inv)\n\u22a2 IsIso \u03b3.\u03c6H"}, {"tactic": "exact IsIso.of_isIso_comp_right _ (LeftHomologyData.homologyIso h\u2082).inv", "annotated_tactic": ["exact <a>IsIso.of_isIso_comp_right</a> _ (<a>LeftHomologyData.homologyIso</a> h\u2082).<a>inv</a>", [{"full_name": "CategoryTheory.IsIso.of_isIso_comp_right", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [466, 9], "def_end_pos": [466, 28]}, {"full_name": "CategoryTheory.ShortComplex.LeftHomologyData.homologyIso", "def_path": "Mathlib/Algebra/Homology/ShortComplex/Homology.lean", "def_pos": [387, 19], "def_end_pos": [387, 47]}, {"full_name": "CategoryTheory.Iso.inv", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [55, 3], "def_end_pos": [55, 6]}]], "state_before": "case mp\nC : Type u_2\ninst\u271d\u2075 : Category.{u_1, u_2} C\ninst\u271d\u2074 : HasZeroMorphisms C\nS\u2081 S\u2082 S\u2083 S\u2084 : ShortComplex C\ninst\u271d\u00b3 : S\u2081.HasHomology\ninst\u271d\u00b2 : S\u2082.HasHomology\ninst\u271d\u00b9 : S\u2083.HasHomology\ninst\u271d : S\u2084.HasHomology\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.LeftHomologyData\nh\u2082 : S\u2082.LeftHomologyData\n\u03b3 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\nh : IsIso (h\u2081.homologyIso.hom \u226b \u03b3.\u03c6H \u226b h\u2082.homologyIso.inv)\nthis : IsIso (\u03b3.\u03c6H \u226b h\u2082.homologyIso.inv)\n\u22a2 IsIso \u03b3.\u03c6H", "state_after": "no goals"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mpr\nC : Type u_2\ninst\u271d\u2075 : Category.{u_1, u_2} C\ninst\u271d\u2074 : HasZeroMorphisms C\nS\u2081 S\u2082 S\u2083 S\u2084 : ShortComplex C\ninst\u271d\u00b3 : S\u2081.HasHomology\ninst\u271d\u00b2 : S\u2082.HasHomology\ninst\u271d\u00b9 : S\u2083.HasHomology\ninst\u271d : S\u2084.HasHomology\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.LeftHomologyData\nh\u2082 : S\u2082.LeftHomologyData\n\u03b3 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\n\u22a2 IsIso \u03b3.\u03c6H \u2192 IsIso (h\u2081.homologyIso.hom \u226b \u03b3.\u03c6H \u226b h\u2082.homologyIso.inv)", "state_after": "case mpr\nC : Type u_2\ninst\u271d\u2075 : Category.{u_1, u_2} C\ninst\u271d\u2074 : HasZeroMorphisms C\nS\u2081 S\u2082 S\u2083 S\u2084 : ShortComplex C\ninst\u271d\u00b3 : S\u2081.HasHomology\ninst\u271d\u00b2 : S\u2082.HasHomology\ninst\u271d\u00b9 : S\u2083.HasHomology\ninst\u271d : S\u2084.HasHomology\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.LeftHomologyData\nh\u2082 : S\u2082.LeftHomologyData\n\u03b3 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\nh : IsIso \u03b3.\u03c6H\n\u22a2 IsIso (h\u2081.homologyIso.hom \u226b \u03b3.\u03c6H \u226b h\u2082.homologyIso.inv)"}, {"tactic": "infer_instance", "annotated_tactic": ["infer_instance", []], "state_before": "case mpr\nC : Type u_2\ninst\u271d\u2075 : Category.{u_1, u_2} C\ninst\u271d\u2074 : HasZeroMorphisms C\nS\u2081 S\u2082 S\u2083 S\u2084 : ShortComplex C\ninst\u271d\u00b3 : S\u2081.HasHomology\ninst\u271d\u00b2 : S\u2082.HasHomology\ninst\u271d\u00b9 : S\u2083.HasHomology\ninst\u271d : S\u2084.HasHomology\n\u03c6 : S\u2081 \u27f6 S\u2082\nh\u2081 : S\u2081.LeftHomologyData\nh\u2082 : S\u2082.LeftHomologyData\n\u03b3 : LeftHomologyMapData \u03c6 h\u2081 h\u2082\nh : IsIso \u03b3.\u03c6H\n\u22a2 IsIso (h\u2081.homologyIso.hom \u226b \u03b3.\u03c6H \u226b h\u2082.homologyIso.inv)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/TuringMachine.lean", "full_name": "Turing.TM1to1.stepAux_read", "start": [1825, 1], "end": [1853, 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": [1295, 5], "def_end_pos": [1295, 12]}, {"full_name": "Turing.TM1to1.readAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1664, 5], "def_end_pos": [1664, 12]}, {"full_name": "Turing.TM1to1.trTape'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1741, 5], "def_end_pos": [1741, 12]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1295, 5], "def_end_pos": [1295, 12]}, {"full_name": "Turing.TM1to1.trTape'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1741, 5], "def_end_pos": [1741, 12]}, {"full_name": "Turing.TM1to1.read", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1682, 5], "def_end_pos": [1682, 9]}, {"full_name": "Turing.TM1to1.stepAux_move", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1704, 9], "def_end_pos": [1704, 21]}, {"full_name": "Turing.TM1to1.trTape'_move_left", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1774, 9], "def_end_pos": [1774, 26]}, {"full_name": "Turing.ListBlank.head_cons", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [261, 9], "def_end_pos": [261, 28]}, {"full_name": "Turing.ListBlank.cons_head_tail", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [273, 9], "def_end_pos": [273, 33]}, {"full_name": "Turing.ListBlank.tail_cons", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [266, 9], "def_end_pos": [266, 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 R.head) 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 R.head)) v (trTape' enc0 (ListBlank.cons R.head L) R.tail)"}, {"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 R.head)) v (trTape' enc0 (ListBlank.cons R.head L) R.tail)", "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.cons a R).head)) v\n        (trTape' enc0 (ListBlank.cons (ListBlank.cons a R).head L) (ListBlank.cons a R).tail)"}, {"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": [261, 9], "def_end_pos": [261, 28]}, {"full_name": "Turing.ListBlank.tail_cons", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [266, 9], "def_end_pos": [266, 28]}, {"full_name": "Turing.TM1to1.trTape'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1741, 5], "def_end_pos": [1741, 12]}, {"full_name": "Turing.ListBlank.cons_bind", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [496, 9], "def_end_pos": [496, 28]}, {"full_name": "Turing.ListBlank.append_assoc", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [469, 9], "def_end_pos": [469, 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.cons a R).head)) v\n        (trTape' enc0 (ListBlank.cons (ListBlank.cons a R).head L) (ListBlank.cons a R).tail)", "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' (L.bind (fun x => (enc x).toList.reverse) \u22ef)\n          (ListBlank.append (enc a).toList (R.bind (fun x => (enc x).toList) \u22ef))) =\n      stepAux (f (enc a)) v\n        (Tape.mk' (ListBlank.append (enc a).toList.reverse (L.bind (fun x => (enc x).toList.reverse) \u22ef))\n          (R.bind (fun x => (enc x).toList) \u22ef))"}, {"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": [1295, 5], "def_end_pos": [1295, 12]}, {"full_name": "Turing.TM1to1.readAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1664, 5], "def_end_pos": [1664, 12]}, {"full_name": "Turing.Tape.mk'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [559, 5], "def_end_pos": [559, 13]}, {"full_name": "Turing.ListBlank.append", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [457, 5], "def_end_pos": [457, 21]}, {"full_name": "Turing.ListBlank.append", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [457, 5], "def_end_pos": [457, 21]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1295, 5], "def_end_pos": [1295, 12]}, {"full_name": "Turing.Tape.mk'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [559, 5], "def_end_pos": [559, 13]}, {"full_name": "Turing.ListBlank.append", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [457, 5], "def_end_pos": [457, 21]}, {"full_name": "List.reverse", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [441, 5], "def_end_pos": [441, 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' (L.bind (fun x => (enc x).toList.reverse) \u22ef)\n          (ListBlank.append (enc a).toList (R.bind (fun x => (enc x).toList) \u22ef))) =\n      stepAux (f (enc a)) v\n        (Tape.mk' (ListBlank.append (enc a).toList.reverse (L.bind (fun x => (enc x).toList.reverse) \u22ef))\n          (R.bind (fun x => (enc x).toList) \u22ef))", "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 : l\u2082.length = i),\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')"}, {"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 : l\u2082.length = i),\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')", "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 : l\u2082.length = i),\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')"}, {"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 : l\u2082.length = i),\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')", "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 : l\u2082.length = i\n\u22a2 stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n    stepAux (f \u27e8l\u2082, h\u271d\u27e9) v (Tape.mk' (ListBlank.append (l\u2082.reverseAux 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 : l\u2082.length = i\n\u22a2 stepAux (readAux i f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n    stepAux (f \u27e8l\u2082, h\u271d\u27e9) v (Tape.mk' (ListBlank.append (l\u2082.reverseAux 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 l\u2082.length \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux l\u2082.length f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n    stepAux (f \u27e8l\u2082, \u22ef\u27e9) v (Tape.mk' (ListBlank.append (l\u2082.reverseAux 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 l\u2082.length \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux l\u2082.length f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n    stepAux (f \u27e8l\u2082, \u22ef\u27e9) v (Tape.mk' (ListBlank.append (l\u2082.reverseAux 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 : List Bool\nf : Vector Bool [].length \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux [].length f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append [] R')) =\n    stepAux (f \u27e8[], \u22ef\u27e9) v (Tape.mk' (ListBlank.append ([].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\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool l\u2082.length \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux l\u2082.length f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f \u27e8l\u2082, \u22ef\u27e9) v (Tape.mk' (ListBlank.append (l\u2082.reverseAux l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (a :: l\u2082).length \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux (a :: l\u2082).length f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append (a :: l\u2082) R')) =\n    stepAux (f \u27e8a :: l\u2082, \u22ef\u27e9) v (Tape.mk' (ListBlank.append ((a :: l\u2082).reverseAux 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": [1295, 5], "def_end_pos": [1295, 12]}, {"full_name": "Turing.TM1to1.readAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1664, 5], "def_end_pos": [1664, 12]}, {"full_name": "Turing.Tape.mk'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [559, 5], "def_end_pos": [559, 13]}, {"full_name": "Turing.ListBlank.cons", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [248, 5], "def_end_pos": [248, 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\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool l\u2082.length \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux l\u2082.length f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f \u27e8l\u2082, \u22ef\u27e9) v (Tape.mk' (ListBlank.append (l\u2082.reverseAux l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (a :: l\u2082).length \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux (a :: l\u2082).length f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append (a :: l\u2082) R')) =\n    stepAux (f \u27e8a :: l\u2082, \u22ef\u27e9) v (Tape.mk' (ListBlank.append ((a :: l\u2082).reverseAux 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\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool l\u2082.length \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux l\u2082.length f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f \u27e8l\u2082, \u22ef\u27e9) v (Tape.mk' (ListBlank.append (l\u2082.reverseAux l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (a :: l\u2082).length \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux (a :: l\u2082).length f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append (a :: l\u2082) R')) =\n    stepAux (readAux l\u2082.length 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\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool l\u2082.length \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux l\u2082.length f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f \u27e8l\u2082, \u22ef\u27e9) v (Tape.mk' (ListBlank.append (l\u2082.reverseAux l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (a :: l\u2082).length \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux l\u2082.length 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 \u27e8a :: l\u2082, \u22ef\u27e9) v (Tape.mk' (ListBlank.append ((a :: l\u2082).reverseAux 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": [457, 5], "def_end_pos": [457, 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\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool l\u2082.length \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux l\u2082.length f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f \u27e8l\u2082, \u22ef\u27e9) v (Tape.mk' (ListBlank.append (l\u2082.reverseAux l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (a :: l\u2082).length \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux l\u2082.length 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 \u27e8a :: l\u2082, \u22ef\u27e9) v (Tape.mk' (ListBlank.append ((a :: l\u2082).reverseAux 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\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool l\u2082.length \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux l\u2082.length f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f \u27e8l\u2082, \u22ef\u27e9) v (Tape.mk' (ListBlank.append (l\u2082.reverseAux l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (a :: l\u2082).length \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (f (a ::\u1d65 \u27e8l\u2082, \u22ef\u27e9)) v (Tape.mk' (ListBlank.append (l\u2082.reverseAux (a :: l\u2081)) L') R') =\n    stepAux (f \u27e8a :: l\u2082, \u22ef\u27e9) v (Tape.mk' (ListBlank.append ((a :: l\u2082).reverseAux 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\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool l\u2082.length \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux l\u2082.length f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f \u27e8l\u2082, \u22ef\u27e9) v (Tape.mk' (ListBlank.append (l\u2082.reverseAux l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (a :: l\u2082).length \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (f (a ::\u1d65 \u27e8l\u2082, \u22ef\u27e9)) v (Tape.mk' (ListBlank.append (l\u2082.reverseAux (a :: l\u2081)) L') R') =\n    stepAux (f \u27e8a :: l\u2082, \u22ef\u27e9) v (Tape.mk' (ListBlank.append ((a :: l\u2082).reverseAux 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": [1682, 5], "def_end_pos": [1682, 9]}, {"full_name": "Turing.TM1to1.stepAux_move", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1704, 9], "def_end_pos": [1704, 21]}, {"full_name": "Turing.TM1to1.trTape'_move_left", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1774, 9], "def_end_pos": [1774, 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 R.head)) v (trTape' enc0 (ListBlank.cons R.head L) R.tail)\n\u22a2 stepAux (read dec f) v (trTape' enc0 L R) = stepAux (f R.head) 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 R.head)) v (trTape' enc0 (ListBlank.cons R.head L) R.tail)\n\u22a2 stepAux (f R.head) v\n      (trTape' enc0 (ListBlank.cons R.head L).tail (ListBlank.cons (ListBlank.cons R.head L).head R.tail)) =\n    stepAux (f R.head) 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": [261, 9], "def_end_pos": [261, 28]}, {"full_name": "Turing.ListBlank.cons_head_tail", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [273, 9], "def_end_pos": [273, 33]}, {"full_name": "Turing.ListBlank.tail_cons", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [266, 9], "def_end_pos": [266, 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 R.head)) v (trTape' enc0 (ListBlank.cons R.head L) R.tail)\n\u22a2 stepAux (f R.head) v\n      (trTape' enc0 (ListBlank.cons R.head L).tail (ListBlank.cons (ListBlank.cons R.head L).head R.tail)) =\n    stepAux (f R.head) 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 : l\u2082.length = i),\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')\n\u22a2 \u2200 (f : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux n f) v\n        (Tape.mk' (L.bind (fun x => (enc x).toList.reverse) \u22ef)\n          (ListBlank.append (enc a).toList (R.bind (fun x => (enc x).toList) \u22ef))) =\n      stepAux (f (enc a)) v\n        (Tape.mk' (ListBlank.append (enc a).toList.reverse (L.bind (fun x => (enc x).toList.reverse) \u22ef))\n          (R.bind (fun x => (enc x).toList) \u22ef))", "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 : l\u2082.length = i),\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')\nf : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux n f) v\n      (Tape.mk' (L.bind (fun x => (enc x).toList.reverse) \u22ef)\n        (ListBlank.append (enc a).toList (R.bind (fun x => (enc x).toList) \u22ef))) =\n    stepAux (f (enc a)) v\n      (Tape.mk' (ListBlank.append (enc a).toList.reverse (L.bind (fun x => (enc x).toList.reverse) \u22ef))\n        (R.bind (fun x => (enc x).toList) \u22ef))"}, {"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": [441, 5], "def_end_pos": [441, 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 : l\u2082.length = i),\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')\nf : Vector Bool n \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux n f) v\n      (Tape.mk' (L.bind (fun x => (enc x).toList.reverse) \u22ef)\n        (ListBlank.append (enc a).toList (R.bind (fun x => (enc x).toList) \u22ef))) =\n    stepAux (f (enc a)) v\n      (Tape.mk' (ListBlank.append (enc a).toList.reverse (L.bind (fun x => (enc x).toList.reverse) \u22ef))\n        (R.bind (fun x => (enc x).toList) \u22ef))", "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 : List Bool\nf : Vector Bool [].length \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux [].length f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append [] R')) =\n    stepAux (f \u27e8[], \u22ef\u27e9) v (Tape.mk' (ListBlank.append ([].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": [1664, 5], "def_end_pos": [1664, 12]}, {"full_name": "Turing.TM1.stepAux", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [1295, 5], "def_end_pos": [1295, 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\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool l\u2082.length \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux l\u2082.length f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f \u27e8l\u2082, \u22ef\u27e9) v (Tape.mk' (ListBlank.append (l\u2082.reverseAux l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (a :: l\u2082).length \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 stepAux (readAux (a :: l\u2082).length f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append (a :: l\u2082) R')) =\n    stepAux (readAux l\u2082.length 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\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool l\u2082.length \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux l\u2082.length f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f \u27e8l\u2082, \u22ef\u27e9) v (Tape.mk' (ListBlank.append (l\u2082.reverseAux l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (a :: l\u2082).length \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 (bif (ListBlank.cons a (ListBlank.append l\u2082 R')).head then\n      stepAux (readAux l\u2082.length 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 l\u2082.length 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 l\u2082.length 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": [261, 9], "def_end_pos": [261, 28]}, {"full_name": "Turing.Tape.move_right_mk'", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [602, 9], "def_end_pos": [602, 28]}, {"full_name": "Turing.ListBlank.tail_cons", "def_path": "Mathlib/Computability/TuringMachine.lean", "def_pos": [266, 9], "def_end_pos": [266, 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\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool l\u2082.length \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux l\u2082.length f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f \u27e8l\u2082, \u22ef\u27e9) v (Tape.mk' (ListBlank.append (l\u2082.reverseAux l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (a :: l\u2082).length \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 (bif (ListBlank.cons a (ListBlank.append l\u2082 R')).head then\n      stepAux (readAux l\u2082.length 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 l\u2082.length 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 l\u2082.length 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\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool l\u2082.length \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux l\u2082.length f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f \u27e8l\u2082, \u22ef\u27e9) v (Tape.mk' (ListBlank.append (l\u2082.reverseAux l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (a :: l\u2082).length \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 (bif a then\n      stepAux (readAux l\u2082.length 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 l\u2082.length 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 l\u2082.length 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\na : Bool\nl\u2082 : List Bool\nIH :\n  \u2200 (l\u2081 : List Bool) (f : Vector Bool l\u2082.length \u2192 Stmt Bool \u039b' \u03c3),\n    stepAux (readAux l\u2082.length f) v (Tape.mk' (ListBlank.append l\u2081 L') (ListBlank.append l\u2082 R')) =\n      stepAux (f \u27e8l\u2082, \u22ef\u27e9) v (Tape.mk' (ListBlank.append (l\u2082.reverseAux l\u2081) L') R')\nl\u2081 : List Bool\nf : Vector Bool (a :: l\u2082).length \u2192 Stmt Bool \u039b' \u03c3\n\u22a2 (bif a then\n      stepAux (readAux l\u2082.length 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 l\u2082.length 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 l\u2082.length 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": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Index.lean", "full_name": "Subgroup.index_iInf_ne_zero", "start": [472, 1], "end": [475, 33], "traced_tactics": [{"tactic": "simp_rw [\u2190 relindex_top_right] at hf \u22a2", "annotated_tactic": ["simp_rw [\u2190 <a>relindex_top_right</a>] at hf \u22a2", [{"full_name": "Subgroup.relindex_top_right", "def_path": "Mathlib/GroupTheory/Index.lean", "def_pos": [247, 9], "def_end_pos": [247, 27]}]], "state_before": "G : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 Subgroup G\nhf : \u2200 (i : \u03b9), (f i).index \u2260 0\n\u22a2 (\u2a05 i, f i).index \u2260 0", "state_after": "G : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 Subgroup G\nhf : \u2200 (i : \u03b9), (f i).relindex \u22a4 \u2260 0\n\u22a2 (\u2a05 i, f i).relindex \u22a4 \u2260 0"}, {"tactic": "exact relindex_iInf_ne_zero hf", "annotated_tactic": ["exact <a>relindex_iInf_ne_zero</a> hf", [{"full_name": "Subgroup.relindex_iInf_ne_zero", "def_path": "Mathlib/GroupTheory/Index.lean", "def_pos": [451, 9], "def_end_pos": [451, 30]}]], "state_before": "G : Type u_1\ninst\u271d\u00b9 : Group G\nH K L : Subgroup G\n\u03b9 : Type u_2\ninst\u271d : Finite \u03b9\nf : \u03b9 \u2192 Subgroup G\nhf : \u2200 (i : \u03b9), (f i).relindex \u22a4 \u2260 0\n\u22a2 (\u2a05 i, f i).relindex \u22a4 \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matrix/Block.lean", "full_name": "Matrix.fromBlocks_submatrix_sum_swap_sum_swap", "start": [176, 1], "end": [178, 85], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "l\u271d : Type u_1\nm\u271d : Type u_2\nn\u271d : Type u_3\no\u271d : Type u_4\np : Type u_5\nq : Type u_6\nm' : o\u271d \u2192 Type u_7\nn' : o\u271d \u2192 Type u_8\np' : o\u271d \u2192 Type u_9\nR : Type u_10\nS : Type u_11\n\u03b1\u271d : Type u_12\n\u03b2 : Type u_13\nl : Type u_14\nm : Type u_15\nn : Type u_16\no : Type u_17\n\u03b1 : Type u_18\nA : Matrix n l \u03b1\nB : Matrix n m \u03b1\nC : Matrix o l \u03b1\nD : Matrix o m \u03b1\n\u22a2 (fromBlocks A B C D).submatrix Sum.swap Sum.swap = fromBlocks D C B A", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/Instances/Sphere.lean", "full_name": "sphere_ext_iff", "start": [390, 1], "end": [391, 55], "traced_tactics": [{"tactic": "simp [Subtype.ext_iff, inner_eq_one_iff_of_norm_one]", "annotated_tactic": ["simp [<a>Subtype.ext_iff</a>, <a>inner_eq_one_iff_of_norm_one</a>]", [{"full_name": "Subtype.ext_iff", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [78, 9], "def_end_pos": [78, 16]}, {"full_name": "inner_eq_one_iff_of_norm_one", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [1703, 9], "def_end_pos": [1703, 37]}]], "state_before": "E : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : InnerProductSpace \u211d E\nu v : \u2191(sphere 0 1)\n\u22a2 u = v \u2194 \u27ea\u2191u, \u2191v\u27eb_\u211d = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/ContDiff/Basic.lean", "full_name": "ContDiffOn.continuousLinearMap_comp", "start": [237, 1], "end": [238, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/String/Lemmas.lean", "full_name": "String.utf8Len_le_of_prefix", "start": [82, 1], "end": [83, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Polynomial/Chebyshev.lean", "full_name": "Polynomial.Chebyshev.T_sub_two", "start": [93, 1], "end": [94, 59], "traced_tactics": [{"tactic": "linear_combination (norm := ring_nf) T_add_two R (n - 2)", "annotated_tactic": ["linear_combination (norm := ring_nf) <a>T_add_two</a> R (n - 2)", [{"full_name": "Polynomial.Chebyshev.T_add_two", "def_path": "Mathlib/RingTheory/Polynomial/Chebyshev.lean", "def_pos": [85, 9], "def_end_pos": [85, 18]}]], "state_before": "R : Type u_1\nS : Type u_2\ninst\u271d\u00b9 : CommRing R\ninst\u271d : CommRing S\nn : \u2124\n\u22a2 T R (n - 2) = 2 * X * T R (n - 1) - T R n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/IsConnected.lean", "full_name": "CategoryTheory.constant_of_preserves_morphisms", "start": [142, 1], "end": [148, 11], "traced_tactics": [{"tactic": "simpa using\n  any_functor_const_on_obj\n    { obj := Discrete.mk \u2218 F\n      map := fun f => eqToHom (by ext; exact h _ _ f) }\n    j j'", "annotated_tactic": ["simpa using\n    <a>any_functor_const_on_obj</a>\n      { obj := <a>Discrete.mk</a> \u2218 F\n        map := fun f => <a>eqToHom</a> (by ext; exact h _ _ f) }\n      j j'", [{"full_name": "CategoryTheory.any_functor_const_on_obj", "def_path": "Mathlib/CategoryTheory/IsConnected.lean", "def_pos": [115, 9], "def_end_pos": [115, 33]}, {"full_name": "CategoryTheory.Discrete.mk", "def_path": "Mathlib/CategoryTheory/DiscreteCategory.lean", "def_pos": [49, 11], "def_end_pos": [49, 19]}, {"full_name": "CategoryTheory.eqToHom", "def_path": "Mathlib/CategoryTheory/EqToHom.lean", "def_pos": [43, 5], "def_end_pos": [43, 12]}]], "state_before": "J : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : IsPreconnected J\n\u03b1 : Type u\u2082\nF : J \u2192 \u03b1\nh : \u2200 (j\u2081 j\u2082 : J), (j\u2081 \u27f6 j\u2082) \u2192 F j\u2081 = F j\u2082\nj j' : J\n\u22a2 F j = F j'", "state_after": "no goals"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "J : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} K\ninst\u271d : IsPreconnected J\n\u03b1 : Type u\u2082\nF : J \u2192 \u03b1\nh : \u2200 (j\u2081 j\u2082 : J), (j\u2081 \u27f6 j\u2082) \u2192 F j\u2081 = F j\u2082\nj j' X\u271d Y\u271d : J\nf : X\u271d \u27f6 Y\u271d\n\u22a2 (Discrete.mk \u2218 F) X\u271d = (Discrete.mk \u2218 F) Y\u271d", "state_after": "case as\nJ : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} J\nK : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, 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"29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Enum.lean", "full_name": "List.getElem?_enumFrom", "start": [20, 1], "end": [26, 83], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\na : \u03b1\nl : List \u03b1\n\u22a2 (enumFrom n (a :: l))[0]? = Option.map (fun a => (n + 0, a)) (a :: l)[0]?", "state_after": "no goals"}, {"tactic": "simp only [enumFrom_cons, getElem?_cons_succ]", "annotated_tactic": ["simp only [<a>enumFrom_cons</a>, <a>getElem?_cons_succ</a>]", [{"full_name": "List.enumFrom_cons", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [1188, 17], "def_end_pos": [1188, 30]}, {"full_name": "List.getElem?_cons_succ", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [169, 17], "def_end_pos": [169, 35]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn : \u2115\na : \u03b1\nl : List 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"https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Exp.lean", "full_name": "Real.isBigO_exp_comp_exp_comp", "start": [402, 1], "end": [405, 88], "traced_tactics": [{"tactic": "simp only [norm_eq_abs, abs_exp, \u2190 exp_sub, isBoundedUnder_le_exp_comp, Pi.sub_def]", "annotated_tactic": ["simp only [<a>norm_eq_abs</a>, <a>abs_exp</a>, \u2190 <a>exp_sub</a>, <a>isBoundedUnder_le_exp_comp</a>, <a>Pi.sub_def</a>]", [{"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1434, 9], "def_end_pos": [1434, 20]}, {"full_name": "Real.abs_exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1206, 9], "def_end_pos": [1206, 16]}, {"full_name": "Real.exp_sub", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [866, 9], "def_end_pos": [866, 16]}, {"full_name": "Real.isBoundedUnder_le_exp_comp", "def_path": "Mathlib/Analysis/SpecialFunctions/Exp.lean", "def_pos": [248, 9], "def_end_pos": [248, 35]}, {"full_name": "Pi.sub_def", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [201, 3], "def_end_pos": [201, 14]}]], "state_before": "\u03b1 : Type u_1\nx y z : \u211d\nl : Filter \u03b1\nf g : \u03b1 \u2192 \u211d\n\u22a2 (IsBoundedUnder (fun x x_1 => x \u2264 x_1) l fun x => \u2016rexp (f x)\u2016 / \u2016rexp (g x)\u2016) \u2194\n    IsBoundedUnder (fun x x_1 => x \u2264 x_1) l (f - g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean", "full_name": "CategoryTheory.Limits.Cotrident.IsColimit.homIso_natural", "start": [396, 1], "end": [400, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Seq/Computation.lean", "full_name": "Computation.tail_empty", "start": [180, 1], "end": [181, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Star/Basic.lean", "full_name": "star_ne_zero", "start": [289, 1], "end": [290, 34], "traced_tactics": [{"tactic": "simp only [ne_eq, star_eq_zero]", "annotated_tactic": ["simp only [<a>ne_eq</a>, <a>star_eq_zero</a>]", [{"full_name": "ne_eq", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [89, 17], "def_end_pos": [89, 22]}, {"full_name": "star_eq_zero", "def_path": "Mathlib/Algebra/Star/Basic.lean", "def_pos": [285, 9], "def_end_pos": [285, 21]}]], "state_before": "R : Type u\ninst\u271d\u00b9 : AddMonoid R\ninst\u271d : StarAddMonoid R\nx : R\n\u22a2 star x \u2260 0 \u2194 x \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicTopology/SimplicialObject.lean", "full_name": "CategoryTheory.CosimplicialObject.\u03b4_comp_\u03c3_of_le", "start": [516, 1], "end": [519, 61], "traced_tactics": [{"tactic": "dsimp [\u03b4, \u03c3]", "annotated_tactic": ["dsimp [<a>\u03b4</a>, <a>\u03c3</a>]", [{"full_name": "CategoryTheory.CosimplicialObject.\u03b4", "def_path": "Mathlib/AlgebraicTopology/SimplicialObject.lean", "def_pos": [454, 5], "def_end_pos": [454, 6]}, {"full_name": "CategoryTheory.CosimplicialObject.\u03c3", "def_path": "Mathlib/AlgebraicTopology/SimplicialObject.lean", "def_pos": [459, 5], "def_end_pos": [459, 6]}]], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 2)\nj : Fin (n + 1)\nH : i \u2264 j.castSucc\n\u22a2 X.\u03b4 i.castSucc \u226b X.\u03c3 j.succ = X.\u03c3 j \u226b X.\u03b4 i", "state_after": "C : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 2)\nj : Fin (n + 1)\nH : i \u2264 j.castSucc\n\u22a2 X.map (SimplexCategory.\u03b4 i.castSucc) \u226b X.map (SimplexCategory.\u03c3 j.succ) =\n    X.map (SimplexCategory.\u03c3 j) \u226b X.map (SimplexCategory.\u03b4 i)"}, {"tactic": "simp only [\u2190 X.map_comp, SimplexCategory.\u03b4_comp_\u03c3_of_le H]", "annotated_tactic": ["simp only [\u2190 X.map_comp, <a>SimplexCategory.\u03b4_comp_\u03c3_of_le</a> H]", [{"full_name": "SimplexCategory.\u03b4_comp_\u03c3_of_le", "def_path": "Mathlib/AlgebraicTopology/SimplexCategory.lean", "def_pos": [273, 9], "def_end_pos": [273, 23]}]], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX : CosimplicialObject C\nn : \u2115\ni : Fin (n + 2)\nj : Fin (n + 1)\nH : i \u2264 j.castSucc\n\u22a2 X.map (SimplexCategory.\u03b4 i.castSucc) \u226b X.map (SimplexCategory.\u03c3 j.succ) =\n    X.map (SimplexCategory.\u03c3 j) \u226b X.map (SimplexCategory.\u03b4 i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Opposite.lean", "full_name": "Opposite.equivToOpposite_coe", "start": [94, 1], "end": [95, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "full_name": "MeasureTheory.Measure.sigmaFinite_of_le", "start": [1056, 1], "end": [1057, 46], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/NaturalOps.lean", "full_name": "Ordinal.toNatOrdinal_eq_zero", "start": [171, 1], "end": [172, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Between.lean", "full_name": "wbtw_vadd_const_iff", "start": [187, 1], "end": [189, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Sets/Compacts.lean", "full_name": "TopologicalSpace.PositiveCompacts.isCompact", "start": [330, 11], "end": [331, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Interval/Finset.lean", "full_name": "Finset.image_add_left_Icc", "start": [74, 1], "end": [75, 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/Algebra/Order/Interval/Finset.lean", "def_pos": [24, 15], "def_end_pos": [24, 31]}, {"full_name": "Finset.map_eq_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [369, 9], "def_end_pos": [369, 21]}, {"full_name": "addLeftEmbedding", "def_path": "Mathlib/Algebra/Group/Embedding.lean", "def_pos": [24, 3], "def_end_pos": [24, 14]}, {"full_name": "Function.Embedding.coeFn_mk", "def_path": "Mathlib/Logic/Embedding/Basic.lean", "def_pos": [129, 9], "def_end_pos": [129, 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 => c + x) (Icc a b) = Icc (c + a) (c + b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Adjoin/Field.lean", "full_name": "IsIntegral.minpoly_splits_tower_top'", "start": [100, 1], "end": [104, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Star/StarAlgHom.lean", "full_name": "StarAlgHom.comp_apply", "start": [499, 1], "end": [500, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Classes/Order.lean", "full_name": "Batteries.OrientedOrd.instOn", "start": [310, 1], "end": [311, 49], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subsemigroup/Operations.lean", "full_name": "Subsemigroup.srange_snd", "start": [918, 1], "end": [919, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Cyclotomic/PrimitiveRoots.lean", "full_name": "IsPrimitiveRoot.norm_eq_neg_one_pow", "start": [289, 1], "end": [291, 98], "traced_tactics": [{"tactic": "rw [h\u03b6.eq_neg_one_of_two_right, show -1 = algebraMap K L (-1) by simp, Algebra.norm_algebraMap]", "annotated_tactic": ["rw [h\u03b6.eq_neg_one_of_two_right, show -1 = <a>algebraMap</a> K L (-1) by simp, <a>Algebra.norm_algebraMap</a>]", [{"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "Algebra.norm_algebraMap", "def_path": "Mathlib/RingTheory/Norm.lean", "def_pos": [105, 19], "def_end_pos": [105, 34]}]], "state_before": "p n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : CommRing L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nh\u03b6 : IsPrimitiveRoot \u03b6 2\ninst\u271d : IsDomain L\n\u22a2 (Algebra.norm K) \u03b6 = (-1) ^ FiniteDimensional.finrank K L", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "p n : \u2115+\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC : Type w\ninst\u271d\u2077 : CommRing A\ninst\u271d\u2076 : CommRing B\ninst\u271d\u2075 : Algebra A B\ninst\u271d\u2074 : IsCyclotomicExtension {n} A B\ninst\u271d\u00b3 : CommRing L\n\u03b6 : L\nh\u03b6\u271d : IsPrimitiveRoot \u03b6 \u2191n\ninst\u271d\u00b2 : Field K\ninst\u271d\u00b9 : Algebra K L\nh\u03b6 : IsPrimitiveRoot \u03b6 2\ninst\u271d : IsDomain L\n\u22a2 -1 = (algebraMap K L) (-1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "full_name": "QPF.suppPreservation_iff_liftpPreservation", "start": [707, 1], "end": [718, 24], "traced_tactics": [{"tactic": "constructor <;> intro h", "annotated_tactic": ["constructor <;> intro h", []], "state_before": "F : Type u \u2192 Type u\nq : QPF F\n\u22a2 SuppPreservation \u2194 LiftpPreservation", "state_after": "case mp\nF : Type u \u2192 Type u\nq : QPF F\nh : SuppPreservation\n\u22a2 LiftpPreservation\n\ncase mpr\nF : Type u \u2192 Type u\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\nq : QPF F\nh : SuppPreservation\n\u22a2 LiftpPreservation", "state_after": "case mp.mk\nF : Type u \u2192 Type u\nq : QPF F\nh : SuppPreservation\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : (P F).B a \u2192 \u03b1\n\u22a2 Liftp p (abs \u27e8a, f\u27e9) \u2194 Liftp p \u27e8a, f\u27e9"}, {"tactic": "have h' := h", "annotated_tactic": ["have h' := h", []], "state_before": "case mp.mk\nF : Type u \u2192 Type u\nq : QPF F\nh : SuppPreservation\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : (P F).B a \u2192 \u03b1\n\u22a2 Liftp p (abs \u27e8a, f\u27e9) \u2194 Liftp p \u27e8a, f\u27e9", "state_after": "case mp.mk\nF : Type u \u2192 Type u\nq : QPF F\nh : SuppPreservation\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : (P F).B a \u2192 \u03b1\nh' : SuppPreservation\n\u22a2 Liftp p (abs \u27e8a, f\u27e9) \u2194 Liftp p \u27e8a, f\u27e9"}, {"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": [699, 9], "def_end_pos": [699, 37]}]], "state_before": "case mp.mk\nF : Type u \u2192 Type u\nq : QPF F\nh : SuppPreservation\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : (P F).B a \u2192 \u03b1\nh' : SuppPreservation\n\u22a2 Liftp p (abs \u27e8a, f\u27e9) \u2194 Liftp p \u27e8a, f\u27e9", "state_after": "case mp.mk\nF : Type u \u2192 Type u\nq : QPF F\nh : SuppPreservation\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : (P F).B a \u2192 \u03b1\nh' : IsUniform\n\u22a2 Liftp p (abs \u27e8a, f\u27e9) \u2194 Liftp p \u27e8a, f\u27e9"}, {"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": [665, 5], "def_end_pos": [665, 21]}, {"full_name": "Functor.supp", "def_path": "Mathlib/Control/Functor.lean", "def_pos": [288, 5], "def_end_pos": [288, 9]}]], "state_before": "case mp.mk\nF : Type u \u2192 Type u\nq : QPF F\nh : SuppPreservation\n\u03b1 : Type u\np : \u03b1 \u2192 Prop\na : (P F).A\nf : (P F).B a \u2192 \u03b1\nh' : IsUniform\n\u22a2 Liftp p (abs \u27e8a, f\u27e9) \u2194 Liftp p \u27e8a, f\u27e9", "state_after": "case mp.mk\nF : Type u \u2192 Type u\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 : (P F).B a \u2192 \u03b1\nh' : IsUniform\n\u22a2 Liftp p (abs \u27e8a, f\u27e9) \u2194 Liftp p \u27e8a, f\u27e9"}, {"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": [678, 9], "def_end_pos": [678, 31]}, {"full_name": "QPF.supp_eq_of_isUniform", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [669, 9], "def_end_pos": [669, 29]}, {"full_name": "PFunctor.liftp_iff'", "def_path": "Mathlib/Data/PFunctor/Univariate/Basic.lean", "def_pos": [211, 9], "def_end_pos": [211, 19]}]], "state_before": "case mp.mk\nF : Type u \u2192 Type u\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 : (P F).B a \u2192 \u03b1\nh' : IsUniform\n\u22a2 Liftp p (abs \u27e8a, f\u27e9) \u2194 Liftp p \u27e8a, f\u27e9", "state_after": "case mp.mk\nF : Type u \u2192 Type u\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 : (P F).B a \u2192 \u03b1\nh' : IsUniform\n\u22a2 (\u2200 u \u2208 f '' univ, p u) \u2194 \u2200 (i : (P F).B 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": [701, 9], "def_end_pos": [701, 19]}, {"full_name": "Set.mem_range", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [160, 17], "def_end_pos": [160, 26]}, {"full_name": "exists_imp", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [200, 9], "def_end_pos": [200, 19]}]], "state_before": "case mp.mk\nF : Type u \u2192 Type u\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 : (P F).B a \u2192 \u03b1\nh' : IsUniform\n\u22a2 (\u2200 u \u2208 f '' univ, p u) \u2194 \u2200 (i : (P F).B a), p (f i)", "state_after": "case mp.mk\nF : Type u \u2192 Type u\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 : (P F).B a \u2192 \u03b1\nh' : IsUniform\n\u22a2 (\u2200 (u : \u03b1) (x : (P F).B a), f x = u \u2192 p u) \u2194 \u2200 (i : (P F).B 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\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 : (P F).B a \u2192 \u03b1\nh' : IsUniform\n\u22a2 (\u2200 (u : \u03b1) (x : (P F).B a), f x = u \u2192 p u) \u2194 \u2200 (i : (P F).B 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\nq : QPF F\nh : LiftpPreservation\n\u22a2 SuppPreservation", "state_after": "case mpr.mk\nF : Type u \u2192 Type u\nq : QPF F\nh : LiftpPreservation\n\u03b1 : Type u\na : (P F).A\nf : (P F).B a \u2192 \u03b1\n\u22a2 supp (abs \u27e8a, f\u27e9) = supp \u27e8a, f\u27e9"}, {"tactic": "simp only [LiftpPreservation] at h", "annotated_tactic": ["simp only [<a>LiftpPreservation</a>] at h", [{"full_name": "QPF.LiftpPreservation", "def_path": "Mathlib/Data/QPF/Univariate/Basic.lean", "def_pos": [660, 5], "def_end_pos": [660, 22]}]], "state_before": "case mpr.mk\nF : Type u \u2192 Type u\nq : QPF F\nh : LiftpPreservation\n\u03b1 : Type u\na : (P F).A\nf : (P F).B a \u2192 \u03b1\n\u22a2 supp (abs \u27e8a, f\u27e9) = supp \u27e8a, f\u27e9", "state_after": "case mpr.mk\nF : Type u \u2192 Type u\nq : QPF F\nh : \u2200 \u2983\u03b1 : Type u\u2984 (p : \u03b1 \u2192 Prop) (x : \u2191(P F) \u03b1), Liftp p (abs x) \u2194 Liftp p x\n\u03b1 : Type u\na : (P F).A\nf : (P F).B a \u2192 \u03b1\n\u22a2 supp (abs \u27e8a, f\u27e9) = supp \u27e8a, f\u27e9"}, {"tactic": "simp only [supp, h]", "annotated_tactic": ["simp only [<a>supp</a>, h]", [{"full_name": "Functor.supp", "def_path": "Mathlib/Control/Functor.lean", "def_pos": [288, 5], "def_end_pos": [288, 9]}]], "state_before": "case mpr.mk\nF : Type u \u2192 Type u\nq : QPF F\nh : \u2200 \u2983\u03b1 : Type u\u2984 (p : \u03b1 \u2192 Prop) (x : \u2191(P F) \u03b1), Liftp p (abs x) \u2194 Liftp p x\n\u03b1 : Type u\na : (P F).A\nf : (P F).B a \u2192 \u03b1\n\u22a2 supp (abs \u27e8a, f\u27e9) = supp \u27e8a, f\u27e9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Basic.lean", "full_name": "convex_Iic", "start": [277, 1], "end": [281, 37], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Subgraph.lean", "full_name": "SimpleGraph.neighborSet_singletonSubgraph", "start": [864, 1], "end": [865, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Acyclic.lean", "full_name": "SimpleGraph.IsAcyclic.path_unique", "start": [88, 1], "end": [115, 64], "traced_tactics": [{"tactic": "obtain \u27e8p, hp\u27e9 := p", "annotated_tactic": ["obtain \u27e8p, hp\u27e9 := p", []], "state_before": "V : Type u\nG\u271d G : SimpleGraph V\nh : G.IsAcyclic\nv w : V\np q : G.Path v w\n\u22a2 p = q", "state_after": "case mk\nV : Type u\nG\u271d G : SimpleGraph V\nh : G.IsAcyclic\nv w : V\nq : G.Path v w\np : G.Walk v w\nhp : p.IsPath\n\u22a2 \u27e8p, hp\u27e9 = q"}, {"tactic": "obtain \u27e8q, hq\u27e9 := q", "annotated_tactic": ["obtain \u27e8q, hq\u27e9 := q", []], "state_before": "case mk\nV : Type u\nG\u271d G : SimpleGraph V\nh : G.IsAcyclic\nv w : V\nq : G.Path v w\np : G.Walk v w\nhp : p.IsPath\n\u22a2 \u27e8p, hp\u27e9 = q", "state_after": "case mk.mk\nV : Type u\nG\u271d G : SimpleGraph V\nh : G.IsAcyclic\nv w : V\np : G.Walk v w\nhp : p.IsPath\nq : G.Walk v w\nhq : q.IsPath\n\u22a2 \u27e8p, hp\u27e9 = \u27e8q, hq\u27e9"}, {"tactic": "rw [Subtype.mk.injEq]", "annotated_tactic": ["rw [Subtype.mk.injEq]", []], "state_before": "case mk.mk\nV : Type u\nG\u271d G : SimpleGraph V\nh : G.IsAcyclic\nv w : V\np : G.Walk v w\nhp : p.IsPath\nq : G.Walk v w\nhq : q.IsPath\n\u22a2 \u27e8p, hp\u27e9 = \u27e8q, hq\u27e9", "state_after": "case mk.mk\nV : Type u\nG\u271d G : SimpleGraph V\nh : G.IsAcyclic\nv w : V\np : G.Walk v w\nhp : p.IsPath\nq : G.Walk v w\nhq : q.IsPath\n\u22a2 p = q"}, {"tactic": "cases (Walk.isPath_iff_eq_nil _).mp hq", "annotated_tactic": ["cases (<a>Walk.isPath_iff_eq_nil</a> _).<a>mp</a> hq", [{"full_name": "SimpleGraph.Walk.isPath_iff_eq_nil", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [1089, 9], "def_end_pos": [1089, 26]}, {"full_name": "Iff.mp", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [118, 3], "def_end_pos": [118, 5]}]], "state_before": "case mk.mk.nil\nV : Type u\nG\u271d G : SimpleGraph V\nh : G.IsAcyclic\nv w u\u271d : V\nhp : nil.IsPath\nq : G.Walk u\u271d u\u271d\nhq : q.IsPath\n\u22a2 nil = q", "state_after": "case mk.mk.nil.refl\nV : Type u\nG\u271d G : SimpleGraph V\nh : G.IsAcyclic\nv w u\u271d : V\nhp hq : nil.IsPath\n\u22a2 nil = nil"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case mk.mk.nil.refl\nV : Type u\nG\u271d G : SimpleGraph V\nh : G.IsAcyclic\nv w u\u271d : V\nhp hq : nil.IsPath\n\u22a2 nil = nil", "state_after": "no goals"}, {"tactic": "rw [isAcyclic_iff_forall_adj_isBridge] at h", "annotated_tactic": ["rw [<a>isAcyclic_iff_forall_adj_isBridge</a>] at h", [{"full_name": "SimpleGraph.isAcyclic_iff_forall_adj_isBridge", "def_path": "Mathlib/Combinatorics/SimpleGraph/Acyclic.lean", "def_pos": [68, 9], "def_end_pos": [68, 42]}]], "state_before": "case mk.mk.cons\nV : Type u\nG\u271d G : SimpleGraph V\nh : G.IsAcyclic\nv w u\u271d v\u271d w\u271d : V\nph : G.Adj u\u271d v\u271d\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nhp : (cons ph p).IsPath\nq : G.Walk u\u271d w\u271d\nhq : q.IsPath\n\u22a2 cons ph p = q", "state_after": "case mk.mk.cons\nV : Type u\nG\u271d G : SimpleGraph V\nh : \u2200 \u2983v w : V\u2984, G.Adj v w \u2192 G.IsBridge s(v, w)\nv w u\u271d v\u271d w\u271d : V\nph : G.Adj u\u271d v\u271d\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nhp : (cons ph p).IsPath\nq : G.Walk u\u271d w\u271d\nhq : q.IsPath\n\u22a2 cons ph p = q"}, {"tactic": "specialize h ph", "annotated_tactic": ["specialize h ph", []], "state_before": "case mk.mk.cons\nV : Type u\nG\u271d G : SimpleGraph V\nh : \u2200 \u2983v w : V\u2984, G.Adj v w \u2192 G.IsBridge s(v, w)\nv w u\u271d v\u271d w\u271d : V\nph : G.Adj u\u271d v\u271d\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nhp : (cons ph p).IsPath\nq : G.Walk u\u271d w\u271d\nhq : q.IsPath\n\u22a2 cons ph p = q", "state_after": "case mk.mk.cons\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d w\u271d : V\nph : G.Adj u\u271d v\u271d\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nhp : (cons ph p).IsPath\nq : G.Walk u\u271d w\u271d\nhq : q.IsPath\nh : G.IsBridge s(u\u271d, v\u271d)\n\u22a2 cons ph p = q"}, {"tactic": "rw [isBridge_iff_adj_and_forall_walk_mem_edges] at h", "annotated_tactic": ["rw [<a>isBridge_iff_adj_and_forall_walk_mem_edges</a>] at h", [{"full_name": "SimpleGraph.isBridge_iff_adj_and_forall_walk_mem_edges", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [2650, 9], "def_end_pos": [2650, 51]}]], "state_before": "case mk.mk.cons\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d w\u271d : V\nph : G.Adj u\u271d v\u271d\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nhp : (cons ph p).IsPath\nq : G.Walk u\u271d w\u271d\nhq : q.IsPath\nh : G.IsBridge s(u\u271d, v\u271d)\n\u22a2 cons ph p = q", "state_after": "case mk.mk.cons\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d w\u271d : V\nph : G.Adj u\u271d v\u271d\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nhp : (cons ph p).IsPath\nq : G.Walk u\u271d w\u271d\nhq : q.IsPath\nh : G.Adj u\u271d v\u271d \u2227 \u2200 (p : G.Walk u\u271d v\u271d), s(u\u271d, v\u271d) \u2208 p.edges\n\u22a2 cons ph p = q"}, {"tactic": "replace h := h.2 (q.append p.reverse)", "annotated_tactic": ["replace h := h.2 (q.append p.reverse)", []], "state_before": "case mk.mk.cons\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d w\u271d : V\nph : G.Adj u\u271d v\u271d\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nhp : (cons ph p).IsPath\nq : G.Walk u\u271d w\u271d\nhq : q.IsPath\nh : G.Adj u\u271d v\u271d \u2227 \u2200 (p : G.Walk u\u271d v\u271d), s(u\u271d, v\u271d) \u2208 p.edges\n\u22a2 cons ph p = q", "state_after": "case mk.mk.cons\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d w\u271d : V\nph : G.Adj u\u271d v\u271d\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nhp : (cons ph p).IsPath\nq : G.Walk u\u271d w\u271d\nhq : q.IsPath\nh : s(u\u271d, v\u271d) \u2208 (q.append p.reverse).edges\n\u22a2 cons ph p = q"}, {"tactic": "simp only [Walk.edges_append, Walk.edges_reverse, List.mem_append, List.mem_reverse] at h", "annotated_tactic": ["simp only [<a>Walk.edges_append</a>, <a>Walk.edges_reverse</a>, <a>List.mem_append</a>, <a>List.mem_reverse</a>] at h", [{"full_name": "SimpleGraph.Walk.edges_append", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [778, 9], "def_end_pos": [778, 21]}, {"full_name": "SimpleGraph.Walk.edges_reverse", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [783, 9], "def_end_pos": [783, 22]}, {"full_name": "List.mem_append", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [1165, 17], "def_end_pos": [1165, 27]}, {"full_name": "List.mem_reverse", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [1496, 17], "def_end_pos": [1496, 28]}]], "state_before": "case mk.mk.cons\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d w\u271d : V\nph : G.Adj u\u271d v\u271d\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nhp : (cons ph p).IsPath\nq : G.Walk u\u271d w\u271d\nhq : q.IsPath\nh : s(u\u271d, v\u271d) \u2208 (q.append p.reverse).edges\n\u22a2 cons ph p = q", "state_after": "case mk.mk.cons\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d w\u271d : V\nph : G.Adj u\u271d v\u271d\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nhp : (cons ph p).IsPath\nq : G.Walk u\u271d w\u271d\nhq : q.IsPath\nh : s(u\u271d, v\u271d) \u2208 q.edges \u2228 s(u\u271d, v\u271d) \u2208 p.edges\n\u22a2 cons ph p = q"}, {"tactic": "cases' h with h h", "annotated_tactic": ["cases' h with h h", []], "state_before": "case mk.mk.cons\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d w\u271d : V\nph : G.Adj u\u271d v\u271d\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nhp : (cons ph p).IsPath\nq : G.Walk u\u271d w\u271d\nhq : q.IsPath\nh : s(u\u271d, v\u271d) \u2208 q.edges \u2228 s(u\u271d, v\u271d) \u2208 p.edges\n\u22a2 cons ph p = q", "state_after": "case mk.mk.cons.inl\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d w\u271d : V\nph : G.Adj u\u271d v\u271d\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nhp : (cons ph p).IsPath\nq : G.Walk u\u271d w\u271d\nhq : q.IsPath\nh : s(u\u271d, v\u271d) \u2208 q.edges\n\u22a2 cons ph p = q\n\ncase mk.mk.cons.inr\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d w\u271d : V\nph : G.Adj u\u271d v\u271d\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nhp : (cons ph p).IsPath\nq : G.Walk u\u271d w\u271d\nhq : q.IsPath\nh : s(u\u271d, v\u271d) \u2208 p.edges\n\u22a2 cons ph p = q"}, {"tactic": "simp [Walk.isPath_def] at hp", "annotated_tactic": ["simp [<a>Walk.isPath_def</a>] at hp", [{"full_name": "SimpleGraph.Walk.isPath_def", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [989, 9], "def_end_pos": [989, 19]}]], "state_before": "case mk.mk.cons.inl.nil\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d : V\nph : G.Adj u\u271d v\u271d\np : G.Walk v\u271d u\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d u\u271d), q.IsPath \u2192 p = q\nhp : (cons ph p).IsPath\nhq : nil.IsPath\nh : s(u\u271d, v\u271d) \u2208 nil.edges\n\u22a2 cons ph p = nil", "state_after": "no goals"}, {"tactic": "rw [Walk.cons_isPath_iff] at hp hq", "annotated_tactic": ["rw [<a>Walk.cons_isPath_iff</a>] at hp hq", [{"full_name": "SimpleGraph.Walk.cons_isPath_iff", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [1079, 9], "def_end_pos": [1079, 24]}]], "state_before": "case mk.mk.cons.inl.cons\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d\u00b9 w\u271d : V\nph : G.Adj u\u271d v\u271d\u00b9\np : G.Walk v\u271d\u00b9 w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d\u00b9 w\u271d), q.IsPath \u2192 p = q\nhp : (cons ph p).IsPath\nv\u271d : V\nh\u271d : G.Adj u\u271d v\u271d\nq : G.Walk v\u271d w\u271d\nhq : (cons h\u271d q).IsPath\nh : s(u\u271d, v\u271d\u00b9) \u2208 (cons h\u271d q).edges\n\u22a2 cons ph p = cons h\u271d q", "state_after": "case mk.mk.cons.inl.cons\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d\u00b9 w\u271d : V\nph : G.Adj u\u271d v\u271d\u00b9\np : G.Walk v\u271d\u00b9 w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d\u00b9 w\u271d), q.IsPath \u2192 p = q\nhp : p.IsPath \u2227 u\u271d \u2209 p.support\nv\u271d : V\nh\u271d : G.Adj u\u271d v\u271d\nq : G.Walk v\u271d w\u271d\nhq : q.IsPath \u2227 u\u271d \u2209 q.support\nh : s(u\u271d, v\u271d\u00b9) \u2208 (cons h\u271d q).edges\n\u22a2 cons ph p = cons h\u271d q"}, {"tactic": "simp only [Walk.edges_cons, List.mem_cons, Sym2.eq_iff, true_and] at h", "annotated_tactic": ["simp only [<a>Walk.edges_cons</a>, <a>List.mem_cons</a>, <a>Sym2.eq_iff</a>, <a>true_and</a>] at h", [{"full_name": "SimpleGraph.Walk.edges_cons", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [761, 9], "def_end_pos": [761, 19]}, {"full_name": "List.mem_cons", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [285, 17], "def_end_pos": [285, 25]}, {"full_name": "Sym2.eq_iff", "def_path": "Mathlib/Data/Sym/Sym2.lean", "def_pos": [188, 9], "def_end_pos": [188, 15]}, {"full_name": "true_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [105, 17], "def_end_pos": [105, 25]}]], "state_before": "case mk.mk.cons.inl.cons\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d\u00b9 w\u271d : V\nph : G.Adj u\u271d v\u271d\u00b9\np : G.Walk v\u271d\u00b9 w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d\u00b9 w\u271d), q.IsPath \u2192 p = q\nhp : p.IsPath \u2227 u\u271d \u2209 p.support\nv\u271d : V\nh\u271d : G.Adj u\u271d v\u271d\nq : G.Walk v\u271d w\u271d\nhq : q.IsPath \u2227 u\u271d \u2209 q.support\nh : s(u\u271d, v\u271d\u00b9) \u2208 (cons h\u271d q).edges\n\u22a2 cons ph p = cons h\u271d q", "state_after": "case mk.mk.cons.inl.cons\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d\u00b9 w\u271d : V\nph : G.Adj u\u271d v\u271d\u00b9\np : G.Walk v\u271d\u00b9 w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d\u00b9 w\u271d), q.IsPath \u2192 p = q\nhp : p.IsPath \u2227 u\u271d \u2209 p.support\nv\u271d : V\nh\u271d : G.Adj u\u271d v\u271d\nq : G.Walk v\u271d w\u271d\nhq : q.IsPath \u2227 u\u271d \u2209 q.support\nh : (v\u271d\u00b9 = v\u271d \u2228 u\u271d = v\u271d \u2227 v\u271d\u00b9 = u\u271d) \u2228 s(u\u271d, v\u271d\u00b9) \u2208 q.edges\n\u22a2 cons ph p = cons h\u271d q"}, {"tactic": "rcases h with (\u27e8h, rfl\u27e9 | \u27e8rfl, rfl\u27e9) | h", "annotated_tactic": ["rcases h with (\u27e8h, rfl\u27e9 | \u27e8rfl, rfl\u27e9) | h", []], "state_before": "case mk.mk.cons.inl.cons\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d\u00b9 w\u271d : V\nph : G.Adj u\u271d v\u271d\u00b9\np : G.Walk v\u271d\u00b9 w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d\u00b9 w\u271d), q.IsPath \u2192 p = q\nhp : p.IsPath \u2227 u\u271d \u2209 p.support\nv\u271d : V\nh\u271d : G.Adj u\u271d v\u271d\nq : G.Walk v\u271d w\u271d\nhq : q.IsPath \u2227 u\u271d \u2209 q.support\nh : (v\u271d\u00b9 = v\u271d \u2228 u\u271d = v\u271d \u2227 v\u271d\u00b9 = u\u271d) \u2228 s(u\u271d, v\u271d\u00b9) \u2208 q.edges\n\u22a2 cons ph p = cons h\u271d q", "state_after": "case mk.mk.cons.inl.cons.inl.inl.refl\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d w\u271d : V\nph : G.Adj u\u271d v\u271d\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nhp : p.IsPath \u2227 u\u271d \u2209 p.support\nh\u271d : G.Adj u\u271d v\u271d\nq : G.Walk v\u271d w\u271d\nhq : q.IsPath \u2227 u\u271d \u2209 q.support\n\u22a2 cons ph p = cons h\u271d q\n\ncase mk.mk.cons.inl.cons.inl.inr.intro\nV : Type u\nG\u271d G : SimpleGraph V\nv w v\u271d w\u271d : V\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nph : G.Adj v\u271d v\u271d\nhp : p.IsPath \u2227 v\u271d \u2209 p.support\nh\u271d : G.Adj v\u271d v\u271d\nq : G.Walk v\u271d w\u271d\nhq : q.IsPath \u2227 v\u271d \u2209 q.support\n\u22a2 cons ph p = cons h\u271d q\n\ncase mk.mk.cons.inl.cons.inr\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d\u00b9 w\u271d : V\nph : G.Adj u\u271d v\u271d\u00b9\np : G.Walk v\u271d\u00b9 w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d\u00b9 w\u271d), q.IsPath \u2192 p = q\nhp : p.IsPath \u2227 u\u271d \u2209 p.support\nv\u271d : V\nh\u271d : G.Adj u\u271d v\u271d\nq : G.Walk v\u271d w\u271d\nhq : q.IsPath \u2227 u\u271d \u2209 q.support\nh : s(u\u271d, v\u271d\u00b9) \u2208 q.edges\n\u22a2 cons ph p = cons h\u271d q"}, {"tactic": "cases ih hp.1 q hq.1", "annotated_tactic": ["cases ih hp.1 q hq.1", []], "state_before": "case mk.mk.cons.inl.cons.inl.inl.refl\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d w\u271d : V\nph : G.Adj u\u271d v\u271d\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nhp : p.IsPath \u2227 u\u271d \u2209 p.support\nh\u271d : G.Adj u\u271d v\u271d\nq : G.Walk v\u271d w\u271d\nhq : q.IsPath \u2227 u\u271d \u2209 q.support\n\u22a2 cons ph p = cons h\u271d q", "state_after": "case mk.mk.cons.inl.cons.inl.inl.refl.refl\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d w\u271d : V\nph : G.Adj u\u271d v\u271d\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nhp : p.IsPath \u2227 u\u271d \u2209 p.support\nh\u271d : G.Adj u\u271d v\u271d\nhq : p.IsPath \u2227 u\u271d \u2209 p.support\n\u22a2 cons ph p = cons h\u271d p"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case mk.mk.cons.inl.cons.inl.inl.refl.refl\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d w\u271d : V\nph : G.Adj u\u271d v\u271d\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nhp : p.IsPath \u2227 u\u271d \u2209 p.support\nh\u271d : G.Adj u\u271d v\u271d\nhq : p.IsPath \u2227 u\u271d \u2209 p.support\n\u22a2 cons ph p = cons h\u271d p", "state_after": "no goals"}, {"tactic": "simp at hq", "annotated_tactic": ["simp at hq", []], "state_before": "case mk.mk.cons.inl.cons.inl.inr.intro\nV : Type u\nG\u271d G : SimpleGraph V\nv w v\u271d w\u271d : V\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nph : G.Adj v\u271d v\u271d\nhp : p.IsPath \u2227 v\u271d \u2209 p.support\nh\u271d : G.Adj v\u271d v\u271d\nq : G.Walk v\u271d w\u271d\nhq : q.IsPath \u2227 v\u271d \u2209 q.support\n\u22a2 cons ph p = cons h\u271d q", "state_after": "no goals"}, {"tactic": "exact absurd (Walk.fst_mem_support_of_mem_edges _ h) hq.2", "annotated_tactic": ["exact <a>absurd</a> (<a>Walk.fst_mem_support_of_mem_edges</a> _ h) hq.2", [{"full_name": "absurd", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [246, 21], "def_end_pos": [246, 27]}, {"full_name": "SimpleGraph.Walk.fst_mem_support_of_mem_edges", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [815, 9], "def_end_pos": [815, 37]}]], "state_before": "case mk.mk.cons.inl.cons.inr\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d\u00b9 w\u271d : V\nph : G.Adj u\u271d v\u271d\u00b9\np : G.Walk v\u271d\u00b9 w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d\u00b9 w\u271d), q.IsPath \u2192 p = q\nhp : p.IsPath \u2227 u\u271d \u2209 p.support\nv\u271d : V\nh\u271d : G.Adj u\u271d v\u271d\nq : G.Walk v\u271d w\u271d\nhq : q.IsPath \u2227 u\u271d \u2209 q.support\nh : s(u\u271d, v\u271d\u00b9) \u2208 q.edges\n\u22a2 cons ph p = cons h\u271d q", "state_after": "no goals"}, {"tactic": "rw [Walk.cons_isPath_iff] at hp", "annotated_tactic": ["rw [<a>Walk.cons_isPath_iff</a>] at hp", [{"full_name": "SimpleGraph.Walk.cons_isPath_iff", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [1079, 9], "def_end_pos": [1079, 24]}]], "state_before": "case mk.mk.cons.inr\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d w\u271d : V\nph : G.Adj u\u271d v\u271d\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nhp : (cons ph p).IsPath\nq : G.Walk u\u271d w\u271d\nhq : q.IsPath\nh : s(u\u271d, v\u271d) \u2208 p.edges\n\u22a2 cons ph p = q", "state_after": "case mk.mk.cons.inr\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d w\u271d : V\nph : G.Adj u\u271d v\u271d\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nhp : p.IsPath \u2227 u\u271d \u2209 p.support\nq : G.Walk u\u271d w\u271d\nhq : q.IsPath\nh : s(u\u271d, v\u271d) \u2208 p.edges\n\u22a2 cons ph p = q"}, {"tactic": "exact absurd (Walk.fst_mem_support_of_mem_edges _ h) hp.2", "annotated_tactic": ["exact <a>absurd</a> (<a>Walk.fst_mem_support_of_mem_edges</a> _ h) hp.2", [{"full_name": "absurd", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [246, 21], "def_end_pos": [246, 27]}, {"full_name": "SimpleGraph.Walk.fst_mem_support_of_mem_edges", "def_path": "Mathlib/Combinatorics/SimpleGraph/Connectivity.lean", "def_pos": [815, 9], "def_end_pos": [815, 37]}]], "state_before": "case mk.mk.cons.inr\nV : Type u\nG\u271d G : SimpleGraph V\nv w u\u271d v\u271d w\u271d : V\nph : G.Adj u\u271d v\u271d\np : G.Walk v\u271d w\u271d\nih : p.IsPath \u2192 \u2200 (q : G.Walk v\u271d w\u271d), q.IsPath \u2192 p = q\nhp : p.IsPath \u2227 u\u271d \u2209 p.support\nq : G.Walk u\u271d w\u271d\nhq : q.IsPath\nh : s(u\u271d, v\u271d) \u2208 p.edges\n\u22a2 cons ph p = q", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/LocalInvariantProperties.lean", "full_name": "StructureGroupoid.LocalInvariantProp.liftPropWithinAt_indep_chart_source_aux", "start": [255, 1], "end": [270, 12], "traced_tactics": [{"tactic": "rw [\u2190 hG.right_invariance (compatible_of_mem_maximalAtlas he he')]", "annotated_tactic": ["rw [\u2190 hG.right_invariance (<a>compatible_of_mem_maximalAtlas</a> he he')]", [{"full_name": "StructureGroupoid.compatible_of_mem_maximalAtlas", "def_path": "Mathlib/Geometry/Manifold/ChartedSpace.lean", "def_pos": [1057, 9], "def_end_pos": [1057, 57]}]], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 P (g \u2218 \u2191e.symm) (\u2191e.symm \u207b\u00b9' s) (\u2191e x) \u2194 P (g \u2218 \u2191e'.symm) (\u2191e'.symm \u207b\u00b9' s) (\u2191e' x)", "state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 P ((g \u2218 \u2191e.symm) \u2218 \u2191(e.symm \u226b\u2095 e').symm) (\u2191(e.symm \u226b\u2095 e').symm \u207b\u00b9' (\u2191e.symm \u207b\u00b9' s)) (\u2191(e.symm \u226b\u2095 e') (\u2191e x)) \u2194\n    P (g \u2218 \u2191e'.symm) (\u2191e'.symm \u207b\u00b9' s) (\u2191e' x)\n\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 \u2191e x \u2208 (e.symm \u226b\u2095 e').source"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 P ((g \u2218 \u2191e.symm) \u2218 \u2191(e.symm \u226b\u2095 e').symm) (\u2191(e.symm \u226b\u2095 e').symm \u207b\u00b9' (\u2191e.symm \u207b\u00b9' s)) (\u2191(e.symm \u226b\u2095 e') (\u2191e x)) \u2194\n    P (g \u2218 \u2191e'.symm) (\u2191e'.symm \u207b\u00b9' s) (\u2191e' x)\n\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 \u2191e x \u2208 (e.symm \u226b\u2095 e').source", "state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 \u2191e x \u2208 (e.symm \u226b\u2095 e').source\n\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 P ((g \u2218 \u2191e.symm) \u2218 \u2191(e.symm \u226b\u2095 e').symm) (\u2191(e.symm \u226b\u2095 e').symm \u207b\u00b9' (\u2191e.symm \u207b\u00b9' s)) (\u2191(e.symm \u226b\u2095 e') (\u2191e x)) \u2194\n    P (g \u2218 \u2191e'.symm) (\u2191e'.symm \u207b\u00b9' s) (\u2191e' x)"}, {"tactic": "simp_rw [PartialHomeomorph.trans_apply, e.left_inv xe]", "annotated_tactic": ["simp_rw [<a>PartialHomeomorph.trans_apply</a>, e.left_inv xe]", [{"full_name": "PartialHomeomorph.trans_apply", "def_path": "Mathlib/Topology/PartialHomeomorph.lean", "def_pos": [863, 9], "def_end_pos": [863, 20]}]], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 P ((g \u2218 \u2191e.symm) \u2218 \u2191(e.symm \u226b\u2095 e').symm) (\u2191(e.symm \u226b\u2095 e').symm \u207b\u00b9' (\u2191e.symm \u207b\u00b9' s)) (\u2191(e.symm \u226b\u2095 e') (\u2191e x)) \u2194\n    P (g \u2218 \u2191e'.symm) (\u2191e'.symm \u207b\u00b9' s) (\u2191e' x)", "state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 P ((g \u2218 \u2191e.symm) \u2218 \u2191(e.symm \u226b\u2095 e').symm) (\u2191(e.symm \u226b\u2095 e').symm \u207b\u00b9' (\u2191e.symm \u207b\u00b9' s)) (\u2191e' x) \u2194\n    P (g \u2218 \u2191e'.symm) (\u2191e'.symm \u207b\u00b9' s) (\u2191e' x)"}, {"tactic": "rw [hG.congr_iff]", "annotated_tactic": ["rw [hG.congr_iff]", []], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 P ((g \u2218 \u2191e.symm) \u2218 \u2191(e.symm \u226b\u2095 e').symm) (\u2191(e.symm \u226b\u2095 e').symm \u207b\u00b9' (\u2191e.symm \u207b\u00b9' s)) (\u2191e' x) \u2194\n    P (g \u2218 \u2191e'.symm) (\u2191e'.symm \u207b\u00b9' s) (\u2191e' x)", "state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 P ?m.19727 (\u2191(e.symm \u226b\u2095 e').symm \u207b\u00b9' (\u2191e.symm \u207b\u00b9' s)) (\u2191e' x) \u2194 P (g \u2218 \u2191e'.symm) (\u2191e'.symm \u207b\u00b9' s) (\u2191e' x)\n\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 (g \u2218 \u2191e.symm) \u2218 \u2191(e.symm \u226b\u2095 e').symm =\u1da0[\ud835\udcdd (\u2191e' x)] ?m.19727\n\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 H \u2192 H'"}, {"tactic": "simp only [xe, xe', mfld_simps]", "annotated_tactic": ["simp only [xe, xe', mfld_simps]", []], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 \u2191e x \u2208 (e.symm \u226b\u2095 e').source", "state_after": "no goals"}, {"tactic": "refine hG.congr_set ?_", "annotated_tactic": ["refine hG.congr_set ?_", []], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 P ?m.19727 (\u2191(e.symm \u226b\u2095 e').symm \u207b\u00b9' (\u2191e.symm \u207b\u00b9' s)) (\u2191e' x) \u2194 P (g \u2218 \u2191e'.symm) (\u2191e'.symm \u207b\u00b9' s) (\u2191e' x)", "state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 \u2191(e.symm \u226b\u2095 e').symm \u207b\u00b9' (\u2191e.symm \u207b\u00b9' s) =\u1da0[\ud835\udcdd (\u2191e' x)] \u2191e'.symm \u207b\u00b9' s"}, {"tactic": "refine (eventually_of_mem ?_ fun y (hy : y \u2208 e'.symm \u207b\u00b9' e.source) \u21a6 ?_).set_eq", "annotated_tactic": ["refine (<a>eventually_of_mem</a> ?_ fun y (hy : y \u2208 e'.symm \u207b\u00b9' e.source) \u21a6 ?_).<a>set_eq</a>", [{"full_name": "Filter.eventually_of_mem", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1116, 9], "def_end_pos": [1116, 26]}, {"full_name": "Filter.Eventually.set_eq", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1491, 30], "def_end_pos": [1491, 47]}]], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 \u2191(e.symm \u226b\u2095 e').symm \u207b\u00b9' (\u2191e.symm \u207b\u00b9' s) =\u1da0[\ud835\udcdd (\u2191e' x)] \u2191e'.symm \u207b\u00b9' s", "state_after": "case refine_1\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 \u2191e'.symm \u207b\u00b9' e.source \u2208 \ud835\udcdd (\u2191e' x)\n\ncase refine_2\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\ny : H\nhy : y \u2208 \u2191e'.symm \u207b\u00b9' e.source\n\u22a2 y \u2208 \u2191(e.symm \u226b\u2095 e').symm \u207b\u00b9' (\u2191e.symm \u207b\u00b9' s) \u2194 y \u2208 \u2191e'.symm \u207b\u00b9' s"}, {"tactic": "simp_rw [mem_preimage, PartialHomeomorph.coe_trans_symm, PartialHomeomorph.symm_symm,\n  Function.comp_apply, e.left_inv hy]", "annotated_tactic": ["simp_rw [<a>mem_preimage</a>, <a>PartialHomeomorph.coe_trans_symm</a>, <a>PartialHomeomorph.symm_symm</a>,\n      <a>Function.comp_apply</a>, e.left_inv hy]", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [120, 9], "def_end_pos": [120, 21]}, {"full_name": "PartialHomeomorph.coe_trans_symm", "def_path": "Mathlib/Topology/PartialHomeomorph.lean", "def_pos": [859, 9], "def_end_pos": [859, 23]}, {"full_name": "PartialHomeomorph.symm_symm", "def_path": "Mathlib/Topology/PartialHomeomorph.lean", "def_pos": [364, 29], "def_end_pos": [364, 38]}, {"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}]], "state_before": "case refine_2\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\ny : H\nhy : y \u2208 \u2191e'.symm \u207b\u00b9' e.source\n\u22a2 y \u2208 \u2191(e.symm \u226b\u2095 e').symm \u207b\u00b9' (\u2191e.symm \u207b\u00b9' s) \u2194 y \u2208 \u2191e'.symm \u207b\u00b9' s", "state_after": "no goals"}, {"tactic": "refine (e'.symm.continuousAt <| e'.mapsTo xe').preimage_mem_nhds (e.open_source.mem_nhds ?_)", "annotated_tactic": ["refine (e'.symm.continuousAt <| e'.mapsTo xe').<a>preimage_mem_nhds</a> (e.open_source.mem_nhds ?_)", [{"full_name": "ContinuousAt.preimage_mem_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1578, 9], "def_end_pos": [1578, 39]}]], "state_before": "case refine_1\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 \u2191e'.symm \u207b\u00b9' e.source \u2208 \ud835\udcdd (\u2191e' x)", "state_after": "case refine_1\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 \u2191e'.symm (\u2191e' x) \u2208 e.source"}, {"tactic": "simp_rw [e'.left_inv xe', xe]", "annotated_tactic": ["simp_rw [e'.left_inv xe', xe]", []], "state_before": "case refine_1\nH : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 \u2191e'.symm (\u2191e' x) \u2208 e.source", "state_after": "no goals"}, {"tactic": "refine ((e'.eventually_nhds' _ xe').mpr <| e.eventually_left_inverse xe).mono fun y hy \u21a6 ?_", "annotated_tactic": ["refine ((e'.eventually_nhds' _ xe').<a>mpr</a> <| e.eventually_left_inverse xe).<a>mono</a> fun y hy \u21a6 ?_", [{"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}, {"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1158, 9], "def_end_pos": [1158, 24]}]], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\n\u22a2 (g \u2218 \u2191e.symm) \u2218 \u2191(e.symm \u226b\u2095 e').symm =\u1da0[\ud835\udcdd (\u2191e' x)] g \u2218 \u2191e'.symm", "state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\ny : H\nhy : \u2191e.symm (\u2191e (\u2191e'.symm y)) = \u2191e'.symm y\n\u22a2 ((g \u2218 \u2191e.symm) \u2218 \u2191(e.symm \u226b\u2095 e').symm) y = (g \u2218 \u2191e'.symm) y"}, {"tactic": "simp only [mfld_simps]", "annotated_tactic": ["simp only [mfld_simps]", []], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\ny : H\nhy : \u2191e.symm (\u2191e (\u2191e'.symm y)) = \u2191e'.symm y\n\u22a2 ((g \u2218 \u2191e.symm) \u2218 \u2191(e.symm \u226b\u2095 e').symm) y = (g \u2218 \u2191e'.symm) y", "state_after": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\ny : H\nhy : \u2191e.symm (\u2191e (\u2191e'.symm y)) = \u2191e'.symm y\n\u22a2 g (\u2191e.symm (\u2191e (\u2191e'.symm y))) = g (\u2191e'.symm y)"}, {"tactic": "rw [hy]", "annotated_tactic": ["rw [hy]", []], "state_before": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\nX : Type u_5\ninst\u271d\u2076 : TopologicalSpace H\ninst\u271d\u2075 : TopologicalSpace M\ninst\u271d\u2074 : ChartedSpace H M\ninst\u271d\u00b3 : TopologicalSpace H'\ninst\u271d\u00b2 : TopologicalSpace M'\ninst\u271d\u00b9 : ChartedSpace H' M'\ninst\u271d : TopologicalSpace X\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : PartialHomeomorph M H\nf f' : PartialHomeomorph M' H'\nP : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop\ng\u271d g' : M \u2192 M'\ns t : Set M\nx : M\nQ : (H \u2192 H) \u2192 Set H \u2192 H \u2192 Prop\nhG : G.LocalInvariantProp G' P\ng : M \u2192 H'\nhe : e \u2208 maximalAtlas M G\nxe : x \u2208 e.source\nhe' : e' \u2208 maximalAtlas M G\nxe' : x \u2208 e'.source\ny : H\nhy : \u2191e.symm (\u2191e (\u2191e'.symm y)) = \u2191e'.symm y\n\u22a2 g (\u2191e.symm (\u2191e (\u2191e'.symm y))) = g (\u2191e'.symm y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Comma/StructuredArrow.lean", "full_name": "CategoryTheory.CostructuredArrow.homMk'_mk_comp", "start": [525, 1], "end": [527, 20], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b9 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d : Category.{v\u2082, u\u2082} D\nT T' T'' : D\nY Y' Y'' : C\nS S' : C \u2964 D\nf : S.obj Y \u27f6 T\ng : Y' \u27f6 Y\ng' : Y'' \u27f6 Y'\n\u22a2 mk (S.map (g' \u226b g) \u226b (mk f).hom) = mk (S.map g' \u226b (mk (S.map g \u226b f)).hom)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "full_name": "Finset.prod_range_div", "start": [1695, 1], "end": [1696, 91], "traced_tactics": [{"tactic": "apply prod_range_induction <;> simp", "annotated_tactic": ["apply <a>prod_range_induction</a> <;> simp", [{"full_name": "Finset.prod_range_induction", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [1682, 9], "def_end_pos": [1682, 29]}]], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : CommMonoid \u03b2\nM : Type u_6\ninst\u271d : CommGroup M\nf : \u2115 \u2192 M\nn : \u2115\n\u22a2 \u220f i \u2208 range n, f (i + 1) / f i = f n / f 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "full_name": "Matrix.Pivot.exists_list_transvec_mul_mul_list_transvec_eq_diagonal_aux", "start": [662, 1], "end": [679, 31], "traced_tactics": [{"tactic": "induction' hn : Fintype.card n with r IH generalizing n M", "annotated_tactic": ["induction' hn : <a>Fintype.card</a> n with r IH generalizing n M", [{"full_name": "Fintype.card", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [63, 5], "def_end_pos": [63, 9]}]], "state_before": "n\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\n\u22a2 \u2203 L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D", "state_after": "case zero\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = 0\n\u22a2 \u2203 L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D\n\ncase succ\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr\u271d : \u2115\nM\u271d : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = r + 1\n\u22a2 \u2203 L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D"}, {"tactic": "refine \u27e8List.nil, List.nil, fun _ => 1, ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>List.nil</a>, <a>List.nil</a>, fun _ => 1, ?_\u27e9", [{"full_name": "List.nil", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2286, 5], "def_end_pos": [2286, 8]}, {"full_name": "List.nil", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2286, 5], "def_end_pos": [2286, 8]}]], "state_before": "case zero\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = 0\n\u22a2 \u2203 L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D", "state_after": "case zero\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = 0\n\u22a2 (List.map toMatrix []).prod * M * (List.map toMatrix []).prod = diagonal fun x => 1"}, {"tactic": "ext i j", "annotated_tactic": ["ext i j", []], "state_before": "case zero\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = 0\n\u22a2 (List.map toMatrix []).prod * M * (List.map toMatrix []).prod = diagonal fun x => 1", "state_after": "case zero.a\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = 0\ni j : n\n\u22a2 ((List.map toMatrix []).prod * M * (List.map toMatrix []).prod) i j = diagonal (fun x => 1) i j"}, {"tactic": "rw [Fintype.card_eq_zero_iff] at hn", "annotated_tactic": ["rw [<a>Fintype.card_eq_zero_iff</a>] at hn", [{"full_name": "Fintype.card_eq_zero_iff", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [537, 9], "def_end_pos": [537, 25]}]], "state_before": "case zero.a\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = 0\ni j : n\n\u22a2 ((List.map toMatrix []).prod * M * (List.map toMatrix []).prod) i j = diagonal (fun x => 1) i j", "state_after": "case zero.a\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : IsEmpty n\ni j : n\n\u22a2 ((List.map toMatrix []).prod * M * (List.map toMatrix []).prod) i j = diagonal (fun x => 1) i j"}, {"tactic": "exact hn.elim' i", "annotated_tactic": ["exact hn.elim' i", []], "state_before": "case zero.a\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr : \u2115\nM\u271d : Matrix (Fin r \u2295 Unit) (Fin r \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : IsEmpty n\ni j : n\n\u22a2 ((List.map toMatrix []).prod * M * (List.map toMatrix []).prod) i j = diagonal (fun x => 1) i j", "state_after": "no goals"}, {"tactic": "have e : n \u2243 Sum (Fin r) Unit := by\n  refine Fintype.equivOfCardEq ?_\n  rw [hn]\n  rw [@Fintype.card_sum (Fin r) Unit _ _]\n  simp", "annotated_tactic": ["have e : n \u2243 <a>Sum</a> (<a>Fin</a> r) <a>Unit</a> := by\n      refine <a>Fintype.equivOfCardEq</a> ?_\n      rw [hn]\n      rw [@<a>Fintype.card_sum</a> (<a>Fin</a> r) <a>Unit</a> _ _]\n      simp", [{"full_name": "Sum", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [130, 11], "def_end_pos": [130, 14]}, {"full_name": "Fin", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1831, 11], "def_end_pos": [1831, 14]}, {"full_name": "Unit", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [142, 8], "def_end_pos": [142, 12]}, {"full_name": "Fintype.equivOfCardEq", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [199, 19], "def_end_pos": [199, 32]}, {"full_name": "Fintype.card_sum", "def_path": "Mathlib/Data/Fintype/Sum.lean", "def_pos": [36, 9], "def_end_pos": [36, 25]}, {"full_name": "Fin", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1831, 11], "def_end_pos": [1831, 14]}, {"full_name": "Unit", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [142, 8], "def_end_pos": [142, 12]}]], "state_before": "case succ\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr\u271d : \u2115\nM\u271d : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = r + 1\n\u22a2 \u2203 L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D", "state_after": "case succ\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr\u271d : \u2115\nM\u271d : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = r + 1\ne : n \u2243 Fin r \u2295 Unit\n\u22a2 \u2203 L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D"}, {"tactic": "apply reindex_exists_list_transvec_mul_mul_list_transvec_eq_diagonal M e", "annotated_tactic": ["apply <a>reindex_exists_list_transvec_mul_mul_list_transvec_eq_diagonal</a> M e", [{"full_name": "Matrix.Pivot.reindex_exists_list_transvec_mul_mul_list_transvec_eq_diagonal", "def_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "def_pos": [639, 9], "def_end_pos": [639, 71]}]], "state_before": "case succ\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr\u271d : \u2115\nM\u271d : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = r + 1\ne : n \u2243 Fin r \u2295 Unit\n\u22a2 \u2203 L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D", "state_after": "case succ\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr\u271d : \u2115\nM\u271d : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = r + 1\ne : n \u2243 Fin r \u2295 Unit\n\u22a2 \u2203 L L' D, (List.map toMatrix L).prod * (reindexAlgEquiv \ud835\udd5c e) M * (List.map toMatrix L').prod = diagonal D"}, {"tactic": "apply\n  exists_list_transvec_mul_mul_list_transvec_eq_diagonal_induction fun N =>\n    IH (Fin r) N (by simp)", "annotated_tactic": ["apply\n      <a>exists_list_transvec_mul_mul_list_transvec_eq_diagonal_induction</a> fun N =>\n        IH (<a>Fin</a> r) N (by simp)", [{"full_name": "Matrix.Pivot.exists_list_transvec_mul_mul_list_transvec_eq_diagonal_induction", "def_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "def_pos": [609, 9], "def_end_pos": [609, 73]}, {"full_name": "Fin", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1831, 11], "def_end_pos": [1831, 14]}]], "state_before": "case succ\nn\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr\u271d : \u2115\nM\u271d : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = r + 1\ne : n \u2243 Fin r \u2295 Unit\n\u22a2 \u2203 L L' D, (List.map toMatrix L).prod * (reindexAlgEquiv \ud835\udd5c e) M * (List.map toMatrix L').prod = diagonal D", "state_after": "no goals"}, {"tactic": "refine Fintype.equivOfCardEq ?_", "annotated_tactic": ["refine <a>Fintype.equivOfCardEq</a> ?_", [{"full_name": "Fintype.equivOfCardEq", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [199, 19], "def_end_pos": [199, 32]}]], "state_before": "n\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr\u271d : \u2115\nM\u271d : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = r + 1\n\u22a2 n \u2243 Fin r \u2295 Unit", "state_after": "n\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr\u271d : \u2115\nM\u271d : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = r + 1\n\u22a2 Fintype.card n = Fintype.card (Fin r \u2295 Unit)"}, {"tactic": "rw [hn]", "annotated_tactic": ["rw [hn]", []], "state_before": "n\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr\u271d : \u2115\nM\u271d : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = r + 1\n\u22a2 Fintype.card n = Fintype.card (Fin r \u2295 Unit)", "state_after": "n\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr\u271d : \u2115\nM\u271d : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = r + 1\n\u22a2 r + 1 = Fintype.card (Fin r \u2295 Unit)"}, {"tactic": "rw [@Fintype.card_sum (Fin r) Unit _ _]", "annotated_tactic": ["rw [@<a>Fintype.card_sum</a> (<a>Fin</a> r) <a>Unit</a> _ _]", [{"full_name": "Fintype.card_sum", "def_path": "Mathlib/Data/Fintype/Sum.lean", "def_pos": [36, 9], "def_end_pos": [36, 25]}, {"full_name": "Fin", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1831, 11], "def_end_pos": [1831, 14]}, {"full_name": "Unit", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [142, 8], "def_end_pos": [142, 12]}]], "state_before": "n\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr\u271d : \u2115\nM\u271d : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = r + 1\n\u22a2 r + 1 = Fintype.card (Fin r \u2295 Unit)", "state_after": "n\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr\u271d : \u2115\nM\u271d : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = r + 1\n\u22a2 r + 1 = Fintype.card (Fin r) + Fintype.card Unit"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "n\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr\u271d : \u2115\nM\u271d : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = r + 1\n\u22a2 r + 1 = Fintype.card (Fin r) + Fintype.card Unit", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "n\u271d : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2077 : Field \ud835\udd5c\ninst\u271d\u2076 : DecidableEq n\u271d\ninst\u271d\u2075 : DecidableEq p\ninst\u271d\u2074 : CommRing R\nr\u271d : \u2115\nM\u271d : Matrix (Fin r\u271d \u2295 Unit) (Fin r\u271d \u2295 Unit) \ud835\udd5c\ninst\u271d\u00b3 : Fintype n\u271d\ninst\u271d\u00b2 : Fintype p\nr : \u2115\nIH :\n  \u2200 (n : Type) [inst : Fintype n] [inst_1 : DecidableEq n] (M : Matrix n n \ud835\udd5c),\n    Fintype.card n = r \u2192 \u2203 L L' D, (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D\nn : Type\ninst\u271d\u00b9 : Fintype n\ninst\u271d : DecidableEq n\nM : Matrix n n \ud835\udd5c\nhn : Fintype.card n = r + 1\ne : n \u2243 Fin r \u2295 Unit\nN : Matrix (Fin r) (Fin r) \ud835\udd5c\n\u22a2 Fintype.card (Fin r) = r", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Exponent.lean", "full_name": "Submonoid.pow_exponent_eq_one", "start": [422, 1], "end": [425, 64], "traced_tactics": [{"tactic": "have := Monoid.pow_exponent_eq_one (\u27e8g, g_in_s\u27e9 : S)", "annotated_tactic": ["have := <a>Monoid.pow_exponent_eq_one</a> (\u27e8g, g_in_s\u27e9 : S)", [{"full_name": "Monoid.pow_exponent_eq_one", "def_path": "Mathlib/GroupTheory/Exponent.lean", "def_pos": [149, 9], "def_end_pos": [149, 28]}]], "state_before": "G : Type u\ninst\u271d : Monoid G\nS : Submonoid G\ng : G\ng_in_s : g \u2208 S\n\u22a2 g ^ exponent \u21a5S = 1", "state_after": "G : Type u\ninst\u271d : Monoid G\nS : Submonoid G\ng : G\ng_in_s : g \u2208 S\nthis : \u27e8g, g_in_s\u27e9 ^ exponent \u21a5S = 1\n\u22a2 g ^ exponent \u21a5S = 1"}, {"tactic": "rwa [SubmonoidClass.mk_pow, \u2190 OneMemClass.coe_eq_one] at this", "annotated_tactic": ["rwa [<a>SubmonoidClass.mk_pow</a>, \u2190 <a>OneMemClass.coe_eq_one</a>] at this", [{"full_name": "SubmonoidClass.mk_pow", "def_path": "Mathlib/Algebra/Group/Submonoid/Operations.lean", "def_pos": [568, 9], "def_end_pos": [568, 15]}, {"full_name": "OneMemClass.coe_eq_one", "def_path": "Mathlib/Algebra/Group/Submonoid/Operations.lean", "def_pos": [528, 9], "def_end_pos": [528, 19]}]], "state_before": "G : Type u\ninst\u271d : Monoid G\nS : Submonoid G\ng : G\ng_in_s : g \u2208 S\nthis : \u27e8g, g_in_s\u27e9 ^ exponent \u21a5S = 1\n\u22a2 g ^ exponent \u21a5S = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "full_name": "Submonoid.closure_induction\u2082", "start": [467, 1], "end": [474, 55], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Action/Defs.lean", "full_name": "Function.End.one_def", "start": [752, 1], "end": [752, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/MonoidLocalization.lean", "full_name": "Submonoid.LocalizationMap.ext", "start": [557, 1], "end": [561, 21], "traced_tactics": [{"tactic": "rcases f with \u27e8\u27e8\u27e9\u27e9", "annotated_tactic": ["rcases f with \u27e8\u27e8\u27e9\u27e9", []], "state_before": "M : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\nf g : S.LocalizationMap N\nh : \u2200 (x : M), f.toMap x = g.toMap x\n\u22a2 f = g", "state_after": "case mk.mk\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\ng : S.LocalizationMap N\ntoOneHom\u271d : OneHom M N\nmap_mul'\u271d : \u2200 (x y : M), toOneHom\u271d.toFun (x * y) = toOneHom\u271d.toFun x * toOneHom\u271d.toFun y\nmap_units'\u271d : \u2200 (y : \u21a5S), IsUnit ((\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun \u2191y)\nsurj'\u271d :\n  \u2200 (z : N),\n    \u2203 x,\n      z * (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun \u2191x.2 =\n        (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun x.1\nexists_of_eq\u271d :\n  \u2200 (x y : M),\n    (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun x =\n        (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun y \u2192\n      \u2203 c, \u2191c * x = \u2191c * y\nh :\n  \u2200 (x : M),\n    { toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d, map_units' := map_units'\u271d, surj' := surj'\u271d,\n            exists_of_eq := exists_of_eq\u271d }.toMap\n        x =\n      g.toMap x\n\u22a2 { toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d, map_units' := map_units'\u271d, surj' := surj'\u271d,\n      exists_of_eq := exists_of_eq\u271d } =\n    g"}, {"tactic": "rcases g with \u27e8\u27e8\u27e9\u27e9", "annotated_tactic": ["rcases g with \u27e8\u27e8\u27e9\u27e9", []], "state_before": "case mk.mk\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\ng : S.LocalizationMap N\ntoOneHom\u271d : OneHom M N\nmap_mul'\u271d : \u2200 (x y : M), toOneHom\u271d.toFun (x * y) = toOneHom\u271d.toFun x * toOneHom\u271d.toFun y\nmap_units'\u271d : \u2200 (y : \u21a5S), IsUnit ((\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun \u2191y)\nsurj'\u271d :\n  \u2200 (z : N),\n    \u2203 x,\n      z * (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun \u2191x.2 =\n        (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun x.1\nexists_of_eq\u271d :\n  \u2200 (x y : M),\n    (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun x =\n        (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun y \u2192\n      \u2203 c, \u2191c * x = \u2191c * y\nh :\n  \u2200 (x : M),\n    { toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d, map_units' := map_units'\u271d, surj' := surj'\u271d,\n            exists_of_eq := exists_of_eq\u271d }.toMap\n        x =\n      g.toMap x\n\u22a2 { toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d, map_units' := map_units'\u271d, surj' := surj'\u271d,\n      exists_of_eq := exists_of_eq\u271d } =\n    g", "state_after": "case mk.mk.mk.mk\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\ntoOneHom\u271d\u00b9 : OneHom M N\nmap_mul'\u271d\u00b9 : \u2200 (x y : M), toOneHom\u271d\u00b9.toFun (x * y) = toOneHom\u271d\u00b9.toFun x * toOneHom\u271d\u00b9.toFun y\nmap_units'\u271d\u00b9 : \u2200 (y : \u21a5S), IsUnit ((\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }).toFun \u2191y)\nsurj'\u271d\u00b9 :\n  \u2200 (z : N),\n    \u2203 x,\n      z * (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }).toFun \u2191x.2 =\n        (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }).toFun x.1\nexists_of_eq\u271d\u00b9 :\n  \u2200 (x y : M),\n    (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }).toFun x =\n        (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }).toFun y \u2192\n      \u2203 c, \u2191c * x = \u2191c * y\ntoOneHom\u271d : OneHom M N\nmap_mul'\u271d : \u2200 (x y : M), toOneHom\u271d.toFun (x * y) = toOneHom\u271d.toFun x * toOneHom\u271d.toFun y\nmap_units'\u271d : \u2200 (y : \u21a5S), IsUnit ((\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun \u2191y)\nsurj'\u271d :\n  \u2200 (z : N),\n    \u2203 x,\n      z * (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun \u2191x.2 =\n        (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun x.1\nexists_of_eq\u271d :\n  \u2200 (x y : M),\n    (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun x =\n        (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun y \u2192\n      \u2203 c, \u2191c * x = \u2191c * y\nh :\n  \u2200 (x : M),\n    { toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9, map_units' := map_units'\u271d\u00b9, surj' := surj'\u271d\u00b9,\n            exists_of_eq := exists_of_eq\u271d\u00b9 }.toMap\n        x =\n      { toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d, map_units' := map_units'\u271d, surj' := surj'\u271d,\n            exists_of_eq := exists_of_eq\u271d }.toMap\n        x\n\u22a2 { toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9, map_units' := map_units'\u271d\u00b9, surj' := surj'\u271d\u00b9,\n      exists_of_eq := exists_of_eq\u271d\u00b9 } =\n    { toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d, map_units' := map_units'\u271d, surj' := surj'\u271d,\n      exists_of_eq := exists_of_eq\u271d }"}, {"tactic": "simp only [mk.injEq, MonoidHom.mk.injEq]", "annotated_tactic": ["simp only [mk.injEq, MonoidHom.mk.injEq]", []], "state_before": "case mk.mk.mk.mk\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\ntoOneHom\u271d\u00b9 : OneHom M N\nmap_mul'\u271d\u00b9 : \u2200 (x y : M), toOneHom\u271d\u00b9.toFun (x * y) = toOneHom\u271d\u00b9.toFun x * toOneHom\u271d\u00b9.toFun y\nmap_units'\u271d\u00b9 : \u2200 (y : \u21a5S), IsUnit ((\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }).toFun \u2191y)\nsurj'\u271d\u00b9 :\n  \u2200 (z : N),\n    \u2203 x,\n      z * (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }).toFun \u2191x.2 =\n        (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }).toFun x.1\nexists_of_eq\u271d\u00b9 :\n  \u2200 (x y : M),\n    (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }).toFun x =\n        (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }).toFun y \u2192\n      \u2203 c, \u2191c * x = \u2191c * y\ntoOneHom\u271d : OneHom M N\nmap_mul'\u271d : \u2200 (x y : M), toOneHom\u271d.toFun (x * y) = toOneHom\u271d.toFun x * toOneHom\u271d.toFun y\nmap_units'\u271d : \u2200 (y : \u21a5S), IsUnit ((\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun \u2191y)\nsurj'\u271d :\n  \u2200 (z : N),\n    \u2203 x,\n      z * (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun \u2191x.2 =\n        (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun x.1\nexists_of_eq\u271d :\n  \u2200 (x y : M),\n    (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun x =\n        (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun y \u2192\n      \u2203 c, \u2191c * x = \u2191c * y\nh :\n  \u2200 (x : M),\n    { toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9, map_units' := map_units'\u271d\u00b9, surj' := surj'\u271d\u00b9,\n            exists_of_eq := exists_of_eq\u271d\u00b9 }.toMap\n        x =\n      { toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d, map_units' := map_units'\u271d, surj' := surj'\u271d,\n            exists_of_eq := exists_of_eq\u271d }.toMap\n        x\n\u22a2 { toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9, map_units' := map_units'\u271d\u00b9, surj' := surj'\u271d\u00b9,\n      exists_of_eq := exists_of_eq\u271d\u00b9 } =\n    { toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d, map_units' := map_units'\u271d, surj' := surj'\u271d,\n      exists_of_eq := exists_of_eq\u271d }", "state_after": "case mk.mk.mk.mk\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\ntoOneHom\u271d\u00b9 : OneHom M N\nmap_mul'\u271d\u00b9 : \u2200 (x y : M), toOneHom\u271d\u00b9.toFun (x * y) = toOneHom\u271d\u00b9.toFun x * toOneHom\u271d\u00b9.toFun y\nmap_units'\u271d\u00b9 : \u2200 (y : \u21a5S), IsUnit ((\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }).toFun \u2191y)\nsurj'\u271d\u00b9 :\n  \u2200 (z : N),\n    \u2203 x,\n      z * (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }).toFun \u2191x.2 =\n        (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }).toFun x.1\nexists_of_eq\u271d\u00b9 :\n  \u2200 (x y : M),\n    (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }).toFun x =\n        (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }).toFun y \u2192\n      \u2203 c, \u2191c * x = \u2191c * y\ntoOneHom\u271d : OneHom M N\nmap_mul'\u271d : \u2200 (x y : M), toOneHom\u271d.toFun (x * y) = toOneHom\u271d.toFun x * toOneHom\u271d.toFun y\nmap_units'\u271d : \u2200 (y : \u21a5S), IsUnit ((\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun \u2191y)\nsurj'\u271d :\n  \u2200 (z : N),\n    \u2203 x,\n      z * (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun \u2191x.2 =\n        (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun x.1\nexists_of_eq\u271d :\n  \u2200 (x y : M),\n    (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun x =\n        (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun y \u2192\n      \u2203 c, \u2191c * x = \u2191c * y\nh :\n  \u2200 (x : M),\n    { toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9, map_units' := map_units'\u271d\u00b9, surj' := surj'\u271d\u00b9,\n            exists_of_eq := exists_of_eq\u271d\u00b9 }.toMap\n        x =\n      { toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d, map_units' := map_units'\u271d, surj' := surj'\u271d,\n            exists_of_eq := exists_of_eq\u271d }.toMap\n        x\n\u22a2 toOneHom\u271d\u00b9 = toOneHom\u271d"}, {"tactic": "exact OneHom.ext h", "annotated_tactic": ["exact <a>OneHom.ext</a> h", [{"full_name": "OneHom.ext", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [588, 9], "def_end_pos": [588, 19]}]], "state_before": "case mk.mk.mk.mk\nM : Type u_1\ninst\u271d\u00b2 : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst\u271d\u00b9 : CommMonoid N\nP : Type u_3\ninst\u271d : CommMonoid P\ntoOneHom\u271d\u00b9 : OneHom M N\nmap_mul'\u271d\u00b9 : \u2200 (x y : M), toOneHom\u271d\u00b9.toFun (x * y) = toOneHom\u271d\u00b9.toFun x * toOneHom\u271d\u00b9.toFun y\nmap_units'\u271d\u00b9 : \u2200 (y : \u21a5S), IsUnit ((\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }).toFun \u2191y)\nsurj'\u271d\u00b9 :\n  \u2200 (z : N),\n    \u2203 x,\n      z * (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }).toFun \u2191x.2 =\n        (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }).toFun x.1\nexists_of_eq\u271d\u00b9 :\n  \u2200 (x y : M),\n    (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }).toFun x =\n        (\u2191{ toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9 }).toFun y \u2192\n      \u2203 c, \u2191c * x = \u2191c * y\ntoOneHom\u271d : OneHom M N\nmap_mul'\u271d : \u2200 (x y : M), toOneHom\u271d.toFun (x * y) = toOneHom\u271d.toFun x * toOneHom\u271d.toFun y\nmap_units'\u271d : \u2200 (y : \u21a5S), IsUnit ((\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun \u2191y)\nsurj'\u271d :\n  \u2200 (z : N),\n    \u2203 x,\n      z * (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun \u2191x.2 =\n        (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun x.1\nexists_of_eq\u271d :\n  \u2200 (x y : M),\n    (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun x =\n        (\u2191{ toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d }).toFun y \u2192\n      \u2203 c, \u2191c * x = \u2191c * y\nh :\n  \u2200 (x : M),\n    { toOneHom := toOneHom\u271d\u00b9, map_mul' := map_mul'\u271d\u00b9, map_units' := map_units'\u271d\u00b9, surj' := surj'\u271d\u00b9,\n            exists_of_eq := exists_of_eq\u271d\u00b9 }.toMap\n        x =\n      { toOneHom := toOneHom\u271d, map_mul' := map_mul'\u271d, map_units' := map_units'\u271d, surj' := surj'\u271d,\n            exists_of_eq := exists_of_eq\u271d }.toMap\n        x\n\u22a2 toOneHom\u271d\u00b9 = toOneHom\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "full_name": "Int.cast_dvd_cast", "start": [140, 1], "end": [141, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "full_name": "Function.surjective_pi_map", "start": [525, 1], "end": [527, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Sigma/Basic.lean", "full_name": "CategoryTheory.Sigma.natTrans_app", "start": [115, 1], "end": [118, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Complex/Basic.lean", "full_name": "Complex.nnnorm_real", "start": [198, 1], "end": [199, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Ring/Canonical.lean", "full_name": "CanonicallyOrderedCommSemiring.mul_pos", "start": [79, 11], "end": [80, 58], "traced_tactics": [{"tactic": "simp only [pos_iff_ne_zero, ne_eq, mul_eq_zero, not_or]", "annotated_tactic": ["simp only [<a>pos_iff_ne_zero</a>, <a>ne_eq</a>, <a>mul_eq_zero</a>, <a>not_or</a>]", [{"full_name": "pos_iff_ne_zero", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [230, 3], "def_end_pos": [230, 14]}, {"full_name": "ne_eq", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [89, 17], "def_end_pos": [89, 22]}, {"full_name": "mul_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [282, 9], "def_end_pos": [282, 20]}, {"full_name": "not_or", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [134, 17], "def_end_pos": [134, 23]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\ninst\u271d : CanonicallyOrderedCommSemiring \u03b1\na b c d : \u03b1\n\u22a2 0 < a * b \u2194 0 < a \u2227 0 < b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Polynomial/Nilpotent.lean", "full_name": "Polynomial.isNilpotent_iff", "start": [71, 1], "end": [89, 53], "traced_tactics": [{"tactic": "refine\n  \u27e8P.recOnHorner (by simp) (fun p r hp\u2080 _ hp hpr i \u21a6 ?_) (fun p _ hnp hpX i \u21a6 ?_), fun h \u21a6 ?_\u27e9", "annotated_tactic": ["refine\n    \u27e8P.recOnHorner (by simp) (fun p r hp\u2080 _ hp hpr i \u21a6 ?_) (fun p _ hnp hpX i \u21a6 ?_), fun h \u21a6 ?_\u27e9", []], "state_before": "R : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\n\u22a2 IsNilpotent P \u2194 \u2200 (i : \u2115), IsNilpotent (P.coeff i)", "state_after": "case refine_1\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nh : \u2200 (i : \u2115), IsNilpotent (P.coeff i)\n\u22a2 IsNilpotent P\n\ncase refine_2\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : p.coeff 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\nhpr : IsNilpotent (p + C r)\ni : \u2115\n\u22a2 IsNilpotent ((p + C r).coeff i)\n\ncase refine_3\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP p : R[X]\nx\u271d : p \u2260 0\nhnp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\nhpX : IsNilpotent (p * X)\ni : \u2115\n\u22a2 IsNilpotent ((p * X).coeff i)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "R : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\n\u22a2 IsNilpotent 0 \u2192 \u2200 (i : \u2115), IsNilpotent (coeff 0 i)", "state_after": "no goals"}, {"tactic": "rw [\u2190 sum_monomial_eq P]", "annotated_tactic": ["rw [\u2190 <a>sum_monomial_eq</a> P]", [{"full_name": "Polynomial.sum_monomial_eq", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [1041, 9], "def_end_pos": [1041, 24]}]], "state_before": "case refine_1\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nh : \u2200 (i : \u2115), IsNilpotent (P.coeff i)\n\u22a2 IsNilpotent P", "state_after": "case refine_1\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nh : \u2200 (i : \u2115), IsNilpotent (P.coeff i)\n\u22a2 IsNilpotent (P.sum fun n a => (monomial n) a)"}, {"tactic": "exact isNilpotent_sum (fun i _ \u21a6 by simpa only [isNilpotent_monomial_iff] using h i)", "annotated_tactic": ["exact <a>isNilpotent_sum</a> (fun i _ \u21a6 by simpa only [<a>isNilpotent_monomial_iff</a>] using h i)", [{"full_name": "isNilpotent_sum", "def_path": "Mathlib/RingTheory/Nilpotent/Basic.lean", "def_pos": [189, 7], "def_end_pos": [189, 22]}, {"full_name": "Polynomial.isNilpotent_monomial_iff", "def_path": "Mathlib/RingTheory/Polynomial/Nilpotent.lean", "def_pos": [45, 15], "def_end_pos": [45, 39]}]], "state_before": "case refine_1\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nh : \u2200 (i : \u2115), IsNilpotent (P.coeff i)\n\u22a2 IsNilpotent (P.sum fun n a => (monomial n) a)", "state_after": "no goals"}, {"tactic": "simpa only [isNilpotent_monomial_iff] using h i", "annotated_tactic": ["simpa only [<a>isNilpotent_monomial_iff</a>] using h i", [{"full_name": "Polynomial.isNilpotent_monomial_iff", "def_path": "Mathlib/RingTheory/Polynomial/Nilpotent.lean", "def_pos": [45, 15], "def_end_pos": [45, 39]}]], "state_before": "R : Type u_1\nr : R\ninst\u271d : CommRing R\nP : R[X]\nh : \u2200 (i : \u2115), IsNilpotent (P.coeff i)\ni : \u2115\nx\u271d : i \u2208 P.support\n\u22a2 IsNilpotent ((fun n a => (monomial n) a) i (P.coeff i))", "state_after": "no goals"}, {"tactic": "have hr : IsNilpotent (C r) := by\n  obtain \u27e8k, hk\u27e9 := hpr\n  replace hp : eval 0 p = 0 := by rwa [coeff_zero_eq_aeval_zero] at hp\u2080\n  refine isNilpotent_C_iff.mpr \u27e8k, ?_\u27e9\n  simpa [coeff_zero_eq_aeval_zero, hp] using congr_arg (fun q \u21a6 coeff q 0) hk", "annotated_tactic": ["have hr : <a>IsNilpotent</a> (<a>C</a> r) := by\n      obtain \u27e8k, hk\u27e9 := hpr\n      replace hp : <a>eval</a> 0 p = 0 := by rwa [<a>coeff_zero_eq_aeval_zero</a>] at hp\u2080\n      refine isNilpotent_C_iff.mpr \u27e8k, ?_\u27e9\n      simpa [<a>coeff_zero_eq_aeval_zero</a>, hp] using <a>congr_arg</a> (fun q \u21a6 <a>coeff</a> q 0) hk", [{"full_name": "IsNilpotent", "def_path": "Mathlib/RingTheory/Nilpotent/Defs.lean", "def_pos": [40, 5], "def_end_pos": [40, 16]}, {"full_name": "Polynomial.C", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [501, 5], "def_end_pos": [501, 6]}, {"full_name": "Polynomial.eval", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [310, 5], "def_end_pos": [310, 9]}, {"full_name": "Polynomial.coeff_zero_eq_aeval_zero", "def_path": "Mathlib/Algebra/Polynomial/AlgebraMap.lean", "def_pos": [340, 9], "def_end_pos": [340, 33]}, {"full_name": "Polynomial.coeff_zero_eq_aeval_zero", "def_path": "Mathlib/Algebra/Polynomial/AlgebraMap.lean", "def_pos": [340, 9], "def_end_pos": [340, 33]}, {"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "Polynomial.coeff", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [663, 5], "def_end_pos": [663, 10]}]], "state_before": "case refine_2\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : p.coeff 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\nhpr : IsNilpotent (p + C r)\ni : \u2115\n\u22a2 IsNilpotent ((p + C r).coeff i)", "state_after": "case refine_2\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : p.coeff 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\nhpr : IsNilpotent (p + C r)\ni : \u2115\nhr : IsNilpotent (C r)\n\u22a2 IsNilpotent ((p + C r).coeff i)"}, {"tactic": "cases' i with i", "annotated_tactic": ["cases' i with i", []], "state_before": "case refine_2\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : p.coeff 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\nhpr : IsNilpotent (p + C r)\ni : \u2115\nhr : IsNilpotent (C r)\n\u22a2 IsNilpotent ((p + C r).coeff i)", "state_after": "case refine_2.zero\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : p.coeff 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\nhpr : IsNilpotent (p + C r)\nhr : IsNilpotent (C r)\n\u22a2 IsNilpotent ((p + C r).coeff 0)\n\ncase refine_2.succ\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : p.coeff 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\nhpr : IsNilpotent (p + C r)\nhr : IsNilpotent (C r)\ni : \u2115\n\u22a2 IsNilpotent ((p + C r).coeff (i + 1))"}, {"tactic": "simp only [coeff_add, coeff_C_succ, add_zero]", "annotated_tactic": ["simp only [<a>coeff_add</a>, <a>coeff_C_succ</a>, <a>add_zero</a>]", [{"full_name": "Polynomial.coeff_add", "def_path": "Mathlib/Algebra/Polynomial/Coeff.lean", "def_pos": [40, 9], "def_end_pos": [40, 18]}, {"full_name": "Polynomial.coeff_C_succ", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [748, 7], "def_end_pos": [748, 19]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [482, 3], "def_end_pos": [482, 14]}]], "state_before": "case refine_2.succ\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : p.coeff 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\nhpr : IsNilpotent (p + C r)\nhr : IsNilpotent (C r)\ni : \u2115\n\u22a2 IsNilpotent ((p + C r).coeff (i + 1))", "state_after": "case refine_2.succ\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : p.coeff 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\nhpr : IsNilpotent (p + C r)\nhr : IsNilpotent (C r)\ni : \u2115\n\u22a2 IsNilpotent (p.coeff (i + 1))"}, {"tactic": "apply hp", "annotated_tactic": ["apply hp", []], "state_before": "case refine_2.succ\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : p.coeff 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\nhpr : IsNilpotent (p + C r)\nhr : IsNilpotent (C r)\ni : \u2115\n\u22a2 IsNilpotent (p.coeff (i + 1))", "state_after": "case refine_2.succ.a\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : p.coeff 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\nhpr : IsNilpotent (p + C r)\nhr : IsNilpotent (C r)\ni : \u2115\n\u22a2 IsNilpotent p"}, {"tactic": "simpa using Commute.isNilpotent_sub (Commute.all _ _) hpr hr", "annotated_tactic": ["simpa using <a>Commute.isNilpotent_sub</a> (<a>Commute.all</a> _ _) hpr hr", [{"full_name": "Commute.isNilpotent_sub", "def_path": "Mathlib/RingTheory/Nilpotent/Basic.lean", "def_pos": [174, 9], "def_end_pos": [174, 24]}, {"full_name": "Commute.all", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [145, 19], "def_end_pos": [145, 22]}]], "state_before": "case refine_2.succ.a\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : p.coeff 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\nhpr : IsNilpotent (p + C r)\nhr : IsNilpotent (C r)\ni : \u2115\n\u22a2 IsNilpotent p", "state_after": "no goals"}, {"tactic": "obtain \u27e8k, hk\u27e9 := hpr", "annotated_tactic": ["obtain \u27e8k, hk\u27e9 := hpr", []], "state_before": "R : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : p.coeff 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\nhpr : IsNilpotent (p + C r)\ni : \u2115\n\u22a2 IsNilpotent (C r)", "state_after": "case intro\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : p.coeff 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\ni k : \u2115\nhk : (p + C r) ^ k = 0\n\u22a2 IsNilpotent (C r)"}, {"tactic": "replace hp : eval 0 p = 0 := by rwa [coeff_zero_eq_aeval_zero] at hp\u2080", "annotated_tactic": ["replace hp : <a>eval</a> 0 p = 0 := by rwa [<a>coeff_zero_eq_aeval_zero</a>] at hp\u2080", [{"full_name": "Polynomial.eval", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [310, 5], "def_end_pos": [310, 9]}, {"full_name": "Polynomial.coeff_zero_eq_aeval_zero", "def_path": "Mathlib/Algebra/Polynomial/AlgebraMap.lean", "def_pos": [340, 9], "def_end_pos": [340, 33]}]], "state_before": "case intro\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : p.coeff 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\ni k : \u2115\nhk : (p + C r) ^ k = 0\n\u22a2 IsNilpotent (C r)", "state_after": "case intro\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : p.coeff 0 = 0\nx\u271d : r \u2260 0\ni k : \u2115\nhk : (p + C r) ^ k = 0\nhp : eval 0 p = 0\n\u22a2 IsNilpotent (C r)"}, {"tactic": "refine isNilpotent_C_iff.mpr \u27e8k, ?_\u27e9", "annotated_tactic": ["refine isNilpotent_C_iff.mpr \u27e8k, ?_\u27e9", []], "state_before": "case intro\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : p.coeff 0 = 0\nx\u271d : r \u2260 0\ni k : \u2115\nhk : (p + C r) ^ k = 0\nhp : eval 0 p = 0\n\u22a2 IsNilpotent (C r)", "state_after": "case intro\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : p.coeff 0 = 0\nx\u271d : r \u2260 0\ni k : \u2115\nhk : (p + C r) ^ k = 0\nhp : eval 0 p = 0\n\u22a2 r ^ k = 0"}, {"tactic": "simpa [coeff_zero_eq_aeval_zero, hp] using congr_arg (fun q \u21a6 coeff q 0) hk", "annotated_tactic": ["simpa [<a>coeff_zero_eq_aeval_zero</a>, hp] using <a>congr_arg</a> (fun q \u21a6 <a>coeff</a> q 0) hk", [{"full_name": "Polynomial.coeff_zero_eq_aeval_zero", "def_path": "Mathlib/Algebra/Polynomial/AlgebraMap.lean", "def_pos": [340, 9], "def_end_pos": [340, 33]}, {"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "Polynomial.coeff", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [663, 5], "def_end_pos": [663, 10]}]], "state_before": "case intro\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : p.coeff 0 = 0\nx\u271d : r \u2260 0\ni k : \u2115\nhk : (p + C r) ^ k = 0\nhp : eval 0 p = 0\n\u22a2 r ^ k = 0", "state_after": "no goals"}, {"tactic": "rwa [coeff_zero_eq_aeval_zero] at hp\u2080", "annotated_tactic": ["rwa [<a>coeff_zero_eq_aeval_zero</a>] at hp\u2080", [{"full_name": "Polynomial.coeff_zero_eq_aeval_zero", "def_path": "Mathlib/Algebra/Polynomial/AlgebraMap.lean", "def_pos": [340, 9], "def_end_pos": [340, 33]}]], "state_before": "R : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : p.coeff 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\ni k : \u2115\nhk : (p + C r) ^ k = 0\n\u22a2 eval 0 p = 0", "state_after": "no goals"}, {"tactic": "simpa [hp\u2080] using hr", "annotated_tactic": ["simpa [hp\u2080] using hr", []], "state_before": "case refine_2.zero\nR : Type u_1\nr\u271d : R\ninst\u271d : CommRing R\nP p : R[X]\nr : R\nhp\u2080 : p.coeff 0 = 0\nx\u271d : r \u2260 0\nhp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\nhpr : IsNilpotent (p + C r)\nhr : IsNilpotent (C r)\n\u22a2 IsNilpotent ((p + C r).coeff 0)", "state_after": "no goals"}, {"tactic": "cases' i with i", "annotated_tactic": ["cases' i with i", []], "state_before": "case refine_3\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP p : R[X]\nx\u271d : p \u2260 0\nhnp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\nhpX : IsNilpotent (p * X)\ni : \u2115\n\u22a2 IsNilpotent ((p * X).coeff i)", "state_after": "case refine_3.zero\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP p : R[X]\nx\u271d : p \u2260 0\nhnp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\nhpX : IsNilpotent (p * X)\n\u22a2 IsNilpotent ((p * X).coeff 0)\n\ncase refine_3.succ\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP p : R[X]\nx\u271d : p \u2260 0\nhnp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\nhpX : IsNilpotent (p * X)\ni : \u2115\n\u22a2 IsNilpotent ((p * X).coeff (i + 1))"}, {"tactic": "simpa using hnp (isNilpotent_mul_X_iff.mp hpX) i", "annotated_tactic": ["simpa using hnp (isNilpotent_mul_X_iff.mp hpX) i", []], "state_before": "case refine_3.succ\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP p : R[X]\nx\u271d : p \u2260 0\nhnp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\nhpX : IsNilpotent (p * X)\ni : \u2115\n\u22a2 IsNilpotent ((p * X).coeff (i + 1))", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case refine_3.zero\nR : Type u_1\nr : R\ninst\u271d : CommRing R\nP p : R[X]\nx\u271d : p \u2260 0\nhnp : IsNilpotent p \u2192 \u2200 (i : \u2115), IsNilpotent (p.coeff i)\nhpX : IsNilpotent (p * X)\n\u22a2 IsNilpotent ((p * X).coeff 0)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "List.toFinset_eq_of_perm", "start": [3280, 1], "end": [3281, 41], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/Pi/Wallis.lean", "full_name": "Real.Wallis.tendsto_W_nhds_pi_div_two", "start": [101, 1], "end": [114, 60], "traced_tactics": [{"tactic": "refine tendsto_of_tendsto_of_tendsto_of_le_of_le ?_ tendsto_const_nhds le_W W_le", "annotated_tactic": ["refine <a>tendsto_of_tendsto_of_tendsto_of_le_of_le</a> ?_ <a>tendsto_const_nhds</a> <a>le_W</a> <a>W_le</a>", [{"full_name": "tendsto_of_tendsto_of_tendsto_of_le_of_le", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [162, 9], "def_end_pos": [162, 50]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [991, 9], "def_end_pos": [991, 27]}, {"full_name": "Real.Wallis.le_W", "def_path": "Mathlib/Data/Real/Pi/Wallis.lean", "def_pos": [91, 9], "def_end_pos": [91, 13]}, {"full_name": "Real.Wallis.W_le", "def_path": "Mathlib/Data/Real/Pi/Wallis.lean", "def_pos": [85, 9], "def_end_pos": [85, 13]}]], "state_before": "\u22a2 Tendsto W atTop (\ud835\udcdd (\u03c0 / 2))", "state_after": "\u22a2 Tendsto (fun i => (2 * \u2191i + 1) / (2 * \u2191i + 2) * (\u03c0 / 2)) atTop (\ud835\udcdd (\u03c0 / 2))"}, {"tactic": "have : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2)) := by rw [sub_zero, one_mul]", "annotated_tactic": ["have : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2)) := by rw [<a>sub_zero</a>, <a>one_mul</a>]", [{"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [489, 3], "def_end_pos": [489, 14]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "\u22a2 Tendsto (fun i => (2 * \u2191i + 1) / (2 * \u2191i + 2) * (\u03c0 / 2)) atTop (\ud835\udcdd (\u03c0 / 2))", "state_after": "this : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))\n\u22a2 Tendsto (fun i => (2 * \u2191i + 1) / (2 * \u2191i + 2) * (\u03c0 / 2)) atTop (\ud835\udcdd (\u03c0 / 2))"}, {"tactic": "rw [this]", "annotated_tactic": ["rw [this]", []], "state_before": "this : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))\n\u22a2 Tendsto (fun i => (2 * \u2191i + 1) / (2 * \u2191i + 2) * (\u03c0 / 2)) atTop (\ud835\udcdd (\u03c0 / 2))", "state_after": "this : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))\n\u22a2 Tendsto (fun i => (2 * \u2191i + 1) / (2 * \u2191i + 2) * (\u03c0 / 2)) atTop (\ud835\udcdd ((1 - 0) * (\u03c0 / 2)))"}, {"tactic": "refine Tendsto.mul ?_ tendsto_const_nhds", "annotated_tactic": ["refine <a>Tendsto.mul</a> ?_ <a>tendsto_const_nhds</a>", [{"full_name": "Filter.Tendsto.mul", "def_path": "Mathlib/Topology/Algebra/Monoid.lean", "def_pos": [129, 9], "def_end_pos": [129, 27]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [991, 9], "def_end_pos": [991, 27]}]], "state_before": "this : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))\n\u22a2 Tendsto (fun i => (2 * \u2191i + 1) / (2 * \u2191i + 2) * (\u03c0 / 2)) atTop (\ud835\udcdd ((1 - 0) * (\u03c0 / 2)))", "state_after": "this : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))\n\u22a2 Tendsto (fun i => (2 * \u2191i + 1) / (2 * \u2191i + 2)) atTop (\ud835\udcdd (1 - 0))"}, {"tactic": "have h : \u2200 n : \u2115, ((2 : \u211d) * n + 1) / (2 * n + 2) = 1 - 1 / (2 * n + 2) := by\n  intro n\n  rw [sub_div' _ _ _ (ne_of_gt (add_pos_of_nonneg_of_pos (mul_nonneg\n    (two_pos : 0 < (2 : \u211d)).le (Nat.cast_nonneg _)) two_pos)), one_mul]\n  congr 1; ring", "annotated_tactic": ["have h : \u2200 n : \u2115, ((2 : \u211d) * n + 1) / (2 * n + 2) = 1 - 1 / (2 * n + 2) := by\n    intro n\n    rw [<a>sub_div'</a> _ _ _ (<a>ne_of_gt</a> (<a>add_pos_of_nonneg_of_pos</a> (<a>mul_nonneg</a>\n      (<a>two_pos</a> : 0 < (2 : \u211d)).<a>le</a> (<a>Nat.cast_nonneg</a> _)) <a>two_pos</a>)), <a>one_mul</a>]\n    congr 1; ring", [{"full_name": "sub_div'", "def_path": "Mathlib/Algebra/Field/Basic.lean", "def_pos": [241, 9], "def_end_pos": [241, 17]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}, {"full_name": "add_pos_of_nonneg_of_pos", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [1095, 24], "def_end_pos": [1095, 48]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [437, 7], "def_end_pos": [437, 17]}, {"full_name": "two_pos", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [113, 7], "def_end_pos": [113, 14]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}, {"full_name": "Nat.cast_nonneg", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [50, 9], "def_end_pos": [50, 20]}, {"full_name": "two_pos", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [113, 7], "def_end_pos": [113, 14]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "this : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))\n\u22a2 Tendsto (fun i => (2 * \u2191i + 1) / (2 * \u2191i + 2)) atTop (\ud835\udcdd (1 - 0))", "state_after": "this : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))\nh : \u2200 (n : \u2115), (2 * \u2191n + 1) / (2 * \u2191n + 2) = 1 - 1 / (2 * \u2191n + 2)\n\u22a2 Tendsto (fun i => (2 * \u2191i + 1) / (2 * \u2191i + 2)) atTop (\ud835\udcdd (1 - 0))"}, {"tactic": "simp_rw [h]", "annotated_tactic": ["simp_rw [h]", []], "state_before": "this : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))\nh : \u2200 (n : \u2115), (2 * \u2191n + 1) / (2 * \u2191n + 2) = 1 - 1 / (2 * \u2191n + 2)\n\u22a2 Tendsto (fun i => (2 * \u2191i + 1) / (2 * \u2191i + 2)) atTop (\ud835\udcdd (1 - 0))", "state_after": "this : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))\nh : \u2200 (n : \u2115), (2 * \u2191n + 1) / (2 * \u2191n + 2) = 1 - 1 / (2 * \u2191n + 2)\n\u22a2 Tendsto (fun i => 1 - 1 / (2 * \u2191i + 2)) atTop (\ud835\udcdd (1 - 0))"}, {"tactic": "refine (tendsto_const_nhds.div_atTop ?_).const_sub _", "annotated_tactic": ["refine (tendsto_const_nhds.div_atTop ?_).<a>const_sub</a> _", [{"full_name": "Filter.Tendsto.const_sub", "def_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "def_pos": [1140, 15], "def_end_pos": [1140, 24]}]], "state_before": "this : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))\nh : \u2200 (n : \u2115), (2 * \u2191n + 1) / (2 * \u2191n + 2) = 1 - 1 / (2 * \u2191n + 2)\n\u22a2 Tendsto (fun i => 1 - 1 / (2 * \u2191i + 2)) atTop (\ud835\udcdd (1 - 0))", "state_after": "this : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))\nh : \u2200 (n : \u2115), (2 * \u2191n + 1) / (2 * \u2191n + 2) = 1 - 1 / (2 * \u2191n + 2)\n\u22a2 Tendsto (fun i => 2 * \u2191i + 2) atTop atTop"}, {"tactic": "refine Tendsto.atTop_add ?_ tendsto_const_nhds", "annotated_tactic": ["refine <a>Tendsto.atTop_add</a> ?_ <a>tendsto_const_nhds</a>", [{"full_name": "Filter.Tendsto.atTop_add", "def_path": "Mathlib/Topology/Order/LeftRightNhds.lean", "def_pos": [354, 9], "def_end_pos": [354, 33]}, {"full_name": "tendsto_const_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [991, 9], "def_end_pos": [991, 27]}]], "state_before": "this : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))\nh : \u2200 (n : \u2115), (2 * \u2191n + 1) / (2 * \u2191n + 2) = 1 - 1 / (2 * \u2191n + 2)\n\u22a2 Tendsto (fun i => 2 * \u2191i + 2) atTop atTop", "state_after": "this : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))\nh : \u2200 (n : \u2115), (2 * \u2191n + 1) / (2 * \u2191n + 2) = 1 - 1 / (2 * \u2191n + 2)\n\u22a2 Tendsto (fun i => 2 * \u2191i) atTop atTop"}, {"tactic": "exact tendsto_natCast_atTop_atTop.const_mul_atTop two_pos", "annotated_tactic": ["exact tendsto_natCast_atTop_atTop.const_mul_atTop <a>two_pos</a>", [{"full_name": "two_pos", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [113, 7], "def_end_pos": [113, 14]}]], "state_before": "this : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))\nh : \u2200 (n : \u2115), (2 * \u2191n + 1) / (2 * \u2191n + 2) = 1 - 1 / (2 * \u2191n + 2)\n\u22a2 Tendsto (fun i => 2 * \u2191i) atTop atTop", "state_after": "no goals"}, {"tactic": "rw [sub_zero, one_mul]", "annotated_tactic": ["rw [<a>sub_zero</a>, <a>one_mul</a>]", [{"full_name": "sub_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [489, 3], "def_end_pos": [489, 14]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "\u22a2 \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "this : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))\n\u22a2 \u2200 (n : \u2115), (2 * \u2191n + 1) / (2 * \u2191n + 2) = 1 - 1 / (2 * \u2191n + 2)", "state_after": "this : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))\nn : \u2115\n\u22a2 (2 * \u2191n + 1) / (2 * \u2191n + 2) = 1 - 1 / (2 * \u2191n + 2)"}, {"tactic": "rw [sub_div' _ _ _ (ne_of_gt (add_pos_of_nonneg_of_pos (mul_nonneg\n  (two_pos : 0 < (2 : \u211d)).le (Nat.cast_nonneg _)) two_pos)), one_mul]", "annotated_tactic": ["rw [<a>sub_div'</a> _ _ _ (<a>ne_of_gt</a> (<a>add_pos_of_nonneg_of_pos</a> (<a>mul_nonneg</a>\n      (<a>two_pos</a> : 0 < (2 : \u211d)).<a>le</a> (<a>Nat.cast_nonneg</a> _)) <a>two_pos</a>)), <a>one_mul</a>]", [{"full_name": "sub_div'", "def_path": "Mathlib/Algebra/Field/Basic.lean", "def_pos": [241, 9], "def_end_pos": [241, 17]}, {"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}, {"full_name": "add_pos_of_nonneg_of_pos", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [1095, 24], "def_end_pos": [1095, 48]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [437, 7], "def_end_pos": [437, 17]}, {"full_name": "two_pos", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [113, 7], "def_end_pos": [113, 14]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}, {"full_name": "Nat.cast_nonneg", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [50, 9], "def_end_pos": [50, 20]}, {"full_name": "two_pos", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [113, 7], "def_end_pos": [113, 14]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "this : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))\nn : \u2115\n\u22a2 (2 * \u2191n + 1) / (2 * \u2191n + 2) = 1 - 1 / (2 * \u2191n + 2)", "state_after": "this : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))\nn : \u2115\n\u22a2 (2 * \u2191n + 1) / (2 * \u2191n + 2) = (2 * \u2191n + 2 - 1) / (2 * \u2191n + 2)"}, {"tactic": "congr 1", "annotated_tactic": ["congr 1", []], "state_before": "this : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))\nn : \u2115\n\u22a2 (2 * \u2191n + 1) / (2 * \u2191n + 2) = (2 * \u2191n + 2 - 1) / (2 * \u2191n + 2)", "state_after": "case e_a\nthis : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))\nn : \u2115\n\u22a2 2 * \u2191n + 1 = 2 * \u2191n + 2 - 1"}, {"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "case e_a\nthis : \ud835\udcdd (\u03c0 / 2) = \ud835\udcdd ((1 - 0) * (\u03c0 / 2))\nn : \u2115\n\u22a2 2 * \u2191n + 1 = 2 * \u2191n + 2 - 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.univ_mem", "start": [147, 1], "end": [148, 14], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean", "full_name": "minpolyDiv_eq_of_isIntegrallyClosed", "start": [128, 1], "end": [133, 59], "traced_tactics": [{"tactic": "delta minpolyDiv", "annotated_tactic": ["delta <a>minpolyDiv</a>", [{"full_name": "minpolyDiv", "def_path": "Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean", "def_pos": [29, 19], "def_end_pos": [29, 29]}]], "state_before": "R : Type u_1\nK : Type u_3\nL : Type ?u.75447\nS : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra K L\nx : S\nhx : IsIntegral R x\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsIntegrallyClosed R\ninst\u271d\u2074 : IsDomain S\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : Algebra K S\ninst\u271d\u00b9 : IsScalarTower R K S\ninst\u271d : IsFractionRing R K\n\u22a2 minpolyDiv R x = minpolyDiv K x", "state_after": "R : Type u_1\nK : Type u_3\nL : Type ?u.75447\nS : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra K L\nx : S\nhx : IsIntegral R x\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsIntegrallyClosed R\ninst\u271d\u2074 : IsDomain S\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : Algebra K S\ninst\u271d\u00b9 : IsScalarTower R K S\ninst\u271d : IsFractionRing R K\n\u22a2 map (algebraMap R S) (minpoly R x) /\u2098 (X - C x) = map (algebraMap K S) (minpoly K x) /\u2098 (X - C x)"}, {"tactic": "rw [IsScalarTower.algebraMap_eq R K S, \u2190 map_map,\n  \u2190 minpoly.isIntegrallyClosed_eq_field_fractions' _ hx]", "annotated_tactic": ["rw [<a>IsScalarTower.algebraMap_eq</a> R K S, \u2190 <a>map_map</a>,\n    \u2190 <a>minpoly.isIntegrallyClosed_eq_field_fractions'</a> _ hx]", [{"full_name": "IsScalarTower.algebraMap_eq", "def_path": "Mathlib/Algebra/Algebra/Tower.lean", "def_pos": [125, 9], "def_end_pos": [125, 22]}, {"full_name": "Polynomial.map_map", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [820, 9], "def_end_pos": [820, 16]}, {"full_name": "minpoly.isIntegrallyClosed_eq_field_fractions'", "def_path": "Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean", "def_pos": [61, 9], "def_end_pos": [61, 47]}]], "state_before": "R : Type u_1\nK : Type u_3\nL : Type ?u.75447\nS : Type u_2\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : Field K\ninst\u271d\u00b9\u2070 : Field L\ninst\u271d\u2079 : CommRing S\ninst\u271d\u2078 : Algebra R S\ninst\u271d\u2077 : Algebra K L\nx : S\nhx : IsIntegral R x\ninst\u271d\u2076 : IsDomain R\ninst\u271d\u2075 : IsIntegrallyClosed R\ninst\u271d\u2074 : IsDomain S\ninst\u271d\u00b3 : Algebra R K\ninst\u271d\u00b2 : Algebra K S\ninst\u271d\u00b9 : IsScalarTower R K S\ninst\u271d : IsFractionRing R K\n\u22a2 map (algebraMap R S) (minpoly R x) /\u2098 (X - C x) = map (algebraMap K S) (minpoly K x) /\u2098 (X - C x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/MonoidAlgebra/Degree.lean", "full_name": "AddMonoidAlgebra.ne_zero_of_supDegree_ne_bot", "start": [265, 1], "end": [265, 100], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean", "full_name": "HasStrictFDerivAt.log", "start": [146, 1], "end": [148, 56], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/MeanInequalitiesPow.lean", "full_name": "NNReal.rpow_arith_mean_le_arith_mean_rpow", "start": [134, 1], "end": [138, 54], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Ray.lean", "full_name": "Module.Ray.units_smul_of_pos", "start": [348, 1], "end": [351, 43], "traced_tactics": [{"tactic": "induction v using Module.Ray.ind", "annotated_tactic": ["induction v using <a>Module.Ray.ind</a>", [{"full_name": "Module.Ray.ind", "def_path": "Mathlib/LinearAlgebra/Ray.lean", "def_pos": [259, 9], "def_end_pos": [259, 23]}]], "state_before": "R : Type u_1\ninst\u271d\u2075 : StrictOrderedCommSemiring R\nM : Type u_2\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : Module R N\n\u03b9 : Type u_4\ninst\u271d : DecidableEq \u03b9\nu : R\u02e3\nhu : 0 < \u2191u\nv : Ray R M\n\u22a2 u \u2022 v = v", "state_after": "case h\nR : Type u_1\ninst\u271d\u2075 : StrictOrderedCommSemiring R\nM : Type u_2\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : Module R N\n\u03b9 : Type u_4\ninst\u271d : DecidableEq \u03b9\nu : R\u02e3\nhu : 0 < \u2191u\nv\u271d : M\nhv\u271d : v\u271d \u2260 0\n\u22a2 u \u2022 rayOfNeZero R v\u271d hv\u271d = rayOfNeZero R v\u271d hv\u271d"}, {"tactic": "rw [smul_rayOfNeZero, ray_eq_iff]", "annotated_tactic": ["rw [<a>smul_rayOfNeZero</a>, <a>ray_eq_iff</a>]", [{"full_name": "smul_rayOfNeZero", "def_path": "Mathlib/LinearAlgebra/Ray.lean", "def_pos": [337, 9], "def_end_pos": [337, 25]}, {"full_name": "ray_eq_iff", "def_path": "Mathlib/LinearAlgebra/Ray.lean", "def_pos": [271, 9], "def_end_pos": [271, 19]}]], "state_before": "case h\nR : Type u_1\ninst\u271d\u2075 : StrictOrderedCommSemiring R\nM : Type u_2\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : Module R N\n\u03b9 : Type u_4\ninst\u271d : DecidableEq \u03b9\nu : R\u02e3\nhu : 0 < \u2191u\nv\u271d : M\nhv\u271d : v\u271d \u2260 0\n\u22a2 u \u2022 rayOfNeZero R v\u271d hv\u271d = rayOfNeZero R v\u271d hv\u271d", "state_after": "case h\nR : Type u_1\ninst\u271d\u2075 : StrictOrderedCommSemiring R\nM : Type u_2\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : Module R N\n\u03b9 : Type u_4\ninst\u271d : DecidableEq \u03b9\nu : R\u02e3\nhu : 0 < \u2191u\nv\u271d : M\nhv\u271d : v\u271d \u2260 0\n\u22a2 SameRay R (u \u2022 v\u271d) v\u271d"}, {"tactic": "exact SameRay.sameRay_pos_smul_left _ hu", "annotated_tactic": ["exact <a>SameRay.sameRay_pos_smul_left</a> _ hu", [{"full_name": "SameRay.sameRay_pos_smul_left", "def_path": "Mathlib/LinearAlgebra/Ray.lean", "def_pos": [137, 7], "def_end_pos": [137, 28]}]], "state_before": "case h\nR : Type u_1\ninst\u271d\u2075 : StrictOrderedCommSemiring R\nM : Type u_2\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : Module R N\n\u03b9 : Type u_4\ninst\u271d : DecidableEq \u03b9\nu : R\u02e3\nhu : 0 < \u2191u\nv\u271d : M\nhv\u271d : v\u271d \u2260 0\n\u22a2 SameRay R (u \u2022 v\u271d) v\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Category/Profinite/Nobeling.lean", "full_name": "Profinite.NobelingProof.projRestricts_eq_id", "start": [156, 1], "end": [158, 92], "traced_tactics": [{"tactic": "ext \u27e8x, y, hy, rfl\u27e9 i", "annotated_tactic": ["ext \u27e8x, y, hy, rfl\u27e9 i", []], "state_before": "I : Type u\ninst\u271d\u2074 : LinearOrder I\ninst\u271d\u00b3 : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\nJ K L : I \u2192 Prop\ninst\u271d\u00b2 : (i : I) \u2192 Decidable (J i)\ninst\u271d\u00b9 : (i : I) \u2192 Decidable (K i)\ninst\u271d : (i : I) \u2192 Decidable (L i)\n\u22a2 ProjRestricts C \u22ef = id", "state_after": "case h.mk.intro.intro.a.h\nI : Type u\ninst\u271d\u2074 : LinearOrder I\ninst\u271d\u00b3 : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\nJ K L : I \u2192 Prop\ninst\u271d\u00b2 : (i : I) \u2192 Decidable (J i)\ninst\u271d\u00b9 : (i : I) \u2192 Decidable (K i)\ninst\u271d : (i : I) \u2192 Decidable (L i)\ny : I \u2192 Bool\nhy : y \u2208 C\ni : I\n\u22a2 \u2191(ProjRestricts C \u22ef \u27e8Proj J y, \u22ef\u27e9) i = \u2191(id \u27e8Proj J y, \u22ef\u27e9) i"}, {"tactic": "simp (config := { contextual := true }) only [\u03c0, Proj, ProjRestricts_coe, id_eq, if_true]", "annotated_tactic": ["simp (config := { contextual := <a>true</a> }) only [<a>\u03c0</a>, <a>Proj</a>, <a>ProjRestricts_coe</a>, <a>id_eq</a>, <a>if_true</a>]", [{"full_name": "Bool.true", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [571, 5], "def_end_pos": [571, 9]}, {"full_name": "Profinite.NobelingProof.\u03c0", "def_path": "Mathlib/Topology/Category/Profinite/Nobeling.lean", "def_pos": [102, 5], "def_end_pos": [102, 6]}, {"full_name": "Profinite.NobelingProof.Proj", "def_path": "Mathlib/Topology/Category/Profinite/Nobeling.lean", "def_pos": [88, 5], "def_end_pos": [88, 9]}, {"full_name": "Profinite.NobelingProof.ProjRestricts_coe", "def_path": "Mathlib/Topology/Category/Profinite/Nobeling.lean", "def_pos": [144, 3], "def_end_pos": [144, 9]}, {"full_name": "id_eq", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [297, 17], "def_end_pos": [297, 22]}, {"full_name": "if_true", "def_path": ".lake/packages/lean4/src/lean/Init/ByCases.lean", "def_pos": [24, 17], "def_end_pos": [24, 24]}]], "state_before": "case h.mk.intro.intro.a.h\nI : Type u\ninst\u271d\u2074 : LinearOrder I\ninst\u271d\u00b3 : IsWellOrder I fun x x_1 => x < x_1\nC : Set (I \u2192 Bool)\nJ K L : I \u2192 Prop\ninst\u271d\u00b2 : (i : I) \u2192 Decidable (J i)\ninst\u271d\u00b9 : (i : I) \u2192 Decidable (K i)\ninst\u271d : (i : I) \u2192 Decidable (L i)\ny : I \u2192 Bool\nhy : y \u2208 C\ni : I\n\u22a2 \u2191(ProjRestricts C \u22ef \u27e8Proj J y, \u22ef\u27e9) i = \u2191(id \u27e8Proj J y, \u22ef\u27e9) i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Control/Applicative.lean", "full_name": "Functor.Comp.map_pure", "start": [89, 1], "end": [90, 22], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "F : Type u \u2192 Type w\nG : Type v \u2192 Type u\ninst\u271d\u00b3 : Applicative F\ninst\u271d\u00b2 : Applicative G\ninst\u271d\u00b9 : LawfulApplicative F\ninst\u271d : LawfulApplicative G\n\u03b1 \u03b2 \u03b3 : Type v\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\n\u22a2 (f <$> pure x).run = (pure (f x)).run", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/Morphisms/ClosedImmersion.lean", "full_name": "AlgebraicGeometry.IsClosedImmersion.respectsIso", "start": [72, 1], "end": [74, 56], "traced_tactics": [{"tactic": 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w\nM' : Type w\u2081\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : LieRing L\ninst\u271d\u00b9\u2070 : LieAlgebra R L\ninst\u271d\u2079 : LieRing L'\ninst\u271d\u2078 : LieAlgebra R L'\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nf : L \u2192\u2097\u2045R\u2046 L'\nI : LieIdeal R L\nJ : LieIdeal R L'\n\u22a2 I \u2264 f.ker \u2194 \u2200 x \u2208 I, f x = 0", "state_after": "case mp\nR : Type u\nL : Type v\nL' : Type w\u2082\nM : Type w\nM' : Type w\u2081\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : LieRing L\ninst\u271d\u00b9\u2070 : LieAlgebra R L\ninst\u271d\u2079 : LieRing L'\ninst\u271d\u2078 : LieAlgebra R L'\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L 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f.ker"}, {"tactic": "specialize h hx", "annotated_tactic": ["specialize h hx", []], "state_before": "case mp\nR : Type u\nL : Type v\nL' : Type w\u2082\nM : Type w\nM' : Type w\u2081\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : LieRing L\ninst\u271d\u00b9\u2070 : LieAlgebra R L\ninst\u271d\u2079 : LieRing L'\ninst\u271d\u2078 : LieAlgebra R L'\ninst\u271d\u2077 : AddCommGroup M\ninst\u271d\u2076 : Module R M\ninst\u271d\u2075 : LieRingModule L M\ninst\u271d\u2074 : LieModule R L M\ninst\u271d\u00b3 : AddCommGroup M'\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : LieRingModule L M'\ninst\u271d : LieModule R L M'\nf : L \u2192\u2097\u2045R\u2046 L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nh : I \u2264 f.ker\nx : L\nhx : x \u2208 I\n\u22a2 f x = 0", "state_after": "case mp\nR : Type u\nL : Type v\nL' : Type w\u2082\nM : Type w\nM' : Type w\u2081\ninst\u271d\u00b9\u00b2 : CommRing R\ninst\u271d\u00b9\u00b9 : LieRing L\ninst\u271d\u00b9\u2070 : LieAlgebra R 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"MeasureTheory.Integrable.hasFiniteIntegral", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [456, 9], "def_end_pos": [456, 37]}]], "state_before": "case refine_2\nb : \u2102\nhb : 0 < b.re\n\u22a2 HasFiniteIntegral (fun x => \u2191x * cexp (-b * \u2191x ^ 2)) volume", "state_after": "case refine_2\nb : \u2102\nhb : 0 < b.re\nthis : HasFiniteIntegral (fun x => x * rexp (-b.re * x ^ 2)) volume\n\u22a2 HasFiniteIntegral (fun x => \u2191x * cexp (-b * \u2191x ^ 2)) volume"}, {"tactic": "rw [\u2190 hasFiniteIntegral_norm_iff] at this \u22a2", "annotated_tactic": ["rw [\u2190 <a>hasFiniteIntegral_norm_iff</a>] at this \u22a2", [{"full_name": "MeasureTheory.hasFiniteIntegral_norm_iff", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [257, 9], "def_end_pos": [257, 35]}]], "state_before": "case refine_2\nb : \u2102\nhb : 0 < b.re\nthis : HasFiniteIntegral (fun x => x * rexp (-b.re * x ^ 2)) volume\n\u22a2 HasFiniteIntegral (fun x => \u2191x * cexp (-b * \u2191x ^ 2)) volume", "state_after": "case refine_2\nb : \u2102\nhb : 0 < b.re\nthis : HasFiniteIntegral (fun a => \u2016a * rexp (-b.re * a ^ 2)\u2016) volume\n\u22a2 HasFiniteIntegral (fun a => \u2016\u2191a * cexp (-b * \u2191a ^ 2)\u2016) volume"}, {"tactic": "convert this", "annotated_tactic": ["convert this", []], "state_before": "case refine_2\nb : \u2102\nhb : 0 < b.re\nthis : HasFiniteIntegral (fun a => \u2016a * rexp (-b.re * a ^ 2)\u2016) volume\n\u22a2 HasFiniteIntegral (fun a => \u2016\u2191a * cexp (-b * \u2191a ^ 2)\u2016) volume", "state_after": "case h.e'_5.h\nb : \u2102\nhb : 0 < b.re\nthis : HasFiniteIntegral (fun a => \u2016a * rexp (-b.re * a ^ 2)\u2016) volume\nx\u271d : \u211d\n\u22a2 \u2016\u2191x\u271d * cexp (-b * \u2191x\u271d ^ 2)\u2016 = \u2016x\u271d * rexp (-b.re * x\u271d ^ 2)\u2016"}, {"tactic": "rw [norm_mul, norm_mul, norm_cexp_neg_mul_sq b, Complex.norm_eq_abs, abs_ofReal, Real.norm_eq_abs,\n  norm_of_nonneg (exp_pos _).le]", "annotated_tactic": ["rw [<a>norm_mul</a>, <a>norm_mul</a>, <a>norm_cexp_neg_mul_sq</a> b, <a>Complex.norm_eq_abs</a>, <a>abs_ofReal</a>, <a>Real.norm_eq_abs</a>,\n    <a>norm_of_nonneg</a> (<a>exp_pos</a> _).<a>le</a>]", [{"full_name": "norm_mul", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [687, 9], "def_end_pos": [687, 17]}, {"full_name": "norm_mul", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [687, 9], "def_end_pos": [687, 17]}, {"full_name": "norm_cexp_neg_mul_sq", "def_path": "Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean", "def_pos": [157, 9], "def_end_pos": [157, 29]}, {"full_name": "Complex.norm_eq_abs", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [55, 9], "def_end_pos": [55, 20]}, {"full_name": "Complex.abs_ofReal", "def_path": "Mathlib/Data/Complex/Abs.lean", "def_pos": [75, 9], "def_end_pos": [75, 19]}, {"full_name": "Real.norm_eq_abs", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1434, 9], "def_end_pos": [1434, 20]}, {"full_name": "Real.norm_of_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [1441, 9], "def_end_pos": [1441, 23]}, {"full_name": "Real.exp_pos", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1197, 9], "def_end_pos": [1197, 16]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}]], "state_before": "case h.e'_5.h\nb : \u2102\nhb : 0 < b.re\nthis : HasFiniteIntegral (fun a => \u2016a * rexp (-b.re * a ^ 2)\u2016) volume\nx\u271d : \u211d\n\u22a2 \u2016\u2191x\u271d * cexp (-b * \u2191x\u271d ^ 2)\u2016 = \u2016x\u271d * rexp (-b.re * x\u271d ^ 2)\u2016", "state_after": "no goals"}, {"tactic": "exact continuous_const.mul (continuous_ofReal.pow 2)", "annotated_tactic": ["exact continuous_const.mul (continuous_ofReal.pow 2)", []], "state_before": "case refine_1\nb : \u2102\nhb : 0 < b.re\n\u22a2 Continuous fun x => -b * \u2191x ^ 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.lintegral_edist_lt_top", "start": [646, 1], "end": [651, 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/Algebra/Group/Pi/Basic.lean", "def_pos": [62, 3], "def_end_pos": [62, 14]}, {"full_name": "MeasureTheory.hasFiniteIntegral_iff_edist", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [119, 9], "def_end_pos": [119, 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 \u03bc\nhg : Integrable g \u03bc\n\u22a2 \u222b\u207b (a : \u03b1), edist (f a) (0 a) \u2202\u03bc < \u22a4 \u2227 \u222b\u207b (a : \u03b1), edist (g a) (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 \u03bc\nhg : Integrable g \u03bc\n\u22a2 HasFiniteIntegral f \u03bc \u2227 HasFiniteIntegral g \u03bc"}, {"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 \u03bc\nhg : Integrable g \u03bc\n\u22a2 HasFiniteIntegral f \u03bc \u2227 HasFiniteIntegral g \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Field/Basic.lean", "full_name": "Commute.div_add_div", "start": [75, 11], "end": [77, 83], "traced_tactics": [{"tactic": "rw [add_div, mul_div_mul_right _ b hd, hbc.eq, hbd.eq, mul_div_mul_right c d hb]", "annotated_tactic": ["rw [<a>add_div</a>, <a>mul_div_mul_right</a> _ b hd, hbc.eq, hbd.eq, <a>mul_div_mul_right</a> c d hb]", [{"full_name": "add_div", "def_path": "Mathlib/Algebra/Field/Basic.lean", "def_pos": [29, 9], "def_end_pos": [29, 16]}, {"full_name": "mul_div_mul_right", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [384, 7], "def_end_pos": [384, 24]}, {"full_name": "mul_div_mul_right", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [384, 7], "def_end_pos": [384, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nK : Type u_3\ninst\u271d : DivisionSemiring \u03b1\na b c d : \u03b1\nhbc : Commute b c\nhbd : Commute b d\nhb : b \u2260 0\nhd : d \u2260 0\n\u22a2 a / b + c / d = (a * d + b * c) / (b * d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/Scheme.lean", "full_name": "AlgebraicGeometry.Scheme.preimage_basicOpen", "start": [489, 1], "end": [491, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.self_eq_append_right", "start": [388, 9], "end": [389, 37], "traced_tactics": [{"tactic": "rw [eq_comm, append_right_eq_self]", "annotated_tactic": ["rw [<a>eq_comm</a>, <a>append_right_eq_self</a>]", [{"full_name": "eq_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}, {"full_name": "List.append_right_eq_self", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [385, 17], "def_end_pos": [385, 37]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 x y : List \u03b1\n\u22a2 x = x ++ y \u2194 y = []", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Padics/PadicIntegers.lean", "full_name": "PadicInt.mk_coe", "start": [178, 1], "end": [178, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Submonoid/Operations.lean", "full_name": "MonoidHom.restrict_mrange", "start": [1048, 1], "end": [1049, 25], "traced_tactics": [{"tactic": "simp [SetLike.ext_iff]", "annotated_tactic": ["simp [<a>SetLike.ext_iff</a>]", [{"full_name": "SetLike.ext_iff", "def_path": "Mathlib/Data/SetLike/Basic.lean", "def_pos": [175, 9], "def_end_pos": [175, 16]}]], "state_before": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst\u271d\u2074 : MulOneClass M\ninst\u271d\u00b3 : MulOneClass N\ninst\u271d\u00b2 : MulOneClass P\nS : Submonoid M\nA : Type u_4\ninst\u271d\u00b9 : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nF : Type u_5\ninst\u271d : FunLike F M N\nmc : MonoidHomClass F M N\nf : M \u2192* N\n\u22a2 mrange (f.restrict S) = map f S", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Add.lean", "full_name": "DifferentiableOn.sub_const", "start": [611, 1], "end": [612, 81], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.Integrable.const_mul", "start": [1210, 1], "end": [1212, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Complex/Basic.lean", "full_name": "Complex.I_mul_re", "start": [328, 1], "end": [328, 57], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "z : \u2102\n\u22a2 (I * z).re = -z.im", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.disjoint_left", "start": [959, 1], "end": [962, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/DirectSum/Finsupp.lean", "full_name": "TensorProduct.finsuppRight_symm_apply_single", "start": [144, 1], "end": [148, 36], "traced_tactics": [{"tactic": "simp [finsuppRight, Finsupp.lsum]", "annotated_tactic": ["simp [<a>finsuppRight</a>, <a>Finsupp.lsum</a>]", [{"full_name": "TensorProduct.finsuppRight", "def_path": "Mathlib/LinearAlgebra/DirectSum/Finsupp.lean", "def_pos": [117, 19], "def_end_pos": [117, 31]}, {"full_name": "Finsupp.lsum", "def_path": "Mathlib/LinearAlgebra/Finsupp.lean", "def_pos": [462, 5], "def_end_pos": [462, 9]}]], "state_before": "R : Type u_1\ninst\u271d\u2075 : CommSemiring R\nM : Type u_2\ninst\u271d\u2074 : AddCommMonoid M\ninst\u271d\u00b3 : Module R M\nN : Type u_3\ninst\u271d\u00b2 : AddCommMonoid N\ninst\u271d\u00b9 : Module R N\n\u03b9 : Type u_4\ninst\u271d : DecidableEq \u03b9\ni : \u03b9\nm : M\nn : N\n\u22a2 (finsuppRight R M N \u03b9).symm (Finsupp.single i (m \u2297\u209c[R] n)) = m \u2297\u209c[R] Finsupp.single i n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/NormalMono/Equalizers.lean", "full_name": "CategoryTheory.NormalEpiCategory.mono_of_zero_kernel", "start": [315, 1], "end": [328, 37], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u_1\ninst\u271d\u2074 : Category.{?u.249922, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\n\u22a2 0 \u226b f = 0", "state_after": "no goals"}, {"tactic": "obtain \u27e8W, w, hw, hl\u27e9 := normalEpiOfEpi (coequalizer.\u03c0 u v)", "annotated_tactic": ["obtain \u27e8W, w, hw, hl\u27e9 := <a>normalEpiOfEpi</a> (<a>coequalizer.\u03c0</a> u v)", [{"full_name": "CategoryTheory.NormalEpiCategory.normalEpiOfEpi", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/NormalMono/Basic.lean", "def_pos": [286, 3], "def_end_pos": [286, 17]}, {"full_name": "CategoryTheory.Limits.coequalizer.\u03c0", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [954, 22], "def_end_pos": [954, 35]}]], "state_before": "C : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 \u22ef)\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\n\u22a2 u = v", "state_after": "case mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 \u22ef)\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\n\u22a2 u = v"}, {"tactic": "obtain \u27e8m, hm\u27e9 := coequalizer.desc' f huv", "annotated_tactic": ["obtain \u27e8m, hm\u27e9 := <a>coequalizer.desc'</a> f huv", [{"full_name": "CategoryTheory.Limits.coequalizer.desc'", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [1011, 19], "def_end_pos": [1011, 36]}]], "state_before": "case mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 \u22ef)\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\n\u22a2 u = v", "state_after": "case mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 \u22ef)\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\n\u22a2 u = v"}, {"tactic": "have reassoced {W : C} (h : coequalizer u v \u27f6 W) : w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h := by\n  rw [\u2190 Category.assoc, eq_whisker hw]", "annotated_tactic": ["have reassoced {W : C} (h : <a>coequalizer</a> u v \u27f6 W) : w \u226b <a>coequalizer.\u03c0</a> u v \u226b h = 0 \u226b h := by\n      rw [\u2190 <a>Category.assoc</a>, <a>eq_whisker</a> hw]", [{"full_name": "CategoryTheory.Limits.coequalizer", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [948, 22], "def_end_pos": [948, 33]}, {"full_name": "CategoryTheory.Limits.coequalizer.\u03c0", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [954, 22], "def_end_pos": [954, 35]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.eq_whisker", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [213, 9], "def_end_pos": [213, 19]}]], "state_before": "case mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 \u22ef)\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\n\u22a2 u = v", "state_after": "case mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 \u22ef)\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nreassoced : \u2200 {W_1 : C} (h : coequalizer u v \u27f6 W_1), w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h\n\u22a2 u = v"}, {"tactic": "have hwf : w \u226b f = 0 := by rw [\u2190 hm, reassoced, zero_comp]", "annotated_tactic": ["have hwf : w \u226b f = 0 := by rw [\u2190 hm, reassoced, <a>zero_comp</a>]", [{"full_name": "CategoryTheory.Limits.zero_comp", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean", "def_pos": [72, 9], "def_end_pos": [72, 18]}]], "state_before": "case mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 \u22ef)\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nreassoced : \u2200 {W_1 : C} (h : coequalizer u v \u27f6 W_1), w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h\n\u22a2 u = v", "state_after": "case mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 \u22ef)\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nreassoced : \u2200 {W_1 : C} (h : coequalizer u v \u27f6 W_1), w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h\nhwf : w \u226b f = 0\n\u22a2 u = v"}, {"tactic": "obtain \u27e8n, hn\u27e9 := KernelFork.IsLimit.lift' l _ hwf", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 := <a>KernelFork.IsLimit.lift'</a> l _ hwf", [{"full_name": "CategoryTheory.Limits.KernelFork.IsLimit.lift'", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean", "def_pos": [136, 5], "def_end_pos": [136, 29]}]], "state_before": "case mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 \u22ef)\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nreassoced : \u2200 {W_1 : C} (h : coequalizer u v \u27f6 W_1), w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h\nhwf : w \u226b f = 0\n\u22a2 u = v", "state_after": "case mk.mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 \u22ef)\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nreassoced : \u2200 {W_1 : C} (h : coequalizer u v \u27f6 W_1), w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h\nhwf : w \u226b f = 0\nn : W \u27f6 (KernelFork.of\u03b9 0 \u22ef).pt\nhn : n \u226b Fork.\u03b9 (KernelFork.of\u03b9 0 \u22ef) = w\n\u22a2 u = v"}, {"tactic": "rw [Fork.\u03b9_of\u03b9, HasZeroMorphisms.comp_zero] at hn", "annotated_tactic": ["rw [<a>Fork.\u03b9_of\u03b9</a>, <a>HasZeroMorphisms.comp_zero</a>] at hn", [{"full_name": "CategoryTheory.Limits.Fork.\u03b9_of\u03b9", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [382, 9], "def_end_pos": [382, 19]}, {"full_name": "CategoryTheory.Limits.HasZeroMorphisms.comp_zero", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean", "def_pos": [54, 3], "def_end_pos": [54, 12]}]], "state_before": "case mk.mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 \u22ef)\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nreassoced : \u2200 {W_1 : C} (h : coequalizer u v \u27f6 W_1), w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h\nhwf : w \u226b f = 0\nn : W \u27f6 (KernelFork.of\u03b9 0 \u22ef).pt\nhn : n \u226b Fork.\u03b9 (KernelFork.of\u03b9 0 \u22ef) = w\n\u22a2 u = v", "state_after": "case mk.mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 \u22ef)\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nreassoced : \u2200 {W_1 : C} (h : coequalizer u v \u27f6 W_1), w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h\nhwf : w \u226b f = 0\nn : W \u27f6 (KernelFork.of\u03b9 0 \u22ef).pt\nhn : 0 = w\n\u22a2 u = v"}, {"tactic": "have : IsIso (coequalizer.\u03c0 u v) := by\n  apply isIso_colimit_cocone_parallelPair_of_eq hn.symm hl", "annotated_tactic": ["have : <a>IsIso</a> (<a>coequalizer.\u03c0</a> u v) := by\n      apply <a>isIso_colimit_cocone_parallelPair_of_eq</a> hn.symm hl", [{"full_name": "CategoryTheory.IsIso", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [272, 7], "def_end_pos": [272, 12]}, {"full_name": "CategoryTheory.Limits.coequalizer.\u03c0", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [954, 22], "def_end_pos": [954, 35]}, {"full_name": "CategoryTheory.Limits.isIso_colimit_cocone_parallelPair_of_eq", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [1064, 9], "def_end_pos": [1064, 48]}]], "state_before": "case mk.mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 \u22ef)\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nreassoced : \u2200 {W_1 : C} (h : coequalizer u v \u27f6 W_1), w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h\nhwf : w \u226b f = 0\nn : W \u27f6 (KernelFork.of\u03b9 0 \u22ef).pt\nhn : 0 = w\n\u22a2 u = v", "state_after": "case mk.mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 \u22ef)\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nreassoced : \u2200 {W_1 : C} (h : coequalizer u v \u27f6 W_1), w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h\nhwf : w \u226b f = 0\nn : W \u27f6 (KernelFork.of\u03b9 0 \u22ef).pt\nhn : 0 = w\nthis : IsIso (coequalizer.\u03c0 u v)\n\u22a2 u = v"}, {"tactic": "apply (cancel_mono (coequalizer.\u03c0 u v)).1", "annotated_tactic": ["apply (<a>cancel_mono</a> (<a>coequalizer.\u03c0</a> u v)).1", [{"full_name": "CategoryTheory.cancel_mono", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [310, 9], "def_end_pos": [310, 20]}, {"full_name": "CategoryTheory.Limits.coequalizer.\u03c0", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [954, 22], "def_end_pos": [954, 35]}]], "state_before": "case mk.mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 \u22ef)\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nreassoced : \u2200 {W_1 : C} (h : coequalizer u v \u27f6 W_1), w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h\nhwf : w \u226b f = 0\nn : W \u27f6 (KernelFork.of\u03b9 0 \u22ef).pt\nhn : 0 = w\nthis : IsIso (coequalizer.\u03c0 u v)\n\u22a2 u = v", "state_after": "case mk.mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 \u22ef)\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nreassoced : \u2200 {W_1 : C} (h : coequalizer u v \u27f6 W_1), w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h\nhwf : w \u226b f = 0\nn : W \u27f6 (KernelFork.of\u03b9 0 \u22ef).pt\nhn : 0 = w\nthis : IsIso (coequalizer.\u03c0 u v)\n\u22a2 u \u226b coequalizer.\u03c0 u v = v \u226b coequalizer.\u03c0 u v"}, {"tactic": "exact coequalizer.condition _ _", "annotated_tactic": ["exact <a>coequalizer.condition</a> _ _", [{"full_name": "CategoryTheory.Limits.coequalizer.condition", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [975, 9], "def_end_pos": [975, 30]}]], "state_before": "case mk.mk.mk\nC : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 \u22ef)\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nreassoced : \u2200 {W_1 : C} (h : coequalizer u v \u27f6 W_1), w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h\nhwf : w \u226b f = 0\nn : W \u27f6 (KernelFork.of\u03b9 0 \u22ef).pt\nhn : 0 = w\nthis : IsIso (coequalizer.\u03c0 u v)\n\u22a2 u \u226b coequalizer.\u03c0 u v = v \u226b coequalizer.\u03c0 u v", "state_after": "no goals"}, {"tactic": "rw [\u2190 Category.assoc, eq_whisker hw]", "annotated_tactic": ["rw [\u2190 <a>Category.assoc</a>, <a>eq_whisker</a> hw]", [{"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.eq_whisker", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [213, 9], "def_end_pos": [213, 19]}]], "state_before": "C : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 \u22ef)\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW\u271d : C\nw : W\u271d \u27f6 X\nhw : w \u226b coequalizer.\u03c0 u v = 0\nhl : IsColimit (CokernelCofork.of\u03c0 (coequalizer.\u03c0 u v) hw)\nm : coequalizer u v \u27f6 Y\nhm : coequalizer.\u03c0 u v \u226b m = f\nW : C\nh : coequalizer u v \u27f6 W\n\u22a2 w \u226b coequalizer.\u03c0 u v \u226b h = 0 \u226b h", "state_after": "no goals"}, {"tactic": "rw [\u2190 hm, reassoced, zero_comp]", "annotated_tactic": ["rw [\u2190 hm, reassoced, <a>zero_comp</a>]", [{"full_name": "CategoryTheory.Limits.zero_comp", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean", "def_pos": [72, 9], "def_end_pos": [72, 18]}]], "state_before": "C : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} C\ninst\u271d\u00b3 : HasZeroMorphisms C\ninst\u271d\u00b2 : HasFiniteCoproducts C\ninst\u271d\u00b9 : HasCokernels C\ninst\u271d : NormalEpiCategory C\nX Y : C\nf : X \u27f6 Y\nZ : C\nl : IsLimit (KernelFork.of\u03b9 0 \u22ef)\nZ\u271d : C\nu v : Z\u271d \u27f6 X\nhuv : u \u226b f = v \u226b f\nW : C\nw : W \u27f6 X\nhw : w \u226b 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: G\nP : G \u2192 Prop\nh_one : P 1\nh_mul : \u2200 (a : G), P a \u2192 P (a * g)\nh_inv : \u2200 (a : G), P a \u2192 P (a * g\u207b\u00b9)\n\u22a2 P (g ^ 0)", "state_after": "no goals"}, {"tactic": "rw [zpow_add_one]", "annotated_tactic": ["rw [<a>zpow_add_one</a>]", [{"full_name": "zpow_add_one", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1161, 7], "def_end_pos": [1161, 19]}]], "state_before": "case hp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nG : Type u_3\nM : Type u_4\ninst\u271d : Group G\na b c d : G\nn\u271d : \u2124\ng : G\nP : G \u2192 Prop\nh_one : P 1\nh_mul : \u2200 (a : G), P a \u2192 P (a * g)\nh_inv : \u2200 (a : G), P a \u2192 P (a * g\u207b\u00b9)\nn : \u2115\nih : P (g ^ \u2191n)\n\u22a2 P (g ^ (\u2191n + 1))", "state_after": "case hp\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nG : Type u_3\nM : Type u_4\ninst\u271d : Group G\na b c d : G\nn\u271d : \u2124\ng : G\nP : G \u2192 Prop\nh_one : P 1\nh_mul : \u2200 (a : G), P a \u2192 P (a * g)\nh_inv : \u2200 (a 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"state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 whiskerRight (M.whiskerLeft f) P =\n    (M.associatorBimod N P).hom \u226b M.whiskerLeft (whiskerRight f P) \u226b (M.associatorBimod N' P).inv", "state_after": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 whiskerRight (M.whiskerLeft f) P =\n    (isoOfIso { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := \u22ef, inv_hom_id := \u22ef } \u22ef\n          \u22ef).hom \u226b\n      M.whiskerLeft (whiskerRight f P) \u226b\n        (isoOfIso\n            { hom := AssociatorBimod.hom M N' P, inv := AssociatorBimod.inv M N' P, hom_inv_id := \u22ef, inv_hom_id := \u22ef } \u22ef\n            \u22ef).inv"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 whiskerRight (M.whiskerLeft f) P =\n    (isoOfIso { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := \u22ef, inv_hom_id := \u22ef } \u22ef\n          \u22ef).hom \u226b\n      M.whiskerLeft (whiskerRight f P) \u226b\n        (isoOfIso\n            { hom := AssociatorBimod.hom M N' P, inv := AssociatorBimod.inv M N' P, hom_inv_id := \u22ef, inv_hom_id := \u22ef } \u22ef\n            \u22ef).inv", "state_after": "case h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (whiskerRight (M.whiskerLeft f) P).hom =\n    ((isoOfIso { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := \u22ef, inv_hom_id := \u22ef }\n            \u22ef \u22ef).hom \u226b\n        M.whiskerLeft (whiskerRight f P) \u226b\n          (isoOfIso\n              { hom := AssociatorBimod.hom M N' P, inv := AssociatorBimod.inv M N' P, hom_inv_id := \u22ef, inv_hom_id := \u22ef }\n              \u22ef \u22ef).inv).hom"}, {"tactic": "apply coequalizer.hom_ext", "annotated_tactic": ["apply <a>coequalizer.hom_ext</a>", [{"full_name": "CategoryTheory.Limits.coequalizer.hom_ext", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [1019, 9], "def_end_pos": [1019, 28]}]], "state_before": "case h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (whiskerRight (M.whiskerLeft f) P).hom =\n    ((isoOfIso { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := \u22ef, inv_hom_id := \u22ef }\n            \u22ef \u22ef).hom \u226b\n        M.whiskerLeft (whiskerRight f P) \u226b\n          (isoOfIso\n              { hom := AssociatorBimod.hom M N' P, inv := AssociatorBimod.inv M N' P, hom_inv_id := \u22ef, inv_hom_id := \u22ef }\n              \u22ef \u22ef).inv).hom", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 coequalizer.\u03c0\n        ({ X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := \u22ef, left_assoc := \u22ef,\n              actRight := TensorBimod.actRight M N, actRight_one := \u22ef, right_assoc := \u22ef, middle_assoc := \u22ef }.actRight \u25b7\n          P.X)\n        ((\u03b1_\n              { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := \u22ef, left_assoc := \u22ef,\n                  actRight := TensorBimod.actRight M N, actRight_one := \u22ef, right_assoc := \u22ef, middle_assoc := \u22ef }.X\n              Y.X P.X).hom \u226b\n          { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := \u22ef, left_assoc := \u22ef,\n                actRight := TensorBimod.actRight M N, actRight_one := \u22ef, right_assoc := \u22ef, middle_assoc := \u22ef }.X \u25c1\n            P.actLeft) \u226b\n      (whiskerRight (M.whiskerLeft f) P).hom =\n    coequalizer.\u03c0\n        ({ X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := \u22ef, left_assoc := \u22ef,\n              actRight := TensorBimod.actRight M N, actRight_one := \u22ef, right_assoc := \u22ef, middle_assoc := \u22ef }.actRight \u25b7\n          P.X)\n        ((\u03b1_\n              { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := \u22ef, left_assoc := \u22ef,\n                  actRight := TensorBimod.actRight M N, actRight_one := \u22ef, right_assoc := \u22ef, middle_assoc := \u22ef }.X\n              Y.X P.X).hom \u226b\n          { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := \u22ef, left_assoc := \u22ef,\n                actRight := TensorBimod.actRight M N, actRight_one := \u22ef, right_assoc := \u22ef, middle_assoc := \u22ef }.X \u25c1\n            P.actLeft) \u226b\n      ((isoOfIso\n              { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := \u22ef, inv_hom_id := \u22ef } \u22ef\n              \u22ef).hom \u226b\n          M.whiskerLeft (whiskerRight f P) \u226b\n            (isoOfIso\n                { hom := AssociatorBimod.hom M N' P, inv := AssociatorBimod.inv M N' P, hom_inv_id := \u22ef,\n                  inv_hom_id := \u22ef }\n                \u22ef \u22ef).inv).hom"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 coequalizer.\u03c0\n        ({ X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := \u22ef, left_assoc := \u22ef,\n              actRight := TensorBimod.actRight M N, actRight_one := \u22ef, right_assoc := \u22ef, middle_assoc := \u22ef }.actRight \u25b7\n          P.X)\n        ((\u03b1_\n              { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := \u22ef, left_assoc := \u22ef,\n                  actRight := TensorBimod.actRight M N, actRight_one := \u22ef, right_assoc := \u22ef, middle_assoc := \u22ef }.X\n              Y.X P.X).hom \u226b\n          { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := \u22ef, left_assoc := \u22ef,\n                actRight := TensorBimod.actRight M N, actRight_one := \u22ef, right_assoc := \u22ef, middle_assoc := \u22ef }.X \u25c1\n            P.actLeft) \u226b\n      (whiskerRight (M.whiskerLeft f) P).hom =\n    coequalizer.\u03c0\n        ({ X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := \u22ef, left_assoc := \u22ef,\n              actRight := TensorBimod.actRight M N, actRight_one := \u22ef, right_assoc := \u22ef, middle_assoc := \u22ef }.actRight \u25b7\n          P.X)\n        ((\u03b1_\n              { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := \u22ef, left_assoc := \u22ef,\n                  actRight := TensorBimod.actRight M N, actRight_one := \u22ef, right_assoc := \u22ef, middle_assoc := \u22ef }.X\n              Y.X P.X).hom \u226b\n          { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := \u22ef, left_assoc := \u22ef,\n                actRight := TensorBimod.actRight M N, actRight_one := \u22ef, right_assoc := \u22ef, middle_assoc := \u22ef }.X \u25c1\n            P.actLeft) \u226b\n      ((isoOfIso\n              { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := \u22ef, inv_hom_id := \u22ef } \u22ef\n              \u22ef).hom \u226b\n          M.whiskerLeft (whiskerRight f P) \u226b\n            (isoOfIso\n                { hom := AssociatorBimod.hom M N' P, inv := AssociatorBimod.inv M N' P, hom_inv_id := \u22ef,\n                  inv_hom_id := \u22ef }\n                \u22ef \u22ef).inv).hom", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 coequalizer.\u03c0 (TensorBimod.actRight M N \u25b7 P.X)\n        ((\u03b1_ (TensorBimod.X M N) Y.X P.X).hom \u226b TensorBimod.X M N \u25c1 P.actLeft) \u226b\n      colimMap\n        (parallelPairHom (TensorBimod.actRight M N \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N) Y.X P.X).hom \u226b TensorBimod.X M N \u25c1 P.actLeft) (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft)\n          (colimMap\n                (parallelPairHom (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) (M.actRight \u25b7 N'.X)\n                  ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) ((M.X \u2297 X.X) \u25c1 f.hom) (M.X \u25c1 f.hom) \u22ef \u22ef) \u25b7\n              Y.X \u25b7\n            P.X)\n          (colimMap\n              (parallelPairHom (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) (M.actRight \u25b7 N'.X)\n                ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) ((M.X \u2297 X.X) \u25c1 f.hom) (M.X \u25c1 f.hom) \u22ef \u22ef) \u25b7\n            P.X)\n          \u22ef \u22ef) =\n    coequalizer.\u03c0 (TensorBimod.actRight M N \u25b7 P.X)\n        ((\u03b1_ (TensorBimod.X M N) Y.X P.X).hom \u226b TensorBimod.X M N \u25c1 P.actLeft) \u226b\n      AssociatorBimod.hom M N P \u226b\n        colimMap\n            (parallelPairHom (M.actRight \u25b7 TensorBimod.X N P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P) (M.actRight \u25b7 TensorBimod.X N' P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P)\n              ((M.X \u2297 X.X) \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              (M.X \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              \u22ef \u22ef) \u226b\n          AssociatorBimod.inv M N' P"}, {"tactic": "slice_lhs 1 2 => rw [\u03b9_colimMap, parallelPairHom_app_one]", "annotated_tactic": ["slice_lhs 1 2 => rw [<a>\u03b9_colimMap</a>, <a>parallelPairHom_app_one</a>]", [{"full_name": "CategoryTheory.Limits.\u03b9_colimMap", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [805, 9], "def_end_pos": [805, 19]}, {"full_name": "CategoryTheory.Limits.parallelPairHom_app_one", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [272, 9], "def_end_pos": [272, 32]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 coequalizer.\u03c0 (TensorBimod.actRight M N \u25b7 P.X)\n        ((\u03b1_ (TensorBimod.X M N) Y.X P.X).hom \u226b TensorBimod.X M N \u25c1 P.actLeft) \u226b\n      colimMap\n        (parallelPairHom (TensorBimod.actRight M N \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N) Y.X P.X).hom \u226b TensorBimod.X M N \u25c1 P.actLeft) (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft)\n          (colimMap\n                (parallelPairHom (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) (M.actRight \u25b7 N'.X)\n                  ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) ((M.X \u2297 X.X) \u25c1 f.hom) (M.X \u25c1 f.hom) \u22ef \u22ef) \u25b7\n              Y.X \u25b7\n            P.X)\n          (colimMap\n              (parallelPairHom (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) (M.actRight \u25b7 N'.X)\n                ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) ((M.X \u2297 X.X) \u25c1 f.hom) (M.X \u25c1 f.hom) \u22ef \u22ef) \u25b7\n            P.X)\n          \u22ef \u22ef) =\n    coequalizer.\u03c0 (TensorBimod.actRight M N \u25b7 P.X)\n        ((\u03b1_ (TensorBimod.X M N) Y.X P.X).hom \u226b TensorBimod.X M N \u25c1 P.actLeft) \u226b\n      AssociatorBimod.hom M N P \u226b\n        colimMap\n            (parallelPairHom (M.actRight \u25b7 TensorBimod.X N P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P) (M.actRight \u25b7 TensorBimod.X N' P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P)\n              ((M.X \u2297 X.X) \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              (M.X \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              \u22ef \u22ef) \u226b\n          AssociatorBimod.inv M N' P", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 colimMap\n          (parallelPairHom (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) (M.actRight \u25b7 N'.X)\n            ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) ((M.X \u2297 X.X) \u25c1 f.hom) (M.X \u25c1 f.hom) \u22ef \u22ef) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    coequalizer.\u03c0 (TensorBimod.actRight M N \u25b7 P.X)\n        ((\u03b1_ (TensorBimod.X M N) Y.X P.X).hom \u226b TensorBimod.X M N \u25c1 P.actLeft) \u226b\n      AssociatorBimod.hom M N P \u226b\n        colimMap\n            (parallelPairHom (M.actRight \u25b7 TensorBimod.X N P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P) (M.actRight \u25b7 TensorBimod.X N' P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P)\n              ((M.X \u2297 X.X) \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              (M.X \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              \u22ef \u22ef) \u226b\n          AssociatorBimod.inv M N' P"}, {"tactic": "dsimp [AssociatorBimod.hom]", "annotated_tactic": ["dsimp [<a>AssociatorBimod.hom</a>]", [{"full_name": "Bimod.AssociatorBimod.hom", "def_path": "Mathlib/CategoryTheory/Monoidal/Bimod.lean", "def_pos": [493, 19], "def_end_pos": [493, 22]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 colimMap\n          (parallelPairHom (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) (M.actRight \u25b7 N'.X)\n            ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) ((M.X \u2297 X.X) \u25c1 f.hom) (M.X \u25c1 f.hom) \u22ef \u22ef) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    coequalizer.\u03c0 (TensorBimod.actRight M N \u25b7 P.X)\n        ((\u03b1_ (TensorBimod.X M N) Y.X P.X).hom \u226b TensorBimod.X M N \u25c1 P.actLeft) \u226b\n      AssociatorBimod.hom M N P \u226b\n        colimMap\n            (parallelPairHom (M.actRight \u25b7 TensorBimod.X N P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P) (M.actRight \u25b7 TensorBimod.X N' P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P)\n              ((M.X \u2297 X.X) \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              (M.X \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              \u22ef \u22ef) \u226b\n          AssociatorBimod.inv M N' P", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 colimMap\n          (parallelPairHom (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) (M.actRight \u25b7 N'.X)\n            ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) ((M.X \u2297 X.X) \u25c1 f.hom) (M.X \u25c1 f.hom) \u22ef \u22ef) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    coequalizer.\u03c0 (TensorBimod.actRight M N \u25b7 P.X)\n        ((\u03b1_ (TensorBimod.X M N) Y.X P.X).hom \u226b TensorBimod.X M N \u25c1 P.actLeft) \u226b\n      coequalizer.desc (AssociatorBimod.homAux M N P) \u22ef \u226b\n        colimMap\n            (parallelPairHom (M.actRight \u25b7 TensorBimod.X N P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P) (M.actRight \u25b7 TensorBimod.X N' P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P)\n              ((M.X \u2297 X.X) \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              (M.X \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              \u22ef \u22ef) \u226b\n          AssociatorBimod.inv M N' P"}, {"tactic": "slice_rhs 1 2 => rw [coequalizer.\u03c0_desc]", "annotated_tactic": ["slice_rhs 1 2 => rw [<a>coequalizer.\u03c0_desc</a>]", [{"full_name": "CategoryTheory.Limits.coequalizer.\u03c0_desc", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [996, 9], "def_end_pos": [996, 27]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 colimMap\n          (parallelPairHom (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) (M.actRight \u25b7 N'.X)\n            ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) ((M.X \u2297 X.X) \u25c1 f.hom) (M.X \u25c1 f.hom) \u22ef \u22ef) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    coequalizer.\u03c0 (TensorBimod.actRight M N \u25b7 P.X)\n        ((\u03b1_ (TensorBimod.X M N) Y.X P.X).hom \u226b TensorBimod.X M N \u25c1 P.actLeft) \u226b\n      coequalizer.desc (AssociatorBimod.homAux M N P) \u22ef \u226b\n        colimMap\n            (parallelPairHom (M.actRight \u25b7 TensorBimod.X N P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P) (M.actRight \u25b7 TensorBimod.X N' P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P)\n              ((M.X \u2297 X.X) \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              (M.X \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              \u22ef \u22ef) \u226b\n          AssociatorBimod.inv M N' P", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 colimMap\n          (parallelPairHom (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) (M.actRight \u25b7 N'.X)\n            ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) ((M.X \u2297 X.X) \u25c1 f.hom) (M.X \u25c1 f.hom) \u22ef \u22ef) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (AssociatorBimod.homAux M N P \u226b\n        colimMap\n          (parallelPairHom (M.actRight \u25b7 TensorBimod.X N P)\n            ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P) (M.actRight \u25b7 TensorBimod.X N' P)\n            ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P)\n            ((M.X \u2297 X.X) \u25c1\n              colimMap\n                (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                  ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n            (M.X \u25c1\n              colimMap\n                (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                  ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n            \u22ef \u22ef)) \u226b\n      AssociatorBimod.inv M N' P"}, {"tactic": "dsimp [AssociatorBimod.homAux]", "annotated_tactic": ["dsimp [<a>AssociatorBimod.homAux</a>]", [{"full_name": "Bimod.AssociatorBimod.homAux", "def_path": "Mathlib/CategoryTheory/Monoidal/Bimod.lean", "def_pos": [476, 19], "def_end_pos": [476, 25]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 colimMap\n          (parallelPairHom (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) (M.actRight \u25b7 N'.X)\n            ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) ((M.X \u2297 X.X) \u25c1 f.hom) (M.X \u25c1 f.hom) \u22ef \u22ef) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (AssociatorBimod.homAux M N P \u226b\n        colimMap\n          (parallelPairHom (M.actRight \u25b7 TensorBimod.X N P)\n            ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P) (M.actRight \u25b7 TensorBimod.X N' P)\n            ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P)\n            ((M.X \u2297 X.X) \u25c1\n              colimMap\n                (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                  ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n            (M.X \u25c1\n              colimMap\n                (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                  ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n            \u22ef \u22ef)) \u226b\n      AssociatorBimod.inv M N' P", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 colimMap\n          (parallelPairHom (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) (M.actRight \u25b7 N'.X)\n            ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) ((M.X \u2297 X.X) \u25c1 f.hom) (M.X \u25c1 f.hom) \u22ef \u22ef) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft)).inv \u226b\n          coequalizer.desc\n            ((\u03b1_ M.X N.X P.X).hom \u226b\n              M.X \u25c1 coequalizer.\u03c0 (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) \u226b\n                coequalizer.\u03c0 (M.actRight \u25b7 TensorBimod.X N P)\n                  ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P))\n            \u22ef) \u226b\n        colimMap\n          (parallelPairHom (M.actRight \u25b7 TensorBimod.X N P)\n            ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P) (M.actRight \u25b7 TensorBimod.X N' P)\n            ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P)\n            ((M.X \u2297 X.X) \u25c1\n              colimMap\n                (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                  ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n            (M.X \u25c1\n              colimMap\n                (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                  ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n            \u22ef \u22ef)) \u226b\n      AssociatorBimod.inv M N' P"}, {"tactic": "refine (cancel_epi ((tensorRight _).map (coequalizer.\u03c0 _ _))).1 ?_", "annotated_tactic": ["refine (<a>cancel_epi</a> ((<a>tensorRight</a> _).<a>map</a> (<a>coequalizer.\u03c0</a> _ _))).1 ?_", [{"full_name": "CategoryTheory.cancel_epi", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 19]}, {"full_name": "CategoryTheory.MonoidalCategory.tensorRight", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [864, 5], "def_end_pos": [864, 16]}, {"full_name": "Prefunctor.map", "def_path": "Mathlib/Combinatorics/Quiver/Basic.lean", "def_pos": [61, 3], "def_end_pos": [61, 6]}, {"full_name": "CategoryTheory.Limits.coequalizer.\u03c0", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [954, 22], "def_end_pos": [954, 35]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 colimMap\n          (parallelPairHom (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) (M.actRight \u25b7 N'.X)\n            ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) ((M.X \u2297 X.X) \u25c1 f.hom) (M.X \u25c1 f.hom) \u22ef \u22ef) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft)).inv \u226b\n          coequalizer.desc\n            ((\u03b1_ M.X N.X P.X).hom \u226b\n              M.X \u25c1 coequalizer.\u03c0 (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) \u226b\n                coequalizer.\u03c0 (M.actRight \u25b7 TensorBimod.X N P)\n                  ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P))\n            \u22ef) \u226b\n        colimMap\n          (parallelPairHom (M.actRight \u25b7 TensorBimod.X N P)\n            ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P) (M.actRight \u25b7 TensorBimod.X N' P)\n            ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P)\n            ((M.X \u2297 X.X) \u25c1\n              colimMap\n                (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                  ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n            (M.X \u25c1\n              colimMap\n                (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                  ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n            \u22ef \u22ef)) \u226b\n      AssociatorBimod.inv M N' P", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (tensorRight P.X).map (coequalizer.\u03c0 (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft)) \u226b\n      colimMap\n            (parallelPairHom (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) (M.actRight \u25b7 N'.X)\n              ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) ((M.X \u2297 X.X) \u25c1 f.hom) (M.X \u25c1 f.hom) \u22ef \u22ef) \u25b7\n          P.X \u226b\n        colimit.\u03b9\n          (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n            ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n          WalkingParallelPair.one =\n    (tensorRight P.X).map (coequalizer.\u03c0 (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft)) \u226b\n      (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft)).inv \u226b\n            coequalizer.desc\n              ((\u03b1_ M.X N.X P.X).hom \u226b\n                M.X \u25c1 coequalizer.\u03c0 (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) \u226b\n                  coequalizer.\u03c0 (M.actRight \u25b7 TensorBimod.X N P)\n                    ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P))\n              \u22ef) \u226b\n          colimMap\n            (parallelPairHom (M.actRight \u25b7 TensorBimod.X N P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P) (M.actRight \u25b7 TensorBimod.X N' P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P)\n              ((M.X \u2297 X.X) \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              (M.X \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              \u22ef \u22ef)) \u226b\n        AssociatorBimod.inv M N' P"}, {"tactic": "rw [tensorRight_map]", "annotated_tactic": ["rw [<a>tensorRight_map</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.tensorRight_map", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [863, 3], "def_end_pos": [863, 9]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (tensorRight P.X).map (coequalizer.\u03c0 (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft)) \u226b\n      colimMap\n            (parallelPairHom (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) (M.actRight \u25b7 N'.X)\n              ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) ((M.X \u2297 X.X) \u25c1 f.hom) (M.X \u25c1 f.hom) \u22ef \u22ef) \u25b7\n          P.X \u226b\n        colimit.\u03b9\n          (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n            ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n          WalkingParallelPair.one =\n    (tensorRight P.X).map (coequalizer.\u03c0 (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft)) \u226b\n      (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft)).inv \u226b\n            coequalizer.desc\n              ((\u03b1_ M.X N.X P.X).hom \u226b\n                M.X \u25c1 coequalizer.\u03c0 (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) \u226b\n                  coequalizer.\u03c0 (M.actRight \u25b7 TensorBimod.X N P)\n                    ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P))\n              \u22ef) \u226b\n          colimMap\n            (parallelPairHom (M.actRight \u25b7 TensorBimod.X N P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P) (M.actRight \u25b7 TensorBimod.X N' P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P)\n              ((M.X \u2297 X.X) \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              (M.X \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              \u22ef \u22ef)) \u226b\n        AssociatorBimod.inv M N' P", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 coequalizer.\u03c0 (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) \u25b7 P.X \u226b\n      colimMap\n            (parallelPairHom (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) (M.actRight \u25b7 N'.X)\n              ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) ((M.X \u2297 X.X) \u25c1 f.hom) (M.X \u25c1 f.hom) \u22ef \u22ef) \u25b7\n          P.X \u226b\n        colimit.\u03b9\n          (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n            ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n          WalkingParallelPair.one =\n    coequalizer.\u03c0 (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) \u25b7 P.X \u226b\n      (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft)).inv \u226b\n            coequalizer.desc\n              ((\u03b1_ M.X N.X P.X).hom \u226b\n                M.X \u25c1 coequalizer.\u03c0 (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) \u226b\n                  coequalizer.\u03c0 (M.actRight \u25b7 TensorBimod.X N P)\n                    ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P))\n              \u22ef) \u226b\n          colimMap\n            (parallelPairHom (M.actRight \u25b7 TensorBimod.X N P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P) (M.actRight \u25b7 TensorBimod.X N' P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P)\n              ((M.X \u2297 X.X) \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              (M.X \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              \u22ef \u22ef)) \u226b\n        AssociatorBimod.inv M N' P"}, {"tactic": "slice_lhs 1 2 => rw [\u2190 comp_whiskerRight, \u03b9_colimMap, parallelPairHom_app_one]", "annotated_tactic": ["slice_lhs 1 2 => rw [\u2190 <a>comp_whiskerRight</a>, <a>\u03b9_colimMap</a>, <a>parallelPairHom_app_one</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.comp_whiskerRight", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [253, 9], "def_end_pos": [253, 26]}, {"full_name": "CategoryTheory.Limits.\u03b9_colimMap", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [805, 9], "def_end_pos": [805, 19]}, {"full_name": "CategoryTheory.Limits.parallelPairHom_app_one", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [272, 9], "def_end_pos": [272, 32]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 coequalizer.\u03c0 (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) \u25b7 P.X \u226b\n      colimMap\n            (parallelPairHom (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) (M.actRight \u25b7 N'.X)\n              ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) ((M.X \u2297 X.X) \u25c1 f.hom) (M.X \u25c1 f.hom) \u22ef \u22ef) \u25b7\n          P.X \u226b\n        colimit.\u03b9\n          (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n            ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n          WalkingParallelPair.one =\n    coequalizer.\u03c0 (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) \u25b7 P.X \u226b\n      (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft)).inv \u226b\n            coequalizer.desc\n              ((\u03b1_ M.X N.X P.X).hom \u226b\n                M.X \u25c1 coequalizer.\u03c0 (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) \u226b\n                  coequalizer.\u03c0 (M.actRight \u25b7 TensorBimod.X N P)\n                    ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P))\n              \u22ef) \u226b\n          colimMap\n            (parallelPairHom (M.actRight \u25b7 TensorBimod.X N P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P) (M.actRight \u25b7 TensorBimod.X N' P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P)\n              ((M.X \u2297 X.X) \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              (M.X \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              \u22ef \u22ef)) \u226b\n        AssociatorBimod.inv M N' P", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    coequalizer.\u03c0 (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) \u25b7 P.X \u226b\n      (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft)).inv \u226b\n            coequalizer.desc\n              ((\u03b1_ M.X N.X P.X).hom \u226b\n                M.X \u25c1 coequalizer.\u03c0 (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) \u226b\n                  coequalizer.\u03c0 (M.actRight \u25b7 TensorBimod.X N P)\n                    ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P))\n              \u22ef) \u226b\n          colimMap\n            (parallelPairHom (M.actRight \u25b7 TensorBimod.X N P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P) (M.actRight \u25b7 TensorBimod.X N' P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P)\n              ((M.X \u2297 X.X) \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              (M.X \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              \u22ef \u22ef)) \u226b\n        AssociatorBimod.inv M N' P"}, {"tactic": "slice_rhs 1 3 => rw [\u03c0_tensor_id_preserves_coequalizer_inv_desc]", "annotated_tactic": ["slice_rhs 1 3 => rw [<a>\u03c0_tensor_id_preserves_coequalizer_inv_desc</a>]", [{"full_name": "\u03c0_tensor_id_preserves_coequalizer_inv_desc", "def_path": "Mathlib/CategoryTheory/Monoidal/Bimod.lean", "def_pos": [59, 9], "def_end_pos": [59, 51]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    coequalizer.\u03c0 (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft) \u25b7 P.X \u226b\n      (((PreservesCoequalizer.iso (tensorRight P.X) (M.actRight \u25b7 N.X) ((\u03b1_ M.X X.X N.X).hom \u226b M.X \u25c1 N.actLeft)).inv \u226b\n            coequalizer.desc\n              ((\u03b1_ M.X N.X P.X).hom \u226b\n                M.X \u25c1 coequalizer.\u03c0 (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) \u226b\n                  coequalizer.\u03c0 (M.actRight \u25b7 TensorBimod.X N P)\n                    ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P))\n              \u22ef) \u226b\n          colimMap\n            (parallelPairHom (M.actRight \u25b7 TensorBimod.X N P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P) (M.actRight \u25b7 TensorBimod.X N' P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P)\n              ((M.X \u2297 X.X) \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              (M.X \u25c1\n                colimMap\n                  (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                    ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n              \u22ef \u22ef)) \u226b\n        AssociatorBimod.inv M N' P", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (((\u03b1_ M.X N.X P.X).hom \u226b\n          M.X \u25c1 coequalizer.\u03c0 (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) \u226b\n            coequalizer.\u03c0 (M.actRight \u25b7 TensorBimod.X N P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P)) \u226b\n        colimMap\n          (parallelPairHom (M.actRight \u25b7 TensorBimod.X N P)\n            ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P) (M.actRight \u25b7 TensorBimod.X N' P)\n            ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P)\n            ((M.X \u2297 X.X) \u25c1\n              colimMap\n                (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                  ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n            (M.X \u25c1\n              colimMap\n                (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                  ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n            \u22ef \u22ef)) \u226b\n      AssociatorBimod.inv M N' P"}, {"tactic": "slice_rhs 3 4 => rw [\u03b9_colimMap, parallelPairHom_app_one]", "annotated_tactic": ["slice_rhs 3 4 => rw [<a>\u03b9_colimMap</a>, <a>parallelPairHom_app_one</a>]", [{"full_name": "CategoryTheory.Limits.\u03b9_colimMap", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [805, 9], "def_end_pos": [805, 19]}, {"full_name": "CategoryTheory.Limits.parallelPairHom_app_one", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [272, 9], "def_end_pos": [272, 32]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (((\u03b1_ M.X N.X P.X).hom \u226b\n          M.X \u25c1 coequalizer.\u03c0 (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) \u226b\n            coequalizer.\u03c0 (M.actRight \u25b7 TensorBimod.X N P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P)) \u226b\n        colimMap\n          (parallelPairHom (M.actRight \u25b7 TensorBimod.X N P)\n            ((\u03b1_ M.X X.X (TensorBimod.X N P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N P) (M.actRight \u25b7 TensorBimod.X N' P)\n            ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P)\n            ((M.X \u2297 X.X) \u25c1\n              colimMap\n                (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                  ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n            (M.X \u25c1\n              colimMap\n                (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                  ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef))\n            \u22ef \u22ef)) \u226b\n      AssociatorBimod.inv M N' P", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (\u03b1_ M.X N.X P.X).hom \u226b\n      M.X \u25c1 coequalizer.\u03c0 (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) \u226b\n        (M.X \u25c1\n              colimMap\n                (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                  ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef) \u226b\n            colimit.\u03b9\n              (parallelPair (M.actRight \u25b7 TensorBimod.X N' P)\n                ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P))\n              WalkingParallelPair.one) \u226b\n          AssociatorBimod.inv M N' P"}, {"tactic": "slice_rhs 2 3 => rw [\u2190 MonoidalCategory.whiskerLeft_comp, \u03b9_colimMap, parallelPairHom_app_one]", "annotated_tactic": ["slice_rhs 2 3 => rw [\u2190 <a>MonoidalCategory.whiskerLeft_comp</a>, <a>\u03b9_colimMap</a>, <a>parallelPairHom_app_one</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.whiskerLeft_comp", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [235, 9], "def_end_pos": [235, 25]}, {"full_name": "CategoryTheory.Limits.\u03b9_colimMap", "def_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "def_pos": [805, 9], "def_end_pos": [805, 19]}, {"full_name": "CategoryTheory.Limits.parallelPairHom_app_one", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [272, 9], "def_end_pos": [272, 32]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (\u03b1_ M.X N.X P.X).hom \u226b\n      M.X \u25c1 coequalizer.\u03c0 (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) \u226b\n        (M.X \u25c1\n              colimMap\n                (parallelPairHom (N.actRight \u25b7 P.X) ((\u03b1_ N.X Y.X P.X).hom \u226b N.X \u25c1 P.actLeft) (N'.actRight \u25b7 P.X)\n                  ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft) (f.hom \u25b7 Y.X \u25b7 P.X) (f.hom \u25b7 P.X) \u22ef \u22ef) \u226b\n            colimit.\u03b9\n              (parallelPair (M.actRight \u25b7 TensorBimod.X N' P)\n                ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P))\n              WalkingParallelPair.one) \u226b\n          AssociatorBimod.inv M N' P", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (\u03b1_ M.X N.X P.X).hom \u226b\n      (M.X \u25c1\n            (f.hom \u25b7 P.X \u226b\n              colimit.\u03b9 (parallelPair (N'.actRight \u25b7 P.X) ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft))\n                WalkingParallelPair.one) \u226b\n          colimit.\u03b9\n            (parallelPair (M.actRight \u25b7 TensorBimod.X N' P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P))\n            WalkingParallelPair.one) \u226b\n        AssociatorBimod.inv M N' P"}, {"tactic": "dsimp [AssociatorBimod.inv]", "annotated_tactic": ["dsimp [<a>AssociatorBimod.inv</a>]", [{"full_name": "Bimod.AssociatorBimod.inv", "def_path": "Mathlib/CategoryTheory/Monoidal/Bimod.lean", "def_pos": [593, 19], "def_end_pos": [593, 22]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (\u03b1_ M.X N.X P.X).hom \u226b\n      (M.X \u25c1\n            (f.hom \u25b7 P.X \u226b\n              colimit.\u03b9 (parallelPair (N'.actRight \u25b7 P.X) ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft))\n                WalkingParallelPair.one) \u226b\n          colimit.\u03b9\n            (parallelPair (M.actRight \u25b7 TensorBimod.X N' P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P))\n            WalkingParallelPair.one) \u226b\n        AssociatorBimod.inv M N' P", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (\u03b1_ M.X N.X P.X).hom \u226b\n      (M.X \u25c1\n            (f.hom \u25b7 P.X \u226b\n              colimit.\u03b9 (parallelPair (N'.actRight \u25b7 P.X) ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft))\n                WalkingParallelPair.one) \u226b\n          colimit.\u03b9\n            (parallelPair (M.actRight \u25b7 TensorBimod.X N' P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P))\n            WalkingParallelPair.one) \u226b\n        coequalizer.desc (AssociatorBimod.invAux M N' P) \u22ef"}, {"tactic": "slice_rhs 3 4 => rw [coequalizer.\u03c0_desc]", "annotated_tactic": ["slice_rhs 3 4 => rw [<a>coequalizer.\u03c0_desc</a>]", [{"full_name": "CategoryTheory.Limits.coequalizer.\u03c0_desc", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "def_pos": [996, 9], "def_end_pos": [996, 27]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (\u03b1_ M.X N.X P.X).hom \u226b\n      (M.X \u25c1\n            (f.hom \u25b7 P.X \u226b\n              colimit.\u03b9 (parallelPair (N'.actRight \u25b7 P.X) ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft))\n                WalkingParallelPair.one) \u226b\n          colimit.\u03b9\n            (parallelPair (M.actRight \u25b7 TensorBimod.X N' P)\n              ((\u03b1_ M.X X.X (TensorBimod.X N' P)).hom \u226b M.X \u25c1 TensorBimod.actLeft N' P))\n            WalkingParallelPair.one) \u226b\n        coequalizer.desc (AssociatorBimod.invAux M N' P) \u22ef", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (\u03b1_ M.X N.X P.X).hom \u226b\n      M.X \u25c1\n          (f.hom \u25b7 P.X \u226b\n            colimit.\u03b9 (parallelPair (N'.actRight \u25b7 P.X) ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft))\n              WalkingParallelPair.one) \u226b\n        AssociatorBimod.invAux M N' P"}, {"tactic": "dsimp [AssociatorBimod.invAux]", "annotated_tactic": ["dsimp [<a>AssociatorBimod.invAux</a>]", [{"full_name": "Bimod.AssociatorBimod.invAux", "def_path": "Mathlib/CategoryTheory/Monoidal/Bimod.lean", "def_pos": [573, 19], "def_end_pos": [573, 25]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (\u03b1_ M.X N.X P.X).hom \u226b\n      M.X \u25c1\n          (f.hom \u25b7 P.X \u226b\n            colimit.\u03b9 (parallelPair (N'.actRight \u25b7 P.X) ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft))\n              WalkingParallelPair.one) \u226b\n        AssociatorBimod.invAux M N' P", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (\u03b1_ M.X N.X P.X).hom \u226b\n      M.X \u25c1\n          (f.hom \u25b7 P.X \u226b\n            colimit.\u03b9 (parallelPair (N'.actRight \u25b7 P.X) ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft))\n              WalkingParallelPair.one) \u226b\n        (PreservesCoequalizer.iso (tensorLeft M.X) (N'.actRight \u25b7 P.X) ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft)).inv \u226b\n          coequalizer.desc\n            ((\u03b1_ M.X N'.X P.X).inv \u226b\n              coequalizer.\u03c0 (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) \u25b7 P.X \u226b\n                coequalizer.\u03c0 (TensorBimod.actRight M N' \u25b7 P.X)\n                  ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n            \u22ef"}, {"tactic": "slice_rhs 2 2 => rw [MonoidalCategory.whiskerLeft_comp]", "annotated_tactic": ["slice_rhs 2 2 => rw [<a>MonoidalCategory.whiskerLeft_comp</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.whiskerLeft_comp", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [235, 9], "def_end_pos": [235, 25]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (\u03b1_ M.X N.X P.X).hom \u226b\n      M.X \u25c1\n          (f.hom \u25b7 P.X \u226b\n            colimit.\u03b9 (parallelPair (N'.actRight \u25b7 P.X) ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft))\n              WalkingParallelPair.one) \u226b\n        (PreservesCoequalizer.iso (tensorLeft M.X) (N'.actRight \u25b7 P.X) ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft)).inv \u226b\n          coequalizer.desc\n            ((\u03b1_ M.X N'.X P.X).inv \u226b\n              coequalizer.\u03c0 (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) \u25b7 P.X \u226b\n                coequalizer.\u03c0 (TensorBimod.actRight M N' \u25b7 P.X)\n                  ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n            \u22ef", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (\u03b1_ M.X N.X P.X).hom \u226b\n      ((M.X \u25c1 f.hom \u25b7 P.X \u226b\n            M.X \u25c1\n              colimit.\u03b9 (parallelPair (N'.actRight \u25b7 P.X) ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft))\n                WalkingParallelPair.one) \u226b\n          (PreservesCoequalizer.iso (tensorLeft M.X) (N'.actRight \u25b7 P.X)\n              ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft)).inv) \u226b\n        coequalizer.desc\n          ((\u03b1_ M.X N'.X P.X).inv \u226b\n            coequalizer.\u03c0 (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) \u25b7 P.X \u226b\n              coequalizer.\u03c0 (TensorBimod.actRight M N' \u25b7 P.X)\n                ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n          \u22ef"}, {"tactic": "slice_rhs 3 5 => rw [id_tensor_\u03c0_preserves_coequalizer_inv_desc]", "annotated_tactic": ["slice_rhs 3 5 => rw [<a>id_tensor_\u03c0_preserves_coequalizer_inv_desc</a>]", [{"full_name": "id_tensor_\u03c0_preserves_coequalizer_inv_desc", "def_path": "Mathlib/CategoryTheory/Monoidal/Bimod.lean", "def_pos": [35, 9], "def_end_pos": [35, 51]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (\u03b1_ M.X N.X P.X).hom \u226b\n      ((M.X \u25c1 f.hom \u25b7 P.X \u226b\n            M.X \u25c1\n              colimit.\u03b9 (parallelPair (N'.actRight \u25b7 P.X) ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft))\n                WalkingParallelPair.one) \u226b\n          (PreservesCoequalizer.iso (tensorLeft M.X) (N'.actRight \u25b7 P.X)\n              ((\u03b1_ N'.X Y.X P.X).hom \u226b N'.X \u25c1 P.actLeft)).inv) \u226b\n        coequalizer.desc\n          ((\u03b1_ M.X N'.X P.X).inv \u226b\n            coequalizer.\u03c0 (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) \u25b7 P.X \u226b\n              coequalizer.\u03c0 (TensorBimod.actRight M N' \u25b7 P.X)\n                ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n          \u22ef", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (\u03b1_ M.X N.X P.X).hom \u226b\n      M.X \u25c1 f.hom \u25b7 P.X \u226b\n        (\u03b1_ M.X N'.X P.X).inv \u226b\n          coequalizer.\u03c0 (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) \u25b7 P.X \u226b\n            coequalizer.\u03c0 (TensorBimod.actRight M N' \u25b7 P.X)\n              ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft)"}, {"tactic": "slice_rhs 2 3 => rw [associator_inv_naturality_middle]", "annotated_tactic": ["slice_rhs 2 3 => rw [<a>associator_inv_naturality_middle</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.associator_inv_naturality_middle", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [474, 9], "def_end_pos": [474, 41]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (\u03b1_ M.X N.X P.X).hom \u226b\n      M.X \u25c1 f.hom \u25b7 P.X \u226b\n        (\u03b1_ M.X N'.X P.X).inv \u226b\n          coequalizer.\u03c0 (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) \u25b7 P.X \u226b\n            coequalizer.\u03c0 (TensorBimod.actRight M N' \u25b7 P.X)\n              ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft)", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (\u03b1_ M.X N.X P.X).hom \u226b\n      (((\u03b1_ M.X N.X P.X).inv \u226b (M.X \u25c1 f.hom) \u25b7 P.X) \u226b\n          coequalizer.\u03c0 (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) \u25b7 P.X) \u226b\n        coequalizer.\u03c0 (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft)"}, {"tactic": "slice_rhs 1 3 => rw [Iso.hom_inv_id_assoc]", "annotated_tactic": ["slice_rhs 1 3 => rw [<a>Iso.hom_inv_id_assoc</a>]", [{"full_name": "CategoryTheory.Iso.hom_inv_id_assoc", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [67, 12], "def_end_pos": [67, 34]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    (\u03b1_ M.X N.X P.X).hom \u226b\n      (((\u03b1_ M.X N.X P.X).inv \u226b (M.X \u25c1 f.hom) \u25b7 P.X) \u226b\n          coequalizer.\u03c0 (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) \u25b7 P.X) \u226b\n        coequalizer.\u03c0 (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft)", "state_after": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    ((M.X \u25c1 f.hom) \u25b7 P.X \u226b coequalizer.\u03c0 (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) \u25b7 P.X) \u226b\n      coequalizer.\u03c0 (TensorBimod.actRight M N' \u25b7 P.X)\n        ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft)"}, {"tactic": "slice_lhs 1 1 => rw [comp_whiskerRight]", "annotated_tactic": ["slice_lhs 1 1 => rw [<a>comp_whiskerRight</a>]", [{"full_name": "CategoryTheory.MonoidalCategory.comp_whiskerRight", "def_path": "Mathlib/CategoryTheory/Monoidal/Category.lean", "def_pos": [253, 9], "def_end_pos": [253, 26]}]], "state_before": "case h.h\nC : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\ninst\u271d\u00b3 : MonoidalCategory C\nA B : Mon_ C\nM\u271d : Bimod A B\ninst\u271d\u00b2 : HasCoequalizers C\ninst\u271d\u00b9 : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorLeft X)\ninst\u271d : (X : C) \u2192 PreservesColimitsOfSize.{0, 0, v\u2081, v\u2081, u\u2081, u\u2081} (tensorRight X)\nW X Y Z : Mon_ C\nM : Bimod W X\nN N' : Bimod X Y\nf : N \u27f6 N'\nP : Bimod Y Z\n\u22a2 (M.X \u25c1 f.hom \u226b\n          colimit.\u03b9 (parallelPair (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft))\n            WalkingParallelPair.one) \u25b7\n        P.X \u226b\n      colimit.\u03b9\n        (parallelPair (TensorBimod.actRight M N' \u25b7 P.X)\n          ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft))\n        WalkingParallelPair.one =\n    ((M.X \u25c1 f.hom) \u25b7 P.X \u226b coequalizer.\u03c0 (M.actRight \u25b7 N'.X) ((\u03b1_ M.X X.X N'.X).hom \u226b M.X \u25c1 N'.actLeft) \u25b7 P.X) \u226b\n      coequalizer.\u03c0 (TensorBimod.actRight M N' \u25b7 P.X)\n        ((\u03b1_ (TensorBimod.X M N') Y.X P.X).hom \u226b TensorBimod.X M N' \u25c1 P.actLeft)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Ordmap/Ordset.lean", "full_name": "Ordnode.Valid'.node4L_lemma\u2082", "start": [1147, 1], "end": [1147, 91], "traced_tactics": [{"tactic": "omega", "annotated_tactic": ["omega", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : Preorder \u03b1\nb c d : \u2115\nmr\u2082 : b + c + 1 \u2264 3 * d\n\u22a2 c \u2264 3 * d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/List/Perm.lean", "full_name": "List.isSubperm_iff", "start": [575, 1], "end": [576, 36], "traced_tactics": [{"tactic": "simp [isSubperm, subperm_ext_iff]", "annotated_tactic": ["simp [<a>isSubperm</a>, <a>subperm_ext_iff</a>]", [{"full_name": "List.isSubperm", "def_path": ".lake/packages/batteries/Batteries/Data/List/Basic.lean", "def_pos": [1383, 5], "def_end_pos": [1383, 14]}, {"full_name": "List.subperm_ext_iff", "def_path": ".lake/packages/batteries/Batteries/Data/List/Perm.lean", "def_pos": [569, 9], "def_end_pos": [569, 24]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nl\u2081 l\u2082 : List \u03b1\n\u22a2 l\u2081.isSubperm l\u2082 = true \u2194 l\u2081 <+~ l\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/GoldenRatio.lean", "full_name": "gold_sub_goldConj", "start": [84, 1], "end": [84, 50], "traced_tactics": [{"tactic": "ring", "annotated_tactic": ["ring", []], "state_before": "\u22a2 \u03c6 - \u03c8 = \u221a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Lifts.lean", "full_name": "Polynomial.lifts_iff_ringHom_rangeS", "start": [69, 1], "end": [70, 71], "traced_tactics": [{"tactic": "simp only [coe_mapRingHom, lifts, Set.mem_range, RingHom.mem_rangeS]", "annotated_tactic": ["simp only [<a>coe_mapRingHom</a>, <a>lifts</a>, <a>Set.mem_range</a>, <a>RingHom.mem_rangeS</a>]", [{"full_name": "Polynomial.coe_mapRingHom", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [775, 9], "def_end_pos": [775, 23]}, {"full_name": "Polynomial.lifts", "def_path": "Mathlib/Algebra/Polynomial/Lifts.lean", "def_pos": [57, 5], "def_end_pos": [57, 10]}, {"full_name": "Set.mem_range", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [160, 17], "def_end_pos": [160, 26]}, {"full_name": "RingHom.mem_rangeS", "def_path": "Mathlib/Algebra/Ring/Subsemiring/Basic.lean", "def_pos": [521, 9], "def_end_pos": [521, 19]}]], "state_before": "R : Type u\ninst\u271d\u00b9 : Semiring R\nS : Type v\ninst\u271d : Semiring S\nf : R \u2192+* S\np : S[X]\n\u22a2 p \u2208 lifts f \u2194 p \u2208 (mapRingHom f).rangeS", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/Adjoin.lean", "full_name": "IntermediateField.adjoin.mono", "start": [368, 1], "end": [369, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/ModelTheory/Algebra/Ring/Basic.lean", "full_name": "FirstOrder.Ring.realize_zero", "start": [195, 1], "end": [196, 44], "traced_tactics": [{"tactic": "simp [zero_def, funMap_zero, constantMap]", "annotated_tactic": ["simp [<a>zero_def</a>, <a>funMap_zero</a>, <a>constantMap</a>]", [{"full_name": "FirstOrder.Ring.zero_def", "def_path": "Mathlib/ModelTheory/Algebra/Ring/Basic.lean", "def_pos": [98, 9], "def_end_pos": [98, 17]}, {"full_name": "FirstOrder.Ring.CompatibleRing.funMap_zero", "def_path": "Mathlib/ModelTheory/Algebra/Ring/Basic.lean", "def_pos": [166, 3], "def_end_pos": [166, 14]}, {"full_name": "FirstOrder.Language.constantMap", "def_path": "Mathlib/ModelTheory/Basic.lean", "def_pos": [366, 5], "def_end_pos": [366, 16]}]], "state_before": "\u03b1 : Type u_1\nR : Type u_2\ninst\u271d\u2075 : Add R\ninst\u271d\u2074 : Mul R\ninst\u271d\u00b3 : Neg R\ninst\u271d\u00b2 : One R\ninst\u271d\u00b9 : Zero R\ninst\u271d : CompatibleRing R\nv : \u03b1 \u2192 R\n\u22a2 Term.realize v 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Basic.lean", "full_name": "Set.subset_inter", "start": [937, 1], "end": [938, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "full_name": "Real.cos_eq_sqrt_one_sub_sin_sq", "start": [549, 1], "end": [551, 88], "traced_tactics": [{"tactic": "rw [\u2190 abs_cos_eq_sqrt_one_sub_sin_sq, abs_of_nonneg (cos_nonneg_of_mem_Icc \u27e8hl, hu\u27e9)]", "annotated_tactic": ["rw [\u2190 <a>abs_cos_eq_sqrt_one_sub_sin_sq</a>, <a>abs_of_nonneg</a> (<a>cos_nonneg_of_mem_Icc</a> \u27e8hl, hu\u27e9)]", [{"full_name": "Real.abs_cos_eq_sqrt_one_sub_sin_sq", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1006, 9], "def_end_pos": [1006, 39]}, {"full_name": "abs_of_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"full_name": "Real.cos_nonneg_of_mem_Icc", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "def_pos": [525, 9], "def_end_pos": [525, 30]}]], "state_before": "x : \u211d\nhl : -(\u03c0 / 2) \u2264 x\nhu : x \u2264 \u03c0 / 2\n\u22a2 cos x = \u221a(1 - sin x ^ 2)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean", "full_name": "Complex.differentiable_sin", "start": [54, 1], "end": [54, 98], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean", "full_name": "ContinuousMultilinearMap.toMultilinearMap_injective", "start": [70, 1], "end": [74, 43], "traced_tactics": [{"tactic": "subst h", "annotated_tactic": ["subst h", []], "state_before": "R : Type u\n\u03b9 : Type v\nn : \u2115\nM : Fin n.succ \u2192 Type w\nM\u2081 : \u03b9 \u2192 Type w\u2081\nM\u2081' : \u03b9 \u2192 Type w\u2081'\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\nM\u2084 : Type w\u2084\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : (i : Fin n.succ) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081' i)\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2084\ninst\u271d\u00b9\u00b9 : (i : Fin n.succ) \u2192 Module R (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 Module R (M\u2081' i)\ninst\u271d\u2078 : Module R M\u2082\ninst\u271d\u2077 : Module R M\u2083\ninst\u271d\u2076 : Module R M\u2084\ninst\u271d\u2075 : (i : Fin n.succ) \u2192 TopologicalSpace (M i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 TopologicalSpace (M\u2081 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 TopologicalSpace (M\u2081' i)\ninst\u271d\u00b2 : TopologicalSpace M\u2082\ninst\u271d\u00b9 : TopologicalSpace M\u2083\ninst\u271d : TopologicalSpace M\u2084\nf\u271d f' : ContinuousMultilinearMap R M\u2081 M\u2082\nf : MultilinearMap R M\u2081 M\u2082\nhf : Continuous f.toFun\ng : MultilinearMap R M\u2081 M\u2082\nhg : Continuous g.toFun\nh : { toMultilinearMap := f, cont := hf }.toMultilinearMap = { toMultilinearMap := g, cont := hg }.toMultilinearMap\n\u22a2 { toMultilinearMap := f, cont := hf } = { toMultilinearMap := g, cont := hg }", "state_after": "R : Type u\n\u03b9 : Type v\nn : \u2115\nM : Fin n.succ \u2192 Type w\nM\u2081 : \u03b9 \u2192 Type w\u2081\nM\u2081' : \u03b9 \u2192 Type w\u2081'\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\nM\u2084 : Type w\u2084\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : (i : Fin n.succ) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081' i)\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2084\ninst\u271d\u00b9\u00b9 : (i : Fin n.succ) \u2192 Module R (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 Module R (M\u2081' i)\ninst\u271d\u2078 : Module R M\u2082\ninst\u271d\u2077 : Module R M\u2083\ninst\u271d\u2076 : Module R M\u2084\ninst\u271d\u2075 : (i : Fin n.succ) \u2192 TopologicalSpace (M i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 TopologicalSpace (M\u2081 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 TopologicalSpace (M\u2081' i)\ninst\u271d\u00b2 : TopologicalSpace M\u2082\ninst\u271d\u00b9 : TopologicalSpace M\u2083\ninst\u271d : TopologicalSpace M\u2084\nf\u271d f' : ContinuousMultilinearMap R M\u2081 M\u2082\nf : MultilinearMap R M\u2081 M\u2082\nhf : Continuous f.toFun\nhg : Continuous { toMultilinearMap := f, cont := hf }.toFun\n\u22a2 { toMultilinearMap := f, cont := hf } =\n    { toMultilinearMap := { toMultilinearMap := f, cont := hf }.toMultilinearMap, cont := hg }"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "R : Type u\n\u03b9 : Type v\nn : \u2115\nM : Fin n.succ \u2192 Type w\nM\u2081 : \u03b9 \u2192 Type w\u2081\nM\u2081' : \u03b9 \u2192 Type w\u2081'\nM\u2082 : Type w\u2082\nM\u2083 : Type w\u2083\nM\u2084 : Type w\u2084\ninst\u271d\u00b9\u2078 : Semiring R\ninst\u271d\u00b9\u2077 : (i : Fin n.succ) \u2192 AddCommMonoid (M i)\ninst\u271d\u00b9\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i)\ninst\u271d\u00b9\u2075 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081' i)\ninst\u271d\u00b9\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b9\u00b3 : AddCommMonoid M\u2083\ninst\u271d\u00b9\u00b2 : AddCommMonoid M\u2084\ninst\u271d\u00b9\u00b9 : (i : Fin n.succ) \u2192 Module R (M i)\ninst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u2079 : (i : \u03b9) \u2192 Module R (M\u2081' i)\ninst\u271d\u2078 : Module R M\u2082\ninst\u271d\u2077 : Module R M\u2083\ninst\u271d\u2076 : Module R M\u2084\ninst\u271d\u2075 : (i : Fin n.succ) \u2192 TopologicalSpace (M i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 TopologicalSpace (M\u2081 i)\ninst\u271d\u00b3 : (i : \u03b9) \u2192 TopologicalSpace (M\u2081' i)\ninst\u271d\u00b2 : TopologicalSpace M\u2082\ninst\u271d\u00b9 : TopologicalSpace M\u2083\ninst\u271d : TopologicalSpace M\u2084\nf\u271d f' : ContinuousMultilinearMap R M\u2081 M\u2082\nf : MultilinearMap R M\u2081 M\u2082\nhf : Continuous f.toFun\nhg : Continuous { toMultilinearMap := f, cont := hf }.toFun\n\u22a2 { toMultilinearMap := f, cont := hf } =\n    { toMultilinearMap := { toMultilinearMap := f, cont := hf }.toMultilinearMap, cont := hg }", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Normed/Order/Lattice.lean", "full_name": "lipschitzWith_posPart", "start": [202, 1], "end": [203, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Ideal/QuotientOperations.lean", "full_name": "Ideal.quotientInfEquivQuotientProd_fst", "start": [280, 1], "end": [283, 34], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Dual.lean", "full_name": "Submodule.dualAnnihilator_sup_eq", "start": [1002, 1], "end": [1004, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/Hom/Esakia.lean", "full_name": "EsakiaHom.cancel_left", "start": [360, 1], "end": [362, 81], "traced_tactics": [{"tactic": "rw [\u2190 comp_apply, h, comp_apply]", "annotated_tactic": ["rw [\u2190 <a>comp_apply</a>, h, <a>comp_apply</a>]", [{"full_name": "EsakiaHom.comp_apply", "def_path": "Mathlib/Topology/Order/Hom/Esakia.lean", "def_pos": [335, 9], "def_end_pos": [335, 19]}, {"full_name": "EsakiaHom.comp_apply", "def_path": "Mathlib/Topology/Order/Hom/Esakia.lean", "def_pos": [335, 9], "def_end_pos": [335, 19]}]], "state_before": "F : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03b3 : Type u_4\n\u03b4 : Type u_5\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : Preorder \u03b1\ninst\u271d\u2075 : TopologicalSpace \u03b2\ninst\u271d\u2074 : Preorder \u03b2\ninst\u271d\u00b3 : TopologicalSpace \u03b3\ninst\u271d\u00b2 : Preorder \u03b3\ninst\u271d\u00b9 : TopologicalSpace \u03b4\ninst\u271d : Preorder \u03b4\ng : EsakiaHom \u03b2 \u03b3\nf\u2081 f\u2082 : EsakiaHom \u03b1 \u03b2\nhg : Injective \u21d1g\nh : g.comp f\u2081 = g.comp f\u2082\na : \u03b1\n\u22a2 g (f\u2081 a) = g (f\u2082 a)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Rat/Defs.lean", "full_name": "Rat.inv_divInt'", "start": [286, 1], "end": [286, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Multiset/Basic.lean", "full_name": "Multiset.rel_of_forall", "start": [2899, 1], "end": [2911, 21], "traced_tactics": [{"tactic": "revert m1", "annotated_tactic": ["revert m1", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm1 m2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nh : \u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 m2 \u2192 r a b\nhc : card m1 = card m2\n\u22a2 Rel r m1 m2", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 m2 \u2192 r a b) \u2192 card m1 = card m2 \u2192 Rel r m1 m2"}, {"tactic": "refine @(m2.induction_on ?_ ?_)", "annotated_tactic": ["refine @(m2.induction_on ?_ ?_)", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 m2 \u2192 r a b) \u2192 card m1 = card m2 \u2192 Rel r m1 m2", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 0 \u2192 r a b) \u2192 card m1 = card 0 \u2192 Rel r m1 0\n\ncase refine_2\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u2200 (a : \u03b1) (s : Multiset \u03b1),\n    (\u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 s \u2192 r a b) \u2192 card m1 = card s \u2192 Rel r m1 s) \u2192\n      \u2200 {m1 : Multiset \u03b1},\n        (\u2200 (a_2 b : \u03b1), a_2 \u2208 m1 \u2192 b \u2208 a ::\u2098 s \u2192 r a_2 b) \u2192 card m1 = card (a ::\u2098 s) \u2192 Rel r m1 (a ::\u2098 s)"}, {"tactic": "intro m _h hc", "annotated_tactic": ["intro m _h hc", []], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 0 \u2192 r a b) \u2192 card m1 = card 0 \u2192 Rel r m1 0", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nm : Multiset \u03b1\n_h : \u2200 (a b : \u03b1), a \u2208 m \u2192 b \u2208 0 \u2192 r a b\nhc : card m = card 0\n\u22a2 Rel r m 0"}, {"tactic": "rw [rel_zero_right, \u2190 card_eq_zero, hc, card_zero]", "annotated_tactic": ["rw [<a>rel_zero_right</a>, \u2190 <a>card_eq_zero</a>, hc, <a>card_zero</a>]", [{"full_name": "Multiset.rel_zero_right", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2822, 9], "def_end_pos": [2822, 23]}, {"full_name": "Multiset.card_eq_zero", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [831, 9], "def_end_pos": [831, 21]}, {"full_name": "Multiset.card_zero", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [775, 9], "def_end_pos": [775, 18]}]], "state_before": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\nm : Multiset \u03b1\n_h : \u2200 (a b : \u03b1), a \u2208 m \u2192 b \u2208 0 \u2192 r a b\nhc : card m = card 0\n\u22a2 Rel r m 0", "state_after": "no goals"}, {"tactic": "intro a t ih m h hc", "annotated_tactic": ["intro a t ih m h hc", []], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u22a2 \u2200 (a : \u03b1) (s : Multiset \u03b1),\n    (\u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 s \u2192 r a b) \u2192 card m1 = card s \u2192 Rel r m1 s) \u2192\n      \u2200 {m1 : Multiset \u03b1},\n        (\u2200 (a_2 b : \u03b1), a_2 \u2208 m1 \u2192 b \u2208 a ::\u2098 s \u2192 r a_2 b) \u2192 card m1 = card (a ::\u2098 s) \u2192 Rel r m1 (a ::\u2098 s)", "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nt : Multiset \u03b1\nih : \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 t \u2192 r a b) \u2192 card m1 = card t \u2192 Rel r m1 t\nm : Multiset \u03b1\nh : \u2200 (a_1 b : \u03b1), a_1 \u2208 m \u2192 b \u2208 a ::\u2098 t \u2192 r a_1 b\nhc : card m = card (a ::\u2098 t)\n\u22a2 Rel r m (a ::\u2098 t)"}, {"tactic": "rw [card_cons] at hc", "annotated_tactic": ["rw [<a>card_cons</a>] at hc", [{"full_name": "Multiset.card_cons", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [788, 9], "def_end_pos": [788, 18]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nt : Multiset \u03b1\nih : \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 t \u2192 r a b) \u2192 card m1 = card t \u2192 Rel r m1 t\nm : Multiset \u03b1\nh : \u2200 (a_1 b : \u03b1), a_1 \u2208 m \u2192 b \u2208 a ::\u2098 t \u2192 r a_1 b\nhc : card m = card (a ::\u2098 t)\n\u22a2 Rel r m (a ::\u2098 t)", "state_after": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nt : Multiset \u03b1\nih : \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 t \u2192 r a b) \u2192 card m1 = card t \u2192 Rel r m1 t\nm : Multiset \u03b1\nh : \u2200 (a_1 b : \u03b1), a_1 \u2208 m \u2192 b \u2208 a ::\u2098 t \u2192 r a_1 b\nhc : card m = card t + 1\n\u22a2 Rel r m (a ::\u2098 t)"}, {"tactic": "obtain \u27e8b, hb\u27e9 := card_pos_iff_exists_mem.1 (show 0 < card m from hc.symm \u25b8 Nat.succ_pos _)", "annotated_tactic": ["obtain \u27e8b, hb\u27e9 := <a>card_pos_iff_exists_mem</a>.1 (show 0 < <a>card</a> m from hc.symm \u25b8 <a>Nat.succ_pos</a> _)", [{"full_name": "Multiset.card_pos_iff_exists_mem", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [839, 9], "def_end_pos": [839, 32]}, {"full_name": "Multiset.card", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [758, 5], "def_end_pos": [758, 9]}, {"full_name": "Nat.succ_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1693, 9], "def_end_pos": [1693, 21]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nt : Multiset \u03b1\nih : \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 t \u2192 r a b) \u2192 card m1 = card t \u2192 Rel r m1 t\nm : Multiset \u03b1\nh : \u2200 (a_1 b : \u03b1), a_1 \u2208 m \u2192 b \u2208 a ::\u2098 t \u2192 r a_1 b\nhc : card m = card t + 1\n\u22a2 Rel r m (a ::\u2098 t)", "state_after": "case refine_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nt : Multiset \u03b1\nih : \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 t \u2192 r a b) \u2192 card m1 = card t \u2192 Rel r m1 t\nm : Multiset \u03b1\nh : \u2200 (a_1 b : \u03b1), a_1 \u2208 m \u2192 b \u2208 a ::\u2098 t \u2192 r a_1 b\nhc : card m = card t + 1\nb : \u03b1\nhb : b \u2208 m\n\u22a2 Rel r m (a ::\u2098 t)"}, {"tactic": "obtain \u27e8m', rfl\u27e9 := exists_cons_of_mem hb", "annotated_tactic": ["obtain \u27e8m', rfl\u27e9 := <a>exists_cons_of_mem</a> hb", [{"full_name": "Multiset.exists_cons_of_mem", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [267, 9], "def_end_pos": [267, 27]}]], "state_before": "case refine_2.intro\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nt : Multiset \u03b1\nih : \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 t \u2192 r a b) \u2192 card m1 = card t \u2192 Rel r m1 t\nm : Multiset \u03b1\nh : \u2200 (a_1 b : \u03b1), a_1 \u2208 m \u2192 b \u2208 a ::\u2098 t \u2192 r a_1 b\nhc : card m = card t + 1\nb : \u03b1\nhb : b \u2208 m\n\u22a2 Rel r m (a ::\u2098 t)", "state_after": "case refine_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nt : Multiset \u03b1\nih : \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 t \u2192 r a b) \u2192 card m1 = card t \u2192 Rel r m1 t\nb : \u03b1\nm' : Multiset \u03b1\nh : \u2200 (a_1 b_1 : \u03b1), a_1 \u2208 b ::\u2098 m' \u2192 b_1 \u2208 a ::\u2098 t \u2192 r a_1 b_1\nhc : card (b ::\u2098 m') = card t + 1\nhb : b \u2208 b ::\u2098 m'\n\u22a2 Rel r (b ::\u2098 m') (a ::\u2098 t)"}, {"tactic": "refine rel_cons_right.mpr \u27e8b, m', h _ _ hb (mem_cons_self _ _), ih ?_ ?_, rfl\u27e9", "annotated_tactic": ["refine rel_cons_right.mpr \u27e8b, m', h _ _ hb (<a>mem_cons_self</a> _ _), ih ?_ ?_, <a>rfl</a>\u27e9", [{"full_name": "Multiset.mem_cons_self", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [258, 9], "def_end_pos": [258, 22]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case refine_2.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nt : Multiset \u03b1\nih : \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 t \u2192 r a b) \u2192 card m1 = card t \u2192 Rel r m1 t\nb : \u03b1\nm' : Multiset \u03b1\nh : \u2200 (a_1 b_1 : \u03b1), a_1 \u2208 b ::\u2098 m' \u2192 b_1 \u2208 a ::\u2098 t \u2192 r a_1 b_1\nhc : card (b ::\u2098 m') = card t + 1\nhb : b \u2208 b ::\u2098 m'\n\u22a2 Rel r (b ::\u2098 m') (a ::\u2098 t)", "state_after": "case refine_2.intro.intro.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nt : Multiset \u03b1\nih : \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 t \u2192 r a b) \u2192 card m1 = card t \u2192 Rel r m1 t\nb : \u03b1\nm' : Multiset \u03b1\nh : \u2200 (a_1 b_1 : \u03b1), a_1 \u2208 b ::\u2098 m' \u2192 b_1 \u2208 a ::\u2098 t \u2192 r a_1 b_1\nhc : card (b ::\u2098 m') = card t + 1\nhb : b \u2208 b ::\u2098 m'\n\u22a2 \u2200 (a b : \u03b1), a \u2208 m' \u2192 b \u2208 t \u2192 r a b\n\ncase refine_2.intro.intro.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nt : Multiset \u03b1\nih : \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 t \u2192 r a b) \u2192 card m1 = card t \u2192 Rel r m1 t\nb : \u03b1\nm' : Multiset \u03b1\nh : \u2200 (a_1 b_1 : \u03b1), a_1 \u2208 b ::\u2098 m' \u2192 b_1 \u2208 a ::\u2098 t \u2192 r a_1 b_1\nhc : card (b ::\u2098 m') = card t + 1\nhb : b \u2208 b ::\u2098 m'\n\u22a2 card m' = card t"}, {"tactic": "exact fun _ _ ha hb => h _ _ (mem_cons_of_mem ha) (mem_cons_of_mem hb)", "annotated_tactic": ["exact fun _ _ ha hb => h _ _ (<a>mem_cons_of_mem</a> ha) (<a>mem_cons_of_mem</a> hb)", [{"full_name": "Multiset.mem_cons_of_mem", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [253, 9], "def_end_pos": [253, 24]}, {"full_name": "Multiset.mem_cons_of_mem", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [253, 9], "def_end_pos": [253, 24]}]], "state_before": "case refine_2.intro.intro.refine_1\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nt : Multiset \u03b1\nih : \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 t \u2192 r a b) \u2192 card m1 = card t \u2192 Rel r m1 t\nb : \u03b1\nm' : Multiset \u03b1\nh : \u2200 (a_1 b_1 : \u03b1), a_1 \u2208 b ::\u2098 m' \u2192 b_1 \u2208 a ::\u2098 t \u2192 r a_1 b_1\nhc : card (b ::\u2098 m') = card t + 1\nhb : b \u2208 b ::\u2098 m'\n\u22a2 \u2200 (a b : \u03b1), a \u2208 m' \u2192 b \u2208 t \u2192 r a b", "state_after": "no goals"}, {"tactic": "simpa using hc", "annotated_tactic": ["simpa using hc", []], "state_before": "case refine_2.intro.intro.refine_2\n\u03b1 : Type u_1\n\u03b2 : Type v\n\u03b3 : Type u_2\n\u03b4 : Type u_3\nr\u271d : \u03b1 \u2192 \u03b2 \u2192 Prop\np : \u03b3 \u2192 \u03b4 \u2192 Prop\nm2 : Multiset \u03b1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nt : Multiset \u03b1\nih : \u2200 {m1 : Multiset \u03b1}, (\u2200 (a b : \u03b1), a \u2208 m1 \u2192 b \u2208 t \u2192 r a b) \u2192 card m1 = card t \u2192 Rel r m1 t\nb : \u03b1\nm' : Multiset \u03b1\nh : \u2200 (a_1 b_1 : \u03b1), a_1 \u2208 b ::\u2098 m' \u2192 b_1 \u2208 a ::\u2098 t \u2192 r a_1 b_1\nhc : card (b ::\u2098 m') = card t + 1\nhb : b \u2208 b ::\u2098 m'\n\u22a2 card m' = card t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Lattice.lean", "full_name": "inf_ind", "start": [822, 1], "end": [823, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "full_name": "MeasureTheory.Measure.QuasiMeasurePreserving.liminf_preimage_iterate_ae_eq", "start": [1872, 1], "end": [1878, 68], "traced_tactics": [{"tactic": "rw [\u2190 ae_eq_set_compl_compl, @Filter.liminf_compl (Set \u03b1)]", "annotated_tactic": ["rw [\u2190 <a>ae_eq_set_compl_compl</a>, @<a>Filter.liminf_compl</a> (<a>Set</a> \u03b1)]", [{"full_name": "MeasureTheory.ae_eq_set_compl_compl", "def_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean", "def_pos": [189, 9], "def_end_pos": [189, 30]}, {"full_name": "Filter.liminf_compl", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [1169, 9], "def_end_pos": [1169, 21]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}]], "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\nf\u271d\u00b9 : \u03b1 \u2192 \u03b2\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 \u03bc \u03bc\nhs : f \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 liminf (fun n => (preimage f)^[n] s) atTop =\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\nf\u271d\u00b9 : \u03b1 \u2192 \u03b2\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 \u03bc \u03bc\nhs : f \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 limsup (compl \u2218 fun n => (preimage f)^[n] s) atTop =\u1d50[\u03bc] s\u1d9c"}, {"tactic": "rw [\u2190 ae_eq_set_compl_compl, \u2190 preimage_compl] at hs", "annotated_tactic": ["rw [\u2190 <a>ae_eq_set_compl_compl</a>, \u2190 <a>preimage_compl</a>] at hs", [{"full_name": "MeasureTheory.ae_eq_set_compl_compl", "def_path": "Mathlib/MeasureTheory/OuterMeasure/AE.lean", "def_pos": [189, 9], "def_end_pos": [189, 30]}, {"full_name": "Set.preimage_compl", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [82, 9], "def_end_pos": [82, 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\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\nf\u271d\u00b9 : \u03b1 \u2192 \u03b2\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 \u03bc \u03bc\nhs : f \u207b\u00b9' s =\u1d50[\u03bc] s\n\u22a2 limsup (compl \u2218 fun n => (preimage f)^[n] s) atTop =\u1d50[\u03bc] s\u1d9c", "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\nf\u271d\u00b9 : \u03b1 \u2192 \u03b2\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 \u03bc \u03bc\nhs : f \u207b\u00b9' s\u1d9c =\u1d50[\u03bc] s\u1d9c\n\u22a2 limsup (compl \u2218 fun n => (preimage f)^[n] s) atTop =\u1d50[\u03bc] s\u1d9c"}, {"tactic": "convert hf.limsup_preimage_iterate_ae_eq hs", "annotated_tactic": ["convert hf.limsup_preimage_iterate_ae_eq 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\nf\u271d\u00b9 : \u03b1 \u2192 \u03b2\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 \u03bc \u03bc\nhs : f \u207b\u00b9' s\u1d9c =\u1d50[\u03bc] s\u1d9c\n\u22a2 limsup (compl \u2218 fun n => (preimage f)^[n] s) atTop =\u1d50[\u03bc] s\u1d9c", "state_after": "case h.e'_4.h.e'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\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\nf\u271d\u00b9 : \u03b1 \u2192 \u03b2\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 \u03bc \u03bc\nhs : f \u207b\u00b9' s\u1d9c =\u1d50[\u03bc] s\u1d9c\nx\u271d : \u2115\n\u22a2 (compl \u2218 fun n => (preimage f)^[n] s) x\u271d = (preimage f)^[x\u271d] s\u1d9c"}, {"tactic": "ext1 n", "annotated_tactic": ["ext1 n", []], "state_before": "case h.e'_4.h.e'_4.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\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\nf\u271d\u00b9 : \u03b1 \u2192 \u03b2\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 \u03bc \u03bc\nhs : f \u207b\u00b9' s\u1d9c =\u1d50[\u03bc] s\u1d9c\nx\u271d : \u2115\n\u22a2 (compl \u2218 fun n => (preimage f)^[n] s) x\u271d = (preimage f)^[x\u271d] s\u1d9c", "state_after": "case h.e'_4.h.e'_4.h.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\nf\u271d\u00b9 : \u03b1 \u2192 \u03b2\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 \u03bc \u03bc\nhs : f \u207b\u00b9' s\u1d9c =\u1d50[\u03bc] s\u1d9c\nx\u271d : \u2115\nn : \u03b1\n\u22a2 n \u2208 (compl \u2218 fun n => (preimage f)^[n] s) x\u271d \u2194 n \u2208 (preimage f)^[x\u271d] s\u1d9c"}, {"tactic": "simp only [\u2190 Set.preimage_iterate_eq, comp_apply, preimage_compl]", "annotated_tactic": ["simp only [\u2190 <a>Set.preimage_iterate_eq</a>, <a>comp_apply</a>, <a>preimage_compl</a>]", [{"full_name": "Set.preimage_iterate_eq", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [157, 9], "def_end_pos": [157, 28]}, {"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}, {"full_name": "Set.preimage_compl", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [82, 9], "def_end_pos": [82, 23]}]], "state_before": "case h.e'_4.h.e'_4.h.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\nf\u271d\u00b9 : \u03b1 \u2192 \u03b2\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 \u03bc \u03bc\nhs : f \u207b\u00b9' s\u1d9c =\u1d50[\u03bc] s\u1d9c\nx\u271d : \u2115\nn : \u03b1\n\u22a2 n \u2208 (compl \u2218 fun n => (preimage f)^[n] s) x\u271d \u2194 n \u2208 (preimage f)^[x\u271d] s\u1d9c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/GradedObject/Unitor.lean", "full_name": "CategoryTheory.GradedObject.mapBifunctorRightUnitor_inv_naturality", "start": [233, 1], "end": [244, 6], "traced_tactics": [{"tactic": "ext j", "annotated_tactic": ["ext j", []], "state_before": "C : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst\u271d\u2077 : Category.{u_6, u_1} C\ninst\u271d\u2076 : Category.{u_5, u_2} D\ninst\u271d\u2075 : Zero I\ninst\u271d\u2074 : DecidableEq I\ninst\u271d\u00b3 : HasInitial C\nF : D \u2964 C \u2964 D\nY : C\ne : F.flip.obj Y \u2245 \ud835\udfed D\ninst\u271d\u00b2 : (X : D) \u2192 PreservesColimit (Functor.empty C) (F.obj X)\np : J \u00d7 I \u2192 J\nhp : \u2200 (j : J), p (j, 0) = j\nX X' : GradedObject J D\n\u03c6 : X \u27f6 X'\ninst\u271d\u00b9 : (((mapBifunctor F J I).obj X).obj ((single\u2080 I).obj Y)).HasMap p\ninst\u271d : (((mapBifunctor F J I).obj X').obj ((single\u2080 I).obj Y)).HasMap p\nY' : ?m.129019\n\u22a2 \u03c6 \u226b (mapBifunctorRightUnitor F Y e p hp X').inv =\n    (mapBifunctorRightUnitor F Y e p hp X).inv \u226b mapBifunctorMapMap F p \u03c6 (\ud835\udfd9 ((single\u2080 I).obj Y))", "state_after": "case h\nC : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst\u271d\u2077 : Category.{u_6, u_1} C\ninst\u271d\u2076 : Category.{u_5, u_2} D\ninst\u271d\u2075 : Zero I\ninst\u271d\u2074 : DecidableEq I\ninst\u271d\u00b3 : HasInitial C\nF : D \u2964 C \u2964 D\nY : C\ne : F.flip.obj Y \u2245 \ud835\udfed D\ninst\u271d\u00b2 : (X : D) \u2192 PreservesColimit (Functor.empty C) (F.obj X)\np : J \u00d7 I \u2192 J\nhp : \u2200 (j : J), p (j, 0) = j\nX X' : GradedObject J D\n\u03c6 : X \u27f6 X'\ninst\u271d\u00b9 : (((mapBifunctor F J I).obj X).obj ((single\u2080 I).obj Y)).HasMap p\ninst\u271d : (((mapBifunctor F J I).obj X').obj ((single\u2080 I).obj Y)).HasMap p\nY' : ?m.129019\nj : J\n\u22a2 (\u03c6 \u226b (mapBifunctorRightUnitor F Y e p hp X').inv) j =\n    ((mapBifunctorRightUnitor F Y e p hp X).inv \u226b mapBifunctorMapMap F p \u03c6 (\ud835\udfd9 ((single\u2080 I).obj Y))) j"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case h\nC : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst\u271d\u2077 : Category.{u_6, u_1} C\ninst\u271d\u2076 : Category.{u_5, u_2} D\ninst\u271d\u2075 : Zero I\ninst\u271d\u2074 : DecidableEq I\ninst\u271d\u00b3 : HasInitial C\nF : D \u2964 C \u2964 D\nY : C\ne : F.flip.obj Y \u2245 \ud835\udfed D\ninst\u271d\u00b2 : (X : D) \u2192 PreservesColimit (Functor.empty C) (F.obj X)\np : J \u00d7 I \u2192 J\nhp : \u2200 (j : J), p (j, 0) = j\nX X' : GradedObject J D\n\u03c6 : X \u27f6 X'\ninst\u271d\u00b9 : (((mapBifunctor F J I).obj X).obj ((single\u2080 I).obj Y)).HasMap p\ninst\u271d : (((mapBifunctor F J I).obj X').obj ((single\u2080 I).obj Y)).HasMap p\nY' : ?m.129019\nj : J\n\u22a2 (\u03c6 \u226b (mapBifunctorRightUnitor F Y e p hp X').inv) j =\n    ((mapBifunctorRightUnitor F Y e p hp X).inv \u226b mapBifunctorMapMap F p \u03c6 (\ud835\udfd9 ((single\u2080 I).obj Y))) j", "state_after": "case h\nC : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst\u271d\u2077 : Category.{u_6, u_1} C\ninst\u271d\u2076 : Category.{u_5, u_2} D\ninst\u271d\u2075 : Zero I\ninst\u271d\u2074 : DecidableEq I\ninst\u271d\u00b3 : HasInitial C\nF : D \u2964 C \u2964 D\nY : C\ne : F.flip.obj Y \u2245 \ud835\udfed D\ninst\u271d\u00b2 : (X : D) \u2192 PreservesColimit (Functor.empty C) (F.obj X)\np : J \u00d7 I \u2192 J\nhp : \u2200 (j : J), p (j, 0) = j\nX X' : GradedObject J D\n\u03c6 : X \u27f6 X'\ninst\u271d\u00b9 : (((mapBifunctor F J I).obj X).obj ((single\u2080 I).obj Y)).HasMap p\ninst\u271d : (((mapBifunctor F J I).obj X').obj ((single\u2080 I).obj Y)).HasMap p\nY' : ?m.129019\nj : J\n\u22a2 \u03c6 j \u226b (mapBifunctorRightUnitor F Y e p hp X').inv j =\n    (mapBifunctorRightUnitor F Y e p hp X).inv j \u226b mapBifunctorMapMap F p \u03c6 (\ud835\udfd9 ((single\u2080 I).obj Y)) j"}, {"tactic": "rw [mapBifunctorRightUnitor_inv_apply, mapBifunctorRightUnitor_inv_apply, assoc, assoc,\n  \u03b9_mapBifunctorMapMap]", "annotated_tactic": ["rw [<a>mapBifunctorRightUnitor_inv_apply</a>, <a>mapBifunctorRightUnitor_inv_apply</a>, <a>assoc</a>, <a>assoc</a>,\n    <a>\u03b9_mapBifunctorMapMap</a>]", [{"full_name": "CategoryTheory.GradedObject.mapBifunctorRightUnitor_inv_apply", "def_path": "Mathlib/CategoryTheory/GradedObject/Unitor.lean", "def_pos": [226, 7], "def_end_pos": [226, 40]}, {"full_name": "CategoryTheory.GradedObject.mapBifunctorRightUnitor_inv_apply", "def_path": "Mathlib/CategoryTheory/GradedObject/Unitor.lean", "def_pos": [226, 7], "def_end_pos": [226, 40]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.GradedObject.\u03b9_mapBifunctorMapMap", "def_path": "Mathlib/CategoryTheory/GradedObject/Bifunctor.lean", "def_pos": [74, 7], "def_end_pos": [74, 27]}]], "state_before": "case h\nC : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst\u271d\u2077 : Category.{u_6, u_1} C\ninst\u271d\u2076 : Category.{u_5, u_2} D\ninst\u271d\u2075 : Zero I\ninst\u271d\u2074 : DecidableEq I\ninst\u271d\u00b3 : HasInitial C\nF : D \u2964 C \u2964 D\nY : C\ne : F.flip.obj Y \u2245 \ud835\udfed D\ninst\u271d\u00b2 : (X : D) \u2192 PreservesColimit (Functor.empty C) (F.obj X)\np : J \u00d7 I \u2192 J\nhp : \u2200 (j : J), p (j, 0) = j\nX X' : GradedObject J D\n\u03c6 : X \u27f6 X'\ninst\u271d\u00b9 : (((mapBifunctor F J I).obj X).obj ((single\u2080 I).obj Y)).HasMap p\ninst\u271d : (((mapBifunctor F J I).obj X').obj ((single\u2080 I).obj Y)).HasMap p\nY' : ?m.129019\nj : J\n\u22a2 \u03c6 j \u226b (mapBifunctorRightUnitor F Y e p hp X').inv j =\n    (mapBifunctorRightUnitor F Y e p hp X).inv j \u226b mapBifunctorMapMap F p \u03c6 (\ud835\udfd9 ((single\u2080 I).obj Y)) j", "state_after": "case h\nC : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst\u271d\u2077 : Category.{u_6, u_1} C\ninst\u271d\u2076 : Category.{u_5, u_2} D\ninst\u271d\u2075 : Zero I\ninst\u271d\u2074 : DecidableEq I\ninst\u271d\u00b3 : HasInitial C\nF : D \u2964 C \u2964 D\nY : C\ne : F.flip.obj Y \u2245 \ud835\udfed D\ninst\u271d\u00b2 : (X : D) \u2192 PreservesColimit (Functor.empty C) (F.obj X)\np : J \u00d7 I \u2192 J\nhp : \u2200 (j : J), p (j, 0) = j\nX X' : GradedObject J D\n\u03c6 : X \u27f6 X'\ninst\u271d\u00b9 : (((mapBifunctor F J I).obj X).obj ((single\u2080 I).obj Y)).HasMap p\ninst\u271d : (((mapBifunctor F J I).obj X').obj ((single\u2080 I).obj Y)).HasMap p\nY' : ?m.129019\nj : J\n\u22a2 \u03c6 j \u226b\n      e.inv.app (X' j) \u226b\n        (F.obj (X' j)).map (singleObjApplyIso 0 Y).inv \u226b \u03b9MapBifunctorMapObj F p X' ((single\u2080 I).obj Y) j 0 j \u22ef =\n    e.inv.app (X j) \u226b\n      (F.obj (X j)).map (singleObjApplyIso 0 Y).inv \u226b\n        (F.map (\u03c6 j)).app ((single\u2080 I).obj Y 0) \u226b\n          (F.obj (X' j)).map (\ud835\udfd9 ((single\u2080 I).obj Y) 0) \u226b \u03b9MapBifunctorMapObj F p X' ((single\u2080 I).obj Y) j 0 j \u22ef"}, {"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "case h\nC : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst\u271d\u2077 : Category.{u_6, u_1} C\ninst\u271d\u2076 : Category.{u_5, u_2} D\ninst\u271d\u2075 : Zero I\ninst\u271d\u2074 : DecidableEq I\ninst\u271d\u00b3 : HasInitial C\nF : D \u2964 C \u2964 D\nY : C\ne : F.flip.obj Y \u2245 \ud835\udfed D\ninst\u271d\u00b2 : (X : D) \u2192 PreservesColimit (Functor.empty C) (F.obj X)\np : J \u00d7 I \u2192 J\nhp : \u2200 (j : J), p (j, 0) = j\nX X' : GradedObject J D\n\u03c6 : X \u27f6 X'\ninst\u271d\u00b9 : (((mapBifunctor F J I).obj X).obj ((single\u2080 I).obj Y)).HasMap p\ninst\u271d : (((mapBifunctor F J I).obj X').obj ((single\u2080 I).obj Y)).HasMap p\nY' : ?m.129019\nj : J\n\u22a2 \u03c6 j \u226b\n      e.inv.app (X' j) \u226b\n        (F.obj (X' j)).map (singleObjApplyIso 0 Y).inv \u226b \u03b9MapBifunctorMapObj F p X' ((single\u2080 I).obj Y) j 0 j \u22ef =\n    e.inv.app (X j) \u226b\n      (F.obj (X j)).map (singleObjApplyIso 0 Y).inv \u226b\n        (F.map (\u03c6 j)).app ((single\u2080 I).obj Y 0) \u226b\n          (F.obj (X' j)).map (\ud835\udfd9 ((single\u2080 I).obj Y) 0) \u226b \u03b9MapBifunctorMapObj F p X' ((single\u2080 I).obj Y) j 0 j \u22ef", "state_after": "case h\nC : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst\u271d\u2077 : Category.{u_6, u_1} C\ninst\u271d\u2076 : Category.{u_5, u_2} D\ninst\u271d\u2075 : Zero I\ninst\u271d\u2074 : DecidableEq I\ninst\u271d\u00b3 : HasInitial C\nF : D \u2964 C \u2964 D\nY : C\ne : F.flip.obj Y \u2245 \ud835\udfed D\ninst\u271d\u00b2 : (X : D) \u2192 PreservesColimit (Functor.empty C) (F.obj X)\np : J \u00d7 I \u2192 J\nhp : \u2200 (j : J), p (j, 0) = j\nX X' : GradedObject J D\n\u03c6 : X \u27f6 X'\ninst\u271d\u00b9 : (((mapBifunctor F J I).obj X).obj ((single\u2080 I).obj Y)).HasMap p\ninst\u271d : (((mapBifunctor F J I).obj X').obj ((single\u2080 I).obj Y)).HasMap p\nY' : ?m.129019\nj : J\n\u22a2 \u03c6 j \u226b\n      e.inv.app (X' j) \u226b\n        (F.obj (X' j)).map (singleObjApplyIso 0 Y).inv \u226b \u03b9MapBifunctorMapObj F p X' ((single\u2080 I).obj Y) j 0 j \u22ef =\n    e.inv.app (X j) \u226b\n      (F.obj (X j)).map (singleObjApplyIso 0 Y).inv \u226b\n        (F.map (\u03c6 j)).app ((single\u2080 I).obj Y 0) \u226b\n          (F.obj (X' j)).map (\ud835\udfd9 ((single\u2080 I).obj Y 0)) \u226b \u03b9MapBifunctorMapObj F p X' ((single\u2080 I).obj Y) j 0 j \u22ef"}, {"tactic": "rw [Functor.map_id, id_comp, NatTrans.naturality_assoc]", "annotated_tactic": ["rw [<a>Functor.map_id</a>, <a>id_comp</a>, <a>NatTrans.naturality_assoc</a>]", [{"full_name": "CategoryTheory.Functor.map_id", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [41, 3], "def_end_pos": [41, 9]}, {"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [177, 3], "def_end_pos": [177, 10]}, {"full_name": "CategoryTheory.NatTrans.naturality_assoc", "def_path": "Mathlib/CategoryTheory/NatTrans.lean", "def_pos": [60, 12], "def_end_pos": [60, 34]}]], "state_before": "case h\nC : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst\u271d\u2077 : Category.{u_6, u_1} C\ninst\u271d\u2076 : Category.{u_5, u_2} D\ninst\u271d\u2075 : Zero I\ninst\u271d\u2074 : DecidableEq I\ninst\u271d\u00b3 : HasInitial C\nF : D \u2964 C \u2964 D\nY : C\ne : F.flip.obj Y \u2245 \ud835\udfed D\ninst\u271d\u00b2 : (X : D) \u2192 PreservesColimit (Functor.empty C) (F.obj X)\np : J \u00d7 I \u2192 J\nhp : \u2200 (j : J), p (j, 0) = j\nX X' : GradedObject J D\n\u03c6 : X \u27f6 X'\ninst\u271d\u00b9 : (((mapBifunctor F J I).obj X).obj ((single\u2080 I).obj Y)).HasMap p\ninst\u271d : (((mapBifunctor F J I).obj X').obj ((single\u2080 I).obj Y)).HasMap p\nY' : ?m.129019\nj : J\n\u22a2 \u03c6 j \u226b\n      e.inv.app (X' j) \u226b\n        (F.obj (X' j)).map (singleObjApplyIso 0 Y).inv \u226b \u03b9MapBifunctorMapObj F p X' ((single\u2080 I).obj Y) j 0 j \u22ef =\n    e.inv.app (X j) \u226b\n      (F.obj (X j)).map (singleObjApplyIso 0 Y).inv \u226b\n        (F.map (\u03c6 j)).app ((single\u2080 I).obj Y 0) \u226b\n          (F.obj (X' j)).map (\ud835\udfd9 ((single\u2080 I).obj Y 0)) \u226b \u03b9MapBifunctorMapObj F p X' ((single\u2080 I).obj Y) j 0 j \u22ef", "state_after": "case h\nC : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst\u271d\u2077 : Category.{u_6, u_1} C\ninst\u271d\u2076 : Category.{u_5, u_2} D\ninst\u271d\u2075 : Zero I\ninst\u271d\u2074 : DecidableEq I\ninst\u271d\u00b3 : HasInitial C\nF : D \u2964 C \u2964 D\nY : C\ne : F.flip.obj Y \u2245 \ud835\udfed D\ninst\u271d\u00b2 : (X : D) \u2192 PreservesColimit (Functor.empty C) (F.obj X)\np : J \u00d7 I \u2192 J\nhp : \u2200 (j : J), p (j, 0) = j\nX X' : GradedObject J D\n\u03c6 : X \u27f6 X'\ninst\u271d\u00b9 : (((mapBifunctor F J I).obj X).obj ((single\u2080 I).obj Y)).HasMap p\ninst\u271d : (((mapBifunctor F J I).obj X').obj ((single\u2080 I).obj Y)).HasMap p\nY' : ?m.129019\nj : J\n\u22a2 \u03c6 j \u226b\n      e.inv.app (X' j) \u226b\n        (F.obj (X' j)).map (singleObjApplyIso 0 Y).inv \u226b \u03b9MapBifunctorMapObj F p X' ((single\u2080 I).obj Y) j 0 j \u22ef =\n    e.inv.app (X j) \u226b\n      (F.map (\u03c6 j)).app Y \u226b\n        (F.obj (X' j)).map (singleObjApplyIso 0 Y).inv \u226b \u03b9MapBifunctorMapObj F p X' ((single\u2080 I).obj Y) j 0 j \u22ef"}, {"tactic": "erw [\u2190 NatTrans.naturality_assoc]", "annotated_tactic": ["erw [\u2190 <a>NatTrans.naturality_assoc</a>]", [{"full_name": "CategoryTheory.NatTrans.naturality_assoc", "def_path": "Mathlib/CategoryTheory/NatTrans.lean", "def_pos": [60, 12], "def_end_pos": [60, 34]}]], "state_before": "case h\nC : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst\u271d\u2077 : Category.{u_6, u_1} C\ninst\u271d\u2076 : Category.{u_5, u_2} D\ninst\u271d\u2075 : Zero I\ninst\u271d\u2074 : DecidableEq I\ninst\u271d\u00b3 : HasInitial C\nF : D \u2964 C \u2964 D\nY : C\ne : F.flip.obj Y \u2245 \ud835\udfed D\ninst\u271d\u00b2 : (X : D) \u2192 PreservesColimit (Functor.empty C) (F.obj X)\np : J \u00d7 I \u2192 J\nhp : \u2200 (j : J), p (j, 0) = j\nX X' : GradedObject J D\n\u03c6 : X \u27f6 X'\ninst\u271d\u00b9 : (((mapBifunctor F J I).obj X).obj ((single\u2080 I).obj Y)).HasMap p\ninst\u271d : (((mapBifunctor F J I).obj X').obj ((single\u2080 I).obj Y)).HasMap p\nY' : ?m.129019\nj : J\n\u22a2 \u03c6 j \u226b\n      e.inv.app (X' j) \u226b\n        (F.obj (X' j)).map (singleObjApplyIso 0 Y).inv \u226b \u03b9MapBifunctorMapObj F p X' ((single\u2080 I).obj Y) j 0 j \u22ef =\n    e.inv.app (X j) \u226b\n      (F.map (\u03c6 j)).app Y \u226b\n        (F.obj (X' j)).map (singleObjApplyIso 0 Y).inv \u226b \u03b9MapBifunctorMapObj F p X' ((single\u2080 I).obj Y) j 0 j \u22ef", "state_after": "case h\nC : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst\u271d\u2077 : Category.{u_6, u_1} C\ninst\u271d\u2076 : Category.{u_5, u_2} D\ninst\u271d\u2075 : Zero I\ninst\u271d\u2074 : DecidableEq I\ninst\u271d\u00b3 : HasInitial C\nF : D \u2964 C \u2964 D\nY : C\ne : F.flip.obj Y \u2245 \ud835\udfed D\ninst\u271d\u00b2 : (X : D) \u2192 PreservesColimit (Functor.empty C) (F.obj X)\np : J \u00d7 I \u2192 J\nhp : \u2200 (j : J), p (j, 0) = j\nX X' : GradedObject J D\n\u03c6 : X \u27f6 X'\ninst\u271d\u00b9 : (((mapBifunctor F J I).obj X).obj ((single\u2080 I).obj Y)).HasMap p\ninst\u271d : (((mapBifunctor F J I).obj X').obj ((single\u2080 I).obj Y)).HasMap p\nY' : ?m.129019\nj : J\n\u22a2 \u03c6 j \u226b\n      e.inv.app (X' j) \u226b\n        (F.obj (X' j)).map (singleObjApplyIso 0 Y).inv \u226b \u03b9MapBifunctorMapObj F p X' ((single\u2080 I).obj Y) j 0 j \u22ef =\n    (\ud835\udfed D).map (\u03c6 j) \u226b\n      e.inv.app (X' j) \u226b\n        (F.obj (X' j)).map (singleObjApplyIso 0 Y).inv \u226b \u03b9MapBifunctorMapObj F p X' ((single\u2080 I).obj Y) j 0 j \u22ef"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case h\nC : Type u_1\nD : Type u_2\nI : Type u_3\nJ : Type u_4\ninst\u271d\u2077 : Category.{u_6, u_1} C\ninst\u271d\u2076 : Category.{u_5, u_2} D\ninst\u271d\u2075 : Zero I\ninst\u271d\u2074 : DecidableEq I\ninst\u271d\u00b3 : HasInitial C\nF : D \u2964 C \u2964 D\nY : C\ne : F.flip.obj Y \u2245 \ud835\udfed D\ninst\u271d\u00b2 : (X : D) \u2192 PreservesColimit (Functor.empty C) (F.obj X)\np : J \u00d7 I \u2192 J\nhp : \u2200 (j : J), p (j, 0) = j\nX X' : GradedObject J D\n\u03c6 : X \u27f6 X'\ninst\u271d\u00b9 : (((mapBifunctor F J I).obj X).obj ((single\u2080 I).obj Y)).HasMap p\ninst\u271d : (((mapBifunctor F J I).obj X').obj ((single\u2080 I).obj Y)).HasMap p\nY' : ?m.129019\nj : J\n\u22a2 \u03c6 j \u226b\n      e.inv.app (X' j) \u226b\n        (F.obj (X' j)).map (singleObjApplyIso 0 Y).inv \u226b \u03b9MapBifunctorMapObj F p X' ((single\u2080 I).obj Y) j 0 j \u22ef =\n    (\ud835\udfed D).map (\u03c6 j) \u226b\n      e.inv.app (X' j) \u226b\n        (F.obj (X' j)).map (singleObjApplyIso 0 Y).inv \u226b \u03b9MapBifunctorMapObj F p X' ((single\u2080 I).obj Y) j 0 j \u22ef", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Function/Iterate.lean", "full_name": "Function.Commute.iterate_self", "start": [166, 1], "end": [167, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": 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(algebraMap T S) p \u2260 0 \u2227 (aeval x) p = 0\n\u22a2 map (algebraMap T S') p \u2260 0 \u2227 (aeval (f x)) p = 0"}, {"tactic": "rw [aeval_algHom, AlgHom.comp_apply, hx.2, _root_.map_zero]", "annotated_tactic": ["rw [<a>aeval_algHom</a>, <a>AlgHom.comp_apply</a>, hx.2, <a>_root_.map_zero</a>]", [{"full_name": "Polynomial.aeval_algHom", "def_path": "Mathlib/Algebra/Polynomial/AlgebraMap.lean", "def_pos": [276, 9], "def_end_pos": [276, 21]}, {"full_name": "AlgHom.comp_apply", "def_path": "Mathlib/Algebra/Algebra/Hom.lean", "def_pos": [324, 9], "def_end_pos": [324, 19]}, {"full_name": "map_zero", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [203, 3], "def_end_pos": [203, 14]}]], "state_before": "R : Type u\nS\u271d : Type v\nT : Type w\na b : R\nn : \u2115\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : IsDomain R\np\u271d q : R[X]\ninst\u271d\u2076 : CommRing T\np : T[X]\nS : Type u_1\nS' : Type u_2\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : IsDomain 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"https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "full_name": "CategoryTheory.Sieve.functor_galoisConnection", "start": [716, 1], "end": [727, 19], "traced_tactics": [{"tactic": "intro R S", "annotated_tactic": ["intro R S", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nS R : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX : C\n\u22a2 GaloisConnection (functorPushforward F) (functorPullback F)", "state_after": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nS\u271d R\u271d : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX : C\nR : Sieve X\nS : Sieve (F.obj X)\n\u22a2 functorPushforward F R \u2264 S \u2194 R \u2264 functorPullback F S"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nS\u271d R\u271d : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX : C\nR : Sieve X\nS : Sieve (F.obj X)\n\u22a2 functorPushforward F R \u2264 S \u2194 R \u2264 functorPullback F S", "state_after": "case mp\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nS\u271d R\u271d : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX : C\nR : Sieve X\nS : Sieve (F.obj X)\n\u22a2 functorPushforward F R \u2264 S \u2192 R \u2264 functorPullback F S\n\ncase mpr\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nS\u271d R\u271d : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX : C\nR : Sieve X\nS : Sieve (F.obj X)\n\u22a2 R \u2264 functorPullback F S \u2192 functorPushforward F R \u2264 S"}, {"tactic": "intro hle X f hf", "annotated_tactic": ["intro hle X f hf", []], "state_before": "case mp\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nS\u271d R\u271d : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX : C\nR : Sieve X\nS : Sieve (F.obj X)\n\u22a2 functorPushforward F R \u2264 S \u2192 R \u2264 functorPullback F S", "state_after": "case mp\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d\u00b9 Y Z : C\nf\u271d : Y \u27f6 X\u271d\u00b9\nS\u271d R\u271d : Sieve X\u271d\u00b9\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX\u271d : C\nR : Sieve X\u271d\nS : Sieve (F.obj X\u271d)\nhle : functorPushforward F R \u2264 S\nX : C\nf : X \u27f6 X\u271d\nhf : R.arrows f\n\u22a2 (functorPullback F S).arrows f"}, {"tactic": "apply hle", "annotated_tactic": ["apply hle", []], "state_before": "case mp\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d\u00b9 Y Z : C\nf\u271d : Y \u27f6 X\u271d\u00b9\nS\u271d R\u271d : Sieve X\u271d\u00b9\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX\u271d : C\nR : Sieve X\u271d\nS : Sieve (F.obj X\u271d)\nhle : functorPushforward F R \u2264 S\nX : C\nf : X \u27f6 X\u271d\nhf : R.arrows f\n\u22a2 (functorPullback F S).arrows f", "state_after": "case mp.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d\u00b9 Y Z : C\nf\u271d : Y \u27f6 X\u271d\u00b9\nS\u271d R\u271d : Sieve X\u271d\u00b9\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX\u271d : C\nR : Sieve X\u271d\nS : Sieve (F.obj X\u271d)\nhle : functorPushforward F R \u2264 S\nX : C\nf : X \u27f6 X\u271d\nhf : R.arrows f\n\u22a2 (functorPushforward F R).arrows (F.map f)"}, {"tactic": "refine \u27e8X, f, \ud835\udfd9 _, hf, ?_\u27e9", "annotated_tactic": ["refine \u27e8X, f, \ud835\udfd9 _, hf, ?_\u27e9", []], "state_before": "case mp.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d\u00b9 Y Z : C\nf\u271d : Y \u27f6 X\u271d\u00b9\nS\u271d R\u271d : Sieve X\u271d\u00b9\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX\u271d : C\nR : Sieve X\u271d\nS : Sieve (F.obj X\u271d)\nhle : functorPushforward F R \u2264 S\nX : C\nf : X \u27f6 X\u271d\nhf : R.arrows f\n\u22a2 (functorPushforward F R).arrows (F.map f)", "state_after": "case mp.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d\u00b9 Y Z : C\nf\u271d : Y \u27f6 X\u271d\u00b9\nS\u271d R\u271d : Sieve X\u271d\u00b9\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX\u271d : C\nR : Sieve X\u271d\nS : Sieve (F.obj X\u271d)\nhle : functorPushforward F R \u2264 S\nX : C\nf : X \u27f6 X\u271d\nhf : R.arrows f\n\u22a2 F.map f = \ud835\udfd9 (F.obj X) \u226b F.map f"}, {"tactic": "rw [id_comp]", "annotated_tactic": ["rw [<a>id_comp</a>]", [{"full_name": "CategoryTheory.Category.id_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [177, 3], "def_end_pos": [177, 10]}]], "state_before": "case mp.a\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d\u00b9 Y Z : C\nf\u271d : Y \u27f6 X\u271d\u00b9\nS\u271d R\u271d : Sieve X\u271d\u00b9\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX\u271d : C\nR : Sieve X\u271d\nS : Sieve (F.obj X\u271d)\nhle : functorPushforward F R \u2264 S\nX : C\nf : X \u27f6 X\u271d\nhf : R.arrows f\n\u22a2 F.map f = \ud835\udfd9 (F.obj X) \u226b F.map f", "state_after": "no goals"}, {"tactic": "rintro hle Y f \u27e8X, g, h, hg, rfl\u27e9", "annotated_tactic": ["rintro hle Y f \u27e8X, g, h, hg, rfl\u27e9", []], "state_before": "case mpr\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d Y Z : C\nf : Y \u27f6 X\u271d\nS\u271d R\u271d : Sieve X\u271d\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX : C\nR : Sieve X\nS : Sieve (F.obj X)\n\u22a2 R \u2264 functorPullback F S \u2192 functorPushforward F R \u2264 S", "state_after": "case mpr.intro.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d\u00b9 Y\u271d Z : C\nf : Y\u271d \u27f6 X\u271d\u00b9\nS\u271d R\u271d : Sieve X\u271d\u00b9\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX\u271d : C\nR : Sieve X\u271d\nS : Sieve (F.obj X\u271d)\nhle : R \u2264 functorPullback F S\nY : D\nX : C\ng : X \u27f6 X\u271d\nh : Y \u27f6 F.obj X\nhg : R.arrows g\n\u22a2 S.arrows (h \u226b F.map g)"}, {"tactic": "apply Sieve.downward_closed S", "annotated_tactic": ["apply <a>Sieve.downward_closed</a> S", [{"full_name": "CategoryTheory.Sieve.downward_closed", "def_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "def_pos": [274, 3], "def_end_pos": [274, 18]}]], "state_before": "case mpr.intro.intro.intro.intro\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d\u00b9 Y\u271d Z : C\nf : Y\u271d \u27f6 X\u271d\u00b9\nS\u271d R\u271d : Sieve X\u271d\u00b9\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX\u271d : C\nR : Sieve X\u271d\nS : Sieve (F.obj X\u271d)\nhle : R \u2264 functorPullback F S\nY : D\nX : C\ng : X \u27f6 X\u271d\nh : Y \u27f6 F.obj X\nhg : R.arrows g\n\u22a2 S.arrows (h \u226b F.map g)", "state_after": "case mpr.intro.intro.intro.intro.x\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d\u00b9 Y\u271d Z : C\nf : Y\u271d \u27f6 X\u271d\u00b9\nS\u271d R\u271d : Sieve X\u271d\u00b9\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX\u271d : C\nR : Sieve X\u271d\nS : Sieve (F.obj X\u271d)\nhle : R \u2264 functorPullback F S\nY : D\nX : C\ng : X \u27f6 X\u271d\nh : Y \u27f6 F.obj X\nhg : R.arrows g\n\u22a2 S.arrows (F.map g)"}, {"tactic": "exact hle g hg", "annotated_tactic": ["exact hle g hg", []], "state_before": "case mpr.intro.intro.intro.intro.x\nC : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\nX\u271d\u00b9 Y\u271d Z : C\nf : Y\u271d \u27f6 X\u271d\u00b9\nS\u271d R\u271d : Sieve X\u271d\u00b9\nE : Type u\u2083\ninst\u271d : Category.{v\u2083, u\u2083} E\nG : D \u2964 E\nX\u271d : C\nR : Sieve X\u271d\nS : Sieve (F.obj X\u271d)\nhle : R \u2264 functorPullback F S\nY : D\nX : C\ng : X \u27f6 X\u271d\nh : Y \u27f6 F.obj X\nhg : R.arrows g\n\u22a2 S.arrows (F.map g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Category/Cat.lean", "full_name": "CategoryTheory.Cat.rightUnitor_inv_app", "start": [129, 1], "end": [130, 6], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "B C : Cat\nF : B \u27f6 C\nX : \u2191B\n\u22a2 F.obj X = (F \u226b \ud835\udfd9 C).obj X", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Extreme.lean", "full_name": "Convex.mem_extremePoints_iff_mem_diff_convexHull_diff", "start": [281, 1], "end": [284, 14], "traced_tactics": [{"tactic": "rw [hA.mem_extremePoints_iff_convex_diff, hA.convex_remove_iff_not_mem_convexHull_remove,\n  mem_diff]", "annotated_tactic": ["rw [hA.mem_extremePoints_iff_convex_diff, hA.convex_remove_iff_not_mem_convexHull_remove,\n    <a>mem_diff</a>]", [{"full_name": "Set.mem_diff", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [93, 17], "def_end_pos": [93, 25]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b9 : Type u_4\n\u03c0 : \u03b9 \u2192 Type u_5\ninst\u271d\u2074 : LinearOrderedRing \ud835\udd5c\ninst\u271d\u00b3 : AddCommGroup E\ninst\u271d\u00b2 : Module \ud835\udd5c E\ninst\u271d\u00b9 : DenselyOrdered \ud835\udd5c\ninst\u271d : NoZeroSMulDivisors \ud835\udd5c E\nA B : Set E\nx : E\nhA : Convex \ud835\udd5c A\n\u22a2 x \u2208 extremePoints \ud835\udd5c A \u2194 x \u2208 A \\ (convexHull \ud835\udd5c) (A \\ {x})", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/SmoothNumbers.lean", "full_name": "Nat.eq_prod_primes_mul_sq_of_mem_smoothNumbers", "start": [443, 1], "end": [456, 50], "traced_tactics": [{"tactic": "obtain \u27e8l, m, H\u2081, H\u2082\u27e9 := sq_mul_squarefree n", "annotated_tactic": ["obtain \u27e8l, m, H\u2081, H\u2082\u27e9 := <a>sq_mul_squarefree</a> n", [{"full_name": "Nat.sq_mul_squarefree", "def_path": "Mathlib/Data/Nat/Squarefree.lean", "def_pos": [363, 9], "def_end_pos": [363, 26]}]], "state_before": "n k : \u2115\nh : n \u2208 k.smoothNumbers\n\u22a2 \u2203 s \u2208 k.primesBelow.powerset, \u2203 m, n = m ^ 2 * s.prod id", "state_after": "case intro.intro.intro\nn k : \u2115\nh : n \u2208 k.smoothNumbers\nl m : \u2115\nH\u2081 : m ^ 2 * l = n\nH\u2082 : Squarefree l\n\u22a2 \u2203 s \u2208 k.primesBelow.powerset, \u2203 m, n = m ^ 2 * s.prod id"}, {"tactic": "have hl : l \u2208 smoothNumbers k := mem_smoothNumbers_of_dvd h (Dvd.intro_left (m ^ 2) H\u2081)", "annotated_tactic": ["have hl : l \u2208 <a>smoothNumbers</a> k := <a>mem_smoothNumbers_of_dvd</a> h (<a>Dvd.intro_left</a> (m ^ 2) H\u2081)", [{"full_name": "Nat.smoothNumbers", "def_path": "Mathlib/NumberTheory/SmoothNumbers.lean", "def_pos": [269, 5], "def_end_pos": [269, 18]}, {"full_name": "Nat.mem_smoothNumbers_of_dvd", "def_path": "Mathlib/NumberTheory/SmoothNumbers.lean", "def_pos": [292, 7], "def_end_pos": [292, 31]}, {"full_name": "Dvd.intro_left", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [183, 9], "def_end_pos": [183, 23]}]], "state_before": "case intro.intro.intro\nn k : \u2115\nh : n \u2208 k.smoothNumbers\nl m : \u2115\nH\u2081 : m ^ 2 * l = n\nH\u2082 : Squarefree l\n\u22a2 \u2203 s \u2208 k.primesBelow.powerset, \u2203 m, n = m ^ 2 * s.prod id", "state_after": "case intro.intro.intro\nn k : \u2115\nh : n \u2208 k.smoothNumbers\nl m : \u2115\nH\u2081 : m ^ 2 * l = n\nH\u2082 : Squarefree l\nhl : l \u2208 k.smoothNumbers\n\u22a2 \u2203 s \u2208 k.primesBelow.powerset, \u2203 m, n = m ^ 2 * s.prod id"}, {"tactic": "refine \u27e8l.factors.toFinset, ?_,  m, ?_\u27e9", "annotated_tactic": ["refine \u27e8l.factors.toFinset, ?_,  m, ?_\u27e9", []], "state_before": "case intro.intro.intro\nn k : \u2115\nh : n \u2208 k.smoothNumbers\nl m : \u2115\nH\u2081 : m ^ 2 * l = n\nH\u2082 : Squarefree l\nhl : l \u2208 k.smoothNumbers\n\u22a2 \u2203 s \u2208 k.primesBelow.powerset, \u2203 m, n = m ^ 2 * s.prod id", "state_after": "case intro.intro.intro.refine_1\nn k : \u2115\nh : n \u2208 k.smoothNumbers\nl m : \u2115\nH\u2081 : m ^ 2 * l = n\nH\u2082 : Squarefree l\nhl : l \u2208 k.smoothNumbers\n\u22a2 l.factors.toFinset \u2208 k.primesBelow.powerset\n\ncase intro.intro.intro.refine_2\nn k : \u2115\nh : n \u2208 k.smoothNumbers\nl m : \u2115\nH\u2081 : m ^ 2 * l = n\nH\u2082 : Squarefree l\nhl : l \u2208 k.smoothNumbers\n\u22a2 n = m ^ 2 * l.factors.toFinset.prod id"}, {"tactic": "rw [\u2190 H\u2081]", "annotated_tactic": ["rw [\u2190 H\u2081]", []], "state_before": "case intro.intro.intro.refine_2\nn k : \u2115\nh : n \u2208 k.smoothNumbers\nl m : \u2115\nH\u2081 : m ^ 2 * l = n\nH\u2082 : Squarefree l\nhl : l \u2208 k.smoothNumbers\n\u22a2 n = m ^ 2 * l.factors.toFinset.prod id", "state_after": "case intro.intro.intro.refine_2\nn k : \u2115\nh : n \u2208 k.smoothNumbers\nl m : \u2115\nH\u2081 : m ^ 2 * l = n\nH\u2082 : Squarefree l\nhl : l \u2208 k.smoothNumbers\n\u22a2 m ^ 2 * l = m ^ 2 * l.factors.toFinset.prod id"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case intro.intro.intro.refine_2\nn k : \u2115\nh : n \u2208 k.smoothNumbers\nl m : \u2115\nH\u2081 : m ^ 2 * l = n\nH\u2082 : Squarefree l\nhl : l \u2208 k.smoothNumbers\n\u22a2 m ^ 2 * l = m ^ 2 * l.factors.toFinset.prod id", "state_after": "case intro.intro.intro.refine_2.e_a\nn k : \u2115\nh : n \u2208 k.smoothNumbers\nl m : \u2115\nH\u2081 : m ^ 2 * l = n\nH\u2082 : Squarefree l\nhl : l \u2208 k.smoothNumbers\n\u22a2 l = l.factors.toFinset.prod id"}, {"tactic": "simp only [toFinset_factors]", "annotated_tactic": ["simp only [<a>toFinset_factors</a>]", [{"full_name": "Nat.toFinset_factors", "def_path": "Mathlib/Data/Nat/PrimeFin.lean", "def_pos": [35, 15], "def_end_pos": [35, 31]}]], "state_before": "case intro.intro.intro.refine_2.e_a\nn k : \u2115\nh : n \u2208 k.smoothNumbers\nl m : \u2115\nH\u2081 : m ^ 2 * l = n\nH\u2082 : Squarefree l\nhl : l \u2208 k.smoothNumbers\n\u22a2 l = l.factors.toFinset.prod id", "state_after": "case intro.intro.intro.refine_2.e_a\nn k : \u2115\nh : n \u2208 k.smoothNumbers\nl m : \u2115\nH\u2081 : m ^ 2 * l = n\nH\u2082 : Squarefree l\nhl : l \u2208 k.smoothNumbers\n\u22a2 l = l.primeFactors.prod id"}, {"tactic": "exact (prod_primeFactors_of_squarefree H\u2082).symm", "annotated_tactic": ["exact (<a>prod_primeFactors_of_squarefree</a> H\u2082).<a>symm</a>", [{"full_name": "Nat.prod_primeFactors_of_squarefree", "def_path": "Mathlib/Data/Nat/Squarefree.lean", "def_pos": [394, 7], "def_end_pos": [394, 38]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case intro.intro.intro.refine_2.e_a\nn k : \u2115\nh : n \u2208 k.smoothNumbers\nl m : \u2115\nH\u2081 : m ^ 2 * l = n\nH\u2082 : Squarefree l\nhl : l \u2208 k.smoothNumbers\n\u22a2 l = l.primeFactors.prod id", "state_after": "no goals"}, {"tactic": "simp only [toFinset_factors, Finset.mem_powerset]", "annotated_tactic": ["simp only [<a>toFinset_factors</a>, <a>Finset.mem_powerset</a>]", [{"full_name": "Nat.toFinset_factors", "def_path": "Mathlib/Data/Nat/PrimeFin.lean", "def_pos": [35, 15], "def_end_pos": [35, 31]}, {"full_name": "Finset.mem_powerset", "def_path": "Mathlib/Data/Finset/Powerset.lean", "def_pos": [34, 9], "def_end_pos": [34, 21]}]], "state_before": "case intro.intro.intro.refine_1\nn k : \u2115\nh : n \u2208 k.smoothNumbers\nl m : \u2115\nH\u2081 : m ^ 2 * l = n\nH\u2082 : Squarefree l\nhl : l \u2208 k.smoothNumbers\n\u22a2 l.factors.toFinset \u2208 k.primesBelow.powerset", "state_after": "case intro.intro.intro.refine_1\nn k : \u2115\nh : n \u2208 k.smoothNumbers\nl m : \u2115\nH\u2081 : m ^ 2 * l = n\nH\u2082 : Squarefree l\nhl : l \u2208 k.smoothNumbers\n\u22a2 l.primeFactors \u2286 k.primesBelow"}, {"tactic": "refine fun p hp \u21a6 mem_primesBelow.mpr \u27e8?_, (mem_primeFactors.mp hp).1\u27e9", "annotated_tactic": ["refine fun p hp \u21a6 mem_primesBelow.mpr \u27e8?_, (mem_primeFactors.mp hp).1\u27e9", []], "state_before": "case intro.intro.intro.refine_1\nn k : \u2115\nh : n \u2208 k.smoothNumbers\nl m : \u2115\nH\u2081 : m ^ 2 * l = n\nH\u2082 : Squarefree l\nhl : l \u2208 k.smoothNumbers\n\u22a2 l.primeFactors \u2286 k.primesBelow", "state_after": "case intro.intro.intro.refine_1\nn k : \u2115\nh : n \u2208 k.smoothNumbers\nl m : \u2115\nH\u2081 : m ^ 2 * l = n\nH\u2082 : Squarefree l\nhl : l \u2208 k.smoothNumbers\np : \u2115\nhp : p \u2208 l.primeFactors\n\u22a2 p < k"}, {"tactic": "rw [mem_primeFactors] at hp", "annotated_tactic": ["rw [<a>mem_primeFactors</a>] at hp", [{"full_name": "Nat.mem_primeFactors", "def_path": "Mathlib/Data/Nat/PrimeFin.lean", "def_pos": [37, 15], "def_end_pos": [37, 31]}]], "state_before": "case intro.intro.intro.refine_1\nn k : \u2115\nh : n \u2208 k.smoothNumbers\nl m : \u2115\nH\u2081 : m ^ 2 * l = n\nH\u2082 : Squarefree l\nhl : l \u2208 k.smoothNumbers\np : \u2115\nhp : p \u2208 l.primeFactors\n\u22a2 p < k", "state_after": "case intro.intro.intro.refine_1\nn k : \u2115\nh : n \u2208 k.smoothNumbers\nl m : \u2115\nH\u2081 : m ^ 2 * l = n\nH\u2082 : Squarefree l\nhl : l \u2208 k.smoothNumbers\np : \u2115\nhp : Prime p \u2227 p \u2223 l \u2227 l \u2260 0\n\u22a2 p < k"}, {"tactic": "exact mem_smoothNumbers'.mp hl p hp.1 hp.2.1", "annotated_tactic": ["exact mem_smoothNumbers'.mp hl p hp.1 hp.2.1", []], "state_before": "case intro.intro.intro.refine_1\nn k : \u2115\nh : n \u2208 k.smoothNumbers\nl m : \u2115\nH\u2081 : m ^ 2 * l = n\nH\u2082 : Squarefree l\nhl : l \u2208 k.smoothNumbers\np : \u2115\nhp : Prime p \u2227 p \u2223 l \u2227 l \u2260 0\n\u22a2 p < k", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/Cardinality.lean", "full_name": "Fintype.isPrimePow_card_of_field", "start": [40, 1], "end": [49, 42], "traced_tactics": [{"tactic": "cases' CharP.exists \u03b1 with p _", "annotated_tactic": ["cases' <a>CharP.exists</a> \u03b1 with p _", [{"full_name": "CharP.exists", "def_path": "Mathlib/Algebra/CharP/Defs.lean", "def_pos": [131, 7], "def_end_pos": [131, 15]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Field \u03b1\n\u22a2 IsPrimePow \u2016\u03b1\u2016", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Field \u03b1\np : \u2115\nh\u271d : CharP \u03b1 p\n\u22a2 IsPrimePow \u2016\u03b1\u2016"}, {"tactic": "haveI hp := Fact.mk (CharP.char_is_prime \u03b1 p)", "annotated_tactic": ["haveI hp := <a>Fact.mk</a> (<a>CharP.char_is_prime</a> \u03b1 p)", [{"full_name": "Fact.mk", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [114, 7], "def_end_pos": [114, 11]}, {"full_name": "CharP.char_is_prime", "def_path": "Mathlib/Algebra/CharP/Basic.lean", "def_pos": [179, 9], "def_end_pos": [179, 22]}]], "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Field \u03b1\np : \u2115\nh\u271d : CharP \u03b1 p\n\u22a2 IsPrimePow \u2016\u03b1\u2016", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Field \u03b1\np : \u2115\nh\u271d : CharP \u03b1 p\nhp : Fact (Nat.Prime p)\n\u22a2 IsPrimePow \u2016\u03b1\u2016"}, {"tactic": "letI : Algebra (ZMod p) \u03b1 := ZMod.algebra _ _", "annotated_tactic": ["letI : <a>Algebra</a> (<a>ZMod</a> p) \u03b1 := <a>ZMod.algebra</a> _ _", [{"full_name": "Algebra", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [101, 7], "def_end_pos": [101, 14]}, {"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [95, 5], "def_end_pos": [95, 9]}, {"full_name": "ZMod.algebra", "def_path": "Mathlib/Data/ZMod/Algebra.lean", "def_pos": [45, 8], "def_end_pos": [45, 15]}]], "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Field \u03b1\np : \u2115\nh\u271d : CharP \u03b1 p\nhp : Fact (Nat.Prime p)\n\u22a2 IsPrimePow \u2016\u03b1\u2016", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Field \u03b1\np : \u2115\nh\u271d : CharP \u03b1 p\nhp : Fact (Nat.Prime p)\nthis : Algebra (ZMod p) \u03b1 := ZMod.algebra \u03b1 p\n\u22a2 IsPrimePow \u2016\u03b1\u2016"}, {"tactic": "let b := IsNoetherian.finsetBasis (ZMod p) \u03b1", "annotated_tactic": ["let b := <a>IsNoetherian.finsetBasis</a> (<a>ZMod</a> p) \u03b1", [{"full_name": "IsNoetherian.finsetBasis", "def_path": "Mathlib/FieldTheory/Finiteness.lean", "def_pos": [90, 19], "def_end_pos": [90, 30]}, {"full_name": "ZMod", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [95, 5], "def_end_pos": [95, 9]}]], "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Field \u03b1\np : \u2115\nh\u271d : CharP \u03b1 p\nhp : Fact (Nat.Prime p)\nthis : Algebra (ZMod p) \u03b1 := ZMod.algebra \u03b1 p\n\u22a2 IsPrimePow \u2016\u03b1\u2016", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Field \u03b1\np : \u2115\nh\u271d : CharP \u03b1 p\nhp : Fact (Nat.Prime p)\nthis : Algebra (ZMod p) \u03b1 := ZMod.algebra \u03b1 p\nb : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex (ZMod p) \u03b1 } (ZMod p) \u03b1 := IsNoetherian.finsetBasis (ZMod p) \u03b1\n\u22a2 IsPrimePow \u2016\u03b1\u2016"}, {"tactic": "rw [Module.card_fintype b, ZMod.card, isPrimePow_pow_iff]", "annotated_tactic": ["rw [<a>Module.card_fintype</a> b, <a>ZMod.card</a>, <a>isPrimePow_pow_iff</a>]", [{"full_name": "Module.card_fintype", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [897, 9], "def_end_pos": [897, 28]}, {"full_name": "ZMod.card", "def_path": "Mathlib/Data/ZMod/Defs.lean", "def_pos": [124, 9], "def_end_pos": [124, 13]}, {"full_name": "isPrimePow_pow_iff", "def_path": "Mathlib/Data/Nat/Factorization/PrimePow.lean", "def_pos": [111, 9], "def_end_pos": [111, 27]}]], "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Field \u03b1\np : \u2115\nh\u271d : CharP \u03b1 p\nhp : Fact (Nat.Prime p)\nthis : Algebra (ZMod p) \u03b1 := ZMod.algebra \u03b1 p\nb : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex (ZMod p) \u03b1 } (ZMod p) \u03b1 := IsNoetherian.finsetBasis (ZMod p) \u03b1\n\u22a2 IsPrimePow \u2016\u03b1\u2016", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Field \u03b1\np : \u2115\nh\u271d : CharP \u03b1 p\nhp : Fact (Nat.Prime p)\nthis : Algebra (ZMod p) \u03b1 := ZMod.algebra \u03b1 p\nb : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex (ZMod p) \u03b1 } (ZMod p) \u03b1 := IsNoetherian.finsetBasis (ZMod p) \u03b1\n\u22a2 IsPrimePow p\n\ncase intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Field \u03b1\np : \u2115\nh\u271d : CharP \u03b1 p\nhp : Fact (Nat.Prime p)\nthis : Algebra (ZMod p) \u03b1 := ZMod.algebra \u03b1 p\nb : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex (ZMod p) \u03b1 } (ZMod p) \u03b1 := IsNoetherian.finsetBasis (ZMod p) \u03b1\n\u22a2 \u2016{ x // x \u2208 IsNoetherian.finsetBasisIndex (ZMod p) \u03b1 }\u2016 \u2260 0"}, {"tactic": "rw [\u2190 FiniteDimensional.finrank_eq_card_basis b]", "annotated_tactic": ["rw [\u2190 <a>FiniteDimensional.finrank_eq_card_basis</a> b]", [{"full_name": "FiniteDimensional.finrank_eq_card_basis", "def_path": "Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean", "def_pos": [417, 9], "def_end_pos": [417, 30]}]], "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Field \u03b1\np : \u2115\nh\u271d : CharP \u03b1 p\nhp : Fact (Nat.Prime p)\nthis : Algebra (ZMod p) \u03b1 := ZMod.algebra \u03b1 p\nb : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex (ZMod p) \u03b1 } (ZMod p) \u03b1 := IsNoetherian.finsetBasis (ZMod p) \u03b1\n\u22a2 \u2016{ x // x \u2208 IsNoetherian.finsetBasisIndex (ZMod p) \u03b1 }\u2016 \u2260 0", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Field \u03b1\np : \u2115\nh\u271d : CharP \u03b1 p\nhp : Fact (Nat.Prime p)\nthis : Algebra (ZMod p) \u03b1 := ZMod.algebra \u03b1 p\nb : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex (ZMod p) \u03b1 } (ZMod p) \u03b1 := IsNoetherian.finsetBasis (ZMod p) \u03b1\n\u22a2 FiniteDimensional.finrank (ZMod p) \u03b1 \u2260 0"}, {"tactic": "exact FiniteDimensional.finrank_pos.ne'", "annotated_tactic": ["exact FiniteDimensional.finrank_pos.ne'", []], "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Field \u03b1\np : \u2115\nh\u271d : CharP \u03b1 p\nhp : Fact (Nat.Prime p)\nthis : Algebra (ZMod p) \u03b1 := ZMod.algebra \u03b1 p\nb : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex (ZMod p) \u03b1 } (ZMod p) \u03b1 := IsNoetherian.finsetBasis (ZMod p) \u03b1\n\u22a2 FiniteDimensional.finrank (ZMod p) \u03b1 \u2260 0", "state_after": "no goals"}, {"tactic": "exact hp.1.isPrimePow", "annotated_tactic": ["exact hp.1.<a>isPrimePow</a>", [{"full_name": "Nat.Prime.isPrimePow", "def_path": "Mathlib/Algebra/IsPrimePow.lean", "def_pos": [80, 9], "def_end_pos": [80, 29]}]], "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Field \u03b1\np : \u2115\nh\u271d : CharP \u03b1 p\nhp : Fact (Nat.Prime p)\nthis : Algebra (ZMod p) \u03b1 := ZMod.algebra \u03b1 p\nb : Basis { x // x \u2208 IsNoetherian.finsetBasisIndex (ZMod p) \u03b1 } (ZMod p) \u03b1 := IsNoetherian.finsetBasis (ZMod p) \u03b1\n\u22a2 IsPrimePow p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Prod.lean", "full_name": "HasFDerivAt.prodMap", "start": [365, 11], "end": [367, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Bases.lean", "full_name": "TopologicalSpace.secondCountableTopology_induced", "start": [828, 1], "end": [833, 35], "traced_tactics": [{"tactic": "rcases @SecondCountableTopology.is_open_generated_countable \u03b2 _ _ with \u27e8b, hb, eq\u27e9", "annotated_tactic": ["rcases @<a>SecondCountableTopology.is_open_generated_countable</a> \u03b2 _ _ with \u27e8b, hb, eq\u27e9", [{"full_name": "SecondCountableTopology.is_open_generated_countable", "def_path": "Mathlib/Topology/Bases.lean", "def_pos": [750, 3], "def_end_pos": [750, 30]}]], "state_before": "\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type u_1\nt : TopologicalSpace \u03b2\ninst\u271d : SecondCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 SecondCountableTopology \u03b1", "state_after": "case intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type u_1\nt : TopologicalSpace \u03b2\ninst\u271d : SecondCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nb : Set (Set \u03b2)\nhb : b.Countable\neq : t = generateFrom b\n\u22a2 SecondCountableTopology \u03b1"}, {"tactic": "letI := t.induced f", "annotated_tactic": ["letI := t.induced f", []], "state_before": "case intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type u_1\nt : TopologicalSpace \u03b2\ninst\u271d : SecondCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nb : Set (Set \u03b2)\nhb : b.Countable\neq : t = generateFrom b\n\u22a2 SecondCountableTopology \u03b1", "state_after": "case intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type u_1\nt : TopologicalSpace \u03b2\ninst\u271d : SecondCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nb : Set (Set \u03b2)\nhb : b.Countable\neq : t = generateFrom b\nthis : TopologicalSpace \u03b1 := induced f t\n\u22a2 SecondCountableTopology \u03b1"}, {"tactic": "refine { is_open_generated_countable := \u27e8preimage f '' b, hb.image _, ?_\u27e9 }", "annotated_tactic": ["refine { is_open_generated_countable := \u27e8<a>preimage</a> f '' b, hb.image _, ?_\u27e9 }", [{"full_name": "Set.preimage", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [113, 5], "def_end_pos": [113, 13]}]], "state_before": "case intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type u_1\nt : TopologicalSpace \u03b2\ninst\u271d : SecondCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nb : Set (Set \u03b2)\nhb : b.Countable\neq : t = generateFrom b\nthis : TopologicalSpace \u03b1 := induced f t\n\u22a2 SecondCountableTopology \u03b1", "state_after": "case intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type u_1\nt : TopologicalSpace \u03b2\ninst\u271d : SecondCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nb : Set (Set \u03b2)\nhb : b.Countable\neq : t = generateFrom b\nthis : TopologicalSpace \u03b1 := induced f t\n\u22a2 induced f t = generateFrom (preimage f '' b)"}, {"tactic": "rw [eq, induced_generateFrom_eq]", "annotated_tactic": ["rw [eq, <a>induced_generateFrom_eq</a>]", [{"full_name": "induced_generateFrom_eq", "def_path": "Mathlib/Topology/Order.lean", "def_pos": [568, 9], "def_end_pos": [568, 32]}]], "state_before": "case intro.intro\n\u03b1 : Type u\nt\u271d : TopologicalSpace \u03b1\n\u03b2 : Type u_1\nt : TopologicalSpace \u03b2\ninst\u271d : SecondCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nb : Set (Set \u03b2)\nhb : b.Countable\neq : t = generateFrom b\nthis : TopologicalSpace \u03b1 := induced f t\n\u22a2 induced f t = generateFrom (preimage f '' b)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ULift.lean", "full_name": "ULift.down_bijective", "start": [128, 1], "end": [129, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/SingleHomology.lean", "full_name": "HomologicalComplex.homology\u03b9_singleObjOpcyclesSelfIso_inv", "start": [103, 1], "end": [108, 82], "traced_tactics": [{"tactic": "rw [\u2190 cancel_epi (singleObjHomologySelfIso _ _ _).inv,\n  singleObjHomologySelfIso_inv_homology\u03b9_assoc, Iso.hom_inv_id, Iso.inv_hom_id]", "annotated_tactic": ["rw [\u2190 <a>cancel_epi</a> (<a>singleObjHomologySelfIso</a> _ _ _).<a>inv</a>,\n    <a>singleObjHomologySelfIso_inv_homology\u03b9_assoc</a>, <a>Iso.hom_inv_id</a>, <a>Iso.inv_hom_id</a>]", [{"full_name": "CategoryTheory.cancel_epi", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 19]}, {"full_name": "HomologicalComplex.singleObjHomologySelfIso", "def_path": "Mathlib/Algebra/Homology/SingleHomology.lean", "def_pos": [60, 19], "def_end_pos": [60, 43]}, {"full_name": "CategoryTheory.Iso.inv", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [55, 3], "def_end_pos": [55, 6]}, {"full_name": "HomologicalComplex.singleObjHomologySelfIso_inv_homology\u03b9_assoc", "def_path": "Mathlib/Algebra/Homology/SingleHomology.lean", "def_pos": [95, 3], "def_end_pos": [95, 25]}, {"full_name": "CategoryTheory.Iso.hom_inv_id", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [57, 3], "def_end_pos": [57, 13]}, {"full_name": "CategoryTheory.Iso.inv_hom_id", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [60, 3], "def_end_pos": [60, 13]}]], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\n\u03b9 : Type u_1\ninst\u271d : DecidableEq \u03b9\nc : ComplexShape \u03b9\nj : \u03b9\nA : C\n\u22a2 ((single C c j).obj A).homology\u03b9 j \u226b (singleObjOpcyclesSelfIso c j A).inv = (singleObjHomologySelfIso c j A).hom", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Localization/Adjunction.lean", "full_name": "CategoryTheory.Adjunction.Localization.\u03b7_app", "start": [65, 1], "end": [73, 84], "traced_tactics": [{"tactic": "letI : Lifting L\u2082 W\u2082 ((F \u22d9 G) \u22d9 L\u2082) (F' \u22d9 G') :=\n  Lifting.mk (CatCommSq.hComp F G L\u2082 L\u2081 L\u2082 F' G').iso'.symm", "annotated_tactic": ["letI : <a>Lifting</a> L\u2082 W\u2082 ((F \u22d9 G) \u22d9 L\u2082) (F' \u22d9 G') :=\n    <a>Lifting.mk</a> (<a>CatCommSq.hComp</a> F G L\u2082 L\u2081 L\u2082 F' G').iso'.symm", [{"full_name": "CategoryTheory.Localization.Lifting", "def_path": "Mathlib/CategoryTheory/Localization/Predicate.lean", "def_pos": [291, 7], "def_end_pos": [291, 14]}, {"full_name": "CategoryTheory.Localization.Lifting.mk", "def_path": "Mathlib/CategoryTheory/Localization/Predicate.lean", "def_pos": [291, 7], "def_end_pos": [291, 14]}, {"full_name": "CategoryTheory.CatCommSq.hComp", "def_path": "Mathlib/CategoryTheory/CatCommSq.lean", "def_pos": [46, 5], "def_end_pos": [46, 10]}]], "state_before": "C\u2081 : Type u_1\nC\u2082 : Type u_2\nD\u2081 : Type u_3\nD\u2082 : Type u_4\ninst\u271d\u2077 : Category.{u_8, u_1} C\u2081\ninst\u271d\u2076 : Category.{u_7, u_2} C\u2082\ninst\u271d\u2075 : Category.{u_6, u_3} D\u2081\ninst\u271d\u2074 : Category.{u_5, u_4} D\u2082\nG : C\u2081 \u2964 C\u2082\nF : C\u2082 \u2964 C\u2081\nadj : G \u22a3 F\nL\u2081 : C\u2081 \u2964 D\u2081\nW\u2081 : MorphismProperty C\u2081\ninst\u271d\u00b3 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\nW\u2082 : MorphismProperty C\u2082\ninst\u271d\u00b2 : L\u2082.IsLocalization W\u2082\nG' : D\u2081 \u2964 D\u2082\nF' : D\u2082 \u2964 D\u2081\ninst\u271d\u00b9 : CatCommSq G L\u2081 L\u2082 G'\ninst\u271d : CatCommSq F L\u2082 L\u2081 F'\nX\u2082 : C\u2082\n\u22a2 (\u03b7 adj L\u2081 L\u2082 W\u2082 G' F').app (L\u2082.obj X\u2082) =\n    G'.map ((CatCommSq.iso F L\u2082 L\u2081 F').inv.app X\u2082) \u226b\n      (CatCommSq.iso G L\u2081 L\u2082 G').inv.app (F.obj X\u2082) \u226b L\u2082.map (adj.counit.app X\u2082)", "state_after": "C\u2081 : Type u_1\nC\u2082 : Type u_2\nD\u2081 : Type u_3\nD\u2082 : Type u_4\ninst\u271d\u2077 : Category.{u_8, u_1} C\u2081\ninst\u271d\u2076 : Category.{u_7, u_2} C\u2082\ninst\u271d\u2075 : Category.{u_6, u_3} D\u2081\ninst\u271d\u2074 : Category.{u_5, u_4} D\u2082\nG : C\u2081 \u2964 C\u2082\nF : C\u2082 \u2964 C\u2081\nadj : G \u22a3 F\nL\u2081 : C\u2081 \u2964 D\u2081\nW\u2081 : MorphismProperty C\u2081\ninst\u271d\u00b3 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\nW\u2082 : MorphismProperty C\u2082\ninst\u271d\u00b2 : L\u2082.IsLocalization W\u2082\nG' : D\u2081 \u2964 D\u2082\nF' : D\u2082 \u2964 D\u2081\ninst\u271d\u00b9 : CatCommSq G L\u2081 L\u2082 G'\ninst\u271d : CatCommSq F L\u2082 L\u2081 F'\nX\u2082 : C\u2082\nthis : Lifting L\u2082 W\u2082 ((F \u22d9 G) \u22d9 L\u2082) (F' \u22d9 G') := { iso' := CatCommSq.iso'.symm }\n\u22a2 (\u03b7 adj L\u2081 L\u2082 W\u2082 G' F').app (L\u2082.obj X\u2082) =\n    G'.map ((CatCommSq.iso F L\u2082 L\u2081 F').inv.app X\u2082) \u226b\n      (CatCommSq.iso G L\u2081 L\u2082 G').inv.app (F.obj X\u2082) \u226b L\u2082.map (adj.counit.app X\u2082)"}, {"tactic": "simp only [\u03b7, liftNatTrans_app, Lifting.iso, Iso.symm, CatCommSq.hComp_iso'_inv_app,\n  whiskerRight_app, Lifting.id_iso', Functor.rightUnitor_inv_app, comp_id, assoc]", "annotated_tactic": ["simp only [<a>\u03b7</a>, <a>liftNatTrans_app</a>, <a>Lifting.iso</a>, <a>Iso.symm</a>, <a>CatCommSq.hComp_iso'_inv_app</a>,\n    <a>whiskerRight_app</a>, <a>Lifting.id_iso'</a>, <a>Functor.rightUnitor_inv_app</a>, <a>comp_id</a>, <a>assoc</a>]", [{"full_name": "CategoryTheory.Adjunction.Localization.\u03b7", "def_path": "Mathlib/CategoryTheory/Localization/Adjunction.lean", "def_pos": [60, 19], "def_end_pos": [60, 20]}, {"full_name": "CategoryTheory.Localization.liftNatTrans_app", "def_path": "Mathlib/CategoryTheory/Localization/Predicate.lean", "def_pos": [340, 9], "def_end_pos": [340, 25]}, {"full_name": "CategoryTheory.Localization.Lifting.iso", "def_path": "Mathlib/CategoryTheory/Localization/Predicate.lean", "def_pos": [297, 5], "def_end_pos": [297, 16]}, {"full_name": "CategoryTheory.Iso.symm", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [94, 5], "def_end_pos": [94, 9]}, {"full_name": "CategoryTheory.CatCommSq.hComp_iso'_inv_app", "def_path": "Mathlib/CategoryTheory/CatCommSq.lean", "def_pos": [45, 23], "def_end_pos": [45, 35]}, {"full_name": "CategoryTheory.whiskerRight_app", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [57, 3], "def_end_pos": [57, 8]}, {"full_name": "CategoryTheory.Localization.Lifting.id_iso'", "def_path": "Mathlib/CategoryTheory/Localization/Predicate.lean", "def_pos": [382, 3], "def_end_pos": [382, 8]}, {"full_name": "CategoryTheory.Functor.rightUnitor_inv_app", "def_path": "Mathlib/CategoryTheory/Whiskering.lean", "def_pos": [262, 3], "def_end_pos": [262, 8]}, {"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [179, 3], "def_end_pos": [179, 10]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}]], "state_before": "C\u2081 : Type u_1\nC\u2082 : Type u_2\nD\u2081 : Type u_3\nD\u2082 : Type u_4\ninst\u271d\u2077 : Category.{u_8, u_1} C\u2081\ninst\u271d\u2076 : Category.{u_7, u_2} C\u2082\ninst\u271d\u2075 : Category.{u_6, u_3} D\u2081\ninst\u271d\u2074 : Category.{u_5, u_4} D\u2082\nG : C\u2081 \u2964 C\u2082\nF : C\u2082 \u2964 C\u2081\nadj : G \u22a3 F\nL\u2081 : C\u2081 \u2964 D\u2081\nW\u2081 : MorphismProperty C\u2081\ninst\u271d\u00b3 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\nW\u2082 : MorphismProperty C\u2082\ninst\u271d\u00b2 : L\u2082.IsLocalization W\u2082\nG' : D\u2081 \u2964 D\u2082\nF' : D\u2082 \u2964 D\u2081\ninst\u271d\u00b9 : CatCommSq G L\u2081 L\u2082 G'\ninst\u271d : CatCommSq F L\u2082 L\u2081 F'\nX\u2082 : C\u2082\nthis : Lifting L\u2082 W\u2082 ((F \u22d9 G) \u22d9 L\u2082) (F' \u22d9 G') := { iso' := CatCommSq.iso'.symm }\n\u22a2 (\u03b7 adj L\u2081 L\u2082 W\u2082 G' F').app (L\u2082.obj X\u2082) =\n    G'.map ((CatCommSq.iso F L\u2082 L\u2081 F').inv.app X\u2082) \u226b\n      (CatCommSq.iso G L\u2081 L\u2082 G').inv.app (F.obj X\u2082) \u226b L\u2082.map (adj.counit.app X\u2082)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/LpSpace.lean", "full_name": "MeasureTheory.Lp.coeFn_const", "start": [906, 1], "end": [907, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Finsupp.lean", "full_name": "Finsupp.lsingle_apply", "start": [224, 1], "end": [225, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "full_name": "ContinuousLinearMap.coe_mul", "start": [855, 1], "end": [856, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ULift.lean", "full_name": "PLift.forall", "start": [68, 1], "end": [69, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/PartENat.lean", "full_name": "PartENat.ofNat_ne_top", "start": [394, 1], "end": [395, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Sum/Basic.lean", "full_name": "Sum.update_inr_comp_inr", "start": [160, 1], "end": [162, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Connected/PathConnected.lean", "full_name": "pathComponent_subset_component", "start": [930, 1], "end": [932, 96], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u271d y\u271d z : X\n\u03b9 : Type u_3\nF : Set X\nx y : X\nh : y \u2208 pathComponent x\n\u22a2 (Joined.somePath h) 0 = x", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "X : Type u_1\nY : Type u_2\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u271d y\u271d z : X\n\u03b9 : Type u_3\nF : Set X\nx y : X\nh : y \u2208 pathComponent x\n\u22a2 (Joined.somePath h) 1 = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/MvPolynomial/Symmetric.lean", "full_name": "MvPolynomial.mem_symmetricSubalgebra", "start": [95, 1], "end": [97, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/QuadraticForm/Basic.lean", "full_name": "QuadraticForm.polar_add_left", "start": [270, 1], "end": [271, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Localization/Finiteness.lean", "full_name": "Module.Finite.of_localizationSpan_finite'", "start": [73, 1], "end": [99, 12], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "R : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\n\u22a2 Finite R M", "state_after": "case out\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\n\u22a2 \u22a4.FG"}, {"tactic": "choose s\u2081 s\u2082 using (fun g \u21a6 (H g).1)", "annotated_tactic": ["choose s\u2081 s\u2082 using (fun g \u21a6 (H g).1)", []], "state_before": "case out\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\n\u22a2 \u22a4.FG", "state_after": "case out\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\n\u22a2 \u22a4.FG"}, {"tactic": "let sf := fun x : t \u21a6\n  IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers x.val) (f x) (s\u2081 x)", "annotated_tactic": ["let sf := fun x : t \u21a6\n    <a>IsLocalizedModule.finsetIntegerMultiple</a> (<a>Submonoid.powers</a> x.val) (f x) (s\u2081 x)", [{"full_name": "IsLocalizedModule.finsetIntegerMultiple", "def_path": "Mathlib/Algebra/Module/LocalizedModuleIntegers.lean", "def_pos": [103, 19], "def_end_pos": [103, 40]}, {"full_name": "Submonoid.powers", "def_path": "Mathlib/Algebra/Group/Submonoid/Membership.lean", "def_pos": [464, 5], "def_end_pos": [464, 11]}]], "state_before": "case out\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\n\u22a2 \u22a4.FG", "state_after": "case out\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\n\u22a2 \u22a4.FG"}, {"tactic": "use t.attach.biUnion sf", "annotated_tactic": ["use t.attach.biUnion sf", []], "state_before": "case out\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\n\u22a2 \u22a4.FG", "state_after": "case h\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\n\u22a2 Submodule.span R \u2191(t.attach.biUnion sf) = \u22a4"}, {"tactic": "rw [Submodule.span_attach_biUnion, eq_top_iff]", "annotated_tactic": ["rw [<a>Submodule.span_attach_biUnion</a>, <a>eq_top_iff</a>]", [{"full_name": "Submodule.span_attach_biUnion", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [333, 9], "def_end_pos": [333, 28]}, {"full_name": "eq_top_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [133, 9], "def_end_pos": [133, 19]}]], "state_before": "case h\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\n\u22a2 Submodule.span R \u2191(t.attach.biUnion sf) = \u22a4", "state_after": "case h\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\n\u22a2 \u22a4 \u2264 \u2a06 x, Submodule.span R \u2191(sf x)"}, {"tactic": "rintro x -", "annotated_tactic": ["rintro x -", []], "state_before": "case h\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\n\u22a2 \u22a4 \u2264 \u2a06 x, Submodule.span R \u2191(sf x)", "state_after": "case h\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\n\u22a2 x \u2208 \u2a06 x, Submodule.span R \u2191(sf x)"}, {"tactic": "refine Submodule.mem_of_span_eq_top_of_smul_pow_mem _ (t : Set R) ht _ (fun r \u21a6 ?_)", "annotated_tactic": ["refine <a>Submodule.mem_of_span_eq_top_of_smul_pow_mem</a> _ (t : <a>Set</a> R) ht _ (fun r \u21a6 ?_)", [{"full_name": "Submodule.mem_of_span_eq_top_of_smul_pow_mem", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [294, 9], "def_end_pos": [294, 43]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}]], "state_before": "case h\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\n\u22a2 x \u2208 \u2a06 x, Submodule.span R \u2191(sf x)", "state_after": "case h\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\nr : \u2191\u2191t\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2a06 x, Submodule.span R \u2191(sf x)"}, {"tactic": "set S : Submonoid R := Submonoid.powers r.val", "annotated_tactic": ["set S : <a>Submonoid</a> R := <a>Submonoid.powers</a> r.val", [{"full_name": "Submonoid", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [92, 11], "def_end_pos": [92, 20]}, {"full_name": "Submonoid.powers", "def_path": "Mathlib/Algebra/Group/Submonoid/Membership.lean", "def_pos": [464, 5], "def_end_pos": [464, 11]}]], "state_before": "case h\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\nr : \u2191\u2191t\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2a06 x, Submodule.span R \u2191(sf x)", "state_after": "case h\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS\u271d : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\nr : \u2191\u2191t\nS : Submonoid R := Submonoid.powers \u2191r\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2a06 x, Submodule.span R \u2191(sf x)"}, {"tactic": "obtain \u27e8\u27e8_, n\u2081, rfl\u27e9, hn\u2081\u27e9 := multiple_mem_span_of_mem_localization_span S (R\u209a r)\n  (s\u2081 r : Set (M\u209a r)) (IsLocalizedModule.mk' (f r) x (1 : S)) (by rw [s\u2082 r]; trivial)", "annotated_tactic": ["obtain \u27e8\u27e8_, n\u2081, rfl\u27e9, hn\u2081\u27e9 := <a>multiple_mem_span_of_mem_localization_span</a> S (R\u209a r)\n    (s\u2081 r : <a>Set</a> (M\u209a r)) (<a>IsLocalizedModule.mk'</a> (f r) x (1 : S)) (by rw [s\u2082 r]; trivial)", [{"full_name": "multiple_mem_span_of_mem_localization_span", "def_path": "Mathlib/RingTheory/LocalProperties.lean", "def_pos": [484, 9], "def_end_pos": [484, 51]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "IsLocalizedModule.mk'", "def_path": "Mathlib/Algebra/Module/LocalizedModule.lean", "def_pos": [960, 19], "def_end_pos": [960, 22]}]], "state_before": "case h\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS\u271d : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\nr : \u2191\u2191t\nS : Submonoid R := Submonoid.powers \u2191r\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2a06 x, Submodule.span R \u2191(sf x)", "state_after": "case h.intro.mk.intro\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS\u271d : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\nr : \u2191\u2191t\nS : Submonoid R := Submonoid.powers \u2191r\nn\u2081 : \u2115\nhn\u2081 : \u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 IsLocalizedModule.mk' (f r) x 1 \u2208 Submodule.span R \u2191(s\u2081 r)\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2a06 x, Submodule.span R \u2191(sf x)"}, {"tactic": "rw [Submonoid.smul_def, \u2190 IsLocalizedModule.mk'_smul, IsLocalizedModule.mk'_one] at hn\u2081", "annotated_tactic": ["rw [<a>Submonoid.smul_def</a>, \u2190 <a>IsLocalizedModule.mk'_smul</a>, <a>IsLocalizedModule.mk'_one</a>] at hn\u2081", [{"full_name": "Submonoid.smul_def", "def_path": "Mathlib/Algebra/Group/Submonoid/Operations.lean", "def_pos": [1461, 22], "def_end_pos": [1461, 30]}, {"full_name": "IsLocalizedModule.mk'_smul", "def_path": "Mathlib/Algebra/Module/LocalizedModule.lean", "def_pos": [964, 9], "def_end_pos": [964, 17]}, {"full_name": "IsLocalizedModule.mk'_one", "def_path": "Mathlib/Algebra/Module/LocalizedModule.lean", "def_pos": [982, 9], "def_end_pos": [982, 16]}]], "state_before": "case h.intro.mk.intro\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS\u271d : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\nr : \u2191\u2191t\nS : Submonoid R := Submonoid.powers \u2191r\nn\u2081 : \u2115\nhn\u2081 : \u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 IsLocalizedModule.mk' (f r) x 1 \u2208 Submodule.span R \u2191(s\u2081 r)\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2a06 x, Submodule.span R \u2191(sf x)", "state_after": "case h.intro.mk.intro\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS\u271d : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\nr : \u2191\u2191t\nS : Submonoid R := Submonoid.powers \u2191r\nn\u2081 : \u2115\nhn\u2081 : (f r) (\u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x) \u2208 Submodule.span R \u2191(s\u2081 r)\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2a06 x, Submodule.span R \u2191(sf x)"}, {"tactic": "obtain \u27e8\u27e8_, n\u2082, rfl\u27e9, hn\u2082\u27e9 := IsLocalizedModule.smul_mem_finsetIntegerMultiple_span\n  S (f r) _ (s\u2081 r) hn\u2081", "annotated_tactic": ["obtain \u27e8\u27e8_, n\u2082, rfl\u27e9, hn\u2082\u27e9 := <a>IsLocalizedModule.smul_mem_finsetIntegerMultiple_span</a>\n    S (f r) _ (s\u2081 r) hn\u2081", [{"full_name": "IsLocalizedModule.smul_mem_finsetIntegerMultiple_span", "def_path": "Mathlib/Algebra/Module/LocalizedModuleIntegers.lean", "def_pos": [120, 9], "def_end_pos": [120, 44]}]], "state_before": "case h.intro.mk.intro\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS\u271d : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\nr : \u2191\u2191t\nS : Submonoid R := Submonoid.powers \u2191r\nn\u2081 : \u2115\nhn\u2081 : (f r) (\u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x) \u2208 Submodule.span R \u2191(s\u2081 r)\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2a06 x, Submodule.span R \u2191(sf x)", "state_after": "case h.intro.mk.intro.intro.mk.intro\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS\u271d : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\nr : \u2191\u2191t\nS : Submonoid R := Submonoid.powers \u2191r\nn\u2081 : \u2115\nhn\u2081 : (f r) (\u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x) \u2208 Submodule.span R \u2191(s\u2081 r)\nn\u2082 : \u2115\nhn\u2082 :\n  \u27e8(fun x => \u2191r ^ x) n\u2082, \u22ef\u27e9 \u2022 \u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x \u2208\n    Submodule.span R \u2191(IsLocalizedModule.finsetIntegerMultiple S (f r) (s\u2081 r))\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2a06 x, Submodule.span R \u2191(sf x)"}, {"tactic": "rw [Submonoid.smul_def] at hn\u2082", "annotated_tactic": ["rw [<a>Submonoid.smul_def</a>] at hn\u2082", [{"full_name": "Submonoid.smul_def", "def_path": "Mathlib/Algebra/Group/Submonoid/Operations.lean", "def_pos": [1461, 22], "def_end_pos": [1461, 30]}]], "state_before": "case h.intro.mk.intro.intro.mk.intro\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS\u271d : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\nr : \u2191\u2191t\nS : Submonoid R := Submonoid.powers \u2191r\nn\u2081 : \u2115\nhn\u2081 : (f r) (\u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x) \u2208 Submodule.span R \u2191(s\u2081 r)\nn\u2082 : \u2115\nhn\u2082 :\n  \u27e8(fun x => \u2191r ^ x) n\u2082, \u22ef\u27e9 \u2022 \u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x \u2208\n    Submodule.span R \u2191(IsLocalizedModule.finsetIntegerMultiple S (f r) (s\u2081 r))\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2a06 x, Submodule.span R \u2191(sf x)", "state_after": "case h.intro.mk.intro.intro.mk.intro\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS\u271d : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\nr : \u2191\u2191t\nS : Submonoid R := Submonoid.powers \u2191r\nn\u2081 : \u2115\nhn\u2081 : (f r) (\u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x) \u2208 Submodule.span R \u2191(s\u2081 r)\nn\u2082 : \u2115\nhn\u2082 :\n  \u2191\u27e8(fun x => \u2191r ^ x) n\u2082, \u22ef\u27e9 \u2022 \u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x \u2208\n    Submodule.span R \u2191(IsLocalizedModule.finsetIntegerMultiple S (f r) (s\u2081 r))\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2a06 x, Submodule.span R \u2191(sf x)"}, {"tactic": "use n\u2082 + n\u2081", "annotated_tactic": ["use n\u2082 + n\u2081", []], "state_before": "case h.intro.mk.intro.intro.mk.intro\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS\u271d : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\nr : \u2191\u2191t\nS : Submonoid R := Submonoid.powers \u2191r\nn\u2081 : \u2115\nhn\u2081 : (f r) (\u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x) \u2208 Submodule.span R \u2191(s\u2081 r)\nn\u2082 : \u2115\nhn\u2082 :\n  \u2191\u27e8(fun x => \u2191r ^ x) n\u2082, \u22ef\u27e9 \u2022 \u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x \u2208\n    Submodule.span R \u2191(IsLocalizedModule.finsetIntegerMultiple S (f r) (s\u2081 r))\n\u22a2 \u2203 n, \u2191r ^ n \u2022 x \u2208 \u2a06 x, Submodule.span R \u2191(sf x)", "state_after": "case h\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS\u271d : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\nr : \u2191\u2191t\nS : Submonoid R := Submonoid.powers \u2191r\nn\u2081 : \u2115\nhn\u2081 : (f r) (\u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x) \u2208 Submodule.span R \u2191(s\u2081 r)\nn\u2082 : \u2115\nhn\u2082 :\n  \u2191\u27e8(fun x => \u2191r ^ x) n\u2082, \u22ef\u27e9 \u2022 \u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x \u2208\n    Submodule.span R \u2191(IsLocalizedModule.finsetIntegerMultiple S (f r) (s\u2081 r))\n\u22a2 \u2191r ^ (n\u2082 + n\u2081) \u2022 x \u2208 \u2a06 x, Submodule.span R \u2191(sf x)"}, {"tactic": "apply le_iSup (fun x : t \u21a6 Submodule.span R (sf x : Set M)) r", "annotated_tactic": ["apply <a>le_iSup</a> (fun x : t \u21a6 <a>Submodule.span</a> R (sf x : <a>Set</a> M)) r", [{"full_name": "le_iSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [702, 9], "def_end_pos": [702, 16]}, {"full_name": "Submodule.span", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [51, 5], "def_end_pos": [51, 9]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}]], "state_before": "case h\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS\u271d : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\nr : \u2191\u2191t\nS : Submonoid R := Submonoid.powers \u2191r\nn\u2081 : \u2115\nhn\u2081 : (f r) (\u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x) \u2208 Submodule.span R \u2191(s\u2081 r)\nn\u2082 : \u2115\nhn\u2082 :\n  \u2191\u27e8(fun x => \u2191r ^ x) n\u2082, \u22ef\u27e9 \u2022 \u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x \u2208\n    Submodule.span R \u2191(IsLocalizedModule.finsetIntegerMultiple S (f r) (s\u2081 r))\n\u22a2 \u2191r ^ (n\u2082 + n\u2081) \u2022 x \u2208 \u2a06 x, Submodule.span R \u2191(sf x)", "state_after": "case h.a\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS\u271d : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\nr : \u2191\u2191t\nS : Submonoid R := Submonoid.powers \u2191r\nn\u2081 : \u2115\nhn\u2081 : (f r) (\u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x) \u2208 Submodule.span R \u2191(s\u2081 r)\nn\u2082 : \u2115\nhn\u2082 :\n  \u2191\u27e8(fun x => \u2191r ^ x) n\u2082, \u22ef\u27e9 \u2022 \u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x \u2208\n    Submodule.span R \u2191(IsLocalizedModule.finsetIntegerMultiple S (f r) (s\u2081 r))\n\u22a2 \u2191r ^ (n\u2082 + n\u2081) \u2022 x \u2208 Submodule.span R \u2191(sf r)"}, {"tactic": "rw [pow_add, mul_smul]", "annotated_tactic": ["rw [<a>pow_add</a>, <a>mul_smul</a>]", [{"full_name": "pow_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [703, 7], "def_end_pos": [703, 14]}, {"full_name": "MulAction.mul_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [114, 3], "def_end_pos": [114, 11]}]], "state_before": "case h.a\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS\u271d : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\nr : \u2191\u2191t\nS : Submonoid R := Submonoid.powers \u2191r\nn\u2081 : \u2115\nhn\u2081 : (f r) (\u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x) \u2208 Submodule.span R \u2191(s\u2081 r)\nn\u2082 : \u2115\nhn\u2082 :\n  \u2191\u27e8(fun x => \u2191r ^ x) n\u2082, \u22ef\u27e9 \u2022 \u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x \u2208\n    Submodule.span R \u2191(IsLocalizedModule.finsetIntegerMultiple S (f r) (s\u2081 r))\n\u22a2 \u2191r ^ (n\u2082 + n\u2081) \u2022 x \u2208 Submodule.span R \u2191(sf r)", "state_after": "case h.a\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS\u271d : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\nr : \u2191\u2191t\nS : Submonoid R := Submonoid.powers \u2191r\nn\u2081 : \u2115\nhn\u2081 : (f r) (\u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x) \u2208 Submodule.span R \u2191(s\u2081 r)\nn\u2082 : \u2115\nhn\u2082 :\n  \u2191\u27e8(fun x => \u2191r ^ x) n\u2082, \u22ef\u27e9 \u2022 \u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x \u2208\n    Submodule.span R \u2191(IsLocalizedModule.finsetIntegerMultiple S (f r) (s\u2081 r))\n\u22a2 \u2191r ^ n\u2082 \u2022 \u2191r ^ n\u2081 \u2022 x \u2208 Submodule.span R \u2191(sf r)"}, {"tactic": "exact hn\u2082", "annotated_tactic": ["exact hn\u2082", []], "state_before": "case h.a\nR : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS\u271d : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\nr : \u2191\u2191t\nS : Submonoid R := Submonoid.powers \u2191r\nn\u2081 : \u2115\nhn\u2081 : (f r) (\u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x) \u2208 Submodule.span R \u2191(s\u2081 r)\nn\u2082 : \u2115\nhn\u2082 :\n  \u2191\u27e8(fun x => \u2191r ^ x) n\u2082, \u22ef\u27e9 \u2022 \u2191\u27e8(fun x => \u2191r ^ x) n\u2081, \u22ef\u27e9 \u2022 x \u2208\n    Submodule.span R \u2191(IsLocalizedModule.finsetIntegerMultiple S (f r) (s\u2081 r))\n\u22a2 \u2191r ^ n\u2082 \u2022 \u2191r ^ n\u2081 \u2022 x \u2208 Submodule.span R \u2191(sf r)", "state_after": "no goals"}, {"tactic": "rw [s\u2082 r]", "annotated_tactic": ["rw [s\u2082 r]", []], "state_before": "R : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS\u271d : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\nr : \u2191\u2191t\nS : Submonoid R := Submonoid.powers \u2191r\n\u22a2 IsLocalizedModule.mk' (f r) x 1 \u2208 Submodule.span (R\u209a r) \u2191(s\u2081 r)", "state_after": "R : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS\u271d : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\nr : \u2191\u2191t\nS : Submonoid R := Submonoid.powers \u2191r\n\u22a2 IsLocalizedModule.mk' (f r) x 1 \u2208 \u22a4"}, {"tactic": "trivial", "annotated_tactic": ["trivial", []], "state_before": "R : Type u\ninst\u271d\u00b9\u2070 : CommRing R\nS\u271d : Submonoid R\nM : Type w\ninst\u271d\u2079 : AddCommMonoid M\ninst\u271d\u2078 : Module R M\nt : Finset R\nht : Ideal.span \u2191t = \u22a4\nM\u209a : { x // x \u2208 t } \u2192 Type u_1\ninst\u271d\u2077 : (g : { x // x \u2208 t }) \u2192 AddCommMonoid (M\u209a g)\ninst\u271d\u2076 : (g : { x // x \u2208 t }) \u2192 Module R (M\u209a g)\nR\u209a : { x // x \u2208 t } \u2192 Type u\ninst\u271d\u2075 : (g : { x // x \u2208 t }) \u2192 CommRing (R\u209a g)\ninst\u271d\u2074 : (g : { x // x \u2208 t }) \u2192 Algebra R (R\u209a g)\ninst\u271d\u00b3 : \u2200 (g : { x // x \u2208 t }), IsLocalization.Away (\u2191g) (R\u209a g)\ninst\u271d\u00b2 : (g : { x // x \u2208 t }) \u2192 Module (R\u209a g) (M\u209a g)\ninst\u271d\u00b9 : \u2200 (g : { x // x \u2208 t }), IsScalarTower R (R\u209a g) (M\u209a g)\nf : (g : { x // x \u2208 t }) \u2192 M \u2192\u2097[R] M\u209a g\ninst\u271d : \u2200 (g : { x // x \u2208 t }), IsLocalizedModule (Submonoid.powers \u2191g) (f g)\nH : \u2200 (g : { x // x \u2208 t }), Finite (R\u209a g) (M\u209a g)\ns\u2081 : (g : { x // x \u2208 t }) \u2192 Finset (M\u209a g)\ns\u2082 : \u2200 (g : { x // x \u2208 t }), Submodule.span (R\u209a g) \u2191(s\u2081 g) = \u22a4\nsf : { x // x \u2208 t } \u2192 Finset M := fun x => IsLocalizedModule.finsetIntegerMultiple (Submonoid.powers \u2191x) (f x) (s\u2081 x)\nx : M\nr : \u2191\u2191t\nS : Submonoid R := Submonoid.powers \u2191r\n\u22a2 IsLocalizedModule.mk' (f r) x 1 \u2208 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Units.lean", "full_name": "Units.val_pow_eq_pow_val", "start": [423, 1], "end": [424, 65], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/NegOnePow.lean", "full_name": "Int.negOnePow_eq_neg_one_iff", "start": [64, 1], "end": [71, 26], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "n : \u2124\n\u22a2 n.negOnePow = -1 \u2194 Odd n", "state_after": "case mp\nn : \u2124\n\u22a2 n.negOnePow = -1 \u2192 Odd n\n\ncase mpr\nn : \u2124\n\u22a2 Odd n \u2192 n.negOnePow = -1"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mp\nn : \u2124\n\u22a2 n.negOnePow = -1 \u2192 Odd n", "state_after": "case mp\nn : \u2124\nh : n.negOnePow = -1\n\u22a2 Odd n"}, {"tactic": "rw [Int.odd_iff_not_even]", "annotated_tactic": ["rw [<a>Int.odd_iff_not_even</a>]", [{"full_name": "Int.odd_iff_not_even", "def_path": "Mathlib/Algebra/Ring/Int.lean", "def_pos": [111, 15], "def_end_pos": [111, 31]}]], "state_before": "case mp\nn : \u2124\nh : n.negOnePow = -1\n\u22a2 Odd n", "state_after": "case mp\nn : \u2124\nh : n.negOnePow = -1\n\u22a2 \u00acEven n"}, {"tactic": "intro h'", "annotated_tactic": ["intro h'", []], "state_before": "case mp\nn : \u2124\nh : n.negOnePow = -1\n\u22a2 \u00acEven n", "state_after": "case mp\nn : \u2124\nh : n.negOnePow = -1\nh' : Even n\n\u22a2 False"}, {"tactic": "rw [negOnePow_even _ h'] at h", "annotated_tactic": ["rw [<a>negOnePow_even</a> _ h'] at h", [{"full_name": "Int.negOnePow_even", "def_path": "Mathlib/Algebra/Ring/NegOnePow.lean", "def_pos": [39, 7], "def_end_pos": [39, 21]}]], "state_before": "case mp\nn : \u2124\nh : n.negOnePow = -1\nh' : Even n\n\u22a2 False", "state_after": "case mp\nn : \u2124\nh : 1 = -1\nh' : Even n\n\u22a2 False"}, {"tactic": "contradiction", "annotated_tactic": ["contradiction", []], "state_before": "case mp\nn : \u2124\nh : 1 = -1\nh' : Even n\n\u22a2 False", "state_after": "no goals"}, {"tactic": "exact negOnePow_odd n", "annotated_tactic": ["exact <a>negOnePow_odd</a> n", [{"full_name": "Int.negOnePow_odd", "def_path": "Mathlib/Algebra/Ring/NegOnePow.lean", "def_pos": [47, 7], "def_end_pos": [47, 20]}]], "state_before": "case mpr\nn : \u2124\n\u22a2 Odd n \u2192 n.negOnePow = -1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean", "full_name": "Rat.normalize_eq_mkRat", "start": [98, 1], "end": [99, 23], "traced_tactics": [{"tactic": "simp [mkRat, den_nz]", "annotated_tactic": ["simp [<a>mkRat</a>, den_nz]", [{"full_name": "mkRat", "def_path": ".lake/packages/batteries/Batteries/Data/Rat/Basic.lean", "def_pos": [83, 5], "def_end_pos": [83, 10]}]], "state_before": "num : Int\nden : Nat\nden_nz : den \u2260 0\n\u22a2 normalize num den den_nz = mkRat num den", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finsupp/Indicator.lean", "full_name": "Finsupp.support_indicator_subset", "start": [66, 1], "end": [70, 38], "traced_tactics": [{"tactic": "intro i hi", "annotated_tactic": ["intro i hi", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1\ni : \u03b9\n\u22a2 \u2191(indicator s f).support \u2286 \u2191s", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1\ni\u271d i : \u03b9\nhi : i \u2208 \u2191(indicator s f).support\n\u22a2 i \u2208 \u2191s"}, {"tactic": "rw [mem_coe, mem_support_iff] at hi", "annotated_tactic": ["rw [<a>mem_coe</a>, <a>mem_support_iff</a>] at hi", [{"full_name": "Finset.mem_coe", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [208, 9], "def_end_pos": [208, 16]}, {"full_name": "Finsupp.mem_support_iff", "def_path": "Mathlib/Data/Finsupp/Defs.lean", "def_pos": [171, 9], "def_end_pos": [171, 24]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1\ni\u271d i : \u03b9\nhi : i \u2208 \u2191(indicator s f).support\n\u22a2 i \u2208 \u2191s", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1\ni\u271d i : \u03b9\nhi : (indicator s f) i \u2260 0\n\u22a2 i \u2208 \u2191s"}, {"tactic": "by_contra h", "annotated_tactic": ["by_contra h", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1\ni\u271d i : \u03b9\nhi : (indicator s f) i \u2260 0\n\u22a2 i \u2208 \u2191s", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1\ni\u271d i : \u03b9\nhi : (indicator s f) i \u2260 0\nh : i \u2209 \u2191s\n\u22a2 False"}, {"tactic": "exact hi (indicator_of_not_mem h _)", "annotated_tactic": ["exact hi (<a>indicator_of_not_mem</a> h _)", [{"full_name": "Finsupp.indicator_of_not_mem", "def_path": "Mathlib/Data/Finsupp/Indicator.lean", "def_pos": [47, 9], "def_end_pos": [47, 29]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d : Zero \u03b1\ns : Finset \u03b9\nf : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1\ni\u271d i : \u03b9\nhi : (indicator s f) i \u2260 0\nh : i \u2209 \u2191s\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GroupPower/IterateHom.lean", "full_name": "zpow_iterate", "start": [132, 1], "end": [137, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/Star/Spectrum.lean", "full_name": "IsSelfAdjoint.mem_spectrum_eq_re", "start": [90, 1], "end": [100, 48], "traced_tactics": [{"tactic": "have hu := exp_mem_unitary_of_mem_skewAdjoint \u2102 (ha.smul_mem_skewAdjoint conj_I)", "annotated_tactic": ["have hu := <a>exp_mem_unitary_of_mem_skewAdjoint</a> \u2102 (ha.smul_mem_skewAdjoint <a>conj_I</a>)", [{"full_name": "NormedSpace.exp_mem_unitary_of_mem_skewAdjoint", "def_path": "Mathlib/Analysis/NormedSpace/Exponential.lean", "def_pos": [516, 9], "def_end_pos": [516, 43]}, {"full_name": "Complex.conj_I", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [577, 9], "def_end_pos": [577, 15]}]], "state_before": "A : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\na : A\nha : IsSelfAdjoint a\nz : \u2102\nhz : z \u2208 spectrum \u2102 a\n\u22a2 z = \u2191z.re", "state_after": "A : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\na : A\nha : IsSelfAdjoint a\nz : \u2102\nhz : z \u2208 spectrum \u2102 a\nhu : NormedSpace.exp \u2102 (I \u2022 a) \u2208 unitary A\n\u22a2 z = \u2191z.re"}, {"tactic": "let Iu := Units.mk0 I I_ne_zero", "annotated_tactic": ["let Iu := <a>Units.mk0</a> <a>I</a> <a>I_ne_zero</a>", [{"full_name": "Units.mk0", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [186, 5], "def_end_pos": [186, 8]}, {"full_name": "Complex.I", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [278, 5], "def_end_pos": [278, 6]}, {"full_name": "Complex.I_ne_zero", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [306, 15], "def_end_pos": [306, 24]}]], "state_before": "A : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\na : A\nha : IsSelfAdjoint a\nz : \u2102\nhz : z \u2208 spectrum \u2102 a\nhu : NormedSpace.exp \u2102 (I \u2022 a) \u2208 unitary A\n\u22a2 z = \u2191z.re", "state_after": "A : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\na : A\nha : IsSelfAdjoint a\nz : \u2102\nhz : z \u2208 spectrum \u2102 a\nhu : NormedSpace.exp \u2102 (I \u2022 a) \u2208 unitary A\nIu : \u2102\u02e3 := Units.mk0 I I_ne_zero\n\u22a2 z = \u2191z.re"}, {"tactic": "have : NormedSpace.exp \u2102 (I \u2022 z) \u2208 spectrum \u2102 (NormedSpace.exp \u2102 (I \u2022 a)) := by\n  simpa only [Units.smul_def, Units.val_mk0] using\n    spectrum.exp_mem_exp (Iu \u2022 a) (smul_mem_smul_iff.mpr hz)", "annotated_tactic": ["have : <a>NormedSpace.exp</a> \u2102 (<a>I</a> \u2022 z) \u2208 <a>spectrum</a> \u2102 (<a>NormedSpace.exp</a> \u2102 (<a>I</a> \u2022 a)) := by\n    simpa only [<a>Units.smul_def</a>, <a>Units.val_mk0</a>] using\n      <a>spectrum.exp_mem_exp</a> (Iu \u2022 a) (smul_mem_smul_iff.mpr hz)", [{"full_name": "NormedSpace.exp", "def_path": "Mathlib/Analysis/NormedSpace/Exponential.lean", "def_pos": [113, 19], "def_end_pos": [113, 22]}, {"full_name": "Complex.I", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [278, 5], "def_end_pos": [278, 6]}, {"full_name": "spectrum", "def_path": "Mathlib/Algebra/Algebra/Spectrum.lean", "def_pos": [68, 5], "def_end_pos": [68, 13]}, {"full_name": "NormedSpace.exp", "def_path": "Mathlib/Analysis/NormedSpace/Exponential.lean", "def_pos": [113, 19], "def_end_pos": [113, 22]}, {"full_name": "Complex.I", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [278, 5], "def_end_pos": [278, 6]}, {"full_name": "Units.smul_def", "def_path": "Mathlib/GroupTheory/GroupAction/Units.lean", "def_pos": [37, 9], "def_end_pos": [37, 17]}, {"full_name": "Units.val_mk0", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [197, 9], "def_end_pos": [197, 16]}, {"full_name": "spectrum.exp_mem_exp", "def_path": "Mathlib/Analysis/NormedSpace/Spectrum.lean", "def_pos": [497, 9], "def_end_pos": [497, 20]}]], "state_before": "A : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\na : A\nha : IsSelfAdjoint a\nz : \u2102\nhz : z \u2208 spectrum \u2102 a\nhu : NormedSpace.exp \u2102 (I \u2022 a) \u2208 unitary A\nIu : \u2102\u02e3 := Units.mk0 I I_ne_zero\n\u22a2 z = \u2191z.re", "state_after": "A : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\na : A\nha : IsSelfAdjoint a\nz : \u2102\nhz : z \u2208 spectrum \u2102 a\nhu : NormedSpace.exp \u2102 (I \u2022 a) \u2208 unitary A\nIu : \u2102\u02e3 := Units.mk0 I I_ne_zero\nthis : NormedSpace.exp \u2102 (I \u2022 z) \u2208 spectrum \u2102 (NormedSpace.exp \u2102 (I \u2022 a))\n\u22a2 z = \u2191z.re"}, {"tactic": "exact Complex.ext (ofReal_re _) <| by\n  simpa only [\u2190 Complex.exp_eq_exp_\u2102, mem_sphere_zero_iff_norm, norm_eq_abs, abs_exp,\n    Real.exp_eq_one_iff, smul_eq_mul, I_mul, neg_eq_zero] using\n    spectrum.subset_circle_of_unitary hu this", "annotated_tactic": ["exact <a>Complex.ext</a> (<a>ofReal_re</a> _) <| by\n    simpa only [\u2190 <a>Complex.exp_eq_exp_\u2102</a>, <a>mem_sphere_zero_iff_norm</a>, <a>norm_eq_abs</a>, <a>abs_exp</a>,\n      <a>Real.exp_eq_one_iff</a>, <a>smul_eq_mul</a>, <a>I_mul</a>, <a>neg_eq_zero</a>] using\n      <a>spectrum.subset_circle_of_unitary</a> hu this", [{"full_name": "Complex.ext", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [60, 9], "def_end_pos": [60, 12]}, {"full_name": "Complex.ofReal_re", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [96, 9], "def_end_pos": [96, 18]}, {"full_name": "Complex.exp_eq_exp_\u2102", "def_path": "Mathlib/Analysis/SpecialFunctions/Exponential.lean", "def_pos": [220, 9], "def_end_pos": [220, 29]}, {"full_name": "mem_sphere_zero_iff_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [673, 3], "def_end_pos": [673, 14]}, {"full_name": "Complex.norm_eq_abs", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [55, 9], "def_end_pos": [55, 20]}, {"full_name": "Complex.abs_exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1750, 9], "def_end_pos": [1750, 16]}, {"full_name": "Real.exp_eq_one_iff", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 23]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [89, 7], "def_end_pos": [89, 18]}, {"full_name": "Complex.I_mul", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [301, 9], "def_end_pos": [301, 14]}, {"full_name": "neg_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [634, 3], "def_end_pos": [634, 14]}, {"full_name": "spectrum.subset_circle_of_unitary", "def_path": "Mathlib/Analysis/NormedSpace/Star/Spectrum.lean", "def_pos": [44, 9], "def_end_pos": [44, 42]}]], "state_before": "A : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\na : A\nha : IsSelfAdjoint a\nz : \u2102\nhz : z \u2208 spectrum \u2102 a\nhu : NormedSpace.exp \u2102 (I \u2022 a) \u2208 unitary A\nIu : \u2102\u02e3 := Units.mk0 I I_ne_zero\nthis : NormedSpace.exp \u2102 (I \u2022 z) \u2208 spectrum \u2102 (NormedSpace.exp \u2102 (I \u2022 a))\n\u22a2 z = \u2191z.re", "state_after": "no goals"}, {"tactic": "simpa only [Units.smul_def, Units.val_mk0] using\n  spectrum.exp_mem_exp (Iu \u2022 a) (smul_mem_smul_iff.mpr hz)", "annotated_tactic": ["simpa only [<a>Units.smul_def</a>, <a>Units.val_mk0</a>] using\n      <a>spectrum.exp_mem_exp</a> (Iu \u2022 a) (smul_mem_smul_iff.mpr hz)", [{"full_name": "Units.smul_def", "def_path": "Mathlib/GroupTheory/GroupAction/Units.lean", "def_pos": [37, 9], "def_end_pos": [37, 17]}, {"full_name": "Units.val_mk0", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [197, 9], "def_end_pos": [197, 16]}, {"full_name": "spectrum.exp_mem_exp", "def_path": "Mathlib/Analysis/NormedSpace/Spectrum.lean", "def_pos": [497, 9], "def_end_pos": [497, 20]}]], "state_before": "A : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\na : A\nha : IsSelfAdjoint a\nz : \u2102\nhz : z \u2208 spectrum \u2102 a\nhu : NormedSpace.exp \u2102 (I \u2022 a) \u2208 unitary A\nIu : \u2102\u02e3 := Units.mk0 I I_ne_zero\n\u22a2 NormedSpace.exp \u2102 (I \u2022 z) \u2208 spectrum \u2102 (NormedSpace.exp \u2102 (I \u2022 a))", "state_after": "no goals"}, {"tactic": "simpa only [\u2190 Complex.exp_eq_exp_\u2102, mem_sphere_zero_iff_norm, norm_eq_abs, abs_exp,\n  Real.exp_eq_one_iff, smul_eq_mul, I_mul, neg_eq_zero] using\n  spectrum.subset_circle_of_unitary hu this", "annotated_tactic": ["simpa only [\u2190 <a>Complex.exp_eq_exp_\u2102</a>, <a>mem_sphere_zero_iff_norm</a>, <a>norm_eq_abs</a>, <a>abs_exp</a>,\n      <a>Real.exp_eq_one_iff</a>, <a>smul_eq_mul</a>, <a>I_mul</a>, <a>neg_eq_zero</a>] using\n      <a>spectrum.subset_circle_of_unitary</a> hu this", [{"full_name": "Complex.exp_eq_exp_\u2102", "def_path": "Mathlib/Analysis/SpecialFunctions/Exponential.lean", "def_pos": [220, 9], "def_end_pos": [220, 29]}, {"full_name": "mem_sphere_zero_iff_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [673, 3], "def_end_pos": [673, 14]}, {"full_name": "Complex.norm_eq_abs", "def_path": "Mathlib/Analysis/Complex/Basic.lean", "def_pos": [55, 9], "def_end_pos": [55, 20]}, {"full_name": "Complex.abs_exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1750, 9], "def_end_pos": [1750, 16]}, {"full_name": "Real.exp_eq_one_iff", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1251, 9], "def_end_pos": [1251, 23]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [89, 7], "def_end_pos": [89, 18]}, {"full_name": "Complex.I_mul", "def_path": "Mathlib/Data/Complex/Basic.lean", "def_pos": [301, 9], "def_end_pos": [301, 14]}, {"full_name": "neg_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [634, 3], "def_end_pos": [634, 14]}, {"full_name": "spectrum.subset_circle_of_unitary", "def_path": "Mathlib/Analysis/NormedSpace/Star/Spectrum.lean", "def_pos": [44, 9], "def_end_pos": [44, 42]}]], "state_before": "A : Type u_1\ninst\u271d\u2075 : NormedRing A\ninst\u271d\u2074 : NormedAlgebra \u2102 A\ninst\u271d\u00b3 : CompleteSpace A\ninst\u271d\u00b2 : StarRing A\ninst\u271d\u00b9 : CstarRing A\ninst\u271d : StarModule \u2102 A\na : A\nha : IsSelfAdjoint a\nz : \u2102\nhz : z \u2208 spectrum \u2102 a\nhu : NormedSpace.exp \u2102 (I \u2022 a) \u2208 unitary A\nIu : \u2102\u02e3 := Units.mk0 I I_ne_zero\nthis : NormedSpace.exp \u2102 (I \u2022 z) \u2208 spectrum \u2102 (NormedSpace.exp \u2102 (I \u2022 a))\n\u22a2 z.im = (\u2191z.re).im", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Preadditive/Biproducts.lean", "full_name": "CategoryTheory.Limits.biproduct.lift_desc", "start": [238, 1], "end": [241, 15], "traced_tactics": [{"tactic": "simp [biproduct.lift_eq, biproduct.desc_eq, comp_sum, sum_comp, biproduct.\u03b9_\u03c0_assoc, comp_dite,\n  dite_comp]", "annotated_tactic": ["simp [<a>biproduct.lift_eq</a>, <a>biproduct.desc_eq</a>, <a>comp_sum</a>, <a>sum_comp</a>, <a>biproduct.\u03b9_\u03c0_assoc</a>, <a>comp_dite</a>,\n    <a>dite_comp</a>]", [{"full_name": "CategoryTheory.Limits.biproduct.lift_eq", "def_path": "Mathlib/CategoryTheory/Preadditive/Biproducts.lean", "def_pos": [224, 9], "def_end_pos": [224, 26]}, {"full_name": "CategoryTheory.Limits.biproduct.desc_eq", "def_path": "Mathlib/CategoryTheory/Preadditive/Biproducts.lean", "def_pos": [231, 9], "def_end_pos": [231, 26]}, {"full_name": "CategoryTheory.Preadditive.comp_sum", "def_path": "Mathlib/CategoryTheory/Preadditive/Basic.lean", "def_pos": [181, 9], "def_end_pos": [181, 17]}, {"full_name": "CategoryTheory.Preadditive.sum_comp", "def_path": "Mathlib/CategoryTheory/Preadditive/Basic.lean", "def_pos": [187, 9], "def_end_pos": [187, 17]}, {"full_name": "CategoryTheory.Limits.biproduct.\u03b9_\u03c0_assoc", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean", "def_pos": [501, 3], "def_end_pos": [501, 10]}, {"full_name": "CategoryTheory.comp_dite", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [270, 9], "def_end_pos": [270, 18]}, {"full_name": "CategoryTheory.dite_comp", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [275, 9], "def_end_pos": [275, 18]}]], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : Preadditive C\nJ : Type\ninst\u271d\u00b9 : Fintype J\nf : J \u2192 C\ninst\u271d : HasBiproduct f\nT U : C\ng : (j : J) \u2192 T \u27f6 f j\nh : (j : J) \u2192 f j \u27f6 U\n\u22a2 lift g \u226b desc h = \u2211 j : J, g j \u226b h j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/ContinuousFunction/Basic.lean", "full_name": "ContinuousMap.coe_restrict", "start": [412, 1], "end": [413, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.Eventually.ne_of_lt", "start": [1753, 1], "end": [1755, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/GroupAction/Basic.lean", "full_name": "MulAction.mem_orbit_of_mem_orbit_submonoid", "start": [115, 1], "end": [118, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/InitialSeg.lean", "full_name": "PrincipalSeg.subrelIso_apply", "start": [414, 1], "end": [416, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean", "full_name": "CliffordAlgebra.evenOdd_mul_le", "start": [58, 1], "end": [65, 36], "traced_tactics": [{"tactic": "simp_rw [evenOdd, Submodule.iSup_eq_span, Submodule.span_mul_span]", "annotated_tactic": ["simp_rw [<a>evenOdd</a>, <a>Submodule.iSup_eq_span</a>, <a>Submodule.span_mul_span</a>]", [{"full_name": "CliffordAlgebra.evenOdd", "def_path": "Mathlib/LinearAlgebra/CliffordAlgebra/Grading.lean", "def_pos": [31, 5], "def_end_pos": [31, 12]}, {"full_name": "Submodule.iSup_eq_span", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [682, 9], "def_end_pos": [682, 21]}, {"full_name": "Submodule.span_mul_span", "def_path": "Mathlib/Algebra/Algebra/Operations.lean", "def_pos": [197, 9], "def_end_pos": [197, 22]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\ni j : ZMod 2\n\u22a2 evenOdd Q i * evenOdd Q j \u2264 evenOdd Q (i + j)", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\ni j : ZMod 2\n\u22a2 Submodule.span R ((\u22c3 i_1, \u2191(LinearMap.range (\u03b9 Q) ^ \u2191i_1)) * \u22c3 i, \u2191(LinearMap.range (\u03b9 Q) ^ \u2191i)) \u2264\n    Submodule.span R (\u22c3 i_1, \u2191(LinearMap.range (\u03b9 Q) ^ \u2191i_1))"}, {"tactic": "apply Submodule.span_mono", "annotated_tactic": ["apply <a>Submodule.span_mono</a>", [{"full_name": "Submodule.span_mono", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [85, 9], "def_end_pos": [85, 18]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\ni j : ZMod 2\n\u22a2 Submodule.span R ((\u22c3 i_1, \u2191(LinearMap.range (\u03b9 Q) ^ \u2191i_1)) * \u22c3 i, \u2191(LinearMap.range (\u03b9 Q) ^ \u2191i)) \u2264\n    Submodule.span R (\u22c3 i_1, \u2191(LinearMap.range (\u03b9 Q) ^ \u2191i_1))", "state_after": "case h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\ni j : ZMod 2\n\u22a2 (\u22c3 i_1, \u2191(LinearMap.range (\u03b9 Q) ^ \u2191i_1)) * \u22c3 i, \u2191(LinearMap.range (\u03b9 Q) ^ \u2191i) \u2286 \u22c3 i_1, \u2191(LinearMap.range (\u03b9 Q) ^ \u2191i_1)"}, {"tactic": "simp_rw [Set.iUnion_mul, Set.mul_iUnion, Set.iUnion_subset_iff, Set.mul_subset_iff]", "annotated_tactic": ["simp_rw [<a>Set.iUnion_mul</a>, <a>Set.mul_iUnion</a>, <a>Set.iUnion_subset_iff</a>, <a>Set.mul_subset_iff</a>]", [{"full_name": "Set.iUnion_mul", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [491, 9], "def_end_pos": [491, 19]}, {"full_name": "Set.mul_iUnion", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [497, 9], "def_end_pos": [497, 19]}, {"full_name": "Set.iUnion_subset_iff", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [258, 9], "def_end_pos": [258, 26]}, {"full_name": "Set.mul_subset_iff", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [437, 9], "def_end_pos": [437, 23]}]], "state_before": "case h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\ni j : ZMod 2\n\u22a2 (\u22c3 i_1, \u2191(LinearMap.range (\u03b9 Q) ^ \u2191i_1)) * \u22c3 i, \u2191(LinearMap.range (\u03b9 Q) ^ \u2191i) \u2286 \u22c3 i_1, \u2191(LinearMap.range (\u03b9 Q) ^ \u2191i_1)", "state_after": "case h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\ni j : ZMod 2\n\u22a2 \u2200 (i_1 : { n // \u2191n = i }) (i_2 : { n // \u2191n = j }),\n    \u2200 x \u2208 \u2191(LinearMap.range (\u03b9 Q) ^ \u2191i_1),\n      \u2200 y \u2208 \u2191(LinearMap.range (\u03b9 Q) ^ \u2191i_2), x * y \u2208 \u22c3 i_3, \u2191(LinearMap.range (\u03b9 Q) ^ \u2191i_3)"}, {"tactic": "rintro \u27e8xi, rfl\u27e9 \u27e8yi, rfl\u27e9 x hx y hy", "annotated_tactic": ["rintro \u27e8xi, rfl\u27e9 \u27e8yi, rfl\u27e9 x hx y hy", []], "state_before": "case h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\ni j : ZMod 2\n\u22a2 \u2200 (i_1 : { n // \u2191n = i }) (i_2 : { n // \u2191n = j }),\n    \u2200 x \u2208 \u2191(LinearMap.range (\u03b9 Q) ^ \u2191i_1),\n      \u2200 y \u2208 \u2191(LinearMap.range (\u03b9 Q) ^ \u2191i_2), x * y \u2208 \u22c3 i_3, \u2191(LinearMap.range (\u03b9 Q) ^ \u2191i_3)", "state_after": "case h.mk.mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nxi yi : \u2115\nx : CliffordAlgebra Q\nhx : x \u2208 \u2191(LinearMap.range (\u03b9 Q) ^ \u2191\u27e8xi, \u22ef\u27e9)\ny : CliffordAlgebra Q\nhy : y \u2208 \u2191(LinearMap.range (\u03b9 Q) ^ \u2191\u27e8yi, \u22ef\u27e9)\n\u22a2 x * y \u2208 \u22c3 i, \u2191(LinearMap.range (\u03b9 Q) ^ \u2191i)"}, {"tactic": "refine Set.mem_iUnion.mpr \u27e8\u27e8xi + yi, Nat.cast_add _ _\u27e9, ?_\u27e9", "annotated_tactic": ["refine Set.mem_iUnion.mpr \u27e8\u27e8xi + yi, <a>Nat.cast_add</a> _ _\u27e9, ?_\u27e9", [{"full_name": "Nat.cast_add", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 17]}]], "state_before": "case h.mk.mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nxi yi : \u2115\nx : CliffordAlgebra Q\nhx : x \u2208 \u2191(LinearMap.range (\u03b9 Q) ^ \u2191\u27e8xi, \u22ef\u27e9)\ny : CliffordAlgebra Q\nhy : y \u2208 \u2191(LinearMap.range (\u03b9 Q) ^ \u2191\u27e8yi, \u22ef\u27e9)\n\u22a2 x * y \u2208 \u22c3 i, \u2191(LinearMap.range (\u03b9 Q) ^ \u2191i)", "state_after": "case h.mk.mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nxi yi : \u2115\nx : CliffordAlgebra Q\nhx : x \u2208 \u2191(LinearMap.range (\u03b9 Q) ^ \u2191\u27e8xi, \u22ef\u27e9)\ny : CliffordAlgebra Q\nhy : y \u2208 \u2191(LinearMap.range (\u03b9 Q) ^ \u2191\u27e8yi, \u22ef\u27e9)\n\u22a2 x * y \u2208 \u2191(LinearMap.range (\u03b9 Q) ^ \u2191\u27e8xi + yi, \u22ef\u27e9)"}, {"tactic": "simp only [Subtype.coe_mk, Nat.cast_add, pow_add]", "annotated_tactic": ["simp only [<a>Subtype.coe_mk</a>, <a>Nat.cast_add</a>, <a>pow_add</a>]", [{"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [108, 9], "def_end_pos": [108, 15]}, {"full_name": "Nat.cast_add", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [159, 9], "def_end_pos": [159, 17]}, {"full_name": "pow_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [703, 7], "def_end_pos": [703, 14]}]], "state_before": "case h.mk.mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nxi yi : \u2115\nx : CliffordAlgebra Q\nhx : x \u2208 \u2191(LinearMap.range (\u03b9 Q) ^ \u2191\u27e8xi, \u22ef\u27e9)\ny : CliffordAlgebra Q\nhy : y \u2208 \u2191(LinearMap.range (\u03b9 Q) ^ \u2191\u27e8yi, \u22ef\u27e9)\n\u22a2 x * y \u2208 \u2191(LinearMap.range (\u03b9 Q) ^ \u2191\u27e8xi + yi, \u22ef\u27e9)", "state_after": "case h.mk.mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nxi yi : \u2115\nx : CliffordAlgebra Q\nhx : x \u2208 \u2191(LinearMap.range (\u03b9 Q) ^ \u2191\u27e8xi, \u22ef\u27e9)\ny : CliffordAlgebra Q\nhy : y \u2208 \u2191(LinearMap.range (\u03b9 Q) ^ \u2191\u27e8yi, \u22ef\u27e9)\n\u22a2 x * y \u2208 \u2191(LinearMap.range (\u03b9 Q) ^ xi * LinearMap.range (\u03b9 Q) ^ yi)"}, {"tactic": "exact Submodule.mul_mem_mul hx hy", "annotated_tactic": ["exact <a>Submodule.mul_mem_mul</a> hx hy", [{"full_name": "Submodule.mul_mem_mul", "def_path": "Mathlib/Algebra/Algebra/Operations.lean", "def_pos": [160, 9], "def_end_pos": [160, 20]}]], "state_before": "case h.mk.mk\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nQ : QuadraticForm R M\nxi yi : \u2115\nx : CliffordAlgebra Q\nhx : x \u2208 \u2191(LinearMap.range (\u03b9 Q) ^ \u2191\u27e8xi, \u22ef\u27e9)\ny : CliffordAlgebra Q\nhy : y \u2208 \u2191(LinearMap.range (\u03b9 Q) ^ \u2191\u27e8yi, \u22ef\u27e9)\n\u22a2 x * y \u2208 \u2191(LinearMap.range (\u03b9 Q) ^ xi * LinearMap.range 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a.zero\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np : R[X]\n\u22a2 \u00ac(X * p).coeff 0 = 0 \u2194 \u2203 a, \u00acp.coeff a = 0 \u2227 a.succ = 0", "state_after": "no goals"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case a.succ.mp\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np : R[X]\na : \u2115\n\u22a2 \u00acp.coeff a = 0 \u2192 \u2203 a_2, \u00acp.coeff a_2 = 0 \u2227 a_2.succ = a + 1", "state_after": "case a.succ.mp\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np : R[X]\na : \u2115\nh : \u00acp.coeff a = 0\n\u22a2 \u2203 a_1, \u00acp.coeff a_1 = 0 \u2227 a_1.succ = a + 1"}, {"tactic": "use a", "annotated_tactic": ["use a", []], "state_before": "case a.succ.mp\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np : R[X]\na : \u2115\nh : \u00acp.coeff a = 0\n\u22a2 \u2203 a_1, \u00acp.coeff a_1 = 0 \u2227 a_1.succ = a + 1", "state_after": "no goals"}, {"tactic": "rintro \u27e8b, \u27e8h1, h2\u27e9\u27e9", "annotated_tactic": ["rintro \u27e8b, \u27e8h1, h2\u27e9\u27e9", []], "state_before": "case a.succ.mpr\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np : R[X]\na : \u2115\n\u22a2 (\u2203 a_1, \u00acp.coeff a_1 = 0 \u2227 a_1.succ = a + 1) \u2192 \u00acp.coeff a = 0", "state_after": "case a.succ.mpr.intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np : R[X]\na b : \u2115\nh1 : \u00acp.coeff b = 0\nh2 : b.succ = a + 1\n\u22a2 \u00acp.coeff a = 0"}, {"tactic": "rw [\u2190 Nat.succ_injective h2]", "annotated_tactic": ["rw [\u2190 <a>Nat.succ_injective</a> h2]", [{"full_name": "Nat.succ_injective", "def_path": "Mathlib/Data/Nat/Defs.lean", "def_pos": [101, 7], "def_end_pos": [101, 21]}]], "state_before": "case a.succ.mpr.intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np : R[X]\na b : \u2115\nh1 : \u00acp.coeff b = 0\nh2 : b.succ = a + 1\n\u22a2 \u00acp.coeff a = 0", "state_after": "case a.succ.mpr.intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np : R[X]\na b : \u2115\nh1 : \u00acp.coeff b = 0\nh2 : b.succ = a + 1\n\u22a2 \u00acp.coeff b = 0"}, {"tactic": "apply h1", "annotated_tactic": ["apply h1", []], "state_before": "case a.succ.mpr.intro.intro\nR : Type u_1\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : IsDomain R\ninst\u271d : NormalizedGCDMonoid R\np : R[X]\na b : \u2115\nh1 : \u00acp.coeff b = 0\nh2 : b.succ = a + 1\n\u22a2 \u00acp.coeff b = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Padics/PadicNumbers.lean", "full_name": "PadicSeq.norm_zero_iff", "start": [121, 1], "end": [134, 19], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "p : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u22a2 f.norm = 0 \u2194 f \u2248 0", "state_after": "case mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u22a2 f.norm = 0 \u2192 f \u2248 0\n\ncase mpr\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u22a2 f \u2248 0 \u2192 f.norm = 0"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u22a2 f.norm = 0 \u2192 f \u2248 0", "state_after": "case mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nh : f.norm = 0\n\u22a2 f \u2248 0"}, {"tactic": "by_contra hf", "annotated_tactic": ["by_contra hf", []], "state_before": "case mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nh : f.norm = 0\n\u22a2 f \u2248 0", "state_after": "case mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nh : f.norm = 0\nhf : \u00acf \u2248 0\n\u22a2 False"}, {"tactic": "unfold norm at h", "annotated_tactic": ["unfold <a>norm</a> at h", [{"full_name": "PadicSeq.norm", "def_path": "Mathlib/NumberTheory/Padics/PadicNumbers.lean", "def_pos": [117, 5], "def_end_pos": [117, 9]}]], "state_before": "case mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nh : f.norm = 0\nhf : \u00acf \u2248 0\n\u22a2 False", "state_after": "case mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nh : (if hf : f \u2248 0 then 0 else padicNorm p (\u2191f (stationaryPoint hf))) = 0\nhf : \u00acf \u2248 0\n\u22a2 False"}, {"tactic": "split_ifs at h", "annotated_tactic": ["split_ifs at h", []], "state_before": "case mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nh : (if hf : f \u2248 0 then 0 else padicNorm p (\u2191f (stationaryPoint hf))) = 0\nhf : \u00acf \u2248 0\n\u22a2 False", "state_after": "case mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nh : padicNorm p (\u2191f (stationaryPoint hf)) = 0\n\u22a2 False"}, {"tactic": "apply hf", "annotated_tactic": ["apply hf", []], "state_before": "case mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nh : padicNorm p (\u2191f (stationaryPoint hf)) = 0\n\u22a2 False", "state_after": "case mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nh : padicNorm p (\u2191f (stationaryPoint hf)) = 0\n\u22a2 f \u2248 0"}, {"tactic": "intro \u03b5 h\u03b5", "annotated_tactic": ["intro \u03b5 h\u03b5", []], "state_before": "case mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nh : padicNorm p (\u2191f (stationaryPoint hf)) = 0\n\u22a2 f \u2248 0", "state_after": "case mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nh : padicNorm p (\u2191f (stationaryPoint hf)) = 0\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 j \u2265 i, padicNorm p (\u2191(f - 0) j) < \u03b5"}, {"tactic": "exists stationaryPoint hf", "annotated_tactic": ["exists <a>stationaryPoint</a> hf", [{"full_name": "PadicSeq.stationaryPoint", "def_path": "Mathlib/NumberTheory/Padics/PadicNumbers.lean", "def_pos": [105, 5], "def_end_pos": [105, 20]}]], "state_before": "case mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nh : padicNorm p (\u2191f (stationaryPoint hf)) = 0\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2203 i, \u2200 j \u2265 i, padicNorm p (\u2191(f - 0) j) < \u03b5", "state_after": "case mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nh : padicNorm p (\u2191f (stationaryPoint hf)) = 0\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2200 j \u2265 stationaryPoint hf, padicNorm p (\u2191(f - 0) j) < \u03b5"}, {"tactic": "intro j hj", "annotated_tactic": ["intro j hj", []], "state_before": "case mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nh : padicNorm p (\u2191f (stationaryPoint hf)) = 0\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\n\u22a2 \u2200 j \u2265 stationaryPoint hf, padicNorm p (\u2191(f - 0) j) < \u03b5", "state_after": "case mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nh : padicNorm p (\u2191f (stationaryPoint hf)) = 0\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nj : \u2115\nhj : j \u2265 stationaryPoint hf\n\u22a2 padicNorm p (\u2191(f - 0) j) < \u03b5"}, {"tactic": "have heq := stationaryPoint_spec hf le_rfl hj", "annotated_tactic": ["have heq := <a>stationaryPoint_spec</a> hf <a>le_rfl</a> hj", [{"full_name": "PadicSeq.stationaryPoint_spec", "def_path": "Mathlib/NumberTheory/Padics/PadicNumbers.lean", "def_pos": [109, 9], "def_end_pos": [109, 29]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}]], "state_before": "case mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nh : padicNorm p (\u2191f (stationaryPoint hf)) = 0\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nj : \u2115\nhj : j \u2265 stationaryPoint hf\n\u22a2 padicNorm p (\u2191(f - 0) j) < \u03b5", "state_after": "case mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nh : padicNorm p (\u2191f (stationaryPoint hf)) = 0\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nj : \u2115\nhj : j \u2265 stationaryPoint hf\nheq : padicNorm p (\u2191f j) = padicNorm p (\u2191f (stationaryPoint hf))\n\u22a2 padicNorm p (\u2191(f - 0) j) < \u03b5"}, {"tactic": "simpa [h, heq]", "annotated_tactic": ["simpa [h, heq]", []], "state_before": "case mp\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nhf : \u00acf \u2248 0\nh : padicNorm p (\u2191f (stationaryPoint hf)) = 0\n\u03b5 : \u211a\nh\u03b5 : \u03b5 > 0\nj : \u2115\nhj : j \u2265 stationaryPoint hf\nheq : padicNorm p (\u2191f j) = padicNorm p (\u2191f (stationaryPoint hf))\n\u22a2 padicNorm p (\u2191(f - 0) j) < \u03b5", "state_after": "no goals"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mpr\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\n\u22a2 f \u2248 0 \u2192 f.norm = 0", "state_after": "case mpr\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nh : f \u2248 0\n\u22a2 f.norm = 0"}, {"tactic": "simp [norm, h]", "annotated_tactic": ["simp [<a>norm</a>, h]", [{"full_name": "PadicSeq.norm", "def_path": "Mathlib/NumberTheory/Padics/PadicNumbers.lean", "def_pos": [117, 5], "def_end_pos": [117, 9]}]], "state_before": "case mpr\np : \u2115\ninst\u271d : Fact (Nat.Prime p)\nf : PadicSeq p\nh : f \u2248 0\n\u22a2 f.norm = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/ShortComplex/Basic.lean", "full_name": "CategoryTheory.ShortComplex.comp_\u03c4\u2082", "start": [109, 1], "end": [110, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Piecewise.lean", "full_name": "Finset.piecewise_univ", "start": [144, 1], "end": [148, 19], "traced_tactics": [{"tactic": "ext i", "annotated_tactic": ["ext i", []], "state_before": "\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Sort u_2\ns : Finset \u03b9\nf\u271d g\u271d : (i : \u03b9) \u2192 \u03c0 i\ninst\u271d\u00b2 : (j : \u03b9) \u2192 Decidable (j \u2208 s)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : (i : \u03b9) \u2192 Decidable (i \u2208 univ)\nf g : (i : \u03b9) \u2192 \u03c0 i\n\u22a2 univ.piecewise f g = f", "state_after": "case h\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Sort u_2\ns : Finset \u03b9\nf\u271d g\u271d : (i : \u03b9) \u2192 \u03c0 i\ninst\u271d\u00b2 : (j : \u03b9) \u2192 Decidable (j \u2208 s)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : (i : \u03b9) \u2192 Decidable (i \u2208 univ)\nf g : (i : \u03b9) \u2192 \u03c0 i\ni : \u03b9\n\u22a2 univ.piecewise f g i = f i"}, {"tactic": "simp [piecewise]", "annotated_tactic": ["simp [<a>piecewise</a>]", [{"full_name": "Finset.piecewise", "def_path": "Mathlib/Data/Finset/Piecewise.lean", "def_pos": [27, 5], "def_end_pos": [27, 14]}]], "state_before": "case h\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Sort u_2\ns : Finset \u03b9\nf\u271d g\u271d : (i : \u03b9) \u2192 \u03c0 i\ninst\u271d\u00b2 : (j : \u03b9) \u2192 Decidable (j \u2208 s)\ninst\u271d\u00b9 : Fintype \u03b9\ninst\u271d : (i : \u03b9) \u2192 Decidable (i \u2208 univ)\nf g : (i : \u03b9) \u2192 \u03c0 i\ni : \u03b9\n\u22a2 univ.piecewise f g i = f i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Hom/Bounded.lean", "full_name": "BotHom.coe_copy", "start": [428, 1], "end": [429, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Chain.lean", "full_name": "ChainClosure.total", "start": [250, 1], "end": [253, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Basic.lean", "full_name": "Convex.lineMap_mem", "start": [517, 1], "end": [519, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.integrable_inv_smul_measure", "start": [586, 1], "end": [588, 66], "traced_tactics": [{"tactic": "simpa using h\u2082", "annotated_tactic": ["simpa using h\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\u00b2 : MeasurableSpace \u03b4\ninst\u271d\u00b9 : NormedAddCommGroup \u03b2\ninst\u271d : NormedAddCommGroup \u03b3\nf : \u03b1 \u2192 \u03b2\nc : \u211d\u22650\u221e\nh\u2081 : c \u2260 0\nh\u2082 : c \u2260 \u22a4\n\u22a2 c\u207b\u00b9 \u2260 0", "state_after": "no goals"}, {"tactic": "simpa using h\u2081", "annotated_tactic": ["simpa using h\u2081", []], "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\nc : \u211d\u22650\u221e\nh\u2081 : c \u2260 0\nh\u2082 : c \u2260 \u22a4\n\u22a2 c\u207b\u00b9 \u2260 \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/CharP/ExpChar.lean", "full_name": "sub_pow_expChar_pow", "start": [253, 1], "end": [257, 56], "traced_tactics": [{"tactic": "cases' hR with _ _ hprime _", "annotated_tactic": ["cases' hR with _ _ hprime _", []], "state_before": "R : Type u\ninst\u271d : CommRing R\nq : \u2115\nhR : ExpChar R q\nn : \u2115\nx y : R\n\u22a2 (x - y) ^ q ^ n = x ^ q ^ n - y ^ q ^ n", "state_after": "case zero\nR : Type u\ninst\u271d\u00b9 : CommRing R\nn : \u2115\nx y : R\ninst\u271d : CharZero R\n\u22a2 (x - y) ^ 1 ^ n = x ^ 1 ^ n - y ^ 1 ^ n\n\ncase prime\nR : Type u\ninst\u271d : CommRing R\nq n : \u2115\nx y : R\nhprime : Nat.Prime q\nhchar\u271d : CharP R q\n\u22a2 (x - y) ^ q ^ n = x ^ q ^ n - y ^ q ^ n"}, {"tactic": "haveI := Fact.mk hprime", "annotated_tactic": ["haveI := <a>Fact.mk</a> hprime", [{"full_name": "Fact.mk", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [114, 7], "def_end_pos": [114, 11]}]], "state_before": "case prime\nR : Type u\ninst\u271d : CommRing R\nq n : \u2115\nx y : R\nhprime : Nat.Prime q\nhchar\u271d : CharP R q\n\u22a2 (x - y) ^ q ^ n = x ^ q ^ n - y ^ q ^ n", "state_after": "case prime\nR : Type u\ninst\u271d : CommRing R\nq n : \u2115\nx y : R\nhprime : Nat.Prime q\nhchar\u271d : CharP R q\nthis : Fact (Nat.Prime q)\n\u22a2 (x - y) ^ q ^ n = x ^ q ^ n - y ^ q ^ n"}, {"tactic": "exact sub_pow_char_pow R x y", "annotated_tactic": ["exact <a>sub_pow_char_pow</a> R x y", [{"full_name": "sub_pow_char_pow", "def_path": "Mathlib/Algebra/CharP/Basic.lean", "def_pos": [134, 9], "def_end_pos": [134, 25]}]], "state_before": "case prime\nR : Type u\ninst\u271d : CommRing R\nq n : \u2115\nx y : R\nhprime : Nat.Prime q\nhchar\u271d : CharP R q\nthis : Fact (Nat.Prime q)\n\u22a2 (x - y) ^ q ^ n = x ^ q ^ n - y ^ q ^ n", "state_after": "no goals"}, {"tactic": "simp only [one_pow, pow_one]", "annotated_tactic": ["simp only [<a>one_pow</a>, <a>pow_one</a>]", [{"full_name": "one_pow", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [696, 39], "def_end_pos": [696, 46]}, {"full_name": "pow_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [663, 7], "def_end_pos": [663, 14]}]], "state_before": "case zero\nR : Type u\ninst\u271d\u00b9 : CommRing R\nn : \u2115\nx y : R\ninst\u271d : CharZero R\n\u22a2 (x - y) ^ 1 ^ n = x ^ 1 ^ n - y ^ 1 ^ n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matrix/RowCol.lean", "full_name": "Matrix.updateRow_ne", "start": [201, 1], "end": [203, 57], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "full_name": "tsub_tsub_eq_min", "start": [495, 1], "end": [496, 70], "traced_tactics": [{"tactic": "rw [tsub_eq_tsub_min _ b, tsub_tsub_cancel_of_le (min_le_left a _)]", "annotated_tactic": ["rw [<a>tsub_eq_tsub_min</a> _ b, <a>tsub_tsub_cancel_of_le</a> (<a>min_le_left</a> a _)]", [{"full_name": "tsub_eq_tsub_min", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [431, 9], "def_end_pos": [431, 25]}, {"full_name": "tsub_tsub_cancel_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [296, 9], "def_end_pos": [296, 31]}, {"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\u00b3 : CanonicallyLinearOrderedAddCommMonoid \u03b1\ninst\u271d\u00b2 : Sub \u03b1\ninst\u271d\u00b9 : OrderedSub \u03b1\na\u271d b\u271d c d : \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\n\u22a2 a - (a - b) = min a b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Image.lean", "full_name": "Set.exists_image_iff", "start": [605, 1], "end": [608, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "full_name": "LinearMap.surjective_of_injective", "start": [627, 1], "end": [631, 48], "traced_tactics": [{"tactic": "have h := rank_range_of_injective _ hinj", "annotated_tactic": ["have h := <a>rank_range_of_injective</a> _ hinj", [{"full_name": "rank_range_of_injective", "def_path": "Mathlib/LinearAlgebra/Dimension/Basic.lean", "def_pos": [312, 9], "def_end_pos": [312, 32]}]], "state_before": "K : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nf : V \u2192\u2097[K] V\nhinj : Injective \u21d1f\n\u22a2 Surjective \u21d1f", "state_after": "K : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nf : V \u2192\u2097[K] V\nhinj : Injective \u21d1f\nh : Module.rank K \u21a5(range f) = Module.rank K V\n\u22a2 Surjective \u21d1f"}, {"tactic": "rw [\u2190 finrank_eq_rank, \u2190 finrank_eq_rank, natCast_inj] at h", "annotated_tactic": ["rw [\u2190 <a>finrank_eq_rank</a>, \u2190 <a>finrank_eq_rank</a>, <a>natCast_inj</a>] at h", [{"full_name": "finrank_eq_rank", "def_path": "Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean", "def_pos": [478, 9], "def_end_pos": [478, 24]}, {"full_name": "finrank_eq_rank", "def_path": "Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean", "def_pos": [478, 9], "def_end_pos": [478, 24]}, {"full_name": "Cardinal.natCast_inj", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1464, 9], "def_end_pos": [1464, 20]}]], "state_before": "K : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nf : V \u2192\u2097[K] V\nhinj : Injective \u21d1f\nh : Module.rank K \u21a5(range f) = Module.rank K V\n\u22a2 Surjective \u21d1f", "state_after": "K : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nf : V \u2192\u2097[K] V\nhinj : Injective \u21d1f\nh : finrank K \u21a5(range f) = finrank K V\n\u22a2 Surjective \u21d1f"}, {"tactic": "exact range_eq_top.1 (eq_top_of_finrank_eq h)", "annotated_tactic": ["exact <a>range_eq_top</a>.1 (<a>eq_top_of_finrank_eq</a> h)", [{"full_name": "LinearMap.range_eq_top", "def_path": "Mathlib/Algebra/Module/Submodule/Range.lean", "def_pos": [100, 9], "def_end_pos": [100, 21]}, {"full_name": "Submodule.eq_top_of_finrank_eq", "def_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "def_pos": [225, 9], "def_end_pos": [225, 46]}]], "state_before": "K : Type u\nV : Type v\ninst\u271d\u2075 : DivisionRing K\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module K V\nV\u2082 : Type v'\ninst\u271d\u00b2 : AddCommGroup V\u2082\ninst\u271d\u00b9 : Module K V\u2082\ninst\u271d : FiniteDimensional K V\nf : V \u2192\u2097[K] V\nhinj : Injective \u21d1f\nh : finrank K \u21a5(range f) = finrank K V\n\u22a2 Surjective \u21d1f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Card.lean", "full_name": "Finset.le_card_of_inj_on_range", "start": [468, 1], "end": [472, 81], "traced_tactics": [{"tactic": "simpa only [mem_range]", "annotated_tactic": ["simpa only [<a>mem_range</a>]", [{"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2935, 9], "def_end_pos": [2935, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t\u271d u : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : \u2115 \u2192 \u03b1\nhf : \u2200 i < n, f i \u2208 s\nf_inj : \u2200 i < n, \u2200 j < n, f i = f j \u2192 i = j\n\u22a2 \u2200 a \u2208 range n, f a \u2208 s", "state_after": "no goals"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t\u271d u : Finset \u03b1\nf\u271d : \u03b1 \u2192 \u03b2\nn : \u2115\nt : Finset \u03b2\nf : \u2115 \u2192 \u03b1\nhf : \u2200 i < n, f i \u2208 s\nf_inj : \u2200 i < n, \u2200 j < n, f i = f j \u2192 i = j\n\u22a2 Set.InjOn f \u2191(range n)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/AlexandrovDiscrete.lean", "full_name": "isClopen_iInter\u2082", "start": [87, 1], "end": [89, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Equiv/Basic.lean", "full_name": "Equiv.piCongr_symm_apply", "start": [1935, 1], "end": [1937, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/Padics/PadicNorm.lean", "full_name": "padicNorm.int_eq_one_iff", "start": [268, 1], "end": [285, 54], "traced_tactics": [{"tactic": "nth_rw 2 [\u2190 pow_one p]", "annotated_tactic": ["nth_rw 2 [\u2190 <a>pow_one</a> p]", [{"full_name": "pow_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [663, 7], "def_end_pos": [663, 14]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\n\u22a2 padicNorm p \u2191m = 1 \u2194 \u00ac\u2191p \u2223 m", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\n\u22a2 padicNorm p \u2191m = 1 \u2194 \u00ac\u2191(p ^ 1) \u2223 m"}, {"tactic": "simp only [dvd_iff_norm_le, Int.cast_natCast, Nat.cast_one, zpow_neg, zpow_one, not_le]", "annotated_tactic": ["simp only [<a>dvd_iff_norm_le</a>, <a>Int.cast_natCast</a>, <a>Nat.cast_one</a>, <a>zpow_neg</a>, <a>zpow_one</a>, <a>not_le</a>]", [{"full_name": "padicNorm.dvd_iff_norm_le", "def_path": "Mathlib/NumberTheory/Padics/PadicNorm.lean", "def_pos": [254, 9], "def_end_pos": [254, 24]}, {"full_name": "Int.cast_natCast", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [66, 9], "def_end_pos": [66, 21]}, {"full_name": "Nat.cast_one", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [154, 9], "def_end_pos": [154, 17]}, {"full_name": "zpow_neg", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [593, 7], "def_end_pos": [593, 15]}, {"full_name": "zpow_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1068, 7], "def_end_pos": [1068, 15]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [375, 9], "def_end_pos": [375, 15]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\n\u22a2 padicNorm p \u2191m = 1 \u2194 \u00ac\u2191(p ^ 1) \u2223 m", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\n\u22a2 padicNorm p \u2191m = 1 \u2194 (\u2191p)\u207b\u00b9 < padicNorm p \u2191m"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\n\u22a2 padicNorm p \u2191m = 1 \u2194 (\u2191p)\u207b\u00b9 < padicNorm p \u2191m", "state_after": "case mp\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\n\u22a2 padicNorm p \u2191m = 1 \u2192 (\u2191p)\u207b\u00b9 < padicNorm p \u2191m\n\ncase mpr\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\n\u22a2 (\u2191p)\u207b\u00b9 < padicNorm p \u2191m \u2192 padicNorm p \u2191m = 1"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mp\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\n\u22a2 padicNorm p \u2191m = 1 \u2192 (\u2191p)\u207b\u00b9 < padicNorm p \u2191m", "state_after": "case mp\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh : padicNorm p \u2191m = 1\n\u22a2 (\u2191p)\u207b\u00b9 < padicNorm p \u2191m"}, {"tactic": "rw [h, inv_lt_one_iff_of_pos] <;> norm_cast", "annotated_tactic": ["rw [h, <a>inv_lt_one_iff_of_pos</a>] <;> norm_cast", [{"full_name": "inv_lt_one_iff_of_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [243, 9], "def_end_pos": [243, 30]}]], "state_before": "case mp\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh : padicNorm p \u2191m = 1\n\u22a2 (\u2191p)\u207b\u00b9 < padicNorm p \u2191m", "state_after": "case mp\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh : padicNorm p \u2191m = 1\n\u22a2 1 < p\n\ncase mp\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh : padicNorm p \u2191m = 1\n\u22a2 0 < p"}, {"tactic": "exact Nat.Prime.one_lt Fact.out", "annotated_tactic": ["exact <a>Nat.Prime.one_lt</a> <a>Fact.out</a>", [{"full_name": "Nat.Prime.one_lt", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [75, 9], "def_end_pos": [75, 21]}, {"full_name": "Fact.out", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [117, 3], "def_end_pos": [117, 6]}]], "state_before": "case mp\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh : padicNorm p \u2191m = 1\n\u22a2 1 < p", "state_after": "no goals"}, {"tactic": "exact Nat.Prime.pos Fact.out", "annotated_tactic": ["exact <a>Nat.Prime.pos</a> <a>Fact.out</a>", [{"full_name": "Nat.Prime.pos", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [65, 9], "def_end_pos": [65, 18]}, {"full_name": "Fact.out", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [117, 3], "def_end_pos": [117, 6]}]], "state_before": "case mp\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh : padicNorm p \u2191m = 1\n\u22a2 0 < p", "state_after": "no goals"}, {"tactic": "simp only [padicNorm]", "annotated_tactic": ["simp only [<a>padicNorm</a>]", [{"full_name": "padicNorm", "def_path": "Mathlib/NumberTheory/Padics/PadicNorm.lean", "def_pos": [42, 5], "def_end_pos": [42, 14]}]], "state_before": "case mpr\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\n\u22a2 (\u2191p)\u207b\u00b9 < padicNorm p \u2191m \u2192 padicNorm p \u2191m = 1", "state_after": "case mpr\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\n\u22a2 ((\u2191p)\u207b\u00b9 < if \u2191m = 0 then 0 else \u2191p ^ (-padicValRat p \u2191m)) \u2192 (if \u2191m = 0 then 0 else \u2191p ^ (-padicValRat p \u2191m)) = 1"}, {"tactic": "split_ifs", "annotated_tactic": ["split_ifs", []], "state_before": "case mpr\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\n\u22a2 ((\u2191p)\u207b\u00b9 < if \u2191m = 0 then 0 else \u2191p ^ (-padicValRat p \u2191m)) \u2192 (if \u2191m = 0 then 0 else \u2191p ^ (-padicValRat p \u2191m)) = 1", "state_after": "case pos\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh\u271d : \u2191m = 0\n\u22a2 (\u2191p)\u207b\u00b9 < 0 \u2192 0 = 1\n\ncase neg\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh\u271d : \u00ac\u2191m = 0\n\u22a2 (\u2191p)\u207b\u00b9 < \u2191p ^ (-padicValRat p \u2191m) \u2192 \u2191p ^ (-padicValRat p \u2191m) = 1"}, {"tactic": "rw [inv_lt_zero, \u2190 Nat.cast_zero, Nat.cast_lt]", "annotated_tactic": ["rw [<a>inv_lt_zero</a>, \u2190 <a>Nat.cast_zero</a>, <a>Nat.cast_lt</a>]", [{"full_name": "inv_lt_zero", "def_path": "Mathlib/Algebra/Order/Field/Defs.lean", "def_pos": [61, 15], "def_end_pos": [61, 26]}, {"full_name": "Nat.cast_zero", "def_path": "Mathlib/Data/Nat/Cast/Defs.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}, {"full_name": "Nat.cast_lt", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [129, 9], "def_end_pos": [129, 16]}]], "state_before": "case pos\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh\u271d : \u2191m = 0\n\u22a2 (\u2191p)\u207b\u00b9 < 0 \u2192 0 = 1", "state_after": "case pos\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh\u271d : \u2191m = 0\n\u22a2 p < 0 \u2192 \u21910 = 1"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case pos\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh\u271d : \u2191m = 0\n\u22a2 p < 0 \u2192 \u21910 = 1", "state_after": "case pos\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh\u271d : \u2191m = 0\nh : p < 0\n\u22a2 \u21910 = 1"}, {"tactic": "exact (Nat.not_lt_zero p h).elim", "annotated_tactic": ["exact (<a>Nat.not_lt_zero</a> p h).<a>elim</a>", [{"full_name": "Nat.not_lt_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1660, 9], "def_end_pos": [1660, 24]}, {"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "case pos\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh\u271d : \u2191m = 0\nh : p < 0\n\u22a2 \u21910 = 1", "state_after": "no goals"}, {"tactic": "have : 1 < (p : \u211a) := by norm_cast; exact Nat.Prime.one_lt (Fact.out : Nat.Prime p)", "annotated_tactic": ["have : 1 < (p : \u211a) := by norm_cast; exact <a>Nat.Prime.one_lt</a> (<a>Fact.out</a> : <a>Nat.Prime</a> p)", [{"full_name": "Nat.Prime.one_lt", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [75, 9], "def_end_pos": [75, 21]}, {"full_name": "Fact.out", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [117, 3], "def_end_pos": [117, 6]}, {"full_name": "Nat.Prime", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [45, 5], "def_end_pos": [45, 10]}]], "state_before": "case neg\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh\u271d : \u00ac\u2191m = 0\n\u22a2 (\u2191p)\u207b\u00b9 < \u2191p ^ (-padicValRat p \u2191m) \u2192 \u2191p ^ (-padicValRat p \u2191m) = 1", "state_after": "case neg\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh\u271d : \u00ac\u2191m = 0\nthis : 1 < \u2191p\n\u22a2 (\u2191p)\u207b\u00b9 < \u2191p ^ (-padicValRat p \u2191m) \u2192 \u2191p ^ (-padicValRat p \u2191m) = 1"}, {"tactic": "rw [\u2190 zpow_neg_one, zpow_lt_iff_lt this]", "annotated_tactic": ["rw [\u2190 <a>zpow_neg_one</a>, <a>zpow_lt_iff_lt</a> this]", [{"full_name": "zpow_neg_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1077, 7], "def_end_pos": [1077, 19]}, {"full_name": "zpow_lt_iff_lt", "def_path": "Mathlib/Algebra/Order/Field/Power.lean", "def_pos": [79, 9], "def_end_pos": [79, 23]}]], "state_before": "case neg\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh\u271d : \u00ac\u2191m = 0\nthis : 1 < \u2191p\n\u22a2 (\u2191p)\u207b\u00b9 < \u2191p ^ (-padicValRat p \u2191m) \u2192 \u2191p ^ (-padicValRat p \u2191m) = 1", "state_after": "case neg\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh\u271d : \u00ac\u2191m = 0\nthis : 1 < \u2191p\n\u22a2 -1 < -padicValRat p \u2191m \u2192 \u2191p ^ (-padicValRat p \u2191m) = 1"}, {"tactic": "have : 0 \u2264 padicValRat p m := by simp only [of_int, Nat.cast_nonneg]", "annotated_tactic": ["have : 0 \u2264 <a>padicValRat</a> p m := by simp only [<a>of_int</a>, <a>Nat.cast_nonneg</a>]", [{"full_name": "padicValRat", "def_path": "Mathlib/NumberTheory/Padics/PadicVal.lean", "def_pos": [200, 5], "def_end_pos": [200, 16]}, {"full_name": "padicValRat.of_int", "def_path": "Mathlib/NumberTheory/Padics/PadicVal.lean", "def_pos": [232, 9], "def_end_pos": [232, 15]}, {"full_name": "Nat.cast_nonneg", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [50, 9], "def_end_pos": [50, 20]}]], "state_before": "case neg\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh\u271d : \u00ac\u2191m = 0\nthis : 1 < \u2191p\n\u22a2 -1 < -padicValRat p \u2191m \u2192 \u2191p ^ (-padicValRat p \u2191m) = 1", "state_after": "case neg\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh\u271d : \u00ac\u2191m = 0\nthis\u271d : 1 < \u2191p\nthis : 0 \u2264 padicValRat p \u2191m\n\u22a2 -1 < -padicValRat p \u2191m \u2192 \u2191p ^ (-padicValRat p \u2191m) = 1"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case neg\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh\u271d : \u00ac\u2191m = 0\nthis\u271d : 1 < \u2191p\nthis : 0 \u2264 padicValRat p \u2191m\n\u22a2 -1 < -padicValRat p \u2191m \u2192 \u2191p ^ (-padicValRat p \u2191m) = 1", "state_after": "case neg\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh\u271d : \u00ac\u2191m = 0\nthis\u271d : 1 < \u2191p\nthis : 0 \u2264 padicValRat p \u2191m\nh : -1 < -padicValRat p \u2191m\n\u22a2 \u2191p ^ (-padicValRat p \u2191m) = 1"}, {"tactic": "rw [\u2190 zpow_zero (p : \u211a), zpow_inj] <;> linarith", "annotated_tactic": ["rw [\u2190 <a>zpow_zero</a> (p : \u211a), <a>zpow_inj</a>] <;> linarith", [{"full_name": "zpow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1015, 50], "def_end_pos": [1015, 59]}, {"full_name": "zpow_inj", "def_path": "Mathlib/Algebra/Order/Field/Power.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}]], "state_before": "case neg\np : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh\u271d : \u00ac\u2191m = 0\nthis\u271d : 1 < \u2191p\nthis : 0 \u2264 padicValRat p \u2191m\nh : -1 < -padicValRat p \u2191m\n\u22a2 \u2191p ^ (-padicValRat p \u2191m) = 1", "state_after": "no goals"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh\u271d : \u00ac\u2191m = 0\n\u22a2 1 < \u2191p", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh\u271d : \u00ac\u2191m = 0\n\u22a2 1 < p"}, {"tactic": "exact Nat.Prime.one_lt (Fact.out : Nat.Prime p)", "annotated_tactic": ["exact <a>Nat.Prime.one_lt</a> (<a>Fact.out</a> : <a>Nat.Prime</a> p)", [{"full_name": "Nat.Prime.one_lt", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [75, 9], "def_end_pos": [75, 21]}, {"full_name": "Fact.out", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [117, 3], "def_end_pos": [117, 6]}, {"full_name": "Nat.Prime", "def_path": "Mathlib/Data/Nat/Prime.lean", "def_pos": [45, 5], "def_end_pos": [45, 10]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh\u271d : \u00ac\u2191m = 0\n\u22a2 1 < p", "state_after": "no goals"}, {"tactic": "simp only [of_int, Nat.cast_nonneg]", "annotated_tactic": ["simp only [<a>of_int</a>, <a>Nat.cast_nonneg</a>]", [{"full_name": "padicValRat.of_int", "def_path": "Mathlib/NumberTheory/Padics/PadicVal.lean", "def_pos": [232, 9], "def_end_pos": [232, 15]}, {"full_name": "Nat.cast_nonneg", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [50, 9], "def_end_pos": [50, 20]}]], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nm : \u2124\nh\u271d : \u00ac\u2191m = 0\nthis : 1 < \u2191p\n\u22a2 0 \u2264 padicValRat p \u2191m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/DropRight.lean", "full_name": "List.rtake_nil", "start": [74, 1], "end": [74, 66], "traced_tactics": [{"tactic": "simp [rtake]", "annotated_tactic": ["simp [<a>rtake</a>]", [{"full_name": "List.rtake", "def_path": "Mathlib/Data/List/DropRight.lean", "def_pos": [69, 5], "def_end_pos": [69, 10]}]], "state_before": "\u03b1 : Type u_1\np : \u03b1 \u2192 Bool\nl : List \u03b1\nn : \u2115\n\u22a2 [].rtake n = []", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/ModelTheory/Substructures.lean", "full_name": "FirstOrder.Language.Substructure.comap_map_comap", "start": [504, 1], "end": [506, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "Submodule.mem_smul_top_iff", "start": [372, 1], "end": [378, 68], "traced_tactics": [{"tactic": "change _ \u2194 N.subtype x \u2208 I \u2022 N", "annotated_tactic": ["change _ \u2194 N.subtype x \u2208 I \u2022 N", []], "state_before": "R : Type u\nM : Type v\nM'\u271d : Type u_1\nF : Type u_2\nG : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid M'\u271d\ninst\u271d\u00b2 : Module R M'\u271d\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nN : Submodule R M\nx : \u21a5N\n\u22a2 x \u2208 I \u2022 \u22a4 \u2194 \u2191x \u2208 I \u2022 N", "state_after": "R : Type u\nM : Type v\nM'\u271d : Type u_1\nF : Type u_2\nG : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid M'\u271d\ninst\u271d\u00b2 : Module R M'\u271d\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nN : Submodule R M\nx : \u21a5N\n\u22a2 x \u2208 I \u2022 \u22a4 \u2194 N.subtype x \u2208 I \u2022 N"}, {"tactic": "have : Submodule.map N.subtype (I \u2022 \u22a4) = I \u2022 N := by\n  rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype]", "annotated_tactic": ["have : <a>Submodule.map</a> N.subtype (I \u2022 \u22a4) = I \u2022 N := by\n    rw [<a>Submodule.map_smul''</a>, <a>Submodule.map_top</a>, <a>Submodule.range_subtype</a>]", [{"full_name": "Submodule.map", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [48, 5], "def_end_pos": [48, 8]}, {"full_name": "Submodule.map_smul''", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [311, 9], "def_end_pos": [311, 19]}, {"full_name": "Submodule.map_top", "def_path": "Mathlib/Algebra/Module/Submodule/Range.lean", "def_pos": [290, 9], "def_end_pos": [290, 16]}, {"full_name": "Submodule.range_subtype", "def_path": "Mathlib/Algebra/Module/Submodule/Range.lean", "def_pos": [294, 9], "def_end_pos": [294, 22]}]], "state_before": "R : Type u\nM : Type v\nM'\u271d : Type u_1\nF : Type u_2\nG : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid M'\u271d\ninst\u271d\u00b2 : Module R M'\u271d\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nN : Submodule R M\nx : \u21a5N\n\u22a2 x \u2208 I \u2022 \u22a4 \u2194 N.subtype x \u2208 I \u2022 N", "state_after": "R : Type u\nM : Type v\nM'\u271d : Type u_1\nF : Type u_2\nG : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid M'\u271d\ninst\u271d\u00b2 : Module R M'\u271d\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nN : Submodule R M\nx : \u21a5N\nthis : map N.subtype (I \u2022 \u22a4) = I \u2022 N\n\u22a2 x \u2208 I \u2022 \u22a4 \u2194 N.subtype x \u2208 I \u2022 N"}, {"tactic": "rw [\u2190 this]", "annotated_tactic": ["rw [\u2190 this]", []], "state_before": "R : Type u\nM : Type v\nM'\u271d : Type u_1\nF : Type u_2\nG : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid M'\u271d\ninst\u271d\u00b2 : Module R M'\u271d\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nN : Submodule R M\nx : \u21a5N\nthis : map N.subtype (I \u2022 \u22a4) = I \u2022 N\n\u22a2 x \u2208 I \u2022 \u22a4 \u2194 N.subtype x \u2208 I \u2022 N", "state_after": "R : Type u\nM : Type v\nM'\u271d : Type u_1\nF : Type u_2\nG : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid M'\u271d\ninst\u271d\u00b2 : Module R M'\u271d\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nN : Submodule R M\nx : \u21a5N\nthis : map N.subtype (I \u2022 \u22a4) = I \u2022 N\n\u22a2 x \u2208 I \u2022 \u22a4 \u2194 N.subtype x \u2208 map N.subtype (I \u2022 \u22a4)"}, {"tactic": "exact (Function.Injective.mem_set_image N.injective_subtype).symm", "annotated_tactic": ["exact (<a>Function.Injective.mem_set_image</a> N.injective_subtype).<a>symm</a>", [{"full_name": "Function.Injective.mem_set_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [218, 9], "def_end_pos": [218, 48]}, {"full_name": "Iff.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [813, 9], "def_end_pos": [813, 17]}]], "state_before": "R : Type u\nM : Type v\nM'\u271d : Type u_1\nF : Type u_2\nG : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid M'\u271d\ninst\u271d\u00b2 : Module R M'\u271d\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nN : Submodule R M\nx : \u21a5N\nthis : map N.subtype (I \u2022 \u22a4) = I \u2022 N\n\u22a2 x \u2208 I \u2022 \u22a4 \u2194 N.subtype x \u2208 map N.subtype (I \u2022 \u22a4)", "state_after": "no goals"}, {"tactic": "rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype]", "annotated_tactic": ["rw [<a>Submodule.map_smul''</a>, <a>Submodule.map_top</a>, <a>Submodule.range_subtype</a>]", [{"full_name": "Submodule.map_smul''", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [311, 9], "def_end_pos": [311, 19]}, {"full_name": "Submodule.map_top", "def_path": "Mathlib/Algebra/Module/Submodule/Range.lean", "def_pos": [290, 9], "def_end_pos": [290, 16]}, {"full_name": "Submodule.range_subtype", "def_path": "Mathlib/Algebra/Module/Submodule/Range.lean", "def_pos": [294, 9], "def_end_pos": [294, 22]}]], "state_before": "R : Type u\nM : Type v\nM'\u271d : Type u_1\nF : Type u_2\nG : Type u_3\ninst\u271d\u2076 : CommSemiring R\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : Module R M\ninst\u271d\u00b3 : AddCommMonoid M'\u271d\ninst\u271d\u00b2 : Module R M'\u271d\nI J : Ideal R\nN\u271d P : Submodule R M\nS : Set R\nT : Set M\nM' : Type w\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nN : Submodule R M\nx : \u21a5N\n\u22a2 map N.subtype (I \u2022 \u22a4) = I \u2022 N", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Semicontinuous.lean", "full_name": "lowerSemicontinuousWithinAt_iff_le_liminf", "start": [320, 1], "end": [328, 69], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CompleteLinearOrder \u03b3\ninst\u271d : DenselyOrdered \u03b3\nf : \u03b1 \u2192 \u03b3\n\u22a2 LowerSemicontinuousWithinAt f s x \u2194 f x \u2264 liminf f (\ud835\udcdd[s] x)", "state_after": "case mp\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CompleteLinearOrder \u03b3\ninst\u271d : DenselyOrdered \u03b3\nf : \u03b1 \u2192 \u03b3\n\u22a2 LowerSemicontinuousWithinAt f s x \u2192 f x \u2264 liminf f (\ud835\udcdd[s] x)\n\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CompleteLinearOrder \u03b3\ninst\u271d : DenselyOrdered \u03b3\nf : \u03b1 \u2192 \u03b3\n\u22a2 f x \u2264 liminf f (\ud835\udcdd[s] x) \u2192 LowerSemicontinuousWithinAt f s x"}, {"tactic": "exact fun hf y ylt => eventually_lt_of_lt_liminf (ylt.trans_le hf)", "annotated_tactic": ["exact fun hf y ylt => <a>eventually_lt_of_lt_liminf</a> (ylt.trans_le hf)", [{"full_name": "Filter.eventually_lt_of_lt_liminf", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [1274, 9], "def_end_pos": [1274, 35]}]], "state_before": "case mpr\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CompleteLinearOrder \u03b3\ninst\u271d : DenselyOrdered \u03b3\nf : \u03b1 \u2192 \u03b3\n\u22a2 f x \u2264 liminf f (\ud835\udcdd[s] x) \u2192 LowerSemicontinuousWithinAt f s x", "state_after": "no goals"}, {"tactic": "intro hf", "annotated_tactic": ["intro hf", []], "state_before": "case mp\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CompleteLinearOrder \u03b3\ninst\u271d : DenselyOrdered \u03b3\nf : \u03b1 \u2192 \u03b3\n\u22a2 LowerSemicontinuousWithinAt f s x \u2192 f x \u2264 liminf f (\ud835\udcdd[s] x)", "state_after": "case mp\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CompleteLinearOrder \u03b3\ninst\u271d : DenselyOrdered \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\n\u22a2 f x \u2264 liminf f (\ud835\udcdd[s] x)"}, {"tactic": "unfold LowerSemicontinuousWithinAt at hf", "annotated_tactic": ["unfold <a>LowerSemicontinuousWithinAt</a> at hf", [{"full_name": "LowerSemicontinuousWithinAt", "def_path": "Mathlib/Topology/Semicontinuous.lean", "def_pos": [84, 5], "def_end_pos": [84, 32]}]], "state_before": "case mp\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CompleteLinearOrder \u03b3\ninst\u271d : DenselyOrdered \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : LowerSemicontinuousWithinAt f s x\n\u22a2 f x \u2264 liminf f (\ud835\udcdd[s] x)", "state_after": "case mp\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CompleteLinearOrder \u03b3\ninst\u271d : DenselyOrdered \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : \u2200 y < f x, \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < f x'\n\u22a2 f x \u2264 liminf f (\ud835\udcdd[s] x)"}, {"tactic": "contrapose! hf", "annotated_tactic": ["contrapose! hf", []], "state_before": "case mp\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CompleteLinearOrder \u03b3\ninst\u271d : DenselyOrdered \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : \u2200 y < f x, \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < f x'\n\u22a2 f x \u2264 liminf f (\ud835\udcdd[s] x)", "state_after": "case mp\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CompleteLinearOrder \u03b3\ninst\u271d : DenselyOrdered \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : liminf f (\ud835\udcdd[s] x) < f x\n\u22a2 \u2203 y < f x, \u00ac\u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < f x'"}, {"tactic": "obtain \u27e8y, lty, ylt\u27e9 := exists_between hf", "annotated_tactic": ["obtain \u27e8y, lty, ylt\u27e9 := <a>exists_between</a> hf", [{"full_name": "exists_between", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [1369, 9], "def_end_pos": [1369, 23]}]], "state_before": "case mp\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CompleteLinearOrder \u03b3\ninst\u271d : DenselyOrdered \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : liminf f (\ud835\udcdd[s] x) < f x\n\u22a2 \u2203 y < f x, \u00ac\u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < f x'", "state_after": "case mp.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CompleteLinearOrder \u03b3\ninst\u271d : DenselyOrdered \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : liminf f (\ud835\udcdd[s] x) < f x\ny : \u03b3\nlty : liminf f (\ud835\udcdd[s] x) < y\nylt : y < f x\n\u22a2 \u2203 y < f x, \u00ac\u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < f x'"}, {"tactic": "use y", "annotated_tactic": ["use y", []], "state_before": "case mp.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CompleteLinearOrder \u03b3\ninst\u271d : DenselyOrdered \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : liminf f (\ud835\udcdd[s] x) < f x\ny : \u03b3\nlty : liminf f (\ud835\udcdd[s] x) < y\nylt : y < f x\n\u22a2 \u2203 y < f x, \u00ac\u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < f x'", "state_after": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CompleteLinearOrder \u03b3\ninst\u271d : DenselyOrdered \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : liminf f (\ud835\udcdd[s] x) < f x\ny : \u03b3\nlty : liminf f (\ud835\udcdd[s] x) < y\nylt : y < f x\n\u22a2 y < f x \u2227 \u00ac\u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < f x'"}, {"tactic": "exact \u27e8ylt, fun h => lty.not_le\n  (le_liminf_of_le (by isBoundedDefault) (h.mono fun _ hx => le_of_lt hx))\u27e9", "annotated_tactic": ["exact \u27e8ylt, fun h => lty.not_le\n      (<a>le_liminf_of_le</a> (by isBoundedDefault) (h.mono fun _ hx => <a>le_of_lt</a> hx))\u27e9", [{"full_name": "Filter.le_liminf_of_le", "def_path": "Mathlib/Order/LiminfLimsup.lean", "def_pos": [545, 9], "def_end_pos": [545, 24]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case h\n\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CompleteLinearOrder \u03b3\ninst\u271d : DenselyOrdered \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : liminf f (\ud835\udcdd[s] x) < f x\ny : \u03b3\nlty : liminf f (\ud835\udcdd[s] x) < y\nylt : y < f x\n\u22a2 y < f x \u2227 \u00ac\u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < f x'", "state_after": "no goals"}, {"tactic": "isBoundedDefault", "annotated_tactic": ["isBoundedDefault", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b3 : TopologicalSpace \u03b1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b2\nf\u271d g : \u03b1 \u2192 \u03b2\nx : \u03b1\ns t : Set \u03b1\ny\u271d z : \u03b2\n\u03b3 : Type u_3\ninst\u271d\u00b9 : CompleteLinearOrder \u03b3\ninst\u271d : DenselyOrdered \u03b3\nf : \u03b1 \u2192 \u03b3\nhf : liminf f (\ud835\udcdd[s] x) < f x\ny : \u03b3\nlty : liminf f (\ud835\udcdd[s] x) < y\nylt : y < f x\nh : \u2200\u1da0 (x' : \u03b1) in \ud835\udcdd[s] x, y < f x'\n\u22a2 IsCoboundedUnder (fun x x_1 => x \u2265 x_1) (\ud835\udcdd[s] x) f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Basic.lean", "full_name": "Finset.filter_and_not", "start": [2814, 1], "end": [2816, 88], "traced_tactics": [{"tactic": "rw [filter_and, filter_not, \u2190 inter_sdiff_assoc, inter_eq_left.2 (filter_subset _ _)]", "annotated_tactic": ["rw [<a>filter_and</a>, <a>filter_not</a>, \u2190 <a>inter_sdiff_assoc</a>, <a>inter_eq_left</a>.2 (<a>filter_subset</a> _ _)]", [{"full_name": "Finset.filter_and", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2804, 9], "def_end_pos": [2804, 19]}, {"full_name": "Finset.filter_not", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2808, 9], "def_end_pos": [2808, 19]}, {"full_name": "Finset.inter_sdiff_assoc", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2135, 7], "def_end_pos": [2135, 24]}, {"full_name": "Finset.inter_eq_left", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1830, 15], "def_end_pos": [1830, 28]}, {"full_name": "Finset.filter_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2586, 9], "def_end_pos": [2586, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\np\u271d q\u271d : \u03b1 \u2192 Prop\ninst\u271d\u2074 : DecidablePred p\u271d\ninst\u271d\u00b3 : DecidablePred q\u271d\ns\u271d : Finset \u03b1\ninst\u271d\u00b2 : DecidableEq \u03b1\ns : Finset \u03b1\np q : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidablePred q\n\u22a2 filter (fun a => p a \u2227 \u00acq a) s = filter p s \\ filter q s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Submodule.lean", "full_name": "LieSubmodule.incl_eq_val", "start": [678, 1], "end": [679, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Vector/Basic.lean", "full_name": "Vector.map\u2082_nil", "start": [135, 1], "end": [136, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Kernel/WithDensity.lean", "full_name": "ProbabilityTheory.kernel.withDensity_sub_add_cancel", "start": [156, 1], "end": [164, 59], "traced_tactics": [{"tactic": "rw [\u2190 withDensity_add_right _ hg]", "annotated_tactic": ["rw [\u2190 <a>withDensity_add_right</a> _ hg]", [{"full_name": "ProbabilityTheory.kernel.withDensity_add_right", "def_path": "Mathlib/Probability/Kernel/WithDensity.lean", "def_pos": [147, 7], "def_end_pos": [147, 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 : \u21a5(kernel \u03b1 \u03b2)\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : IsSFiniteKernel \u03ba\nf g : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (Function.uncurry f)\nhg : Measurable (Function.uncurry g)\nhfg : \u2200 (a : \u03b1), g a \u2264\u1da0[ae (\u03ba a)] f a\n\u22a2 (withDensity \u03ba fun a x => f a x - g a x) + withDensity \u03ba g = withDensity \u03ba f", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : \u21a5(kernel \u03b1 \u03b2)\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : IsSFiniteKernel \u03ba\nf g : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (Function.uncurry f)\nhg : Measurable (Function.uncurry g)\nhfg : \u2200 (a : \u03b1), g a \u2264\u1da0[ae (\u03ba a)] f a\n\u22a2 withDensity \u03ba ((fun a x => f a x - g a x) + g) = withDensity \u03ba f\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 : \u21a5(kernel \u03b1 \u03b2)\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : IsSFiniteKernel \u03ba\nf g : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (Function.uncurry f)\nhg : Measurable (Function.uncurry g)\nhfg : \u2200 (a : \u03b1), g a \u2264\u1da0[ae (\u03ba a)] f a\n\u22a2 Measurable (Function.uncurry fun a x => f a x - g a x)"}, {"tactic": "swap", "annotated_tactic": ["swap", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : \u21a5(kernel \u03b1 \u03b2)\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : IsSFiniteKernel \u03ba\nf g : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (Function.uncurry f)\nhg : Measurable (Function.uncurry g)\nhfg : \u2200 (a : \u03b1), g a \u2264\u1da0[ae (\u03ba a)] f a\n\u22a2 withDensity \u03ba ((fun a x => f a x - g a x) + g) = withDensity \u03ba f\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 : \u21a5(kernel \u03b1 \u03b2)\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : IsSFiniteKernel \u03ba\nf g : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (Function.uncurry f)\nhg : Measurable (Function.uncurry g)\nhfg : \u2200 (a : \u03b1), g a \u2264\u1da0[ae (\u03ba a)] f a\n\u22a2 Measurable (Function.uncurry fun a x => f a x - g a x)", "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 : \u21a5(kernel \u03b1 \u03b2)\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : IsSFiniteKernel \u03ba\nf g : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (Function.uncurry f)\nhg : Measurable (Function.uncurry g)\nhfg : \u2200 (a : \u03b1), g a \u2264\u1da0[ae (\u03ba a)] f a\n\u22a2 Measurable (Function.uncurry fun a x => f a x - g a x)\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 : \u21a5(kernel \u03b1 \u03b2)\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : IsSFiniteKernel \u03ba\nf g : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (Function.uncurry f)\nhg : Measurable (Function.uncurry g)\nhfg : \u2200 (a : \u03b1), g a \u2264\u1da0[ae (\u03ba a)] f a\n\u22a2 withDensity \u03ba ((fun a x => f a x - g a x) + g) = withDensity \u03ba f"}, {"tactic": "refine withDensity_congr_ae \u03ba ((hf.sub hg).add hg) hf (fun a \u21a6 ?_)", "annotated_tactic": ["refine <a>withDensity_congr_ae</a> \u03ba ((hf.sub hg).<a>add</a> hg) hf (fun a \u21a6 ?_)", [{"full_name": "ProbabilityTheory.kernel.withDensity_congr_ae", "def_path": "Mathlib/Probability/Kernel/WithDensity.lean", "def_pos": [74, 14], "def_end_pos": [74, 34]}, {"full_name": "Measurable.add", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [139, 3], "def_end_pos": [139, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : \u21a5(kernel \u03b1 \u03b2)\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : IsSFiniteKernel \u03ba\nf g : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (Function.uncurry f)\nhg : Measurable (Function.uncurry g)\nhfg : \u2200 (a : \u03b1), g a \u2264\u1da0[ae (\u03ba a)] f a\n\u22a2 withDensity \u03ba ((fun a x => f a x - g a x) + g) = withDensity \u03ba f", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : \u21a5(kernel \u03b1 \u03b2)\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : IsSFiniteKernel \u03ba\nf g : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (Function.uncurry f)\nhg : Measurable (Function.uncurry g)\nhfg : \u2200 (a : \u03b1), g a \u2264\u1da0[ae (\u03ba a)] f a\na : \u03b1\n\u22a2 ((fun a x => f a x - g a x) + g) a =\u1da0[ae (\u03ba a)] f a"}, {"tactic": "filter_upwards [hfg a] with x hx", "annotated_tactic": ["filter_upwards [hfg a] with x hx", []], "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 : \u21a5(kernel \u03b1 \u03b2)\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : IsSFiniteKernel \u03ba\nf g : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (Function.uncurry f)\nhg : Measurable (Function.uncurry g)\nhfg : \u2200 (a : \u03b1), g a \u2264\u1da0[ae (\u03ba a)] f a\na : \u03b1\n\u22a2 ((fun a x => f a x - g a x) + g) a =\u1da0[ae (\u03ba a)] f a", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : \u21a5(kernel \u03b1 \u03b2)\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : IsSFiniteKernel \u03ba\nf g : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (Function.uncurry f)\nhg : Measurable (Function.uncurry g)\nhfg : \u2200 (a : \u03b1), g a \u2264\u1da0[ae (\u03ba a)] f a\na : \u03b1\nx : \u03b2\nhx : g a x \u2264 f a x\n\u22a2 ((fun a x => f a x - g a x) + g) a x = f a x"}, {"tactic": "rwa [Pi.add_apply, Pi.add_apply, tsub_add_cancel_iff_le]", "annotated_tactic": ["rwa [<a>Pi.add_apply</a>, <a>Pi.add_apply</a>, <a>tsub_add_cancel_iff_le</a>]", [{"full_name": "Pi.add_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [96, 3], "def_end_pos": [96, 14]}, {"full_name": "Pi.add_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [96, 3], "def_end_pos": [96, 14]}, {"full_name": "tsub_add_cancel_iff_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [326, 9], "def_end_pos": [326, 31]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b9 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\n\u03ba : \u21a5(kernel \u03b1 \u03b2)\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : IsSFiniteKernel \u03ba\nf g : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (Function.uncurry f)\nhg : Measurable (Function.uncurry g)\nhfg : \u2200 (a : \u03b1), g a \u2264\u1da0[ae (\u03ba a)] f a\na : \u03b1\nx : \u03b2\nhx : g a x \u2264 f a x\n\u22a2 ((fun a x => f a x - g a x) + g) a x = f a x", "state_after": "no goals"}, {"tactic": "exact hf.sub hg", "annotated_tactic": ["exact hf.sub hg", []], "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 : \u21a5(kernel \u03b1 \u03b2)\nf\u271d : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\ninst\u271d : IsSFiniteKernel \u03ba\nf g : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e\nhf : Measurable (Function.uncurry f)\nhg : Measurable (Function.uncurry g)\nhfg : \u2200 (a : \u03b1), g a \u2264\u1da0[ae (\u03ba a)] f a\n\u22a2 Measurable (Function.uncurry fun a x => f a x - g a x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/NhdsSet.lean", "full_name": "nhdsSet_interior", "start": [119, 1], "end": [120, 29], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Category/Profinite/Basic.lean", "full_name": "Profinite.forget_ContinuousMap_mk", "start": [87, 1], "end": [90, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Asymptotics/Theta.lean", "full_name": "Asymptotics.IsTheta.isBigO_symm", "start": [57, 1], "end": [57, 61], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/Pullbacks.lean", "full_name": "AlgebraicGeometry.Scheme.Pullback.gluedLift_p1", "start": [314, 1], "end": [320, 32], "traced_tactics": [{"tactic": "rw [\u2190 cancel_epi (\ud835\udcb0.pullbackCover s.fst).fromGlued]", "annotated_tactic": ["rw [\u2190 <a>cancel_epi</a> (\ud835\udcb0.pullbackCover s.fst).<a>fromGlued</a>]", [{"full_name": "CategoryTheory.cancel_epi", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 19]}, {"full_name": "AlgebraicGeometry.Scheme.OpenCover.fromGlued", "def_path": "Mathlib/AlgebraicGeometry/Gluing.lean", "def_pos": [358, 5], "def_end_pos": [358, 14]}]], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : X.OpenCover\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (\ud835\udcb0.map i \u226b f) g\ns : PullbackCone f g\n\u22a2 gluedLift \ud835\udcb0 f g s \u226b p1 \ud835\udcb0 f g = s.fst", "state_after": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : X.OpenCover\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (\ud835\udcb0.map i \u226b f) g\ns : PullbackCone f g\n\u22a2 (\ud835\udcb0.pullbackCover s.fst).fromGlued \u226b gluedLift \ud835\udcb0 f g s \u226b p1 \ud835\udcb0 f g = (\ud835\udcb0.pullbackCover s.fst).fromGlued \u226b s.fst"}, {"tactic": "apply Multicoequalizer.hom_ext", "annotated_tactic": ["apply <a>Multicoequalizer.hom_ext</a>", [{"full_name": "CategoryTheory.Limits.Multicoequalizer.hom_ext", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean", "def_pos": [885, 9], "def_end_pos": [885, 16]}]], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : X.OpenCover\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (\ud835\udcb0.map i \u226b f) g\ns : PullbackCone f g\n\u22a2 (\ud835\udcb0.pullbackCover s.fst).fromGlued \u226b gluedLift \ud835\udcb0 f g s \u226b p1 \ud835\udcb0 f g = (\ud835\udcb0.pullbackCover s.fst).fromGlued \u226b s.fst", "state_after": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : X.OpenCover\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (\ud835\udcb0.map i \u226b f) g\ns : PullbackCone f g\n\u22a2 \u2200 (b : (\ud835\udcb0.pullbackCover s.fst).gluedCover.diagram.R),\n    Multicoequalizer.\u03c0 (\ud835\udcb0.pullbackCover s.fst).gluedCover.diagram b \u226b\n        (\ud835\udcb0.pullbackCover s.fst).fromGlued \u226b gluedLift \ud835\udcb0 f g s \u226b p1 \ud835\udcb0 f g =\n      Multicoequalizer.\u03c0 (\ud835\udcb0.pullbackCover s.fst).gluedCover.diagram b \u226b (\ud835\udcb0.pullbackCover s.fst).fromGlued \u226b s.fst"}, {"tactic": "intro b", "annotated_tactic": ["intro b", []], "state_before": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : X.OpenCover\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (\ud835\udcb0.map i \u226b f) g\ns : PullbackCone f g\n\u22a2 \u2200 (b : (\ud835\udcb0.pullbackCover s.fst).gluedCover.diagram.R),\n    Multicoequalizer.\u03c0 (\ud835\udcb0.pullbackCover s.fst).gluedCover.diagram b \u226b\n        (\ud835\udcb0.pullbackCover s.fst).fromGlued \u226b gluedLift \ud835\udcb0 f g s \u226b p1 \ud835\udcb0 f g =\n      Multicoequalizer.\u03c0 (\ud835\udcb0.pullbackCover s.fst).gluedCover.diagram b \u226b (\ud835\udcb0.pullbackCover s.fst).fromGlued \u226b s.fst", "state_after": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : X.OpenCover\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (\ud835\udcb0.map i \u226b f) g\ns : PullbackCone f g\nb : (\ud835\udcb0.pullbackCover s.fst).gluedCover.diagram.R\n\u22a2 Multicoequalizer.\u03c0 (\ud835\udcb0.pullbackCover s.fst).gluedCover.diagram b \u226b\n      (\ud835\udcb0.pullbackCover s.fst).fromGlued \u226b gluedLift \ud835\udcb0 f g s \u226b p1 \ud835\udcb0 f g =\n    Multicoequalizer.\u03c0 (\ud835\udcb0.pullbackCover s.fst).gluedCover.diagram b \u226b (\ud835\udcb0.pullbackCover s.fst).fromGlued \u226b s.fst"}, {"tactic": "simp_rw [OpenCover.fromGlued, Multicoequalizer.\u03c0_desc_assoc, gluedLift, \u2190 Category.assoc]", "annotated_tactic": ["simp_rw [<a>OpenCover.fromGlued</a>, <a>Multicoequalizer.\u03c0_desc_assoc</a>, <a>gluedLift</a>, \u2190 <a>Category.assoc</a>]", [{"full_name": "AlgebraicGeometry.Scheme.OpenCover.fromGlued", "def_path": "Mathlib/AlgebraicGeometry/Gluing.lean", "def_pos": [358, 5], "def_end_pos": [358, 14]}, {"full_name": "CategoryTheory.Limits.Multicoequalizer.\u03c0_desc_assoc", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean", "def_pos": [877, 3], "def_end_pos": [877, 10]}, {"full_name": "AlgebraicGeometry.Scheme.Pullback.gluedLift", "def_path": "Mathlib/AlgebraicGeometry/Pullbacks.lean", "def_pos": [296, 5], "def_end_pos": [296, 14]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}]], "state_before": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : X.OpenCover\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (\ud835\udcb0.map i \u226b f) g\ns : PullbackCone f g\nb : (\ud835\udcb0.pullbackCover s.fst).gluedCover.diagram.R\n\u22a2 Multicoequalizer.\u03c0 (\ud835\udcb0.pullbackCover s.fst).gluedCover.diagram b \u226b\n      (\ud835\udcb0.pullbackCover s.fst).fromGlued \u226b gluedLift \ud835\udcb0 f g s \u226b p1 \ud835\udcb0 f g =\n    Multicoequalizer.\u03c0 (\ud835\udcb0.pullbackCover s.fst).gluedCover.diagram b \u226b (\ud835\udcb0.pullbackCover s.fst).fromGlued \u226b s.fst", "state_after": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : X.OpenCover\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (\ud835\udcb0.map i \u226b f) g\ns : PullbackCone f g\nb : (\ud835\udcb0.pullbackCover s.fst).gluedCover.diagram.R\n\u22a2 ((\ud835\udcb0.pullbackCover s.fst).map b \u226b\n        (\ud835\udcb0.pullbackCover s.fst).glueMorphisms\n          (fun i =>\n            ((pullbackSymmetry s.fst (\ud835\udcb0.map i)).hom \u226b\n                pullback.map (\ud835\udcb0.map i) s.fst (\ud835\udcb0.map i \u226b f) g (\ud835\udfd9 (\ud835\udcb0.obj i)) s.snd f \u22ef \u22ef) \u226b\n              (gluing \ud835\udcb0 f g).\u03b9 i)\n          \u22ef) \u226b\n      p1 \ud835\udcb0 f g =\n    (\ud835\udcb0.pullbackCover s.fst).map b \u226b s.fst"}, {"tactic": "simp_rw [(\ud835\udcb0.pullbackCover s.fst).\u03b9_glueMorphisms]", "annotated_tactic": ["simp_rw [(\ud835\udcb0.pullbackCover s.fst).<a>\u03b9_glueMorphisms</a>]", [{"full_name": "AlgebraicGeometry.Scheme.OpenCover.\u03b9_glueMorphisms", "def_path": "Mathlib/AlgebraicGeometry/Gluing.lean", "def_pos": [466, 9], "def_end_pos": [466, 24]}]], "state_before": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : X.OpenCover\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (\ud835\udcb0.map i \u226b f) g\ns : PullbackCone f g\nb : (\ud835\udcb0.pullbackCover s.fst).gluedCover.diagram.R\n\u22a2 ((\ud835\udcb0.pullbackCover s.fst).map b \u226b\n        (\ud835\udcb0.pullbackCover s.fst).glueMorphisms\n          (fun i =>\n            ((pullbackSymmetry s.fst (\ud835\udcb0.map i)).hom \u226b\n                pullback.map (\ud835\udcb0.map i) s.fst (\ud835\udcb0.map i \u226b f) g (\ud835\udfd9 (\ud835\udcb0.obj i)) s.snd f \u22ef \u22ef) \u226b\n              (gluing \ud835\udcb0 f g).\u03b9 i)\n          \u22ef) \u226b\n      p1 \ud835\udcb0 f g =\n    (\ud835\udcb0.pullbackCover s.fst).map b \u226b s.fst", "state_after": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : X.OpenCover\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (\ud835\udcb0.map i \u226b f) g\ns : PullbackCone f g\nb : (\ud835\udcb0.pullbackCover s.fst).gluedCover.diagram.R\n\u22a2 (((pullbackSymmetry s.fst (\ud835\udcb0.map b)).hom \u226b pullback.map (\ud835\udcb0.map b) s.fst (\ud835\udcb0.map b \u226b f) g (\ud835\udfd9 (\ud835\udcb0.obj b)) s.snd f \u22ef \u22ef) \u226b\n        (gluing \ud835\udcb0 f g).\u03b9 b) \u226b\n      p1 \ud835\udcb0 f g =\n    (\ud835\udcb0.pullbackCover s.fst).map b \u226b s.fst"}, {"tactic": "simp [p1, pullback.condition]", "annotated_tactic": ["simp [<a>p1</a>, <a>pullback.condition</a>]", [{"full_name": "AlgebraicGeometry.Scheme.Pullback.p1", "def_path": "Mathlib/AlgebraicGeometry/Pullbacks.lean", "def_pos": [238, 5], "def_end_pos": [238, 7]}, {"full_name": "CategoryTheory.Limits.pullback.condition", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean", "def_pos": [1245, 9], "def_end_pos": [1245, 27]}]], "state_before": "case h\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nX Y Z : Scheme\n\ud835\udcb0 : X.OpenCover\nf : X \u27f6 Z\ng : Y \u27f6 Z\ninst\u271d : \u2200 (i : \ud835\udcb0.J), HasPullback (\ud835\udcb0.map i \u226b f) g\ns : PullbackCone f g\nb : (\ud835\udcb0.pullbackCover s.fst).gluedCover.diagram.R\n\u22a2 (((pullbackSymmetry s.fst (\ud835\udcb0.map b)).hom \u226b pullback.map (\ud835\udcb0.map b) s.fst (\ud835\udcb0.map b \u226b f) g (\ud835\udfd9 (\ud835\udcb0.obj b)) s.snd f \u22ef \u22ef) \u226b\n        (gluing \ud835\udcb0 f g).\u03b9 b) \u226b\n      p1 \ud835\udcb0 f g =\n    (\ud835\udcb0.pullbackCover s.fst).map b \u226b s.fst", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/Support.lean", "full_name": "Equiv.Perm.Disjoint.inv_right", "start": [99, 1], "end": [100, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Tactic/Ring/Basic.lean", "full_name": "Mathlib.Tactic.Ring.neg_zero", "start": [556, 1], "end": [556, 56], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "u : Lean.Level\nR\u271d : Type ?u.118732\n\u03b1 : Q(Type u)\ns\u03b1 : Q(CommSemiring \u00ab$\u03b1\u00bb)\ninst\u271d\u00b9 : CommSemiring R\u271d\nR : Type u_1\ninst\u271d : Ring R\n\u22a2 -0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/TensorPower.lean", "full_name": "TensorPower.cast_tprod", "start": [108, 1], "end": [110, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Relation.lean", "full_name": "Relation.transGen_eq_self", "start": [494, 1], "end": [499, 64], "traced_tactics": [{"tactic": "induction h with\n| single hc => exact hc\n| tail _ hcd hac => exact trans hac hcd", "annotated_tactic": ["induction h with\n      | <a>single</a> hc => exact hc\n      | <a>tail</a> _ hcd hac => exact trans hac hcd", [{"full_name": "Relation.TransGen.single", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [268, 5], "def_end_pos": [268, 11]}, {"full_name": "Relation.TransGen.tail", "def_path": "Mathlib/Logic/Relation.lean", "def_pos": [269, 5], "def_end_pos": [269, 9]}]], "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\u03b6 : Type u_6\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d b\u271d c d : \u03b1\ntrans : Transitive r\na b : \u03b1\nh : TransGen r a b\n\u22a2 r a b", "state_after": "no goals"}, {"tactic": "exact hc", "annotated_tactic": ["exact hc", []], "state_before": "case single\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\u03b6 : Type u_6\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d b\u271d\u00b9 c d : \u03b1\ntrans : Transitive r\na b b\u271d : \u03b1\nhc : r a b\u271d\n\u22a2 r a b\u271d", "state_after": "no goals"}, {"tactic": "exact trans hac hcd", "annotated_tactic": ["exact trans hac hcd", []], "state_before": "case tail\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\u03b6 : Type u_6\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\na\u271d\u00b9 b\u271d\u00b9 c d : \u03b1\ntrans : Transitive r\na b b\u271d c\u271d : \u03b1\na\u271d : TransGen r a b\u271d\nhcd : r b\u271d c\u271d\nhac : r a b\u271d\n\u22a2 r a c\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Factorization/Basic.lean", "full_name": "Nat.factorization_lt", "start": [393, 1], "end": [397, 73], "traced_tactics": [{"tactic": "by_cases pp : p.Prime", "annotated_tactic": ["by_cases pp : p.Prime", []], "state_before": "a b m n\u271d p\u271d n p : \u2115\nhn : n \u2260 0\n\u22a2 n.factorization p < n", "state_after": "case pos\na b m n\u271d p\u271d n p : \u2115\nhn : n \u2260 0\npp : Prime p\n\u22a2 n.factorization p < n\n\ncase neg\na b m n\u271d p\u271d n p : \u2115\nhn : n \u2260 0\npp : \u00acPrime p\n\u22a2 n.factorization p < n"}, {"tactic": "exact (pow_lt_pow_iff_right pp.one_lt).1 <| (ord_proj_le p hn).trans_lt <|\n  lt_pow_self pp.one_lt _", "annotated_tactic": ["exact (<a>pow_lt_pow_iff_right</a> pp.one_lt).1 <| (<a>ord_proj_le</a> p hn).<a>trans_lt</a> <|\n      <a>lt_pow_self</a> pp.one_lt _", [{"full_name": "pow_lt_pow_iff_right", "def_path": "Mathlib/Algebra/Order/Ring/Basic.lean", "def_pos": [155, 7], "def_end_pos": [155, 27]}, {"full_name": "Nat.ord_proj_le", "def_path": "Mathlib/Data/Nat/Factorization/Basic.lean", "def_pos": [357, 9], "def_end_pos": [357, 20]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [119, 7], "def_end_pos": [119, 21]}, {"full_name": "Nat.lt_pow_self", "def_path": "Mathlib/Data/Nat/Defs.lean", "def_pos": [774, 7], "def_end_pos": [774, 18]}]], "state_before": "case pos\na b m n\u271d p\u271d n p : \u2115\nhn : n \u2260 0\npp : Prime p\n\u22a2 n.factorization p < n", "state_after": "no goals"}, {"tactic": "simpa only [factorization_eq_zero_of_non_prime n pp] using hn.bot_lt", "annotated_tactic": ["simpa only [<a>factorization_eq_zero_of_non_prime</a> n pp] using hn.bot_lt", [{"full_name": "Nat.factorization_eq_zero_of_non_prime", "def_path": "Mathlib/Data/Nat/Factorization/Basic.lean", "def_pos": [139, 9], "def_end_pos": [139, 43]}]], "state_before": "case neg\na b m n\u271d p\u271d n p : \u2115\nhn : n \u2260 0\npp : \u00acPrime p\n\u22a2 n.factorization p < n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "full_name": "MeasureTheory.lintegral_prod_swap", "start": [932, 1], "end": [934, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/AtTopBot.lean", "full_name": "Filter.tendsto_atBot_atBot", "start": [1410, 1], "end": [1412, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Setoid/Partition.lean", "full_name": "Setoid.empty_not_mem_classes", "start": [102, 1], "end": [103, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Cardinal/ENat.lean", "full_name": "Cardinal.ofENat_mul_aleph0", "start": [270, 1], "end": [273, 59], "traced_tactics": [{"tactic": "induction m with\n| top => exact aleph0_mul_aleph0\n| coe m => rw [ofENat_nat, nat_mul_aleph0 (mod_cast hm)]", "annotated_tactic": ["induction m with\n  | top => exact <a>aleph0_mul_aleph0</a>\n  | coe m => rw [<a>ofENat_nat</a>, <a>nat_mul_aleph0</a> (mod_cast hm)]", [{"full_name": "Cardinal.aleph0_mul_aleph0", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1771, 9], "def_end_pos": [1771, 26]}, {"full_name": "Cardinal.ofENat_nat", "def_path": "Mathlib/SetTheory/Cardinal/ENat.lean", "def_pos": [46, 26], "def_end_pos": [46, 36]}, {"full_name": "Cardinal.nat_mul_aleph0", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1776, 9], "def_end_pos": [1776, 23]}]], "state_before": "m : \u2115\u221e\nhm : m \u2260 0\n\u22a2 \u2191m * \u2135\u2080 = \u2135\u2080", "state_after": "no goals"}, {"tactic": "exact aleph0_mul_aleph0", "annotated_tactic": ["exact <a>aleph0_mul_aleph0</a>", [{"full_name": "Cardinal.aleph0_mul_aleph0", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1771, 9], "def_end_pos": [1771, 26]}]], "state_before": "case top\nhm : \u22a4 \u2260 0\n\u22a2 \u2191\u22a4 * \u2135\u2080 = \u2135\u2080", "state_after": "no goals"}, {"tactic": "rw [ofENat_nat, nat_mul_aleph0 (mod_cast hm)]", "annotated_tactic": ["rw [<a>ofENat_nat</a>, <a>nat_mul_aleph0</a> (mod_cast hm)]", [{"full_name": "Cardinal.ofENat_nat", "def_path": "Mathlib/SetTheory/Cardinal/ENat.lean", "def_pos": [46, 26], "def_end_pos": [46, 36]}, {"full_name": "Cardinal.nat_mul_aleph0", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1776, 9], "def_end_pos": [1776, 23]}]], "state_before": "case coe\nm : \u2115\nhm : \u2191m \u2260 0\n\u22a2 \u2191\u2191m * \u2135\u2080 = \u2135\u2080", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Field/Basic.lean", "full_name": "ofLex_ratCast", "start": [394, 1], "end": [394, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/ComposableArrows.lean", "full_name": "CategoryTheory.ComposableArrows.ext\u2082", "start": [603, 1], "end": [607, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/Hyperreal.lean", "full_name": "Hyperreal.st_eq_sSup", "start": [324, 1], "end": [334, 31], "traced_tactics": [{"tactic": "rcases _root_.em (Infinite x) with (hx|hx)", "annotated_tactic": ["rcases <a>_root_.em</a> (<a>Infinite</a> x) with (hx|hx)", [{"full_name": "em", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [195, 7], "def_end_pos": [195, 9]}, {"full_name": "Hyperreal.Infinite", "def_path": "Mathlib/Data/Real/Hyperreal.lean", "def_pos": [246, 5], "def_end_pos": [246, 13]}]], "state_before": "x : \u211d*\n\u22a2 x.st = sSup {y | \u2191y < x}", "state_after": "case inl\nx : \u211d*\nhx : x.Infinite\n\u22a2 x.st = sSup {y | \u2191y < x}\n\ncase inr\nx : \u211d*\nhx : \u00acx.Infinite\n\u22a2 x.st = sSup {y | \u2191y < x}"}, {"tactic": "rw [hx.st_eq]", "annotated_tactic": ["rw [hx.st_eq]", []], "state_before": "case inl\nx : \u211d*\nhx : x.Infinite\n\u22a2 x.st = sSup {y | \u2191y < x}", "state_after": "case inl\nx : \u211d*\nhx : x.Infinite\n\u22a2 0 = sSup {y | \u2191y < x}"}, {"tactic": "cases hx with\n| inl hx =>\n  convert Real.sSup_univ.symm\n  exact Set.eq_univ_of_forall hx\n| inr hx =>\n  convert Real.sSup_empty.symm\n  exact Set.eq_empty_of_forall_not_mem fun y hy \u21a6 hy.out.not_lt (hx _)", "annotated_tactic": ["cases hx with\n    | <a>inl</a> hx =>\n      convert Real.sSup_univ.symm\n      exact <a>Set.eq_univ_of_forall</a> hx\n    | <a>inr</a> hx =>\n      convert Real.sSup_empty.symm\n      exact <a>Set.eq_empty_of_forall_not_mem</a> fun y hy \u21a6 hy.out.not_lt (hx _)", [{"full_name": "Or.inl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [534, 5], "def_end_pos": [534, 8]}, {"full_name": "Set.eq_univ_of_forall", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [672, 9], "def_end_pos": [672, 26]}, {"full_name": "Or.inr", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [536, 5], "def_end_pos": [536, 8]}, {"full_name": "Set.eq_empty_of_forall_not_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [556, 9], "def_end_pos": [556, 35]}]], "state_before": "case inl\nx : \u211d*\nhx : x.Infinite\n\u22a2 0 = sSup {y | \u2191y < x}", "state_after": "no goals"}, {"tactic": "convert Real.sSup_univ.symm", "annotated_tactic": ["convert Real.sSup_univ.symm", []], "state_before": "case inl.inl\nx : \u211d*\nhx : x.InfinitePos\n\u22a2 0 = sSup {y | \u2191y < x}", "state_after": "case h.e'_3.h.e'_3\nx : \u211d*\nhx : x.InfinitePos\n\u22a2 {y | \u2191y < x} = Set.univ"}, {"tactic": "exact Set.eq_univ_of_forall hx", "annotated_tactic": ["exact <a>Set.eq_univ_of_forall</a> hx", [{"full_name": "Set.eq_univ_of_forall", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [672, 9], "def_end_pos": [672, 26]}]], "state_before": "case h.e'_3.h.e'_3\nx : \u211d*\nhx : x.InfinitePos\n\u22a2 {y | \u2191y < x} = Set.univ", "state_after": "no goals"}, {"tactic": "convert Real.sSup_empty.symm", "annotated_tactic": ["convert Real.sSup_empty.symm", []], "state_before": "case inl.inr\nx : \u211d*\nhx : x.InfiniteNeg\n\u22a2 0 = sSup {y | \u2191y < x}", "state_after": "case h.e'_3.h.e'_3\nx : \u211d*\nhx : x.InfiniteNeg\n\u22a2 {y | \u2191y < x} = \u2205"}, {"tactic": "exact Set.eq_empty_of_forall_not_mem fun y hy \u21a6 hy.out.not_lt (hx _)", "annotated_tactic": ["exact <a>Set.eq_empty_of_forall_not_mem</a> fun y hy \u21a6 hy.out.not_lt (hx _)", [{"full_name": "Set.eq_empty_of_forall_not_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [556, 9], "def_end_pos": [556, 35]}]], "state_before": "case h.e'_3.h.e'_3\nx : \u211d*\nhx : x.InfiniteNeg\n\u22a2 {y | \u2191y < x} = \u2205", "state_after": "no goals"}, {"tactic": "exact (isSt_sSup hx).st_eq", "annotated_tactic": ["exact (<a>isSt_sSup</a> hx).<a>st_eq</a>", [{"full_name": "Hyperreal.isSt_sSup", "def_path": "Mathlib/Data/Real/Hyperreal.lean", "def_pos": [300, 9], "def_end_pos": [300, 18]}, {"full_name": "Hyperreal.IsSt.st_eq", "def_path": "Mathlib/Data/Real/Hyperreal.lean", "def_pos": [282, 9], "def_end_pos": [282, 19]}]], "state_before": "case inr\nx : \u211d*\nhx : \u00acx.Infinite\n\u22a2 x.st = sSup {y | \u2191y < x}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/lpSpace.lean", "full_name": "lp.infty_coeFn_mul", "start": [793, 1], "end": [794, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "full_name": "one_le_of_le_mul_left", "start": [442, 1], "end": [445, 53], "traced_tactics": [{"tactic": "simpa only [one_mul]", "annotated_tactic": ["simpa only [<a>one_mul</a>]", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b2 : MulOneClass \u03b1\ninst\u271d\u00b9 : LE \u03b1\ninst\u271d : ContravariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\na b : \u03b1\nh : b \u2264 a * b\n\u22a2 1 * ?m.21939 \u2264 a * ?m.21939", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Complex/Abs.lean", "full_name": "Complex.re_neg_ne_zero_of_one_lt_re", "start": [362, 1], "end": [363, 58], "traced_tactics": [{"tactic": "linarith", "annotated_tactic": ["linarith", []], "state_before": "s : \u2102\nhs : 1 < s.re\n\u22a2 -s.re < 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/AddConstMap/Basic.lean", "full_name": "AddConstMapClass.map_sub_one", "start": [170, 1], "end": [172, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Basic.lean", "full_name": "interior_inter", "start": [315, 1], "end": [318, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Dual.lean", "full_name": "Subspace.dualEquivDual_def", "start": [1167, 1], "end": [1169, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/Irrational.lean", "full_name": "Irrational.of_int_mul", "start": [393, 1], "end": [394, 40], "traced_tactics": [{"tactic": "rwa [cast_intCast]", "annotated_tactic": ["rwa [<a>cast_intCast</a>]", [{"full_name": "Rat.cast_intCast", "def_path": "Mathlib/Data/Rat/Cast/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 21]}]], "state_before": "q : \u211a\nx y : \u211d\nm : \u2124\nh : Irrational (\u2191m * x)\n\u22a2 Irrational (\u2191\u2191m * x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Submodule/LinearMap.lean", "full_name": "Submodule.coe_inclusion", "start": [313, 1], "end": [314, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean", "full_name": "MeasureTheory.integral_comp_smul_deriv_Ioi", "start": [1093, 1], "end": [1117, 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": [782, 9], "def_end_pos": [782, 46]}]], "state_before": "E : Type u_1\nf\u271d : \u211d \u2192 E\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) volume\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\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ioi a) volume\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": [676, 9], "def_end_pos": [676, 46]}, {"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3094, 9], "def_end_pos": [3094, 19]}]], "state_before": "E : Type u_1\nf\u271d : \u211d \u2192 E\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ioi a) volume\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\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ioi a) volume\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 \u2202volume))\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/Order/Interval/Set/Basic.lean", "def_pos": [79, 5], "def_end_pos": [79, 8]}, {"full_name": "Set.Ici", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [74, 5], "def_end_pos": [74, 8]}, {"full_name": "IsPreconnected.intermediate_value_Ici", "def_path": "Mathlib/Topology/Order/IntermediateValue.lean", "def_pos": [155, 9], "def_end_pos": [155, 46]}, {"full_name": "isPreconnected_Ici", "def_path": "Mathlib/Topology/Order/IntermediateValue.lean", "def_pos": [426, 9], "def_end_pos": [426, 27]}, {"full_name": "Set.left_mem_Ici", "def_path": "Mathlib/Order/Interval/Set/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\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ioi a) volume\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 \u2202volume))\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\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ioi a) volume\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 \u2202volume))\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": [676, 9], "def_end_pos": [676, 46]}, {"full_name": "Filter.tendsto_id", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3094, 9], "def_end_pos": [3094, 19]}, {"full_name": "Filter.Tendsto.comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3098, 9], "def_end_pos": [3098, 21]}]], "state_before": "E : Type u_1\nf\u271d : \u211d \u2192 E\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ioi a) volume\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 \u2202volume))\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\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ioi a) volume\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 \u2202volume))\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 \u2202volume))\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": [1508, 9], "def_end_pos": [1508, 28]}, {"full_name": "Filter.Tendsto.congr'", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [3075, 9], "def_end_pos": [3075, 23]}, {"full_name": "Filter.eventuallyEq_of_mem", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1505, 9], "def_end_pos": [1505, 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\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ioi a) volume\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 \u2202volume))\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 \u2202volume))\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/Order/Interval/Set/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": [1142, 3], "def_end_pos": [1142, 6]}, {"full_name": "Max.max", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1129, 3], "def_end_pos": [1129, 6]}, {"full_name": "Set.Ioi", "def_path": "Mathlib/Order/Interval/Set/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/Order/Interval/Set/Basic.lean", "def_pos": [545, 9], "def_end_pos": [545, 28]}]], "state_before": "E : Type u_1\nf\u271d : \u211d \u2192 E\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) volume\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\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) volume\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/Order/Interval/Set/Basic.lean", "def_pos": [74, 5], "def_end_pos": [74, 8]}, {"full_name": "Set.uIcc_of_le", "def_path": "Mathlib/Order/Interval/Set/UnorderedInterval.lean", "def_pos": [71, 7], "def_end_pos": [71, 17]}, {"full_name": "Set.Icc_subset_Ici_self", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [484, 9], "def_end_pos": [484, 28]}]], "state_before": "E : Type u_1\nf\u271d : \u211d \u2192 E\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) volume\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\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) volume\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": [1476, 9], "def_end_pos": [1476, 36]}, {"full_name": "Set.mem_of_mem_of_subset", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [220, 9], "def_end_pos": [220, 29]}, {"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [291, 9], "def_end_pos": [291, 21]}, {"full_name": "Set.image_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [291, 9], "def_end_pos": [291, 21]}]], "state_before": "E : Type u_1\nf\u271d : \u211d \u2192 E\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) volume\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\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) volume\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\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) volume\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\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) volume\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\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) volume\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/Order/Interval/Set/Basic.lean", "def_pos": [545, 9], "def_end_pos": [545, 28]}]], "state_before": "E : Type u_1\nf\u271d : \u211d \u2192 E\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) volume\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/Order/Interval/Set/UnorderedInterval.lean", "def_pos": [71, 7], "def_end_pos": [71, 17]}]], "state_before": "E : Type u_1\nf\u271d : \u211d \u2192 E\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) volume\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\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) volume\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/Order/Interval/Set/Basic.lean", "def_pos": [484, 9], "def_end_pos": [484, 28]}]], "state_before": "E : Type u_1\nf\u271d : \u211d \u2192 E\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) volume\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\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) volume\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\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) volume\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/Order/Interval/Set/Basic.lean", "def_pos": [545, 9], "def_end_pos": [545, 28]}]], "state_before": "case refine_1\nE : Type u_1\nf\u271d : \u211d \u2192 E\ninst\u271d\u00b9 : NormedAddCommGroup E\ninst\u271d : NormedSpace \u211d 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 \u2208 Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) volume\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/Order/Interval/Set/UnorderedInterval.lean", "def_pos": [71, 7], "def_end_pos": [71, 17]}]], "state_before": "case refine_2\nE : Type u_1\nf\u271d : \u211d \u2192 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"https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Regular/IsSMulRegular.lean", "full_name": "isSMulRegular_of_isSMulRegular_on_submodule_on_quotient", "start": [125, 1], "end": [128, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Parity.lean", "full_name": "Nat.even_add'", "start": [304, 1], "end": [305, 65], "traced_tactics": [{"tactic": "rw [even_add, even_iff_not_odd, even_iff_not_odd, not_iff_not]", "annotated_tactic": ["rw [<a>even_add</a>, <a>even_iff_not_odd</a>, <a>even_iff_not_odd</a>, <a>not_iff_not</a>]", [{"full_name": "Nat.even_add", "def_path": "Mathlib/Algebra/Group/Nat.lean", "def_pos": [108, 23], "def_end_pos": [108, 31]}, {"full_name": "Nat.even_iff_not_odd", "def_path": "Mathlib/Algebra/Ring/Parity.lean", "def_pos": [265, 7], "def_end_pos": [265, 23]}, 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"full_name": "Complex.not_le_iff", "start": [87, 1], "end": [88, 34], "traced_tactics": [{"tactic": "rw [le_def, not_and_or, not_le]", "annotated_tactic": ["rw [<a>le_def</a>, <a>not_and_or</a>, <a>not_le</a>]", [{"full_name": "Complex.le_def", "def_path": "Mathlib/Data/Complex/Order.lean", "def_pos": [55, 9], "def_end_pos": [55, 15]}, {"full_name": "not_and_or", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [481, 9], "def_end_pos": [481, 19]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [375, 9], "def_end_pos": [375, 15]}]], "state_before": "z w : \u2102\n\u22a2 \u00acz \u2264 w \u2194 w.re < z.re \u2228 z.im \u2260 w.im", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/LinearMap/Polynomial.lean", "full_name": "LinearMap.nilRank_le_card", "start": [482, 1], "end": [486, 20], "traced_tactics": [{"tactic": "apply Polynomial.natTrailingDegree_le_of_ne_zero", "annotated_tactic": ["apply <a>Polynomial.natTrailingDegree_le_of_ne_zero</a>", [{"full_name": "Polynomial.natTrailingDegree_le_of_ne_zero", "def_path": "Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean", "def_pos": [138, 9], "def_end_pos": [138, 40]}]], "state_before": "R : Type u_1\nL : Type u_2\nM : Type u_3\nn : Type u_4\n\u03b9 : Type u_5\n\u03b9' : Type u_6\n\u03b9M : Type u_7\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : AddCommGroup L\ninst\u271d\u00b9\u00b2 : Module R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\n\u03c6 : L \u2192\u2097[R] Module.End R M\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fintype \u03b9'\ninst\u271d\u2077 : Fintype \u03b9M\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9'\ninst\u271d\u2074 : Module.Free R M\ninst\u271d\u00b3 : Module.Finite R M\nb\u271d : Basis \u03b9 R L\ninst\u271d\u00b2 : Module.Finite R L\ninst\u271d\u00b9 : Module.Free R L\ninst\u271d : Nontrivial R\nb : Basis \u03b9 R M\n\u22a2 \u03c6.nilRank \u2264 Fintype.card \u03b9", "state_after": "case h\nR : Type u_1\nL : Type u_2\nM : Type u_3\nn : Type u_4\n\u03b9 : Type u_5\n\u03b9' : Type u_6\n\u03b9M : Type u_7\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : AddCommGroup L\ninst\u271d\u00b9\u00b2 : Module R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\n\u03c6 : L \u2192\u2097[R] Module.End R M\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fintype \u03b9'\ninst\u271d\u2077 : Fintype \u03b9M\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9'\ninst\u271d\u2074 : Module.Free R M\ninst\u271d\u00b3 : Module.Finite R M\nb\u271d : Basis \u03b9 R L\ninst\u271d\u00b2 : Module.Finite R L\ninst\u271d\u00b9 : Module.Free R L\ninst\u271d : Nontrivial R\nb : Basis \u03b9 R M\n\u22a2 (\u03c6.polyCharpoly (chooseBasis R L)).coeff (Fintype.card \u03b9) 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: Module.Free R M\ninst\u271d\u00b3 : Module.Finite R M\nb\u271d : Basis \u03b9 R L\ninst\u271d\u00b2 : Module.Finite R L\ninst\u271d\u00b9 : Module.Free R L\ninst\u271d : Nontrivial R\nb : Basis \u03b9 R M\n\u22a2 (\u03c6.polyCharpoly (chooseBasis R L)).coeff (Fintype.card \u03b9) \u2260 0", "state_after": "case h\nR : Type u_1\nL : Type u_2\nM : Type u_3\nn : Type u_4\n\u03b9 : Type u_5\n\u03b9' : Type u_6\n\u03b9M : Type u_7\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : AddCommGroup L\ninst\u271d\u00b9\u00b2 : Module R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\n\u03c6 : L \u2192\u2097[R] Module.End R M\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fintype \u03b9'\ninst\u271d\u2077 : Fintype \u03b9M\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9'\ninst\u271d\u2074 : Module.Free R M\ninst\u271d\u00b3 : Module.Finite R M\nb\u271d : Basis \u03b9 R L\ninst\u271d\u00b2 : Module.Finite R L\ninst\u271d\u00b9 : Module.Free R L\ninst\u271d : Nontrivial R\nb : Basis \u03b9 R M\n\u22a2 1 \u2260 0"}, {"tactic": "apply one_ne_zero", "annotated_tactic": ["apply <a>one_ne_zero</a>", [{"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [58, 15], "def_end_pos": [58, 26]}]], "state_before": "case h\nR : Type u_1\nL : Type u_2\nM : Type u_3\nn : Type u_4\n\u03b9 : Type u_5\n\u03b9' : Type u_6\n\u03b9M : Type u_7\ninst\u271d\u00b9\u2074 : CommRing R\ninst\u271d\u00b9\u00b3 : AddCommGroup L\ninst\u271d\u00b9\u00b2 : Module R L\ninst\u271d\u00b9\u00b9 : AddCommGroup M\ninst\u271d\u00b9\u2070 : Module R M\n\u03c6 : L \u2192\u2097[R] Module.End R M\ninst\u271d\u2079 : Fintype \u03b9\ninst\u271d\u2078 : Fintype \u03b9'\ninst\u271d\u2077 : Fintype \u03b9M\ninst\u271d\u2076 : DecidableEq \u03b9\ninst\u271d\u2075 : DecidableEq \u03b9'\ninst\u271d\u2074 : Module.Free R M\ninst\u271d\u00b3 : Module.Finite R M\nb\u271d : Basis \u03b9 R L\ninst\u271d\u00b2 : Module.Finite R L\ninst\u271d\u00b9 : Module.Free R L\ninst\u271d : Nontrivial R\nb : Basis \u03b9 R M\n\u22a2 1 \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Category/Profinite/Basic.lean", "full_name": "Profinite.isClosedMap", "start": [318, 1], "end": [319, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Tilted.lean", "full_name": "MeasureTheory.setLIntegral_tilted'", "start": [144, 1], "end": [160, 9], "traced_tactics": [{"tactic": "by_cases hf : AEMeasurable f \u03bc", "annotated_tactic": ["by_cases hf : <a>AEMeasurable</a> f \u03bc", [{"full_name": "AEMeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [418, 5], "def_end_pos": [418, 17]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc.tilted f = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : AEMeasurable f \u03bc\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc.tilted f = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : \u00acAEMeasurable f \u03bc\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc.tilted f = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc"}, {"tactic": "rw [Measure.tilted, setLIntegral_withDensity_eq_setLIntegral_mul_non_measurable\u2080]", "annotated_tactic": ["rw [<a>Measure.tilted</a>, <a>setLIntegral_withDensity_eq_setLIntegral_mul_non_measurable\u2080</a>]", [{"full_name": "MeasureTheory.Measure.tilted", "def_path": "Mathlib/MeasureTheory/Measure/Tilted.lean", "def_pos": [38, 5], "def_end_pos": [38, 19]}, {"full_name": "MeasureTheory.setLIntegral_withDensity_eq_setLIntegral_mul_non_measurable\u2080", "def_path": "Mathlib/MeasureTheory/Measure/WithDensity.lean", "def_pos": [521, 9], "def_end_pos": [521, 69]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : AEMeasurable f \u03bc\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc.tilted f = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : AEMeasurable f \u03bc\n\u22a2 \u222b\u207b (a : \u03b1) in s, ((fun x => ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc)) * g) a \u2202\u03bc =\n    \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc\n\ncase pos.hf\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : AEMeasurable f \u03bc\n\u22a2 AEMeasurable (fun x => ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc)) (\u03bc.restrict s)\n\ncase pos.hs\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : AEMeasurable f \u03bc\n\u22a2 MeasurableSet s\n\ncase pos.h'f\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : AEMeasurable f \u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc.restrict s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) < \u22a4"}, {"tactic": "simp only [Pi.mul_apply]", "annotated_tactic": ["simp only [<a>Pi.mul_apply</a>]", [{"full_name": "Pi.mul_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [97, 9], "def_end_pos": [97, 18]}]], "state_before": "case pos\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : AEMeasurable f \u03bc\n\u22a2 \u222b\u207b (a : \u03b1) in s, ((fun x => ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc)) * g) a \u2202\u03bc =\n    \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "refine AEMeasurable.restrict ?_", "annotated_tactic": ["refine <a>AEMeasurable.restrict</a> ?_", [{"full_name": "AEMeasurable.restrict", "def_path": "Mathlib/MeasureTheory/Measure/AEMeasurable.lean", "def_pos": [313, 9], "def_end_pos": [313, 30]}]], "state_before": "case pos.hf\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : AEMeasurable f \u03bc\n\u22a2 AEMeasurable (fun x => ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc)) (\u03bc.restrict s)", "state_after": "case pos.hf\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : AEMeasurable f \u03bc\n\u22a2 AEMeasurable (fun x => ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc)) \u03bc"}, {"tactic": "exact ((measurable_exp.comp_aemeasurable hf).div_const _).ennreal_ofReal", "annotated_tactic": ["exact ((measurable_exp.comp_aemeasurable hf).<a>div_const</a> _).<a>ennreal_ofReal</a>", [{"full_name": "AEMeasurable.div_const", "def_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "def_pos": [319, 9], "def_end_pos": [319, 31]}, {"full_name": "AEMeasurable.ennreal_ofReal", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean", "def_pos": [199, 7], "def_end_pos": [199, 34]}]], "state_before": "case pos.hf\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : AEMeasurable f \u03bc\n\u22a2 AEMeasurable (fun x => ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc)) \u03bc", "state_after": "no goals"}, {"tactic": "exact hs", "annotated_tactic": ["exact hs", []], "state_before": "case pos.hs\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : AEMeasurable f \u03bc\n\u22a2 MeasurableSet s", "state_after": "no goals"}, {"tactic": "filter_upwards", "annotated_tactic": ["filter_upwards", []], "state_before": "case pos.h'f\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : AEMeasurable f \u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc.restrict s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) < \u22a4", "state_after": "case pos.h'f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : AEMeasurable f \u03bc\n\u22a2 \u2200 (a : \u03b1), ENNReal.ofReal (rexp (f a) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) < \u22a4"}, {"tactic": "simp only [ENNReal.ofReal_lt_top, implies_true]", "annotated_tactic": ["simp only [<a>ENNReal.ofReal_lt_top</a>, <a>implies_true</a>]", [{"full_name": "ENNReal.ofReal_lt_top", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [337, 17], "def_end_pos": [337, 30]}, {"full_name": "implies_true", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [132, 17], "def_end_pos": [132, 29]}]], "state_before": "case pos.h'f.h\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : AEMeasurable f \u03bc\n\u22a2 \u2200 (a : \u03b1), ENNReal.ofReal (rexp (f a) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) < \u22a4", "state_after": "no goals"}, {"tactic": "have hf' : \u00ac Integrable (fun x \u21a6 exp (f x)) \u03bc := by\n  exact fun h \u21a6 hf (aemeasurable_of_aemeasurable_exp h.1.aemeasurable)", "annotated_tactic": ["have hf' : \u00ac <a>Integrable</a> (fun x \u21a6 <a>exp</a> (f x)) \u03bc := by\n      exact fun h \u21a6 hf (<a>aemeasurable_of_aemeasurable_exp</a> h.1.<a>aemeasurable</a>)", [{"full_name": "MeasureTheory.Integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [438, 5], "def_end_pos": [438, 15]}, {"full_name": "Real.exp", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [114, 12], "def_end_pos": [114, 15]}, {"full_name": "Real.aemeasurable_of_aemeasurable_exp", "def_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean", "def_pos": [47, 7], "def_end_pos": [47, 39]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1246, 19], "def_end_pos": [1246, 31]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : \u00acAEMeasurable f \u03bc\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc.tilted f = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : \u00acAEMeasurable f \u03bc\nhf' : \u00acIntegrable (fun x => rexp (f x)) \u03bc\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc.tilted f = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc"}, {"tactic": "simp only [hf, not_false_eq_true, tilted_of_not_aemeasurable, Measure.restrict_zero,\n  lintegral_zero_measure]", "annotated_tactic": ["simp only [hf, <a>not_false_eq_true</a>, <a>tilted_of_not_aemeasurable</a>, <a>Measure.restrict_zero</a>,\n      <a>lintegral_zero_measure</a>]", [{"full_name": "not_false_eq_true", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [134, 17], "def_end_pos": [134, 34]}, {"full_name": "MeasureTheory.tilted_of_not_aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/Tilted.lean", "def_pos": [47, 7], "def_end_pos": [47, 33]}, {"full_name": "MeasureTheory.Measure.restrict_zero", "def_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "def_pos": [165, 9], "def_end_pos": [165, 22]}, {"full_name": "MeasureTheory.lintegral_zero_measure", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [666, 9], "def_end_pos": [666, 31]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : \u00acAEMeasurable f \u03bc\nhf' : \u00acIntegrable (fun x => rexp (f x)) \u03bc\n\u22a2 \u222b\u207b (x : \u03b1) in s, g x \u2202\u03bc.tilted f = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : \u00acAEMeasurable f \u03bc\nhf' : \u00acIntegrable (fun x => rexp (f x)) \u03bc\n\u22a2 0 = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc"}, {"tactic": "rw [integral_undef hf']", "annotated_tactic": ["rw [<a>integral_undef</a> hf']", [{"full_name": "MeasureTheory.integral_undef", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [833, 9], "def_end_pos": [833, 23]}]], "state_before": "case neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : \u00acAEMeasurable f \u03bc\nhf' : \u00acIntegrable (fun x => rexp (f x)) \u03bc\n\u22a2 0 = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / \u222b (x : \u03b1), rexp (f x) \u2202\u03bc) * g x \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : \u00acAEMeasurable f \u03bc\nhf' : \u00acIntegrable (fun x => rexp (f x)) \u03bc\n\u22a2 0 = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / 0) * g x \u2202\u03bc"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case neg\n\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : \u00acAEMeasurable f \u03bc\nhf' : \u00acIntegrable (fun x => rexp (f x)) \u03bc\n\u22a2 0 = \u222b\u207b (x : \u03b1) in s, ENNReal.ofReal (rexp (f x) / 0) * g x \u2202\u03bc", "state_after": "no goals"}, {"tactic": "exact fun h \u21a6 hf (aemeasurable_of_aemeasurable_exp h.1.aemeasurable)", "annotated_tactic": ["exact fun h \u21a6 hf (<a>aemeasurable_of_aemeasurable_exp</a> h.1.<a>aemeasurable</a>)", [{"full_name": "Real.aemeasurable_of_aemeasurable_exp", "def_path": "Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean", "def_pos": [47, 7], "def_end_pos": [47, 39]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1246, 19], "def_end_pos": [1246, 31]}]], "state_before": "\u03b1 : Type u_1\nm\u03b1 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf\u271d f : \u03b1 \u2192 \u211d\ng : \u03b1 \u2192 \u211d\u22650\u221e\ns : Set \u03b1\nhs : MeasurableSet s\nhf : \u00acAEMeasurable f \u03bc\n\u22a2 \u00acIntegrable (fun x => rexp (f x)) \u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Ideal/Operations.lean", "full_name": "Ideal.mem_radical_of_pow_mem", "start": [982, 1], "end": [984, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean", "full_name": "NumberField.Units.dirichletUnitTheorem.exists_unit", "start": [280, 1], "end": [308, 99], "traced_tactics": [{"tactic": "obtain \u27e8B, hB\u27e9 : \u2203 B : \u2115, minkowskiBound K 1 < (convexBodyLTFactor K) * B := by\n  conv => congr; ext; rw [mul_comm]\n  exact ENNReal.exists_nat_mul_gt (ENNReal.coe_ne_zero.mpr (convexBodyLTFactor_ne_zero K))\n    (ne_of_lt (minkowskiBound_lt_top K 1))", "annotated_tactic": ["obtain \u27e8B, hB\u27e9 : \u2203 B : \u2115, <a>minkowskiBound</a> K 1 < (<a>convexBodyLTFactor</a> K) * B := by\n    conv => congr; ext; rw [<a>mul_comm</a>]\n    exact <a>ENNReal.exists_nat_mul_gt</a> (ENNReal.coe_ne_zero.mpr (<a>convexBodyLTFactor_ne_zero</a> K))\n      (<a>ne_of_lt</a> (<a>minkowskiBound_lt_top</a> K 1))", [{"full_name": "NumberField.mixedEmbedding.minkowskiBound", "def_path": "Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean", "def_pos": [469, 19], "def_end_pos": [469, 33]}, {"full_name": "NumberField.mixedEmbedding.convexBodyLTFactor", "def_path": "Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean", "def_pos": [96, 22], "def_end_pos": [96, 40]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "ENNReal.exists_nat_mul_gt", "def_path": "Mathlib/Data/ENNReal/Inv.lean", "def_pos": [577, 9], "def_end_pos": [577, 26]}, {"full_name": "NumberField.mixedEmbedding.convexBodyLTFactor_ne_zero", "def_path": "Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean", "def_pos": [99, 9], "def_end_pos": [99, 35]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "NumberField.mixedEmbedding.minkowskiBound_lt_top", "def_path": "Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean", "def_pos": [486, 9], "def_end_pos": [486, 30]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nw\u2081 : InfinitePlace K\n\u22a2 \u2203 u, \u2200 (w : InfinitePlace K), w \u2260 w\u2081 \u2192 Real.log (w ((algebraMap (\ud835\udcde K) K) \u2191u)) < 0", "state_after": "case intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\n\u22a2 \u2203 u, \u2200 (w : InfinitePlace K), w \u2260 w\u2081 \u2192 Real.log (w ((algebraMap (\ud835\udcde K) K) \u2191u)) < 0"}, {"tactic": "rsuffices \u27e8n, m, hnm, h\u27e9 : \u2203 n m, n < m \u2227\n    (Ideal.span ({ (seq K w\u2081 hB n : \ud835\udcde K) }) = Ideal.span ({ (seq K w\u2081 hB m : \ud835\udcde K) }))", "annotated_tactic": ["rsuffices \u27e8n, m, hnm, h\u27e9 : \u2203 n m, n < m \u2227\n      (<a>Ideal.span</a> ({ (<a>seq</a> K w\u2081 hB n : \ud835\udcde K) }) = <a>Ideal.span</a> ({ (<a>seq</a> K w\u2081 hB m : \ud835\udcde K) }))", [{"full_name": "Ideal.span", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [115, 5], "def_end_pos": [115, 9]}, {"full_name": "NumberField.Units.dirichletUnitTheorem.seq", "def_path": "Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean", "def_pos": [235, 5], "def_end_pos": [235, 8]}, {"full_name": "Ideal.span", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [115, 5], "def_end_pos": [115, 9]}, {"full_name": "NumberField.Units.dirichletUnitTheorem.seq", "def_path": "Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean", "def_pos": [235, 5], "def_end_pos": [235, 8]}]], "state_before": "case intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\n\u22a2 \u2203 u, \u2200 (w : InfinitePlace K), w \u2260 w\u2081 \u2192 Real.log (w ((algebraMap (\ud835\udcde K) K) \u2191u)) < 0", "state_after": "case intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nn m : \u2115\nhnm : n < m\nh : Ideal.span {\u2191(seq K w\u2081 hB n)} = Ideal.span {\u2191(seq K w\u2081 hB m)}\n\u22a2 \u2203 u, \u2200 (w : InfinitePlace K), w \u2260 w\u2081 \u2192 Real.log (w ((algebraMap (\ud835\udcde K) K) \u2191u)) < 0\n\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\n\u22a2 \u2203 n m, n < m \u2227 Ideal.span {\u2191(seq K w\u2081 hB n)} = Ideal.span {\u2191(seq K w\u2081 hB m)}"}, {"tactic": "refine Set.Finite.exists_lt_map_eq_of_forall_mem\n  (t := { I : Ideal (\ud835\udcde K) | 1 \u2264 Ideal.absNorm I \u2227 Ideal.absNorm I \u2264 B })\n  (fun n => ?_) ?_", "annotated_tactic": ["refine <a>Set.Finite.exists_lt_map_eq_of_forall_mem</a>\n    (t := { I : <a>Ideal</a> (\ud835\udcde K) | 1 \u2264 <a>Ideal.absNorm</a> I \u2227 <a>Ideal.absNorm</a> I \u2264 B })\n    (fun n => ?_) ?_", [{"full_name": "Set.Finite.exists_lt_map_eq_of_forall_mem", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [1508, 9], "def_end_pos": [1508, 46]}, {"full_name": "Ideal", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [41, 8], "def_end_pos": [41, 13]}, {"full_name": "Ideal.absNorm", "def_path": "Mathlib/RingTheory/Ideal/Norm.lean", "def_pos": [251, 19], "def_end_pos": [251, 32]}, {"full_name": "Ideal.absNorm", "def_path": "Mathlib/RingTheory/Ideal/Norm.lean", "def_pos": [251, 19], "def_end_pos": [251, 32]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\n\u22a2 \u2203 n m, n < m \u2227 Ideal.span {\u2191(seq K w\u2081 hB n)} = Ideal.span {\u2191(seq K w\u2081 hB m)}", "state_after": "case refine_1\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nn : \u2115\n\u22a2 Ideal.span {\u2191(seq K w\u2081 hB n)} \u2208 {I | 1 \u2264 Ideal.absNorm I \u2227 Ideal.absNorm I \u2264 B}\n\ncase refine_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\n\u22a2 {I | 1 \u2264 Ideal.absNorm I \u2227 Ideal.absNorm I \u2264 B}.Finite"}, {"tactic": "conv => congr; ext; rw [mul_comm]", "annotated_tactic": ["conv => congr; ext; rw [<a>mul_comm</a>]", [{"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nw\u2081 : InfinitePlace K\n\u22a2 \u2203 B, minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B", "state_after": "K : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nw\u2081 : InfinitePlace K\n\u22a2 \u2203 x, minkowskiBound K 1 < \u2191x * \u2191(convexBodyLTFactor K)"}, {"tactic": "exact ENNReal.exists_nat_mul_gt (ENNReal.coe_ne_zero.mpr (convexBodyLTFactor_ne_zero K))\n  (ne_of_lt (minkowskiBound_lt_top K 1))", "annotated_tactic": ["exact <a>ENNReal.exists_nat_mul_gt</a> (ENNReal.coe_ne_zero.mpr (<a>convexBodyLTFactor_ne_zero</a> K))\n      (<a>ne_of_lt</a> (<a>minkowskiBound_lt_top</a> K 1))", [{"full_name": "ENNReal.exists_nat_mul_gt", "def_path": "Mathlib/Data/ENNReal/Inv.lean", "def_pos": [577, 9], "def_end_pos": [577, 26]}, {"full_name": "NumberField.mixedEmbedding.convexBodyLTFactor_ne_zero", "def_path": "Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean", "def_pos": [99, 9], "def_end_pos": [99, 35]}, {"full_name": "ne_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [101, 9], "def_end_pos": [101, 17]}, {"full_name": "NumberField.mixedEmbedding.minkowskiBound_lt_top", "def_path": "Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean", "def_pos": [486, 9], "def_end_pos": [486, 30]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nw\u2081 : InfinitePlace K\n\u22a2 \u2203 x, minkowskiBound K 1 < \u2191x * \u2191(convexBodyLTFactor K)", "state_after": "no goals"}, {"tactic": "have hu := Ideal.span_singleton_eq_span_singleton.mp h", "annotated_tactic": ["have hu := Ideal.span_singleton_eq_span_singleton.mp h", []], "state_before": "case intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nn m : \u2115\nhnm : n < m\nh : Ideal.span {\u2191(seq K w\u2081 hB n)} = Ideal.span {\u2191(seq K w\u2081 hB m)}\n\u22a2 \u2203 u, \u2200 (w : InfinitePlace K), w \u2260 w\u2081 \u2192 Real.log (w ((algebraMap (\ud835\udcde K) K) \u2191u)) < 0", "state_after": "case intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nn m : \u2115\nhnm : n < m\nh : Ideal.span {\u2191(seq K w\u2081 hB n)} = Ideal.span {\u2191(seq K w\u2081 hB m)}\nhu : Associated \u2191(seq K w\u2081 hB n) \u2191(seq K w\u2081 hB m)\n\u22a2 \u2203 u, \u2200 (w : InfinitePlace K), w \u2260 w\u2081 \u2192 Real.log (w ((algebraMap (\ud835\udcde K) K) \u2191u)) < 0"}, {"tactic": "refine \u27e8hu.choose, fun w hw => Real.log_neg ?_ ?_\u27e9", "annotated_tactic": ["refine \u27e8hu.choose, fun w hw => <a>Real.log_neg</a> ?_ ?_\u27e9", [{"full_name": "Real.log_neg", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [190, 9], "def_end_pos": [190, 16]}]], "state_before": "case intro.intro.intro.intro\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nn m : \u2115\nhnm : n < m\nh : Ideal.span {\u2191(seq K w\u2081 hB n)} = Ideal.span {\u2191(seq K w\u2081 hB m)}\nhu : Associated \u2191(seq K w\u2081 hB n) \u2191(seq K w\u2081 hB m)\n\u22a2 \u2203 u, \u2200 (w : InfinitePlace K), w \u2260 w\u2081 \u2192 Real.log (w ((algebraMap (\ud835\udcde K) K) \u2191u)) < 0", "state_after": "case intro.intro.intro.intro.refine_1\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nn m : \u2115\nhnm : n < m\nh : Ideal.span {\u2191(seq K w\u2081 hB n)} = Ideal.span {\u2191(seq K w\u2081 hB m)}\nhu : Associated \u2191(seq K w\u2081 hB n) \u2191(seq K w\u2081 hB m)\nw : InfinitePlace K\nhw : w \u2260 w\u2081\n\u22a2 0 < w ((algebraMap (\ud835\udcde K) K) \u2191(Exists.choose hu))\n\ncase intro.intro.intro.intro.refine_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nn m : \u2115\nhnm : n < m\nh : Ideal.span {\u2191(seq K w\u2081 hB n)} = Ideal.span {\u2191(seq K w\u2081 hB m)}\nhu : Associated \u2191(seq K w\u2081 hB n) \u2191(seq K w\u2081 hB m)\nw : InfinitePlace K\nhw : w \u2260 w\u2081\n\u22a2 w ((algebraMap (\ud835\udcde K) K) \u2191(Exists.choose hu)) < 1"}, {"tactic": "exact pos_iff.mpr (coe_ne_zero _)", "annotated_tactic": ["exact pos_iff.mpr (<a>coe_ne_zero</a> _)", [{"full_name": "NumberField.Units.coe_ne_zero", "def_path": "Mathlib/NumberTheory/NumberField/Units/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 20]}]], "state_before": "case intro.intro.intro.intro.refine_1\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nn m : \u2115\nhnm : n < m\nh : Ideal.span {\u2191(seq K w\u2081 hB n)} = Ideal.span {\u2191(seq K w\u2081 hB m)}\nhu : Associated \u2191(seq K w\u2081 hB n) \u2191(seq K w\u2081 hB m)\nw : InfinitePlace K\nhw : w \u2260 w\u2081\n\u22a2 0 < w ((algebraMap (\ud835\udcde K) K) \u2191(Exists.choose hu))", "state_after": "no goals"}, {"tactic": "calc\n  _ = w (algebraMap (\ud835\udcde K) K (seq K w\u2081 hB m) * (algebraMap (\ud835\udcde K) K (seq K w\u2081 hB n))\u207b\u00b9) := by\n    rw [\u2190 congr_arg (algebraMap (\ud835\udcde K) K) hu.choose_spec, mul_comm, map_mul (algebraMap _ _),\n    \u2190 mul_assoc, inv_mul_cancel (seq_ne_zero K w\u2081 hB n), one_mul]\n  _ = w (algebraMap (\ud835\udcde K) K (seq K w\u2081 hB m)) * w (algebraMap (\ud835\udcde K) K (seq K w\u2081 hB n))\u207b\u00b9 :=\n    _root_.map_mul _ _ _\n  _ < 1 := by\n    rw [map_inv\u2080, mul_inv_lt_iff (pos_iff.mpr (seq_ne_zero K w\u2081 hB n)), mul_one]\n    exact seq_decreasing K w\u2081 hB hnm w hw", "annotated_tactic": ["calc\n        _ = w (<a>algebraMap</a> (\ud835\udcde K) K (<a>seq</a> K w\u2081 hB m) * (<a>algebraMap</a> (\ud835\udcde K) K (<a>seq</a> K w\u2081 hB n))\u207b\u00b9) := by\n          rw [\u2190 <a>congr_arg</a> (<a>algebraMap</a> (\ud835\udcde K) K) hu.choose_spec, <a>mul_comm</a>, <a>map_mul</a> (<a>algebraMap</a> _ _),\n          \u2190 <a>mul_assoc</a>, <a>inv_mul_cancel</a> (<a>seq_ne_zero</a> K w\u2081 hB n), <a>one_mul</a>]\n        _ = w (<a>algebraMap</a> (\ud835\udcde K) K (<a>seq</a> K w\u2081 hB m)) * w (<a>algebraMap</a> (\ud835\udcde K) K (<a>seq</a> K w\u2081 hB n))\u207b\u00b9 :=\n          <a>_root_.map_mul</a> _ _ _\n        _ < 1 := by\n          rw [<a>map_inv\u2080</a>, <a>mul_inv_lt_iff</a> (pos_iff.mpr (<a>seq_ne_zero</a> K w\u2081 hB n)), <a>mul_one</a>]\n          exact <a>seq_decreasing</a> K w\u2081 hB hnm w hw", [{"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "NumberField.Units.dirichletUnitTheorem.seq", "def_path": "Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean", "def_pos": [235, 5], "def_end_pos": [235, 8]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "NumberField.Units.dirichletUnitTheorem.seq", "def_path": "Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean", "def_pos": [235, 5], "def_end_pos": [235, 8]}, {"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "map_mul", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [309, 9], "def_end_pos": [309, 16]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "inv_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/NeZero.lean", "def_pos": [55, 9], "def_end_pos": [55, 23]}, {"full_name": "NumberField.Units.dirichletUnitTheorem.seq_ne_zero", "def_path": "Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean", "def_pos": [241, 9], "def_end_pos": [241, 20]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "NumberField.Units.dirichletUnitTheorem.seq", "def_path": "Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean", "def_pos": [235, 5], "def_end_pos": [235, 8]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "NumberField.Units.dirichletUnitTheorem.seq", "def_path": "Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean", "def_pos": [235, 5], "def_end_pos": [235, 8]}, {"full_name": "map_mul", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [309, 9], "def_end_pos": [309, 16]}, {"full_name": "map_inv\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [64, 9], "def_end_pos": [64, 17]}, {"full_name": "mul_inv_lt_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [125, 9], "def_end_pos": [125, 23]}, {"full_name": "NumberField.Units.dirichletUnitTheorem.seq_ne_zero", "def_path": "Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean", "def_pos": [241, 9], "def_end_pos": [241, 20]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "NumberField.Units.dirichletUnitTheorem.seq_decreasing", "def_path": "Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean", "def_pos": [249, 9], "def_end_pos": [249, 23]}]], "state_before": "case intro.intro.intro.intro.refine_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nn m : \u2115\nhnm : n < m\nh : Ideal.span {\u2191(seq K w\u2081 hB n)} = Ideal.span {\u2191(seq K w\u2081 hB m)}\nhu : Associated \u2191(seq K w\u2081 hB n) \u2191(seq K w\u2081 hB m)\nw : InfinitePlace K\nhw : w \u2260 w\u2081\n\u22a2 w ((algebraMap (\ud835\udcde K) K) \u2191(Exists.choose hu)) < 1", "state_after": "no goals"}, {"tactic": "rw [\u2190 congr_arg (algebraMap (\ud835\udcde K) K) hu.choose_spec, mul_comm, map_mul (algebraMap _ _),\n\u2190 mul_assoc, inv_mul_cancel (seq_ne_zero K w\u2081 hB n), one_mul]", "annotated_tactic": ["rw [\u2190 <a>congr_arg</a> (<a>algebraMap</a> (\ud835\udcde K) K) hu.choose_spec, <a>mul_comm</a>, <a>map_mul</a> (<a>algebraMap</a> _ _),\n          \u2190 <a>mul_assoc</a>, <a>inv_mul_cancel</a> (<a>seq_ne_zero</a> K w\u2081 hB n), <a>one_mul</a>]", [{"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}, {"full_name": "map_mul", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [309, 9], "def_end_pos": [309, 16]}, {"full_name": "algebraMap", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [110, 5], "def_end_pos": [110, 15]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "inv_mul_cancel", "def_path": "Mathlib/Algebra/GroupWithZero/NeZero.lean", "def_pos": [55, 9], "def_end_pos": [55, 23]}, {"full_name": "NumberField.Units.dirichletUnitTheorem.seq_ne_zero", "def_path": "Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean", "def_pos": [241, 9], "def_end_pos": [241, 20]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nn m : \u2115\nhnm : n < m\nh : Ideal.span {\u2191(seq K w\u2081 hB n)} = Ideal.span {\u2191(seq K w\u2081 hB m)}\nhu : Associated \u2191(seq K w\u2081 hB n) \u2191(seq K w\u2081 hB m)\nw : InfinitePlace K\nhw : w \u2260 w\u2081\n\u22a2 w ((algebraMap (\ud835\udcde K) K) \u2191(Exists.choose hu)) =\n    w ((algebraMap (\ud835\udcde K) K) \u2191(seq K w\u2081 hB m) * ((algebraMap (\ud835\udcde K) K) \u2191(seq K w\u2081 hB n))\u207b\u00b9)", "state_after": "no goals"}, {"tactic": "rw [map_inv\u2080, mul_inv_lt_iff (pos_iff.mpr (seq_ne_zero K w\u2081 hB n)), mul_one]", "annotated_tactic": ["rw [<a>map_inv\u2080</a>, <a>mul_inv_lt_iff</a> (pos_iff.mpr (<a>seq_ne_zero</a> K w\u2081 hB n)), <a>mul_one</a>]", [{"full_name": "map_inv\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [64, 9], "def_end_pos": [64, 17]}, {"full_name": "mul_inv_lt_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [125, 9], "def_end_pos": [125, 23]}, {"full_name": "NumberField.Units.dirichletUnitTheorem.seq_ne_zero", "def_path": "Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean", "def_pos": [241, 9], "def_end_pos": [241, 20]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nn m : \u2115\nhnm : n < m\nh : Ideal.span {\u2191(seq K w\u2081 hB n)} = Ideal.span {\u2191(seq K w\u2081 hB m)}\nhu : Associated \u2191(seq K w\u2081 hB n) \u2191(seq K w\u2081 hB m)\nw : InfinitePlace K\nhw : w \u2260 w\u2081\n\u22a2 w ((algebraMap (\ud835\udcde K) K) \u2191(seq K w\u2081 hB m)) * w ((algebraMap (\ud835\udcde K) K) \u2191(seq K w\u2081 hB n))\u207b\u00b9 < 1", "state_after": "K : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nn m : \u2115\nhnm : n < m\nh : Ideal.span {\u2191(seq K w\u2081 hB n)} = Ideal.span {\u2191(seq K w\u2081 hB m)}\nhu : Associated \u2191(seq K w\u2081 hB n) \u2191(seq K w\u2081 hB m)\nw : InfinitePlace K\nhw : w \u2260 w\u2081\n\u22a2 w ((algebraMap (\ud835\udcde K) K) \u2191(seq K w\u2081 hB m)) < w ((algebraMap (\ud835\udcde K) K) \u2191(seq K w\u2081 hB n))"}, {"tactic": "exact seq_decreasing K w\u2081 hB hnm w hw", "annotated_tactic": ["exact <a>seq_decreasing</a> K w\u2081 hB hnm w hw", [{"full_name": "NumberField.Units.dirichletUnitTheorem.seq_decreasing", "def_path": "Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean", "def_pos": [249, 9], "def_end_pos": [249, 23]}]], "state_before": "K : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nn m : \u2115\nhnm : n < m\nh : Ideal.span {\u2191(seq K w\u2081 hB n)} = Ideal.span {\u2191(seq K w\u2081 hB m)}\nhu : Associated \u2191(seq K w\u2081 hB n) \u2191(seq K w\u2081 hB m)\nw : InfinitePlace K\nhw : w \u2260 w\u2081\n\u22a2 w ((algebraMap (\ud835\udcde K) K) \u2191(seq K w\u2081 hB m)) < w ((algebraMap (\ud835\udcde K) K) \u2191(seq K w\u2081 hB n))", "state_after": "no goals"}, {"tactic": "rw [Set.mem_setOf_eq, Ideal.absNorm_span_singleton]", "annotated_tactic": ["rw [<a>Set.mem_setOf_eq</a>, <a>Ideal.absNorm_span_singleton</a>]", [{"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [79, 29], "def_end_pos": [79, 41]}, {"full_name": "Ideal.absNorm_span_singleton", "def_path": "Mathlib/RingTheory/Ideal/Norm.lean", "def_pos": [365, 9], "def_end_pos": [365, 31]}]], "state_before": "case refine_1\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nn : \u2115\n\u22a2 Ideal.span {\u2191(seq K w\u2081 hB n)} \u2208 {I | 1 \u2264 Ideal.absNorm I \u2227 Ideal.absNorm I \u2264 B}", "state_after": "case refine_1\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nn : \u2115\n\u22a2 1 \u2264 ((Algebra.norm \u2124) \u2191(seq K w\u2081 hB n)).natAbs \u2227 ((Algebra.norm \u2124) \u2191(seq K w\u2081 hB n)).natAbs \u2264 B"}, {"tactic": "refine \u27e8?_, seq_norm_le K w\u2081 hB n\u27e9", "annotated_tactic": ["refine \u27e8?_, <a>seq_norm_le</a> K w\u2081 hB n\u27e9", [{"full_name": "NumberField.Units.dirichletUnitTheorem.seq_norm_le", "def_path": "Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean", "def_pos": [265, 9], "def_end_pos": [265, 20]}]], "state_before": "case refine_1\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nn : \u2115\n\u22a2 1 \u2264 ((Algebra.norm \u2124) \u2191(seq K w\u2081 hB n)).natAbs \u2227 ((Algebra.norm \u2124) \u2191(seq K w\u2081 hB n)).natAbs \u2264 B", "state_after": "case refine_1\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nn : \u2115\n\u22a2 1 \u2264 ((Algebra.norm \u2124) \u2191(seq K w\u2081 hB n)).natAbs"}, {"tactic": "exact Nat.one_le_iff_ne_zero.mpr (Int.natAbs_ne_zero.mpr (seq_norm_ne_zero K w\u2081 hB n))", "annotated_tactic": ["exact Nat.one_le_iff_ne_zero.mpr (Int.natAbs_ne_zero.mpr (<a>seq_norm_ne_zero</a> K w\u2081 hB n))", [{"full_name": "NumberField.Units.dirichletUnitTheorem.seq_norm_ne_zero", "def_path": "Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean", "def_pos": [245, 9], "def_end_pos": [245, 25]}]], "state_before": "case refine_1\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nn : \u2115\n\u22a2 1 \u2264 ((Algebra.norm \u2124) \u2191(seq K w\u2081 hB n)).natAbs", "state_after": "no goals"}, {"tactic": "rw [show { I : Ideal (\ud835\udcde K) | 1 \u2264 Ideal.absNorm I \u2227 Ideal.absNorm I \u2264 B } =\n      (\u22c3 n \u2208 Set.Icc 1 B, { I : Ideal (\ud835\udcde K) | Ideal.absNorm I = n }) by ext; simp]", "annotated_tactic": ["rw [show { I : <a>Ideal</a> (\ud835\udcde K) | 1 \u2264 <a>Ideal.absNorm</a> I \u2227 <a>Ideal.absNorm</a> I \u2264 B } =\n          (\u22c3 n \u2208 <a>Set.Icc</a> 1 B, { I : <a>Ideal</a> (\ud835\udcde K) | <a>Ideal.absNorm</a> I = n }) by ext; simp]", [{"full_name": "Ideal", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [41, 8], "def_end_pos": [41, 13]}, {"full_name": "Ideal.absNorm", "def_path": "Mathlib/RingTheory/Ideal/Norm.lean", "def_pos": [251, 19], "def_end_pos": [251, 32]}, {"full_name": "Ideal.absNorm", "def_path": "Mathlib/RingTheory/Ideal/Norm.lean", "def_pos": [251, 19], "def_end_pos": [251, 32]}, {"full_name": "Set.Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}, {"full_name": "Ideal", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [41, 8], "def_end_pos": [41, 13]}, {"full_name": "Ideal.absNorm", "def_path": "Mathlib/RingTheory/Ideal/Norm.lean", "def_pos": [251, 19], "def_end_pos": [251, 32]}]], "state_before": "case refine_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\n\u22a2 {I | 1 \u2264 Ideal.absNorm I \u2227 Ideal.absNorm I \u2264 B}.Finite", "state_after": "case refine_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\n\u22a2 (\u22c3 n \u2208 Set.Icc 1 B, {I | Ideal.absNorm I = n}).Finite"}, {"tactic": "exact Set.Finite.biUnion (Set.finite_Icc _ _) (fun n hn => Ideal.finite_setOf_absNorm_eq hn.1)", "annotated_tactic": ["exact <a>Set.Finite.biUnion</a> (<a>Set.finite_Icc</a> _ _) (fun n hn => <a>Ideal.finite_setOf_absNorm_eq</a> hn.1)", [{"full_name": "Set.Finite.biUnion", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [801, 9], "def_end_pos": [801, 23]}, {"full_name": "Set.finite_Icc", "def_path": "Mathlib/Order/Interval/Finset/Defs.lean", "def_pos": [523, 9], "def_end_pos": [523, 19]}, {"full_name": "Ideal.finite_setOf_absNorm_eq", "def_path": "Mathlib/RingTheory/Ideal/Norm.lean", "def_pos": [457, 9], "def_end_pos": [457, 32]}]], "state_before": "case refine_2\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\n\u22a2 (\u22c3 n \u2208 Set.Icc 1 B, {I | Ideal.absNorm I = n}).Finite", "state_after": "no goals"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "K : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\n\u22a2 {I | 1 \u2264 Ideal.absNorm I \u2227 Ideal.absNorm I \u2264 B} = \u22c3 n \u2208 Set.Icc 1 B, {I | Ideal.absNorm I = n}", "state_after": "case h\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nx\u271d : Ideal (\ud835\udcde K)\n\u22a2 x\u271d \u2208 {I | 1 \u2264 Ideal.absNorm I \u2227 Ideal.absNorm I \u2264 B} \u2194 x\u271d \u2208 \u22c3 n \u2208 Set.Icc 1 B, {I | Ideal.absNorm I = n}"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\nK : Type u_1\ninst\u271d\u00b9 : Field K\ninst\u271d : NumberField K\nw\u2081\u271d : InfinitePlace K\nB\u271d : \u2115\nhB\u271d : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\u271d\nw\u2081 : InfinitePlace K\nB : \u2115\nhB : minkowskiBound K 1 < \u2191(convexBodyLTFactor K) * \u2191B\nx\u271d : Ideal (\ud835\udcde K)\n\u22a2 x\u271d \u2208 {I | 1 \u2264 Ideal.absNorm I \u2227 Ideal.absNorm I \u2264 B} \u2194 x\u271d \u2208 \u22c3 n \u2208 Set.Icc 1 B, {I | Ideal.absNorm I = n}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Basic.lean", "full_name": "List.bind_ret_eq_map", "start": [1567, 1], "end": [1568, 23], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean", "full_name": "CategoryTheory.BraidedCategory.hexagon_forward_inv", "start": [181, 1], "end": [184, 7], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\ninst\u271d\u00b2 : Category.{v, u} C\ninst\u271d\u00b9 : MonoidalCategory C\ninst\u271d : BraidedCategory C\nX Y Z : C\n\u22a2 (\u03b1_ Y Z X).inv \u226b (\u03b2_ X (Y \u2297 Z)).inv \u226b (\u03b1_ X Y Z).inv = Y \u25c1 (\u03b2_ X Z).inv \u226b (\u03b1_ Y X Z).inv \u226b (\u03b2_ X Y).inv \u25b7 Z", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Dart.lean", "full_name": "SimpleGraph.Dart.symm_symm", "start": [94, 1], "end": [95, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/ConditionalProbability.lean", "full_name": "ProbabilityTheory.absolutelyContinuous_cond_univ", "start": [122, 1], "end": [125, 24], "traced_tactics": [{"tactic": "rw [cond, restrict_univ]", "annotated_tactic": ["rw [<a>cond</a>, <a>restrict_univ</a>]", [{"full_name": "ProbabilityTheory.cond", "def_path": "Mathlib/Probability/ConditionalProbability.lean", "def_pos": [73, 5], "def_end_pos": [73, 9]}, {"full_name": "MeasureTheory.Measure.restrict_univ", "def_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "def_pos": [239, 9], "def_end_pos": [239, 22]}]], "state_before": "\u03a9 : Type u_1\n\u03a9' : Type u_2\n\u03b1 : Type u_3\nm : MeasurableSpace \u03a9\nm' : MeasurableSpace \u03a9'\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\n\u22a2 \u03bc \u226a \u03bc[|univ]", "state_after": "\u03a9 : Type u_1\n\u03a9' : Type u_2\n\u03b1 : Type u_3\nm : MeasurableSpace \u03a9\nm' : MeasurableSpace \u03a9'\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\n\u22a2 \u03bc \u226a (\u03bc univ)\u207b\u00b9 \u2022 \u03bc"}, {"tactic": "refine absolutelyContinuous_smul ?_", "annotated_tactic": ["refine <a>absolutelyContinuous_smul</a> ?_", [{"full_name": "MeasureTheory.Measure.absolutelyContinuous_smul", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1712, 7], "def_end_pos": [1712, 32]}]], "state_before": "\u03a9 : Type u_1\n\u03a9' : Type u_2\n\u03b1 : Type u_3\nm : MeasurableSpace \u03a9\nm' : MeasurableSpace \u03a9'\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\n\u22a2 \u03bc \u226a (\u03bc univ)\u207b\u00b9 \u2022 \u03bc", "state_after": "\u03a9 : Type u_1\n\u03a9' : Type u_2\n\u03b1 : Type u_3\nm : MeasurableSpace \u03a9\nm' : MeasurableSpace \u03a9'\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\n\u22a2 (\u03bc univ)\u207b\u00b9 \u2260 0"}, {"tactic": "simp [measure_ne_top]", "annotated_tactic": ["simp [<a>measure_ne_top</a>]", [{"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "def_pos": [61, 9], "def_end_pos": [61, 23]}]], "state_before": "\u03a9 : Type u_1\n\u03a9' : Type u_2\n\u03b1 : Type u_3\nm : MeasurableSpace \u03a9\nm' : MeasurableSpace \u03a9'\n\u03bc : Measure \u03a9\ns t : Set \u03a9\ninst\u271d : IsFiniteMeasure \u03bc\n\u22a2 (\u03bc univ)\u207b\u00b9 \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Multiplicity.lean", "full_name": "multiplicity.le_multiplicity_map", "start": [254, 1], "end": [256, 82], "traced_tactics": [{"tactic": "rw [\u2190 map_pow]", "annotated_tactic": ["rw [\u2190 <a>map_pow</a>]", [{"full_name": "map_pow", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [473, 9], "def_end_pos": [473, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Monoid \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d\u00b2 : DecidableRel fun x x_1 => x \u2223 x_1\nF : Type u_3\ninst\u271d\u00b9 : FunLike F \u03b1 \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : F\na b : \u03b1\nn : \u2115\n\u22a2 a ^ n \u2223 b \u2192 f a ^ n \u2223 f b", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Monoid \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d\u00b2 : DecidableRel fun x x_1 => x \u2223 x_1\nF : Type u_3\ninst\u271d\u00b9 : FunLike F \u03b1 \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : F\na b : \u03b1\nn : \u2115\n\u22a2 a ^ n \u2223 b \u2192 f (a ^ n) \u2223 f b"}, {"tactic": "exact map_dvd f", "annotated_tactic": ["exact <a>map_dvd</a> f", [{"full_name": "map_dvd", "def_path": "Mathlib/Algebra/Divisibility/Basic.lean", "def_pos": [101, 9], "def_end_pos": [101, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u2075 : Monoid \u03b1\ninst\u271d\u2074 : Monoid \u03b2\ninst\u271d\u00b3 : DecidableRel fun x x_1 => x \u2223 x_1\ninst\u271d\u00b2 : DecidableRel fun x x_1 => x \u2223 x_1\nF : Type u_3\ninst\u271d\u00b9 : FunLike F \u03b1 \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nf : F\na b : \u03b1\nn : \u2115\n\u22a2 a ^ n \u2223 b \u2192 f (a ^ n) \u2223 f b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Finsupp.lean", "full_name": "LinearMap.leftInverse_splittingOfFinsuppSurjective", "start": [1392, 1], "end": [1394, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/Arithmetic.lean", "full_name": "Ordinal.le_succ_of_isLimit", "start": [270, 1], "end": [271, 52], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Maps.lean", "full_name": "IsClosedMap.of_nonempty", "start": [477, 1], "end": [481, 21], "traced_tactics": [{"tactic": "intro s hs", "annotated_tactic": ["intro s hs", []], "state_before": "X : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b9 : Type u_4\nf : X \u2192 Y\ng : Y \u2192 Z\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nh : \u2200 (s : Set X), IsClosed s \u2192 s.Nonempty \u2192 IsClosed (f '' s)\n\u22a2 IsClosedMap f", "state_after": "X : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b9 : Type u_4\nf : X \u2192 Y\ng : Y \u2192 Z\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nh : \u2200 (s : Set X), IsClosed s \u2192 s.Nonempty \u2192 IsClosed (f '' s)\ns : Set X\nhs : IsClosed s\n\u22a2 IsClosed (f '' s)"}, {"tactic": "rcases eq_empty_or_nonempty s with h2s | h2s", "annotated_tactic": ["rcases <a>eq_empty_or_nonempty</a> s with h2s | h2s", [{"full_name": "Set.eq_empty_or_nonempty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [605, 9], "def_end_pos": [605, 29]}]], "state_before": "X : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b9 : Type u_4\nf : X \u2192 Y\ng : Y \u2192 Z\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nh : \u2200 (s : Set X), IsClosed s \u2192 s.Nonempty \u2192 IsClosed (f '' s)\ns : Set X\nhs : IsClosed s\n\u22a2 IsClosed (f '' s)", "state_after": "case inl\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b9 : Type u_4\nf : X \u2192 Y\ng : Y \u2192 Z\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nh : \u2200 (s : Set X), IsClosed s \u2192 s.Nonempty \u2192 IsClosed (f '' s)\ns : Set X\nhs : IsClosed s\nh2s : s = \u2205\n\u22a2 IsClosed (f '' s)\n\ncase inr\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b9 : Type u_4\nf : X \u2192 Y\ng : Y \u2192 Z\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nh : \u2200 (s : Set X), IsClosed s \u2192 s.Nonempty \u2192 IsClosed (f '' s)\ns : Set X\nhs : IsClosed s\nh2s : s.Nonempty\n\u22a2 IsClosed (f '' s)"}, {"tactic": "simp_rw [h2s, image_empty, isClosed_empty]", "annotated_tactic": ["simp_rw [h2s, <a>image_empty</a>, <a>isClosed_empty</a>]", [{"full_name": "Set.image_empty", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [309, 9], "def_end_pos": [309, 20]}, {"full_name": "isClosed_empty", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [173, 17], "def_end_pos": [173, 31]}]], "state_before": "case inl\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b9 : Type u_4\nf : X \u2192 Y\ng : Y \u2192 Z\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nh : \u2200 (s : Set X), IsClosed s \u2192 s.Nonempty \u2192 IsClosed (f '' s)\ns : Set X\nhs : IsClosed s\nh2s : s = \u2205\n\u22a2 IsClosed (f '' s)", "state_after": "no goals"}, {"tactic": "exact h s hs h2s", "annotated_tactic": ["exact h s hs h2s", []], "state_before": "case inr\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b9 : Type u_4\nf : X \u2192 Y\ng : Y \u2192 Z\ninst\u271d\u00b2 : TopologicalSpace X\ninst\u271d\u00b9 : TopologicalSpace Y\ninst\u271d : TopologicalSpace Z\nh : \u2200 (s : Set X), IsClosed s \u2192 s.Nonempty \u2192 IsClosed (f '' s)\ns : Set X\nhs : IsClosed s\nh2s : s.Nonempty\n\u22a2 IsClosed (f '' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/TypeVec.lean", "full_name": "TypeVec.dropFun_prod", "start": [730, 1], "end": [733, 44], "traced_tactics": [{"tactic": "ext i : 2", "annotated_tactic": ["ext i : 2", []], "state_before": "n : \u2115\n\u03b1 \u03b1' \u03b2 \u03b2' : TypeVec.{u_1} (n + 1)\nf : \u03b1 \u27f9 \u03b2\nf' : \u03b1' \u27f9 \u03b2'\n\u22a2 dropFun (f \u2297' f') = (dropFun f \u2297' dropFun f')", "state_after": "case a.h\nn : \u2115\n\u03b1 \u03b1' \u03b2 \u03b2' : TypeVec.{u_1} (n + 1)\nf : \u03b1 \u27f9 \u03b2\nf' : \u03b1' \u27f9 \u03b2'\ni : Fin2 n\nx\u271d : (\u03b1 \u2297 \u03b1').drop i\n\u22a2 dropFun (f \u2297' f') i x\u271d = (dropFun f \u2297' dropFun f') i x\u271d"}, {"tactic": "induction i <;> simp [dropFun, *] <;> rfl", "annotated_tactic": ["induction i <;> simp [<a>dropFun</a>, *] <;> rfl", [{"full_name": "TypeVec.dropFun", "def_path": "Mathlib/Data/TypeVec.lean", "def_pos": [158, 5], "def_end_pos": [158, 12]}]], "state_before": "case a.h\nn : \u2115\n\u03b1 \u03b1' \u03b2 \u03b2' : TypeVec.{u_1} (n + 1)\nf : \u03b1 \u27f9 \u03b2\nf' : \u03b1' \u27f9 \u03b2'\ni : Fin2 n\nx\u271d : (\u03b1 \u2297 \u03b1').drop i\n\u22a2 dropFun (f \u2297' f') i x\u271d = (dropFun f \u2297' dropFun f') i x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/ContDiff/Basic.lean", "full_name": "ContDiff.div_const", "start": [1576, 1], "end": [1577, 96], "traced_tactics": [{"tactic": "simpa only [div_eq_mul_inv] using hf.mul contDiff_const", "annotated_tactic": ["simpa only [<a>div_eq_mul_inv</a>] using hf.mul <a>contDiff_const</a>", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1059, 9], "def_end_pos": [1059, 23]}, {"full_name": "contDiff_const", "def_path": "Mathlib/Analysis/Calculus/ContDiff/Basic.lean", "def_pos": [86, 9], "def_end_pos": [86, 23]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u00b9\u2075 : NormedAddCommGroup D\ninst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u00b9\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u00b9\u00b9 : NormedAddCommGroup F\ninst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u2079 : NormedAddCommGroup G\ninst\u271d\u2078 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u2077 : NormedAddCommGroup X\ninst\u271d\u2076 : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf\u271d f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc\u271d : F\nb : E \u00d7 F \u2192 G\nm n\u271d : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\n\ud835\udd38 : Type u_3\n\ud835\udd38' : Type u_4\n\u03b9 : Type u_5\n\ud835\udd5c' : Type u_6\ninst\u271d\u2075 : NormedRing \ud835\udd38\ninst\u271d\u2074 : NormedAlgebra \ud835\udd5c \ud835\udd38\ninst\u271d\u00b3 : NormedCommRing \ud835\udd38'\ninst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd38'\ninst\u271d\u00b9 : NormedField \ud835\udd5c'\ninst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd5c'\nf : E \u2192 \ud835\udd5c'\nn : \u2115\u221e\nhf : ContDiff \ud835\udd5c n f\nc : \ud835\udd5c'\n\u22a2 ContDiff \ud835\udd5c n fun x => f x / c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean", "full_name": "AffineMap.image_vsub_image", "start": [495, 1], "end": [505, 30], "traced_tactics": [{"tactic": "ext v", "annotated_tactic": ["ext v", []], "state_before": "k : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\ns t : Set P1\nf : P1 \u2192\u1d43[k] P2\n\u22a2 \u21d1f '' s -\u1d65 \u21d1f '' t = \u21d1f.linear '' (s -\u1d65 t)", "state_after": "case h\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\ns t : Set P1\nf : P1 \u2192\u1d43[k] P2\nv : V2\n\u22a2 v \u2208 \u21d1f '' s -\u1d65 \u21d1f '' t \u2194 v \u2208 \u21d1f.linear '' (s -\u1d65 t)"}, {"tactic": "simp only [(Set.mem_vsub), Set.mem_image,\n  exists_exists_and_eq_and, exists_and_left, \u2190 f.linearMap_vsub]", "annotated_tactic": ["simp only [(<a>Set.mem_vsub</a>), <a>Set.mem_image</a>,\n    <a>exists_exists_and_eq_and</a>, <a>exists_and_left</a>, \u2190 f.linearMap_vsub]", [{"full_name": "Set.mem_vsub", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [616, 9], "def_end_pos": [616, 17]}, {"full_name": "Set.mem_image", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [127, 9], "def_end_pos": [127, 18]}, {"full_name": "exists_exists_and_eq_and", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [794, 17], "def_end_pos": [794, 41]}, {"full_name": "exists_and_left", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [288, 17], "def_end_pos": [288, 32]}]], "state_before": "case h\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\ns t : Set P1\nf : P1 \u2192\u1d43[k] P2\nv : V2\n\u22a2 v \u2208 \u21d1f '' s -\u1d65 \u21d1f '' t \u2194 v \u2208 \u21d1f.linear '' (s -\u1d65 t)", "state_after": "case h\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\ns t : Set P1\nf : P1 \u2192\u1d43[k] P2\nv : V2\n\u22a2 (\u2203 a \u2208 s, \u2203 a_1 \u2208 t, f.linear (a -\u1d65 a_1) = v) \u2194 \u2203 x, (\u2203 x_1 \u2208 s, \u2203 y \u2208 t, x_1 -\u1d65 y = x) \u2227 f.linear x = v"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\ns t : Set P1\nf : P1 \u2192\u1d43[k] P2\nv : V2\n\u22a2 (\u2203 a \u2208 s, \u2203 a_1 \u2208 t, f.linear (a -\u1d65 a_1) = v) \u2194 \u2203 x, (\u2203 x_1 \u2208 s, \u2203 y \u2208 t, x_1 -\u1d65 y = x) \u2227 f.linear x = v", "state_after": "case h.mp\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\ns t : Set P1\nf : P1 \u2192\u1d43[k] P2\nv : V2\n\u22a2 (\u2203 a \u2208 s, \u2203 a_1 \u2208 t, f.linear (a -\u1d65 a_1) = v) \u2192 \u2203 x, (\u2203 x_1 \u2208 s, \u2203 y \u2208 t, x_1 -\u1d65 y = x) \u2227 f.linear x = v\n\ncase h.mpr\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\ns t : Set P1\nf : P1 \u2192\u1d43[k] P2\nv : V2\n\u22a2 (\u2203 x, (\u2203 x_1 \u2208 s, \u2203 y \u2208 t, x_1 -\u1d65 y = x) \u2227 f.linear x = v) \u2192 \u2203 a \u2208 s, \u2203 a_1 \u2208 t, f.linear (a -\u1d65 a_1) = v"}, {"tactic": "rintro \u27e8x, hx, y, hy, hv\u27e9", "annotated_tactic": ["rintro \u27e8x, hx, y, hy, hv\u27e9", []], "state_before": "case h.mp\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\ns t : Set P1\nf : P1 \u2192\u1d43[k] P2\nv : V2\n\u22a2 (\u2203 a \u2208 s, \u2203 a_1 \u2208 t, f.linear (a -\u1d65 a_1) = v) \u2192 \u2203 x, (\u2203 x_1 \u2208 s, \u2203 y \u2208 t, x_1 -\u1d65 y = x) \u2227 f.linear x = v", "state_after": "case h.mp.intro.intro.intro.intro\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\ns t : Set P1\nf : P1 \u2192\u1d43[k] P2\nv : V2\nx : P1\nhx : x \u2208 s\ny : P1\nhy : y \u2208 t\nhv : f.linear (x -\u1d65 y) = v\n\u22a2 \u2203 x, (\u2203 x_1 \u2208 s, \u2203 y \u2208 t, x_1 -\u1d65 y = x) \u2227 f.linear x = v"}, {"tactic": "exact \u27e8x -\u1d65 y, \u27e8x, hx, y, hy, rfl\u27e9, hv\u27e9", "annotated_tactic": ["exact \u27e8x -\u1d65 y, \u27e8x, hx, y, hy, <a>rfl</a>\u27e9, hv\u27e9", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case h.mp.intro.intro.intro.intro\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\ns t : Set P1\nf : P1 \u2192\u1d43[k] P2\nv : V2\nx : P1\nhx : x \u2208 s\ny : P1\nhy : y \u2208 t\nhv : f.linear (x -\u1d65 y) = v\n\u22a2 \u2203 x, (\u2203 x_1 \u2208 s, \u2203 y \u2208 t, x_1 -\u1d65 y = x) \u2227 f.linear x = v", "state_after": "no goals"}, {"tactic": "rintro \u27e8-, \u27e8x, hx, y, hy, rfl\u27e9, rfl\u27e9", "annotated_tactic": ["rintro \u27e8-, \u27e8x, hx, y, hy, rfl\u27e9, rfl\u27e9", []], "state_before": "case h.mpr\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\ns t : Set P1\nf : P1 \u2192\u1d43[k] P2\nv : V2\n\u22a2 (\u2203 x, (\u2203 x_1 \u2208 s, \u2203 y \u2208 t, x_1 -\u1d65 y = x) \u2227 f.linear x = v) \u2192 \u2203 a \u2208 s, \u2203 a_1 \u2208 t, f.linear (a -\u1d65 a_1) = v", "state_after": "case h.mpr.intro.intro.intro.intro.intro.intro\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\ns t : Set P1\nf : P1 \u2192\u1d43[k] P2\nx : P1\nhx : x \u2208 s\ny : P1\nhy : y \u2208 t\n\u22a2 \u2203 a \u2208 s, \u2203 a_1 \u2208 t, f.linear (a -\u1d65 a_1) = f.linear (x -\u1d65 y)"}, {"tactic": "exact \u27e8x, hx, y, hy, rfl\u27e9", "annotated_tactic": ["exact \u27e8x, hx, y, hy, <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case h.mpr.intro.intro.intro.intro.intro.intro\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst\u271d\u00b9\u00b2 : Ring k\ninst\u271d\u00b9\u00b9 : AddCommGroup V1\ninst\u271d\u00b9\u2070 : Module k V1\ninst\u271d\u2079 : AffineSpace V1 P1\ninst\u271d\u2078 : AddCommGroup V2\ninst\u271d\u2077 : Module k V2\ninst\u271d\u2076 : AffineSpace V2 P2\ninst\u271d\u2075 : AddCommGroup V3\ninst\u271d\u2074 : Module k V3\ninst\u271d\u00b3 : AffineSpace V3 P3\ninst\u271d\u00b2 : AddCommGroup V4\ninst\u271d\u00b9 : Module k V4\ninst\u271d : AffineSpace V4 P4\ns t : Set P1\nf : P1 \u2192\u1d43[k] P2\nx : P1\nhx : x \u2208 s\ny : P1\nhy : y \u2208 t\n\u22a2 \u2203 a \u2208 s, \u2203 a_1 \u2208 t, f.linear (a -\u1d65 a_1) = f.linear (x -\u1d65 y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "full_name": "Real.log_of_pos", "start": [49, 1], "end": [52, 22], "traced_tactics": [{"tactic": "rw [log_of_ne_zero hx.ne']", "annotated_tactic": ["rw [<a>log_of_ne_zero</a> hx.ne']", [{"full_name": "Real.log_of_ne_zero", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [45, 9], "def_end_pos": [45, 23]}]], "state_before": "x y : \u211d\nhx : 0 < x\n\u22a2 log x = expOrderIso.symm \u27e8x, hx\u27e9", "state_after": "x y : \u211d\nhx : 0 < x\n\u22a2 expOrderIso.symm \u27e8|x|, \u22ef\u27e9 = expOrderIso.symm \u27e8x, hx\u27e9"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "x y : \u211d\nhx : 0 < x\n\u22a2 expOrderIso.symm \u27e8|x|, \u22ef\u27e9 = expOrderIso.symm \u27e8x, hx\u27e9", "state_after": "case h.e_6.h.e_val\nx y : \u211d\nhx : 0 < x\n\u22a2 |x| = x"}, {"tactic": "exact abs_of_pos hx", "annotated_tactic": ["exact <a>abs_of_pos</a> hx", [{"full_name": "abs_of_pos", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [101, 3], "def_end_pos": [101, 14]}]], "state_before": "case h.e_6.h.e_val\nx y : \u211d\nhx : 0 < x\n\u22a2 |x| = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/ScottTopology.lean", "full_name": "Topology.WithScott.toScott_inj", "start": [314, 1], "end": [315, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/MeanInequalitiesPow.lean", "full_name": "NNReal.pow_arith_mean_le_arith_mean_pow", "start": [119, 1], "end": [123, 53], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Basic.lean", "full_name": "Convex.is_linear_preimage", "start": [206, 1], "end": [208, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ENNReal/Operations.lean", "full_name": "ENNReal.pow_eq_top_iff", "start": [267, 9], "end": [272, 51], "traced_tactics": [{"tactic": "rcases n.eq_zero_or_pos with rfl | (hn : 0 < n)", "annotated_tactic": ["rcases n.eq_zero_or_pos with rfl | (hn : 0 < n)", []], "state_before": "a b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_1\nn : \u2115\n\u22a2 a ^ n = \u22a4 \u2194 a = \u22a4 \u2227 n \u2260 0", "state_after": "case inl\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_1\n\u22a2 a ^ 0 = \u22a4 \u2194 a = \u22a4 \u2227 0 \u2260 0\n\ncase inr\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_1\nn : \u2115\nhn : 0 < n\n\u22a2 a ^ n = \u22a4 \u2194 a = \u22a4 \u2227 n \u2260 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case inl\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_1\n\u22a2 a ^ 0 = \u22a4 \u2194 a = \u22a4 \u2227 0 \u2260 0", "state_after": "no goals"}, {"tactic": "induction a", "annotated_tactic": ["induction a", []], "state_before": "case inr\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_1\nn : \u2115\nhn : 0 < n\n\u22a2 a ^ n = \u22a4 \u2194 a = \u22a4 \u2227 n \u2260 0", "state_after": "case inr.top\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_1\nn : \u2115\nhn : 0 < n\n\u22a2 \u22a4 ^ n = \u22a4 \u2194 \u22a4 = \u22a4 \u2227 n \u2260 0\n\ncase inr.coe\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_1\nn : \u2115\nhn : 0 < n\nx\u271d : \u211d\u22650\n\u22a2 \u2191x\u271d ^ n = \u22a4 \u2194 \u2191x\u271d = \u22a4 \u2227 n \u2260 0"}, {"tactic": "simp only [Ne, hn.ne', top_pow hn, not_false_eq_true, and_self]", "annotated_tactic": ["simp only [<a>Ne</a>, hn.ne', <a>top_pow</a> hn, <a>not_false_eq_true</a>, <a>and_self</a>]", [{"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "ENNReal.top_pow", "def_path": "Mathlib/Data/ENNReal/Operations.lean", "def_pos": [222, 9], "def_end_pos": [222, 16]}, {"full_name": "not_false_eq_true", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [134, 17], "def_end_pos": [134, 34]}, {"full_name": "and_self", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [111, 17], "def_end_pos": [111, 25]}]], "state_before": "case inr.top\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_1\nn : \u2115\nhn : 0 < n\n\u22a2 \u22a4 ^ n = \u22a4 \u2194 \u22a4 = \u22a4 \u2227 n \u2260 0", "state_after": "no goals"}, {"tactic": "simp only [\u2190 coe_pow, coe_ne_top, false_and]", "annotated_tactic": ["simp only [\u2190 <a>coe_pow</a>, <a>coe_ne_top</a>, <a>false_and</a>]", [{"full_name": "ENNReal.coe_pow", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [418, 26], "def_end_pos": [418, 33]}, {"full_name": "ENNReal.coe_ne_top", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [325, 17], "def_end_pos": [325, 27]}, {"full_name": "false_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [110, 17], "def_end_pos": [110, 26]}]], "state_before": "case inr.coe\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b1 : Type u_1\nn : \u2115\nhn : 0 < n\nx\u271d : \u211d\u22650\n\u22a2 \u2191x\u271d ^ n = \u22a4 \u2194 \u2191x\u271d = \u22a4 \u2227 n \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/BooleanRing.lean", "full_name": "BoundedLatticeHom.asBoolRing_id", "start": [556, 1], "end": [557, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/DFinsupp/Basic.lean", "full_name": "DFinsupp.coe_neg", "start": [299, 1], "end": [299, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/HasLimits.lean", "full_name": "CategoryTheory.Limits.hasColimitsOfSizeOfUnivLE", "start": [1217, 1], "end": [1220, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/Star/ContinuousFunctionalCalculus.lean", "full_name": "elementalStarAlgebra.continuous_characterSpaceToSpectrum", "start": [237, 1], "end": [240, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Category/UniformSpace.lean", "full_name": "UniformSpaceCat.coe_mk", "start": [84, 1], "end": [86, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean", "full_name": "Path.Homotopy.transReflReparamAux_zero", "start": [143, 1], "end": [144, 33], "traced_tactics": [{"tactic": "norm_num [transReflReparamAux]", "annotated_tactic": ["norm_num [<a>transReflReparamAux</a>]", [{"full_name": "Path.Homotopy.transReflReparamAux", "def_path": "Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean", "def_pos": [125, 5], "def_end_pos": [125, 24]}]], "state_before": "X : Type u\nY : Type v\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\nx\u2080 x\u2081 : X\n\u22a2 transReflReparamAux 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/LpEquiv.lean", "full_name": "coe_algEquiv_lpBCF", "start": [226, 1], "end": [227, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Rotate.lean", "full_name": "List.rotate_eq_iff", "start": [340, 1], "end": [348, 33], "traced_tactics": [{"tactic": "rw [\u2190 @rotate_eq_rotate _ l _ n, rotate_rotate, \u2190 rotate_mod l', add_mod]", "annotated_tactic": ["rw [\u2190 @<a>rotate_eq_rotate</a> _ l _ n, <a>rotate_rotate</a>, \u2190 <a>rotate_mod</a> l', <a>add_mod</a>]", [{"full_name": "List.rotate_eq_rotate", "def_path": "Mathlib/Data/List/Rotate.lean", "def_pos": [336, 9], "def_end_pos": [336, 25]}, {"full_name": "List.rotate_rotate", "def_path": "Mathlib/Data/List/Rotate.lean", "def_pos": [162, 9], "def_end_pos": [162, 22]}, {"full_name": "List.rotate_mod", "def_path": "Mathlib/Data/List/Rotate.lean", "def_pos": [37, 9], "def_end_pos": [37, 19]}, {"full_name": "Nat.add_mod", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Lemmas.lean", "def_pos": [572, 9], "def_end_pos": [572, 16]}]], "state_before": "\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\n\u22a2 l.rotate n = l' \u2194 l = l'.rotate (l'.length - n % l'.length)", "state_after": "\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\n\u22a2 l.rotate n = l' \u2194 l.rotate n = l'.rotate (((l'.length - n % l'.length) % l'.length + n % l'.length) % l'.length)"}, {"tactic": "rcases l'.length.zero_le.eq_or_lt with hl | hl", "annotated_tactic": ["rcases l'.length.zero_le.eq_or_lt with hl | hl", []], "state_before": "\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\n\u22a2 l.rotate n = l' \u2194 l.rotate n = l'.rotate (((l'.length - n % l'.length) % l'.length + n % l'.length) % l'.length)", "state_after": "case inl\n\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\nhl : 0 = l'.length\n\u22a2 l.rotate n = l' \u2194 l.rotate n = l'.rotate (((l'.length - n % l'.length) % l'.length + n % l'.length) % l'.length)\n\ncase inr\n\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\nhl : 0 < l'.length\n\u22a2 l.rotate n = l' \u2194 l.rotate n = l'.rotate (((l'.length - n % l'.length) % l'.length + n % l'.length) % l'.length)"}, {"tactic": "rw [eq_nil_of_length_eq_zero hl.symm, rotate_nil]", "annotated_tactic": ["rw [<a>eq_nil_of_length_eq_zero</a> hl.symm, <a>rotate_nil</a>]", [{"full_name": "List.eq_nil_of_length_eq_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [93, 9], "def_end_pos": [93, 33]}, {"full_name": "List.rotate_nil", "def_path": "Mathlib/Data/List/Rotate.lean", "def_pos": [41, 9], "def_end_pos": [41, 19]}]], "state_before": "case inl\n\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\nhl : 0 = l'.length\n\u22a2 l.rotate n = l' \u2194 l.rotate n = l'.rotate (((l'.length - n % l'.length) % l'.length + n % l'.length) % l'.length)", "state_after": "no goals"}, {"tactic": "rcases (Nat.zero_le (n % l'.length)).eq_or_lt with hn | hn", "annotated_tactic": ["rcases (<a>Nat.zero_le</a> (n % l'.length)).<a>eq_or_lt</a> with hn | hn", [{"full_name": "Nat.zero_le", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [1663, 9], "def_end_pos": [1663, 20]}, {"full_name": "LE.le.eq_or_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [395, 7], "def_end_pos": [395, 21]}]], "state_before": "case inr\n\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\nhl : 0 < l'.length\n\u22a2 l.rotate n = l' \u2194 l.rotate n = l'.rotate (((l'.length - n % l'.length) % l'.length + n % l'.length) % l'.length)", "state_after": "case inr.inl\n\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\nhl : 0 < l'.length\nhn : 0 = n % l'.length\n\u22a2 l.rotate n = l' \u2194 l.rotate n = l'.rotate (((l'.length - n % l'.length) % l'.length + n % l'.length) % l'.length)\n\ncase inr.inr\n\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\nhl : 0 < l'.length\nhn : 0 < n % l'.length\n\u22a2 l.rotate n = l' \u2194 l.rotate n = l'.rotate (((l'.length - n % l'.length) % l'.length + n % l'.length) % l'.length)"}, {"tactic": "simp [\u2190 hn]", "annotated_tactic": ["simp [\u2190 hn]", []], "state_before": "case inr.inl\n\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\nhl : 0 < l'.length\nhn : 0 = n % l'.length\n\u22a2 l.rotate n = l' \u2194 l.rotate n = l'.rotate (((l'.length - n % l'.length) % l'.length + n % l'.length) % l'.length)", "state_after": "no goals"}, {"tactic": "rw [mod_eq_of_lt (Nat.sub_lt hl hn), Nat.sub_add_cancel, mod_self, rotate_zero]", "annotated_tactic": ["rw [<a>mod_eq_of_lt</a> (<a>Nat.sub_lt</a> hl hn), <a>Nat.sub_add_cancel</a>, <a>mod_self</a>, <a>rotate_zero</a>]", [{"full_name": "Nat.mod_eq_of_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [131, 9], "def_end_pos": [131, 21]}, {"full_name": "Nat.sub_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [296, 9], "def_end_pos": [296, 15]}, {"full_name": "Nat.sub_add_cancel", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [893, 27], "def_end_pos": [893, 41]}, {"full_name": "Nat.mod_self", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [174, 17], "def_end_pos": [174, 25]}, {"full_name": "List.rotate_zero", "def_path": "Mathlib/Data/List/Rotate.lean", "def_pos": [45, 9], "def_end_pos": [45, 20]}]], "state_before": "case inr.inr\n\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\nhl : 0 < l'.length\nhn : 0 < n % l'.length\n\u22a2 l.rotate n = l' \u2194 l.rotate n = l'.rotate (((l'.length - n % l'.length) % l'.length + n % l'.length) % l'.length)", "state_after": "case inr.inr\n\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\nhl : 0 < l'.length\nhn : 0 < n % l'.length\n\u22a2 n % l'.length \u2264 l'.length"}, {"tactic": "exact (Nat.mod_lt _ hl).le", "annotated_tactic": ["exact (<a>Nat.mod_lt</a> _ hl).<a>le</a>", [{"full_name": "Nat.mod_lt", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Div.lean", "def_pos": [142, 9], "def_end_pos": [142, 15]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}]], "state_before": "case inr.inr\n\u03b1 : Type u\nl l' : List \u03b1\nn : \u2115\nhl : 0 < l'.length\nhn : 0 < n % l'.length\n\u22a2 n % l'.length \u2264 l'.length", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Image.lean", "full_name": "Function.Injective.mem_finset_image", "start": [398, 1], "end": [402, 20], "traced_tactics": [{"tactic": "refine \u27e8fun h => ?_, Finset.mem_image_of_mem f\u27e9", "annotated_tactic": ["refine \u27e8fun h => ?_, <a>Finset.mem_image_of_mem</a> f\u27e9", [{"full_name": "Finset.mem_image_of_mem", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [361, 9], "def_end_pos": [361, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nf g : \u03b1 \u2192 \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\na : \u03b1\nb c : \u03b2\nhf : Injective f\n\u22a2 f a \u2208 image f s \u2194 a \u2208 s", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nf g : \u03b1 \u2192 \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\na : \u03b1\nb c : \u03b2\nhf : Injective f\nh : f a \u2208 image f s\n\u22a2 a \u2208 s"}, {"tactic": "obtain \u27e8y, hy, heq\u27e9 := mem_image.1 h", "annotated_tactic": ["obtain \u27e8y, hy, heq\u27e9 := <a>mem_image</a>.1 h", [{"full_name": "Finset.mem_image", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [357, 9], "def_end_pos": [357, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nf g : \u03b1 \u2192 \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\na : \u03b1\nb c : \u03b2\nhf : Injective f\nh : f a \u2208 image f s\n\u22a2 a \u2208 s", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nf g : \u03b1 \u2192 \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\na : \u03b1\nb c : \u03b2\nhf : Injective f\nh : f a \u2208 image f s\ny : \u03b1\nhy : y \u2208 s\nheq : f y = f a\n\u22a2 a \u2208 s"}, {"tactic": "exact hf heq \u25b8 hy", "annotated_tactic": ["exact hf heq \u25b8 hy", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nf g : \u03b1 \u2192 \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\na : \u03b1\nb c : \u03b2\nhf : Injective f\nh : f a \u2208 image f s\ny : \u03b1\nhy : y \u2208 s\nheq : f y = f a\n\u22a2 a \u2208 s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/ENNReal/Real.lean", "full_name": "ENNReal.iSup_zero_eq_zero", "start": [673, 1], "end": [673, 75], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b9 : Sort u_1\n\u22a2 \u2a06 x, 0 = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Function/Basic.lean", "full_name": "Function.Surjective.right_cancellable", "start": [234, 11], "end": [236, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matrix/Notation.lean", "full_name": "Matrix.vec2_add", "start": [510, 1], "end": [511, 53], "traced_tactics": [{"tactic": "rw [cons_add_cons, cons_add_cons, empty_add_empty]", "annotated_tactic": ["rw [<a>cons_add_cons</a>, <a>cons_add_cons</a>, <a>empty_add_empty</a>]", [{"full_name": "Matrix.cons_add_cons", "def_path": "Mathlib/Data/Fin/VecNotation.lean", "def_pos": [469, 9], "def_end_pos": [469, 22]}, {"full_name": "Matrix.cons_add_cons", "def_path": "Mathlib/Data/Fin/VecNotation.lean", "def_pos": [469, 9], "def_end_pos": [469, 22]}, {"full_name": "Matrix.empty_add_empty", "def_path": "Mathlib/Data/Fin/VecNotation.lean", "def_pos": [450, 9], "def_end_pos": [450, 24]}]], "state_before": "\u03b1 : Type u\no n m : \u2115\nm' : Type u\u2098\nn' : Type u\u2099\no' : Type u\u2092\na b : \u2115\ninst\u271d : Add \u03b1\na\u2080 a\u2081 b\u2080 b\u2081 : \u03b1\n\u22a2 ![a\u2080, a\u2081] + ![b\u2080, b\u2081] = ![a\u2080 + b\u2080, a\u2081 + b\u2081]", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Fintype/Card.lean", "full_name": "Fintype.card_range_le", "start": [509, 1], "end": [511, 99], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : Fintype \u03b1\u271d\ninst\u271d\u00b2 : Fintype \u03b2\u271d\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(Set.range f)\na : \u03b1\n\u22a2 f a \u2208 Set.range f", "state_after": "no goals"}, {"tactic": "simpa using ha", "annotated_tactic": ["simpa using ha", []], "state_before": "\u03b1\u271d : Type u_1\n\u03b2\u271d : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u00b3 : Fintype \u03b1\u271d\ninst\u271d\u00b2 : Fintype \u03b2\u271d\n\u03b1 : Type u_4\n\u03b2 : Type u_5\nf : \u03b1 \u2192 \u03b2\ninst\u271d\u00b9 : Fintype \u03b1\ninst\u271d : Fintype \u2191(Set.range f)\nx\u271d : \u2191(Set.range f)\nval\u271d : \u03b2\na : \u03b1\nha : f a = val\u271d\n\u22a2 (fun a => \u27e8f a, \u22ef\u27e9) a = \u27e8val\u271d, \u22ef\u27e9", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RepresentationTheory/Action/Limits.lean", "full_name": "Action.add_hom", "start": [280, 1], "end": [281, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Nth.lean", "full_name": "Nat.nth_true", "start": [419, 9], "end": [419, 95], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/PowerSeries/Inverse.lean", "full_name": "PowerSeries.invOfUnit_eq'", "start": [172, 1], "end": [174, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Eval.lean", "full_name": "Polynomial.mul_comp", "start": [648, 1], "end": [650, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SetFamily/Shadow.lean", "full_name": "Finset.exists_subset_of_mem_shadow", "start": [177, 1], "end": [180, 17], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Nilpotent.lean", "full_name": "nilpotent_iff_lowerCentralSeries", "start": [342, 1], "end": [350, 86], "traced_tactics": [{"tactic": "rw [nilpotent_iff_finite_descending_central_series]", "annotated_tactic": ["rw [<a>nilpotent_iff_finite_descending_central_series</a>]", [{"full_name": "nilpotent_iff_finite_descending_central_series", "def_path": "Mathlib/GroupTheory/Nilpotent.lean", "def_pos": [262, 9], "def_end_pos": [262, 55]}]], "state_before": "G : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : H.Normal\n\u22a2 Group.IsNilpotent G \u2194 \u2203 n, lowerCentralSeries G n = \u22a5", "state_after": "G : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : H.Normal\n\u22a2 (\u2203 n H, IsDescendingCentralSeries H \u2227 H n = \u22a5) \u2194 \u2203 n, lowerCentralSeries G n = \u22a5"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "G : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : H.Normal\n\u22a2 (\u2203 n H, IsDescendingCentralSeries H \u2227 H n = \u22a5) \u2194 \u2203 n, lowerCentralSeries G n = \u22a5", "state_after": "case mp\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : H.Normal\n\u22a2 (\u2203 n H, IsDescendingCentralSeries H \u2227 H n = \u22a5) \u2192 \u2203 n, lowerCentralSeries G n = \u22a5\n\ncase mpr\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : H.Normal\n\u22a2 (\u2203 n, lowerCentralSeries G n = \u22a5) \u2192 \u2203 n H, IsDescendingCentralSeries H \u2227 H n = \u22a5"}, {"tactic": "rintro \u27e8n, H, \u27e8h0, hs\u27e9, hn\u27e9", "annotated_tactic": ["rintro \u27e8n, H, \u27e8h0, hs\u27e9, hn\u27e9", []], "state_before": "case mp\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : H.Normal\n\u22a2 (\u2203 n H, IsDescendingCentralSeries H \u2227 H n = \u22a5) \u2192 \u2203 n, lowerCentralSeries G n = \u22a5", "state_after": "case mp.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d : Subgroup G\ninst\u271d : H\u271d.Normal\nn : \u2115\nH : \u2115 \u2192 Subgroup G\nhn : H n = \u22a5\nh0 : H 0 = \u22a4\nhs : \u2200 (x : G) (n : \u2115), x \u2208 H n \u2192 \u2200 (g : G), x * g * x\u207b\u00b9 * g\u207b\u00b9 \u2208 H (n + 1)\n\u22a2 \u2203 n, lowerCentralSeries G n = \u22a5"}, {"tactic": "use n", "annotated_tactic": ["use n", []], "state_before": "case mp.intro.intro.intro.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d : Subgroup G\ninst\u271d : H\u271d.Normal\nn : \u2115\nH : \u2115 \u2192 Subgroup G\nhn : H n = \u22a5\nh0 : H 0 = \u22a4\nhs : \u2200 (x : G) (n : \u2115), x \u2208 H n \u2192 \u2200 (g : G), x * g * x\u207b\u00b9 * g\u207b\u00b9 \u2208 H (n + 1)\n\u22a2 \u2203 n, lowerCentralSeries G n = \u22a5", "state_after": "case h\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d : Subgroup G\ninst\u271d : H\u271d.Normal\nn : \u2115\nH : \u2115 \u2192 Subgroup G\nhn : H n = \u22a5\nh0 : H 0 = \u22a4\nhs : \u2200 (x : G) (n : \u2115), x \u2208 H n \u2192 \u2200 (g : G), x * g * x\u207b\u00b9 * g\u207b\u00b9 \u2208 H (n + 1)\n\u22a2 lowerCentralSeries G n = \u22a5"}, {"tactic": "rw [eq_bot_iff, \u2190 hn]", "annotated_tactic": ["rw [<a>eq_bot_iff</a>, \u2190 hn]", [{"full_name": "eq_bot_iff", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [331, 9], "def_end_pos": [331, 19]}]], "state_before": "case h\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d : Subgroup G\ninst\u271d : H\u271d.Normal\nn : \u2115\nH : \u2115 \u2192 Subgroup G\nhn : H n = \u22a5\nh0 : H 0 = \u22a4\nhs : \u2200 (x : G) (n : \u2115), x \u2208 H n \u2192 \u2200 (g : G), x * g * x\u207b\u00b9 * g\u207b\u00b9 \u2208 H (n + 1)\n\u22a2 lowerCentralSeries G n = \u22a5", "state_after": "case h\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d : Subgroup G\ninst\u271d : H\u271d.Normal\nn : \u2115\nH : \u2115 \u2192 Subgroup G\nhn : H n = \u22a5\nh0 : H 0 = \u22a4\nhs : \u2200 (x : G) (n : \u2115), x \u2208 H n \u2192 \u2200 (g : G), x * g * x\u207b\u00b9 * g\u207b\u00b9 \u2208 H (n + 1)\n\u22a2 lowerCentralSeries G n \u2264 H n"}, {"tactic": "exact descending_central_series_ge_lower H \u27e8h0, hs\u27e9 n", "annotated_tactic": ["exact <a>descending_central_series_ge_lower</a> H \u27e8h0, hs\u27e9 n", [{"full_name": "descending_central_series_ge_lower", "def_path": "Mathlib/GroupTheory/Nilpotent.lean", "def_pos": [333, 9], "def_end_pos": [333, 43]}]], "state_before": "case h\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH\u271d : Subgroup G\ninst\u271d : H\u271d.Normal\nn : \u2115\nH : \u2115 \u2192 Subgroup G\nhn : H n = \u22a5\nh0 : H 0 = \u22a4\nhs : \u2200 (x : G) (n : \u2115), x \u2208 H n \u2192 \u2200 (g : G), x * g * x\u207b\u00b9 * g\u207b\u00b9 \u2208 H (n + 1)\n\u22a2 lowerCentralSeries G n \u2264 H n", "state_after": "no goals"}, {"tactic": "rintro \u27e8n, hn\u27e9", "annotated_tactic": ["rintro \u27e8n, hn\u27e9", []], "state_before": "case mpr\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : H.Normal\n\u22a2 (\u2203 n, lowerCentralSeries G n = \u22a5) \u2192 \u2203 n H, IsDescendingCentralSeries H \u2227 H n = \u22a5", "state_after": "case mpr.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : H.Normal\nn : \u2115\nhn : lowerCentralSeries G n = \u22a5\n\u22a2 \u2203 n H, IsDescendingCentralSeries H \u2227 H n = \u22a5"}, {"tactic": "exact \u27e8n, lowerCentralSeries G, lowerCentralSeries_isDescendingCentralSeries, hn\u27e9", "annotated_tactic": ["exact \u27e8n, <a>lowerCentralSeries</a> G, <a>lowerCentralSeries_isDescendingCentralSeries</a>, hn\u27e9", [{"full_name": "lowerCentralSeries", "def_path": "Mathlib/GroupTheory/Nilpotent.lean", "def_pos": [280, 5], "def_end_pos": [280, 23]}, {"full_name": "lowerCentralSeries_isDescendingCentralSeries", "def_path": "Mathlib/GroupTheory/Nilpotent.lean", "def_pos": [324, 9], "def_end_pos": [324, 53]}]], "state_before": "case mpr.intro\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : H.Normal\nn : \u2115\nhn : lowerCentralSeries G n = \u22a5\n\u22a2 \u2203 n H, IsDescendingCentralSeries H \u2227 H n = \u22a5", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/BooleanGenerators.lean", "full_name": "IsCompactlyGenerated.BooleanGenerators.mem_of_isAtom_of_le_sSup_atoms", "start": [105, 1], "end": [114, 33], "traced_tactics": [{"tactic": "obtain \u27e8T, hT, rfl\u27e9 := hS.atomistic a haS", "annotated_tactic": ["obtain \u27e8T, hT, rfl\u27e9 := hS.atomistic a haS", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nS : Set \u03b1\nhS : BooleanGenerators S\na : \u03b1\nha : IsAtom a\nhaS : a \u2264 sSup S\n\u22a2 a \u2208 S", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nS : Set \u03b1\nhS : BooleanGenerators S\nT : Set \u03b1\nhT : T \u2286 S\nha : IsAtom (sSup T)\nhaS : sSup T \u2264 sSup S\n\u22a2 sSup T \u2208 S"}, {"tactic": "obtain rfl | \u27e8a, haT\u27e9 := T.eq_empty_or_nonempty", "annotated_tactic": ["obtain rfl | \u27e8a, haT\u27e9 := T.eq_empty_or_nonempty", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nS : Set \u03b1\nhS : BooleanGenerators S\nT : Set \u03b1\nhT : T \u2286 S\nha : IsAtom (sSup T)\nhaS : sSup T \u2264 sSup S\n\u22a2 sSup T \u2208 S", "state_after": "case intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nS : Set \u03b1\nhS : BooleanGenerators S\nhT : \u2205 \u2286 S\nha : IsAtom (sSup \u2205)\nhaS : sSup \u2205 \u2264 sSup S\n\u22a2 sSup \u2205 \u2208 S\n\ncase intro.intro.inr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nS : Set \u03b1\nhS : BooleanGenerators S\nT : Set \u03b1\nhT : T \u2286 S\nha : IsAtom (sSup T)\nhaS : sSup T \u2264 sSup S\na : \u03b1\nhaT : a \u2208 T\n\u22a2 sSup T \u2208 S"}, {"tactic": "suffices sSup T = a from this \u25b8 hT haT", "annotated_tactic": ["suffices <a>sSup</a> T = a from this \u25b8 hT haT", [{"full_name": "SupSet.sSup", "def_path": "Mathlib/Order/SetNotation.lean", "def_pos": [42, 3], "def_end_pos": [42, 7]}]], "state_before": "case intro.intro.inr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nS : Set \u03b1\nhS : BooleanGenerators S\nT : Set \u03b1\nhT : T \u2286 S\nha : IsAtom (sSup T)\nhaS : sSup T \u2264 sSup S\na : \u03b1\nhaT : a \u2208 T\n\u22a2 sSup T \u2208 S", "state_after": "case intro.intro.inr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nS : Set \u03b1\nhS : BooleanGenerators S\nT : Set \u03b1\nhT : T \u2286 S\nha : IsAtom (sSup T)\nhaS : sSup T \u2264 sSup S\na : \u03b1\nhaT : a \u2208 T\n\u22a2 sSup T = a"}, {"tactic": "have : a \u2264 sSup T := le_sSup haT", "annotated_tactic": ["have : a \u2264 <a>sSup</a> T := <a>le_sSup</a> haT", [{"full_name": "SupSet.sSup", "def_path": "Mathlib/Order/SetNotation.lean", "def_pos": [42, 3], "def_end_pos": [42, 7]}, {"full_name": "le_sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [74, 9], "def_end_pos": [74, 16]}]], "state_before": "case intro.intro.inr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nS : Set \u03b1\nhS : BooleanGenerators S\nT : Set \u03b1\nhT : T \u2286 S\nha : IsAtom (sSup T)\nhaS : sSup T \u2264 sSup S\na : \u03b1\nhaT : a \u2208 T\n\u22a2 sSup T = a", "state_after": "case intro.intro.inr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nS : Set \u03b1\nhS : BooleanGenerators S\nT : Set \u03b1\nhT : T \u2286 S\nha : IsAtom (sSup T)\nhaS : sSup T \u2264 sSup S\na : \u03b1\nhaT : a \u2208 T\nthis : a \u2264 sSup T\n\u22a2 sSup T = a"}, {"tactic": "rwa [ha.le_iff_eq, eq_comm] at this", "annotated_tactic": ["rwa [ha.le_iff_eq, <a>eq_comm</a>] at this", [{"full_name": "eq_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}]], "state_before": "case intro.intro.inr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nS : Set \u03b1\nhS : BooleanGenerators S\nT : Set \u03b1\nhT : T \u2286 S\nha : IsAtom (sSup T)\nhaS : sSup T \u2264 sSup S\na : \u03b1\nhaT : a \u2208 T\nthis : a \u2264 sSup T\n\u22a2 sSup T = a", "state_after": "case intro.intro.inr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nS : Set \u03b1\nhS : BooleanGenerators S\nT : Set \u03b1\nhT : T \u2286 S\nha : IsAtom (sSup T)\nhaS : sSup T \u2264 sSup S\na : \u03b1\nhaT : a \u2208 T\nthis : a \u2264 sSup T\n\u22a2 a \u2260 \u22a5"}, {"tactic": "exact (hS.isAtom a (hT haT)).1", "annotated_tactic": ["exact (hS.isAtom a (hT haT)).1", []], "state_before": "case intro.intro.inr.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nS : Set \u03b1\nhS : BooleanGenerators S\nT : Set \u03b1\nhT : T \u2286 S\nha : IsAtom (sSup T)\nhaS : sSup T \u2264 sSup S\na : \u03b1\nhaT : a \u2208 T\nthis : a \u2264 sSup T\n\u22a2 a \u2260 \u22a5", "state_after": "no goals"}, {"tactic": "simp only [sSup_empty] at ha", "annotated_tactic": ["simp only [<a>sSup_empty</a>] at ha", [{"full_name": "sSup_empty", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [419, 9], "def_end_pos": [419, 19]}]], "state_before": "case intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nS : Set \u03b1\nhS : BooleanGenerators S\nhT : \u2205 \u2286 S\nha : IsAtom (sSup \u2205)\nhaS : sSup \u2205 \u2264 sSup S\n\u22a2 sSup \u2205 \u2208 S", "state_after": "case intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nS : Set \u03b1\nhS : BooleanGenerators S\nhT : \u2205 \u2286 S\nhaS : sSup \u2205 \u2264 sSup S\nha : IsAtom \u22a5\n\u22a2 sSup \u2205 \u2208 S"}, {"tactic": "exact (ha.1 rfl).elim", "annotated_tactic": ["exact (ha.1 <a>rfl</a>).<a>elim</a>", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "case intro.intro.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nS : Set \u03b1\nhS : BooleanGenerators S\nhT : \u2205 \u2286 S\nhaS : sSup \u2205 \u2264 sSup S\nha : IsAtom \u22a5\n\u22a2 sSup \u2205 \u2208 S", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/PNat/Find.lean", "full_name": "PNat.find_min'", "start": [65, 11], "end": [66, 45], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "full_name": "smul_sphere", "start": [435, 1], "end": [439, 30], "traced_tactics": [{"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": [212, 9], "def_end_pos": [212, 17]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nc : \ud835\udd5c\nx : E\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 c \u2022 sphere x r = sphere (c \u2022 x) (\u2016c\u2016 * r)", "state_after": "case inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 0 \u2022 sphere x r = sphere (0 \u2022 x) (\u20160\u2016 * r)\n\ncase inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nc : \ud835\udd5c\nx : E\nr : \u211d\nhr : 0 \u2264 r\nhc : c \u2260 0\n\u22a2 c \u2022 sphere x r = sphere (c \u2022 x) (\u2016c\u2016 * r)"}, {"tactic": "simp [zero_smul_set, Set.singleton_zero, hr]", "annotated_tactic": ["simp [<a>zero_smul_set</a>, <a>Set.singleton_zero</a>, hr]", [{"full_name": "Set.zero_smul_set", "def_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "def_pos": [835, 17], "def_end_pos": [835, 30]}, {"full_name": "Set.singleton_zero", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [93, 3], "def_end_pos": [93, 14]}]], "state_before": "case inl\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nx : E\nr : \u211d\nhr : 0 \u2264 r\n\u22a2 0 \u2022 sphere x r = sphere (0 \u2022 x) (\u20160\u2016 * r)", "state_after": "no goals"}, {"tactic": "exact smul_sphere' hc x r", "annotated_tactic": ["exact <a>smul_sphere'</a> hc x r", [{"full_name": "smul_sphere'", "def_path": "Mathlib/Analysis/NormedSpace/Pointwise.lean", "def_pos": [95, 9], "def_end_pos": [95, 21]}]], "state_before": "case inr\n\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2074 : NormedField \ud835\udd5c\ninst\u271d\u00b3 : NormedAddCommGroup E\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c E\ninst\u271d\u00b9 : NormedSpace \u211d E\ninst\u271d : Nontrivial E\nc : \ud835\udd5c\nx : E\nr : \u211d\nhr : 0 \u2264 r\nhc : c \u2260 0\n\u22a2 c \u2022 sphere x r = sphere (c \u2022 x) (\u2016c\u2016 * r)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/WithBot.lean", "full_name": "WithTop.recTopCoe_top", "start": [680, 1], "end": [682, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Idempotents/Karoubi.lean", "full_name": "CategoryTheory.Idempotents.Karoubi.id_eq", "start": [125, 1], "end": [125, 75], "traced_tactics": [{"tactic": "repeat' rw [P.idem]", "annotated_tactic": ["repeat' rw [P.idem]", []], "state_before": "C : Type u_1\ninst\u271d : Category.{?u.14710, u_1} C\nP : Karoubi C\n\u22a2 P.p = P.p \u226b P.p \u226b P.p", "state_after": "no goals"}, {"tactic": "rw [P.idem]", "annotated_tactic": ["rw [P.idem]", []], "state_before": "C : Type u_1\ninst\u271d : Category.{?u.14710, u_1} C\nP : Karoubi C\n\u22a2 P.p = P.p \u226b P.p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Lattice.lean", "full_name": "inf_le_inf_right", "start": [447, 1], "end": [448, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/PowerSeries/Order.lean", "full_name": "PowerSeries.order_add_of_order_eq", "start": [185, 1], "end": [192, 61], "traced_tactics": [{"tactic": "refine le_antisymm ?_ (le_order_add _ _)", "annotated_tactic": ["refine <a>le_antisymm</a> ?_ (<a>le_order_add</a> _ _)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "PowerSeries.le_order_add", "def_path": "Mathlib/RingTheory/PowerSeries/Order.lean", "def_pos": [162, 9], "def_end_pos": [162, 21]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 \u03c8 : R\u27e6X\u27e7\nh : \u03c6.order \u2260 \u03c8.order\n\u22a2 (\u03c6 + \u03c8).order = \u03c6.order \u2293 \u03c8.order", "state_after": "R : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 \u03c8 : R\u27e6X\u27e7\nh : \u03c6.order \u2260 \u03c8.order\n\u22a2 (\u03c6 + \u03c8).order \u2264 \u03c6.order \u2293 \u03c8.order"}, {"tactic": "by_cases H\u2081 : order \u03c6 < order \u03c8", "annotated_tactic": ["by_cases H\u2081 : <a>order</a> \u03c6 < <a>order</a> \u03c8", [{"full_name": "PowerSeries.order", "def_path": "Mathlib/RingTheory/PowerSeries/Order.lean", "def_pos": [56, 5], "def_end_pos": [56, 10]}, {"full_name": "PowerSeries.order", "def_path": "Mathlib/RingTheory/PowerSeries/Order.lean", "def_pos": [56, 5], "def_end_pos": [56, 10]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 \u03c8 : R\u27e6X\u27e7\nh : \u03c6.order \u2260 \u03c8.order\n\u22a2 (\u03c6 + \u03c8).order \u2264 \u03c6.order \u2293 \u03c8.order", "state_after": "case pos\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 \u03c8 : R\u27e6X\u27e7\nh : \u03c6.order \u2260 \u03c8.order\nH\u2081 : \u03c6.order < \u03c8.order\n\u22a2 (\u03c6 + \u03c8).order \u2264 \u03c6.order \u2293 \u03c8.order\n\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 \u03c8 : R\u27e6X\u27e7\nh : \u03c6.order \u2260 \u03c8.order\nH\u2081 : \u00ac\u03c6.order < \u03c8.order\n\u22a2 (\u03c6 + \u03c8).order \u2264 \u03c6.order \u2293 \u03c8.order"}, {"tactic": "by_cases H\u2082 : order \u03c8 < order \u03c6", "annotated_tactic": ["by_cases H\u2082 : <a>order</a> \u03c8 < <a>order</a> \u03c6", [{"full_name": "PowerSeries.order", "def_path": "Mathlib/RingTheory/PowerSeries/Order.lean", "def_pos": [56, 5], "def_end_pos": [56, 10]}, {"full_name": "PowerSeries.order", "def_path": "Mathlib/RingTheory/PowerSeries/Order.lean", "def_pos": [56, 5], "def_end_pos": [56, 10]}]], "state_before": "case neg\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 \u03c8 : R\u27e6X\u27e7\nh : \u03c6.order \u2260 \u03c8.order\nH\u2081 : \u00ac\u03c6.order < \u03c8.order\n\u22a2 (\u03c6 + \u03c8).order \u2264 \u03c6.order \u2293 \u03c8.order", "state_after": "case pos\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 \u03c8 : R\u27e6X\u27e7\nh : \u03c6.order \u2260 \u03c8.order\nH\u2081 : \u00ac\u03c6.order < \u03c8.order\nH\u2082 : \u03c8.order < \u03c6.order\n\u22a2 (\u03c6 + \u03c8).order \u2264 \u03c6.order \u2293 \u03c8.order\n\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 \u03c8 : R\u27e6X\u27e7\nh : \u03c6.order \u2260 \u03c8.order\nH\u2081 : \u00ac\u03c6.order < \u03c8.order\nH\u2082 : \u00ac\u03c8.order < \u03c6.order\n\u22a2 (\u03c6 + \u03c8).order \u2264 \u03c6.order \u2293 \u03c8.order"}, {"tactic": "exfalso", "annotated_tactic": ["exfalso", []], "state_before": "case neg\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 \u03c8 : R\u27e6X\u27e7\nh : \u03c6.order \u2260 \u03c8.order\nH\u2081 : \u00ac\u03c6.order < \u03c8.order\nH\u2082 : \u00ac\u03c8.order < \u03c6.order\n\u22a2 (\u03c6 + \u03c8).order \u2264 \u03c6.order \u2293 \u03c8.order", "state_after": "case neg\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 \u03c8 : R\u27e6X\u27e7\nh : \u03c6.order \u2260 \u03c8.order\nH\u2081 : \u00ac\u03c6.order < \u03c8.order\nH\u2082 : \u00ac\u03c8.order < \u03c6.order\n\u22a2 False"}, {"tactic": "exact h (le_antisymm (not_lt.1 H\u2082) (not_lt.1 H\u2081))", "annotated_tactic": ["exact h (<a>le_antisymm</a> (<a>not_lt</a>.1 H\u2082) (<a>not_lt</a>.1 H\u2081))", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [370, 9], "def_end_pos": [370, 15]}, {"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [370, 9], "def_end_pos": [370, 15]}]], "state_before": "case neg\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 \u03c8 : R\u27e6X\u27e7\nh : \u03c6.order \u2260 \u03c8.order\nH\u2081 : \u00ac\u03c6.order < \u03c8.order\nH\u2082 : \u00ac\u03c8.order < \u03c6.order\n\u22a2 False", "state_after": "no goals"}, {"tactic": "apply order_add_of_order_eq.aux _ _ h H\u2081", "annotated_tactic": ["apply <a>order_add_of_order_eq.aux</a> _ _ h H\u2081", [{"full_name": "_private.Mathlib.RingTheory.PowerSeries.Order.0.PowerSeries.order_add_of_order_eq.aux", "def_path": "Mathlib/RingTheory/PowerSeries/Order.lean", "def_pos": [167, 17], "def_end_pos": [167, 42]}]], "state_before": "case pos\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 \u03c8 : R\u27e6X\u27e7\nh : \u03c6.order \u2260 \u03c8.order\nH\u2081 : \u03c6.order < \u03c8.order\n\u22a2 (\u03c6 + \u03c8).order \u2264 \u03c6.order \u2293 \u03c8.order", "state_after": "no goals"}, {"tactic": "simpa only [add_comm, inf_comm] using order_add_of_order_eq.aux _ _ h.symm H\u2082", "annotated_tactic": ["simpa only [<a>add_comm</a>, <a>inf_comm</a>] using <a>order_add_of_order_eq.aux</a> _ _ h.symm H\u2082", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "inf_comm", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [461, 9], "def_end_pos": [461, 17]}, {"full_name": "_private.Mathlib.RingTheory.PowerSeries.Order.0.PowerSeries.order_add_of_order_eq.aux", "def_path": "Mathlib/RingTheory/PowerSeries/Order.lean", "def_pos": [167, 17], "def_end_pos": [167, 42]}]], "state_before": "case pos\nR : Type u_1\ninst\u271d : Semiring R\n\u03c6\u271d \u03c6 \u03c8 : R\u27e6X\u27e7\nh : \u03c6.order \u2260 \u03c8.order\nH\u2081 : \u00ac\u03c6.order < \u03c8.order\nH\u2082 : \u03c8.order < \u03c6.order\n\u22a2 (\u03c6 + \u03c8).order \u2264 \u03c6.order \u2293 \u03c8.order", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.image2_iInter\u2082_subset_right", "start": [1912, 1], "end": [1915, 59], "traced_tactics": [{"tactic": "simp_rw [image2_subset_iff, mem_iInter]", "annotated_tactic": ["simp_rw [<a>image2_subset_iff</a>, <a>mem_iInter</a>]", [{"full_name": "Set.image2_subset_iff", "def_path": "Mathlib/Data/Set/NAry.lean", "def_pos": [64, 9], "def_end_pos": [64, 26]}, {"full_name": "Set.mem_iInter", "def_path": "Mathlib/Order/SetNotation.lean", "def_pos": [274, 9], "def_end_pos": [274, 19]}]], "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\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns\u271d : Set \u03b1\nt\u271d : Set \u03b2\ns : Set \u03b1\nt : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b2\n\u22a2 image2 f s (\u22c2 i, \u22c2 j, t i j) \u2286 \u22c2 i, \u22c2 j, image2 f s (t i j)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns\u271d : Set \u03b1\nt\u271d : Set \u03b2\ns : Set \u03b1\nt : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b2\n\u22a2 \u2200 x \u2208 s, \u2200 (y : \u03b2), (\u2200 (i : \u03b9) (i_1 : \u03ba i), y \u2208 t i i_1) \u2192 \u2200 (i : \u03b9) (i_1 : \u03ba i), f x y \u2208 image2 f s (t i i_1)"}, {"tactic": "exact fun x hx y hy i j => mem_image2_of_mem hx (hy _ _)", "annotated_tactic": ["exact fun x hx y hy i j => <a>mem_image2_of_mem</a> hx (hy _ _)", [{"full_name": "Set.mem_image2_of_mem", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [330, 9], "def_end_pos": [330, 26]}]], "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\n\u03b9\u2082 : Sort u_6\n\u03ba : \u03b9 \u2192 Sort u_7\n\u03ba\u2081 : \u03b9 \u2192 Sort u_8\n\u03ba\u2082 : \u03b9 \u2192 Sort u_9\n\u03ba' : \u03b9' \u2192 Sort u_10\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns\u271d : Set \u03b1\nt\u271d : Set \u03b2\ns : Set \u03b1\nt : (i : \u03b9) \u2192 \u03ba i \u2192 Set \u03b2\n\u22a2 \u2200 x \u2208 s, \u2200 (y : \u03b2), (\u2200 (i : \u03b9) (i_1 : \u03ba i), y \u2208 t i i_1) \u2192 \u2200 (i : \u03b9) (i_1 : \u03ba i), f x y \u2208 image2 f s (t i i_1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/Cauchy.lean", "full_name": "CauchySeq.comp_injective", "start": [237, 1], "end": [239, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Asymptotics/Asymptotics.lean", "full_name": "Asymptotics.IsBigOWith.prod_left_snd", "start": [1030, 1], "end": [1032, 69], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "full_name": "Polynomial.le_natDegree_of_ne_zero", "start": [180, 1], "end": [184, 16], "traced_tactics": [{"tactic": "rw [\u2190 Nat.cast_le (\u03b1 := WithBot \u2115), \u2190 degree_eq_natDegree]", "annotated_tactic": ["rw [\u2190 <a>Nat.cast_le</a> (\u03b1 := <a>WithBot</a> \u2115), \u2190 <a>degree_eq_natDegree</a>]", [{"full_name": "Nat.cast_le", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [124, 9], "def_end_pos": [124, 16]}, {"full_name": "WithBot", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [29, 5], "def_end_pos": [29, 12]}, {"full_name": "Polynomial.degree_eq_natDegree", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [132, 9], "def_end_pos": [132, 28]}]], "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : p.coeff n \u2260 0\n\u22a2 n \u2264 p.natDegree", "state_after": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : p.coeff n \u2260 0\n\u22a2 \u2191n \u2264 p.degree\n\nR : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : p.coeff n \u2260 0\n\u22a2 p \u2260 0"}, {"tactic": "exact le_degree_of_ne_zero h", "annotated_tactic": ["exact <a>le_degree_of_ne_zero</a> h", [{"full_name": "Polynomial.le_degree_of_ne_zero", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [175, 9], "def_end_pos": [175, 29]}]], "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : p.coeff n \u2260 0\n\u22a2 \u2191n \u2264 p.degree", "state_after": "no goals"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : p.coeff n \u2260 0\n\u22a2 p \u2260 0", "state_after": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\nq r : R[X]\nh : coeff 0 n \u2260 0\n\u22a2 False"}, {"tactic": "exact h rfl", "annotated_tactic": ["exact h <a>rfl</a>", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "R : Type u\nS : Type v\na b c d : R\nn m : \u2115\ninst\u271d : Semiring R\nq r : R[X]\nh : coeff 0 n \u2260 0\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.inf_map", "start": [358, 1], "end": [359, 11], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/LocalExtr.lean", "full_name": "isLocalExtrOn_const", "start": [196, 1], "end": [197, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "full_name": "csSup_Icc", "start": [766, 1], "end": [767, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Orientation.lean", "full_name": "Orientation.eq_or_eq_neg", "start": [369, 1], "end": [380, 95], "traced_tactics": [{"tactic": "have e := (finBasis R M).reindex (Fintype.equivFinOfCardEq h).symm", "annotated_tactic": ["have e := (<a>finBasis</a> R M).<a>reindex</a> (<a>Fintype.equivFinOfCardEq</a> h).<a>symm</a>", [{"full_name": "FiniteDimensional.finBasis", "def_path": "Mathlib/LinearAlgebra/Dimension/Free.lean", "def_pos": [212, 19], "def_end_pos": [212, 27]}, {"full_name": "Basis.reindex", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [413, 5], "def_end_pos": [413, 12]}, {"full_name": "Fintype.equivFinOfCardEq", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [178, 19], "def_end_pos": [178, 35]}, {"full_name": "Equiv.symm", "def_path": "Mathlib/Logic/Equiv/Defs.lean", "def_pos": [162, 15], "def_end_pos": [162, 19]}]], "state_before": "R : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx\u2081 x\u2082 : Orientation R M \u03b9\nh : Fintype.card \u03b9 = finrank R M\n\u22a2 x\u2081 = x\u2082 \u2228 x\u2081 = -x\u2082", "state_after": "R : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx\u2081 x\u2082 : Orientation R M \u03b9\nh : Fintype.card \u03b9 = finrank R M\ne : Basis \u03b9 R M\n\u22a2 x\u2081 = x\u2082 \u2228 x\u2081 = -x\u2082"}, {"tactic": "letI := Classical.decEq \u03b9", "annotated_tactic": ["letI := <a>Classical.decEq</a> \u03b9", [{"full_name": "Classical.decEq", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1020, 19], "def_end_pos": [1020, 24]}]], "state_before": "R : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx\u2081 x\u2082 : Orientation R M \u03b9\nh : Fintype.card \u03b9 = finrank R M\ne : Basis \u03b9 R M\n\u22a2 x\u2081 = x\u2082 \u2228 x\u2081 = -x\u2082", "state_after": "R : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx\u2081 x\u2082 : Orientation R M \u03b9\nh : Fintype.card \u03b9 = finrank R M\ne : Basis \u03b9 R M\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\n\u22a2 x\u2081 = x\u2082 \u2228 x\u2081 = -x\u2082"}, {"tactic": "have orientation_neg_neg :\n    \u2200 f : Basis \u03b9 R M, - -Basis.orientation f = Basis.orientation f := by\n  #adaptation_note\n  \n  set_option maxSynthPendingDepth 2 in simp", "annotated_tactic": ["have orientation_neg_neg :\n      \u2200 f : <a>Basis</a> \u03b9 R M, - -<a>Basis.orientation</a> f = <a>Basis.orientation</a> f := by\n    #adaptation_note\n    /-- `set_option maxSynthPendingDepth 2` required after https://github.com/leanprover/lean4/pull/4119 -/\n    set_option maxSynthPendingDepth 2 in simp", [{"full_name": "Basis", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [88, 11], "def_end_pos": [88, 16]}, {"full_name": "Basis.orientation", "def_path": "Mathlib/LinearAlgebra/Orientation.lean", "def_pos": [177, 15], "def_end_pos": [177, 26]}, {"full_name": "Basis.orientation", "def_path": "Mathlib/LinearAlgebra/Orientation.lean", "def_pos": [177, 15], "def_end_pos": [177, 26]}]], "state_before": "R : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx\u2081 x\u2082 : Orientation R M \u03b9\nh : Fintype.card \u03b9 = finrank R M\ne : Basis \u03b9 R M\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\n\u22a2 x\u2081 = x\u2082 \u2228 x\u2081 = -x\u2082", "state_after": "R : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx\u2081 x\u2082 : Orientation R M \u03b9\nh : Fintype.card \u03b9 = finrank R M\ne : Basis \u03b9 R M\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\norientation_neg_neg : \u2200 (f : Basis \u03b9 R M), - -f.orientation = f.orientation\n\u22a2 x\u2081 = x\u2082 \u2228 x\u2081 = -x\u2082"}, {"tactic": "rcases e.orientation_eq_or_eq_neg x\u2081 with (h\u2081 | h\u2081) <;>\n  rcases e.orientation_eq_or_eq_neg x\u2082 with (h\u2082 | h\u2082) <;> simp [h\u2081, h\u2082, orientation_neg_neg]", "annotated_tactic": ["rcases e.orientation_eq_or_eq_neg x\u2081 with (h\u2081 | h\u2081) <;>\n    rcases e.orientation_eq_or_eq_neg x\u2082 with (h\u2082 | h\u2082) <;> simp [h\u2081, h\u2082, orientation_neg_neg]", []], "state_before": "R : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx\u2081 x\u2082 : Orientation R M \u03b9\nh : Fintype.card \u03b9 = finrank R M\ne : Basis \u03b9 R M\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\norientation_neg_neg : \u2200 (f : Basis \u03b9 R M), - -f.orientation = f.orientation\n\u22a2 x\u2081 = x\u2082 \u2228 x\u2081 = -x\u2082", "state_after": "no goals"}, {"tactic": "set_option maxSynthPendingDepth 2 in simp", "annotated_tactic": ["set_option maxSynthPendingDepth 2 in simp", []], "state_before": "R : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx\u2081 x\u2082 : Orientation R M \u03b9\nh : Fintype.card \u03b9 = finrank R M\ne : Basis \u03b9 R M\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\n\u22a2 \u2200 (f : Basis \u03b9 R M), - -f.orientation = f.orientation", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "R : Type u_1\ninst\u271d\u00b3 : LinearOrderedField R\nM : Type u_2\ninst\u271d\u00b2 : AddCommGroup M\ninst\u271d\u00b9 : Module R M\n\u03b9 : Type u_3\ninst\u271d : Fintype \u03b9\n_i : FiniteDimensional R M\nx\u2081 x\u2082 : Orientation R M \u03b9\nh : Fintype.card \u03b9 = finrank R M\ne : Basis \u03b9 R M\nthis : DecidableEq \u03b9 := Classical.decEq \u03b9\n\u22a2 \u2200 (f : Basis \u03b9 R M), - -f.orientation = f.orientation", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Compactness/Compact.lean", "full_name": "IsCompact.nhdsSet_prod_eq", "start": [748, 1], "end": [751, 18], "traced_tactics": [{"tactic": "simp_rw [hs.nhdsSet_prod_eq_biSup, ht.prod_nhdsSet_eq_biSup, nhdsSet, sSup_image, biSup_prod,\n  nhds_prod_eq]", "annotated_tactic": ["simp_rw [hs.nhdsSet_prod_eq_biSup, ht.prod_nhdsSet_eq_biSup, <a>nhdsSet</a>, <a>sSup_image</a>, <a>biSup_prod</a>,\n    <a>nhds_prod_eq</a>]", [{"full_name": "nhdsSet", "def_path": "Mathlib/Topology/Defs/Filter.lean", "def_pos": [147, 5], "def_end_pos": [147, 12]}, {"full_name": "sSup_image", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1349, 9], "def_end_pos": [1349, 19]}, {"full_name": "biSup_prod", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [1537, 9], "def_end_pos": [1537, 19]}, {"full_name": "nhds_prod_eq", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [549, 9], "def_end_pos": [549, 21]}]], "state_before": "X : Type u\nY : Type v\n\u03b9 : Type u_1\ninst\u271d\u00b9 : TopologicalSpace X\ninst\u271d : TopologicalSpace Y\ns t\u271d : Set X\nt : Set Y\nhs : IsCompact s\nht : IsCompact t\n\u22a2 \ud835\udcdd\u02e2 (s \u00d7\u02e2 t) = \ud835\udcdd\u02e2 s \u00d7\u02e2 \ud835\udcdd\u02e2 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Monotone/Basic.lean", "full_name": "monotone_app", "start": [374, 1], "end": [375, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/SymmDiff.lean", "full_name": "bihimp_himp_left", "start": [604, 1], "end": [605, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/ModelTheory/Substructures.lean", "full_name": "FirstOrder.Language.Substructure.lift_card_closure_le_card_term", "start": [319, 1], "end": [322, 34], "traced_tactics": [{"tactic": "rw [\u2190 SetLike.coe_sort_coe, coe_closure_eq_range_term_realize]", "annotated_tactic": ["rw [\u2190 <a>SetLike.coe_sort_coe</a>, <a>coe_closure_eq_range_term_realize</a>]", [{"full_name": "SetLike.coe_sort_coe", "def_path": "Mathlib/Data/SetLike/Basic.lean", "def_pos": [137, 9], "def_end_pos": [137, 21]}, {"full_name": "FirstOrder.Language.Substructure.coe_closure_eq_range_term_realize", "def_path": "Mathlib/ModelTheory/Substructures.lean", "def_pos": [296, 9], "def_end_pos": [296, 42]}]], "state_before": "L : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : L.Structure M\ninst\u271d\u00b9 : L.Structure N\ninst\u271d : L.Structure P\nS : L.Substructure M\ns : Set M\n\u22a2 lift.{max u w, w} #\u21a5((closure L).toFun s) \u2264 #(L.Term \u2191s)", "state_after": "L : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : L.Structure M\ninst\u271d\u00b9 : L.Structure N\ninst\u271d : L.Structure P\nS : L.Substructure M\ns : Set M\n\u22a2 lift.{max u w, w} #\u2191(range (Term.realize Subtype.val)) \u2264 #(L.Term \u2191s)"}, {"tactic": "rw [\u2190 Cardinal.lift_id'.{w, max u w} #(L.Term s)]", "annotated_tactic": ["rw [\u2190 <a>Cardinal.lift_id'</a>.{w, max u w} #(L.Term s)]", [{"full_name": "Cardinal.lift_id'", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [218, 9], "def_end_pos": [218, 17]}]], "state_before": "L : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : L.Structure M\ninst\u271d\u00b9 : L.Structure N\ninst\u271d : L.Structure P\nS : L.Substructure M\ns : Set M\n\u22a2 lift.{max u w, w} #\u2191(range (Term.realize Subtype.val)) \u2264 #(L.Term \u2191s)", "state_after": "L : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : L.Structure M\ninst\u271d\u00b9 : L.Structure N\ninst\u271d : L.Structure P\nS : L.Substructure M\ns : Set M\n\u22a2 lift.{max u w, w} #\u2191(range (Term.realize Subtype.val)) \u2264 lift.{w, max u w} #(L.Term \u2191s)"}, {"tactic": "exact Cardinal.mk_range_le_lift", "annotated_tactic": ["exact <a>Cardinal.mk_range_le_lift</a>", [{"full_name": "Cardinal.mk_range_le_lift", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1958, 9], "def_end_pos": [1958, 25]}]], "state_before": "L : Language\nM : Type w\nN : Type u_1\nP : Type u_2\ninst\u271d\u00b2 : L.Structure M\ninst\u271d\u00b9 : L.Structure N\ninst\u271d : L.Structure P\nS : L.Substructure M\ns : Set M\n\u22a2 lift.{max u w, w} #\u2191(range (Term.realize Subtype.val)) \u2264 lift.{w, max u w} #(L.Term \u2191s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "full_name": "Set.exists_lt_mem_inter_of_not_pairwiseDisjoint", "start": [448, 1], "end": [454, 44], "traced_tactics": [{"tactic": "obtain \u27e8i, hi, j, hj, hne, x, hx\u2081, hx\u2082\u27e9 := exists_ne_mem_inter_of_not_pairwiseDisjoint h", "annotated_tactic": ["obtain \u27e8i, hi, j, hj, hne, x, hx\u2081, hx\u2082\u27e9 := <a>exists_ne_mem_inter_of_not_pairwiseDisjoint</a> h", [{"full_name": "Set.exists_ne_mem_inter_of_not_pairwiseDisjoint", "def_path": "Mathlib/Data/Set/Pairwise/Basic.lean", "def_pos": [437, 7], "def_end_pos": [437, 50]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b9\nt : Set \u03b9'\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u00acs.PairwiseDisjoint f\n\u22a2 \u2203 i \u2208 s, \u2203 j \u2208 s, i < j \u2227 \u2203 x, x \u2208 f i \u2229 f j", "state_after": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b9\nt : Set \u03b9'\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u00acs.PairwiseDisjoint f\ni : \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 s\nhne : i \u2260 j\nx : \u03b1\nhx\u2081 : x \u2208 f i\nhx\u2082 : x \u2208 f j\n\u22a2 \u2203 i \u2208 s, \u2203 j \u2208 s, i < j \u2227 \u2203 x, x \u2208 f i \u2229 f j"}, {"tactic": "cases' lt_or_lt_iff_ne.mpr hne with h_lt h_lt", "annotated_tactic": ["cases' lt_or_lt_iff_ne.mpr hne with h_lt h_lt", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\n\u03b9' : Type u_5\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b9\nt : Set \u03b9'\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u00acs.PairwiseDisjoint f\ni : \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 s\nhne : i \u2260 j\nx : \u03b1\nhx\u2081 : x \u2208 f i\nhx\u2082 : x \u2208 f j\n\u22a2 \u2203 i \u2208 s, \u2203 j \u2208 s, i < j \u2227 \u2203 x, x \u2208 f i \u2229 f j", "state_after": "case intro.intro.intro.intro.intro.intro.intro.inl\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\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b9\nt : Set \u03b9'\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u00acs.PairwiseDisjoint f\ni : \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 s\nhne : i \u2260 j\nx : \u03b1\nhx\u2081 : x \u2208 f i\nhx\u2082 : x \u2208 f j\nh_lt : i < j\n\u22a2 \u2203 i \u2208 s, \u2203 j \u2208 s, i < j \u2227 \u2203 x, x \u2208 f i \u2229 f j\n\ncase intro.intro.intro.intro.intro.intro.intro.inr\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\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b9\nt : Set \u03b9'\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u00acs.PairwiseDisjoint f\ni : \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 s\nhne : i \u2260 j\nx : \u03b1\nhx\u2081 : x \u2208 f i\nhx\u2082 : x \u2208 f j\nh_lt : j < i\n\u22a2 \u2203 i \u2208 s, \u2203 j \u2208 s, i < j \u2227 \u2203 x, x \u2208 f i \u2229 f j"}, {"tactic": "exact \u27e8i, hi, j, hj, h_lt, x, hx\u2081, hx\u2082\u27e9", "annotated_tactic": ["exact \u27e8i, hi, j, hj, h_lt, x, hx\u2081, hx\u2082\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.inl\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\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b9\nt : Set \u03b9'\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u00acs.PairwiseDisjoint f\ni : \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 s\nhne : i \u2260 j\nx : \u03b1\nhx\u2081 : x \u2208 f i\nhx\u2082 : x \u2208 f j\nh_lt : i < j\n\u22a2 \u2203 i \u2208 s, \u2203 j \u2208 s, i < j \u2227 \u2203 x, x \u2208 f i \u2229 f j", "state_after": "no goals"}, {"tactic": "exact \u27e8j, hj, i, hi, h_lt, x, hx\u2082, hx\u2081\u27e9", "annotated_tactic": ["exact \u27e8j, hj, i, hi, h_lt, x, hx\u2082, hx\u2081\u27e9", []], "state_before": "case intro.intro.intro.intro.intro.intro.intro.inr\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\nr p q : \u03b1 \u2192 \u03b1 \u2192 Prop\ns : Set \u03b9\nt : Set \u03b9'\ninst\u271d : LinearOrder \u03b9\nf : \u03b9 \u2192 Set \u03b1\nh : \u00acs.PairwiseDisjoint f\ni : \u03b9\nhi : i \u2208 s\nj : \u03b9\nhj : j \u2208 s\nhne : i \u2260 j\nx : \u03b1\nhx\u2081 : x \u2208 f i\nhx\u2082 : x \u2208 f j\nh_lt : j < i\n\u22a2 \u2203 i \u2208 s, \u2203 j \u2208 s, i < j \u2227 \u2203 x, x \u2208 f i \u2229 f j", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "full_name": "nnnorm_zpow", "start": [769, 1], "end": [770, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Category/Ring/Basic.lean", "full_name": "CommRingCat.RingEquiv_coe_eq", "start": [537, 1], "end": [541, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.encard_univ_coe", "start": [66, 9], "end": [67, 63], "traced_tactics": [{"tactic": "rw [encard, encard, PartENat.card_congr (Equiv.Set.univ \u2191s)]", "annotated_tactic": ["rw [<a>encard</a>, <a>encard</a>, <a>PartENat.card_congr</a> (<a>Equiv.Set.univ</a> \u2191s)]", [{"full_name": "Set.encard", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [64, 19], "def_end_pos": [64, 25]}, {"full_name": "Set.encard", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [64, 19], "def_end_pos": [64, 25]}, {"full_name": "PartENat.card_congr", "def_path": "Mathlib/SetTheory/Cardinal/Finite.lean", "def_pos": [238, 9], "def_end_pos": [238, 19]}, {"full_name": "Equiv.Set.univ", "def_path": "Mathlib/Logic/Equiv/Set.lean", "def_pos": [211, 15], "def_end_pos": [211, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ns\u271d t s : Set \u03b1\n\u22a2 univ.encard = s.encard", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Measurable.lean", "full_name": "aemeasurable_deriv", "start": [432, 1], "end": [434, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Homology/HomologicalComplex.lean", "full_name": "HomologicalComplex.kernel_eq_kernel", "start": [411, 1], "end": [414, 34], "traced_tactics": [{"tactic": "rw [\u2190 d_comp_eqToHom C r r']", "annotated_tactic": ["rw [\u2190 <a>d_comp_eqToHom</a> C r r']", [{"full_name": "HomologicalComplex.d_comp_eqToHom", "def_path": "Mathlib/Algebra/Homology/HomologicalComplex.lean", "def_pos": [395, 9], "def_end_pos": [395, 23]}]], "state_before": "\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b2 : Category.{v, u} V\ninst\u271d\u00b9 : HasZeroMorphisms V\nc : ComplexShape \u03b9\nC : HomologicalComplex V c\ninst\u271d : HasKernels V\ni j j' : \u03b9\nr : c.Rel i j\nr' : c.Rel i j'\n\u22a2 kernelSubobject (C.d i j) = kernelSubobject (C.d i j')", "state_after": "\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b2 : Category.{v, u} V\ninst\u271d\u00b9 : HasZeroMorphisms V\nc : ComplexShape \u03b9\nC : HomologicalComplex V c\ninst\u271d : HasKernels V\ni j j' : \u03b9\nr : c.Rel i j\nr' : c.Rel i j'\n\u22a2 kernelSubobject (C.d i j' \u226b eqToHom \u22ef) = kernelSubobject (C.d i j')"}, {"tactic": "apply kernelSubobject_comp_mono", "annotated_tactic": ["apply <a>kernelSubobject_comp_mono</a>", [{"full_name": "CategoryTheory.Limits.kernelSubobject_comp_mono", "def_path": "Mathlib/CategoryTheory/Subobject/Limits.lean", "def_pos": [232, 9], "def_end_pos": [232, 34]}]], "state_before": "\u03b9 : Type u_1\nV : Type u\ninst\u271d\u00b2 : Category.{v, u} V\ninst\u271d\u00b9 : HasZeroMorphisms V\nc : ComplexShape \u03b9\nC : HomologicalComplex V c\ninst\u271d : HasKernels V\ni j j' : \u03b9\nr : c.Rel i j\nr' : c.Rel i j'\n\u22a2 kernelSubobject (C.d i j' \u226b eqToHom \u22ef) = kernelSubobject (C.d i j')", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Simple.lean", "full_name": "CategoryTheory.simple_of_isSimpleOrder_subobject", "start": [237, 1], "end": [248, 52], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "C : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\n\u22a2 Simple X", "state_after": "case mono_isIso_iff_nonzero\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\n\u22a2 \u2200 {Y : C} (f : Y \u27f6 X) [inst : Mono f], IsIso f \u2194 f \u2260 0"}, {"tactic": "intros Y f hf", "annotated_tactic": ["intros Y f hf", []], "state_before": "case mono_isIso_iff_nonzero\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\n\u22a2 \u2200 {Y : C} (f : Y \u27f6 X) [inst : Mono f], IsIso f \u2194 f \u2260 0", "state_after": "case mono_isIso_iff_nonzero\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\n\u22a2 IsIso f \u2194 f \u2260 0"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case mono_isIso_iff_nonzero\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\n\u22a2 IsIso f \u2194 f \u2260 0", "state_after": "case mono_isIso_iff_nonzero.mp\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\n\u22a2 IsIso f \u2192 f \u2260 0\n\ncase mono_isIso_iff_nonzero.mpr\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\n\u22a2 f \u2260 0 \u2192 IsIso f"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "case mono_isIso_iff_nonzero.mp\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\n\u22a2 IsIso f \u2192 f \u2260 0", "state_after": "case mono_isIso_iff_nonzero.mp\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\ni : IsIso f\n\u22a2 f \u2260 0"}, {"tactic": "rw [Subobject.isIso_iff_mk_eq_top] at i", "annotated_tactic": ["rw [<a>Subobject.isIso_iff_mk_eq_top</a>] at i", [{"full_name": "CategoryTheory.Subobject.isIso_iff_mk_eq_top", "def_path": "Mathlib/CategoryTheory/Subobject/Lattice.lean", "def_pos": [258, 9], "def_end_pos": [258, 28]}]], "state_before": "case mono_isIso_iff_nonzero.mp\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\ni : IsIso f\n\u22a2 f \u2260 0", "state_after": "case mono_isIso_iff_nonzero.mp\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\ni : mk f = \u22a4\n\u22a2 f \u2260 0"}, {"tactic": "intro w", "annotated_tactic": ["intro w", []], "state_before": "case mono_isIso_iff_nonzero.mp\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\ni : mk f = \u22a4\n\u22a2 f \u2260 0", "state_after": "case mono_isIso_iff_nonzero.mp\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\ni : mk f = \u22a4\nw : f = 0\n\u22a2 False"}, {"tactic": "rw [\u2190 Subobject.mk_eq_bot_iff_zero] at w", "annotated_tactic": ["rw [\u2190 <a>Subobject.mk_eq_bot_iff_zero</a>] at w", [{"full_name": "CategoryTheory.Subobject.mk_eq_bot_iff_zero", "def_path": "Mathlib/CategoryTheory/Subobject/Lattice.lean", "def_pos": [354, 9], "def_end_pos": [354, 27]}]], "state_before": "case mono_isIso_iff_nonzero.mp\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\ni : mk f = \u22a4\nw : f = 0\n\u22a2 False", "state_after": "case mono_isIso_iff_nonzero.mp\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\ni : mk f = \u22a4\nw : mk f = \u22a5\n\u22a2 False"}, {"tactic": "exact IsSimpleOrder.bot_ne_top (w.symm.trans i)", "annotated_tactic": ["exact <a>IsSimpleOrder.bot_ne_top</a> (w.symm.trans i)", [{"full_name": "IsSimpleOrder.bot_ne_top", "def_path": "Mathlib/Order/Atoms.lean", "def_pos": [544, 9], "def_end_pos": [544, 33]}]], "state_before": "case mono_isIso_iff_nonzero.mp\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\ni : mk f = \u22a4\nw : mk f = \u22a5\n\u22a2 False", "state_after": "no goals"}, {"tactic": "intro i", "annotated_tactic": ["intro i", []], "state_before": "case mono_isIso_iff_nonzero.mpr\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\n\u22a2 f \u2260 0 \u2192 IsIso f", "state_after": "case mono_isIso_iff_nonzero.mpr\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\ni : f \u2260 0\n\u22a2 IsIso f"}, {"tactic": "rcases IsSimpleOrder.eq_bot_or_eq_top (Subobject.mk f) with (h | h)", "annotated_tactic": ["rcases <a>IsSimpleOrder.eq_bot_or_eq_top</a> (<a>Subobject.mk</a> f) with (h | h)", [{"full_name": "IsSimpleOrder.eq_bot_or_eq_top", "def_path": "Mathlib/Order/Atoms.lean", "def_pos": [528, 3], "def_end_pos": [528, 19]}, {"full_name": "CategoryTheory.Subobject.mk", "def_path": "Mathlib/CategoryTheory/Subobject/Basic.lean", "def_pos": [110, 5], "def_end_pos": [110, 7]}]], "state_before": "case mono_isIso_iff_nonzero.mpr\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\ni : f \u2260 0\n\u22a2 IsIso f", "state_after": "case mono_isIso_iff_nonzero.mpr.inl\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\ni : f \u2260 0\nh : mk f = \u22a5\n\u22a2 IsIso f\n\ncase mono_isIso_iff_nonzero.mpr.inr\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\ni : f \u2260 0\nh : mk f = \u22a4\n\u22a2 IsIso f"}, {"tactic": "rw [Subobject.mk_eq_bot_iff_zero] at h", "annotated_tactic": ["rw [<a>Subobject.mk_eq_bot_iff_zero</a>] at h", [{"full_name": "CategoryTheory.Subobject.mk_eq_bot_iff_zero", "def_path": "Mathlib/CategoryTheory/Subobject/Lattice.lean", "def_pos": [354, 9], "def_end_pos": [354, 27]}]], "state_before": "case mono_isIso_iff_nonzero.mpr.inl\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\ni : f \u2260 0\nh : mk f = \u22a5\n\u22a2 IsIso f", "state_after": "case mono_isIso_iff_nonzero.mpr.inl\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\ni : f \u2260 0\nh : f = 0\n\u22a2 IsIso f"}, {"tactic": "exact False.elim (i h)", "annotated_tactic": ["exact <a>False.elim</a> (i h)", [{"full_name": "False.elim", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [236, 21], "def_end_pos": [236, 31]}]], "state_before": "case mono_isIso_iff_nonzero.mpr.inl\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\ni : f \u2260 0\nh : f = 0\n\u22a2 IsIso f", "state_after": "no goals"}, {"tactic": "exact (Subobject.isIso_iff_mk_eq_top _).mpr h", "annotated_tactic": ["exact (<a>Subobject.isIso_iff_mk_eq_top</a> _).<a>mpr</a> h", [{"full_name": "CategoryTheory.Subobject.isIso_iff_mk_eq_top", "def_path": "Mathlib/CategoryTheory/Subobject/Lattice.lean", "def_pos": [258, 9], "def_end_pos": [258, 28]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}]], "state_before": "case mono_isIso_iff_nonzero.mpr.inr\nC : Type u\ninst\u271d\u00b3 : Category.{v, u} C\ninst\u271d\u00b2 : HasZeroMorphisms C\ninst\u271d\u00b9 : HasZeroObject C\nX : C\ninst\u271d : IsSimpleOrder (Subobject X)\nY : C\nf : Y \u27f6 X\nhf : Mono f\ni : f \u2260 0\nh : mk f = \u22a4\n\u22a2 IsIso f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Prod.lean", "full_name": "LinearMap.range_coprod", "start": [441, 1], "end": [442, 43], "traced_tactics": [{"tactic": "simp [mem_sup]", "annotated_tactic": ["simp [<a>mem_sup</a>]", [{"full_name": "Submodule.mem_sup", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [425, 9], "def_end_pos": [425, 16]}]], "state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM\u2082 : Type w\nV\u2082 : Type w'\nM\u2083 : Type y\nV\u2083 : Type y'\nM\u2084 : Type z\n\u03b9 : Type x\nM\u2085 : Type u_1\nM\u2086 : Type u_2\ninst\u271d\u2078 : Semiring R\ninst\u271d\u2077 : AddCommMonoid M\ninst\u271d\u2076 : AddCommMonoid M\u2082\ninst\u271d\u2075 : AddCommMonoid M\u2083\ninst\u271d\u2074 : AddCommMonoid M\u2084\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M\u2082\ninst\u271d\u00b9 : Module R M\u2083\ninst\u271d : Module R M\u2084\nf : M \u2192\u2097[R] M\u2083\ng : M\u2082 \u2192\u2097[R] M\u2083\nx : M\u2083\n\u22a2 x \u2208 range (f.coprod g) \u2194 x \u2208 range f \u2294 range g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/InfiniteSum/Real.lean", "full_name": "summable_of_sum_le", "start": [93, 1], "end": [96, 91], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Generator.lean", "full_name": "CategoryTheory.IsCodetecting.isCoseparating", "start": [159, 1], "end": [161, 90], "traced_tactics": [{"tactic": "simpa only [\u2190 isSeparating_op_iff, \u2190 isDetecting_op_iff] using IsDetecting.isSeparating", "annotated_tactic": ["simpa only [\u2190 <a>isSeparating_op_iff</a>, \u2190 <a>isDetecting_op_iff</a>] using <a>IsDetecting.isSeparating</a>", [{"full_name": "CategoryTheory.isSeparating_op_iff", "def_path": "Mathlib/CategoryTheory/Generator.lean", "def_pos": [93, 9], "def_end_pos": [93, 28]}, {"full_name": "CategoryTheory.isDetecting_op_iff", "def_path": "Mathlib/CategoryTheory/Generator.lean", "def_pos": [117, 9], "def_end_pos": [117, 27]}, {"full_name": "CategoryTheory.IsDetecting.isSeparating", "def_path": "Mathlib/CategoryTheory/Generator.lean", "def_pos": [151, 9], "def_end_pos": [151, 33]}]], "state_before": "C : Type u\u2081\ninst\u271d\u00b2 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b9 : Category.{v\u2082, u\u2082} D\ninst\u271d : HasCoequalizers C\n\ud835\udca2 : Set C\n\u22a2 IsCodetecting \ud835\udca2 \u2192 IsCoseparating \ud835\udca2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "full_name": "Subgroup.comap_top", "start": [1582, 1], "end": [1583, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Cardinal/Cofinality.lean", "full_name": "Cardinal.sum_lt_of_isRegular", "start": [1114, 1], "end": [1116, 56], "traced_tactics": [{"tactic": "rwa [lift_id]", "annotated_tactic": ["rwa [<a>lift_id</a>]", [{"full_name": "Cardinal.lift_id", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [224, 9], "def_end_pos": [224, 16]}]], "state_before": "\u03b1 : Type u_1\nr : \u03b1 \u2192 \u03b1 \u2192 Prop\n\u03b9 : Type u\nf : \u03b9 \u2192 Cardinal.{u}\nc : Cardinal.{u}\nhc : c.IsRegular\nh\u03b9 : #\u03b9 < c\n\u22a2 lift.{u, u} #\u03b9 < c", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/TensorPower.lean", "full_name": "TensorPower.mul_one", "start": [181, 1], "end": [192, 58], "traced_tactics": [{"tactic": "rw [gMul_def, gOne_def]", "annotated_tactic": ["rw [<a>gMul_def</a>, <a>gOne_def</a>]", [{"full_name": "TensorPower.gMul_def", "def_path": "Mathlib/LinearAlgebra/TensorPower.lean", "def_pos": [91, 9], "def_end_pos": [91, 17]}, {"full_name": "TensorPower.gOne_def", "def_path": "Mathlib/LinearAlgebra/TensorPower.lean", "def_pos": [74, 9], "def_end_pos": [74, 17]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\na : \u2a02[R]^n M\n\u22a2 (cast R M \u22ef) (GradedMonoid.GMul.mul a GradedMonoid.GOne.one) = a", "state_after": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\na : \u2a02[R]^n M\n\u22a2 (cast R M \u22ef) (mulEquiv (a \u2297\u209c[R] (tprod R) Fin.elim0)) = a"}, {"tactic": "induction a using PiTensorProduct.induction_on with\n| smul_tprod r a =>\n  rw [\u2190 TensorProduct.smul_tmul', LinearEquiv.map_smul, LinearEquiv.map_smul, \u2190 gMul_def,\n    tprod_mul_tprod R a _, cast_tprod]\n  congr 2 with i\n  rw [Fin.append_elim0]\n  refine congr_arg a (Fin.ext ?_)\n  simp\n| add x y hx hy =>\n  rw [TensorProduct.add_tmul, map_add, map_add, hx, hy]", "annotated_tactic": ["induction a using <a>PiTensorProduct.induction_on</a> with\n  | smul_tprod r a =>\n    rw [\u2190 <a>TensorProduct.smul_tmul'</a>, <a>LinearEquiv.map_smul</a>, <a>LinearEquiv.map_smul</a>, \u2190 <a>gMul_def</a>,\n      <a>tprod_mul_tprod</a> R a _, <a>cast_tprod</a>]\n    congr 2 with i\n    rw [<a>Fin.append_elim0</a>]\n    refine <a>congr_arg</a> a (<a>Fin.ext</a> ?_)\n    simp\n  | add x y hx hy =>\n    rw [<a>TensorProduct.add_tmul</a>, <a>map_add</a>, <a>map_add</a>, hx, hy]", [{"full_name": "PiTensorProduct.induction_on", "def_path": "Mathlib/LinearAlgebra/PiTensorProduct.lean", "def_pos": [380, 19], "def_end_pos": [380, 31]}, {"full_name": "TensorProduct.smul_tmul'", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [368, 9], "def_end_pos": [368, 19]}, {"full_name": "LinearEquiv.map_smul", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}, {"full_name": "LinearEquiv.map_smul", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}, {"full_name": "TensorPower.gMul_def", "def_path": "Mathlib/LinearAlgebra/TensorPower.lean", "def_pos": [91, 9], "def_end_pos": [91, 17]}, {"full_name": "TensorPower.tprod_mul_tprod", "def_path": "Mathlib/LinearAlgebra/TensorPower.lean", "def_pos": [154, 9], "def_end_pos": [154, 24]}, {"full_name": "TensorPower.cast_tprod", "def_path": "Mathlib/LinearAlgebra/TensorPower.lean", "def_pos": [108, 9], "def_end_pos": [108, 19]}, {"full_name": "Fin.append_elim0", "def_path": "Mathlib/Data/Fin/Tuple/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 21]}, {"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "Fin.ext", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean", "def_pos": [40, 16], "def_end_pos": [40, 19]}, {"full_name": "TensorProduct.add_tmul", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [194, 9], "def_end_pos": [194, 17]}, {"full_name": "map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [308, 3], "def_end_pos": [308, 14]}, {"full_name": "map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [308, 3], "def_end_pos": [308, 14]}]], "state_before": "R : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\na : \u2a02[R]^n M\n\u22a2 (cast R M \u22ef) (mulEquiv (a \u2297\u209c[R] (tprod R) Fin.elim0)) = a", "state_after": "no goals"}, {"tactic": "rw [\u2190 TensorProduct.smul_tmul', LinearEquiv.map_smul, LinearEquiv.map_smul, \u2190 gMul_def,\n  tprod_mul_tprod R a _, cast_tprod]", "annotated_tactic": ["rw [\u2190 <a>TensorProduct.smul_tmul'</a>, <a>LinearEquiv.map_smul</a>, <a>LinearEquiv.map_smul</a>, \u2190 <a>gMul_def</a>,\n      <a>tprod_mul_tprod</a> R a _, <a>cast_tprod</a>]", [{"full_name": "TensorProduct.smul_tmul'", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [368, 9], "def_end_pos": [368, 19]}, {"full_name": "LinearEquiv.map_smul", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}, {"full_name": "LinearEquiv.map_smul", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [504, 9], "def_end_pos": [504, 17]}, {"full_name": "TensorPower.gMul_def", "def_path": "Mathlib/LinearAlgebra/TensorPower.lean", "def_pos": [91, 9], "def_end_pos": [91, 17]}, {"full_name": "TensorPower.tprod_mul_tprod", "def_path": "Mathlib/LinearAlgebra/TensorPower.lean", "def_pos": [154, 9], "def_end_pos": [154, 24]}, {"full_name": "TensorPower.cast_tprod", "def_path": "Mathlib/LinearAlgebra/TensorPower.lean", "def_pos": [108, 9], "def_end_pos": [108, 19]}]], "state_before": "case smul_tprod\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\nr : R\na : Fin n \u2192 M\n\u22a2 (cast R M \u22ef) (mulEquiv ((r \u2022 (tprod R) a) \u2297\u209c[R] (tprod R) Fin.elim0)) = r \u2022 (tprod R) a", "state_after": "case smul_tprod\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\nr : R\na : Fin n \u2192 M\n\u22a2 r \u2022 (tprod R) (Fin.append a Fin.elim0 \u2218 Fin.cast \u22ef) = r \u2022 (tprod R) a"}, {"tactic": "congr 2 with i", "annotated_tactic": ["congr 2 with i", []], "state_before": "case smul_tprod\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\nr : R\na : Fin n \u2192 M\n\u22a2 r \u2022 (tprod R) (Fin.append a Fin.elim0 \u2218 Fin.cast \u22ef) = r \u2022 (tprod R) a", "state_after": "case smul_tprod.e_a.h.e_6.h.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\nr : R\na : Fin n \u2192 M\ni : Fin n\n\u22a2 (Fin.append a Fin.elim0 \u2218 Fin.cast \u22ef) i = a i"}, {"tactic": "rw [Fin.append_elim0]", "annotated_tactic": ["rw [<a>Fin.append_elim0</a>]", [{"full_name": "Fin.append_elim0", "def_path": "Mathlib/Data/Fin/Tuple/Basic.lean", "def_pos": [366, 9], "def_end_pos": [366, 21]}]], "state_before": "case smul_tprod.e_a.h.e_6.h.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\nr : R\na : Fin n \u2192 M\ni : Fin n\n\u22a2 (Fin.append a Fin.elim0 \u2218 Fin.cast \u22ef) i = a i", "state_after": "case smul_tprod.e_a.h.e_6.h.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\nr : R\na : Fin n \u2192 M\ni : Fin n\n\u22a2 ((a \u2218 Fin.cast \u22ef) \u2218 Fin.cast \u22ef) i = a i"}, {"tactic": "refine congr_arg a (Fin.ext ?_)", "annotated_tactic": ["refine <a>congr_arg</a> a (<a>Fin.ext</a> ?_)", [{"full_name": "congr_arg", "def_path": ".lake/packages/batteries/Batteries/Logic.lean", "def_pos": [76, 7], "def_end_pos": [76, 16]}, {"full_name": "Fin.ext", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Fin/Lemmas.lean", "def_pos": [40, 16], "def_end_pos": [40, 19]}]], "state_before": "case smul_tprod.e_a.h.e_6.h.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\nr : R\na : Fin n \u2192 M\ni : Fin n\n\u22a2 ((a \u2218 Fin.cast \u22ef) \u2218 Fin.cast \u22ef) i = a i", "state_after": "case smul_tprod.e_a.h.e_6.h.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\nr : R\na : Fin n \u2192 M\ni : Fin n\n\u22a2 \u2191(Fin.cast \u22ef (Fin.cast \u22ef i)) = \u2191i"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case smul_tprod.e_a.h.e_6.h.h\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\nr : R\na : Fin n \u2192 M\ni : Fin n\n\u22a2 \u2191(Fin.cast \u22ef (Fin.cast \u22ef i)) = \u2191i", "state_after": "no goals"}, {"tactic": "rw [TensorProduct.add_tmul, map_add, map_add, hx, hy]", "annotated_tactic": ["rw [<a>TensorProduct.add_tmul</a>, <a>map_add</a>, <a>map_add</a>, hx, hy]", [{"full_name": "TensorProduct.add_tmul", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [194, 9], "def_end_pos": [194, 17]}, {"full_name": "map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [308, 3], "def_end_pos": [308, 14]}, {"full_name": "map_add", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [308, 3], "def_end_pos": [308, 14]}]], "state_before": "case add\nR : Type u_1\nM : Type u_2\ninst\u271d\u00b2 : CommSemiring R\ninst\u271d\u00b9 : AddCommMonoid M\ninst\u271d : Module R M\nn : \u2115\nx y : \u2a02[R] (i : Fin n), M\nhx : (cast R M \u22ef) (mulEquiv (x \u2297\u209c[R] (tprod R) Fin.elim0)) = x\nhy : (cast R M \u22ef) (mulEquiv (y \u2297\u209c[R] (tprod R) Fin.elim0)) = y\n\u22a2 (cast R M \u22ef) (mulEquiv ((x + y) \u2297\u209c[R] (tprod R) Fin.elim0)) = x + y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/EMetricSpace/Basic.lean", "full_name": "edist_pi_le_iff", "start": [501, 1], "end": [503, 74], "traced_tactics": [{"tactic": "simp only [Finset.mem_univ, forall_const]", "annotated_tactic": ["simp only [<a>Finset.mem_univ</a>, <a>forall_const</a>]", [{"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [75, 9], "def_end_pos": [75, 17]}, {"full_name": "forall_const", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [265, 17], "def_end_pos": [265, 29]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nX : Type u_1\ninst\u271d\u00b2 : PseudoEMetricSpace \u03b1\n\u03c0 : \u03b2 \u2192 Type u_2\ninst\u271d\u00b9 : Fintype \u03b2\ninst\u271d : (b : \u03b2) \u2192 EDist (\u03c0 b)\nf g : (b : \u03b2) \u2192 \u03c0 b\nd : \u211d\u22650\u221e\n\u22a2 (\u2200 b \u2208 Finset.univ, edist (f b) (g b) \u2264 d) \u2194 \u2200 (b : \u03b2), edist (f b) (g b) \u2264 d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Final.lean", "full_name": "CategoryTheory.Functor.cofinal_of_isTerminal_colimit_comp_yoneda", "start": [433, 1], "end": [438, 91], "traced_tactics": [{"tactic": "refine cofinal_of_colimit_comp_coyoneda_iso_pUnit _ (fun d => ?_)", "annotated_tactic": ["refine <a>cofinal_of_colimit_comp_coyoneda_iso_pUnit</a> _ (fun d => ?_)", [{"full_name": "CategoryTheory.Functor.cofinal_of_colimit_comp_coyoneda_iso_pUnit", "def_path": "Mathlib/CategoryTheory/Limits/Final.lean", "def_pos": [411, 9], "def_end_pos": [411, 51]}]], "state_before": "C : Type v\ninst\u271d\u00b9 : Category.{v, v} C\nD : Type u\u2081\ninst\u271d : Category.{v, u\u2081} D\nF : C \u2964 D\nh : IsTerminal (colimit (F \u22d9 yoneda))\n\u22a2 F.Final", "state_after": "C : Type v\ninst\u271d\u00b9 : Category.{v, v} C\nD : Type u\u2081\ninst\u271d : Category.{v, u\u2081} D\nF : C \u2964 D\nh : IsTerminal (colimit (F \u22d9 yoneda))\nd : D\n\u22a2 colimit (F \u22d9 coyoneda.obj { unop := d }) \u2245 PUnit.{v + 1}"}, {"tactic": "refine Types.isTerminalEquivIsoPUnit _ ?_", "annotated_tactic": ["refine <a>Types.isTerminalEquivIsoPUnit</a> _ ?_", [{"full_name": "CategoryTheory.Limits.Types.isTerminalEquivIsoPUnit", "def_path": "Mathlib/CategoryTheory/Limits/Shapes/Types.lean", "def_pos": [129, 19], "def_end_pos": [129, 42]}]], "state_before": "C : Type v\ninst\u271d\u00b9 : Category.{v, v} C\nD : Type u\u2081\ninst\u271d : Category.{v, u\u2081} D\nF : C \u2964 D\nh : IsTerminal (colimit (F \u22d9 yoneda))\nd : D\n\u22a2 colimit (F \u22d9 coyoneda.obj { unop := d }) \u2245 PUnit.{v + 1}", "state_after": "C : Type v\ninst\u271d\u00b9 : Category.{v, v} C\nD : Type u\u2081\ninst\u271d : Category.{v, u\u2081} D\nF : C \u2964 D\nh : IsTerminal (colimit (F \u22d9 yoneda))\nd : D\n\u22a2 IsTerminal (colimit (F \u22d9 coyoneda.obj { unop := d }))"}, {"tactic": "let b := IsTerminal.isTerminalObj ((evaluation _ _).obj (Opposite.op d)) _ h", "annotated_tactic": ["let b := <a>IsTerminal.isTerminalObj</a> ((<a>evaluation</a> _ _).<a>obj</a> (<a>Opposite.op</a> d)) _ h", [{"full_name": "CategoryTheory.Limits.IsTerminal.isTerminalObj", "def_path": "Mathlib/CategoryTheory/Limits/Preserves/Shapes/Terminal.lean", "def_pos": [46, 5], "def_end_pos": [46, 29]}, {"full_name": "CategoryTheory.evaluation", "def_path": "Mathlib/CategoryTheory/Products/Basic.lean", "def_pos": [187, 5], "def_end_pos": [187, 15]}, {"full_name": "Prefunctor.obj", "def_path": "Mathlib/Combinatorics/Quiver/Basic.lean", "def_pos": [59, 3], "def_end_pos": [59, 6]}, {"full_name": "Opposite.op", "def_path": "Mathlib/Data/Opposite.lean", "def_pos": [37, 3], "def_end_pos": [37, 5]}]], "state_before": "C : Type v\ninst\u271d\u00b9 : Category.{v, v} C\nD : Type u\u2081\ninst\u271d : Category.{v, u\u2081} D\nF : C \u2964 D\nh : IsTerminal (colimit (F \u22d9 yoneda))\nd : D\n\u22a2 IsTerminal (colimit (F \u22d9 coyoneda.obj { unop := d }))", "state_after": "C : Type v\ninst\u271d\u00b9 : Category.{v, v} C\nD : Type u\u2081\ninst\u271d : Category.{v, u\u2081} D\nF : C \u2964 D\nh : IsTerminal (colimit (F \u22d9 yoneda))\nd : D\nb : IsTerminal (((evaluation D\u1d52\u1d56 (Type v)).obj { unop := d }).obj (colimit (F \u22d9 yoneda))) :=\n  IsTerminal.isTerminalObj ((evaluation D\u1d52\u1d56 (Type v)).obj { unop := d }) (colimit (F \u22d9 yoneda)) h\n\u22a2 IsTerminal (colimit (F \u22d9 coyoneda.obj { unop := d }))"}, {"tactic": "exact b.ofIso <| preservesColimitIso ((evaluation _ _).obj (Opposite.op d)) (F \u22d9 yoneda)", "annotated_tactic": ["exact b.ofIso <| <a>preservesColimitIso</a> ((<a>evaluation</a> _ _).<a>obj</a> (<a>Opposite.op</a> d)) (F \u22d9 <a>yoneda</a>)", [{"full_name": "CategoryTheory.preservesColimitIso", "def_path": "Mathlib/CategoryTheory/Limits/Preserves/Limits.lean", "def_pos": [125, 5], "def_end_pos": [125, 24]}, {"full_name": "CategoryTheory.evaluation", "def_path": "Mathlib/CategoryTheory/Products/Basic.lean", "def_pos": [187, 5], "def_end_pos": [187, 15]}, {"full_name": "Prefunctor.obj", "def_path": "Mathlib/Combinatorics/Quiver/Basic.lean", "def_pos": [59, 3], "def_end_pos": [59, 6]}, {"full_name": "Opposite.op", "def_path": "Mathlib/Data/Opposite.lean", "def_pos": [37, 3], "def_end_pos": [37, 5]}, {"full_name": "CategoryTheory.yoneda", "def_path": "Mathlib/CategoryTheory/Yoneda.lean", "def_pos": [38, 5], "def_end_pos": [38, 11]}]], "state_before": "C : Type v\ninst\u271d\u00b9 : Category.{v, v} C\nD : Type u\u2081\ninst\u271d : Category.{v, u\u2081} D\nF : C \u2964 D\nh : IsTerminal (colimit (F \u22d9 yoneda))\nd : D\nb : IsTerminal (((evaluation D\u1d52\u1d56 (Type v)).obj { unop := d }).obj (colimit (F \u22d9 yoneda))) :=\n  IsTerminal.isTerminalObj ((evaluation D\u1d52\u1d56 (Type v)).obj { unop := d }) (colimit (F \u22d9 yoneda)) h\n\u22a2 IsTerminal (colimit (F \u22d9 coyoneda.obj { unop := d }))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean", "full_name": "Polynomial.dvd_pow_natDegree_of_aeval_eq_zero", "start": [182, 1], "end": [186, 30], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Sites/LocallySurjective.lean", "full_name": "CategoryTheory.Presheaf.imageSieve_app", "start": [65, 1], "end": [70, 31], "traced_tactics": [{"tactic": "ext V i", "annotated_tactic": ["ext V i", []], "state_before": "C : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\nU : C\ns : (forget A).obj (F.obj { unop := U })\n\u22a2 imageSieve f ((f.app { unop := U }) s) = \u22a4", "state_after": "case h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\nU : C\ns : (forget A).obj (F.obj { unop := U })\nV : C\ni : V \u27f6 U\n\u22a2 (imageSieve f ((f.app { unop := U }) s)).arrows i \u2194 \u22a4.arrows i"}, {"tactic": "simp only [Sieve.top_apply, iff_true_iff, imageSieve_apply]", "annotated_tactic": ["simp only [<a>Sieve.top_apply</a>, <a>iff_true_iff</a>, <a>imageSieve_apply</a>]", [{"full_name": "CategoryTheory.Sieve.top_apply", "def_path": "Mathlib/CategoryTheory/Sites/Sieves.lean", "def_pos": [394, 9], "def_end_pos": [394, 18]}, {"full_name": "iff_true_iff", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [163, 9], "def_end_pos": [163, 21]}, {"full_name": "CategoryTheory.Presheaf.imageSieve_apply", "def_path": "Mathlib/CategoryTheory/Sites/LocallySurjective.lean", "def_pos": [46, 3], "def_end_pos": [46, 8]}]], "state_before": "case h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\nU : C\ns : (forget A).obj (F.obj { unop := U })\nV : C\ni : V \u27f6 U\n\u22a2 (imageSieve f ((f.app { unop := U }) s)).arrows i \u2194 \u22a4.arrows i", "state_after": "case h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\nU : C\ns : (forget A).obj (F.obj { unop := U })\nV : C\ni : V \u27f6 U\n\u22a2 \u2203 t, (f.app { unop := V }) t = (G.map i.op) ((f.app { unop := U }) s)"}, {"tactic": "have := elementwise_of% (f.naturality i.op)", "annotated_tactic": ["have := elementwise_of% (f.naturality i.op)", []], "state_before": "case h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\nU : C\ns : (forget A).obj (F.obj { unop := U })\nV : C\ni : V \u27f6 U\n\u22a2 \u2203 t, (f.app { unop := V }) t = (G.map i.op) ((f.app { unop := U }) s)", "state_after": "case h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\nU : C\ns : (forget A).obj (F.obj { unop := U })\nV : C\ni : V \u27f6 U\nthis :\n  \u2200 (x : (forget A).obj (F.obj { unop := U })),\n    (f.app { unop := V }) ((F.map i.op) x) = (G.map i.op) ((f.app { unop := U }) x)\n\u22a2 \u2203 t, (f.app { unop := V }) t = (G.map i.op) ((f.app { unop := U }) s)"}, {"tactic": "exact \u27e8F.map i.op s, this s\u27e9", "annotated_tactic": ["exact \u27e8F.map i.op s, this s\u27e9", []], "state_before": "case h\nC : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst\u271d\u00b9 : Category.{v', u'} A\ninst\u271d : ConcreteCategory A\nF G : C\u1d52\u1d56 \u2964 A\nf : F \u27f6 G\nU : C\ns : (forget A).obj (F.obj { unop := U })\nV : C\ni : V \u27f6 U\nthis :\n  \u2200 (x : (forget A).obj (F.obj { unop := U })),\n    (f.app { unop := V }) ((F.map i.op) x) = (G.map i.op) ((f.app { unop := U }) x)\n\u22a2 \u2203 t, (f.app { unop := V }) t = (G.map i.op) ((f.app { unop := U }) s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Floor.lean", "full_name": "Nat.lt_ceil", "start": [290, 1], "end": [291, 33], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean", "full_name": "ContinuousMultilinearMap.range_toUniformOnFun", "start": [34, 1], "end": [47, 79], "traced_tactics": [{"tactic": "ext f", "annotated_tactic": ["ext f", []], "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nE : \u03b9 \u2192 Type u_3\nF : Type u_4\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommGroup (E i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : TopologicalSpace F\n\u22a2 range toUniformOnFun =\n    {f |\n      Continuous ((toFun {s | IsVonNBounded \ud835\udd5c s}) f) \u2227\n        (\u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n            (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (x + y)) =\n              (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x) + (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i y)) \u2227\n          \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n            (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (c \u2022 x)) =\n              c \u2022 (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x)}", "state_after": "case h\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nE : \u03b9 \u2192 Type u_3\nF : Type u_4\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommGroup (E i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : TopologicalSpace F\nf : ((i : \u03b9) \u2192 E i) \u2192\u1d64[{s | IsVonNBounded \ud835\udd5c s}] F\n\u22a2 f \u2208 range toUniformOnFun \u2194\n    f \u2208\n      {f |\n        Continuous ((toFun {s | IsVonNBounded \ud835\udd5c s}) f) \u2227\n          (\u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n              (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (x + y)) =\n                (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x) + (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i y)) \u2227\n            \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n              (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (c \u2022 x)) =\n                c \u2022 (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x)}"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case h\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nE : \u03b9 \u2192 Type u_3\nF : Type u_4\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommGroup (E i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : TopologicalSpace F\nf : ((i : \u03b9) \u2192 E i) \u2192\u1d64[{s | IsVonNBounded \ud835\udd5c s}] F\n\u22a2 f \u2208 range toUniformOnFun \u2194\n    f \u2208\n      {f |\n        Continuous ((toFun {s | IsVonNBounded \ud835\udd5c s}) f) \u2227\n          (\u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n              (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (x + y)) =\n                (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x) + (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i y)) \u2227\n            \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n              (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (c \u2022 x)) =\n                c \u2022 (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x)}", "state_after": "case h.mp\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nE : \u03b9 \u2192 Type u_3\nF : Type u_4\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommGroup (E i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : TopologicalSpace F\nf : ((i : \u03b9) \u2192 E i) \u2192\u1d64[{s | IsVonNBounded \ud835\udd5c s}] F\n\u22a2 f \u2208 range toUniformOnFun \u2192\n    f \u2208\n      {f |\n        Continuous ((toFun {s | IsVonNBounded \ud835\udd5c s}) f) \u2227\n          (\u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n              (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (x + y)) =\n                (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x) + (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i y)) \u2227\n            \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n              (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (c \u2022 x)) =\n                c \u2022 (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x)}\n\ncase h.mpr\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nE : \u03b9 \u2192 Type u_3\nF : Type u_4\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommGroup (E i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : TopologicalSpace F\nf : ((i : \u03b9) \u2192 E i) \u2192\u1d64[{s | IsVonNBounded \ud835\udd5c s}] F\n\u22a2 f \u2208\n      {f |\n        Continuous ((toFun {s | IsVonNBounded \ud835\udd5c s}) f) \u2227\n          (\u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n              (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (x + y)) =\n                (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x) + (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i y)) \u2227\n            \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n              (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (c \u2022 x)) =\n                c \u2022 (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x)} \u2192\n    f \u2208 range toUniformOnFun"}, {"tactic": "rintro \u27e8f, rfl\u27e9", "annotated_tactic": ["rintro \u27e8f, rfl\u27e9", []], "state_before": "case h.mp\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nE : \u03b9 \u2192 Type u_3\nF : Type u_4\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommGroup (E i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : TopologicalSpace F\nf : ((i : \u03b9) \u2192 E i) \u2192\u1d64[{s | IsVonNBounded \ud835\udd5c s}] F\n\u22a2 f \u2208 range toUniformOnFun \u2192\n    f \u2208\n      {f |\n        Continuous ((toFun {s | IsVonNBounded \ud835\udd5c s}) f) \u2227\n          (\u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n              (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (x + y)) =\n                (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x) + (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i y)) \u2227\n            \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n              (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (c \u2022 x)) =\n                c \u2022 (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x)}", "state_after": "case h.mp.intro\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nE : \u03b9 \u2192 Type u_3\nF : Type u_4\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommGroup (E i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : TopologicalSpace F\nf : ContinuousMultilinearMap \ud835\udd5c E F\n\u22a2 f.toUniformOnFun \u2208\n    {f |\n      Continuous ((toFun {s | IsVonNBounded \ud835\udd5c s}) f) \u2227\n        (\u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n            (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (x + y)) =\n              (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x) + (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i y)) \u2227\n          \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n            (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (c \u2022 x)) =\n              c \u2022 (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x)}"}, {"tactic": "exact \u27e8f.cont, f.map_add, f.map_smul\u27e9", "annotated_tactic": ["exact \u27e8f.cont, f.map_add, f.map_smul\u27e9", []], "state_before": "case h.mp.intro\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nE : \u03b9 \u2192 Type u_3\nF : Type u_4\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommGroup (E i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : TopologicalSpace F\nf : ContinuousMultilinearMap \ud835\udd5c E F\n\u22a2 f.toUniformOnFun \u2208\n    {f |\n      Continuous ((toFun {s | IsVonNBounded \ud835\udd5c s}) f) \u2227\n        (\u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n            (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (x + y)) =\n              (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x) + (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i y)) \u2227\n          \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n            (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (c \u2022 x)) =\n              c \u2022 (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x)}", "state_after": "no goals"}, {"tactic": "rintro \u27e8hcont, hadd, hsmul\u27e9", "annotated_tactic": ["rintro \u27e8hcont, hadd, hsmul\u27e9", []], "state_before": "case h.mpr\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nE : \u03b9 \u2192 Type u_3\nF : Type u_4\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommGroup (E i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : TopologicalSpace F\nf : ((i : \u03b9) \u2192 E i) \u2192\u1d64[{s | IsVonNBounded \ud835\udd5c s}] F\n\u22a2 f \u2208\n      {f |\n        Continuous ((toFun {s | IsVonNBounded \ud835\udd5c s}) f) \u2227\n          (\u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n              (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (x + y)) =\n                (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x) + (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i y)) \u2227\n            \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n              (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (c \u2022 x)) =\n                c \u2022 (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x)} \u2192\n    f \u2208 range toUniformOnFun", "state_after": "case h.mpr.intro.intro\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nE : \u03b9 \u2192 Type u_3\nF : Type u_4\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommGroup (E i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : TopologicalSpace F\nf : ((i : \u03b9) \u2192 E i) \u2192\u1d64[{s | IsVonNBounded \ud835\udd5c s}] F\nhcont : Continuous ((toFun {s | IsVonNBounded \ud835\udd5c s}) f)\nhadd :\n  \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n    (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (x + y)) =\n      (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x) + (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i y)\nhsmul :\n  \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n    (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (c \u2022 x)) = c \u2022 (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x)\n\u22a2 f \u2208 range toUniformOnFun"}, {"tactic": "exact \u27e8\u27e8\u27e8f, by intro; convert hadd, by intro; convert hsmul\u27e9, hcont\u27e9, rfl\u27e9", "annotated_tactic": ["exact \u27e8\u27e8\u27e8f, by intro; convert hadd, by intro; convert hsmul\u27e9, hcont\u27e9, <a>rfl</a>\u27e9", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case h.mpr.intro.intro\n\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nE : \u03b9 \u2192 Type u_3\nF : Type u_4\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommGroup (E i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : TopologicalSpace F\nf : ((i : \u03b9) \u2192 E i) \u2192\u1d64[{s | IsVonNBounded \ud835\udd5c s}] F\nhcont : Continuous ((toFun {s | IsVonNBounded \ud835\udd5c s}) f)\nhadd :\n  \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n    (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (x + y)) =\n      (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x) + (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i y)\nhsmul :\n  \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n    (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (c \u2022 x)) = c \u2022 (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x)\n\u22a2 f \u2208 range toUniformOnFun", "state_after": "no goals"}, {"tactic": "intro", "annotated_tactic": ["intro", []], "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nE : \u03b9 \u2192 Type u_3\nF : Type u_4\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommGroup (E i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : TopologicalSpace F\nf : ((i : \u03b9) \u2192 E i) \u2192\u1d64[{s | IsVonNBounded \ud835\udd5c s}] F\nhcont : Continuous ((toFun {s | IsVonNBounded \ud835\udd5c s}) f)\nhadd :\n  \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n    (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (x + y)) =\n      (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x) + (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i y)\nhsmul :\n  \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n    (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (c \u2022 x)) = c \u2022 (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x)\n\u22a2 \u2200 [inst : DecidableEq \u03b9] (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n    f (update m i (x + y)) = f (update m i x) + f (update m i y)", "state_after": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nE : \u03b9 \u2192 Type u_3\nF : Type u_4\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommGroup (E i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : Module \ud835\udd5c F\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : TopologicalSpace F\nf : ((i : \u03b9) \u2192 E i) \u2192\u1d64[{s | IsVonNBounded \ud835\udd5c s}] F\nhcont : Continuous ((toFun {s | IsVonNBounded \ud835\udd5c s}) f)\nhadd :\n  \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n    (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (x + y)) =\n      (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x) + (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i y)\nhsmul :\n  \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n    (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (c \u2022 x)) = c \u2022 (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x)\ninst\u271d : DecidableEq \u03b9\n\u22a2 \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i), f (update m i (x + y)) = f (update m i x) + f (update m i y)"}, {"tactic": "convert hadd", "annotated_tactic": ["convert hadd", []], "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nE : \u03b9 \u2192 Type u_3\nF : Type u_4\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommGroup (E i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : Module \ud835\udd5c F\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : TopologicalSpace F\nf : ((i : \u03b9) \u2192 E i) \u2192\u1d64[{s | IsVonNBounded \ud835\udd5c s}] F\nhcont : Continuous ((toFun {s | IsVonNBounded \ud835\udd5c s}) f)\nhadd :\n  \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n    (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (x + y)) =\n      (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x) + (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i y)\nhsmul :\n  \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n    (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (c \u2022 x)) = c \u2022 (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x)\ninst\u271d : DecidableEq \u03b9\n\u22a2 \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i), f (update m i (x + y)) = f (update m i x) + f (update m i y)", "state_after": "no goals"}, {"tactic": "intro", "annotated_tactic": ["intro", []], "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nE : \u03b9 \u2192 Type u_3\nF : Type u_4\ninst\u271d\u2077 : NormedField \ud835\udd5c\ninst\u271d\u2076 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 AddCommGroup (E i)\ninst\u271d\u2074 : (i : \u03b9) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u00b3 : AddCommGroup F\ninst\u271d\u00b2 : Module \ud835\udd5c F\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : TopologicalSpace F\nf : ((i : \u03b9) \u2192 E i) \u2192\u1d64[{s | IsVonNBounded \ud835\udd5c s}] F\nhcont : Continuous ((toFun {s | IsVonNBounded \ud835\udd5c s}) f)\nhadd :\n  \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n    (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (x + y)) =\n      (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x) + (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i y)\nhsmul :\n  \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n    (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (c \u2022 x)) = c \u2022 (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x)\n\u22a2 \u2200 [inst : DecidableEq \u03b9] (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i), f (update m i (c \u2022 x)) = c \u2022 f (update m i x)", "state_after": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nE : \u03b9 \u2192 Type u_3\nF : Type u_4\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommGroup (E i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : Module \ud835\udd5c F\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : TopologicalSpace F\nf : ((i : \u03b9) \u2192 E i) \u2192\u1d64[{s | IsVonNBounded \ud835\udd5c s}] F\nhcont : Continuous ((toFun {s | IsVonNBounded \ud835\udd5c s}) f)\nhadd :\n  \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n    (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (x + y)) =\n      (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x) + (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i y)\nhsmul :\n  \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n    (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (c \u2022 x)) = c \u2022 (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x)\ninst\u271d : DecidableEq \u03b9\n\u22a2 \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i), f (update m i (c \u2022 x)) = c \u2022 f (update m i x)"}, {"tactic": "convert hsmul", "annotated_tactic": ["convert hsmul", []], "state_before": "\ud835\udd5c : Type u_1\n\u03b9 : Type u_2\nE : \u03b9 \u2192 Type u_3\nF : Type u_4\ninst\u271d\u2078 : NormedField \ud835\udd5c\ninst\u271d\u2077 : (i : \u03b9) \u2192 TopologicalSpace (E i)\ninst\u271d\u2076 : (i : \u03b9) \u2192 AddCommGroup (E i)\ninst\u271d\u2075 : (i : \u03b9) \u2192 Module \ud835\udd5c (E i)\ninst\u271d\u2074 : AddCommGroup F\ninst\u271d\u00b3 : Module \ud835\udd5c F\ninst\u271d\u00b2 : DecidableEq \u03b9\ninst\u271d\u00b9 : TopologicalSpace F\nf : ((i : \u03b9) \u2192 E i) \u2192\u1d64[{s | IsVonNBounded \ud835\udd5c s}] F\nhcont : Continuous ((toFun {s | IsVonNBounded \ud835\udd5c s}) f)\nhadd :\n  \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (x y : E i),\n    (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (x + y)) =\n      (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x) + (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i y)\nhsmul :\n  \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i),\n    (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i (c \u2022 x)) = c \u2022 (toFun {s | IsVonNBounded \ud835\udd5c s}) f (update m i x)\ninst\u271d : DecidableEq \u03b9\n\u22a2 \u2200 (m : (i : \u03b9) \u2192 E i) (i : \u03b9) (c : \ud835\udd5c) (x : E i), f (update m i (c \u2022 x)) = c \u2022 f (update m i x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/ArithmeticFunction.lean", "full_name": "ArithmeticFunction.ppow_zero", "start": [540, 1], "end": [540, 89], "traced_tactics": [{"tactic": "rw [ppow, dif_pos rfl]", "annotated_tactic": ["rw [<a>ppow</a>, <a>dif_pos</a> <a>rfl</a>]", [{"full_name": "ArithmeticFunction.ppow", "def_path": "Mathlib/NumberTheory/ArithmeticFunction.lean", "def_pos": [535, 5], "def_end_pos": [535, 9]}, {"full_name": "dif_pos", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [949, 9], "def_end_pos": [949, 16]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\nf : ArithmeticFunction R\n\u22a2 f.ppow 0 = \u2191\u03b6", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Pointwise.lean", "full_name": "Filter.isUnit_pure", "start": [904, 1], "end": [905, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/WithBot.lean", "full_name": "WithBot.monotone_map_iff", "start": [394, 1], "end": [395, 43], "traced_tactics": [{"tactic": "simp [Monotone]", "annotated_tactic": ["simp [<a>Monotone</a>]", [{"full_name": "Monotone", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [77, 5], "def_end_pos": [77, 13]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\na b : \u03b1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : Preorder \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 ((Monotone fun a => map f \u2191a) \u2227 \u2200 (x : \u03b1), map f \u22a5 \u2264 map f \u2191x) \u2194 Monotone f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/NaturalOps.lean", "full_name": "NatOrdinal.toOrdinal_symm_eq", "start": [79, 1], "end": [80, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Instances/AddCircle.lean", "full_name": "AddCircle.coe_image_Ioc_eq", "start": [323, 1], "end": [325, 42], "traced_tactics": [{"tactic": "rw [image_eq_range]", "annotated_tactic": ["rw [<a>image_eq_range</a>]", [{"full_name": "Set.image_eq_range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [1055, 9], "def_end_pos": [1055, 23]}]], "state_before": "\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : LinearOrderedAddCommGroup \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b9 : OrderTopology \ud835\udd5c\np : \ud835\udd5c\nhp : Fact (0 < p)\na : \ud835\udd5c\ninst\u271d : Archimedean \ud835\udd5c\n\u22a2 QuotientAddGroup.mk '' Ioc a (a + p) = univ", "state_after": "\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : LinearOrderedAddCommGroup \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b9 : OrderTopology \ud835\udd5c\np : \ud835\udd5c\nhp : Fact (0 < p)\na : \ud835\udd5c\ninst\u271d : Archimedean \ud835\udd5c\n\u22a2 (range fun x => \u2191\u2191x) = univ"}, {"tactic": "exact (equivIoc p a).symm.range_eq_univ", "annotated_tactic": ["exact (<a>equivIoc</a> p a).symm.range_eq_univ", [{"full_name": "AddCircle.equivIoc", "def_path": "Mathlib/Topology/Instances/AddCircle.lean", "def_pos": [195, 5], "def_end_pos": [195, 13]}]], "state_before": "\ud835\udd5c : Type u_1\nB : Type u_2\ninst\u271d\u00b3 : LinearOrderedAddCommGroup \ud835\udd5c\ninst\u271d\u00b2 : TopologicalSpace \ud835\udd5c\ninst\u271d\u00b9 : OrderTopology \ud835\udd5c\np : \ud835\udd5c\nhp : Fact (0 < p)\na : \ud835\udd5c\ninst\u271d : Archimedean \ud835\udd5c\n\u22a2 (range fun x => \u2191\u2191x) = univ", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/UpperLower/Hom.lean", "full_name": "UpperSet.icisSupHom_apply", "start": [61, 1], "end": [62, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/OrderClosed.lean", "full_name": "le_of_tendsto", "start": [131, 1], "end": [133, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Connected/PathConnected.lean", "full_name": "isPathConnected_iff", "start": [972, 1], "end": [976, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean", "full_name": "SimpleGraph.ComponentCompl.nonempty", "start": [116, 11], "end": [117, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "full_name": "Polynomial.cyclotomic_prime_pow_mul_X_pow_sub_one", "start": [579, 1], "end": [582, 92], "traced_tactics": [{"tactic": "rw [cyclotomic_prime_pow_eq_geom_sum hn.out, geom_sum_mul, \u2190 pow_mul, pow_succ, mul_comm]", "annotated_tactic": ["rw [<a>cyclotomic_prime_pow_eq_geom_sum</a> hn.out, <a>geom_sum_mul</a>, \u2190 <a>pow_mul</a>, <a>pow_succ</a>, <a>mul_comm</a>]", [{"full_name": "Polynomial.cyclotomic_prime_pow_eq_geom_sum", "def_path": "Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean", "def_pos": [559, 9], "def_end_pos": [559, 41]}, {"full_name": "geom_sum_mul", "def_path": "Mathlib/Algebra/GeomSum.lean", "def_pos": [230, 9], "def_end_pos": [230, 21]}, {"full_name": "pow_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [713, 32], "def_end_pos": [713, 39]}, {"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [657, 9], "def_end_pos": [657, 17]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "R : Type u_1\ninst\u271d : CommRing R\np k : \u2115\nhn : Fact (Nat.Prime p)\n\u22a2 cyclotomic (p ^ (k + 1)) R * (X ^ p ^ k - 1) = X ^ p ^ (k + 1) - 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/SetToL1.lean", "full_name": "MeasureTheory.SimpleFunc.setToSimpleFunc_nonneg", "start": [528, 1], "end": [536, 7], "traced_tactics": [{"tactic": "refine sum_nonneg fun i hi => hT_nonneg _ i ?_", "annotated_tactic": ["refine <a>sum_nonneg</a> fun i hi => hT_nonneg _ i ?_", [{"full_name": "Finset.sum_nonneg", "def_path": "Mathlib/Algebra/Order/BigOperators/Group/Finset.lean", "def_pos": [136, 15], "def_end_pos": [136, 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\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\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b2 : NormedSpace \u211d G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nm : MeasurableSpace \u03b1\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nhT_nonneg : \u2200 (s : Set \u03b1) (x : G'), 0 \u2264 x \u2192 0 \u2264 (T s) x\nf : \u03b1 \u2192\u209b G'\nhf : 0 \u2264 f\n\u22a2 0 \u2264 setToSimpleFunc T f", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\nG : Type u_5\n\ud835\udd5c : Type u_6\np : \u211d\u22650\u221e\ninst\u271d\u00b9\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\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b2 : NormedSpace \u211d G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nm : MeasurableSpace \u03b1\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nhT_nonneg : \u2200 (s : Set \u03b1) (x : G'), 0 \u2264 x \u2192 0 \u2264 (T s) x\nf : \u03b1 \u2192\u209b G'\nhf : 0 \u2264 f\ni : G'\nhi : i \u2208 f.range\n\u22a2 0 \u2264 i"}, {"tactic": "rw [mem_range] at hi", "annotated_tactic": ["rw [<a>mem_range</a>] at hi", [{"full_name": "MeasureTheory.SimpleFunc.mem_range", "def_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "def_pos": [106, 9], "def_end_pos": [106, 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\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b2 : NormedSpace \u211d G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nm : MeasurableSpace \u03b1\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nhT_nonneg : \u2200 (s : Set \u03b1) (x : G'), 0 \u2264 x \u2192 0 \u2264 (T s) x\nf : \u03b1 \u2192\u209b G'\nhf : 0 \u2264 f\ni : G'\nhi : i \u2208 f.range\n\u22a2 0 \u2264 i", "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\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b2 : NormedSpace \u211d G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nm : MeasurableSpace \u03b1\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nhT_nonneg : \u2200 (s : Set \u03b1) (x : G'), 0 \u2264 x \u2192 0 \u2264 (T s) x\nf : \u03b1 \u2192\u209b G'\nhf : 0 \u2264 f\ni : G'\nhi : i \u2208 Set.range \u2191f\n\u22a2 0 \u2264 i"}, {"tactic": "obtain \u27e8y, hy\u27e9 := Set.mem_range.mp hi", "annotated_tactic": ["obtain \u27e8y, hy\u27e9 := Set.mem_range.mp hi", []], "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\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b2 : NormedSpace \u211d G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nm : MeasurableSpace \u03b1\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nhT_nonneg : \u2200 (s : Set \u03b1) (x : G'), 0 \u2264 x \u2192 0 \u2264 (T s) x\nf : \u03b1 \u2192\u209b G'\nhf : 0 \u2264 f\ni : G'\nhi : i \u2208 Set.range \u2191f\n\u22a2 0 \u2264 i", "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\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\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b2 : NormedSpace \u211d G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nm : MeasurableSpace \u03b1\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nhT_nonneg : \u2200 (s : Set \u03b1) (x : G'), 0 \u2264 x \u2192 0 \u2264 (T s) x\nf : \u03b1 \u2192\u209b G'\nhf : 0 \u2264 f\ni : G'\nhi : i \u2208 Set.range \u2191f\ny : \u03b1\nhy : \u2191f y = i\n\u22a2 0 \u2264 i"}, {"tactic": "rw [\u2190 hy]", "annotated_tactic": ["rw [\u2190 hy]", []], "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\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\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b2 : NormedSpace \u211d G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nm : MeasurableSpace \u03b1\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nhT_nonneg : \u2200 (s : Set \u03b1) (x : G'), 0 \u2264 x \u2192 0 \u2264 (T s) x\nf : \u03b1 \u2192\u209b G'\nhf : 0 \u2264 f\ni : G'\nhi : i \u2208 Set.range \u2191f\ny : \u03b1\nhy : \u2191f y = i\n\u22a2 0 \u2264 i", "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\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\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b2 : NormedSpace \u211d G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nm : MeasurableSpace \u03b1\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nhT_nonneg : \u2200 (s : Set \u03b1) (x : G'), 0 \u2264 x \u2192 0 \u2264 (T s) x\nf : \u03b1 \u2192\u209b G'\nhf : 0 \u2264 f\ni : G'\nhi : i \u2208 Set.range \u2191f\ny : \u03b1\nhy : \u2191f y = i\n\u22a2 0 \u2264 \u2191f y"}, {"tactic": "refine le_trans ?_ (hf y)", "annotated_tactic": ["refine <a>le_trans</a> ?_ (hf y)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}]], "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\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\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b2 : NormedSpace \u211d G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nm : MeasurableSpace \u03b1\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nhT_nonneg : \u2200 (s : Set \u03b1) (x : G'), 0 \u2264 x \u2192 0 \u2264 (T s) x\nf : \u03b1 \u2192\u209b G'\nhf : 0 \u2264 f\ni : G'\nhi : i \u2208 Set.range \u2191f\ny : \u03b1\nhy : \u2191f y = i\n\u22a2 0 \u2264 \u2191f y", "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\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\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b2 : NormedSpace \u211d G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nm : MeasurableSpace \u03b1\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nhT_nonneg : \u2200 (s : Set \u03b1) (x : G'), 0 \u2264 x \u2192 0 \u2264 (T s) x\nf : \u03b1 \u2192\u209b G'\nhf : 0 \u2264 f\ni : G'\nhi : i \u2208 Set.range \u2191f\ny : \u03b1\nhy : \u2191f y = i\n\u22a2 0 \u2264 \u21910 y"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "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\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\u271d : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nG' : Type u_7\nG'' : Type u_8\ninst\u271d\u00b3 : NormedLatticeAddCommGroup G''\ninst\u271d\u00b2 : NormedSpace \u211d G''\ninst\u271d\u00b9 : NormedLatticeAddCommGroup G'\ninst\u271d : NormedSpace \u211d G'\nm : MeasurableSpace \u03b1\nT : Set \u03b1 \u2192 G' \u2192L[\u211d] G''\nhT_nonneg : \u2200 (s : Set \u03b1) (x : G'), 0 \u2264 x \u2192 0 \u2264 (T s) x\nf : \u03b1 \u2192\u209b G'\nhf : 0 \u2264 f\ni : G'\nhi : i \u2208 Set.range \u2191f\ny : \u03b1\nhy : \u2191f y = i\n\u22a2 0 \u2264 \u21910 y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/MorphismProperty/Basic.lean", "full_name": "CategoryTheory.MorphismProperty.map_le_iff", "start": [198, 1], "end": [205, 46], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nP : MorphismProperty C\nF : C \u2964 D\nQ : MorphismProperty D\nhQ : Q.RespectsIso\n\u22a2 P.map F \u2264 Q \u2194 P \u2264 Q.inverseImage F", "state_after": "case mp\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nP : MorphismProperty C\nF : C \u2964 D\nQ : MorphismProperty D\nhQ : Q.RespectsIso\n\u22a2 P.map F \u2264 Q \u2192 P \u2264 Q.inverseImage F\n\ncase mpr\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nP : MorphismProperty C\nF : C \u2964 D\nQ : MorphismProperty D\nhQ : Q.RespectsIso\n\u22a2 P \u2264 Q.inverseImage F \u2192 P.map F \u2264 Q"}, {"tactic": "intro h X Y f hf", "annotated_tactic": ["intro h X Y f hf", []], "state_before": "case mp\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nP : MorphismProperty C\nF : C \u2964 D\nQ : MorphismProperty D\nhQ : Q.RespectsIso\n\u22a2 P.map F \u2264 Q \u2192 P \u2264 Q.inverseImage F", "state_after": "case mp\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nP : MorphismProperty C\nF : C \u2964 D\nQ : MorphismProperty D\nhQ : Q.RespectsIso\nh : P.map F \u2264 Q\nX Y : C\nf : X \u27f6 Y\nhf : P f\n\u22a2 Q.inverseImage F f"}, {"tactic": "exact h (F.map f) (map_mem_map P F f hf)", "annotated_tactic": ["exact h (F.map f) (<a>map_mem_map</a> P F f hf)", [{"full_name": "CategoryTheory.MorphismProperty.map_mem_map", "def_path": "Mathlib/CategoryTheory/MorphismProperty/Basic.lean", "def_pos": [96, 7], "def_end_pos": [96, 18]}]], "state_before": "case mp\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nP : MorphismProperty C\nF : C \u2964 D\nQ : MorphismProperty D\nhQ : Q.RespectsIso\nh : P.map F \u2264 Q\nX Y : C\nf : X \u27f6 Y\nhf : P f\n\u22a2 Q.inverseImage F f", "state_after": "no goals"}, {"tactic": "intro h X Y f \u27e8X', Y', f', hf', \u27e8e\u27e9\u27e9", "annotated_tactic": ["intro h X Y f \u27e8X', Y', f', hf', \u27e8e\u27e9\u27e9", []], "state_before": "case mpr\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nP : MorphismProperty C\nF : C \u2964 D\nQ : MorphismProperty D\nhQ : Q.RespectsIso\n\u22a2 P \u2264 Q.inverseImage F \u2192 P.map F \u2264 Q", "state_after": "case mpr\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nP : MorphismProperty C\nF : C \u2964 D\nQ : MorphismProperty D\nhQ : Q.RespectsIso\nh : P \u2264 Q.inverseImage F\nX Y : D\nf : X \u27f6 Y\nX' Y' : C\nf' : X' \u27f6 Y'\nhf' : P f'\ne : Arrow.mk (F.map f') \u2245 Arrow.mk f\n\u22a2 Q f"}, {"tactic": "exact (hQ.arrow_mk_iso_iff e).1 (h _ hf')", "annotated_tactic": ["exact (hQ.arrow_mk_iso_iff e).1 (h _ hf')", []], "state_before": "case mpr\nC : Type u\ninst\u271d\u00b9 : Category.{v, u} C\nD : Type u_1\ninst\u271d : Category.{u_2, u_1} D\nP : MorphismProperty C\nF : C \u2964 D\nQ : MorphismProperty D\nhQ : Q.RespectsIso\nh : P \u2264 Q.inverseImage F\nX Y : D\nf : X \u27f6 Y\nX' Y' : C\nf' : X' \u27f6 Y'\nhf' : P f'\ne : Arrow.mk (F.map f') \u2245 Arrow.mk f\n\u22a2 Q f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Basic.lean", "full_name": "mul_mul_mul_comm", "start": [196, 1], "end": [197, 39], "traced_tactics": [{"tactic": "simp only [mul_left_comm, mul_assoc]", "annotated_tactic": ["simp only [<a>mul_left_comm</a>, <a>mul_assoc</a>]", [{"full_name": "mul_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [184, 9], "def_end_pos": [184, 22]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nG : Type u_3\nM : Type u_4\ninst\u271d : CommSemigroup G\na b c d : G\n\u22a2 a * b * (c * d) = a * c * (b * d)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "full_name": "MeasureTheory.Integrable.pos_part", "start": [1137, 1], "end": [1140, 35], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Process/Stopping.lean", "full_name": "MeasureTheory.condexp_min_stopping_time_ae_eq_restrict_le", "start": [1203, 1], "end": [1215, 84], "traced_tactics": [{"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": [373, 9], "def_end_pos": [373, 27]}, {"full_name": "Filter.EventuallyEq.symm", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1530, 9], "def_end_pos": [1530, 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 : atTop.IsCountablyGenerated\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 : SigmaFinite (\u03bc.trim \u22ef)\nthis : SigmaFinite (\u03bc.trim \u22ef)\n\u22a2 \u03bc[f|\u22ef.measurableSpace] =\u1da0[ae (\u03bc.restrict {x | \u03c4 x \u2264 \u03c3 x})] \u03bc[f|h\u03c4.measurableSpace]", "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 : atTop.IsCountablyGenerated\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 : SigmaFinite (\u03bc.trim \u22ef)\nthis : SigmaFinite (\u03bc.trim \u22ef)\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": [905, 9], "def_end_pos": [905, 19]}, {"full_name": "MeasureTheory.IsStoppingTime.measurableSet_inter_le_iff", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [650, 9], "def_end_pos": [650, 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 : atTop.IsCountablyGenerated\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 : SigmaFinite (\u03bc.trim \u22ef)\nthis : SigmaFinite (\u03bc.trim \u22ef)\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 : atTop.IsCountablyGenerated\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 : SigmaFinite (\u03bc.trim \u22ef)\nthis : SigmaFinite (\u03bc.trim \u22ef)\nt : Set \u03a9\n\u22a2 IsStoppingTime \u2131 \u03c3"}, {"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 : atTop.IsCountablyGenerated\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 : SigmaFinite (\u03bc.trim \u22ef)\nthis : SigmaFinite (\u03bc.trim \u22ef)\nt : Set \u03a9\n\u22a2 IsStoppingTime \u2131 \u03c3", "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": [586, 9], "def_end_pos": [586, 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 : atTop.IsCountablyGenerated\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 : SigmaFinite (\u03bc.trim \u22ef)\n\u22a2 \u22ef.measurableSpace \u2264 h\u03c4.measurableSpace", "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 : atTop.IsCountablyGenerated\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 : SigmaFinite (\u03bc.trim \u22ef)\n\u22a2 ?h\u03c4.measurableSpace \u2293 ?h\u03c0.measurableSpace \u2264 h\u03c4.measurableSpace\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 : atTop.IsCountablyGenerated\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 : SigmaFinite (\u03bc.trim \u22ef)\n\u22a2 IsStoppingTime \u2131 \u03c4\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 : atTop.IsCountablyGenerated\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 : SigmaFinite (\u03bc.trim \u22ef)\n\u22a2 IsStoppingTime \u2131 \u03c3"}, {"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": [358, 9], "def_end_pos": [358, 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 : atTop.IsCountablyGenerated\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 : SigmaFinite (\u03bc.trim \u22ef)\n\u22a2 ?h\u03c4.measurableSpace \u2293 ?h\u03c0.measurableSpace \u2264 h\u03c4.measurableSpace", "state_after": "no goals"}, {"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 : atTop.IsCountablyGenerated\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 : SigmaFinite (\u03bc.trim \u22ef)\n\u22a2 IsStoppingTime \u2131 \u03c3", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/Enumerative/Composition.lean", "full_name": "Composition.one_le_blocks'", "start": [174, 1], "end": [175, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean", "full_name": "CategoryTheory.Limits.prod.diag_map", "start": [810, 1], "end": [811, 50], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\ninst\u271d\u00b2 : Category.{v, u} C\nX\u271d Y\u271d X Y : C\nf : X \u27f6 Y\ninst\u271d\u00b9 : HasBinaryProduct X X\ninst\u271d : HasBinaryProduct Y Y\n\u22a2 diag X \u226b map f f = f \u226b diag Y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.hausdorffMeasure_measurePreserving_piFinTwo", "start": [1037, 1], "end": [1041, 66], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/FiniteDimensional.lean", "full_name": "FiniteDimensional.finrank_of_infinite_dimensional", "start": [183, 1], "end": [184, 44], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Perm.lean", "full_name": "List.bind_append_perm", "start": [518, 1], "end": [523, 40], "traced_tactics": [{"tactic": "induction' l with a l IH <;> simp", "annotated_tactic": ["induction' l with a l IH <;> simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d l\u2081 l\u2082 : List \u03b1\na : \u03b1\nl : List \u03b1\nf g : \u03b1 \u2192 List \u03b2\n\u22a2 l.bind f ++ l.bind g ~ l.bind fun x => f x ++ g x", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d : \u03b1\nf g : \u03b1 \u2192 List \u03b2\na : \u03b1\nl : List \u03b1\nIH : l.bind f ++ l.bind g ~ l.bind fun x => f x ++ g x\n\u22a2 f a ++ (l.bind f ++ (g a ++ l.bind g)) ~ f a ++ (g a ++ l.bind fun x => f x ++ g x)"}, {"tactic": "refine (Perm.trans ?_ (IH.append_left _)).append_left _", "annotated_tactic": ["refine (<a>Perm.trans</a> ?_ (IH.append_left _)).<a>append_left</a> _", [{"full_name": "List.Perm.trans", "def_path": ".lake/packages/batteries/Batteries/Data/List/Basic.lean", "def_pos": [1358, 5], "def_end_pos": [1358, 10]}, {"full_name": "List.Perm.append_left", "def_path": ".lake/packages/batteries/Batteries/Data/List/Perm.lean", "def_pos": [85, 9], "def_end_pos": [85, 25]}]], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d : \u03b1\nf g : \u03b1 \u2192 List \u03b2\na : \u03b1\nl : List \u03b1\nIH : l.bind f ++ l.bind g ~ l.bind fun x => f x ++ g x\n\u22a2 f a ++ (l.bind f ++ (g a ++ l.bind g)) ~ f a ++ (g a ++ l.bind fun x => f x ++ g x)", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d : \u03b1\nf g : \u03b1 \u2192 List \u03b2\na : \u03b1\nl : List \u03b1\nIH : l.bind f ++ l.bind g ~ l.bind fun x => f x ++ g x\n\u22a2 l.bind f ++ (g a ++ l.bind g) ~ g a ++ (l.bind f ++ l.bind g)"}, {"tactic": "rw [\u2190 append_assoc, \u2190 append_assoc]", "annotated_tactic": ["rw [\u2190 <a>append_assoc</a>, \u2190 <a>append_assoc</a>]", [{"full_name": "List.append_assoc", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [501, 17], "def_end_pos": [501, 29]}, {"full_name": "List.append_assoc", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [501, 17], "def_end_pos": [501, 29]}]], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d : \u03b1\nf g : \u03b1 \u2192 List \u03b2\na : \u03b1\nl : List \u03b1\nIH : l.bind f ++ l.bind g ~ l.bind fun x => f x ++ g x\n\u22a2 l.bind f ++ (g a ++ l.bind g) ~ g a ++ (l.bind f ++ l.bind g)", "state_after": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d : \u03b1\nf g : \u03b1 \u2192 List \u03b2\na : \u03b1\nl : List \u03b1\nIH : l.bind f ++ l.bind g ~ l.bind fun x => f x ++ g x\n\u22a2 l.bind f ++ g a ++ l.bind g ~ g a ++ l.bind f ++ l.bind g"}, {"tactic": "exact perm_append_comm.append_right _", "annotated_tactic": ["exact perm_append_comm.append_right _", []], "state_before": "case cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nl\u271d l\u2081 l\u2082 : List \u03b1\na\u271d : \u03b1\nf g : \u03b1 \u2192 List \u03b2\na : \u03b1\nl : List \u03b1\nIH : l.bind f ++ l.bind g ~ l.bind fun x => f x ++ g x\n\u22a2 l.bind f ++ g a ++ l.bind g ~ g a ++ l.bind f ++ l.bind g", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/IdealOperations.lean", "full_name": "LieIdeal.comap_bracket_le", "start": [284, 1], "end": [286, 94], "traced_tactics": [{"tactic": "rw [\u2190 map_le_iff_le_comap]", "annotated_tactic": ["rw [\u2190 <a>map_le_iff_le_comap</a>]", [{"full_name": "LieIdeal.map_le_iff_le_comap", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [1043, 9], "def_end_pos": [1043, 28]}]], "state_before": "R : Type u\nL : Type v\nL' : Type w\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\nf : L \u2192\u2097\u2045R\u2046 L'\nI : LieIdeal R L\nJ J\u2081 J\u2082 : LieIdeal R L'\n\u22a2 \u2045comap f J\u2081, comap f J\u2082\u2046 \u2264 comap f \u2045J\u2081, J\u2082\u2046", "state_after": "R : Type u\nL : Type v\nL' : Type w\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\nf : L \u2192\u2097\u2045R\u2046 L'\nI : LieIdeal R L\nJ J\u2081 J\u2082 : LieIdeal R L'\n\u22a2 map f \u2045comap f J\u2081, comap f J\u2082\u2046 \u2264 \u2045J\u2081, J\u2082\u2046"}, {"tactic": "exact le_trans (map_bracket_le f) (LieSubmodule.mono_lie _ _ _ _ map_comap_le map_comap_le)", "annotated_tactic": ["exact <a>le_trans</a> (<a>map_bracket_le</a> f) (<a>LieSubmodule.mono_lie</a> _ _ _ _ <a>map_comap_le</a> <a>map_comap_le</a>)", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "LieIdeal.map_bracket_le", "def_path": "Mathlib/Algebra/Lie/IdealOperations.lean", "def_pos": [261, 9], "def_end_pos": [261, 23]}, {"full_name": "LieSubmodule.mono_lie", "def_path": "Mathlib/Algebra/Lie/IdealOperations.lean", "def_pos": [148, 9], "def_end_pos": [148, 17]}, {"full_name": "LieIdeal.map_comap_le", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [1060, 9], "def_end_pos": [1060, 21]}, {"full_name": "LieIdeal.map_comap_le", "def_path": "Mathlib/Algebra/Lie/Submodule.lean", "def_pos": [1060, 9], "def_end_pos": [1060, 21]}]], "state_before": "R : Type u\nL : Type v\nL' : Type w\u2082\ninst\u271d\u2074 : CommRing R\ninst\u271d\u00b3 : LieRing L\ninst\u271d\u00b2 : LieAlgebra R L\ninst\u271d\u00b9 : LieRing L'\ninst\u271d : LieAlgebra R L'\nf : L \u2192\u2097\u2045R\u2046 L'\nI : LieIdeal R L\nJ J\u2081 J\u2082 : LieIdeal R L'\n\u22a2 map f \u2045comap f J\u2081, comap f J\u2082\u2046 \u2264 \u2045J\u2081, J\u2082\u2046", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Matrix/Invertible.lean", "full_name": "Matrix.isUnit_conjTranspose", "start": [78, 1], "end": [78, 73], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Sigma.lean", "full_name": "List.nil_kunion", "start": [693, 1], "end": [694, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Ideal/Over.lean", "full_name": "Ideal.isMaximal_of_isIntegral_of_isMaximal_comap'", "start": [258, 1], "end": [262, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Eval.lean", "full_name": "Polynomial.coe_eval\u2082RingHom", "start": [280, 1], "end": [281, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Commute/Basic.lean", "full_name": "pow_inv_comm", "start": [139, 1], "end": [140, 40], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/AList.lean", "full_name": "AList.union_comm_of_disjoint", "start": [514, 1], "end": [536, 21], "traced_tactics": [{"tactic": "intros", "annotated_tactic": ["intros", []], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\n\u22a2 \u2200 (x : \u03b1) (y : \u03b2 x), y \u2208 dlookup x (s\u2081 \u222a s\u2082).entries \u2194 y \u2208 dlookup x (s\u2082 \u222a s\u2081).entries", "state_after": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\n\u22a2 y\u271d \u2208 dlookup x\u271d (s\u2081 \u222a s\u2082).entries \u2194 y\u271d \u2208 dlookup x\u271d (s\u2082 \u222a s\u2081).entries"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\n\u22a2 y\u271d \u2208 dlookup x\u271d (s\u2081 \u222a s\u2082).entries \u2194 y\u271d \u2208 dlookup x\u271d (s\u2082 \u222a s\u2081).entries", "state_after": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\n\u22a2 dlookup x\u271d s\u2081.entries = some y\u271d \u2228 x\u271d \u2209 s\u2081.entries.keys \u2227 dlookup x\u271d s\u2082.entries = some y\u271d \u2194\n    dlookup x\u271d s\u2082.entries = some y\u271d \u2228 x\u271d \u2209 s\u2082.entries.keys \u2227 dlookup x\u271d s\u2081.entries = some y\u271d"}, {"tactic": "constructor <;> intro h'", "annotated_tactic": ["constructor <;> intro h'", []], "state_before": "\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\n\u22a2 dlookup x\u271d s\u2081.entries = some y\u271d \u2228 x\u271d \u2209 s\u2081.entries.keys \u2227 dlookup x\u271d s\u2082.entries = some y\u271d \u2194\n    dlookup x\u271d s\u2082.entries = some y\u271d \u2228 x\u271d \u2209 s\u2082.entries.keys \u2227 dlookup x\u271d s\u2081.entries = some y\u271d", "state_after": "case mp\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2081.entries = some y\u271d \u2228 x\u271d \u2209 s\u2081.entries.keys \u2227 dlookup x\u271d s\u2082.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2082.entries = some y\u271d \u2228 x\u271d \u2209 s\u2082.entries.keys \u2227 dlookup x\u271d s\u2081.entries = some y\u271d\n\ncase mpr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2082.entries = some y\u271d \u2228 x\u271d \u2209 s\u2082.entries.keys \u2227 dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2081.entries = some y\u271d \u2228 x\u271d \u2209 s\u2081.entries.keys \u2227 dlookup x\u271d s\u2082.entries = some y\u271d"}, {"tactic": "cases' h' with h' h'", "annotated_tactic": ["cases' h' with h' h'", []], "state_before": "case mp\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2081.entries = some y\u271d \u2228 x\u271d \u2209 s\u2081.entries.keys \u2227 dlookup x\u271d s\u2082.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2082.entries = some y\u271d \u2228 x\u271d \u2209 s\u2082.entries.keys \u2227 dlookup x\u271d s\u2081.entries = some y\u271d", "state_after": "case mp.inl\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2082.entries = some y\u271d \u2228 x\u271d \u2209 s\u2082.entries.keys \u2227 dlookup x\u271d s\u2081.entries = some y\u271d\n\ncase mp.inr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : x\u271d \u2209 s\u2081.entries.keys \u2227 dlookup x\u271d s\u2082.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2082.entries = some y\u271d \u2228 x\u271d \u2209 s\u2082.entries.keys \u2227 dlookup x\u271d s\u2081.entries = some y\u271d"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "case mp.inl\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2082.entries = some y\u271d \u2228 x\u271d \u2209 s\u2082.entries.keys \u2227 dlookup x\u271d s\u2081.entries = some y\u271d", "state_after": "case mp.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 x\u271d \u2209 s\u2082.entries.keys \u2227 dlookup x\u271d s\u2081.entries = some y\u271d"}, {"tactic": "refine \u27e8?_, h'\u27e9", "annotated_tactic": ["refine \u27e8?_, h'\u27e9", []], "state_before": "case mp.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 x\u271d \u2209 s\u2082.entries.keys \u2227 dlookup x\u271d s\u2081.entries = some y\u271d", "state_after": "case mp.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 x\u271d \u2209 s\u2082.entries.keys"}, {"tactic": "apply h", "annotated_tactic": ["apply h", []], "state_before": "case mp.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 x\u271d \u2209 s\u2082.entries.keys", "state_after": "case mp.inl.h.a\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 x\u271d \u2208 s\u2081.keys"}, {"tactic": "rw [keys, \u2190 List.dlookup_isSome, h']", "annotated_tactic": ["rw [<a>keys</a>, \u2190 <a>List.dlookup_isSome</a>, h']", [{"full_name": "AList.keys", "def_path": "Mathlib/Data/List/AList.lean", "def_pos": [79, 5], "def_end_pos": [79, 9]}, {"full_name": "List.dlookup_isSome", "def_path": "Mathlib/Data/List/Sigma.lean", "def_pos": [188, 9], "def_end_pos": [188, 23]}]], "state_before": "case mp.inl.h.a\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 x\u271d \u2208 s\u2081.keys", "state_after": "case mp.inl.h.a\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 (some y\u271d).isSome = true"}, {"tactic": "exact rfl", "annotated_tactic": ["exact <a>rfl</a>", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case mp.inl.h.a\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 (some y\u271d).isSome = true", "state_after": "no goals"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "case mp.inr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : x\u271d \u2209 s\u2081.entries.keys \u2227 dlookup x\u271d s\u2082.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2082.entries = some y\u271d \u2228 x\u271d \u2209 s\u2082.entries.keys \u2227 dlookup x\u271d s\u2081.entries = some y\u271d", "state_after": "case mp.inr.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : x\u271d \u2209 s\u2081.entries.keys \u2227 dlookup x\u271d s\u2082.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2082.entries = some y\u271d"}, {"tactic": "rw [h'.2]", "annotated_tactic": ["rw [h'.2]", []], "state_before": "case mp.inr.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : x\u271d \u2209 s\u2081.entries.keys \u2227 dlookup x\u271d s\u2082.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2082.entries = some y\u271d", "state_after": "no goals"}, {"tactic": "cases' h' with h' h'", "annotated_tactic": ["cases' h' with h' h'", []], "state_before": "case mpr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2082.entries = some y\u271d \u2228 x\u271d \u2209 s\u2082.entries.keys \u2227 dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2081.entries = some y\u271d \u2228 x\u271d \u2209 s\u2081.entries.keys \u2227 dlookup x\u271d s\u2082.entries = some y\u271d", "state_after": "case mpr.inl\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2082.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2081.entries = some y\u271d \u2228 x\u271d \u2209 s\u2081.entries.keys \u2227 dlookup x\u271d s\u2082.entries = some y\u271d\n\ncase mpr.inr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : x\u271d \u2209 s\u2082.entries.keys \u2227 dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2081.entries = some y\u271d \u2228 x\u271d \u2209 s\u2081.entries.keys \u2227 dlookup x\u271d s\u2082.entries = some y\u271d"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "case mpr.inl\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2082.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2081.entries = some y\u271d \u2228 x\u271d \u2209 s\u2081.entries.keys \u2227 dlookup x\u271d s\u2082.entries = some y\u271d", "state_after": "case mpr.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2082.entries = some y\u271d\n\u22a2 x\u271d \u2209 s\u2081.entries.keys \u2227 dlookup x\u271d s\u2082.entries = some y\u271d"}, {"tactic": "refine \u27e8?_, h'\u27e9", "annotated_tactic": ["refine \u27e8?_, h'\u27e9", []], "state_before": "case mpr.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2082.entries = some y\u271d\n\u22a2 x\u271d \u2209 s\u2081.entries.keys \u2227 dlookup x\u271d s\u2082.entries = some y\u271d", "state_after": "case mpr.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2082.entries = some y\u271d\n\u22a2 x\u271d \u2209 s\u2081.entries.keys"}, {"tactic": "intro h''", "annotated_tactic": ["intro h''", []], "state_before": "case mpr.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2082.entries = some y\u271d\n\u22a2 x\u271d \u2209 s\u2081.entries.keys", "state_after": "case mpr.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2082.entries = some y\u271d\nh'' : x\u271d \u2208 s\u2081.entries.keys\n\u22a2 False"}, {"tactic": "apply h _ h''", "annotated_tactic": ["apply h _ h''", []], "state_before": "case mpr.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2082.entries = some y\u271d\nh'' : x\u271d \u2208 s\u2081.entries.keys\n\u22a2 False", "state_after": "case mpr.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2082.entries = some y\u271d\nh'' : x\u271d \u2208 s\u2081.entries.keys\n\u22a2 x\u271d \u2208 s\u2082.keys"}, {"tactic": "rw [keys, \u2190 List.dlookup_isSome, h']", "annotated_tactic": ["rw [<a>keys</a>, \u2190 <a>List.dlookup_isSome</a>, h']", [{"full_name": "AList.keys", "def_path": "Mathlib/Data/List/AList.lean", "def_pos": [79, 5], "def_end_pos": [79, 9]}, {"full_name": "List.dlookup_isSome", "def_path": "Mathlib/Data/List/Sigma.lean", "def_pos": [188, 9], "def_end_pos": [188, 23]}]], "state_before": "case mpr.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2082.entries = some y\u271d\nh'' : x\u271d \u2208 s\u2081.entries.keys\n\u22a2 x\u271d \u2208 s\u2082.keys", "state_after": "case mpr.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2082.entries = some y\u271d\nh'' : x\u271d \u2208 s\u2081.entries.keys\n\u22a2 (some y\u271d).isSome = true"}, {"tactic": "exact rfl", "annotated_tactic": ["exact <a>rfl</a>", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case mpr.inl.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : dlookup x\u271d s\u2082.entries = some y\u271d\nh'' : x\u271d \u2208 s\u2081.entries.keys\n\u22a2 (some y\u271d).isSome = true", "state_after": "no goals"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "case mpr.inr\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : x\u271d \u2209 s\u2082.entries.keys \u2227 dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2081.entries = some y\u271d \u2228 x\u271d \u2209 s\u2081.entries.keys \u2227 dlookup x\u271d s\u2082.entries = some y\u271d", "state_after": "case mpr.inr.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : x\u271d \u2209 s\u2082.entries.keys \u2227 dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2081.entries = some y\u271d"}, {"tactic": "rw [h'.2]", "annotated_tactic": ["rw [h'.2]", []], "state_before": "case mpr.inr.h\n\u03b1 : Type u\n\u03b2 : \u03b1 \u2192 Type v\ninst\u271d : DecidableEq \u03b1\ns\u2081 s\u2082 : AList \u03b2\nh : s\u2081.Disjoint s\u2082\nx\u271d : \u03b1\ny\u271d : \u03b2 x\u271d\nh' : x\u271d \u2209 s\u2082.entries.keys \u2227 dlookup x\u271d s\u2081.entries = some y\u271d\n\u22a2 dlookup x\u271d s\u2081.entries = some y\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Kernel/Disintegration/Density.lean", "full_name": "ProbabilityTheory.kernel.integrable_densityProcess", "start": [189, 1], "end": [195, 76], "traced_tactics": [{"tactic": "rw [\u2190 mem\u2112p_one_iff_integrable]", "annotated_tactic": ["rw [\u2190 <a>mem\u2112p_one_iff_integrable</a>]", [{"full_name": "MeasureTheory.mem\u2112p_one_iff_integrable", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [442, 9], "def_end_pos": [442, 33]}]], "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\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\nh\u03ba\u03bd : fst \u03ba \u2264 \u03bd\ninst\u271d : IsFiniteKernel \u03bd\nn : \u2115\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 Integrable (fun x => densityProcess \u03ba \u03bd n a x s) (\u03bd a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\nh\u03ba\u03bd : fst \u03ba \u2264 \u03bd\ninst\u271d : IsFiniteKernel \u03bd\nn : \u2115\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 Mem\u2112p (fun x => densityProcess \u03ba \u03bd n a x s) 1 (\u03bd a)"}, {"tactic": "refine \u27e8Measurable.aestronglyMeasurable ?_, ?_\u27e9", "annotated_tactic": ["refine \u27e8<a>Measurable.aestronglyMeasurable</a> ?_, ?_\u27e9", [{"full_name": "Measurable.aestronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1322, 9], "def_end_pos": [1322, 47]}]], "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\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\nh\u03ba\u03bd : fst \u03ba \u2264 \u03bd\ninst\u271d : IsFiniteKernel \u03bd\nn : \u2115\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 Mem\u2112p (fun x => densityProcess \u03ba \u03bd n a x s) 1 (\u03bd a)", "state_after": "case refine_1\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\nh\u03ba\u03bd : fst \u03ba \u2264 \u03bd\ninst\u271d : IsFiniteKernel \u03bd\nn : \u2115\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 Measurable fun x => densityProcess \u03ba \u03bd n a x s\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\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\nh\u03ba\u03bd : fst \u03ba \u2264 \u03bd\ninst\u271d : IsFiniteKernel \u03bd\nn : \u2115\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 snorm (fun x => densityProcess \u03ba \u03bd n a x s) 1 (\u03bd a) < \u22a4"}, {"tactic": "exact measurable_densityProcess_right \u03ba \u03bd n a hs", "annotated_tactic": ["exact <a>measurable_densityProcess_right</a> \u03ba \u03bd n a hs", [{"full_name": "ProbabilityTheory.kernel.measurable_densityProcess_right", "def_path": "Mathlib/Probability/Kernel/Disintegration/Density.lean", "def_pos": [139, 7], "def_end_pos": [139, 38]}]], "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\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\nh\u03ba\u03bd : fst \u03ba \u2264 \u03bd\ninst\u271d : IsFiniteKernel \u03bd\nn : \u2115\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 Measurable fun x => densityProcess \u03ba \u03bd n a x s", "state_after": "no goals"}, {"tactic": "exact (snorm_densityProcess_le h\u03ba\u03bd n a s).trans_lt (measure_lt_top _ _)", "annotated_tactic": ["exact (<a>snorm_densityProcess_le</a> h\u03ba\u03bd n a s).<a>trans_lt</a> (<a>measure_lt_top</a> _ _)", [{"full_name": "ProbabilityTheory.kernel.snorm_densityProcess_le", "def_path": "Mathlib/Probability/Kernel/Disintegration/Density.lean", "def_pos": [182, 7], "def_end_pos": [182, 30]}, {"full_name": "LE.le.trans_lt", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [119, 7], "def_end_pos": [119, 21]}, {"full_name": "MeasureTheory.measure_lt_top", "def_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "def_pos": [52, 9], "def_end_pos": [52, 23]}]], "state_before": "case refine_2\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b9 : CountablyGenerated \u03b3\n\u03ba : \u21a5(kernel \u03b1 (\u03b3 \u00d7 \u03b2))\n\u03bd : \u21a5(kernel \u03b1 \u03b3)\nh\u03ba\u03bd : fst \u03ba \u2264 \u03bd\ninst\u271d : IsFiniteKernel \u03bd\nn : \u2115\na : \u03b1\ns : Set \u03b2\nhs : MeasurableSet s\n\u22a2 snorm (fun x => densityProcess \u03ba \u03bd n a x s) 1 (\u03bd a) < \u22a4", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "full_name": "tsub_right_inj", "start": [399, 1], "end": [401, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean", "full_name": "WeierstrassCurve.Affine.addPolynomial_slope", "start": [553, 1], "end": [579, 78], "traced_tactics": [{"tactic": "rw [addPolynomial_eq, neg_inj, Cubic.prod_X_sub_C_eq, Cubic.toPoly_injective]", "annotated_tactic": ["rw [<a>addPolynomial_eq</a>, <a>neg_inj</a>, <a>Cubic.prod_X_sub_C_eq</a>, <a>Cubic.toPoly_injective</a>]", [{"full_name": "WeierstrassCurve.Affine.addPolynomial_eq", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean", "def_pos": [390, 7], "def_end_pos": [390, 23]}, {"full_name": "neg_inj", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [406, 3], "def_end_pos": [406, 14]}, {"full_name": "Cubic.prod_X_sub_C_eq", "def_path": "Mathlib/Algebra/CubicDiscriminant.lean", "def_pos": [75, 9], "def_end_pos": [75, 24]}, {"full_name": "Cubic.toPoly_injective", "def_path": "Mathlib/Algebra/CubicDiscriminant.lean", "def_pos": [133, 9], "def_end_pos": [133, 25]}]], "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : W.Equation x\u2081 y\u2081\nh\u2082 : W.Equation x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\n\u22a2 W.addPolynomial x\u2081 y\u2081 (W.slope x\u2081 x\u2082 y\u2081 y\u2082) =\n    -((X - C x\u2081) * (X - C x\u2082) * (X - C (W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082))))", "state_after": "R : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : W.Equation x\u2081 y\u2081\nh\u2082 : W.Equation x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\n\u22a2 { a := 1, b := -W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 - W.a\u2081 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 + W.a\u2082,\n      c := 2 * x\u2081 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * W.slope x\u2081 x\u2082 y\u2081 y\u2082 + (-W.a\u2081 * y\u2081 + W.a\u2084),\n      d :=\n        -x\u2081 ^ 2 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * W.slope x\u2081 x\u2082 y\u2081 y\u2082 -\n          (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n    { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082)),\n      c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082) + x\u2082 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082),\n      d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082)) }"}, {"tactic": "by_cases hx : x\u2081 = x\u2082", "annotated_tactic": ["by_cases hx : x\u2081 = x\u2082", []], "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : W.Equation x\u2081 y\u2081\nh\u2082 : W.Equation x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\n\u22a2 { a := 1, b := -W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 - W.a\u2081 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 + W.a\u2082,\n      c := 2 * x\u2081 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * W.slope x\u2081 x\u2082 y\u2081 y\u2082 + (-W.a\u2081 * y\u2081 + W.a\u2084),\n      d :=\n        -x\u2081 ^ 2 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * W.slope x\u2081 x\u2082 y\u2081 y\u2082 -\n          (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n    { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082)),\n      c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082) + x\u2082 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082),\n      d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082)) }", "state_after": "case pos\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : W.Equation x\u2081 y\u2081\nh\u2082 : W.Equation x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : x\u2081 = x\u2082\n\u22a2 { a := 1, b := -W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 - W.a\u2081 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 + W.a\u2082,\n      c := 2 * x\u2081 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * W.slope x\u2081 x\u2082 y\u2081 y\u2082 + (-W.a\u2081 * y\u2081 + W.a\u2084),\n      d :=\n        -x\u2081 ^ 2 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * W.slope x\u2081 x\u2082 y\u2081 y\u2082 -\n          (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n    { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082)),\n      c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082) + x\u2082 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082),\n      d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082)) }\n\ncase neg\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : W.Equation x\u2081 y\u2081\nh\u2082 : W.Equation x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 = x\u2082\n\u22a2 { a := 1, b := -W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 - W.a\u2081 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 + W.a\u2082,\n      c := 2 * x\u2081 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * W.slope x\u2081 x\u2082 y\u2081 y\u2082 + (-W.a\u2081 * y\u2081 + W.a\u2084),\n      d :=\n        -x\u2081 ^ 2 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * W.slope x\u2081 x\u2082 y\u2081 y\u2082 -\n          (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n    { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082)),\n      c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082) + x\u2082 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082),\n      d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082)) }"}, {"tactic": "rcases hx, Y_eq_of_Y_ne h\u2081 h\u2082 hx (hxy hx) with \u27e8rfl, rfl\u27e9", "annotated_tactic": ["rcases hx, <a>Y_eq_of_Y_ne</a> h\u2081 h\u2082 hx (hxy hx) with \u27e8rfl, rfl\u27e9", [{"full_name": "WeierstrassCurve.Affine.Y_eq_of_Y_ne", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean", "def_pos": [547, 7], "def_end_pos": [547, 19]}]], "state_before": "case pos\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : W.Equation x\u2081 y\u2081\nh\u2082 : W.Equation x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : x\u2081 = x\u2082\n\u22a2 { a := 1, b := -W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 - W.a\u2081 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 + W.a\u2082,\n      c := 2 * x\u2081 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * W.slope x\u2081 x\u2082 y\u2081 y\u2082 + (-W.a\u2081 * y\u2081 + W.a\u2084),\n      d :=\n        -x\u2081 ^ 2 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * W.slope x\u2081 x\u2082 y\u2081 y\u2082 -\n          (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n    { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082)),\n      c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082) + x\u2082 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082),\n      d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082)) }", "state_after": "case pos\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : W.Equation x\u2081 y\u2081\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 W.negY x\u2081 y\u2081\n\u22a2 { a := 1, b := -W.slope x\u2081 x\u2081 y\u2081 y\u2081 ^ 2 - W.a\u2081 * W.slope x\u2081 x\u2081 y\u2081 y\u2081 + W.a\u2082,\n      c := 2 * x\u2081 * W.slope x\u2081 x\u2081 y\u2081 y\u2081 ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * W.slope x\u2081 x\u2081 y\u2081 y\u2081 + (-W.a\u2081 * y\u2081 + W.a\u2084),\n      d :=\n        -x\u2081 ^ 2 * W.slope x\u2081 x\u2081 y\u2081 y\u2081 ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * W.slope x\u2081 x\u2081 y\u2081 y\u2081 -\n          (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n    { a := 1, b := -(x\u2081 + x\u2081 + W.addX x\u2081 x\u2081 (W.slope x\u2081 x\u2081 y\u2081 y\u2081)),\n      c := x\u2081 * x\u2081 + x\u2081 * W.addX x\u2081 x\u2081 (W.slope x\u2081 x\u2081 y\u2081 y\u2081) + x\u2081 * W.addX x\u2081 x\u2081 (W.slope x\u2081 x\u2081 y\u2081 y\u2081),\n      d := -(x\u2081 * x\u2081 * W.addX x\u2081 x\u2081 (W.slope x\u2081 x\u2081 y\u2081 y\u2081)) }"}, {"tactic": "rw [equation_iff] at h\u2081 h\u2082", "annotated_tactic": ["rw [<a>equation_iff</a>] at h\u2081 h\u2082", [{"full_name": "WeierstrassCurve.Affine.equation_iff", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean", "def_pos": [199, 7], "def_end_pos": [199, 19]}]], "state_before": "case pos\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : W.Equation x\u2081 y\u2081\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 W.negY x\u2081 y\u2081\n\u22a2 { a := 1, b := -W.slope x\u2081 x\u2081 y\u2081 y\u2081 ^ 2 - W.a\u2081 * W.slope x\u2081 x\u2081 y\u2081 y\u2081 + W.a\u2082,\n      c := 2 * x\u2081 * W.slope x\u2081 x\u2081 y\u2081 y\u2081 ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * W.slope x\u2081 x\u2081 y\u2081 y\u2081 + (-W.a\u2081 * y\u2081 + W.a\u2084),\n      d :=\n        -x\u2081 ^ 2 * W.slope x\u2081 x\u2081 y\u2081 y\u2081 ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * W.slope x\u2081 x\u2081 y\u2081 y\u2081 -\n          (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n    { a := 1, b := -(x\u2081 + x\u2081 + W.addX x\u2081 x\u2081 (W.slope x\u2081 x\u2081 y\u2081 y\u2081)),\n      c := x\u2081 * x\u2081 + x\u2081 * W.addX x\u2081 x\u2081 (W.slope x\u2081 x\u2081 y\u2081 y\u2081) + x\u2081 * W.addX x\u2081 x\u2081 (W.slope x\u2081 x\u2081 y\u2081 y\u2081),\n      d := -(x\u2081 * x\u2081 * W.addX x\u2081 x\u2081 (W.slope x\u2081 x\u2081 y\u2081 y\u2081)) }", "state_after": "case pos\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 W.negY x\u2081 y\u2081\n\u22a2 { a := 1, b := -W.slope x\u2081 x\u2081 y\u2081 y\u2081 ^ 2 - W.a\u2081 * W.slope x\u2081 x\u2081 y\u2081 y\u2081 + W.a\u2082,\n      c := 2 * x\u2081 * W.slope x\u2081 x\u2081 y\u2081 y\u2081 ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * W.slope x\u2081 x\u2081 y\u2081 y\u2081 + (-W.a\u2081 * y\u2081 + W.a\u2084),\n      d :=\n        -x\u2081 ^ 2 * W.slope x\u2081 x\u2081 y\u2081 y\u2081 ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * W.slope x\u2081 x\u2081 y\u2081 y\u2081 -\n          (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n    { a := 1, b := -(x\u2081 + x\u2081 + W.addX x\u2081 x\u2081 (W.slope x\u2081 x\u2081 y\u2081 y\u2081)),\n      c := x\u2081 * x\u2081 + x\u2081 * W.addX x\u2081 x\u2081 (W.slope x\u2081 x\u2081 y\u2081 y\u2081) + x\u2081 * W.addX x\u2081 x\u2081 (W.slope x\u2081 x\u2081 y\u2081 y\u2081),\n      d := -(x\u2081 * x\u2081 * W.addX x\u2081 x\u2081 (W.slope x\u2081 x\u2081 y\u2081 y\u2081)) }"}, {"tactic": "rw [slope_of_Y_ne rfl <| hxy rfl]", "annotated_tactic": ["rw [<a>slope_of_Y_ne</a> <a>rfl</a> <| hxy <a>rfl</a>]", [{"full_name": "WeierstrassCurve.Affine.slope_of_Y_ne", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean", "def_pos": [516, 7], "def_end_pos": [516, 20]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case pos\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 W.negY x\u2081 y\u2081\n\u22a2 { a := 1, b := -W.slope x\u2081 x\u2081 y\u2081 y\u2081 ^ 2 - W.a\u2081 * W.slope x\u2081 x\u2081 y\u2081 y\u2081 + W.a\u2082,\n      c := 2 * x\u2081 * W.slope x\u2081 x\u2081 y\u2081 y\u2081 ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * W.slope x\u2081 x\u2081 y\u2081 y\u2081 + (-W.a\u2081 * y\u2081 + W.a\u2084),\n      d :=\n        -x\u2081 ^ 2 * W.slope x\u2081 x\u2081 y\u2081 y\u2081 ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * W.slope x\u2081 x\u2081 y\u2081 y\u2081 -\n          (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n    { a := 1, b := -(x\u2081 + x\u2081 + W.addX x\u2081 x\u2081 (W.slope x\u2081 x\u2081 y\u2081 y\u2081)),\n      c := x\u2081 * x\u2081 + x\u2081 * W.addX x\u2081 x\u2081 (W.slope x\u2081 x\u2081 y\u2081 y\u2081) + x\u2081 * W.addX x\u2081 x\u2081 (W.slope x\u2081 x\u2081 y\u2081 y\u2081),\n      d := -(x\u2081 * x\u2081 * W.addX x\u2081 x\u2081 (W.slope x\u2081 x\u2081 y\u2081 y\u2081)) }", "state_after": "case pos\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 W.negY x\u2081 y\u2081\n\u22a2 { a := 1,\n      b :=\n        -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 -\n            W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n          W.a\u2082,\n      c :=\n        2 * x\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n            (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n          (-W.a\u2081 * y\u2081 + W.a\u2084),\n      d :=\n        -x\u2081 ^ 2 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n            (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) -\n          (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n    { a := 1, b := -(x\u2081 + x\u2081 + W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))),\n      c :=\n        x\u2081 * x\u2081 + x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n          x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)),\n      d := -(x\u2081 * x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))) }"}, {"tactic": "rw [negY, \u2190 sub_ne_zero] at hxy", "annotated_tactic": ["rw [<a>negY</a>, \u2190 <a>sub_ne_zero</a>] at hxy", [{"full_name": "WeierstrassCurve.Affine.negY", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean", "def_pos": [345, 5], "def_end_pos": [345, 9]}, {"full_name": "sub_ne_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1082, 3], "def_end_pos": [1082, 14]}]], "state_before": "case pos\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 \u2260 W.negY x\u2081 y\u2081\n\u22a2 { a := 1,\n      b :=\n        -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 -\n            W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n          W.a\u2082,\n      c :=\n        2 * x\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n            (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n          (-W.a\u2081 * y\u2081 + W.a\u2084),\n      d :=\n        -x\u2081 ^ 2 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n            (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) -\n          (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n    { a := 1, b := -(x\u2081 + x\u2081 + W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))),\n      c :=\n        x\u2081 * x\u2081 + x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n          x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)),\n      d := -(x\u2081 * x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))) }", "state_after": "case pos\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 { a := 1,\n      b :=\n        -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 -\n            W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n          W.a\u2082,\n      c :=\n        2 * x\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n            (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n          (-W.a\u2081 * y\u2081 + W.a\u2084),\n      d :=\n        -x\u2081 ^ 2 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n            (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) -\n          (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n    { a := 1, b := -(x\u2081 + x\u2081 + W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))),\n      c :=\n        x\u2081 * x\u2081 + x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n          x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)),\n      d := -(x\u2081 * x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))) }"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "case pos\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 { a := 1,\n      b :=\n        -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 -\n            W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n          W.a\u2082,\n      c :=\n        2 * x\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n            (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n          (-W.a\u2081 * y\u2081 + W.a\u2084),\n      d :=\n        -x\u2081 ^ 2 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n            (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) -\n          (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n    { a := 1, b := -(x\u2081 + x\u2081 + W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))),\n      c :=\n        x\u2081 * x\u2081 + x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n          x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)),\n      d := -(x\u2081 * x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))) }", "state_after": "case pos.a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 { a := 1,\n        b :=\n          -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 -\n              W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            W.a\u2082,\n        c :=\n          2 * x\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n              (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            (-W.a\u2081 * y\u2081 + W.a\u2084),\n        d :=\n          -x\u2081 ^ 2 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n              (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) -\n            (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.a =\n    { a := 1, b := -(x\u2081 + x\u2081 + W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))),\n        c :=\n          x\u2081 * x\u2081 + x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)),\n        d := -(x\u2081 * x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))) }.a\n\ncase pos.b\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 { a := 1,\n        b :=\n          -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 -\n              W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            W.a\u2082,\n        c :=\n          2 * x\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n              (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            (-W.a\u2081 * y\u2081 + W.a\u2084),\n        d :=\n          -x\u2081 ^ 2 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n              (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) -\n            (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.b =\n    { a := 1, b := -(x\u2081 + x\u2081 + W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))),\n        c :=\n          x\u2081 * x\u2081 + x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)),\n        d := -(x\u2081 * x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))) }.b\n\ncase pos.c\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 { a := 1,\n        b :=\n          -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 -\n              W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            W.a\u2082,\n        c :=\n          2 * x\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n              (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            (-W.a\u2081 * y\u2081 + W.a\u2084),\n        d :=\n          -x\u2081 ^ 2 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n              (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) -\n            (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.c =\n    { a := 1, b := -(x\u2081 + x\u2081 + W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))),\n        c :=\n          x\u2081 * x\u2081 + x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)),\n        d := -(x\u2081 * x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))) }.c\n\ncase pos.d\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 { a := 1,\n        b :=\n          -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 -\n              W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            W.a\u2082,\n        c :=\n          2 * x\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n              (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            (-W.a\u2081 * y\u2081 + W.a\u2084),\n        d :=\n          -x\u2081 ^ 2 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n              (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) -\n            (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.d =\n    { a := 1, b := -(x\u2081 + x\u2081 + W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))),\n        c :=\n          x\u2081 * x\u2081 + x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)),\n        d := -(x\u2081 * x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))) }.d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case pos.a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 { a := 1,\n        b :=\n          -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 -\n              W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            W.a\u2082,\n        c :=\n          2 * x\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n              (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            (-W.a\u2081 * y\u2081 + W.a\u2084),\n        d :=\n          -x\u2081 ^ 2 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n              (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) -\n            (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.a =\n    { a := 1, b := -(x\u2081 + x\u2081 + W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))),\n        c :=\n          x\u2081 * x\u2081 + x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)),\n        d := -(x\u2081 * x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))) }.a", "state_after": "no goals"}, {"tactic": "simp only [addX]", "annotated_tactic": ["simp only [<a>addX</a>]", [{"full_name": "WeierstrassCurve.Affine.addX", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean", "def_pos": [405, 5], "def_end_pos": [405, 9]}]], "state_before": "case pos.b\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 { a := 1,\n        b :=\n          -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 -\n              W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            W.a\u2082,\n        c :=\n          2 * x\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n              (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            (-W.a\u2081 * y\u2081 + W.a\u2084),\n        d :=\n          -x\u2081 ^ 2 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n              (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) -\n            (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.b =\n    { a := 1, b := -(x\u2081 + x\u2081 + W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))),\n        c :=\n          x\u2081 * x\u2081 + x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)),\n        d := -(x\u2081 * x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))) }.b", "state_after": "case pos.b\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 -\n        W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n      W.a\u2082 =\n    -(x\u2081 + x\u2081 +\n        (((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n                W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) -\n              W.a\u2082 -\n            x\u2081 -\n          x\u2081))"}, {"tactic": "ring1", "annotated_tactic": ["ring1", []], "state_before": "case pos.b\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 -\n        W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n      W.a\u2082 =\n    -(x\u2081 + x\u2081 +\n        (((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n                W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) -\n              W.a\u2082 -\n            x\u2081 -\n          x\u2081))", "state_after": "no goals"}, {"tactic": "field_simp [hxy rfl]", "annotated_tactic": ["field_simp [hxy <a>rfl</a>]", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case pos.c\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 { a := 1,\n        b :=\n          -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 -\n              W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            W.a\u2082,\n        c :=\n          2 * x\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n              (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            (-W.a\u2081 * y\u2081 + W.a\u2084),\n        d :=\n          -x\u2081 ^ 2 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n              (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) -\n            (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.c =\n    { a := 1, b := -(x\u2081 + x\u2081 + W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))),\n        c :=\n          x\u2081 * x\u2081 + x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)),\n        d := -(x\u2081 * x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))) }.c", "state_after": "case pos.c\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 2 * x\u2081 * (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) +\n        (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) *\n          (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 +\n      (-(W.a\u2081 * y\u2081) + W.a\u2084) * ((y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083))) =\n    x\u2081 * x\u2081 * ((y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083))) +\n        x\u2081 *\n          ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) +\n                  W.a\u2081 * (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 -\n                (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * W.a\u2082 -\n              (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * x\u2081 -\n            (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * x\u2081) +\n      x\u2081 *\n        ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) +\n                W.a\u2081 * (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 -\n              (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * W.a\u2082 -\n            (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * x\u2081 -\n          (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * x\u2081)"}, {"tactic": "ring1", "annotated_tactic": ["ring1", []], "state_before": "case pos.c\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 2 * x\u2081 * (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) +\n        (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) *\n          (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 +\n      (-(W.a\u2081 * y\u2081) + W.a\u2084) * ((y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083))) =\n    x\u2081 * x\u2081 * ((y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083))) +\n        x\u2081 *\n          ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) +\n                  W.a\u2081 * (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 -\n                (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * W.a\u2082 -\n              (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * x\u2081 -\n            (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * x\u2081) +\n      x\u2081 *\n        ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) +\n                W.a\u2081 * (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 -\n              (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * W.a\u2082 -\n            (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * x\u2081 -\n          (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * x\u2081)", "state_after": "no goals"}, {"tactic": "linear_combination (norm := (field_simp [hxy rfl]; ring1)) -h\u2081", "annotated_tactic": ["linear_combination (norm := (field_simp [hxy <a>rfl</a>]; ring1)) -h\u2081", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case pos.d\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 { a := 1,\n        b :=\n          -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 -\n              W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            W.a\u2082,\n        c :=\n          2 * x\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n              (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            (-W.a\u2081 * y\u2081 + W.a\u2084),\n        d :=\n          -x\u2081 ^ 2 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n              (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) -\n            (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.d =\n    { a := 1, b := -(x\u2081 + x\u2081 + W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))),\n        c :=\n          x\u2081 * x\u2081 + x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n            x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)),\n        d := -(x\u2081 * x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))) }.d", "state_after": "no goals"}, {"tactic": "field_simp [hxy rfl]", "annotated_tactic": ["field_simp [hxy <a>rfl</a>]", [{"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 { a := 1,\n            b :=\n              -((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 -\n                  W.a\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n                W.a\u2082,\n            c :=\n              2 * x\u2081 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n                  (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) *\n                    ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n                (-W.a\u2081 * y\u2081 + W.a\u2084),\n            d :=\n              -x\u2081 ^ 2 * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) ^ 2 +\n                  (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) -\n                (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.d -\n        { a := 1,\n            b := -(x\u2081 + x\u2081 + W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))),\n            c :=\n              x\u2081 * x\u2081 + x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)) +\n                x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081)),\n            d := -(x\u2081 * x\u2081 * W.addX x\u2081 x\u2081 ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) / (y\u2081 - W.negY x\u2081 y\u2081))) }.d -\n      (-(y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081) - -(x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086)) =\n    0", "state_after": "case a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 -(x\u2081 ^ 2 * (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083))) +\n            (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) *\n              (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 -\n          (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) +\n        x\u2081 * x\u2081 *\n          ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) +\n                  W.a\u2081 * (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 -\n                (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * W.a\u2082 -\n              (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * x\u2081 -\n            (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * x\u2081) -\n      (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) *\n        (-(W.a\u2083 * y\u2081) + (-(W.a\u2081 * x\u2081 * y\u2081) + -y\u2081 ^ 2) - (-W.a\u2086 + (-(W.a\u2084 * x\u2081) + (-(W.a\u2082 * x\u2081 ^ 2) + -x\u2081 ^ 3)))) =\n    0"}, {"tactic": "ring1", "annotated_tactic": ["ring1", []], "state_before": "case a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 y\u2081 : F\nh\u2081 h\u2082 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nhxy : x\u2081 = x\u2081 \u2192 y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083) \u2260 0\n\u22a2 -(x\u2081 ^ 2 * (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083))) +\n            (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) *\n              (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 -\n          (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) +\n        x\u2081 * x\u2081 *\n          ((3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) +\n                  W.a\u2081 * (3 * x\u2081 ^ 2 + 2 * W.a\u2082 * x\u2081 + W.a\u2084 - W.a\u2081 * y\u2081) * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 -\n                (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * W.a\u2082 -\n              (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * x\u2081 -\n            (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) * x\u2081) -\n      (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) ^ 2 * (y\u2081 - (-y\u2081 - W.a\u2081 * x\u2081 - W.a\u2083)) *\n        (-(W.a\u2083 * y\u2081) + (-(W.a\u2081 * x\u2081 * y\u2081) + -y\u2081 ^ 2) - (-W.a\u2086 + (-(W.a\u2084 * x\u2081) + (-(W.a\u2082 * x\u2081 ^ 2) + -x\u2081 ^ 3)))) =\n    0", "state_after": "no goals"}, {"tactic": "rw [equation_iff] at h\u2081 h\u2082", "annotated_tactic": ["rw [<a>equation_iff</a>] at h\u2081 h\u2082", [{"full_name": "WeierstrassCurve.Affine.equation_iff", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean", "def_pos": [199, 7], "def_end_pos": [199, 19]}]], "state_before": "case neg\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : W.Equation x\u2081 y\u2081\nh\u2082 : W.Equation x\u2082 y\u2082\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 = x\u2082\n\u22a2 { a := 1, b := -W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 - W.a\u2081 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 + W.a\u2082,\n      c := 2 * x\u2081 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * W.slope x\u2081 x\u2082 y\u2081 y\u2082 + (-W.a\u2081 * y\u2081 + W.a\u2084),\n      d :=\n        -x\u2081 ^ 2 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * W.slope x\u2081 x\u2082 y\u2081 y\u2082 -\n          (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n    { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082)),\n      c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082) + x\u2082 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082),\n      d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082)) }", "state_after": "case neg\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 = x\u2082\n\u22a2 { a := 1, b := -W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 - W.a\u2081 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 + W.a\u2082,\n      c := 2 * x\u2081 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * W.slope x\u2081 x\u2082 y\u2081 y\u2082 + (-W.a\u2081 * y\u2081 + W.a\u2084),\n      d :=\n        -x\u2081 ^ 2 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * W.slope x\u2081 x\u2082 y\u2081 y\u2082 -\n          (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n    { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082)),\n      c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082) + x\u2082 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082),\n      d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082)) }"}, {"tactic": "rw [slope_of_X_ne hx]", "annotated_tactic": ["rw [<a>slope_of_X_ne</a> hx]", [{"full_name": "WeierstrassCurve.Affine.slope_of_X_ne", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean", "def_pos": [524, 7], "def_end_pos": [524, 20]}]], "state_before": "case neg\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 = x\u2082\n\u22a2 { a := 1, b := -W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 - W.a\u2081 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 + W.a\u2082,\n      c := 2 * x\u2081 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * W.slope x\u2081 x\u2082 y\u2081 y\u2082 + (-W.a\u2081 * y\u2081 + W.a\u2084),\n      d :=\n        -x\u2081 ^ 2 * W.slope x\u2081 x\u2082 y\u2081 y\u2082 ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * W.slope x\u2081 x\u2082 y\u2081 y\u2082 -\n          (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n    { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082)),\n      c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082) + x\u2082 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082),\n      d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 (W.slope x\u2081 x\u2082 y\u2081 y\u2082)) }", "state_after": "case neg\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 = x\u2082\n\u22a2 { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n      c :=\n        2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n          (-W.a\u2081 * y\u2081 + W.a\u2084),\n      d :=\n        -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n          (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n    { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n      c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n      d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }"}, {"tactic": "rw [\u2190 sub_eq_zero] at hx", "annotated_tactic": ["rw [\u2190 <a>sub_eq_zero</a>] at hx", [{"full_name": "sub_eq_zero", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1070, 3], "def_end_pos": [1070, 14]}]], "state_before": "case neg\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 = x\u2082\n\u22a2 { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n      c :=\n        2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n          (-W.a\u2081 * y\u2081 + W.a\u2084),\n      d :=\n        -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n          (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n    { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n      c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n      d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }", "state_after": "case neg\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n      c :=\n        2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n          (-W.a\u2081 * y\u2081 + W.a\u2084),\n      d :=\n        -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n          (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n    { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n      c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n      d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }"}, {"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "case neg\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n      c :=\n        2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n          (-W.a\u2081 * y\u2081 + W.a\u2084),\n      d :=\n        -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n          (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) } =\n    { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n      c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n      d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }", "state_after": "case neg.a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n        c :=\n          2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n            (-W.a\u2081 * y\u2081 + W.a\u2084),\n        d :=\n          -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n            (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.a =\n    { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n        c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n        d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.a\n\ncase neg.b\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n        c :=\n          2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n            (-W.a\u2081 * y\u2081 + W.a\u2084),\n        d :=\n          -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n            (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.b =\n    { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n        c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n        d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.b\n\ncase neg.c\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n        c :=\n          2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n            (-W.a\u2081 * y\u2081 + W.a\u2084),\n        d :=\n          -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n            (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.c =\n    { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n        c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n        d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.c\n\ncase neg.d\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n        c :=\n          2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n            (-W.a\u2081 * y\u2081 + W.a\u2084),\n        d :=\n          -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n            (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.d =\n    { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n        c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n        d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.d"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case neg.a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n        c :=\n          2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n            (-W.a\u2081 * y\u2081 + W.a\u2084),\n        d :=\n          -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n            (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.a =\n    { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n        c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n        d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.a", "state_after": "no goals"}, {"tactic": "simp only [addX]", "annotated_tactic": ["simp only [<a>addX</a>]", [{"full_name": "WeierstrassCurve.Affine.addX", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean", "def_pos": [405, 5], "def_end_pos": [405, 9]}]], "state_before": "case neg.b\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n        c :=\n          2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n            (-W.a\u2081 * y\u2081 + W.a\u2084),\n        d :=\n          -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n            (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.b =\n    { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n        c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n        d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.b", "state_after": "case neg.b\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082 =\n    -(x\u2081 + x\u2082 + (((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) - W.a\u2082 - x\u2081 - x\u2082))"}, {"tactic": "ring1", "annotated_tactic": ["ring1", []], "state_before": "case neg.b\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082 =\n    -(x\u2081 + x\u2082 + (((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) - W.a\u2082 - x\u2081 - x\u2082))", "state_after": "no goals"}, {"tactic": "apply mul_right_injective\u2080 hx", "annotated_tactic": ["apply <a>mul_right_injective\u2080</a> hx", [{"full_name": "mul_right_injective\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 29]}]], "state_before": "case neg.c\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n        c :=\n          2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n            (-W.a\u2081 * y\u2081 + W.a\u2084),\n        d :=\n          -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n            (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.c =\n    { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n        c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n        d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.c", "state_after": "case neg.c.a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 (fun x => (x\u2081 - x\u2082) * x)\n      { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n          c :=\n            2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n              (-W.a\u2081 * y\u2081 + W.a\u2084),\n          d :=\n            -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n              (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.c =\n    (fun x => (x\u2081 - x\u2082) * x)\n      { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n          c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n          d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.c"}, {"tactic": "linear_combination (norm := (field_simp [hx]; ring1)) h\u2082 - h\u2081", "annotated_tactic": ["linear_combination (norm := (field_simp [hx]; ring1)) h\u2082 - h\u2081", []], "state_before": "case neg.c.a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 (fun x => (x\u2081 - x\u2082) * x)\n      { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n          c :=\n            2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n              (-W.a\u2081 * y\u2081 + W.a\u2084),\n          d :=\n            -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n              (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.c =\n    (fun x => (x\u2081 - x\u2082) * x)\n      { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n          c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n          d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.c", "state_after": "no goals"}, {"tactic": "field_simp [hx]", "annotated_tactic": ["field_simp [hx]", []], "state_before": "case a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 (fun x => (x\u2081 - x\u2082) * x)\n          { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n              c :=\n                2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n                  (-W.a\u2081 * y\u2081 + W.a\u2084),\n              d :=\n                -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n                  (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.c -\n        (fun x => (x\u2081 - x\u2082) * x)\n          { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n              c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n              d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.c -\n      (y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 - (y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081) -\n        (x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086 - (x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086))) =\n    0", "state_after": "case a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 (x\u2081 - x\u2082) *\n          (2 * x\u2081 * (y\u2081 - y\u2082) ^ 2 * (x\u2081 - x\u2082) + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * (y\u2081 - y\u2082) * (x\u2081 - x\u2082) ^ 2 +\n            (-(W.a\u2081 * y\u2081) + W.a\u2084) * ((x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082))) -\n        (x\u2081 - x\u2082) *\n          (x\u2081 * x\u2082 * ((x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082)) +\n              x\u2081 *\n                ((y\u2081 - y\u2082) ^ 2 * (x\u2081 - x\u2082) + W.a\u2081 * (y\u2081 - y\u2082) * (x\u2081 - x\u2082) ^ 2 - (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * W.a\u2082 -\n                    (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * x\u2081 -\n                  (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * x\u2082) +\n            x\u2082 *\n              ((y\u2081 - y\u2082) ^ 2 * (x\u2081 - x\u2082) + W.a\u2081 * (y\u2081 - y\u2082) * (x\u2081 - x\u2082) ^ 2 - (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * W.a\u2082 -\n                  (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * x\u2081 -\n                (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * x\u2082)) -\n      (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) *\n        (y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 - (y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081) -\n          (x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 - (x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081))) =\n    0"}, {"tactic": "ring1", "annotated_tactic": ["ring1", []], "state_before": "case a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 (x\u2081 - x\u2082) *\n          (2 * x\u2081 * (y\u2081 - y\u2082) ^ 2 * (x\u2081 - x\u2082) + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * (y\u2081 - y\u2082) * (x\u2081 - x\u2082) ^ 2 +\n            (-(W.a\u2081 * y\u2081) + W.a\u2084) * ((x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082))) -\n        (x\u2081 - x\u2082) *\n          (x\u2081 * x\u2082 * ((x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082)) +\n              x\u2081 *\n                ((y\u2081 - y\u2082) ^ 2 * (x\u2081 - x\u2082) + W.a\u2081 * (y\u2081 - y\u2082) * (x\u2081 - x\u2082) ^ 2 - (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * W.a\u2082 -\n                    (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * x\u2081 -\n                  (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * x\u2082) +\n            x\u2082 *\n              ((y\u2081 - y\u2082) ^ 2 * (x\u2081 - x\u2082) + W.a\u2081 * (y\u2081 - y\u2082) * (x\u2081 - x\u2082) ^ 2 - (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * W.a\u2082 -\n                  (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * x\u2081 -\n                (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * x\u2082)) -\n      (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) *\n        (y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 - (y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081) -\n          (x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 - (x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081))) =\n    0", "state_after": "no goals"}, {"tactic": "apply mul_right_injective\u2080 hx", "annotated_tactic": ["apply <a>mul_right_injective\u2080</a> hx", [{"full_name": "mul_right_injective\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 29]}]], "state_before": "case neg.d\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n        c :=\n          2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n            (-W.a\u2081 * y\u2081 + W.a\u2084),\n        d :=\n          -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n            (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.d =\n    { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n        c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n        d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.d", "state_after": "case neg.d.a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 (fun x => (x\u2081 - x\u2082) * x)\n      { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n          c :=\n            2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n              (-W.a\u2081 * y\u2081 + W.a\u2084),\n          d :=\n            -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n              (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.d =\n    (fun x => (x\u2081 - x\u2082) * x)\n      { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n          c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n          d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.d"}, {"tactic": "linear_combination (norm := (field_simp [hx]; ring1)) x\u2082 * h\u2081 - x\u2081 * h\u2082", "annotated_tactic": ["linear_combination (norm := (field_simp [hx]; ring1)) x\u2082 * h\u2081 - x\u2081 * h\u2082", []], "state_before": "case neg.d.a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 (fun x => (x\u2081 - x\u2082) * x)\n      { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n          c :=\n            2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n              (-W.a\u2081 * y\u2081 + W.a\u2084),\n          d :=\n            -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n              (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.d =\n    (fun x => (x\u2081 - x\u2082) * x)\n      { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n          c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n          d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.d", "state_after": "no goals"}, {"tactic": "field_simp [hx]", "annotated_tactic": ["field_simp [hx]", []], "state_before": "case a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 (fun x => (x\u2081 - x\u2082) * x)\n          { a := 1, b := -((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 - W.a\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + W.a\u2082,\n              c :=\n                2 * x\u2081 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (W.a\u2081 * x\u2081 - 2 * y\u2081 - W.a\u2083) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) +\n                  (-W.a\u2081 * y\u2081 + W.a\u2084),\n              d :=\n                -x\u2081 ^ 2 * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) ^ 2 + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) -\n                  (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086) }.d -\n        (fun x => (x\u2081 - x\u2082) * x)\n          { a := 1, b := -(x\u2081 + x\u2082 + W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))),\n              c := x\u2081 * x\u2082 + x\u2081 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)) + x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082)),\n              d := -(x\u2081 * x\u2082 * W.addX x\u2081 x\u2082 ((y\u2081 - y\u2082) / (x\u2081 - x\u2082))) }.d -\n      (x\u2082 * (y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081) - x\u2081 * (y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082) -\n        (x\u2082 * (x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086) - x\u2081 * (x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086))) =\n    0", "state_after": "case a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 (x\u2081 - x\u2082) *\n          (-(x\u2081 ^ 2 * (y\u2081 - y\u2082) ^ 2 * (x\u2081 - x\u2082)) + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * (y\u2081 - y\u2082) * (x\u2081 - x\u2082) ^ 2 -\n            (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086)) +\n        (x\u2081 - x\u2082) *\n          (x\u2081 * x\u2082 *\n            ((y\u2081 - y\u2082) ^ 2 * (x\u2081 - x\u2082) + W.a\u2081 * (y\u2081 - y\u2082) * (x\u2081 - x\u2082) ^ 2 - (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * W.a\u2082 -\n                (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * x\u2081 -\n              (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * x\u2082)) -\n      (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) *\n        (x\u2082 * (y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081) - x\u2081 * (y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082) -\n          (x\u2082 * (x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086) - x\u2081 * (x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086))) =\n    0"}, {"tactic": "ring1", "annotated_tactic": ["ring1", []], "state_before": "case a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW\u271d : Affine R\nF : Type u\ninst\u271d : Field F\nW : Affine F\nx\u2081 x\u2082 y\u2081 y\u2082 : F\nh\u2081 : y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081 = x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086\nh\u2082 : y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082 = x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086\nhxy : x\u2081 = x\u2082 \u2192 y\u2081 \u2260 W.negY x\u2082 y\u2082\nhx : \u00acx\u2081 - x\u2082 = 0\n\u22a2 (x\u2081 - x\u2082) *\n          (-(x\u2081 ^ 2 * (y\u2081 - y\u2082) ^ 2 * (x\u2081 - x\u2082)) + (2 * x\u2081 * y\u2081 + W.a\u2083 * x\u2081) * (y\u2081 - y\u2082) * (x\u2081 - x\u2082) ^ 2 -\n            (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * (y\u2081 ^ 2 + W.a\u2083 * y\u2081 - W.a\u2086)) +\n        (x\u2081 - x\u2082) *\n          (x\u2081 * x\u2082 *\n            ((y\u2081 - y\u2082) ^ 2 * (x\u2081 - x\u2082) + W.a\u2081 * (y\u2081 - y\u2082) * (x\u2081 - x\u2082) ^ 2 - (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * W.a\u2082 -\n                (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * x\u2081 -\n              (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) * x\u2082)) -\n      (x\u2081 - x\u2082) ^ 2 * (x\u2081 - x\u2082) *\n        (x\u2082 * (y\u2081 ^ 2 + W.a\u2081 * x\u2081 * y\u2081 + W.a\u2083 * y\u2081) - x\u2081 * (y\u2082 ^ 2 + W.a\u2081 * x\u2082 * y\u2082 + W.a\u2083 * y\u2082) -\n          (x\u2082 * (x\u2081 ^ 3 + W.a\u2082 * x\u2081 ^ 2 + W.a\u2084 * x\u2081 + W.a\u2086) - x\u2081 * (x\u2082 ^ 3 + W.a\u2082 * x\u2082 ^ 2 + W.a\u2084 * x\u2082 + W.a\u2086))) =\n    0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean", "full_name": "WeierstrassCurve.Jacobian.negDblY_of_Z_eq_zero", "start": [675, 1], "end": [679, 95], "traced_tactics": [{"tactic": "linear_combination (norm :=\n    (rw [negDblY, dblU_of_Z_eq_zero hPz, dblX_of_Z_eq_zero hP hPz, negY_of_Z_eq_zero hPz]; ring1))\n  (8 * (equation_of_Z_eq_zero hPz).mp hP - 12 * P x ^ 3) * (equation_of_Z_eq_zero hPz).mp hP", "annotated_tactic": ["linear_combination (norm :=\n      (rw [<a>negDblY</a>, <a>dblU_of_Z_eq_zero</a> hPz, <a>dblX_of_Z_eq_zero</a> hP hPz, <a>negY_of_Z_eq_zero</a> hPz]; ring1))\n    (8 * (<a>equation_of_Z_eq_zero</a> hPz).<a>mp</a> hP - 12 * P x ^ 3) * (<a>equation_of_Z_eq_zero</a> hPz).<a>mp</a> hP", [{"full_name": "WeierstrassCurve.Jacobian.negDblY", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean", "def_pos": [668, 19], "def_end_pos": [668, 26]}, {"full_name": "WeierstrassCurve.Jacobian.dblU_of_Z_eq_zero", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean", "def_pos": [571, 7], "def_end_pos": [571, 24]}, {"full_name": "WeierstrassCurve.Jacobian.dblX_of_Z_eq_zero", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean", "def_pos": [642, 7], "def_end_pos": [642, 24]}, {"full_name": "WeierstrassCurve.Jacobian.negY_of_Z_eq_zero", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean", "def_pos": [486, 7], "def_end_pos": [486, 24]}, {"full_name": "WeierstrassCurve.Jacobian.equation_of_Z_eq_zero", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean", "def_pos": [273, 7], "def_end_pos": [273, 28]}, {"full_name": "Iff.mp", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [118, 3], "def_end_pos": [118, 5]}, {"full_name": "WeierstrassCurve.Jacobian.equation_of_Z_eq_zero", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean", "def_pos": [273, 7], "def_end_pos": [273, 28]}, {"full_name": "Iff.mp", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [118, 3], "def_end_pos": [118, 5]}]], "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\nW' : Jacobian R\nF : Type v\ninst\u271d : Field F\nW : Jacobian F\nP : Fin 3 \u2192 R\nhP : W'.Equation P\nhPz : P z = 0\n\u22a2 W'.negDblY P = -(P x ^ 2) ^ 3", "state_after": "no goals"}, {"tactic": "rw [negDblY, dblU_of_Z_eq_zero hPz, dblX_of_Z_eq_zero hP hPz, negY_of_Z_eq_zero hPz]", "annotated_tactic": ["rw [<a>negDblY</a>, <a>dblU_of_Z_eq_zero</a> hPz, <a>dblX_of_Z_eq_zero</a> hP hPz, <a>negY_of_Z_eq_zero</a> hPz]", [{"full_name": "WeierstrassCurve.Jacobian.negDblY", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean", "def_pos": [668, 19], "def_end_pos": [668, 26]}, {"full_name": "WeierstrassCurve.Jacobian.dblU_of_Z_eq_zero", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean", "def_pos": [571, 7], "def_end_pos": [571, 24]}, {"full_name": "WeierstrassCurve.Jacobian.dblX_of_Z_eq_zero", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean", "def_pos": [642, 7], "def_end_pos": [642, 24]}, {"full_name": "WeierstrassCurve.Jacobian.negY_of_Z_eq_zero", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean", "def_pos": [486, 7], "def_end_pos": [486, 24]}]], "state_before": "case a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW' : Jacobian R\nF : Type v\ninst\u271d : Field F\nW : Jacobian F\nP : Fin 3 \u2192 R\nhP : W'.Equation P\nhPz : P z = 0\n\u22a2 W'.negDblY P - -(P x ^ 2) ^ 3 - ((8 * P y ^ 2 - 12 * P x ^ 3) * P y ^ 2 - (8 * P x ^ 3 - 12 * P x ^ 3) * P x ^ 3) = 0", "state_after": "case a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW' : Jacobian R\nF : Type v\ninst\u271d : Field F\nW : Jacobian F\nP : Fin 3 \u2192 R\nhP : W'.Equation P\nhPz : P z = 0\n\u22a2 -(-3 * P x ^ 2) * ((P x ^ 2) ^ 2 - P x * (P y - -P y) ^ 2) + P y * (P y - -P y) ^ 3 - -(P x ^ 2) ^ 3 -\n      ((8 * P y ^ 2 - 12 * P x ^ 3) * P y ^ 2 - (8 * P x ^ 3 - 12 * P x ^ 3) * P x ^ 3) =\n    0"}, {"tactic": "ring1", "annotated_tactic": ["ring1", []], "state_before": "case a\nR : Type u\ninst\u271d\u00b9 : CommRing R\nW' : Jacobian R\nF : Type v\ninst\u271d : Field F\nW : Jacobian F\nP : Fin 3 \u2192 R\nhP : W'.Equation P\nhPz : P z = 0\n\u22a2 -(-3 * P x ^ 2) * ((P x ^ 2) ^ 2 - P x * (P y - -P y) ^ 2) + P y * (P y - -P y) ^ 3 - -(P x ^ 2) ^ 3 -\n      ((8 * P y ^ 2 - 12 * P x ^ 3) * P y ^ 2 - (8 * P x ^ 3 - 12 * P x ^ 3) * P x ^ 3) =\n    0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Interval/Set/Basic.lean", "full_name": "Set.Iic_subset_Iic_union_Ioc", "start": [1441, 1], "end": [1442, 74], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "full_name": "Real.Angle.sin_eq_iff_eq_or_add_eq_pi", "start": [352, 1], "end": [354, 44], "traced_tactics": [{"tactic": "induction \u03c8 using Real.Angle.induction_on", "annotated_tactic": ["induction \u03c8 using <a>Real.Angle.induction_on</a>", [{"full_name": "Real.Angle.induction_on", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [74, 19], "def_end_pos": [74, 31]}]], "state_before": "\u03b8 \u03c8 : Angle\n\u22a2 \u03b8.sin = \u03c8.sin \u2194 \u03b8 = \u03c8 \u2228 \u03b8 + \u03c8 = \u2191\u03c0", "state_after": "case h\n\u03b8 : Angle\nx\u271d : \u211d\n\u22a2 \u03b8.sin = (\u2191x\u271d).sin \u2194 \u03b8 = \u2191x\u271d \u2228 \u03b8 + \u2191x\u271d = \u2191\u03c0"}, {"tactic": "exact sin_eq_real_sin_iff_eq_or_add_eq_pi", "annotated_tactic": ["exact <a>sin_eq_real_sin_iff_eq_or_add_eq_pi</a>", [{"full_name": "Real.Angle.sin_eq_real_sin_iff_eq_or_add_eq_pi", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean", "def_pos": [346, 9], "def_end_pos": [346, 44]}]], "state_before": "case h\n\u03b8 : Angle\nx\u271d : \u211d\n\u22a2 \u03b8.sin = (\u2191x\u271d).sin \u2194 \u03b8 = \u2191x\u271d \u2228 \u03b8 + \u2191x\u271d = \u2191\u03c0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Computability/Primrec.lean", "full_name": "Primrec.option_bind\u2081", "start": [627, 1], "end": [628, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/Algebra/LieGroup.lean", "full_name": "SmoothAt.div\u2080", "start": [366, 1], "end": [368, 28], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "full_name": "MulHom.ext", "start": [594, 1], "end": [595, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/IsEmpty.lean", "full_name": "IsEmpty.exists_iff", "start": [133, 1], "end": [134, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Euclidean/Angle/Sphere.lean", "full_name": "EuclideanGeometry.cospherical_of_two_zsmul_oangle_eq_of_not_collinear", "start": [365, 1], "end": [379, 30], "traced_tactics": [{"tactic": "have hn' : \u00acCollinear \u211d ({p\u2081, p\u2083, p\u2084} : Set P) := by\n  rwa [\u2190 collinear_iff_of_two_zsmul_oangle_eq h]", "annotated_tactic": ["have hn' : \u00ac<a>Collinear</a> \u211d ({p\u2081, p\u2083, p\u2084} : <a>Set</a> P) := by\n    rwa [\u2190 <a>collinear_iff_of_two_zsmul_oangle_eq</a> h]", [{"full_name": "Collinear", "def_path": "Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean", "def_pos": [381, 5], "def_end_pos": [381, 14]}, {"full_name": "Set", "def_path": "Mathlib/Init/Set.lean", "def_pos": [47, 5], "def_end_pos": [47, 8]}, {"full_name": "EuclideanGeometry.collinear_iff_of_two_zsmul_oangle_eq", "def_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean", "def_pos": [250, 9], "def_end_pos": [250, 45]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\n\u22a2 Cospherical {p\u2081, p\u2082, p\u2083, p\u2084}", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\n\u22a2 Cospherical {p\u2081, p\u2082, p\u2083, p\u2084}"}, {"tactic": "let t\u2081 : Affine.Triangle \u211d P := \u27e8![p\u2081, p\u2082, p\u2084], affineIndependent_iff_not_collinear_set.2 hn\u27e9", "annotated_tactic": ["let t\u2081 : <a>Affine.Triangle</a> \u211d P := \u27e8![p\u2081, p\u2082, p\u2084], <a>affineIndependent_iff_not_collinear_set</a>.2 hn\u27e9", [{"full_name": "Affine.Triangle", "def_path": "Mathlib/LinearAlgebra/AffineSpace/Independent.lean", "def_pos": [804, 8], "def_end_pos": [804, 16]}, {"full_name": "affineIndependent_iff_not_collinear_set", "def_path": "Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean", "def_pos": [524, 9], "def_end_pos": [524, 48]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\n\u22a2 Cospherical {p\u2081, p\u2082, p\u2083, p\u2084}", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], independent := \u22ef }\n\u22a2 Cospherical {p\u2081, p\u2082, p\u2083, p\u2084}"}, {"tactic": "let t\u2082 : Affine.Triangle \u211d P := \u27e8![p\u2081, p\u2083, p\u2084], affineIndependent_iff_not_collinear_set.2 hn'\u27e9", "annotated_tactic": ["let t\u2082 : <a>Affine.Triangle</a> \u211d P := \u27e8![p\u2081, p\u2083, p\u2084], <a>affineIndependent_iff_not_collinear_set</a>.2 hn'\u27e9", [{"full_name": "Affine.Triangle", "def_path": "Mathlib/LinearAlgebra/AffineSpace/Independent.lean", "def_pos": [804, 8], "def_end_pos": [804, 16]}, {"full_name": "affineIndependent_iff_not_collinear_set", "def_path": "Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean", "def_pos": [524, 9], "def_end_pos": [524, 48]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], independent := \u22ef }\n\u22a2 Cospherical {p\u2081, p\u2082, p\u2083, p\u2084}", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], independent := \u22ef }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], independent := \u22ef }\n\u22a2 Cospherical {p\u2081, p\u2082, p\u2083, p\u2084}"}, {"tactic": "rw [cospherical_iff_exists_sphere]", "annotated_tactic": ["rw [<a>cospherical_iff_exists_sphere</a>]", [{"full_name": "EuclideanGeometry.cospherical_iff_exists_sphere", "def_path": "Mathlib/Geometry/Euclidean/Sphere/Basic.lean", "def_pos": [160, 9], "def_end_pos": [160, 38]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], independent := \u22ef }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], independent := \u22ef }\n\u22a2 Cospherical {p\u2081, p\u2082, p\u2083, p\u2084}", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], independent := \u22ef }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], independent := \u22ef }\n\u22a2 \u2203 s, {p\u2081, p\u2082, p\u2083, p\u2084} \u2286 Metric.sphere s.center s.radius"}, {"tactic": "refine \u27e8t\u2082.circumsphere, ?_\u27e9", "annotated_tactic": ["refine \u27e8t\u2082.circumsphere, ?_\u27e9", []], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], independent := \u22ef }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], independent := \u22ef }\n\u22a2 \u2203 s, {p\u2081, p\u2082, p\u2083, p\u2084} \u2286 Metric.sphere s.center s.radius", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], independent := \u22ef }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], independent := \u22ef }\n\u22a2 {p\u2081, p\u2082, p\u2083, p\u2084} \u2286 Metric.sphere (Affine.Simplex.circumsphere t\u2082).center (Affine.Simplex.circumsphere t\u2082).radius"}, {"tactic": "simp_rw [Set.insert_subset_iff, Set.singleton_subset_iff]", "annotated_tactic": ["simp_rw [<a>Set.insert_subset_iff</a>, <a>Set.singleton_subset_iff</a>]", [{"full_name": "Set.insert_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1124, 9], "def_end_pos": [1124, 26]}, {"full_name": "Set.singleton_subset_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1287, 9], "def_end_pos": [1287, 29]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], independent := \u22ef }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], independent := \u22ef }\n\u22a2 {p\u2081, p\u2082, p\u2083, p\u2084} \u2286 Metric.sphere (Affine.Simplex.circumsphere t\u2082).center (Affine.Simplex.circumsphere t\u2082).radius", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], independent := \u22ef }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], independent := \u22ef }\n\u22a2 p\u2081 \u2208 Metric.sphere (Affine.Simplex.circumsphere t\u2082).center (Affine.Simplex.circumsphere t\u2082).radius \u2227\n    p\u2082 \u2208 Metric.sphere (Affine.Simplex.circumsphere t\u2082).center (Affine.Simplex.circumsphere t\u2082).radius \u2227\n      p\u2083 \u2208 Metric.sphere (Affine.Simplex.circumsphere t\u2082).center (Affine.Simplex.circumsphere t\u2082).radius \u2227\n        p\u2084 \u2208 Metric.sphere (Affine.Simplex.circumsphere t\u2082).center (Affine.Simplex.circumsphere t\u2082).radius"}, {"tactic": "refine \u27e8t\u2082.mem_circumsphere 0, ?_, t\u2082.mem_circumsphere 1, t\u2082.mem_circumsphere 2\u27e9", "annotated_tactic": ["refine \u27e8t\u2082.mem_circumsphere 0, ?_, t\u2082.mem_circumsphere 1, t\u2082.mem_circumsphere 2\u27e9", []], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], independent := \u22ef }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], independent := \u22ef }\n\u22a2 p\u2081 \u2208 Metric.sphere (Affine.Simplex.circumsphere t\u2082).center (Affine.Simplex.circumsphere t\u2082).radius \u2227\n    p\u2082 \u2208 Metric.sphere (Affine.Simplex.circumsphere t\u2082).center (Affine.Simplex.circumsphere t\u2082).radius \u2227\n      p\u2083 \u2208 Metric.sphere (Affine.Simplex.circumsphere t\u2082).center (Affine.Simplex.circumsphere t\u2082).radius \u2227\n        p\u2084 \u2208 Metric.sphere (Affine.Simplex.circumsphere t\u2082).center (Affine.Simplex.circumsphere t\u2082).radius", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], independent := \u22ef }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], independent := \u22ef }\n\u22a2 p\u2082 \u2208 Metric.sphere (Affine.Simplex.circumsphere t\u2082).center (Affine.Simplex.circumsphere t\u2082).radius"}, {"tactic": "rw [Affine.Triangle.circumsphere_eq_circumsphere_of_eq_of_eq_of_two_zsmul_oangle_eq\n  (by decide : (0 : Fin 3) \u2260 1) (by decide : (0 : Fin 3) \u2260 2) (by decide)\n  (show t\u2082.points 0 = t\u2081.points 0 from rfl) rfl h.symm]", "annotated_tactic": ["rw [<a>Affine.Triangle.circumsphere_eq_circumsphere_of_eq_of_eq_of_two_zsmul_oangle_eq</a>\n    (by decide : (0 : Fin 3) \u2260 1) (by decide : (0 : Fin 3) \u2260 2) (by decide)\n    (show t\u2082.points 0 = t\u2081.points 0 from <a>rfl</a>) <a>rfl</a> h.symm]", [{"full_name": "Affine.Triangle.circumsphere_eq_circumsphere_of_eq_of_eq_of_two_zsmul_oangle_eq", "def_path": "Mathlib/Geometry/Euclidean/Angle/Sphere.lean", "def_pos": [313, 9], "def_end_pos": [313, 72]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], independent := \u22ef }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], independent := \u22ef }\n\u22a2 p\u2082 \u2208 Metric.sphere (Affine.Simplex.circumsphere t\u2082).center (Affine.Simplex.circumsphere t\u2082).radius", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], independent := \u22ef }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], independent := \u22ef }\n\u22a2 p\u2082 \u2208 Metric.sphere (Affine.Simplex.circumsphere t\u2081).center (Affine.Simplex.circumsphere t\u2081).radius"}, {"tactic": "exact t\u2081.mem_circumsphere 1", "annotated_tactic": ["exact t\u2081.mem_circumsphere 1", []], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], independent := \u22ef }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], independent := \u22ef }\n\u22a2 p\u2082 \u2208 Metric.sphere (Affine.Simplex.circumsphere t\u2081).center (Affine.Simplex.circumsphere t\u2081).radius", "state_after": "no goals"}, {"tactic": "rwa [\u2190 collinear_iff_of_two_zsmul_oangle_eq h]", "annotated_tactic": ["rwa [\u2190 <a>collinear_iff_of_two_zsmul_oangle_eq</a> h]", [{"full_name": "EuclideanGeometry.collinear_iff_of_two_zsmul_oangle_eq", "def_path": "Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean", "def_pos": [250, 9], "def_end_pos": [250, 45]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\n\u22a2 \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], independent := \u22ef }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], independent := \u22ef }\n\u22a2 0 \u2260 1", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], independent := \u22ef }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], independent := \u22ef }\n\u22a2 0 \u2260 2", "state_after": "no goals"}, {"tactic": "decide", "annotated_tactic": ["decide", []], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : InnerProductSpace \u211d V\ninst\u271d\u00b2 : MetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\nhd2 : Fact (finrank \u211d V = 2)\ninst\u271d : Module.Oriented \u211d V (Fin 2)\np\u2081 p\u2082 p\u2083 p\u2084 : P\nh : 2 \u2022 \u2221 p\u2081 p\u2082 p\u2084 = 2 \u2022 \u2221 p\u2081 p\u2083 p\u2084\nhn : \u00acCollinear \u211d {p\u2081, p\u2082, p\u2084}\nhn' : \u00acCollinear \u211d {p\u2081, p\u2083, p\u2084}\nt\u2081 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2082, p\u2084], independent := \u22ef }\nt\u2082 : Affine.Triangle \u211d P := { points := ![p\u2081, p\u2083, p\u2084], independent := \u22ef }\n\u22a2 1 \u2260 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subsemigroup/Basic.lean", "full_name": "MulHom.eq_of_eqOn_top", "start": [492, 1], "end": [493, 25], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Algebra/Group/Basic.lean", "full_name": "Filter.Tendsto.inv", "start": [253, 1], "end": [255, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subgroup/MulOpposite.lean", "full_name": "Subgroup.op_inf", "start": [114, 1], "end": [114, 90], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Group/Arithmetic.lean", "full_name": "AEMeasurable.mul", "start": [154, 1], "end": [156, 51], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finsupp/Lex.lean", "full_name": "Finsupp.lex_lt_of_lt", 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Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x : List \u03b1\nf : \u03b1 \u2192 \u03b2\nl : Language \u03b1\n\u22a2 (map f) (\u2a06 i, l ^ i) = \u2a06 i, (map f) l ^ i"}, {"tactic": "simp_rw [\u2190 map_pow]", "annotated_tactic": ["simp_rw [\u2190 <a>map_pow</a>]", [{"full_name": "map_pow", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [473, 9], "def_end_pos": [473, 16]}]], "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\nf : \u03b1 \u2192 \u03b2\nl : Language \u03b1\n\u22a2 (map f) (\u2a06 i, l ^ i) = \u2a06 i, (map f) l ^ i", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nl\u271d m : Language \u03b1\na b x : List \u03b1\nf : \u03b1 \u2192 \u03b2\nl : Language \u03b1\n\u22a2 (map f) (\u2a06 i, l ^ i) = \u2a06 i, (map f) (l ^ i)"}, {"tactic": "exact image_iUnion", "annotated_tactic": ["exact <a>image_iUnion</a>", [{"full_name": "Set.image_iUnion", 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l\u2081.length = l\u2082.length \u2227 \u2200 {a : \u03b1} {b : \u03b2}, (a, b) \u2208 l\u2081.zip l\u2082 \u2192 R a b\n\u22a2 Forall\u2082 R l\u2081 l\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\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u2081 : List \u03b1\nl\u2082 : List \u03b2\nh\u2081 : l\u2081.length = l\u2082.length\nh\u2082 : \u2200 {a : \u03b1} {b : \u03b2}, (a, b) \u2208 l\u2081.zip l\u2082 \u2192 R a b\n\u22a2 Forall\u2082 R l\u2081 l\u2082"}, {"tactic": "induction' l\u2081 with a l\u2081 IH generalizing l\u2082", "annotated_tactic": ["induction' l\u2081 with a l\u2081 IH generalizing l\u2082", []], "state_before": "case intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u2081 : List \u03b1\nl\u2082 : List \u03b2\nh\u2081 : l\u2081.length = l\u2082.length\nh\u2082 : \u2200 {a : \u03b1} {b : \u03b2}, (a, b) \u2208 l\u2081.zip l\u2082 \u2192 R a b\n\u22a2 Forall\u2082 R l\u2081 l\u2082", "state_after": "case intro.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u2082 : List \u03b2\nh\u2081 : [].length = l\u2082.length\nh\u2082 : \u2200 {a : \u03b1} {b : \u03b2}, (a, b) \u2208 [].zip l\u2082 \u2192 R a b\n\u22a2 Forall\u2082 R [] l\u2082\n\ncase intro.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b2}, l\u2081.length = l\u2082.length \u2192 (\u2200 {a : \u03b1} {b : \u03b2}, (a, b) \u2208 l\u2081.zip l\u2082 \u2192 R a b) \u2192 Forall\u2082 R l\u2081 l\u2082\nl\u2082 : List \u03b2\nh\u2081 : (a :: l\u2081).length = l\u2082.length\nh\u2082 : \u2200 {a_1 : \u03b1} {b : \u03b2}, (a_1, b) \u2208 (a :: l\u2081).zip l\u2082 \u2192 R a_1 b\n\u22a2 Forall\u2082 R (a :: l\u2081) l\u2082"}, {"tactic": "cases length_eq_zero.1 h\u2081.symm", "annotated_tactic": ["cases <a>length_eq_zero</a>.1 h\u2081.symm", [{"full_name": "List.length_eq_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [100, 17], "def_end_pos": [100, 31]}]], "state_before": "case intro.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\nl\u2082 : List \u03b2\nh\u2081 : [].length = l\u2082.length\nh\u2082 : \u2200 {a : \u03b1} {b : \u03b2}, (a, b) \u2208 [].zip l\u2082 \u2192 R a b\n\u22a2 Forall\u2082 R [] l\u2082", "state_after": "case intro.nil.refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\nh\u2081 : [].length = [].length\nh\u2082 : \u2200 {a : \u03b1} {b : \u03b2}, (a, b) \u2208 [].zip [] \u2192 R a b\n\u22a2 Forall\u2082 R [] []"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case intro.nil.refl\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\nh\u2081 : [].length = [].length\nh\u2082 : \u2200 {a : \u03b1} {b : \u03b2}, (a, b) \u2208 [].zip [] \u2192 R a b\n\u22a2 Forall\u2082 R [] []", "state_after": "no goals"}, {"tactic": "cases' l\u2082 with b l\u2082", "annotated_tactic": ["cases' l\u2082 with b l\u2082", []], "state_before": "case intro.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b2}, l\u2081.length = l\u2082.length \u2192 (\u2200 {a : \u03b1} {b : \u03b2}, (a, b) \u2208 l\u2081.zip l\u2082 \u2192 R a b) \u2192 Forall\u2082 R l\u2081 l\u2082\nl\u2082 : List \u03b2\nh\u2081 : (a :: l\u2081).length = l\u2082.length\nh\u2082 : \u2200 {a_1 : \u03b1} {b : \u03b2}, (a_1, b) \u2208 (a :: l\u2081).zip l\u2082 \u2192 R a_1 b\n\u22a2 Forall\u2082 R (a :: l\u2081) l\u2082", "state_after": "case intro.cons.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b2}, l\u2081.length = l\u2082.length \u2192 (\u2200 {a : \u03b1} {b : \u03b2}, (a, b) \u2208 l\u2081.zip l\u2082 \u2192 R a b) \u2192 Forall\u2082 R l\u2081 l\u2082\nh\u2081 : (a :: l\u2081).length = [].length\nh\u2082 : \u2200 {a_1 : \u03b1} {b : \u03b2}, (a_1, b) \u2208 (a :: l\u2081).zip [] \u2192 R a_1 b\n\u22a2 Forall\u2082 R (a :: l\u2081) []\n\ncase intro.cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b2}, l\u2081.length = l\u2082.length \u2192 (\u2200 {a : \u03b1} {b : \u03b2}, (a, b) \u2208 l\u2081.zip l\u2082 \u2192 R a b) \u2192 Forall\u2082 R l\u2081 l\u2082\nb : \u03b2\nl\u2082 : List \u03b2\nh\u2081 : (a :: l\u2081).length = (b :: l\u2082).length\nh\u2082 : \u2200 {a_1 : \u03b1} {b_1 : \u03b2}, (a_1, b_1) \u2208 (a :: l\u2081).zip (b :: l\u2082) \u2192 R a_1 b_1\n\u22a2 Forall\u2082 R (a :: l\u2081) (b :: l\u2082)"}, {"tactic": "simp at h\u2081", "annotated_tactic": ["simp at h\u2081", []], "state_before": "case intro.cons.nil\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b2}, l\u2081.length = l\u2082.length \u2192 (\u2200 {a : \u03b1} {b : \u03b2}, (a, b) \u2208 l\u2081.zip l\u2082 \u2192 R a b) \u2192 Forall\u2082 R l\u2081 l\u2082\nh\u2081 : (a :: l\u2081).length = [].length\nh\u2082 : \u2200 {a_1 : \u03b1} {b : \u03b2}, (a_1, b) \u2208 (a :: l\u2081).zip [] \u2192 R a_1 b\n\u22a2 Forall\u2082 R (a :: l\u2081) []", "state_after": "no goals"}, {"tactic": "simp only [length_cons, succ.injEq] at h\u2081", "annotated_tactic": ["simp only [<a>length_cons</a>, <a>succ.injEq</a>] at h\u2081", [{"full_name": "List.length_cons", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [70, 17], "def_end_pos": [70, 28]}, {"full_name": "Nat.succ.injEq", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1323, 9], "def_end_pos": [1323, 23]}]], "state_before": "case intro.cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b2}, l\u2081.length = l\u2082.length \u2192 (\u2200 {a : \u03b1} {b : \u03b2}, (a, b) \u2208 l\u2081.zip l\u2082 \u2192 R a b) \u2192 Forall\u2082 R l\u2081 l\u2082\nb : \u03b2\nl\u2082 : List \u03b2\nh\u2081 : (a :: l\u2081).length = (b :: l\u2082).length\nh\u2082 : \u2200 {a_1 : \u03b1} {b_1 : \u03b2}, (a_1, b_1) \u2208 (a :: l\u2081).zip (b :: l\u2082) \u2192 R a_1 b_1\n\u22a2 Forall\u2082 R (a :: l\u2081) (b :: l\u2082)", "state_after": "case intro.cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b2}, l\u2081.length = l\u2082.length \u2192 (\u2200 {a : \u03b1} {b : \u03b2}, (a, b) \u2208 l\u2081.zip l\u2082 \u2192 R a b) \u2192 Forall\u2082 R l\u2081 l\u2082\nb : \u03b2\nl\u2082 : List \u03b2\nh\u2082 : \u2200 {a_1 : \u03b1} {b_1 : \u03b2}, (a_1, b_1) \u2208 (a :: l\u2081).zip (b :: l\u2082) \u2192 R a_1 b_1\nh\u2081 : l\u2081.length = l\u2082.length\n\u22a2 Forall\u2082 R (a :: l\u2081) (b :: l\u2082)"}, {"tactic": "exact Forall\u2082.cons (h\u2082 <| by simp [zip])\n  (IH h\u2081 fun h => h\u2082 <| by\n    simp only [zip, zipWith, find?, mem_cons, Prod.mk.injEq]; right\n    simpa [zip] using h)", "annotated_tactic": ["exact <a>Forall\u2082.cons</a> (h\u2082 <| by simp [<a>zip</a>])\n          (IH h\u2081 fun h => h\u2082 <| by\n            simp only [<a>zip</a>, <a>zipWith</a>, <a>find?</a>, <a>mem_cons</a>, Prod.mk.injEq]; right\n            simpa [<a>zip</a>] using h)", [{"full_name": "List.Forall\u2082.cons", "def_path": ".lake/packages/batteries/Batteries/Data/List/Basic.lean", "def_pos": [564, 5], "def_end_pos": [564, 9]}, {"full_name": "List.zip", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [1106, 5], "def_end_pos": [1106, 8]}, {"full_name": "List.zip", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [1106, 5], "def_end_pos": [1106, 8]}, {"full_name": "List.zipWith", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [1090, 19], "def_end_pos": [1090, 26]}, {"full_name": "List.find?", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [980, 5], "def_end_pos": [980, 10]}, {"full_name": "List.mem_cons", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [285, 17], "def_end_pos": [285, 25]}, {"full_name": "List.zip", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [1106, 5], "def_end_pos": [1106, 8]}]], "state_before": "case intro.cons.cons\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b2}, l\u2081.length = l\u2082.length \u2192 (\u2200 {a : \u03b1} {b : \u03b2}, (a, b) \u2208 l\u2081.zip l\u2082 \u2192 R a b) \u2192 Forall\u2082 R l\u2081 l\u2082\nb : \u03b2\nl\u2082 : List \u03b2\nh\u2082 : \u2200 {a_1 : \u03b1} {b_1 : \u03b2}, (a_1, b_1) \u2208 (a :: l\u2081).zip (b :: l\u2082) \u2192 R a_1 b_1\nh\u2081 : l\u2081.length = l\u2082.length\n\u22a2 Forall\u2082 R (a :: l\u2081) (b :: l\u2082)", "state_after": "no goals"}, {"tactic": "simp [zip]", "annotated_tactic": ["simp [<a>zip</a>]", [{"full_name": "List.zip", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [1106, 5], "def_end_pos": [1106, 8]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b2}, l\u2081.length = l\u2082.length \u2192 (\u2200 {a : \u03b1} {b : \u03b2}, (a, b) \u2208 l\u2081.zip l\u2082 \u2192 R a b) \u2192 Forall\u2082 R l\u2081 l\u2082\nb : \u03b2\nl\u2082 : List \u03b2\nh\u2082 : \u2200 {a_1 : \u03b1} {b_1 : \u03b2}, (a_1, b_1) \u2208 (a :: l\u2081).zip (b :: l\u2082) \u2192 R a_1 b_1\nh\u2081 : l\u2081.length = l\u2082.length\n\u22a2 (a, b) \u2208 (a :: l\u2081).zip (b :: l\u2082)", "state_after": "no goals"}, {"tactic": "simp only [zip, zipWith, find?, mem_cons, Prod.mk.injEq]", "annotated_tactic": ["simp only [<a>zip</a>, <a>zipWith</a>, <a>find?</a>, <a>mem_cons</a>, Prod.mk.injEq]", [{"full_name": "List.zip", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [1106, 5], "def_end_pos": [1106, 8]}, {"full_name": "List.zipWith", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [1090, 19], "def_end_pos": [1090, 26]}, {"full_name": "List.find?", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [980, 5], "def_end_pos": [980, 10]}, {"full_name": "List.mem_cons", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Lemmas.lean", "def_pos": [285, 17], "def_end_pos": [285, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b2}, l\u2081.length = l\u2082.length \u2192 (\u2200 {a : \u03b1} {b : \u03b2}, (a, b) \u2208 l\u2081.zip l\u2082 \u2192 R a b) \u2192 Forall\u2082 R l\u2081 l\u2082\nb : \u03b2\nl\u2082 : List \u03b2\nh\u2082 : \u2200 {a_1 : \u03b1} {b_1 : \u03b2}, (a_1, b_1) \u2208 (a :: l\u2081).zip (b :: l\u2082) \u2192 R a_1 b_1\nh\u2081 : l\u2081.length = l\u2082.length\na\u271d : \u03b1\nb\u271d : \u03b2\nh : (a\u271d, b\u271d) \u2208 l\u2081.zip l\u2082\n\u22a2 (a\u271d, b\u271d) \u2208 (a :: l\u2081).zip (b :: l\u2082)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b2}, l\u2081.length = l\u2082.length \u2192 (\u2200 {a : \u03b1} {b : \u03b2}, (a, b) \u2208 l\u2081.zip l\u2082 \u2192 R a b) \u2192 Forall\u2082 R l\u2081 l\u2082\nb : \u03b2\nl\u2082 : List \u03b2\nh\u2082 : \u2200 {a_1 : \u03b1} {b_1 : \u03b2}, (a_1, b_1) \u2208 (a :: l\u2081).zip (b :: l\u2082) \u2192 R a_1 b_1\nh\u2081 : l\u2081.length = l\u2082.length\na\u271d : \u03b1\nb\u271d : \u03b2\nh : (a\u271d, b\u271d) \u2208 l\u2081.zip l\u2082\n\u22a2 a\u271d = a \u2227 b\u271d = b \u2228 (a\u271d, b\u271d) \u2208 zipWith Prod.mk l\u2081 l\u2082"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b2}, l\u2081.length = l\u2082.length \u2192 (\u2200 {a : \u03b1} {b : \u03b2}, (a, b) \u2208 l\u2081.zip l\u2082 \u2192 R a b) \u2192 Forall\u2082 R l\u2081 l\u2082\nb : \u03b2\nl\u2082 : List \u03b2\nh\u2082 : \u2200 {a_1 : \u03b1} {b_1 : \u03b2}, (a_1, b_1) \u2208 (a :: l\u2081).zip (b :: l\u2082) \u2192 R a_1 b_1\nh\u2081 : l\u2081.length = l\u2082.length\na\u271d : \u03b1\nb\u271d : \u03b2\nh : (a\u271d, b\u271d) \u2208 l\u2081.zip l\u2082\n\u22a2 a\u271d = a \u2227 b\u271d = b \u2228 (a\u271d, b\u271d) \u2208 zipWith Prod.mk l\u2081 l\u2082", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b2}, l\u2081.length = l\u2082.length \u2192 (\u2200 {a : \u03b1} {b : \u03b2}, (a, b) \u2208 l\u2081.zip l\u2082 \u2192 R a b) \u2192 Forall\u2082 R l\u2081 l\u2082\nb : \u03b2\nl\u2082 : List \u03b2\nh\u2082 : \u2200 {a_1 : \u03b1} {b_1 : \u03b2}, (a_1, b_1) \u2208 (a :: l\u2081).zip (b :: l\u2082) \u2192 R a_1 b_1\nh\u2081 : l\u2081.length = l\u2082.length\na\u271d : \u03b1\nb\u271d : \u03b2\nh : (a\u271d, b\u271d) \u2208 l\u2081.zip l\u2082\n\u22a2 (a\u271d, b\u271d) \u2208 zipWith Prod.mk l\u2081 l\u2082"}, {"tactic": "simpa [zip] using h", "annotated_tactic": ["simpa [<a>zip</a>] using h", [{"full_name": "List.zip", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [1106, 5], "def_end_pos": [1106, 8]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nR S : \u03b1 \u2192 \u03b2 \u2192 Prop\nP : \u03b3 \u2192 \u03b4 \u2192 Prop\nR\u2090 : \u03b1 \u2192 \u03b1 \u2192 Prop\na : \u03b1\nl\u2081 : List \u03b1\nIH : \u2200 {l\u2082 : List \u03b2}, l\u2081.length = l\u2082.length \u2192 (\u2200 {a : \u03b1} {b : \u03b2}, (a, b) \u2208 l\u2081.zip l\u2082 \u2192 R a b) \u2192 Forall\u2082 R l\u2081 l\u2082\nb : \u03b2\nl\u2082 : List \u03b2\nh\u2082 : \u2200 {a_1 : \u03b1} {b_1 : \u03b2}, (a_1, b_1) \u2208 (a :: l\u2081).zip (b :: l\u2082) \u2192 R a_1 b_1\nh\u2081 : l\u2081.length = l\u2082.length\na\u271d : \u03b1\nb\u271d : \u03b2\nh : (a\u271d, b\u271d) \u2208 l\u2081.zip l\u2082\n\u22a2 (a\u271d, b\u271d) \u2208 zipWith Prod.mk l\u2081 l\u2082", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "full_name": "Submodule.map_equiv_eq_comap_symm", "start": [552, 1], "end": [554, 79], "traced_tactics": [{"tactic": "rw [mem_map_equiv, mem_comap, LinearEquiv.coe_coe]", "annotated_tactic": ["rw [<a>mem_map_equiv</a>, <a>mem_comap</a>, <a>LinearEquiv.coe_coe</a>]", [{"full_name": "Submodule.mem_map_equiv", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [543, 9], "def_end_pos": [543, 22]}, {"full_name": "Submodule.mem_comap", "def_path": "Mathlib/Algebra/Module/Submodule/Map.lean", "def_pos": [195, 9], "def_end_pos": [195, 18]}, {"full_name": "LinearEquiv.coe_coe", "def_path": "Mathlib/Algebra/Module/Equiv.lean", "def_pos": [222, 9], "def_end_pos": [222, 16]}]], "state_before": "R : Type u_1\nR\u2081 : Type u_2\nR\u2082 : Type u_3\nR\u2083 : Type u_4\nM : Type u_5\nM\u2081 : Type u_6\nM\u2082 : Type u_7\nM\u2083 : Type u_8\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : Semiring R\u2082\ninst\u271d\u2075 : AddCommMonoid M\ninst\u271d\u2074 : AddCommMonoid M\u2082\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R\u2082 M\u2082\n\u03c4\u2081\u2082 : R \u2192+* R\u2082\n\u03c4\u2082\u2081 : R\u2082 \u2192+* R\ninst\u271d\u00b9 : RingHomInvPair \u03c4\u2081\u2082 \u03c4\u2082\u2081\ninst\u271d : RingHomInvPair \u03c4\u2082\u2081 \u03c4\u2081\u2082\np : Submodule R M\nq : Submodule R\u2082 M\u2082\ne : M \u2243\u209b\u2097[\u03c4\u2081\u2082] M\u2082\nK : Submodule R M\nx\u271d : M\u2082\n\u22a2 x\u271d \u2208 map (\u2191e) K \u2194 x\u271d \u2208 comap (\u2191e.symm) K", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Minimal.lean", "full_name": "maximals_swap", "start": [74, 1], "end": [75, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Basic.lean", "full_name": "imp_iff_not_or", "start": [386, 1], "end": [386, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Holor.lean", "full_name": "Holor.sum_unitVec_mul_slice", "start": [243, 1], "end": [258, 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": [203, 9], "def_end_pos": [203, 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 \u2208 (Finset.range d).attach, unitVec d \u2191i \u2297 x.slice \u2191i \u22ef = 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 (\u2211 i \u2208 (Finset.range d).attach, unitVec d \u2191i \u2297 x.slice \u2191i \u22ef).slice = x.slice"}, {"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 (\u2211 i \u2208 (Finset.range d).attach, unitVec d \u2191i \u2297 x.slice \u2191i \u22ef).slice = x.slice", "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 i \u2208 (Finset.range d).attach, unitVec d \u2191i \u2297 x.slice \u2191i \u22ef).slice i hid = x.slice 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": [231, 9], "def_end_pos": [231, 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 i \u2208 (Finset.range d).attach, unitVec d \u2191i \u2297 x.slice \u2191i \u22ef).slice i hid = x.slice 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 \u2208 (Finset.range d).attach, (unitVec d \u2191x_1 \u2297 x.slice \u2191x_1 \u22ef).slice i hid = x.slice 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": [215, 9], "def_end_pos": [215, 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 \u2208 (Finset.range d).attach, (unitVec d \u2191x_1 \u2297 x.slice \u2191x_1 \u22ef).slice i hid = x.slice 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 \u2208 (Finset.range d).attach, if i = \u2191x_1 then x.slice \u2191x_1 \u22ef else 0) = x.slice 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/Group/Finset.lean", "def_pos": [998, 3], "def_end_pos": [998, 14]}, {"full_name": "Subtype.mk", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [583, 11], "def_end_pos": [583, 18]}, {"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2935, 9], "def_end_pos": [2935, 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 \u2208 (Finset.range d).attach, if i = \u2191x_1 then x.slice \u2191x_1 \u22ef else 0) = x.slice 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\u27e8i, \u22ef\u27e9 then x.slice \u2191\u27e8i, \u22ef\u27e9 \u22ef else 0) = x.slice 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 \u2208 (Finset.range d).attach, b \u2260 \u27e8i, \u22ef\u27e9 \u2192 (if i = \u2191b then x.slice \u2191b \u22ef 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 \u27e8i, \u22ef\u27e9 \u2209 (Finset.range d).attach \u2192 (if i = \u2191\u27e8i, \u22ef\u27e9 then x.slice \u2191\u27e8i, \u22ef\u27e9 \u22ef else 0) = 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\u27e8i, \u22ef\u27e9 then x.slice \u2191\u27e8i, \u22ef\u27e9 \u22ef else 0) = x.slice 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": [2925, 5], "def_end_pos": [2925, 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 \u2208 (Finset.range d).attach, b \u2260 \u27e8i, \u22ef\u27e9 \u2192 (if i = \u2191b then x.slice \u2191b \u22ef 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.range d).attach\nhbi : b \u2260 \u27e8i, \u22ef\u27e9\n\u22a2 (if i = \u2191b then x.slice \u2191b \u22ef else 0) = 0"}, {"tactic": "have hbi' : i \u2260 b := by simpa only [Ne, Subtype.ext_iff, Subtype.coe_mk] using hbi.symm", "annotated_tactic": ["have hbi' : i \u2260 b := by simpa only [<a>Ne</a>, <a>Subtype.ext_iff</a>, <a>Subtype.coe_mk</a>] using hbi.symm", [{"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "Subtype.ext_iff", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [78, 9], "def_end_pos": [78, 16]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [108, 9], "def_end_pos": [108, 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.range d).attach\nhbi : b \u2260 \u27e8i, \u22ef\u27e9\n\u22a2 (if i = \u2191b then x.slice \u2191b \u22ef 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.range d).attach\nhbi : b \u2260 \u27e8i, \u22ef\u27e9\nhbi' : i \u2260 \u2191b\n\u22a2 (if i = \u2191b then x.slice \u2191b \u22ef 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.range d).attach\nhbi : b \u2260 \u27e8i, \u22ef\u27e9\nhbi' : i \u2260 \u2191b\n\u22a2 (if i = \u2191b then x.slice \u2191b \u22ef else 0) = 0", "state_after": "no goals"}, {"tactic": "simpa only [Ne, Subtype.ext_iff, Subtype.coe_mk] using hbi.symm", "annotated_tactic": ["simpa only [<a>Ne</a>, <a>Subtype.ext_iff</a>, <a>Subtype.coe_mk</a>] using hbi.symm", [{"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "Subtype.ext_iff", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [78, 9], "def_end_pos": [78, 16]}, {"full_name": "Subtype.coe_mk", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [108, 9], "def_end_pos": [108, 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.range d).attach\nhbi : b \u2260 \u27e8i, \u22ef\u27e9\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": [583, 11], "def_end_pos": [583, 18]}, {"full_name": "Finset.attach", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2482, 5], "def_end_pos": [2482, 11]}, {"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2925, 5], "def_end_pos": [2925, 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 \u27e8i, \u22ef\u27e9 \u2209 (Finset.range d).attach \u2192 (if i = \u2191\u27e8i, \u22ef\u27e9 then x.slice \u2191\u27e8i, \u22ef\u27e9 \u22ef else 0) = 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' : \u27e8i, \u22ef\u27e9 \u2209 (Finset.range d).attach\n\u22a2 (if i = \u2191\u27e8i, \u22ef\u27e9 then x.slice \u2191\u27e8i, \u22ef\u27e9 \u22ef else 0) = 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' : \u27e8i, \u22ef\u27e9 \u2209 (Finset.range d).attach\n\u22a2 (if i = \u2191\u27e8i, \u22ef\u27e9 then x.slice \u2191\u27e8i, \u22ef\u27e9 \u22ef else 0) = 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' : \u27e8i, \u22ef\u27e9 \u2209 (Finset.range d).attach\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": [246, 21], "def_end_pos": [246, 27]}, {"full_name": "Finset.mem_attach", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2501, 9], "def_end_pos": [2501, 19]}]], "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' : \u27e8i, \u22ef\u27e9 \u2209 (Finset.range d).attach\n\u22a2 False", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Int/Order/Lemmas.lean", "full_name": "Int.eq_zero_of_abs_lt_dvd", "start": [54, 1], "end": [60, 42], "traced_tactics": [{"tactic": "obtain rfl | hm := eq_or_ne m 0", "annotated_tactic": ["obtain rfl | hm := <a>eq_or_ne</a> m 0", [{"full_name": "eq_or_ne", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [212, 9], "def_end_pos": [212, 17]}]], "state_before": "a b : \u2124\nn : \u2115\nm x : \u2124\nh1 : m \u2223 x\nh2 : |x| < m\n\u22a2 x = 0", "state_after": "case inl\na b : \u2124\nn : \u2115\nx : \u2124\nh1 : 0 \u2223 x\nh2 : |x| < 0\n\u22a2 x = 0\n\ncase inr\na b : \u2124\nn : \u2115\nm x : \u2124\nh1 : m \u2223 x\nh2 : |x| < m\nhm : m \u2260 0\n\u22a2 x = 0"}, {"tactic": "rcases h1 with \u27e8d, rfl\u27e9", "annotated_tactic": ["rcases h1 with \u27e8d, rfl\u27e9", []], "state_before": "case inr\na b : \u2124\nn : \u2115\nm x : \u2124\nh1 : m \u2223 x\nh2 : |x| < m\nhm : m \u2260 0\n\u22a2 x = 0", "state_after": "case inr.intro\na b : \u2124\nn : \u2115\nm : \u2124\nhm : m \u2260 0\nd : \u2124\nh2 : |m * d| < m\n\u22a2 m * d = 0"}, {"tactic": "apply mul_eq_zero_of_right", "annotated_tactic": ["apply <a>mul_eq_zero_of_right</a>", [{"full_name": "mul_eq_zero_of_right", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [274, 9], "def_end_pos": [274, 29]}]], "state_before": "case inr.intro\na b : \u2124\nn : \u2115\nm : \u2124\nhm : m \u2260 0\nd : \u2124\nh2 : |m * d| < m\n\u22a2 m * d = 0", "state_after": "case inr.intro.h\na b : \u2124\nn : \u2115\nm : \u2124\nhm : m \u2260 0\nd : \u2124\nh2 : |m * d| < m\n\u22a2 d = 0"}, {"tactic": "rw [\u2190 abs_lt_one_iff, \u2190 mul_lt_iff_lt_one_right (abs_pos.mpr hm), \u2190 abs_mul]", "annotated_tactic": ["rw [\u2190 <a>abs_lt_one_iff</a>, \u2190 <a>mul_lt_iff_lt_one_right</a> (abs_pos.mpr hm), \u2190 <a>abs_mul</a>]", [{"full_name": "Int.abs_lt_one_iff", "def_path": "Mathlib/Algebra/Order/Group/Int.lean", "def_pos": [88, 9], "def_end_pos": [88, 23]}, {"full_name": "mul_lt_iff_lt_one_right", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [687, 9], "def_end_pos": [687, 32]}, {"full_name": "abs_mul", "def_path": "Mathlib/Algebra/Order/Ring/Abs.lean", "def_pos": [48, 7], "def_end_pos": [48, 14]}]], "state_before": "case inr.intro.h\na b : \u2124\nn : \u2115\nm : \u2124\nhm : m \u2260 0\nd : \u2124\nh2 : |m * d| < m\n\u22a2 d = 0", "state_after": "case inr.intro.h\na b : \u2124\nn : \u2115\nm : \u2124\nhm : m \u2260 0\nd : \u2124\nh2 : |m * d| < m\n\u22a2 |m * d| < |m|"}, {"tactic": "exact lt_of_lt_of_le h2 (le_abs_self m)", "annotated_tactic": ["exact <a>lt_of_lt_of_le</a> h2 (<a>le_abs_self</a> m)", [{"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": "le_abs_self", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [73, 3], "def_end_pos": [73, 14]}]], "state_before": "case inr.intro.h\na b : \u2124\nn : \u2115\nm : \u2124\nhm : m \u2260 0\nd : \u2124\nh2 : |m * d| < m\n\u22a2 |m * d| < |m|", "state_after": "no goals"}, {"tactic": "exact Int.zero_dvd.1 h1", "annotated_tactic": ["exact <a>Int.zero_dvd</a>.1 h1", [{"full_name": "Int.zero_dvd", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean", "def_pos": [52, 27], "def_end_pos": [52, 35]}]], "state_before": "case inl\na b : \u2124\nn : \u2115\nx : \u2124\nh1 : 0 \u2223 x\nh2 : |x| < 0\n\u22a2 x = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/FreeMonoid/Count.lean", "full_name": "FreeMonoid.count_of", "start": [84, 1], "end": [86, 81], "traced_tactics": [{"tactic": "simp [count, countP_of, Pi.mulSingle_apply, eq_comm, Bool.beq_eq_decide_eq]", "annotated_tactic": ["simp [<a>count</a>, <a>countP_of</a>, <a>Pi.mulSingle_apply</a>, <a>eq_comm</a>, <a>Bool.beq_eq_decide_eq</a>]", [{"full_name": "FreeMonoid.count", "def_path": "Mathlib/Algebra/FreeMonoid/Count.lean", "def_pos": [77, 5], "def_end_pos": [77, 10]}, {"full_name": "FreeMonoid.countP_of", "def_path": "Mathlib/Algebra/FreeMonoid/Count.lean", "def_pos": [67, 9], "def_end_pos": [67, 18]}, {"full_name": "Pi.mulSingle_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [386, 9], "def_end_pos": [386, 24]}, {"full_name": "eq_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}, {"full_name": "Bool.beq_eq_decide_eq", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Bool.lean", "def_pos": [234, 9], "def_end_pos": [234, 25]}]], "state_before": "\u03b1 : Type u_1\np : \u03b1 \u2192 Prop\ninst\u271d\u00b9 : DecidablePred p\ninst\u271d : DecidableEq \u03b1\nx y : \u03b1\n\u22a2 (count x) (of y) = Pi.mulSingle x (Multiplicative.ofAdd 1) y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Bornology/Hom.lean", "full_name": "LocallyBoundedMap.ofMapBounded_apply", "start": [128, 1], "end": [129, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Martingale/OptionalStopping.lean", "full_name": "MeasureTheory.smul_le_stoppedValue_hitting", "start": [112, 1], "end": [133, 69], "traced_tactics": [{"tactic": "have hn : Set.Icc 0 n = {k | k \u2264 n} := by ext x; simp", "annotated_tactic": ["have hn : <a>Set.Icc</a> 0 n = {k | k \u2264 n} := by ext x; simp", [{"full_name": "Set.Icc", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [59, 5], "def_end_pos": [59, 8]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 \u03b5 \u2022 \u03bc {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9} \u2264\n    ENNReal.ofReal\n      (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9}, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc)", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\n\u22a2 \u03b5 \u2022 \u03bc {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9} \u2264\n    ENNReal.ofReal\n      (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9}, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc)"}, {"tactic": "have : \u2200 \u03c9, ((\u03b5 : \u211d) \u2264 (range (n + 1)).sup' nonempty_range_succ fun k => f k \u03c9) \u2192\n    (\u03b5 : \u211d) \u2264 stoppedValue f (hitting f {y : \u211d | \u2191\u03b5 \u2264 y} 0 n) \u03c9 := by\n  intro x hx\n  simp_rw [le_sup'_iff, mem_range, Nat.lt_succ_iff] at hx\n  refine stoppedValue_hitting_mem ?_\n  simp only [Set.mem_setOf_eq, exists_prop, hn]\n  exact\n    let \u27e8j, hj\u2081, hj\u2082\u27e9 := hx\n    \u27e8j, hj\u2081, hj\u2082\u27e9", "annotated_tactic": ["have : \u2200 \u03c9, ((\u03b5 : \u211d) \u2264 (<a>range</a> (n + 1)).<a>sup'</a> <a>nonempty_range_succ</a> fun k => f k \u03c9) \u2192\n      (\u03b5 : \u211d) \u2264 <a>stoppedValue</a> f (<a>hitting</a> f {y : \u211d | \u2191\u03b5 \u2264 y} 0 n) \u03c9 := by\n    intro x hx\n    simp_rw [<a>le_sup'_iff</a>, <a>mem_range</a>, <a>Nat.lt_succ_iff</a>] at hx\n    refine <a>stoppedValue_hitting_mem</a> ?_\n    simp only [<a>Set.mem_setOf_eq</a>, <a>exists_prop</a>, hn]\n    exact\n      let \u27e8j, hj\u2081, hj\u2082\u27e9 := hx\n      \u27e8j, hj\u2081, hj\u2082\u27e9", [{"full_name": "Finset.range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2925, 5], "def_end_pos": [2925, 10]}, {"full_name": "Finset.sup'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [814, 5], "def_end_pos": [814, 9]}, {"full_name": "Finset.nonempty_range_succ", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [3005, 9], "def_end_pos": [3005, 28]}, {"full_name": "MeasureTheory.stoppedValue", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [762, 5], "def_end_pos": [762, 17]}, {"full_name": "MeasureTheory.hitting", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [52, 19], "def_end_pos": [52, 26]}, {"full_name": "Finset.le_sup'_iff", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1302, 9], "def_end_pos": [1302, 20]}, {"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2935, 9], "def_end_pos": [2935, 18]}, {"full_name": "Nat.lt_succ_iff", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [573, 19], "def_end_pos": [573, 30]}, {"full_name": "MeasureTheory.stoppedValue_hitting_mem", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [248, 9], "def_end_pos": [248, 33]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [79, 29], "def_end_pos": [79, 41]}, {"full_name": "exists_prop", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [307, 17], "def_end_pos": [307, 28]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\n\u22a2 \u03b5 \u2022 \u03bc {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9} \u2264\n    ENNReal.ofReal\n      (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9}, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc)", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis : \u2200 (\u03c9 : \u03a9), (\u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9) \u2192 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\n\u22a2 \u03b5 \u2022 \u03bc {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9} \u2264\n    ENNReal.ofReal\n      (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9}, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc)"}, {"tactic": "have h := setIntegral_ge_of_const_le (measurableSet_le measurable_const\n  (Finset.measurable_range_sup'' fun n _ => (hsub.stronglyMeasurable n).measurable.le (\ud835\udca2.le n)))\n    (measure_ne_top _ _) this (Integrable.integrableOn (hsub.integrable_stoppedValue\n      (hitting_isStoppingTime hsub.adapted measurableSet_Ici) hitting_le))", "annotated_tactic": ["have h := <a>setIntegral_ge_of_const_le</a> (<a>measurableSet_le</a> <a>measurable_const</a>\n    (<a>Finset.measurable_range_sup''</a> fun n _ => (hsub.stronglyMeasurable n).measurable.le (\ud835\udca2.le n)))\n      (<a>measure_ne_top</a> _ _) this (<a>Integrable.integrableOn</a> (hsub.integrable_stoppedValue\n        (<a>hitting_isStoppingTime</a> hsub.adapted <a>measurableSet_Ici</a>) <a>hitting_le</a>))", [{"full_name": "MeasureTheory.setIntegral_ge_of_const_le", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [865, 9], "def_end_pos": [865, 35]}, {"full_name": "measurableSet_le", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean", "def_pos": [167, 9], "def_end_pos": [167, 25]}, {"full_name": "measurable_const", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [586, 9], "def_end_pos": [586, 25]}, {"full_name": "Finset.measurable_range_sup''", "def_path": "Mathlib/MeasureTheory/Order/Lattice.lean", "def_pos": [256, 9], "def_end_pos": [256, 38]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "def_pos": [61, 9], "def_end_pos": [61, 23]}, {"full_name": "MeasureTheory.Integrable.integrableOn", "def_path": "Mathlib/MeasureTheory/Integral/IntegrableOn.lean", "def_pos": [156, 9], "def_end_pos": [156, 32]}, {"full_name": "MeasureTheory.hitting_isStoppingTime", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [231, 9], "def_end_pos": [231, 31]}, {"full_name": "measurableSet_Ici", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean", "def_pos": [111, 9], "def_end_pos": [111, 26]}, {"full_name": "MeasureTheory.hitting_le", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [78, 9], "def_end_pos": [78, 19]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis : \u2200 (\u03c9 : \u03a9), (\u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9) \u2192 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\n\u22a2 \u03b5 \u2022 \u03bc {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9} \u2264\n    ENNReal.ofReal\n      (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9}, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc)", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis : \u2200 (\u03c9 : \u03a9), (\u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9) \u2192 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\nh :\n  \u2191\u03b5 * (\u03bc {a | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k a}).toReal \u2264\n    \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k a}, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc\n\u22a2 \u03b5 \u2022 \u03bc {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9} \u2264\n    ENNReal.ofReal\n      (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9}, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc)"}, {"tactic": "rw [ENNReal.le_ofReal_iff_toReal_le, ENNReal.toReal_smul]", "annotated_tactic": ["rw [<a>ENNReal.le_ofReal_iff_toReal_le</a>, <a>ENNReal.toReal_smul</a>]", [{"full_name": "ENNReal.le_ofReal_iff_toReal_le", "def_path": "Mathlib/Data/ENNReal/Real.lean", "def_pos": [332, 9], "def_end_pos": [332, 32]}, {"full_name": "ENNReal.toReal_smul", "def_path": "Mathlib/Data/ENNReal/Real.lean", "def_pos": [462, 9], "def_end_pos": [462, 20]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis : \u2200 (\u03c9 : \u03a9), (\u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9) \u2192 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\nh :\n  \u2191\u03b5 * (\u03bc {a | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k a}).toReal \u2264\n    \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k a}, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc\n\u22a2 \u03b5 \u2022 \u03bc {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9} \u2264\n    ENNReal.ofReal\n      (\u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9}, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc)", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis : \u2200 (\u03c9 : \u03a9), (\u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9) \u2192 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\nh :\n  \u2191\u03b5 * (\u03bc {a | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k a}).toReal \u2264\n    \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k a}, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc\n\u22a2 \u03b5 \u2022 (\u03bc {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9}).toReal \u2264\n    \u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9}, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc\n\ncase ha\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis : \u2200 (\u03c9 : \u03a9), (\u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9) \u2192 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\nh :\n  \u2191\u03b5 * (\u03bc {a | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k a}).toReal \u2264\n    \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k a}, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc\n\u22a2 \u03b5 \u2022 \u03bc {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9} \u2260 \u22a4\n\ncase hb\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis : \u2200 (\u03c9 : \u03a9), (\u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9) \u2192 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\nh :\n  \u2191\u03b5 * (\u03bc {a | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k a}).toReal \u2264\n    \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k a}, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc\n\u22a2 0 \u2264 \u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9}, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\n\u22a2 Set.Icc 0 n = {k | k \u2264 n}", "state_after": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn x : \u2115\n\u22a2 x \u2208 Set.Icc 0 n \u2194 x \u2208 {k | k \u2264 n}"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn x : \u2115\n\u22a2 x \u2208 Set.Icc 0 n \u2194 x \u2208 {k | k \u2264 n}", "state_after": "no goals"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\n\u22a2 \u2200 (\u03c9 : \u03a9), (\u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9) \u2192 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nx : \u03a9\nhx : \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k x\n\u22a2 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x"}, {"tactic": "simp_rw [le_sup'_iff, mem_range, Nat.lt_succ_iff] at hx", "annotated_tactic": ["simp_rw [<a>le_sup'_iff</a>, <a>mem_range</a>, <a>Nat.lt_succ_iff</a>] at hx", [{"full_name": "Finset.le_sup'_iff", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1302, 9], "def_end_pos": [1302, 20]}, {"full_name": "Finset.mem_range", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2935, 9], "def_end_pos": [2935, 18]}, {"full_name": "Nat.lt_succ_iff", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [573, 19], "def_end_pos": [573, 30]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nx : \u03a9\nhx : \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k x\n\u22a2 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nx : \u03a9\nhx : \u2203 b \u2264 n, \u2191\u03b5 \u2264 f b x\n\u22a2 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x"}, {"tactic": "refine stoppedValue_hitting_mem ?_", "annotated_tactic": ["refine <a>stoppedValue_hitting_mem</a> ?_", [{"full_name": "MeasureTheory.stoppedValue_hitting_mem", "def_path": "Mathlib/Probability/Process/HittingTime.lean", "def_pos": [248, 9], "def_end_pos": [248, 33]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nx : \u03a9\nhx : \u2203 b \u2264 n, \u2191\u03b5 \u2264 f b x\n\u22a2 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nx : \u03a9\nhx : \u2203 b \u2264 n, \u2191\u03b5 \u2264 f b x\n\u22a2 \u2203 j \u2208 Set.Icc 0 n, f j x \u2208 {y | \u2191\u03b5 \u2264 y}"}, {"tactic": "simp only [Set.mem_setOf_eq, exists_prop, hn]", "annotated_tactic": ["simp only [<a>Set.mem_setOf_eq</a>, <a>exists_prop</a>, hn]", [{"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [79, 29], "def_end_pos": [79, 41]}, {"full_name": "exists_prop", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [307, 17], "def_end_pos": [307, 28]}]], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nx : \u03a9\nhx : \u2203 b \u2264 n, \u2191\u03b5 \u2264 f b x\n\u22a2 \u2203 j \u2208 Set.Icc 0 n, f j x \u2208 {y | \u2191\u03b5 \u2264 y}", "state_after": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nx : \u03a9\nhx : \u2203 b \u2264 n, \u2191\u03b5 \u2264 f b x\n\u22a2 \u2203 j \u2264 n, \u2191\u03b5 \u2264 f j x"}, {"tactic": "exact\n  let \u27e8j, hj\u2081, hj\u2082\u27e9 := hx\n  \u27e8j, hj\u2081, hj\u2082\u27e9", "annotated_tactic": ["exact\n      let \u27e8j, hj\u2081, hj\u2082\u27e9 := hx\n      \u27e8j, hj\u2081, hj\u2082\u27e9", []], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nx : \u03a9\nhx : \u2203 b \u2264 n, \u2191\u03b5 \u2264 f b x\n\u22a2 \u2203 j \u2264 n, \u2191\u03b5 \u2264 f j x", "state_after": "no goals"}, {"tactic": "exact h", "annotated_tactic": ["exact h", []], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis : \u2200 (\u03c9 : \u03a9), (\u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9) \u2192 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\nh :\n  \u2191\u03b5 * (\u03bc {a | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k a}).toReal \u2264\n    \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k a}, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc\n\u22a2 \u03b5 \u2022 (\u03bc {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9}).toReal \u2264\n    \u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9}, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc", "state_after": "no goals"}, {"tactic": "exact ENNReal.mul_ne_top (by simp) (measure_ne_top _ _)", "annotated_tactic": ["exact <a>ENNReal.mul_ne_top</a> (by simp) (<a>measure_ne_top</a> _ _)", [{"full_name": "ENNReal.mul_ne_top", "def_path": "Mathlib/Data/ENNReal/Operations.lean", "def_pos": [232, 9], "def_end_pos": [232, 19]}, {"full_name": "MeasureTheory.measure_ne_top", "def_path": "Mathlib/MeasureTheory/Measure/Typeclasses.lean", "def_pos": [61, 9], "def_end_pos": [61, 23]}]], "state_before": "case ha\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis : \u2200 (\u03c9 : \u03a9), (\u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9) \u2192 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\nh :\n  \u2191\u03b5 * (\u03bc {a | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k a}).toReal \u2264\n    \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k a}, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc\n\u22a2 \u03b5 \u2022 \u03bc {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9} \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis : \u2200 (\u03c9 : \u03a9), (\u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9) \u2192 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\nh :\n  \u2191\u03b5 * (\u03bc {a | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k a}).toReal \u2264\n    \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k a}, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc\n\u22a2 \u2191ENNReal.ofNNRealHom.toMonoidWithZeroHom \u03b5 \u2260 \u22a4", "state_after": "no goals"}, {"tactic": "exact le_trans (mul_nonneg \u03b5.coe_nonneg ENNReal.toReal_nonneg) h", "annotated_tactic": ["exact <a>le_trans</a> (<a>mul_nonneg</a> \u03b5.coe_nonneg <a>ENNReal.toReal_nonneg</a>) h", [{"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "mul_nonneg", "def_path": "Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean", "def_pos": [437, 7], "def_end_pos": [437, 17]}, {"full_name": "ENNReal.toReal_nonneg", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [247, 17], "def_end_pos": [247, 30]}]], "state_before": "case hb\n\u03a9 : Type u_1\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\n\ud835\udca2 : Filtration \u2115 m0\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\n\u03c4 \u03c0 : \u03a9 \u2192 \u2115\ninst\u271d : IsFiniteMeasure \u03bc\nhsub : Submartingale f \ud835\udca2 \u03bc\n\u03b5 : \u211d\u22650\nn : \u2115\nhn : Set.Icc 0 n = {k | k \u2264 n}\nthis : \u2200 (\u03c9 : \u03a9), (\u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9) \u2192 \u2191\u03b5 \u2264 stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9\nh :\n  \u2191\u03b5 * (\u03bc {a | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k a}).toReal \u2264\n    \u222b (x : \u03a9) in {a | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k a}, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) x \u2202\u03bc\n\u22a2 0 \u2264 \u222b (\u03c9 : \u03a9) in {\u03c9 | \u2191\u03b5 \u2264 (range (n + 1)).sup' \u22ef fun k => f k \u03c9}, stoppedValue f (hitting f {y | \u2191\u03b5 \u2264 y} 0 n) \u03c9 \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "full_name": "Matrix.mul_transvection_apply_same", "start": [125, 1], "end": [127, 48], "traced_tactics": [{"tactic": "simp [transvection, Matrix.mul_add, mul_comm]", "annotated_tactic": ["simp [<a>transvection</a>, <a>Matrix.mul_add</a>, <a>mul_comm</a>]", [{"full_name": "Matrix.transvection", "def_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "def_pos": [82, 5], "def_end_pos": [82, 17]}, {"full_name": "Matrix.mul_add", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [1049, 19], "def_end_pos": [1049, 26]}, {"full_name": "mul_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [332, 9], "def_end_pos": [332, 17]}]], "state_before": "n : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2074 : Field \ud835\udd5c\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : DecidableEq p\ninst\u271d\u00b9 : CommRing R\ni j : n\ninst\u271d : Fintype n\na : n\nc : R\nM : Matrix n n R\n\u22a2 (M * transvection i j c) a j = M a j + c * M a i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/FiberedCategory/BasedCategory.lean", "full_name": "CategoryTheory.BasedFunctor.w_obj", "start": [98, 1], "end": [100, 31], "traced_tactics": [{"tactic": "rw [\u2190 Functor.comp_obj, F.w]", "annotated_tactic": ["rw [\u2190 <a>Functor.comp_obj</a>, F.w]", [{"full_name": "CategoryTheory.Functor.comp_obj", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [110, 9], "def_end_pos": [110, 12]}]], "state_before": "\ud835\udcae : Type u\u2081\ninst\u271d : Category.{v\u2081, u\u2081} \ud835\udcae\n\ud835\udcb3 : BasedCategory \ud835\udcae\n\ud835\udcb4 : BasedCategory \ud835\udcae\nF : \ud835\udcb3 \u2964\u1d47 \ud835\udcb4\na : \ud835\udcb3.obj\n\u22a2 \ud835\udcb4.p.obj (F.obj a) = \ud835\udcb3.p.obj a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Factorization/Root.lean", "full_name": "Nat.floorRoot_pow_dvd", "start": [95, 1], "end": [95, 87], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Basis.lean", "full_name": "Basis.mem_submodule_iff", "start": [582, 1], "end": [587, 58], "traced_tactics": [{"tactic": "conv_lhs =>\n  rw [\u2190 P.range_subtype, \u2190 Submodule.map_top, \u2190 b.span_eq, Submodule.map_span, \u2190 Set.range_comp,\n      \u2190 Finsupp.range_total]", "annotated_tactic": ["conv_lhs =>\n    rw [\u2190 P.range_subtype, \u2190 <a>Submodule.map_top</a>, \u2190 b.span_eq, <a>Submodule.map_span</a>, \u2190 <a>Set.range_comp</a>,\n        \u2190 <a>Finsupp.range_total</a>]", [{"full_name": "Submodule.map_top", "def_path": "Mathlib/Algebra/Module/Submodule/Range.lean", "def_pos": [290, 9], "def_end_pos": [290, 16]}, {"full_name": "Submodule.map_span", "def_path": "Mathlib/LinearAlgebra/Span.lean", "def_pos": [132, 9], "def_end_pos": [132, 17]}, {"full_name": "Set.range_comp", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [731, 9], "def_end_pos": [731, 19]}, {"full_name": "Finsupp.range_total", "def_path": "Mathlib/LinearAlgebra/Finsupp.lean", "def_pos": [723, 9], "def_end_pos": [723, 20]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nR : Type u_3\nR\u2082 : Type u_4\nK : Type u_5\nM : Type u_6\nM' : Type u_7\nM'' : Type u_8\nV : Type u\nV' : Type u_9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nb\u271d b\u2081 : Basis \u03b9 R M\ni : \u03b9\nc : R\nx\u271d : M\nP : Submodule R M\nb : Basis \u03b9 R \u21a5P\nx : M\n\u22a2 x \u2208 P \u2194 \u2203 c, x = c.sum fun i x => x \u2022 \u2191(b i)", "state_after": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nR : Type u_3\nR\u2082 : Type u_4\nK : Type u_5\nM : Type u_6\nM' : Type u_7\nM'' : Type u_8\nV : Type u\nV' : Type u_9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nb\u271d b\u2081 : Basis \u03b9 R M\ni : \u03b9\nc : R\nx\u271d : M\nP : Submodule R M\nb : Basis \u03b9 R \u21a5P\nx : M\n\u22a2 x \u2208 LinearMap.range (Finsupp.total \u03b9 M R (\u21d1P.subtype \u2218 \u21d1b)) \u2194 \u2203 c, x = c.sum fun i x => x \u2022 \u2191(b i)"}, {"tactic": "simp [@eq_comm _ x, Function.comp, Finsupp.total_apply]", "annotated_tactic": ["simp [@<a>eq_comm</a> _ x, <a>Function.comp</a>, <a>Finsupp.total_apply</a>]", [{"full_name": "eq_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [811, 9], "def_end_pos": [811, 16]}, {"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": "Finsupp.total_apply", "def_path": "Mathlib/LinearAlgebra/Finsupp.lean", "def_pos": [664, 9], "def_end_pos": [664, 20]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\nR : Type u_3\nR\u2082 : Type u_4\nK : Type u_5\nM : Type u_6\nM' : Type u_7\nM'' : Type u_8\nV : Type u\nV' : Type u_9\ninst\u271d\u2074 : Semiring R\ninst\u271d\u00b3 : AddCommMonoid M\ninst\u271d\u00b2 : Module R M\ninst\u271d\u00b9 : AddCommMonoid M'\ninst\u271d : Module R M'\nb\u271d b\u2081 : Basis \u03b9 R M\ni : \u03b9\nc : R\nx\u271d : M\nP : Submodule R M\nb : Basis \u03b9 R \u21a5P\nx : M\n\u22a2 x \u2208 LinearMap.range (Finsupp.total \u03b9 M R (\u21d1P.subtype \u2218 \u21d1b)) \u2194 \u2203 c, x = c.sum fun i x => x \u2022 \u2191(b i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/HahnSeries/Summable.lean", "full_name": "HahnSeries.isPWO_iUnion_support_powers", "start": [96, 1], "end": [106, 94], "traced_tactics": [{"tactic": "apply (x.isWF_support.isPWO.addSubmonoid_closure _).mono _", "annotated_tactic": ["apply (x.isWF_support.isPWO.addSubmonoid_closure _).<a>mono</a> _", [{"full_name": "Set.IsPWO.mono", "def_path": "Mathlib/Order/WellFoundedSet.lean", "def_pos": [436, 16], "def_end_pos": [436, 26]}]], "state_before": "\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < (addVal \u0393 R) x\n\u22a2 (\u22c3 n, (x ^ n).support).IsPWO", "state_after": "\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < (addVal \u0393 R) x\n\u22a2 \u2200 x_1 \u2208 x.support, 0 \u2264 x_1\n\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < (addVal \u0393 R) x\n\u22a2 \u22c3 n, (x ^ n).support \u2286 \u2191(AddSubmonoid.closure x.support)"}, {"tactic": "refine Set.iUnion_subset fun n => ?_", "annotated_tactic": ["refine <a>Set.iUnion_subset</a> fun n => ?_", [{"full_name": "Set.iUnion_subset", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [237, 9], "def_end_pos": [237, 22]}]], "state_before": "\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < (addVal \u0393 R) x\n\u22a2 \u22c3 n, (x ^ n).support \u2286 \u2191(AddSubmonoid.closure x.support)", "state_after": "\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < (addVal \u0393 R) x\nn : \u2115\n\u22a2 (x ^ n).support \u2286 \u2191(AddSubmonoid.closure x.support)"}, {"tactic": "induction' n with n ih <;> intro g hn", "annotated_tactic": ["induction' n with n ih <;> intro g hn", []], "state_before": "\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < (addVal \u0393 R) x\nn : \u2115\n\u22a2 (x ^ n).support \u2286 \u2191(AddSubmonoid.closure x.support)", "state_after": "case zero\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < (addVal \u0393 R) x\ng : \u0393\nhn : g \u2208 (x ^ 0).support\n\u22a2 g \u2208 \u2191(AddSubmonoid.closure x.support)\n\ncase succ\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < (addVal \u0393 R) x\nn : \u2115\nih : (x ^ n).support \u2286 \u2191(AddSubmonoid.closure x.support)\ng : \u0393\nhn : g \u2208 (x ^ (n + 1)).support\n\u22a2 g \u2208 \u2191(AddSubmonoid.closure x.support)"}, {"tactic": "exact fun g hg => WithTop.coe_le_coe.1 (le_trans (le_of_lt hx) (addVal_le_of_coeff_ne_zero hg))", "annotated_tactic": ["exact fun g hg => <a>WithTop.coe_le_coe</a>.1 (<a>le_trans</a> (<a>le_of_lt</a> hx) (<a>addVal_le_of_coeff_ne_zero</a> hg))", [{"full_name": "WithTop.coe_le_coe", "def_path": "Mathlib/Order/WithBot.lean", "def_pos": [898, 9], "def_end_pos": [898, 19]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}, {"full_name": "HahnSeries.addVal_le_of_coeff_ne_zero", "def_path": "Mathlib/RingTheory/HahnSeries/Summable.lean", "def_pos": [89, 9], "def_end_pos": [89, 35]}]], "state_before": "\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < (addVal \u0393 R) x\n\u22a2 \u2200 x_1 \u2208 x.support, 0 \u2264 x_1", "state_after": "no goals"}, {"tactic": "simp only [Nat.zero_eq, pow_zero, support_one, Set.mem_singleton_iff] at hn", "annotated_tactic": ["simp only [<a>Nat.zero_eq</a>, <a>pow_zero</a>, <a>support_one</a>, <a>Set.mem_singleton_iff</a>] at hn", [{"full_name": "Nat.zero_eq", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [106, 17], "def_end_pos": [106, 24]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [651, 9], "def_end_pos": [651, 17]}, {"full_name": "HahnSeries.support_one", "def_path": "Mathlib/RingTheory/HahnSeries/Multiplication.lean", "def_pos": [60, 9], "def_end_pos": [60, 20]}, {"full_name": "Set.mem_singleton_iff", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1230, 9], "def_end_pos": [1230, 26]}]], "state_before": "case zero\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < (addVal \u0393 R) x\ng : \u0393\nhn : g \u2208 (x ^ 0).support\n\u22a2 g \u2208 \u2191(AddSubmonoid.closure x.support)", "state_after": "case zero\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < (addVal \u0393 R) x\ng : \u0393\nhn : g = 0\n\u22a2 g \u2208 \u2191(AddSubmonoid.closure x.support)"}, {"tactic": "rw [hn, SetLike.mem_coe]", "annotated_tactic": ["rw [hn, <a>SetLike.mem_coe</a>]", [{"full_name": "SetLike.mem_coe", "def_path": "Mathlib/Data/SetLike/Basic.lean", "def_pos": [180, 9], "def_end_pos": [180, 16]}]], "state_before": "case zero\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < (addVal \u0393 R) x\ng : \u0393\nhn : g = 0\n\u22a2 g \u2208 \u2191(AddSubmonoid.closure x.support)", "state_after": "case zero\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < (addVal \u0393 R) x\ng : \u0393\nhn : g = 0\n\u22a2 0 \u2208 AddSubmonoid.closure x.support"}, {"tactic": "exact AddSubmonoid.zero_mem _", "annotated_tactic": ["exact <a>AddSubmonoid.zero_mem</a> _", [{"full_name": "AddSubmonoid.zero_mem", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [226, 3], "def_end_pos": [226, 14]}]], "state_before": "case zero\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < (addVal \u0393 R) x\ng : \u0393\nhn : g = 0\n\u22a2 0 \u2208 AddSubmonoid.closure x.support", "state_after": "no goals"}, {"tactic": "obtain \u27e8i, hi, j, hj, rfl\u27e9 := support_mul_subset_add_support hn", "annotated_tactic": ["obtain \u27e8i, hi, j, hj, rfl\u27e9 := <a>support_mul_subset_add_support</a> hn", [{"full_name": "HahnSeries.support_mul_subset_add_support", "def_path": "Mathlib/RingTheory/HahnSeries/Multiplication.lean", "def_pos": [311, 9], "def_end_pos": [311, 39]}]], "state_before": "case succ\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < (addVal \u0393 R) x\nn : \u2115\nih : (x ^ n).support \u2286 \u2191(AddSubmonoid.closure x.support)\ng : \u0393\nhn : g \u2208 (x ^ (n + 1)).support\n\u22a2 g \u2208 \u2191(AddSubmonoid.closure x.support)", "state_after": "case succ.intro.intro.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < (addVal \u0393 R) x\nn : \u2115\nih : (x ^ n).support \u2286 \u2191(AddSubmonoid.closure x.support)\ni : \u0393\nhi : i \u2208 (npowRec n x).support\nj : \u0393\nhj : j \u2208 x.support\nhn : (fun x x_1 => x + x_1) i j \u2208 (x ^ (n + 1)).support\n\u22a2 (fun x x_1 => x + x_1) i j \u2208 \u2191(AddSubmonoid.closure x.support)"}, {"tactic": "exact SetLike.mem_coe.2 (AddSubmonoid.add_mem _ (ih hi) (AddSubmonoid.subset_closure hj))", "annotated_tactic": ["exact <a>SetLike.mem_coe</a>.2 (<a>AddSubmonoid.add_mem</a> _ (ih hi) (<a>AddSubmonoid.subset_closure</a> hj))", [{"full_name": "SetLike.mem_coe", "def_path": "Mathlib/Data/SetLike/Basic.lean", "def_pos": [180, 9], "def_end_pos": [180, 16]}, {"full_name": "AddSubmonoid.add_mem", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [233, 3], "def_end_pos": [233, 14]}, {"full_name": "AddSubmonoid.subset_closure", "def_path": "Mathlib/Algebra/Group/Submonoid/Basic.lean", "def_pos": [397, 3], "def_end_pos": [397, 14]}]], "state_before": "case succ.intro.intro.intro.intro\n\u0393 : Type u_1\nR : Type u_2\ninst\u271d\u00b2 : LinearOrderedCancelAddCommMonoid \u0393\ninst\u271d\u00b9 : Ring R\ninst\u271d : IsDomain R\nx : HahnSeries \u0393 R\nhx : 0 < (addVal \u0393 R) x\nn : \u2115\nih : (x ^ n).support \u2286 \u2191(AddSubmonoid.closure x.support)\ni : \u0393\nhi : i \u2208 (npowRec n x).support\nj : \u0393\nhj : j \u2208 x.support\nhn : (fun x x_1 => x + x_1) i j \u2208 (x ^ (n + 1)).support\n\u22a2 (fun x x_1 => x + x_1) i j \u2208 \u2191(AddSubmonoid.closure x.support)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Instances/AddCircle.lean", "full_name": "AddCircle.homeomorphAddCircle_symm_apply_mk", "start": [372, 1], "end": [374, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/EssentiallySmall.lean", "full_name": "CategoryTheory.Limits.hasLimitsOfShape_of_essentiallySmall", "start": [31, 1], "end": [33, 80], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean", "full_name": "DifferentiableAt.sinh", "start": [1146, 1], "end": [1148, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/Equicontinuity.lean", "full_name": "equicontinuous_empty", "start": [227, 1], "end": [230, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Closed/Ideal.lean", "full_name": "CategoryTheory.bijection_natural", "start": [224, 1], "end": [235, 68], "traced_tactics": [{"tactic": "dsimp [bijection]", "annotated_tactic": ["dsimp [<a>bijection</a>]", [{"full_name": "CategoryTheory.bijection", "def_path": "Mathlib/CategoryTheory/Closed/Ideal.lean", "def_pos": [180, 19], "def_end_pos": [180, 28]}]], "state_before": "C : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2075 : Category.{v\u2081, u\u2082} D\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nX X' : D\nf : (reflector i).obj (A \u2a2f B) \u27f6 X\ng : X \u27f6 X'\n\u22a2 (bijection i A B X') (f \u226b g) = (bijection i A B X) f \u226b g", "state_after": "C : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2075 : Category.{v\u2081, u\u2082} D\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nX X' : D\nf : (reflector i).obj (A \u2a2f B) \u27f6 X\ng : X \u27f6 X'\n\u22a2 i.fullyFaithfulOfReflective.preimage\n      (prodComparison i ((reflector i).obj A) ((reflector i).obj B) \u226b\n        ((exp.adjunction (i.obj ((reflector i).obj A))).homEquiv (i.obj ((reflector i).obj B)) (i.obj X')).symm\n            ((unitCompPartialBijective B \u22ef)\n              (((exp.adjunction (i.obj ((reflector i).obj A))).homEquiv B (i.obj X'))\n                (prod.lift prod.snd prod.fst \u226b\n                  ((exp.adjunction B).homEquiv (i.obj ((reflector i).obj A)) (i.obj X')).symm\n                      ((unitCompPartialBijective A \u22ef)\n                        (((exp.adjunction B).homEquiv A (i.obj X'))\n                          (prod.lift prod.snd prod.fst \u226b\n                            ((reflectorAdjunction i).homEquiv (A \u2a2f B) X') (f \u226b g) \u226b \ud835\udfd9 (i.obj X')))) \u226b\n                    \ud835\udfd9 (i.obj X')))) \u226b\n          \ud835\udfd9 (i.obj X')) =\n    i.fullyFaithfulOfReflective.preimage\n        (prodComparison i ((reflector i).obj A) ((reflector i).obj B) \u226b\n          ((exp.adjunction (i.obj ((reflector i).obj A))).homEquiv (i.obj ((reflector i).obj B)) (i.obj X)).symm\n              ((unitCompPartialBijective B \u22ef)\n                (((exp.adjunction (i.obj ((reflector i).obj A))).homEquiv B (i.obj X))\n                  (prod.lift prod.snd prod.fst \u226b\n                    ((exp.adjunction B).homEquiv (i.obj ((reflector i).obj A)) (i.obj X)).symm\n                        ((unitCompPartialBijective A \u22ef)\n                          (((exp.adjunction B).homEquiv A (i.obj X))\n                            (prod.lift prod.snd prod.fst \u226b\n                              ((reflectorAdjunction i).homEquiv (A \u2a2f B) X) f \u226b \ud835\udfd9 (i.obj X)))) \u226b\n                      \ud835\udfd9 (i.obj X)))) \u226b\n            \ud835\udfd9 (i.obj X)) \u226b\n      g"}, {"tactic": "erw [homEquiv_symm_apply_eq, homEquiv_symm_apply_eq, homEquiv_apply_eq, homEquiv_apply_eq,\n  homEquiv_symm_apply_eq, homEquiv_symm_apply_eq, homEquiv_apply_eq, homEquiv_apply_eq]", "annotated_tactic": ["erw [<a>homEquiv_symm_apply_eq</a>, <a>homEquiv_symm_apply_eq</a>, <a>homEquiv_apply_eq</a>, <a>homEquiv_apply_eq</a>,\n    <a>homEquiv_symm_apply_eq</a>, <a>homEquiv_symm_apply_eq</a>, <a>homEquiv_apply_eq</a>, <a>homEquiv_apply_eq</a>]", [{"full_name": "CategoryTheory.CartesianClosed.homEquiv_symm_apply_eq", "def_path": "Mathlib/CategoryTheory/Closed/Cartesian.lean", "def_pos": [181, 9], "def_end_pos": [181, 31]}, {"full_name": "CategoryTheory.CartesianClosed.homEquiv_symm_apply_eq", "def_path": "Mathlib/CategoryTheory/Closed/Cartesian.lean", "def_pos": [181, 9], "def_end_pos": [181, 31]}, {"full_name": "CategoryTheory.CartesianClosed.homEquiv_apply_eq", "def_path": "Mathlib/CategoryTheory/Closed/Cartesian.lean", "def_pos": [175, 9], "def_end_pos": [175, 26]}, {"full_name": "CategoryTheory.CartesianClosed.homEquiv_apply_eq", "def_path": "Mathlib/CategoryTheory/Closed/Cartesian.lean", "def_pos": [175, 9], "def_end_pos": [175, 26]}, {"full_name": "CategoryTheory.CartesianClosed.homEquiv_symm_apply_eq", "def_path": "Mathlib/CategoryTheory/Closed/Cartesian.lean", "def_pos": [181, 9], "def_end_pos": [181, 31]}, {"full_name": "CategoryTheory.CartesianClosed.homEquiv_symm_apply_eq", "def_path": "Mathlib/CategoryTheory/Closed/Cartesian.lean", "def_pos": [181, 9], "def_end_pos": [181, 31]}, {"full_name": "CategoryTheory.CartesianClosed.homEquiv_apply_eq", "def_path": "Mathlib/CategoryTheory/Closed/Cartesian.lean", "def_pos": [175, 9], "def_end_pos": [175, 26]}, {"full_name": "CategoryTheory.CartesianClosed.homEquiv_apply_eq", "def_path": "Mathlib/CategoryTheory/Closed/Cartesian.lean", "def_pos": [175, 9], "def_end_pos": [175, 26]}]], "state_before": "C : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2075 : Category.{v\u2081, u\u2082} D\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nX X' : D\nf : (reflector i).obj (A \u2a2f B) \u27f6 X\ng : X \u27f6 X'\n\u22a2 i.fullyFaithfulOfReflective.preimage\n      (prodComparison i ((reflector i).obj A) ((reflector i).obj B) \u226b\n        ((exp.adjunction (i.obj ((reflector i).obj A))).homEquiv (i.obj ((reflector i).obj B)) (i.obj X')).symm\n            ((unitCompPartialBijective B \u22ef)\n              (((exp.adjunction (i.obj ((reflector i).obj A))).homEquiv B (i.obj X'))\n                (prod.lift prod.snd prod.fst \u226b\n                  ((exp.adjunction B).homEquiv (i.obj ((reflector i).obj A)) (i.obj X')).symm\n                      ((unitCompPartialBijective A \u22ef)\n                        (((exp.adjunction B).homEquiv A (i.obj X'))\n                          (prod.lift prod.snd prod.fst \u226b\n                            ((reflectorAdjunction i).homEquiv (A \u2a2f B) X') (f \u226b g) \u226b \ud835\udfd9 (i.obj X')))) \u226b\n                    \ud835\udfd9 (i.obj X')))) \u226b\n          \ud835\udfd9 (i.obj X')) =\n    i.fullyFaithfulOfReflective.preimage\n        (prodComparison i ((reflector i).obj A) ((reflector i).obj B) \u226b\n          ((exp.adjunction (i.obj ((reflector i).obj A))).homEquiv (i.obj ((reflector i).obj B)) (i.obj X)).symm\n              ((unitCompPartialBijective B \u22ef)\n                (((exp.adjunction (i.obj ((reflector i).obj A))).homEquiv B (i.obj X))\n                  (prod.lift prod.snd prod.fst \u226b\n                    ((exp.adjunction B).homEquiv (i.obj ((reflector i).obj A)) (i.obj X)).symm\n                        ((unitCompPartialBijective A \u22ef)\n                          (((exp.adjunction B).homEquiv A (i.obj X))\n                            (prod.lift prod.snd prod.fst \u226b\n                              ((reflectorAdjunction i).homEquiv (A \u2a2f B) X) f \u226b \ud835\udfd9 (i.obj X)))) \u226b\n                      \ud835\udfd9 (i.obj X)))) \u226b\n            \ud835\udfd9 (i.obj X)) \u226b\n      g", "state_after": "C : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2075 : Category.{v\u2081, u\u2082} D\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nX X' : D\nf : (reflector i).obj (A \u2a2f B) \u27f6 X\ng : X \u27f6 X'\n\u22a2 i.fullyFaithfulOfReflective.preimage\n      (prodComparison i ((reflector i).obj A) ((reflector i).obj B) \u226b\n        uncurry\n            ((unitCompPartialBijective B \u22ef)\n              (curry\n                (prod.lift prod.snd prod.fst \u226b\n                  uncurry\n                      ((unitCompPartialBijective A \u22ef)\n                        (curry\n                          (prod.lift prod.snd prod.fst \u226b\n                            ((reflectorAdjunction i).homEquiv (A \u2a2f B) X') (f \u226b g) \u226b \ud835\udfd9 (i.obj X')))) \u226b\n                    \ud835\udfd9 (i.obj X')))) \u226b\n          \ud835\udfd9 (i.obj X')) =\n    i.fullyFaithfulOfReflective.preimage\n        (prodComparison i ((reflector i).obj A) ((reflector i).obj B) \u226b\n          uncurry\n              ((unitCompPartialBijective B \u22ef)\n                (curry\n                  (prod.lift prod.snd prod.fst \u226b\n                    uncurry\n                        ((unitCompPartialBijective A \u22ef)\n                          (curry\n                            (prod.lift prod.snd prod.fst \u226b\n                              ((reflectorAdjunction i).homEquiv (A \u2a2f B) X) f \u226b \ud835\udfd9 (i.obj X)))) \u226b\n                      \ud835\udfd9 (i.obj X)))) \u226b\n            \ud835\udfd9 (i.obj X)) \u226b\n      g"}, {"tactic": "apply i.map_injective", "annotated_tactic": ["apply i.map_injective", []], "state_before": "C : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2075 : Category.{v\u2081, u\u2082} D\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nX X' : D\nf : (reflector i).obj (A \u2a2f B) \u27f6 X\ng : X \u27f6 X'\n\u22a2 i.fullyFaithfulOfReflective.preimage\n      (prodComparison i ((reflector i).obj A) ((reflector i).obj B) \u226b\n        uncurry\n            ((unitCompPartialBijective B \u22ef)\n              (curry\n                (prod.lift prod.snd prod.fst \u226b\n                  uncurry\n                      ((unitCompPartialBijective A \u22ef)\n                        (curry\n                          (prod.lift prod.snd prod.fst \u226b\n                            ((reflectorAdjunction i).homEquiv (A \u2a2f B) X') (f \u226b g) \u226b \ud835\udfd9 (i.obj X')))) \u226b\n                    \ud835\udfd9 (i.obj X')))) \u226b\n          \ud835\udfd9 (i.obj X')) =\n    i.fullyFaithfulOfReflective.preimage\n        (prodComparison i ((reflector i).obj A) ((reflector i).obj B) \u226b\n          uncurry\n              ((unitCompPartialBijective B \u22ef)\n                (curry\n                  (prod.lift prod.snd prod.fst \u226b\n                    uncurry\n                        ((unitCompPartialBijective A \u22ef)\n                          (curry\n                            (prod.lift prod.snd prod.fst \u226b\n                              ((reflectorAdjunction i).homEquiv (A \u2a2f B) X) f \u226b \ud835\udfd9 (i.obj X)))) \u226b\n                      \ud835\udfd9 (i.obj X)))) \u226b\n            \ud835\udfd9 (i.obj X)) \u226b\n      g", "state_after": "case a\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2075 : Category.{v\u2081, u\u2082} D\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nX X' : D\nf : (reflector i).obj (A \u2a2f B) \u27f6 X\ng : X \u27f6 X'\n\u22a2 i.map\n      (i.fullyFaithfulOfReflective.preimage\n        (prodComparison i ((reflector i).obj A) ((reflector i).obj B) \u226b\n          uncurry\n              ((unitCompPartialBijective B \u22ef)\n                (curry\n                  (prod.lift prod.snd prod.fst \u226b\n                    uncurry\n                        ((unitCompPartialBijective A \u22ef)\n                          (curry\n                            (prod.lift prod.snd prod.fst \u226b\n                              ((reflectorAdjunction i).homEquiv (A \u2a2f B) X') (f \u226b g) \u226b \ud835\udfd9 (i.obj X')))) \u226b\n                      \ud835\udfd9 (i.obj X')))) \u226b\n            \ud835\udfd9 (i.obj X'))) =\n    i.map\n      (i.fullyFaithfulOfReflective.preimage\n          (prodComparison i ((reflector i).obj A) ((reflector i).obj B) \u226b\n            uncurry\n                ((unitCompPartialBijective B \u22ef)\n                  (curry\n                    (prod.lift prod.snd prod.fst \u226b\n                      uncurry\n                          ((unitCompPartialBijective A \u22ef)\n                            (curry\n                              (prod.lift prod.snd prod.fst \u226b\n                                ((reflectorAdjunction i).homEquiv (A \u2a2f B) X) f \u226b \ud835\udfd9 (i.obj X)))) \u226b\n                        \ud835\udfd9 (i.obj X)))) \u226b\n              \ud835\udfd9 (i.obj X)) \u226b\n        g)"}, {"tactic": "rw [Functor.FullyFaithful.map_preimage, i.map_comp, Functor.FullyFaithful.map_preimage,\n  comp_id, comp_id, comp_id, comp_id, comp_id,\n  comp_id, Adjunction.homEquiv_naturality_right, \u2190 assoc, curry_natural_right _ (i.map g),\n  unitCompPartialBijective_natural, uncurry_natural_right, \u2190 assoc, curry_natural_right,\n  unitCompPartialBijective_natural, uncurry_natural_right, assoc]", "annotated_tactic": ["rw [<a>Functor.FullyFaithful.map_preimage</a>, i.map_comp, <a>Functor.FullyFaithful.map_preimage</a>,\n    <a>comp_id</a>, <a>comp_id</a>, <a>comp_id</a>, <a>comp_id</a>, <a>comp_id</a>,\n    <a>comp_id</a>, <a>Adjunction.homEquiv_naturality_right</a>, \u2190 <a>assoc</a>, <a>curry_natural_right</a> _ (i.map g),\n    <a>unitCompPartialBijective_natural</a>, <a>uncurry_natural_right</a>, \u2190 <a>assoc</a>, <a>curry_natural_right</a>,\n    <a>unitCompPartialBijective_natural</a>, <a>uncurry_natural_right</a>, <a>assoc</a>]", [{"full_name": "CategoryTheory.Functor.FullyFaithful.map_preimage", "def_path": "Mathlib/CategoryTheory/Functor/FullyFaithful.lean", "def_pos": [135, 3], "def_end_pos": [135, 15]}, {"full_name": "CategoryTheory.Functor.FullyFaithful.map_preimage", "def_path": "Mathlib/CategoryTheory/Functor/FullyFaithful.lean", "def_pos": [135, 3], "def_end_pos": [135, 15]}, {"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [179, 3], "def_end_pos": [179, 10]}, {"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [179, 3], "def_end_pos": [179, 10]}, {"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [179, 3], "def_end_pos": [179, 10]}, {"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [179, 3], "def_end_pos": [179, 10]}, {"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [179, 3], "def_end_pos": [179, 10]}, {"full_name": "CategoryTheory.Category.comp_id", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [179, 3], "def_end_pos": [179, 10]}, {"full_name": "CategoryTheory.Adjunction.homEquiv_naturality_right", "def_path": "Mathlib/CategoryTheory/Adjunction/Basic.lean", "def_pos": [172, 9], "def_end_pos": [172, 34]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.CartesianClosed.curry_natural_right", "def_path": "Mathlib/CategoryTheory/Closed/Cartesian.lean", "def_pos": [193, 9], "def_end_pos": [193, 28]}, {"full_name": "CategoryTheory.unitCompPartialBijective_natural", "def_path": "Mathlib/CategoryTheory/Adjunction/Reflective.lean", "def_pos": [159, 9], "def_end_pos": [159, 41]}, {"full_name": "CategoryTheory.CartesianClosed.uncurry_natural_right", "def_path": "Mathlib/CategoryTheory/Closed/Cartesian.lean", "def_pos": [199, 9], "def_end_pos": [199, 30]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.CartesianClosed.curry_natural_right", "def_path": "Mathlib/CategoryTheory/Closed/Cartesian.lean", "def_pos": [193, 9], "def_end_pos": [193, 28]}, {"full_name": "CategoryTheory.unitCompPartialBijective_natural", "def_path": "Mathlib/CategoryTheory/Adjunction/Reflective.lean", "def_pos": [159, 9], "def_end_pos": [159, 41]}, {"full_name": "CategoryTheory.CartesianClosed.uncurry_natural_right", "def_path": "Mathlib/CategoryTheory/Closed/Cartesian.lean", "def_pos": [199, 9], "def_end_pos": [199, 30]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}]], "state_before": "case a\nC : Type u\u2081\nD : Type u\u2082\ninst\u271d\u2076 : Category.{v\u2081, u\u2081} C\ninst\u271d\u2075 : Category.{v\u2081, u\u2082} D\ni : D \u2964 C\ninst\u271d\u2074 : HasFiniteProducts C\ninst\u271d\u00b3 : Reflective i\ninst\u271d\u00b2 : CartesianClosed C\ninst\u271d\u00b9 : HasFiniteProducts D\ninst\u271d : ExponentialIdeal i\nA B : C\nX X' : D\nf : (reflector i).obj (A \u2a2f B) \u27f6 X\ng : X \u27f6 X'\n\u22a2 i.map\n      (i.fullyFaithfulOfReflective.preimage\n        (prodComparison i ((reflector i).obj A) ((reflector i).obj B) \u226b\n          uncurry\n              ((unitCompPartialBijective B \u22ef)\n                (curry\n                  (prod.lift prod.snd prod.fst \u226b\n                    uncurry\n                        ((unitCompPartialBijective A \u22ef)\n                          (curry\n                            (prod.lift prod.snd prod.fst \u226b\n                              ((reflectorAdjunction i).homEquiv (A \u2a2f B) X') (f \u226b g) \u226b \ud835\udfd9 (i.obj X')))) \u226b\n                      \ud835\udfd9 (i.obj X')))) \u226b\n            \ud835\udfd9 (i.obj X'))) =\n    i.map\n      (i.fullyFaithfulOfReflective.preimage\n          (prodComparison i ((reflector i).obj A) ((reflector i).obj B) \u226b\n            uncurry\n                ((unitCompPartialBijective B \u22ef)\n                  (curry\n                    (prod.lift prod.snd prod.fst \u226b\n                      uncurry\n                          ((unitCompPartialBijective A \u22ef)\n                            (curry\n                              (prod.lift prod.snd prod.fst \u226b\n                                ((reflectorAdjunction i).homEquiv (A \u2a2f B) X) f \u226b \ud835\udfd9 (i.obj X)))) \u226b\n                        \ud835\udfd9 (i.obj X)))) \u226b\n              \ud835\udfd9 (i.obj X)) \u226b\n        g)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Bicategory/Coherence.lean", "full_name": "CategoryTheory.FreeBicategory.preinclusion_map\u2082", "start": [94, 1], "end": [98, 39], "traced_tactics": [{"tactic": "rcases \u03b7 with \u27e8\u27e8\u27e9\u27e9", "annotated_tactic": ["rcases \u03b7 with \u27e8\u27e8\u27e9\u27e9", []], "state_before": "B : Type u\ninst\u271d : Quiver B\na b : B\nf g : Discrete (Path a b)\n\u03b7 : f \u27f6 g\n\u22a2 (preinclusion B).map\u2082 \u03b7 = eqToHom \u22ef", "state_after": "case up.up\nB : Type u\ninst\u271d : Quiver B\na b : B\nf g : Discrete (Path a b)\ndown\u271d : f.as = g.as\n\u22a2 (preinclusion B).map\u2082 { down := { down := down\u271d } } = eqToHom \u22ef"}, {"tactic": "cases Discrete.ext _ _ (by assumption)", "annotated_tactic": ["cases <a>Discrete.ext</a> _ _ (by assumption)", [{"full_name": "CategoryTheory.Discrete.ext", "def_path": "Mathlib/CategoryTheory/DiscreteCategory.lean", "def_pos": [48, 3], "def_end_pos": [48, 6]}]], "state_before": "case up.up\nB : Type u\ninst\u271d : Quiver B\na b : B\nf g : Discrete (Path a b)\ndown\u271d : f.as = g.as\n\u22a2 (preinclusion B).map\u2082 { down := { down := down\u271d } } = eqToHom \u22ef", "state_after": "case up.up.refl\nB : Type u\ninst\u271d : Quiver B\na b : B\nf : Discrete (Path a b)\ndown\u271d : f.as = f.as\n\u22a2 (preinclusion B).map\u2082 { down := { down := down\u271d } } = eqToHom \u22ef"}, {"tactic": "convert (inclusionPath a b).map_id _", "annotated_tactic": ["convert (<a>inclusionPath</a> a b).<a>map_id</a> _", [{"full_name": "CategoryTheory.FreeBicategory.inclusionPath", "def_path": "Mathlib/CategoryTheory/Bicategory/Coherence.lean", "def_pos": [73, 5], "def_end_pos": [73, 18]}, {"full_name": "CategoryTheory.Functor.map_id", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [41, 3], "def_end_pos": [41, 9]}]], "state_before": "case up.up.refl\nB : Type u\ninst\u271d : Quiver B\na b : B\nf : Discrete (Path a b)\ndown\u271d : f.as = f.as\n\u22a2 (preinclusion B).map\u2082 { down := { down := down\u271d } } = eqToHom \u22ef", "state_after": "no goals"}, {"tactic": "assumption", "annotated_tactic": ["assumption", []], "state_before": "B : Type u\ninst\u271d : Quiver B\na b : B\nf g : Discrete (Path a b)\ndown\u271d : f.as = g.as\n\u22a2 ?m.3775.as = ?m.3776.as", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Laurent.lean", "full_name": "LaurentPolynomial.T_mul", "start": [339, 1], "end": [340, 21], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "full_name": "MeasureTheory.lintegral_iUnion_le", "start": [1409, 1], "end": [1412, 50], "traced_tactics": [{"tactic": "rw [\u2190 lintegral_sum_measure]", "annotated_tactic": ["rw [\u2190 <a>lintegral_sum_measure</a>]", [{"full_name": "MeasureTheory.lintegral_sum_measure", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [631, 9], "def_end_pos": [631, 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\ns : \u03b2 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1) in \u22c3 i, s i, f a \u2202\u03bc \u2264 \u2211' (i : \u03b2), \u222b\u207b (a : \u03b1) in s i, 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 : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\ns : \u03b2 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1) in \u22c3 i, s i, f a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a \u2202sum fun i => \u03bc.restrict (s i)"}, {"tactic": "exact lintegral_mono' restrict_iUnion_le le_rfl", "annotated_tactic": ["exact <a>lintegral_mono'</a> <a>restrict_iUnion_le</a> <a>le_rfl</a>", [{"full_name": "MeasureTheory.lintegral_mono'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [89, 9], "def_end_pos": [89, 24]}, {"full_name": "MeasureTheory.Measure.restrict_iUnion_le", "def_path": "Mathlib/MeasureTheory/Measure/Restrict.lean", "def_pos": [537, 9], "def_end_pos": [537, 27]}, {"full_name": "le_rfl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [50, 9], "def_end_pos": [50, 15]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\ninst\u271d : Countable \u03b2\ns : \u03b2 \u2192 Set \u03b1\nf : \u03b1 \u2192 \u211d\u22650\u221e\n\u22a2 \u222b\u207b (a : \u03b1) in \u22c3 i, s i, f a \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), f a \u2202sum fun i => \u03bc.restrict (s i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean", "full_name": "CategoryTheory.Limits.equalizer.fork_\u03b9", "start": [786, 1], "end": [787, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Defs.lean", "full_name": "Nat.lt_pow_self", "start": [774, 1], "end": [779, 48], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "a b c d m n k : \u2115\np q : \u2115 \u2192 Prop\nha : 1 < a\n\u22a2 0 < a ^ 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/BooleanGenerators.lean", "full_name": "IsCompactlyGenerated.BooleanGenerators.complementedLattice_of_sSup_eq_top", "start": [148, 1], "end": [151, 43], "traced_tactics": [{"tactic": "let _i := hS.distribLattice_of_sSup_eq_top h", "annotated_tactic": ["let _i := hS.distribLattice_of_sSup_eq_top h", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nS : Set \u03b1\nhS : BooleanGenerators S\nh : sSup S = \u22a4\n\u22a2 ComplementedLattice \u03b1", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nS : Set \u03b1\nhS : BooleanGenerators S\nh : sSup S = \u22a4\n_i : DistribLattice \u03b1 := hS.distribLattice_of_sSup_eq_top h\n\u22a2 ComplementedLattice \u03b1"}, {"tactic": "have _i\u2081 := isAtomistic_of_sSup_eq_top hS h", "annotated_tactic": ["have _i\u2081 := <a>isAtomistic_of_sSup_eq_top</a> hS h", [{"full_name": "IsCompactlyGenerated.BooleanGenerators.isAtomistic_of_sSup_eq_top", "def_path": "Mathlib/Order/BooleanGenerators.lean", "def_pos": [100, 7], "def_end_pos": [100, 33]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nS : Set \u03b1\nhS : BooleanGenerators S\nh : sSup S = \u22a4\n_i : DistribLattice \u03b1 := hS.distribLattice_of_sSup_eq_top h\n\u22a2 ComplementedLattice \u03b1", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nS : Set \u03b1\nhS : BooleanGenerators S\nh : sSup S = \u22a4\n_i : DistribLattice \u03b1 := hS.distribLattice_of_sSup_eq_top h\n_i\u2081 : IsAtomistic \u03b1\n\u22a2 ComplementedLattice \u03b1"}, {"tactic": "apply complementedLattice_of_isAtomistic", "annotated_tactic": ["apply <a>complementedLattice_of_isAtomistic</a>", [{"full_name": "complementedLattice_of_isAtomistic", "def_path": "Mathlib/Order/CompactlyGenerated/Basic.lean", "def_pos": [654, 9], "def_end_pos": [654, 43]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsCompactlyGenerated \u03b1\nS : Set \u03b1\nhS : BooleanGenerators S\nh : sSup S = \u22a4\n_i : DistribLattice \u03b1 := hS.distribLattice_of_sSup_eq_top h\n_i\u2081 : IsAtomistic \u03b1\n\u22a2 ComplementedLattice \u03b1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Interval/Set/Instances.lean", "full_name": "Set.Icc.nonneg", "start": [107, 1], "end": [108, 8], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Cardinal/Cofinality.lean", "full_name": "Cardinal.IsStrongLimit.two_power_lt", "start": [861, 1], "end": [862, 8], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Pointwise/SMul.lean", "full_name": "Set.smul_iInter\u2082_subset", "start": [283, 1], "end": [285, 36], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/AEEqFun.lean", "full_name": "MeasureTheory.AEEqFun.coeFn_one", "start": [651, 1], "end": [652, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Init/Algebra/Classes.lean", "full_name": "StrictWeakOrder.esymm", "start": [398, 1], "end": [398, 68], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean", "full_name": "Real.hasDerivAt_negMulLog", "start": [113, 1], "end": [116, 12], "traced_tactics": [{"tactic": "rw [\u2190 deriv_negMulLog hx, hasDerivAt_deriv_iff]", "annotated_tactic": ["rw [\u2190 <a>deriv_negMulLog</a> hx, <a>hasDerivAt_deriv_iff</a>]", [{"full_name": "Real.deriv_negMulLog", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean", "def_pos": [109, 7], "def_end_pos": [109, 22]}, {"full_name": "hasDerivAt_deriv_iff", "def_path": "Mathlib/Analysis/Calculus/Deriv/Basic.lean", "def_pos": [438, 9], "def_end_pos": [438, 29]}]], "state_before": "x : \u211d\nhx : x \u2260 0\n\u22a2 HasDerivAt negMulLog (-log x - 1) x", "state_after": "x : \u211d\nhx : x \u2260 0\n\u22a2 DifferentiableAt \u211d negMulLog x"}, {"tactic": "refine DifferentiableOn.differentiableAt differentiableOn_negMulLog ?_", "annotated_tactic": ["refine <a>DifferentiableOn.differentiableAt</a> <a>differentiableOn_negMulLog</a> ?_", [{"full_name": "DifferentiableOn.differentiableAt", "def_path": "Mathlib/Analysis/Calculus/FDeriv/Basic.lean", "def_pos": [564, 9], "def_end_pos": [564, 42]}, {"full_name": "Real.differentiableOn_negMulLog", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean", "def_pos": [106, 7], "def_end_pos": [106, 33]}]], "state_before": "x : \u211d\nhx : x \u2260 0\n\u22a2 DifferentiableAt \u211d negMulLog x", "state_after": "x : \u211d\nhx : x \u2260 0\n\u22a2 {0}\u1d9c \u2208 \ud835\udcdd x"}, {"tactic": "simp [hx]", "annotated_tactic": ["simp [hx]", []], "state_before": "x : \u211d\nhx : x \u2260 0\n\u22a2 {0}\u1d9c \u2208 \ud835\udcdd x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Instances/CantorSet.lean", "full_name": "preCantorSet_zero", "start": [33, 1], "end": [33, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/GroupAction/SubMulAction.lean", "full_name": "SubMulAction.val_smul", "start": [239, 1], "end": [240, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Functor/FullyFaithful.lean", "full_name": "CategoryTheory.isIso_of_fully_faithful", "start": [228, 1], "end": [229, 89], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d\u00b2 : F.Full\ninst\u271d\u00b9 : F.Faithful\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsIso (F.map f)\n\u22a2 F.map (f \u226b F.preimage (inv (F.map f))) = F.map (\ud835\udfd9 X)", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\u2081\ninst\u271d\u2074 : Category.{v\u2081, u\u2081} C\nD : Type u\u2082\ninst\u271d\u00b3 : Category.{v\u2082, u\u2082} D\nF : C \u2964 D\ninst\u271d\u00b2 : F.Full\ninst\u271d\u00b9 : F.Faithful\nX Y : C\nf : X \u27f6 Y\ninst\u271d : IsIso (F.map f)\n\u22a2 F.map (F.preimage (inv (F.map f)) \u226b f) = F.map (\ud835\udfd9 Y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Preadditive/AdditiveFunctor.lean", "full_name": "CategoryTheory.AdditiveFunctor.ofExact_map", "start": [359, 1], "end": [361, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/EReal.lean", "full_name": "EReal.coe_lt_top", "start": [299, 1], "end": [300, 47], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Group/Multiset.lean", "full_name": "Multiset.snd_prod", "start": [270, 1], "end": [271, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean", "full_name": "Polynomial.natTrailingDegree_eq_support_min'", "start": [314, 1], "end": [316, 94], "traced_tactics": [{"tactic": "rw [natTrailingDegree, trailingDegree, \u2190 Finset.coe_min', ENat.some_eq_coe, ENat.toNat_coe]", "annotated_tactic": ["rw [<a>natTrailingDegree</a>, <a>trailingDegree</a>, \u2190 <a>Finset.coe_min'</a>, <a>ENat.some_eq_coe</a>, <a>ENat.toNat_coe</a>]", [{"full_name": "Polynomial.natTrailingDegree", "def_path": "Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean", "def_pos": [56, 5], "def_end_pos": [56, 22]}, {"full_name": "Polynomial.trailingDegree", "def_path": "Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean", "def_pos": [46, 5], "def_end_pos": [46, 19]}, {"full_name": "Finset.coe_min'", "def_path": "Mathlib/Data/Finset/Lattice.lean", "def_pos": [1760, 9], "def_end_pos": [1760, 17]}, {"full_name": "ENat.some_eq_coe", "def_path": "Mathlib/Data/ENat/Basic.lean", "def_pos": [60, 17], "def_end_pos": [60, 28]}, {"full_name": "ENat.toNat_coe", "def_path": "Mathlib/Data/ENat/Basic.lean", "def_pos": [117, 9], "def_end_pos": [117, 18]}]], "state_before": "R : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\nh : p \u2260 0\n\u22a2 p.natTrailingDegree = p.support.min' \u22ef", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/ModEq.lean", "full_name": "Nat.modEq_and_modEq_iff_modEq_mul", "start": [402, 1], "end": [409, 49], "traced_tactics": [{"tactic": "rw [Nat.modEq_iff_dvd, Nat.modEq_iff_dvd, \u2190 Int.dvd_natAbs, Int.natCast_dvd_natCast,\n  \u2190 Int.dvd_natAbs, Int.natCast_dvd_natCast] at h", "annotated_tactic": ["rw [<a>Nat.modEq_iff_dvd</a>, <a>Nat.modEq_iff_dvd</a>, \u2190 <a>Int.dvd_natAbs</a>, <a>Int.natCast_dvd_natCast</a>,\n      \u2190 <a>Int.dvd_natAbs</a>, <a>Int.natCast_dvd_natCast</a>] at h", [{"full_name": "Nat.modEq_iff_dvd", "def_path": "Mathlib/Data/Nat/ModEq.lean", "def_pos": [89, 9], "def_end_pos": [89, 22]}, {"full_name": "Nat.modEq_iff_dvd", "def_path": "Mathlib/Data/Nat/ModEq.lean", "def_pos": [89, 9], "def_end_pos": [89, 22]}, {"full_name": "Int.dvd_natAbs", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean", "def_pos": [107, 9], "def_end_pos": [107, 19]}, {"full_name": "Int.natCast_dvd_natCast", "def_path": "Mathlib/Data/Int/Defs.lean", "def_pos": [654, 20], "def_end_pos": [654, 39]}, {"full_name": "Int.dvd_natAbs", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean", "def_pos": [107, 9], "def_end_pos": [107, 19]}, {"full_name": "Int.natCast_dvd_natCast", "def_path": "Mathlib/Data/Int/Defs.lean", "def_pos": [654, 20], "def_end_pos": [654, 39]}]], "state_before": "m\u271d n\u271d a\u271d b\u271d c d a b m n : \u2115\nhmn : m.Coprime n\nh : a \u2261 b [MOD m] \u2227 a \u2261 b [MOD n]\n\u22a2 a \u2261 b [MOD m * n]", "state_after": "m\u271d n\u271d a\u271d b\u271d c d a b m n : \u2115\nhmn : m.Coprime n\nh : m \u2223 (\u2191b - \u2191a).natAbs \u2227 n \u2223 (\u2191b - \u2191a).natAbs\n\u22a2 a \u2261 b [MOD m * n]"}, {"tactic": "rw [Nat.modEq_iff_dvd, \u2190 Int.dvd_natAbs, Int.natCast_dvd_natCast]", "annotated_tactic": ["rw [<a>Nat.modEq_iff_dvd</a>, \u2190 <a>Int.dvd_natAbs</a>, <a>Int.natCast_dvd_natCast</a>]", [{"full_name": "Nat.modEq_iff_dvd", "def_path": "Mathlib/Data/Nat/ModEq.lean", "def_pos": [89, 9], "def_end_pos": [89, 22]}, {"full_name": "Int.dvd_natAbs", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Int/DivModLemmas.lean", "def_pos": [107, 9], "def_end_pos": [107, 19]}, {"full_name": "Int.natCast_dvd_natCast", "def_path": "Mathlib/Data/Int/Defs.lean", "def_pos": [654, 20], "def_end_pos": [654, 39]}]], "state_before": "m\u271d n\u271d a\u271d b\u271d c d a b m n : \u2115\nhmn : m.Coprime n\nh : m \u2223 (\u2191b - \u2191a).natAbs \u2227 n \u2223 (\u2191b - \u2191a).natAbs\n\u22a2 a \u2261 b [MOD m * n]", "state_after": "m\u271d n\u271d a\u271d b\u271d c d a b m n : \u2115\nhmn : m.Coprime n\nh : m \u2223 (\u2191b - \u2191a).natAbs \u2227 n \u2223 (\u2191b - \u2191a).natAbs\n\u22a2 m * n \u2223 (\u2191b - \u2191a).natAbs"}, {"tactic": "exact hmn.mul_dvd_of_dvd_of_dvd h.1 h.2", "annotated_tactic": ["exact hmn.mul_dvd_of_dvd_of_dvd h.1 h.2", []], "state_before": "m\u271d n\u271d a\u271d b\u271d c d a b m n : \u2115\nhmn : m.Coprime n\nh : m \u2223 (\u2191b - \u2191a).natAbs \u2227 n \u2223 (\u2191b - \u2191a).natAbs\n\u22a2 m * n \u2223 (\u2191b - \u2191a).natAbs", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Algebra/Bilinear.lean", "full_name": "LinearMap.mulRight_mul", "start": [146, 1], "end": [148, 52], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "R : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : NonUnitalSemiring A\ninst\u271d\u00b2 : Module R A\ninst\u271d\u00b9 : SMulCommClass R A A\ninst\u271d : IsScalarTower R A A\na b : A\n\u22a2 mulRight R (a * b) = mulRight R b \u2218\u2097 mulRight R a", "state_after": "case h\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : NonUnitalSemiring A\ninst\u271d\u00b2 : Module R A\ninst\u271d\u00b9 : SMulCommClass R A A\ninst\u271d : IsScalarTower R A A\na b x\u271d : A\n\u22a2 (mulRight R (a * b)) x\u271d = (mulRight R b \u2218\u2097 mulRight R a) x\u271d"}, {"tactic": "simp only [mulRight_apply, comp_apply, mul_assoc]", "annotated_tactic": ["simp only [<a>mulRight_apply</a>, <a>comp_apply</a>, <a>mul_assoc</a>]", [{"full_name": "LinearMap.mulRight_apply", "def_path": "Mathlib/Algebra/Algebra/Bilinear.lean", "def_pos": [82, 9], "def_end_pos": [82, 23]}, {"full_name": "LinearMap.comp_apply", "def_path": "Mathlib/Algebra/Module/LinearMap/Defs.lean", "def_pos": [552, 9], "def_end_pos": [552, 19]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "case h\nR : Type u_1\nA : Type u_2\ninst\u271d\u2074 : CommSemiring R\ninst\u271d\u00b3 : NonUnitalSemiring A\ninst\u271d\u00b2 : Module R A\ninst\u271d\u00b9 : SMulCommClass R A A\ninst\u271d : IsScalarTower R A A\na b x\u271d : A\n\u22a2 (mulRight R (a * b)) x\u271d = (mulRight R b \u2218\u2097 mulRight R a) x\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Minimal.lean", "full_name": "Set.Subsingleton.minimals_eq", "start": [199, 1], "end": [200, 16], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/DirectLimit.lean", "full_name": "AddCommGroup.DirectLimit.congr_apply_of", "start": [523, 1], "end": [528, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/PiL2.lean", "full_name": "EuclideanSpace.piLpCongrLeft_single", "start": [325, 1], "end": [329, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Lagrange.lean", "full_name": "Lagrange.nodal_empty", "start": [520, 1], "end": [521, 6], "traced_tactics": [{"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "R : Type u_1\ninst\u271d : CommRing R\n\u03b9 : Type u_2\ns : Finset \u03b9\nv : \u03b9 \u2192 R\n\u22a2 nodal \u2205 v = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Galois/GaloisObjects.lean", "full_name": "CategoryTheory.PreGaloisCategory.evaluationEquivOfIsGalois_apply", "start": [110, 1], "end": [113, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Pointwise.lean", "full_name": "Filter.vsub_pure", "start": [1168, 1], "end": [1169, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "full_name": "mul_eq_mul_iff_eq_and_eq", "start": [319, 16], "end": [324, 94], "traced_tactics": [{"tactic": "rw [le_antisymm_iff, eq_true (mul_le_mul' hac hbd), true_and, mul_le_mul_iff_of_ge hac hbd]", "annotated_tactic": ["rw [<a>le_antisymm_iff</a>, <a>eq_true</a> (<a>mul_le_mul'</a> hac hbd), <a>true_and</a>, <a>mul_le_mul_iff_of_ge</a> hac hbd]", [{"full_name": "le_antisymm_iff", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [194, 9], "def_end_pos": [194, 24]}, {"full_name": "eq_true", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [15, 9], "def_end_pos": [15, 16]}, {"full_name": "mul_le_mul'", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [206, 9], "def_end_pos": [206, 20]}, {"full_name": "true_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [105, 17], "def_end_pos": [105, 25]}, {"full_name": "mul_le_mul_iff_of_ge", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [307, 22], "def_end_pos": [307, 42]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b3 : Mul \u03b1\ninst\u271d\u00b2 : PartialOrder \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c d : \u03b1\nhac : a \u2264 c\nhbd : b \u2264 d\nthis\u271d : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nthis : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\n\u22a2 a * b = c * d \u2194 a = c \u2227 b = d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/AList.lean", "full_name": "AList.keys_singleton", "start": [141, 1], "end": [142, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/StdBasis.lean", "full_name": "LinearMap.proj_comp_stdBasis", "start": [84, 1], "end": [85, 36], "traced_tactics": [{"tactic": "rw [stdBasis_eq_pi_diag, proj_pi]", "annotated_tactic": ["rw [<a>stdBasis_eq_pi_diag</a>, <a>proj_pi</a>]", [{"full_name": "LinearMap.stdBasis_eq_pi_diag", "def_path": "Mathlib/LinearAlgebra/StdBasis.lean", "def_pos": [73, 9], "def_end_pos": [73, 28]}, {"full_name": "LinearMap.proj_pi", "def_path": "Mathlib/LinearAlgebra/Pi.lean", "def_pos": [96, 9], "def_end_pos": [96, 16]}]], "state_before": "R : Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b3 : Semiring R\n\u03c6 : \u03b9 \u2192 Type u_3\ninst\u271d\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (\u03c6 i)\ninst\u271d\u00b9 : (i : \u03b9) \u2192 Module R (\u03c6 i)\ninst\u271d : DecidableEq \u03b9\ni j : \u03b9\n\u22a2 proj i \u2218\u2097 stdBasis R \u03c6 j = diag j i", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/DenselyOrdered.lean", "full_name": "frontier_Ioc", "start": [202, 1], "end": [203, 72], "traced_tactics": [{"tactic": "rw [frontier, closure_Ioc h.ne, interior_Ioc, Icc_diff_Ioo_same h.le]", "annotated_tactic": ["rw [<a>frontier</a>, <a>closure_Ioc</a> h.ne, <a>interior_Ioc</a>, <a>Icc_diff_Ioo_same</a> h.le]", [{"full_name": "frontier", "def_path": "Mathlib/Topology/Defs/Basic.lean", "def_pos": [121, 5], "def_end_pos": [121, 13]}, {"full_name": "closure_Ioc", "def_path": "Mathlib/Topology/Order/DenselyOrdered.lean", "def_pos": [66, 9], "def_end_pos": [66, 20]}, {"full_name": "interior_Ioc", "def_path": "Mathlib/Topology/Order/DenselyOrdered.lean", "def_pos": [120, 9], "def_end_pos": [120, 21]}, {"full_name": "Set.Icc_diff_Ioo_same", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [860, 9], "def_end_pos": [860, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d\u2074 : TopologicalSpace \u03b1\ninst\u271d\u00b3 : LinearOrder \u03b1\ninst\u271d\u00b2 : OrderTopology \u03b1\ninst\u271d\u00b9 : DenselyOrdered \u03b1\na\u271d b\u271d : \u03b1\ns : Set \u03b1\ninst\u271d : NoMaxOrder \u03b1\na b : \u03b1\nh : a < b\n\u22a2 frontier (Ioc a b) = {a, b}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "full_name": "OrderIso.map_ciSup_set", "start": [1515, 1], "end": [1517, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "iInf_and", "start": [1281, 1], "end": [1282, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Analysis/Topology.lean", "full_name": "LocallyFinite.Realizer.to_locallyFinite", "start": [248, 1], "end": [250, 93], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean", "full_name": "CircleDeg1Lift.semiconjBy_iff_semiconj", "start": [271, 1], "end": [273, 10], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "full_name": "MvQPF.wEquiv.refl", "start": [120, 1], "end": [121, 74], "traced_tactics": [{"tactic": "apply q.P.w_cases _ x", "annotated_tactic": ["apply q.P.w_cases _ x", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : (P F).W \u03b1\n\u22a2 WEquiv x x", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : (P F).W \u03b1\n\u22a2 \u2200 (a : (P F).A) (f' : (P F).drop.B a \u27f9 \u03b1) (f : (P F).last.B a \u2192 (P F).W \u03b1),\n    WEquiv ((P F).wMk a f' f) ((P F).wMk a f' f)"}, {"tactic": "intro a f' f", "annotated_tactic": ["intro a f' f", []], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : (P F).W \u03b1\n\u22a2 \u2200 (a : (P F).A) (f' : (P F).drop.B a \u27f9 \u03b1) (f : (P F).last.B a \u2192 (P F).W \u03b1),\n    WEquiv ((P F).wMk a f' f) ((P F).wMk a f' f)", "state_after": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : (P F).W \u03b1\na : (P F).A\nf' : (P F).drop.B a \u27f9 \u03b1\nf : (P F).last.B a \u2192 (P F).W \u03b1\n\u22a2 WEquiv ((P F).wMk a f' f) ((P F).wMk a f' f)"}, {"tactic": "exact WEquiv.abs a f' f a f' f rfl", "annotated_tactic": ["exact <a>WEquiv.abs</a> a f' f a f' f <a>rfl</a>", [{"full_name": "MvQPF.WEquiv.abs", "def_path": "Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean", "def_pos": [84, 5], "def_end_pos": [84, 8]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "n : \u2115\nF : TypeVec.{u} (n + 1) \u2192 Type u\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : (P F).W \u03b1\na : (P F).A\nf' : (P F).drop.B a \u27f9 \u03b1\nf : (P F).last.B a \u2192 (P F).W \u03b1\n\u22a2 WEquiv ((P F).wMk a f' f) ((P F).wMk a f' f)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Lex.lean", "full_name": "List.Lex.nil_left_or_eq_nil", "start": [65, 1], "end": [68, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Quaternion.lean", "full_name": "Quaternion.normSq_eq_zero", "start": [1375, 1], "end": [1379, 48], "traced_tactics": [{"tactic": "refine \u27e8fun h => ?_, fun h => h.symm \u25b8 normSq.map_zero\u27e9", "annotated_tactic": ["refine \u27e8fun h => ?_, fun h => h.symm \u25b8 normSq.map_zero\u27e9", []], "state_before": "R : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\n\u22a2 normSq a = 0 \u2194 a = 0", "state_after": "R : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : normSq a = 0\n\u22a2 a = 0"}, {"tactic": "rw [normSq_def', add_eq_zero_iff', add_eq_zero_iff', add_eq_zero_iff'] at h", "annotated_tactic": ["rw [<a>normSq_def'</a>, <a>add_eq_zero_iff'</a>, <a>add_eq_zero_iff'</a>, <a>add_eq_zero_iff'</a>] at h", [{"full_name": "Quaternion.normSq_def'", "def_path": "Mathlib/Algebra/Quaternion.lean", "def_pos": [1304, 9], "def_end_pos": [1304, 20]}, {"full_name": "add_eq_zero_iff'", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [1179, 3], "def_end_pos": [1179, 14]}, {"full_name": "add_eq_zero_iff'", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [1179, 3], "def_end_pos": [1179, 14]}, {"full_name": "add_eq_zero_iff'", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [1179, 3], "def_end_pos": [1179, 14]}]], "state_before": "R : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : normSq a = 0\n\u22a2 a = 0", "state_after": "R : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : ((a.re ^ 2 = 0 \u2227 a.imI ^ 2 = 0) \u2227 a.imJ ^ 2 = 0) \u2227 a.imK ^ 2 = 0\n\u22a2 a = 0\n\ncase ha\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : (a.re ^ 2 + a.imI ^ 2 = 0 \u2227 a.imJ ^ 2 = 0) \u2227 a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.re ^ 2\n\ncase hb\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : (a.re ^ 2 + a.imI ^ 2 = 0 \u2227 a.imJ ^ 2 = 0) \u2227 a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.imI ^ 2\n\ncase ha\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 = 0 \u2227 a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.re ^ 2 + a.imI ^ 2\n\ncase hb\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 = 0 \u2227 a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.imJ ^ 2\n\ncase ha\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2\n\ncase hb\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.imK ^ 2"}, {"tactic": "all_goals apply_rules [sq_nonneg, add_nonneg]", "annotated_tactic": ["all_goals apply_rules [<a>sq_nonneg</a>, <a>add_nonneg</a>]", [{"full_name": "sq_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [1160, 7], "def_end_pos": [1160, 16]}, {"full_name": "add_nonneg", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [1092, 24], "def_end_pos": [1092, 34]}]], "state_before": "case ha\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : (a.re ^ 2 + a.imI ^ 2 = 0 \u2227 a.imJ ^ 2 = 0) \u2227 a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.re ^ 2\n\ncase hb\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : (a.re ^ 2 + a.imI ^ 2 = 0 \u2227 a.imJ ^ 2 = 0) \u2227 a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.imI ^ 2\n\ncase ha\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 = 0 \u2227 a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.re ^ 2 + a.imI ^ 2\n\ncase hb\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 = 0 \u2227 a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.imJ ^ 2\n\ncase ha\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2\n\ncase hb\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.imK ^ 2", "state_after": "no goals"}, {"tactic": "exact ext a 0 (pow_eq_zero h.1.1.1) (pow_eq_zero h.1.1.2) (pow_eq_zero h.1.2) (pow_eq_zero h.2)", "annotated_tactic": ["exact <a>ext</a> a 0 (<a>pow_eq_zero</a> h.1.1.1) (<a>pow_eq_zero</a> h.1.1.2) (<a>pow_eq_zero</a> h.1.2) (<a>pow_eq_zero</a> h.2)", [{"full_name": "Quaternion.ext", "def_path": "Mathlib/Algebra/Quaternion.lean", "def_pos": [856, 9], "def_end_pos": [856, 12]}, {"full_name": "pow_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [186, 7], "def_end_pos": [186, 18]}, {"full_name": "pow_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [186, 7], "def_end_pos": [186, 18]}, {"full_name": "pow_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [186, 7], "def_end_pos": [186, 18]}, {"full_name": "pow_eq_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [186, 7], "def_end_pos": [186, 18]}]], "state_before": "R : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : ((a.re ^ 2 = 0 \u2227 a.imI ^ 2 = 0) \u2227 a.imJ ^ 2 = 0) \u2227 a.imK ^ 2 = 0\n\u22a2 a = 0", "state_after": "no goals"}, {"tactic": "apply_rules [sq_nonneg, add_nonneg]", "annotated_tactic": ["apply_rules [<a>sq_nonneg</a>, <a>add_nonneg</a>]", [{"full_name": "sq_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [1160, 7], "def_end_pos": [1160, 16]}, {"full_name": "add_nonneg", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [1092, 24], "def_end_pos": [1092, 34]}]], "state_before": "case hb\nR : Type u_1\ninst\u271d : LinearOrderedCommRing R\na : \u210d[R]\nh : a.re ^ 2 + a.imI ^ 2 + a.imJ ^ 2 + a.imK ^ 2 = 0\n\u22a2 0 \u2264 a.imK ^ 2", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/Star/Multiplier.lean", "full_name": "DoubleCentralizer.natCast_toProd", "start": [236, 1], "end": [237, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "full_name": "MonoidHom.ker_snd", "start": [2818, 1], "end": [2819, 89], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Pi.lean", "full_name": "Filter.map_eval_pi", "start": [193, 1], "end": [199, 57], "traced_tactics": [{"tactic": "refine le_antisymm (tendsto_eval_pi f i) fun s hs => ?_", "annotated_tactic": ["refine <a>le_antisymm</a> (<a>tendsto_eval_pi</a> f i) fun s hs => ?_", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "Filter.tendsto_eval_pi", "def_path": "Mathlib/Order/Filter/Pi.lean", "def_pos": [47, 9], "def_end_pos": [47, 24]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf\u271d f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns : (i : \u03b9) \u2192 Set (\u03b1 i)\np : (i : \u03b9) \u2192 \u03b1 i \u2192 Prop\nf : (i : \u03b9) \u2192 Filter (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), (f i).NeBot\ni : \u03b9\n\u22a2 map (eval i) (pi f) = f i", "state_after": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf\u271d f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\np : (i : \u03b9) \u2192 \u03b1 i \u2192 Prop\nf : (i : \u03b9) \u2192 Filter (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), (f i).NeBot\ni : \u03b9\ns : Set (\u03b1 i)\nhs : s \u2208 map (eval i) (pi f)\n\u22a2 s \u2208 f i"}, {"tactic": "rcases mem_pi.1 (mem_map.1 hs) with \u27e8I, hIf, t, htf, hI\u27e9", "annotated_tactic": ["rcases <a>mem_pi</a>.1 (<a>mem_map</a>.1 hs) with \u27e8I, hIf, t, htf, hI\u27e9", [{"full_name": "Filter.mem_pi", "def_path": "Mathlib/Order/Filter/Pi.lean", "def_pos": [80, 9], "def_end_pos": [80, 15]}, {"full_name": "Filter.mem_map", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1930, 9], "def_end_pos": [1930, 16]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf\u271d f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\np : (i : \u03b9) \u2192 \u03b1 i \u2192 Prop\nf : (i : \u03b9) \u2192 Filter (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), (f i).NeBot\ni : \u03b9\ns : Set (\u03b1 i)\nhs : s \u2208 map (eval i) (pi f)\n\u22a2 s \u2208 f i", "state_after": "case intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf\u271d f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\np : (i : \u03b9) \u2192 \u03b1 i \u2192 Prop\nf : (i : \u03b9) \u2192 Filter (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), (f i).NeBot\ni : \u03b9\ns : Set (\u03b1 i)\nhs : s \u2208 map (eval i) (pi f)\nI : Set \u03b9\nhIf : I.Finite\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhI : I.pi t \u2286 eval i \u207b\u00b9' s\n\u22a2 s \u2208 f i"}, {"tactic": "rw [\u2190 image_subset_iff] at hI", "annotated_tactic": ["rw [\u2190 <a>image_subset_iff</a>] at hI", [{"full_name": "Set.image_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [481, 9], "def_end_pos": [481, 25]}]], "state_before": "case intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf\u271d f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\np : (i : \u03b9) \u2192 \u03b1 i \u2192 Prop\nf : (i : \u03b9) \u2192 Filter (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), (f i).NeBot\ni : \u03b9\ns : Set (\u03b1 i)\nhs : s \u2208 map (eval i) (pi f)\nI : Set \u03b9\nhIf : I.Finite\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhI : I.pi t \u2286 eval i \u207b\u00b9' s\n\u22a2 s \u2208 f i", "state_after": "case intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf\u271d f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\np : (i : \u03b9) \u2192 \u03b1 i \u2192 Prop\nf : (i : \u03b9) \u2192 Filter (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), (f i).NeBot\ni : \u03b9\ns : Set (\u03b1 i)\nhs : s \u2208 map (eval i) (pi f)\nI : Set \u03b9\nhIf : I.Finite\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhI : eval i '' I.pi t \u2286 s\n\u22a2 s \u2208 f i"}, {"tactic": "refine mem_of_superset (htf i) ((subset_eval_image_pi ?_ _).trans hI)", "annotated_tactic": ["refine <a>mem_of_superset</a> (htf i) ((<a>subset_eval_image_pi</a> ?_ _).<a>trans</a> hI)", [{"full_name": "Filter.mem_of_superset", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 24]}, {"full_name": "Set.subset_eval_image_pi", "def_path": "Mathlib/Data/Set/Prod.lean", "def_pos": [908, 9], "def_end_pos": [908, 29]}, {"full_name": "HasSubset.Subset.trans", "def_path": "Mathlib/Order/RelClasses.lean", "def_pos": [655, 7], "def_end_pos": [655, 29]}]], "state_before": "case intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf\u271d f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\np : (i : \u03b9) \u2192 \u03b1 i \u2192 Prop\nf : (i : \u03b9) \u2192 Filter (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), (f i).NeBot\ni : \u03b9\ns : Set (\u03b1 i)\nhs : s \u2208 map (eval i) (pi f)\nI : Set \u03b9\nhIf : I.Finite\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhI : eval i '' I.pi t \u2286 s\n\u22a2 s \u2208 f i", "state_after": "case intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf\u271d f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\np : (i : \u03b9) \u2192 \u03b1 i \u2192 Prop\nf : (i : \u03b9) \u2192 Filter (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), (f i).NeBot\ni : \u03b9\ns : Set (\u03b1 i)\nhs : s \u2208 map (eval i) (pi f)\nI : Set \u03b9\nhIf : I.Finite\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhI : eval i '' I.pi t \u2286 s\n\u22a2 (I.pi t).Nonempty"}, {"tactic": "exact nonempty_of_mem (pi_mem_pi hIf fun i _ => htf i)", "annotated_tactic": ["exact <a>nonempty_of_mem</a> (<a>pi_mem_pi</a> hIf fun i _ => htf i)", [{"full_name": "Filter.nonempty_of_mem", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [712, 9], "def_end_pos": [712, 24]}, {"full_name": "Filter.pi_mem_pi", "def_path": "Mathlib/Order/Filter/Pi.lean", "def_pos": [74, 9], "def_end_pos": [74, 18]}]], "state_before": "case intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\nf\u271d f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i)\ns\u271d : (i : \u03b9) \u2192 Set (\u03b1 i)\np : (i : \u03b9) \u2192 \u03b1 i \u2192 Prop\nf : (i : \u03b9) \u2192 Filter (\u03b1 i)\ninst\u271d : \u2200 (i : \u03b9), (f i).NeBot\ni : \u03b9\ns : Set (\u03b1 i)\nhs : s \u2208 map (eval i) (pi f)\nI : Set \u03b9\nhIf : I.Finite\nt : (i : \u03b9) \u2192 Set (\u03b1 i)\nhtf : \u2200 (i : \u03b9), t i \u2208 f i\nhI : eval i '' I.pi t \u2286 s\n\u22a2 (I.pi t).Nonempty", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/NNRat/Defs.lean", "full_name": "NNRat.coe_mk", "start": [80, 1], "end": [80, 77], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/SetTheory/Ordinal/FixedPoint.lean", "full_name": "Ordinal.derivFamily_succ", "start": [168, 1], "end": [170, 26], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Card.lean", "full_name": "Set.encard_univ", "start": [69, 1], "end": [71, 54], "traced_tactics": [{"tactic": "rw [encard, PartENat.card_congr (Equiv.Set.univ \u03b1)]", "annotated_tactic": ["rw [<a>encard</a>, <a>PartENat.card_congr</a> (<a>Equiv.Set.univ</a> \u03b1)]", [{"full_name": "Set.encard", "def_path": "Mathlib/Data/Set/Card.lean", "def_pos": [64, 19], "def_end_pos": [64, 25]}, {"full_name": "PartENat.card_congr", "def_path": "Mathlib/SetTheory/Cardinal/Finite.lean", "def_pos": [238, 9], "def_end_pos": [238, 19]}, {"full_name": "Equiv.Set.univ", "def_path": "Mathlib/Logic/Equiv/Set.lean", "def_pos": [211, 15], "def_end_pos": [211, 19]}]], "state_before": "\u03b1\u271d : Type u_1\n\u03b2 : Type u_2\ns t : Set \u03b1\u271d\n\u03b1 : Type u_3\n\u22a2 univ.encard = PartENat.withTopEquiv (PartENat.card \u03b1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/OfNorm.lean", "full_name": "InnerProductSpaceable.rat_prop", "start": [269, 9], "end": [276, 70], "traced_tactics": [{"tactic": "intro x y", "annotated_tactic": ["intro x y", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : RCLike \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : InnerProductSpaceable E\nr : \u211a\n\u22a2 InnerProductSpaceable.innerProp' E \u2191r", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : RCLike \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : InnerProductSpaceable E\nr : \u211a\nx y : E\n\u22a2 inner_ \ud835\udd5c (\u2191r \u2022 x) y = (starRingEnd \ud835\udd5c) \u2191r * inner_ \ud835\udd5c x y"}, {"tactic": "have : (r.den : \ud835\udd5c) \u2260 0 := by\n  haveI : CharZero \ud835\udd5c := RCLike.charZero_rclike\n  exact mod_cast r.pos.ne'", "annotated_tactic": ["have : (r.den : \ud835\udd5c) \u2260 0 := by\n    haveI : <a>CharZero</a> \ud835\udd5c := <a>RCLike.charZero_rclike</a>\n    exact mod_cast r.pos.ne'", [{"full_name": "CharZero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [42, 7], "def_end_pos": [42, 15]}, {"full_name": "RCLike.charZero_rclike", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [281, 28], "def_end_pos": [281, 43]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : RCLike \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : InnerProductSpaceable E\nr : \u211a\nx y : E\n\u22a2 inner_ \ud835\udd5c (\u2191r \u2022 x) y = (starRingEnd \ud835\udd5c) \u2191r * inner_ \ud835\udd5c x y", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : RCLike \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : InnerProductSpaceable E\nr : \u211a\nx y : E\nthis : \u2191r.den \u2260 0\n\u22a2 inner_ \ud835\udd5c (\u2191r \u2022 x) y = (starRingEnd \ud835\udd5c) \u2191r * inner_ \ud835\udd5c x y"}, {"tactic": "rw [\u2190 r.num_div_den, \u2190 mul_right_inj' this, \u2190 nat r.den _ y, smul_smul, Rat.cast_div]", "annotated_tactic": ["rw [\u2190 r.num_div_den, \u2190 <a>mul_right_inj'</a> this, \u2190 <a>nat</a> r.den _ y, <a>smul_smul</a>, <a>Rat.cast_div</a>]", [{"full_name": "mul_right_inj'", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [132, 9], "def_end_pos": [132, 23]}, {"full_name": "InnerProductSpaceable.nat", "def_path": "Mathlib/Analysis/InnerProductSpace/OfNorm.lean", "def_pos": [243, 9], "def_end_pos": [243, 12]}, {"full_name": "smul_smul", "def_path": "Mathlib/Algebra/Group/Action/Defs.lean", "def_pos": [446, 7], "def_end_pos": [446, 16]}, {"full_name": "Rat.cast_div", "def_path": "Mathlib/Data/Rat/Cast/CharZero.lean", "def_pos": [109, 9], "def_end_pos": [109, 17]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : RCLike \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : InnerProductSpaceable E\nr : \u211a\nx y : E\nthis : \u2191r.den \u2260 0\n\u22a2 inner_ \ud835\udd5c (\u2191r \u2022 x) y = (starRingEnd \ud835\udd5c) \u2191r * inner_ \ud835\udd5c x y", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : RCLike \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : InnerProductSpaceable E\nr : \u211a\nx y : E\nthis : \u2191r.den \u2260 0\n\u22a2 inner_ \ud835\udd5c ((\u2191r.den * (\u2191\u2191r.num / \u2191\u2191r.den)) \u2022 x) y = \u2191r.den * ((starRingEnd \ud835\udd5c) (\u2191\u2191r.num / \u2191\u2191r.den) * inner_ \ud835\udd5c x y)"}, {"tactic": "simp only [map_natCast, Rat.cast_natCast, map_intCast, Rat.cast_intCast, map_div\u2080]", "annotated_tactic": ["simp only [<a>map_natCast</a>, <a>Rat.cast_natCast</a>, <a>map_intCast</a>, <a>Rat.cast_intCast</a>, <a>map_div\u2080</a>]", [{"full_name": "map_natCast", "def_path": "Mathlib/Data/Nat/Cast/Basic.lean", "def_pos": [184, 9], "def_end_pos": [184, 20]}, {"full_name": "Rat.cast_natCast", "def_path": "Mathlib/Data/Rat/Cast/Defs.lean", "def_pos": [120, 9], "def_end_pos": [120, 21]}, {"full_name": "map_intCast", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [417, 9], "def_end_pos": [417, 20]}, {"full_name": "Rat.cast_intCast", "def_path": "Mathlib/Data/Rat/Cast/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 21]}, {"full_name": "map_div\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : RCLike \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : InnerProductSpaceable E\nr : \u211a\nx y : E\nthis : \u2191r.den \u2260 0\n\u22a2 inner_ \ud835\udd5c ((\u2191r.den * (\u2191\u2191r.num / \u2191\u2191r.den)) \u2022 x) y = \u2191r.den * ((starRingEnd \ud835\udd5c) (\u2191\u2191r.num / \u2191\u2191r.den) * inner_ \ud835\udd5c x y)", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : RCLike \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : InnerProductSpaceable E\nr : \u211a\nx y : E\nthis : \u2191r.den \u2260 0\n\u22a2 inner_ \ud835\udd5c ((\u2191r.den * (\u2191r.num / \u2191r.den)) \u2022 x) y = \u2191r.den * (\u2191r.num / \u2191r.den * inner_ \ud835\udd5c x y)"}, {"tactic": "rw [\u2190 mul_assoc, mul_div_cancel\u2080 _ this, int_prop _ x, map_intCast]", "annotated_tactic": ["rw [\u2190 <a>mul_assoc</a>, <a>mul_div_cancel\u2080</a> _ this, <a>int_prop</a> _ x, <a>map_intCast</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "mul_div_cancel\u2080", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [516, 7], "def_end_pos": [516, 22]}, {"full_name": "_private.Mathlib.Analysis.InnerProductSpace.OfNorm.0.InnerProductSpaceable.int_prop", "def_path": "Mathlib/Analysis/InnerProductSpace/OfNorm.lean", "def_pos": [254, 17], "def_end_pos": [254, 25]}, {"full_name": "map_intCast", "def_path": "Mathlib/Data/Int/Cast/Lemmas.lean", "def_pos": [417, 9], "def_end_pos": [417, 20]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : RCLike \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : InnerProductSpaceable E\nr : \u211a\nx y : E\nthis : \u2191r.den \u2260 0\n\u22a2 inner_ \ud835\udd5c ((\u2191r.den * (\u2191r.num / \u2191r.den)) \u2022 x) y = \u2191r.den * (\u2191r.num / \u2191r.den * inner_ \ud835\udd5c x y)", "state_after": "no goals"}, {"tactic": "haveI : CharZero \ud835\udd5c := RCLike.charZero_rclike", "annotated_tactic": ["haveI : <a>CharZero</a> \ud835\udd5c := <a>RCLike.charZero_rclike</a>", [{"full_name": "CharZero", "def_path": "Mathlib/Algebra/CharZero/Defs.lean", "def_pos": [42, 7], "def_end_pos": [42, 15]}, {"full_name": "RCLike.charZero_rclike", "def_path": "Mathlib/Analysis/RCLike/Basic.lean", "def_pos": [281, 28], "def_end_pos": [281, 43]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : RCLike \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : InnerProductSpaceable E\nr : \u211a\nx y : E\n\u22a2 \u2191r.den \u2260 0", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : RCLike \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : InnerProductSpaceable E\nr : \u211a\nx y : E\nthis : CharZero \ud835\udd5c\n\u22a2 \u2191r.den \u2260 0"}, {"tactic": "exact mod_cast r.pos.ne'", "annotated_tactic": ["exact mod_cast r.pos.ne'", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b3 : RCLike \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b2 : NormedAddCommGroup E\ninst\u271d\u00b9 : NormedSpace \ud835\udd5c E\ninst\u271d : InnerProductSpaceable E\nr : \u211a\nx y : E\nthis : CharZero \ud835\udd5c\n\u22a2 \u2191r.den \u2260 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean", "full_name": "AlgebraicTopology.DoldKan.\u0393\u2082N\u2082ToKaroubiIso_hom_app", "start": [180, 1], "end": [183, 26], "traced_tactics": [{"tactic": "simp [\u0393\u2082N\u2082ToKaroubiIso]", "annotated_tactic": ["simp [<a>\u0393\u2082N\u2082ToKaroubiIso</a>]", [{"full_name": "AlgebraicTopology.DoldKan.\u0393\u2082N\u2082ToKaroubiIso", "def_path": "Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean", "def_pos": [175, 5], "def_end_pos": [175, 21]}]], "state_before": "C : Type u_1\ninst\u271d\u00b2 : Category.{u_2, u_1} C\ninst\u271d\u00b9 : Preadditive C\ninst\u271d : HasFiniteCoproducts C\nX : SimplicialObject C\n\u22a2 \u0393\u2082N\u2082ToKaroubiIso.hom.app X = \u0393\u2082.map (toKaroubiCompN\u2082IsoN\u2081.hom.app X)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean", "full_name": "WeierstrassCurve.Jacobian.equation_iff", "start": [258, 1], "end": [261, 33], "traced_tactics": [{"tactic": "rw [Equation, eval_polynomial]", "annotated_tactic": ["rw [<a>Equation</a>, <a>eval_polynomial</a>]", [{"full_name": "WeierstrassCurve.Jacobian.Equation", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean", "def_pos": [255, 5], "def_end_pos": [255, 13]}, {"full_name": "WeierstrassCurve.Jacobian.eval_polynomial", "def_path": "Mathlib/AlgebraicGeometry/EllipticCurve/Jacobian.lean", "def_pos": [239, 7], "def_end_pos": [239, 22]}]], "state_before": "R : Type u\ninst\u271d\u00b9 : CommRing R\nW' : Jacobian R\nF : Type v\ninst\u271d : Field F\nW : Jacobian F\nP : Fin 3 \u2192 R\n\u22a2 W'.Equation P \u2194\n    P y ^ 2 + W'.a\u2081 * P x * P y * P z + W'.a\u2083 * P y * P z ^ 3 -\n        (P x ^ 3 + W'.a\u2082 * P x ^ 2 * P z ^ 2 + W'.a\u2084 * P x * P z ^ 4 + W'.a\u2086 * P z ^ 6) =\n      0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Lattice.lean", "full_name": "MonotoneOn.map_inf", "start": [1132, 1], "end": [1134, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Manifold/PartitionOfUnity.lean", "full_name": "SmoothPartitionOfUnity.IsSubordinate.contMDiff_finsum_smul", "start": [313, 1], "end": [316, 88], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Real/Pi/Wallis.lean", "full_name": "Real.Wallis.W_eq_factorial_ratio", "start": [62, 1], "end": [75, 12], "traced_tactics": [{"tactic": "induction' n with n IH", "annotated_tactic": ["induction' n with n IH", []], "state_before": "n : \u2115\n\u22a2 W n = 2 ^ (4 * n) * \u2191n ! ^ 4 / (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1))", "state_after": "case zero\n\n\u22a2 W 0 = 2 ^ (4 * 0) * \u21910! ^ 4 / (\u2191(2 * 0)! ^ 2 * (2 * \u21910 + 1))\n\ncase succ\nn : \u2115\nIH : W n = 2 ^ (4 * n) * \u2191n ! ^ 4 / (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1))\n\u22a2 W (n + 1) = 2 ^ (4 * (n + 1)) * \u2191(n + 1)! ^ 4 / (\u2191(2 * (n + 1))! ^ 2 * (2 * \u2191(n + 1) + 1))"}, {"tactic": "simp only [W, prod_range_zero, Nat.factorial_zero, mul_zero, pow_zero,\n  algebraMap.coe_one, one_pow, mul_one, algebraMap.coe_zero, zero_add, div_self, Ne,\n  one_ne_zero, not_false_iff]", "annotated_tactic": ["simp only [<a>W</a>, <a>prod_range_zero</a>, <a>Nat.factorial_zero</a>, <a>mul_zero</a>, <a>pow_zero</a>,\n      <a>algebraMap.coe_one</a>, <a>one_pow</a>, <a>mul_one</a>, <a>algebraMap.coe_zero</a>, <a>zero_add</a>, <a>div_self</a>, <a>Ne</a>,\n      <a>one_ne_zero</a>, <a>not_false_iff</a>]", [{"full_name": "Real.Wallis.W", "def_path": "Mathlib/Data/Real/Pi/Wallis.lean", "def_pos": [46, 19], "def_end_pos": [46, 20]}, {"full_name": "Finset.prod_range_zero", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [1562, 9], "def_end_pos": [1562, 24]}, {"full_name": "Nat.factorial_zero", "def_path": "Mathlib/Data/Nat/Factorial/Basic.lean", "def_pos": [43, 17], "def_end_pos": [43, 31]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [39, 3], "def_end_pos": [39, 11]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [651, 9], "def_end_pos": [651, 17]}, {"full_name": "algebraMap.coe_one", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [135, 9], "def_end_pos": [135, 16]}, {"full_name": "one_pow", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [696, 39], "def_end_pos": [696, 46]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}, {"full_name": "algebraMap.coe_zero", "def_path": "Mathlib/Algebra/Algebra/Defs.lean", "def_pos": [130, 9], "def_end_pos": [130, 17]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [476, 3], "def_end_pos": [476, 14]}, {"full_name": "div_self", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [295, 15], "def_end_pos": [295, 23]}, {"full_name": "Ne", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [689, 18], "def_end_pos": [689, 20]}, {"full_name": "one_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [58, 15], "def_end_pos": [58, 26]}, {"full_name": "not_false_iff", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1371, 9], "def_end_pos": [1371, 22]}]], "state_before": "case zero\n\n\u22a2 W 0 = 2 ^ (4 * 0) * \u21910! ^ 4 / (\u2191(2 * 0)! ^ 2 * (2 * \u21910 + 1))", "state_after": "case zero\n\n\u22a2 1 = 1 * \u21911 ^ 4 / (\u21911 ^ 2 * (2 * \u21910 + 1))"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "case zero\n\n\u22a2 1 = 1 * \u21911 ^ 4 / (\u21911 ^ 2 * (2 * \u21910 + 1))", "state_after": "no goals"}, {"tactic": "unfold W at IH \u22a2", "annotated_tactic": ["unfold <a>W</a> at IH \u22a2", [{"full_name": "Real.Wallis.W", "def_path": "Mathlib/Data/Real/Pi/Wallis.lean", "def_pos": [46, 19], "def_end_pos": [46, 20]}]], "state_before": "case succ\nn : \u2115\nIH : W n = 2 ^ (4 * n) * \u2191n ! ^ 4 / (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1))\n\u22a2 W (n + 1) = 2 ^ (4 * (n + 1)) * \u2191(n + 1)! ^ 4 / (\u2191(2 * (n + 1))! ^ 2 * (2 * \u2191(n + 1) + 1))", "state_after": "case succ\nn : \u2115\nIH :\n  \u220f i \u2208 range n, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3)) =\n    2 ^ (4 * n) * \u2191n ! ^ 4 / (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1))\n\u22a2 \u220f i \u2208 range (n + 1), (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3)) =\n    2 ^ (4 * (n + 1)) * \u2191(n + 1)! ^ 4 / (\u2191(2 * (n + 1))! ^ 2 * (2 * \u2191(n + 1) + 1))"}, {"tactic": "rw [prod_range_succ, IH, _root_.div_mul_div_comm, _root_.div_mul_div_comm]", "annotated_tactic": ["rw [<a>prod_range_succ</a>, IH, <a>_root_.div_mul_div_comm</a>, <a>_root_.div_mul_div_comm</a>]", [{"full_name": "Finset.prod_range_succ", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [1520, 9], "def_end_pos": [1520, 24]}, {"full_name": "div_mul_div_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [829, 9], "def_end_pos": [829, 25]}, {"full_name": "div_mul_div_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [829, 9], "def_end_pos": [829, 25]}]], "state_before": "case succ\nn : \u2115\nIH :\n  \u220f i \u2208 range n, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3)) =\n    2 ^ (4 * n) * \u2191n ! ^ 4 / (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1))\n\u22a2 \u220f i \u2208 range (n + 1), (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3)) =\n    2 ^ (4 * (n + 1)) * \u2191(n + 1)! ^ 4 / (\u2191(2 * (n + 1))! ^ 2 * (2 * \u2191(n + 1) + 1))", "state_after": "case succ\nn : \u2115\nIH :\n  \u220f i \u2208 range n, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3)) =\n    2 ^ (4 * n) * \u2191n ! ^ 4 / (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1))\n\u22a2 2 ^ (4 * n) * \u2191n ! ^ 4 * ((2 * \u2191n + 2) * (2 * \u2191n + 2)) /\n      (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1) * ((2 * \u2191n + 1) * (2 * \u2191n + 3))) =\n    2 ^ (4 * (n + 1)) * \u2191(n + 1)! ^ 4 / (\u2191(2 * (n + 1))! ^ 2 * (2 * \u2191(n + 1) + 1))"}, {"tactic": "refine (div_eq_div_iff ?_ ?_).mpr ?_", "annotated_tactic": ["refine (<a>div_eq_div_iff</a> ?_ ?_).<a>mpr</a> ?_", [{"full_name": "div_eq_div_iff", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Basic.lean", "def_pos": [529, 22], "def_end_pos": [529, 36]}, {"full_name": "Iff.mpr", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [120, 3], "def_end_pos": [120, 6]}]], "state_before": "case succ\nn : \u2115\nIH :\n  \u220f i \u2208 range n, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3)) =\n    2 ^ (4 * n) * \u2191n ! ^ 4 / (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1))\n\u22a2 2 ^ (4 * n) * \u2191n ! ^ 4 * ((2 * \u2191n + 2) * (2 * \u2191n + 2)) /\n      (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1) * ((2 * \u2191n + 1) * (2 * \u2191n + 3))) =\n    2 ^ (4 * (n + 1)) * \u2191(n + 1)! ^ 4 / (\u2191(2 * (n + 1))! ^ 2 * (2 * \u2191(n + 1) + 1))", "state_after": "case succ.refine_1\nn : \u2115\nIH :\n  \u220f i \u2208 range n, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3)) =\n    2 ^ (4 * n) * \u2191n ! ^ 4 / (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1))\n\u22a2 \u2191(2 * n)! ^ 2 * (2 * \u2191n + 1) * ((2 * \u2191n + 1) * (2 * \u2191n + 3)) \u2260 0\n\ncase succ.refine_2\nn : \u2115\nIH :\n  \u220f i \u2208 range n, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3)) =\n    2 ^ (4 * n) * \u2191n ! ^ 4 / (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1))\n\u22a2 \u2191(2 * (n + 1))! ^ 2 * (2 * \u2191(n + 1) + 1) \u2260 0\n\ncase succ.refine_3\nn : \u2115\nIH :\n  \u220f i \u2208 range n, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3)) =\n    2 ^ (4 * n) * \u2191n ! ^ 4 / (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1))\n\u22a2 2 ^ (4 * n) * \u2191n ! ^ 4 * ((2 * \u2191n + 2) * (2 * \u2191n + 2)) * (\u2191(2 * (n + 1))! ^ 2 * (2 * \u2191(n + 1) + 1)) =\n    2 ^ (4 * (n + 1)) * \u2191(n + 1)! ^ 4 * (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1) * ((2 * \u2191n + 1) * (2 * \u2191n + 3)))"}, {"tactic": "any_goals exact ne_of_gt (by positivity)", "annotated_tactic": ["any_goals exact <a>ne_of_gt</a> (by positivity)", [{"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}]], "state_before": "case succ.refine_1\nn : \u2115\nIH :\n  \u220f i \u2208 range n, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3)) =\n    2 ^ (4 * n) * \u2191n ! ^ 4 / (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1))\n\u22a2 \u2191(2 * n)! ^ 2 * (2 * \u2191n + 1) * ((2 * \u2191n + 1) * (2 * \u2191n + 3)) \u2260 0\n\ncase succ.refine_2\nn : \u2115\nIH :\n  \u220f i \u2208 range n, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3)) =\n    2 ^ (4 * n) * \u2191n ! ^ 4 / (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1))\n\u22a2 \u2191(2 * (n + 1))! ^ 2 * (2 * \u2191(n + 1) + 1) \u2260 0\n\ncase succ.refine_3\nn : \u2115\nIH :\n  \u220f i \u2208 range n, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3)) =\n    2 ^ (4 * n) * \u2191n ! ^ 4 / (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1))\n\u22a2 2 ^ (4 * n) * \u2191n ! ^ 4 * ((2 * \u2191n + 2) * (2 * \u2191n + 2)) * (\u2191(2 * (n + 1))! ^ 2 * (2 * \u2191(n + 1) + 1)) =\n    2 ^ (4 * (n + 1)) * \u2191(n + 1)! ^ 4 * (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1) * ((2 * \u2191n + 1) * (2 * \u2191n + 3)))", "state_after": "case succ.refine_3\nn : \u2115\nIH :\n  \u220f i \u2208 range n, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3)) =\n    2 ^ (4 * n) * \u2191n ! ^ 4 / (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1))\n\u22a2 2 ^ (4 * n) * \u2191n ! ^ 4 * ((2 * \u2191n + 2) * (2 * \u2191n + 2)) * (\u2191(2 * (n + 1))! ^ 2 * (2 * \u2191(n + 1) + 1)) =\n    2 ^ (4 * (n + 1)) * \u2191(n + 1)! ^ 4 * (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1) * ((2 * \u2191n + 1) * (2 * \u2191n + 3)))"}, {"tactic": "simp_rw [Nat.mul_succ, Nat.factorial_succ, pow_succ]", "annotated_tactic": ["simp_rw [<a>Nat.mul_succ</a>, <a>Nat.factorial_succ</a>, <a>pow_succ</a>]", [{"full_name": "Nat.mul_succ", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [200, 9], "def_end_pos": [200, 17]}, {"full_name": "Nat.factorial_succ", "def_path": "Mathlib/Data/Nat/Factorial/Basic.lean", "def_pos": [47, 9], "def_end_pos": [47, 23]}, {"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [657, 9], "def_end_pos": [657, 17]}]], "state_before": "case succ.refine_3\nn : \u2115\nIH :\n  \u220f i \u2208 range n, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3)) =\n    2 ^ (4 * n) * \u2191n ! ^ 4 / (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1))\n\u22a2 2 ^ (4 * n) * \u2191n ! ^ 4 * ((2 * \u2191n + 2) * (2 * \u2191n + 2)) * (\u2191(2 * (n + 1))! ^ 2 * (2 * \u2191(n + 1) + 1)) =\n    2 ^ (4 * (n + 1)) * \u2191(n + 1)! ^ 4 * (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1) * ((2 * \u2191n + 1) * (2 * \u2191n + 3)))", "state_after": "case succ.refine_3\nn : \u2115\nIH :\n  \u220f i \u2208 range n, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3)) =\n    2 ^ (4 * n) * \u2191n ! ^ 4 / (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1))\n\u22a2 2 ^ (4 * n) * (\u2191n ! ^ 0 * \u2191n ! * \u2191n ! * \u2191n ! * \u2191n !) * ((2 * \u2191n + 2) * (2 * \u2191n + 2)) *\n      (\u2191((2 * n + 1 + 1) * ((2 * n + 1) * (2 * n)!)) ^ 0 * \u2191((2 * n + 1 + 1) * ((2 * n + 1) * (2 * n)!)) *\n          \u2191((2 * n + 1 + 1) * ((2 * n + 1) * (2 * n)!)) *\n        (2 * \u2191(n + 1) + 1)) =\n    2 ^ (4 * n) * 2 * 2 * 2 * 2 *\n        (\u2191((n + 1) * n !) ^ 0 * \u2191((n + 1) * n !) * \u2191((n + 1) * n !) * \u2191((n + 1) * n !) * \u2191((n + 1) * n !)) *\n      (\u2191(2 * n)! ^ 0 * \u2191(2 * n)! * \u2191(2 * n)! * (2 * \u2191n + 1) * ((2 * \u2191n + 1) * (2 * \u2191n + 3)))"}, {"tactic": "push_cast", "annotated_tactic": ["push_cast", []], "state_before": "case succ.refine_3\nn : \u2115\nIH :\n  \u220f i \u2208 range n, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3)) =\n    2 ^ (4 * n) * \u2191n ! ^ 4 / (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1))\n\u22a2 2 ^ (4 * n) * (\u2191n ! ^ 0 * \u2191n ! * \u2191n ! * \u2191n ! * \u2191n !) * ((2 * \u2191n + 2) * (2 * \u2191n + 2)) *\n      (\u2191((2 * n + 1 + 1) * ((2 * n + 1) * (2 * n)!)) ^ 0 * \u2191((2 * n + 1 + 1) * ((2 * n + 1) * (2 * n)!)) *\n          \u2191((2 * n + 1 + 1) * ((2 * n + 1) * (2 * n)!)) *\n        (2 * \u2191(n + 1) + 1)) =\n    2 ^ (4 * n) * 2 * 2 * 2 * 2 *\n        (\u2191((n + 1) * n !) ^ 0 * \u2191((n + 1) * n !) * \u2191((n + 1) * n !) * \u2191((n + 1) * n !) * \u2191((n + 1) * n !)) *\n      (\u2191(2 * n)! ^ 0 * \u2191(2 * n)! * \u2191(2 * n)! * (2 * \u2191n + 1) * ((2 * \u2191n + 1) * (2 * \u2191n + 3)))", "state_after": "case succ.refine_3\nn : \u2115\nIH :\n  \u220f i \u2208 range n, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3)) =\n    2 ^ (4 * n) * \u2191n ! ^ 4 / (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1))\n\u22a2 2 ^ (4 * n) * (\u2191n ! ^ 0 * \u2191n ! * \u2191n ! * \u2191n ! * \u2191n !) * ((2 * \u2191n + 2) * (2 * \u2191n + 2)) *\n      (((2 * \u2191n + 1 + 1) * ((2 * \u2191n + 1) * \u2191(2 * n)!)) ^ 0 * ((2 * \u2191n + 1 + 1) * ((2 * \u2191n + 1) * \u2191(2 * n)!)) *\n          ((2 * \u2191n + 1 + 1) * ((2 * \u2191n + 1) * \u2191(2 * n)!)) *\n        (2 * (\u2191n + 1) + 1)) =\n    2 ^ (4 * n) * 2 * 2 * 2 * 2 *\n        (((\u2191n + 1) * \u2191n !) ^ 0 * ((\u2191n + 1) * \u2191n !) * ((\u2191n + 1) * \u2191n !) * ((\u2191n + 1) * \u2191n !) * ((\u2191n + 1) * \u2191n !)) *\n      (\u2191(2 * n)! ^ 0 * \u2191(2 * n)! * \u2191(2 * n)! * (2 * \u2191n + 1) * ((2 * \u2191n + 1) * (2 * \u2191n + 3)))"}, {"tactic": "ring_nf", "annotated_tactic": ["ring_nf", []], "state_before": "case succ.refine_3\nn : \u2115\nIH :\n  \u220f i \u2208 range n, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3)) =\n    2 ^ (4 * n) * \u2191n ! ^ 4 / (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1))\n\u22a2 2 ^ (4 * n) * (\u2191n ! ^ 0 * \u2191n ! * \u2191n ! * \u2191n ! * \u2191n !) * ((2 * \u2191n + 2) * (2 * \u2191n + 2)) *\n      (((2 * \u2191n + 1 + 1) * ((2 * \u2191n + 1) * \u2191(2 * n)!)) ^ 0 * ((2 * \u2191n + 1 + 1) * ((2 * \u2191n + 1) * \u2191(2 * n)!)) *\n          ((2 * \u2191n + 1 + 1) * ((2 * \u2191n + 1) * \u2191(2 * n)!)) *\n        (2 * (\u2191n + 1) + 1)) =\n    2 ^ (4 * n) * 2 * 2 * 2 * 2 *\n        (((\u2191n + 1) * \u2191n !) ^ 0 * ((\u2191n + 1) * \u2191n !) * ((\u2191n + 1) * \u2191n !) * ((\u2191n + 1) * \u2191n !) * ((\u2191n + 1) * \u2191n !)) *\n      (\u2191(2 * n)! ^ 0 * \u2191(2 * n)! * \u2191(2 * n)! * (2 * \u2191n + 1) * ((2 * \u2191n + 1) * (2 * \u2191n + 3)))", "state_after": "no goals"}, {"tactic": "exact ne_of_gt (by positivity)", "annotated_tactic": ["exact <a>ne_of_gt</a> (by positivity)", [{"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}]], "state_before": "case succ.refine_2\nn : \u2115\nIH :\n  \u220f i \u2208 range n, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3)) =\n    2 ^ (4 * n) * \u2191n ! ^ 4 / (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1))\n\u22a2 \u2191(2 * (n + 1))! ^ 2 * (2 * \u2191(n + 1) + 1) \u2260 0", "state_after": "no goals"}, {"tactic": "positivity", "annotated_tactic": ["positivity", []], "state_before": "n : \u2115\nIH :\n  \u220f i \u2208 range n, (2 * \u2191i + 2) / (2 * \u2191i + 1) * ((2 * \u2191i + 2) / (2 * \u2191i + 3)) =\n    2 ^ (4 * n) * \u2191n ! ^ 4 / (\u2191(2 * n)! ^ 2 * (2 * \u2191n + 1))\n\u22a2 0 < \u2191(2 * (n + 1))! ^ 2 * (2 * \u2191(n + 1) + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/UniformSpace/Basic.lean", "full_name": "uniformContinuous\u2082_curry", "start": [1723, 1], "end": [1725, 41], "traced_tactics": [{"tactic": "rw [UniformContinuous\u2082, uncurry_curry]", "annotated_tactic": ["rw [<a>UniformContinuous\u2082</a>, <a>uncurry_curry</a>]", [{"full_name": "UniformContinuous\u2082", "def_path": "Mathlib/Topology/UniformSpace/Basic.lean", "def_pos": [1709, 5], "def_end_pos": [1709, 23]}, {"full_name": "Function.uncurry_curry", "def_path": "Mathlib/Init/Function.lean", "def_pos": [240, 9], "def_end_pos": [240, 22]}]], "state_before": "\u03b1 : Type ua\n\u03b2 : Type ub\n\u03b3 : Type uc\n\u03b4 : Type ud\n\u03b9 : Sort u_1\n\u03b4' : Type u_2\ninst\u271d\u2074 : UniformSpace \u03b1\ninst\u271d\u00b3 : UniformSpace \u03b2\ninst\u271d\u00b2 : UniformSpace \u03b3\ninst\u271d\u00b9 : UniformSpace \u03b4\ninst\u271d : UniformSpace \u03b4'\nf : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\n\u22a2 UniformContinuous\u2082 (curry f) \u2194 UniformContinuous f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/Perm/Basic.lean", "full_name": "Equiv.Perm.sumCongr_inv", "start": [203, 1], "end": [205, 20], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/WittVector/Truncated.lean", "full_name": "TruncatedWittVector.out_injective", "start": [118, 1], "end": [122, 36], "traced_tactics": [{"tactic": "intro x y h", "annotated_tactic": ["intro x y h", []], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\n\u22a2 Injective out", "state_after": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx y : TruncatedWittVector p n R\nh : x.out = y.out\n\u22a2 x = y"}, {"tactic": "ext i", "annotated_tactic": ["ext i", []], "state_before": "p : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx y : TruncatedWittVector p n R\nh : x.out = y.out\n\u22a2 x = y", "state_after": "case h\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx y : TruncatedWittVector p n R\nh : x.out = y.out\ni : Fin n\n\u22a2 coeff i x = coeff i y"}, {"tactic": "rw [WittVector.ext_iff] at h", "annotated_tactic": ["rw [<a>WittVector.ext_iff</a>] at h", [{"full_name": "WittVector.ext_iff", "def_path": "Mathlib/RingTheory/WittVector/Defs.lean", "def_pos": [81, 9], "def_end_pos": [81, 16]}]], "state_before": "case h\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx y : TruncatedWittVector p n R\nh : x.out = y.out\ni : Fin n\n\u22a2 coeff i x = coeff i y", "state_after": "case h\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx y : TruncatedWittVector p n R\nh : \u2200 (n_1 : \u2115), x.out.coeff n_1 = y.out.coeff n_1\ni : Fin n\n\u22a2 coeff i x = coeff i y"}, {"tactic": "simpa only [coeff_out] using h \u2191i", "annotated_tactic": ["simpa only [<a>coeff_out</a>] using h \u2191i", [{"full_name": "TruncatedWittVector.coeff_out", "def_path": "Mathlib/RingTheory/WittVector/Truncated.lean", "def_pos": [114, 9], "def_end_pos": [114, 18]}]], "state_before": "case h\np : \u2115\nhp : Fact (Nat.Prime p)\nn : \u2115\nR : Type u_1\ninst\u271d : CommRing R\nx y : TruncatedWittVector p n R\nh : \u2200 (n_1 : \u2115), x.out.coeff n_1 = y.out.coeff n_1\ni : Fin n\n\u22a2 coeff i x = coeff i y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/InnerProductSpace/l2Space.lean", "full_name": "HilbertBasis.coe_mkOfOrthogonalEqBot", "start": [554, 11], "end": [556, 27], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Image.lean", "full_name": "Finset.mem_image", "start": [357, 1], "end": [358, 75], "traced_tactics": [{"tactic": "simp only [mem_def, image_val, mem_dedup, Multiset.mem_map, exists_prop]", "annotated_tactic": ["simp only [<a>mem_def</a>, <a>image_val</a>, <a>mem_dedup</a>, <a>Multiset.mem_map</a>, <a>exists_prop</a>]", [{"full_name": "Finset.mem_def", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [175, 9], "def_end_pos": [175, 16]}, {"full_name": "Finset.image_val", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [345, 9], "def_end_pos": [345, 18]}, {"full_name": "Multiset.mem_dedup", "def_path": "Mathlib/Data/Multiset/Dedup.lean", "def_pos": [40, 9], "def_end_pos": [40, 18]}, {"full_name": "Multiset.mem_map", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1277, 9], "def_end_pos": [1277, 16]}, {"full_name": "exists_prop", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [307, 17], "def_end_pos": [307, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b2\nf g : \u03b1 \u2192 \u03b2\ns : Finset \u03b1\nt : Finset \u03b2\na : \u03b1\nb c : \u03b2\n\u22a2 b \u2208 image f s \u2194 \u2203 a \u2208 s, f a = b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Basic.lean", "full_name": "Filter.nonempty_of_neBot", "start": [724, 1], "end": [725, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Set/Lattice.lean", "full_name": "Set.iUnion\u2082_mono", "start": [335, 1], "end": [337, 15], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Monotone/Monovary.lean", "full_name": "MonovaryOn.dual", "start": [221, 1], "end": [222, 8], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "full_name": "TensorProduct.map_pow", "start": [919, 11], "end": [923, 39], "traced_tactics": [{"tactic": "induction' n with n ih", "annotated_tactic": ["induction' n with n ih", []], "state_before": "R : Type u_1\ninst\u271d\u00b2\u2070 : CommSemiring R\nR' : Type u_2\ninst\u271d\u00b9\u2079 : Monoid R'\nR'' : Type u_3\ninst\u271d\u00b9\u2078 : Semiring R''\nM : Type u_4\nN : Type u_5\nP : Type u_6\nQ : Type u_7\nS : Type u_8\nT : Type u_9\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : AddCommMonoid N\ninst\u271d\u00b9\u2075 : AddCommMonoid P\ninst\u271d\u00b9\u2074 : AddCommMonoid Q\ninst\u271d\u00b9\u00b3 : AddCommMonoid S\ninst\u271d\u00b9\u00b2 : AddCommMonoid T\ninst\u271d\u00b9\u00b9 : Module R M\ninst\u271d\u00b9\u2070 : Module R N\ninst\u271d\u2079 : Module R P\ninst\u271d\u2078 : Module R Q\ninst\u271d\u2077 : Module R S\ninst\u271d\u2076 : Module R T\ninst\u271d\u2075 : DistribMulAction R' M\ninst\u271d\u2074 : Module R'' M\nP' : Type u_10\nQ' : Type u_11\ninst\u271d\u00b3 : AddCommMonoid P'\ninst\u271d\u00b2 : Module R P'\ninst\u271d\u00b9 : AddCommMonoid Q'\ninst\u271d : Module R Q'\nf : M \u2192\u2097[R] M\ng : N \u2192\u2097[R] N\nn : \u2115\n\u22a2 map f g ^ n = map (f ^ n) (g ^ n)", "state_after": "case zero\nR : Type u_1\ninst\u271d\u00b2\u2070 : CommSemiring R\nR' : Type u_2\ninst\u271d\u00b9\u2079 : Monoid R'\nR'' : Type u_3\ninst\u271d\u00b9\u2078 : Semiring R''\nM : Type u_4\nN : Type u_5\nP : Type u_6\nQ : Type u_7\nS : Type u_8\nT : Type u_9\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : AddCommMonoid N\ninst\u271d\u00b9\u2075 : AddCommMonoid P\ninst\u271d\u00b9\u2074 : AddCommMonoid Q\ninst\u271d\u00b9\u00b3 : AddCommMonoid S\ninst\u271d\u00b9\u00b2 : AddCommMonoid T\ninst\u271d\u00b9\u00b9 : Module R M\ninst\u271d\u00b9\u2070 : Module R N\ninst\u271d\u2079 : Module R P\ninst\u271d\u2078 : Module R Q\ninst\u271d\u2077 : Module R S\ninst\u271d\u2076 : Module R T\ninst\u271d\u2075 : DistribMulAction R' M\ninst\u271d\u2074 : Module R'' M\nP' : Type u_10\nQ' : Type u_11\ninst\u271d\u00b3 : AddCommMonoid P'\ninst\u271d\u00b2 : Module R P'\ninst\u271d\u00b9 : AddCommMonoid Q'\ninst\u271d : Module R Q'\nf : M \u2192\u2097[R] M\ng : N \u2192\u2097[R] N\n\u22a2 map f g ^ 0 = map (f ^ 0) (g ^ 0)\n\ncase succ\nR : Type u_1\ninst\u271d\u00b2\u2070 : CommSemiring R\nR' : Type u_2\ninst\u271d\u00b9\u2079 : Monoid R'\nR'' : Type u_3\ninst\u271d\u00b9\u2078 : Semiring R''\nM : Type u_4\nN : Type u_5\nP : Type u_6\nQ : Type u_7\nS : Type u_8\nT : Type u_9\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : AddCommMonoid N\ninst\u271d\u00b9\u2075 : AddCommMonoid P\ninst\u271d\u00b9\u2074 : AddCommMonoid Q\ninst\u271d\u00b9\u00b3 : AddCommMonoid S\ninst\u271d\u00b9\u00b2 : AddCommMonoid T\ninst\u271d\u00b9\u00b9 : Module R M\ninst\u271d\u00b9\u2070 : Module R N\ninst\u271d\u2079 : Module R P\ninst\u271d\u2078 : Module R Q\ninst\u271d\u2077 : Module R S\ninst\u271d\u2076 : Module R T\ninst\u271d\u2075 : DistribMulAction R' M\ninst\u271d\u2074 : Module R'' M\nP' : Type u_10\nQ' : Type u_11\ninst\u271d\u00b3 : AddCommMonoid P'\ninst\u271d\u00b2 : Module R P'\ninst\u271d\u00b9 : AddCommMonoid Q'\ninst\u271d : Module R Q'\nf : M \u2192\u2097[R] M\ng : N \u2192\u2097[R] N\nn : \u2115\nih : map f g ^ n = map (f ^ n) (g ^ n)\n\u22a2 map f g ^ (n + 1) = map (f ^ (n + 1)) (g ^ (n + 1))"}, {"tactic": "simp only [Nat.zero_eq, pow_zero, map_one]", "annotated_tactic": ["simp only [<a>Nat.zero_eq</a>, <a>pow_zero</a>, <a>map_one</a>]", [{"full_name": "Nat.zero_eq", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [106, 17], "def_end_pos": [106, 24]}, {"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [651, 9], "def_end_pos": [651, 17]}, {"full_name": "TensorProduct.map_one", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [909, 9], "def_end_pos": [909, 16]}]], "state_before": "case zero\nR : Type u_1\ninst\u271d\u00b2\u2070 : CommSemiring R\nR' : Type u_2\ninst\u271d\u00b9\u2079 : Monoid R'\nR'' : Type u_3\ninst\u271d\u00b9\u2078 : Semiring R''\nM : Type u_4\nN : Type u_5\nP : Type u_6\nQ : Type u_7\nS : Type u_8\nT : Type u_9\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : AddCommMonoid N\ninst\u271d\u00b9\u2075 : AddCommMonoid P\ninst\u271d\u00b9\u2074 : AddCommMonoid Q\ninst\u271d\u00b9\u00b3 : AddCommMonoid S\ninst\u271d\u00b9\u00b2 : AddCommMonoid T\ninst\u271d\u00b9\u00b9 : Module R M\ninst\u271d\u00b9\u2070 : Module R N\ninst\u271d\u2079 : Module R P\ninst\u271d\u2078 : Module R Q\ninst\u271d\u2077 : Module R S\ninst\u271d\u2076 : Module R T\ninst\u271d\u2075 : DistribMulAction R' M\ninst\u271d\u2074 : Module R'' M\nP' : Type u_10\nQ' : Type u_11\ninst\u271d\u00b3 : AddCommMonoid P'\ninst\u271d\u00b2 : Module R P'\ninst\u271d\u00b9 : AddCommMonoid Q'\ninst\u271d : Module R Q'\nf : M \u2192\u2097[R] M\ng : N \u2192\u2097[R] N\n\u22a2 map f g ^ 0 = map (f ^ 0) (g ^ 0)", "state_after": "no goals"}, {"tactic": "simp only [pow_succ', ih, map_mul]", "annotated_tactic": ["simp only [<a>pow_succ'</a>, ih, <a>map_mul</a>]", [{"full_name": "pow_succ'", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [667, 34], "def_end_pos": [667, 43]}, {"full_name": "TensorProduct.map_mul", "def_path": "Mathlib/LinearAlgebra/TensorProduct/Basic.lean", "def_pos": [913, 9], "def_end_pos": [913, 16]}]], "state_before": "case succ\nR : Type u_1\ninst\u271d\u00b2\u2070 : CommSemiring R\nR' : Type u_2\ninst\u271d\u00b9\u2079 : Monoid R'\nR'' : Type u_3\ninst\u271d\u00b9\u2078 : Semiring R''\nM : Type u_4\nN : Type u_5\nP : Type u_6\nQ : Type u_7\nS : Type u_8\nT : Type u_9\ninst\u271d\u00b9\u2077 : AddCommMonoid M\ninst\u271d\u00b9\u2076 : AddCommMonoid N\ninst\u271d\u00b9\u2075 : AddCommMonoid P\ninst\u271d\u00b9\u2074 : AddCommMonoid Q\ninst\u271d\u00b9\u00b3 : AddCommMonoid S\ninst\u271d\u00b9\u00b2 : AddCommMonoid T\ninst\u271d\u00b9\u00b9 : Module R M\ninst\u271d\u00b9\u2070 : Module R N\ninst\u271d\u2079 : Module R P\ninst\u271d\u2078 : Module R Q\ninst\u271d\u2077 : Module R S\ninst\u271d\u2076 : Module R T\ninst\u271d\u2075 : DistribMulAction R' M\ninst\u271d\u2074 : Module R'' M\nP' : Type u_10\nQ' : Type u_11\ninst\u271d\u00b3 : AddCommMonoid P'\ninst\u271d\u00b2 : Module R P'\ninst\u271d\u00b9 : AddCommMonoid Q'\ninst\u271d : Module R Q'\nf : M \u2192\u2097[R] M\ng : N \u2192\u2097[R] N\nn : \u2115\nih : map f g ^ n = map (f ^ n) (g ^ n)\n\u22a2 map f g ^ (n + 1) = map (f ^ (n + 1)) (g ^ (n + 1))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Card.lean", "full_name": "Finset.filter_card_eq", "start": [332, 1], "end": [335, 13], "traced_tactics": [{"tactic": "rw [\u2190 eq_of_subset_of_card_le (s.filter_subset p) h.ge, mem_filter] at hx", "annotated_tactic": ["rw [\u2190 <a>eq_of_subset_of_card_le</a> (s.filter_subset p) h.ge, <a>mem_filter</a>] at hx", [{"full_name": "Finset.eq_of_subset_of_card_le", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [316, 9], "def_end_pos": [316, 32]}, {"full_name": "Finset.mem_filter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [2593, 9], "def_end_pos": [2593, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t u : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : (filter p s).card = s.card\nx : \u03b1\nhx : x \u2208 s\n\u22a2 p x", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t u : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : (filter p s).card = s.card\nx : \u03b1\nhx : x \u2208 s \u2227 p x\n\u22a2 p x"}, {"tactic": "exact hx.2", "annotated_tactic": ["exact hx.2", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nR : Type u_3\ns t u : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nn : \u2115\np : \u03b1 \u2192 Prop\ninst\u271d : DecidablePred p\nh : (filter p s).card = s.card\nx : \u03b1\nhx : x \u2208 s \u2227 p x\n\u22a2 p x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/NormedSpace/LinearIsometry.lean", "full_name": "LinearIsometryEquiv.toContinuousLinearEquiv_inj", "start": [717, 1], "end": [719, 43], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/VectorMeasure.lean", "full_name": "MeasureTheory.VectorMeasure.MutuallySingular.zero_left", "start": [1171, 1], "end": [1172, 18], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Convex/Quasiconvex.lean", "full_name": "quasilinearOn_iff_monotoneOn_or_antitoneOn", "start": [245, 1], "end": [248, 70], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Kernel/IntegralCompProd.lean", "full_name": "ProbabilityTheory.hasFiniteIntegral_compProd_iff'", "start": [107, 1], "end": [120, 53], "traced_tactics": [{"tactic": "rw [hasFiniteIntegral_congr h1f.ae_eq_mk,\n  hasFiniteIntegral_compProd_iff h1f.stronglyMeasurable_mk]", "annotated_tactic": ["rw [<a>hasFiniteIntegral_congr</a> h1f.ae_eq_mk,\n    <a>hasFiniteIntegral_compProd_iff</a> h1f.stronglyMeasurable_mk]", [{"full_name": "MeasureTheory.hasFiniteIntegral_congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}, {"full_name": "ProbabilityTheory.hasFiniteIntegral_compProd_iff", "def_path": "Mathlib/Probability/Kernel/IntegralCompProd.lean", "def_pos": [88, 9], "def_end_pos": [88, 39]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : \u21a5(kernel \u03b1 \u03b2)\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : \u21a5(kernel (\u03b1 \u00d7 \u03b2) \u03b3)\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f ((\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 HasFiniteIntegral f ((\u03ba \u2297\u2096 \u03b7) a) \u2194\n    (\u2200\u1d50 (x : \u03b2) \u2202\u03ba a, HasFiniteIntegral (fun y => f (x, y)) (\u03b7 (a, x))) \u2227\n      HasFiniteIntegral (fun x => \u222b (y : \u03b3), \u2016f (x, y)\u2016 \u2202\u03b7 (a, x)) (\u03ba a)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : \u21a5(kernel \u03b1 \u03b2)\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : \u21a5(kernel (\u03b1 \u00d7 \u03b2) \u03b3)\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f ((\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 (\u2200\u1d50 (x : \u03b2) \u2202\u03ba a, HasFiniteIntegral (fun y => AEStronglyMeasurable.mk f h1f (x, y)) (\u03b7 (a, x))) \u2227\n      HasFiniteIntegral (fun x => \u222b (y : \u03b3), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u03b7 (a, x)) (\u03ba a) \u2194\n    (\u2200\u1d50 (x : \u03b2) \u2202\u03ba a, HasFiniteIntegral (fun y => f (x, y)) (\u03b7 (a, x))) \u2227\n      HasFiniteIntegral (fun x => \u222b (y : \u03b3), \u2016f (x, y)\u2016 \u2202\u03b7 (a, x)) (\u03ba a)"}, {"tactic": "apply and_congr", "annotated_tactic": ["apply <a>and_congr</a>", [{"full_name": "and_congr", "def_path": ".lake/packages/lean4/src/lean/Init/PropLemmas.lean", "def_pos": [43, 9], "def_end_pos": [43, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : \u21a5(kernel \u03b1 \u03b2)\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : \u21a5(kernel (\u03b1 \u00d7 \u03b2) \u03b3)\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f ((\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 (\u2200\u1d50 (x : \u03b2) \u2202\u03ba a, HasFiniteIntegral (fun y => AEStronglyMeasurable.mk f h1f (x, y)) (\u03b7 (a, x))) \u2227\n      HasFiniteIntegral (fun x => \u222b (y : \u03b3), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u03b7 (a, x)) (\u03ba a) \u2194\n    (\u2200\u1d50 (x : \u03b2) \u2202\u03ba a, HasFiniteIntegral (fun y => f (x, y)) (\u03b7 (a, x))) \u2227\n      HasFiniteIntegral (fun x => \u222b (y : \u03b3), \u2016f (x, y)\u2016 \u2202\u03b7 (a, x)) (\u03ba a)", "state_after": "case h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : \u21a5(kernel \u03b1 \u03b2)\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : \u21a5(kernel (\u03b1 \u00d7 \u03b2) \u03b3)\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f ((\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 (\u2200\u1d50 (x : \u03b2) \u2202\u03ba a, HasFiniteIntegral (fun y => AEStronglyMeasurable.mk f h1f (x, y)) (\u03b7 (a, x))) \u2194\n    \u2200\u1d50 (x : \u03b2) \u2202\u03ba a, HasFiniteIntegral (fun y => f (x, y)) (\u03b7 (a, x))\n\ncase h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : \u21a5(kernel \u03b1 \u03b2)\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : \u21a5(kernel (\u03b1 \u00d7 \u03b2) \u03b3)\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f ((\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 HasFiniteIntegral (fun x => \u222b (y : \u03b3), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u03b7 (a, x)) (\u03ba a) \u2194\n    HasFiniteIntegral (fun x => \u222b (y : \u03b3), \u2016f (x, y)\u2016 \u2202\u03b7 (a, x)) (\u03ba a)"}, {"tactic": "apply eventually_congr", "annotated_tactic": ["apply <a>eventually_congr</a>", [{"full_name": "Filter.eventually_congr", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1179, 9], "def_end_pos": [1179, 25]}]], "state_before": "case h\u2081\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : \u21a5(kernel \u03b1 \u03b2)\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : \u21a5(kernel (\u03b1 \u00d7 \u03b2) \u03b3)\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f ((\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 (\u2200\u1d50 (x : \u03b2) \u2202\u03ba a, HasFiniteIntegral (fun y => AEStronglyMeasurable.mk f h1f (x, y)) (\u03b7 (a, x))) \u2194\n    \u2200\u1d50 (x : \u03b2) \u2202\u03ba a, HasFiniteIntegral (fun y => f (x, y)) (\u03b7 (a, x))", "state_after": "case h\u2081.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : \u21a5(kernel \u03b1 \u03b2)\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : \u21a5(kernel (\u03b1 \u00d7 \u03b2) \u03b3)\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f ((\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202\u03ba a,\n    HasFiniteIntegral (fun y => AEStronglyMeasurable.mk f h1f (x, y)) (\u03b7 (a, x)) \u2194\n      HasFiniteIntegral (fun y => f (x, y)) (\u03b7 (a, x))"}, {"tactic": "filter_upwards [ae_ae_of_ae_compProd h1f.ae_eq_mk.symm] with x hx using\n  hasFiniteIntegral_congr hx", "annotated_tactic": ["filter_upwards [<a>ae_ae_of_ae_compProd</a> h1f.ae_eq_mk.symm] with x hx using\n      <a>hasFiniteIntegral_congr</a> hx", [{"full_name": "ProbabilityTheory.kernel.ae_ae_of_ae_compProd", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [317, 9], "def_end_pos": [317, 29]}, {"full_name": "MeasureTheory.hasFiniteIntegral_congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}]], "state_before": "case h\u2081.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : \u21a5(kernel \u03b1 \u03b2)\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : \u21a5(kernel (\u03b1 \u00d7 \u03b2) \u03b3)\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f ((\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 \u2200\u1d50 (x : \u03b2) \u2202\u03ba a,\n    HasFiniteIntegral (fun y => AEStronglyMeasurable.mk f h1f (x, y)) (\u03b7 (a, x)) \u2194\n      HasFiniteIntegral (fun y => f (x, y)) (\u03b7 (a, x))", "state_after": "no goals"}, {"tactic": "apply hasFiniteIntegral_congr", "annotated_tactic": ["apply <a>hasFiniteIntegral_congr</a>", [{"full_name": "MeasureTheory.hasFiniteIntegral_congr", "def_path": "Mathlib/MeasureTheory/Function/L1Space.lean", "def_pos": [163, 9], "def_end_pos": [163, 32]}]], "state_before": "case h\u2082\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : \u21a5(kernel \u03b1 \u03b2)\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : \u21a5(kernel (\u03b1 \u00d7 \u03b2) \u03b3)\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f ((\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 HasFiniteIntegral (fun x => \u222b (y : \u03b3), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u03b7 (a, x)) (\u03ba a) \u2194\n    HasFiniteIntegral (fun x => \u222b (y : \u03b3), \u2016f (x, y)\u2016 \u2202\u03b7 (a, x)) (\u03ba a)", "state_after": "case h\u2082.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : \u21a5(kernel \u03b1 \u03b2)\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : \u21a5(kernel (\u03b1 \u00d7 \u03b2) \u03b3)\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f ((\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 (fun x => \u222b (y : \u03b3), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u03b7 (a, x)) =\u1da0[ae (\u03ba a)] fun x =>\n    \u222b (y : \u03b3), \u2016f (x, y)\u2016 \u2202\u03b7 (a, x)"}, {"tactic": "filter_upwards [ae_ae_of_ae_compProd h1f.ae_eq_mk.symm] with _ hx using\n  integral_congr_ae (EventuallyEq.fun_comp hx _)", "annotated_tactic": ["filter_upwards [<a>ae_ae_of_ae_compProd</a> h1f.ae_eq_mk.symm] with _ hx using\n      <a>integral_congr_ae</a> (<a>EventuallyEq.fun_comp</a> hx _)", [{"full_name": "ProbabilityTheory.kernel.ae_ae_of_ae_compProd", "def_path": "Mathlib/Probability/Kernel/Composition.lean", "def_pos": [317, 9], "def_end_pos": [317, 29]}, {"full_name": "MeasureTheory.integral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Bochner.lean", "def_pos": [938, 9], "def_end_pos": [938, 26]}, {"full_name": "Filter.EventuallyEq.fun_comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1554, 9], "def_end_pos": [1554, 30]}]], "state_before": "case h\u2082.h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nE : Type u_4\nm\u03b1 : MeasurableSpace \u03b1\nm\u03b2 : MeasurableSpace \u03b2\nm\u03b3 : MeasurableSpace \u03b3\ninst\u271d\u00b2 : NormedAddCommGroup E\n\u03ba : \u21a5(kernel \u03b1 \u03b2)\ninst\u271d\u00b9 : IsSFiniteKernel \u03ba\n\u03b7 : \u21a5(kernel (\u03b1 \u00d7 \u03b2) \u03b3)\ninst\u271d : IsSFiniteKernel \u03b7\na : \u03b1\nf : \u03b2 \u00d7 \u03b3 \u2192 E\nh1f : AEStronglyMeasurable f ((\u03ba \u2297\u2096 \u03b7) a)\n\u22a2 (fun x => \u222b (y : \u03b3), \u2016AEStronglyMeasurable.mk f h1f (x, y)\u2016 \u2202\u03b7 (a, x)) =\u1da0[ae (\u03ba a)] fun x =>\n    \u222b (y : \u03b3), \u2016f (x, y)\u2016 \u2202\u03b7 (a, x)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Extr.lean", "full_name": "Filter.EventuallyLE.isMinFilter", "start": [654, 1], "end": [657, 62], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean", "full_name": "SimpleGraph.componentComplMk_mem", "start": [67, 1], "end": [68, 12], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/RelClasses.lean", "full_name": "ssubset_of_subset_of_ne", "start": [768, 1], "end": [769, 48], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Noetherian.lean", "full_name": "isNoetherianRing_of_surjective", "start": [626, 1], "end": [629, 82], "traced_tactics": [{"tactic": "rw [isNoetherianRing_iff, isNoetherian_iff_wellFounded] at H \u22a2", "annotated_tactic": ["rw [<a>isNoetherianRing_iff</a>, <a>isNoetherian_iff_wellFounded</a>] at H \u22a2", [{"full_name": "isNoetherianRing_iff", "def_path": "Mathlib/RingTheory/Noetherian.lean", "def_pos": [538, 9], "def_end_pos": [538, 29]}, {"full_name": "isNoetherian_iff_wellFounded", "def_path": "Mathlib/RingTheory/Noetherian.lean", "def_pos": [313, 9], "def_end_pos": [313, 37]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9 : Ring R\nS : Type u_2\ninst\u271d : Ring S\nf : R \u2192+* S\nhf : Surjective \u21d1f\nH : IsNoetherianRing R\n\u22a2 IsNoetherianRing S", "state_after": "R : Type u_1\ninst\u271d\u00b9 : Ring R\nS : Type u_2\ninst\u271d : Ring S\nf : R \u2192+* S\nhf : Surjective \u21d1f\nH : WellFounded fun x x_1 => x > x_1\n\u22a2 WellFounded fun x x_1 => x > x_1"}, {"tactic": "exact OrderEmbedding.wellFounded (Ideal.orderEmbeddingOfSurjective f hf).dual H", "annotated_tactic": ["exact <a>OrderEmbedding.wellFounded</a> (<a>Ideal.orderEmbeddingOfSurjective</a> f hf).<a>dual</a> H", [{"full_name": "OrderEmbedding.wellFounded", "def_path": "Mathlib/Order/Hom/Basic.lean", "def_pos": [660, 19], "def_end_pos": [660, 30]}, {"full_name": "Ideal.orderEmbeddingOfSurjective", "def_path": "Mathlib/RingTheory/Ideal/Maps.lean", "def_pos": [441, 5], "def_end_pos": [441, 31]}, {"full_name": "OrderEmbedding.dual", "def_path": "Mathlib/Order/Hom/Basic.lean", "def_pos": [670, 15], "def_end_pos": [670, 19]}]], "state_before": "R : Type u_1\ninst\u271d\u00b9 : Ring R\nS : Type u_2\ninst\u271d : Ring S\nf : R \u2192+* S\nhf : Surjective \u21d1f\nH : WellFounded fun x x_1 => x > x_1\n\u22a2 WellFounded fun x x_1 => x > x_1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Ring/Parity.lean", "full_name": "even_bit0", "start": [91, 1], "end": [91, 60], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/GCDMonoid/Basic.lean", "full_name": "dvd_normalize_iff", "start": [200, 1], "end": [201, 22], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "full_name": "bot_eq_one", "start": [211, 1], "end": [212, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean", "full_name": "RingHom.PropertyIsLocal.affine_openCover_TFAE", "start": [346, 1], "end": [370, 14], "traced_tactics": [{"tactic": "tfae_have 1 \u2192 4", "annotated_tactic": ["tfae_have 1 \u2192 4", []], "state_before": "P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\n\u22a2 [sourceAffineLocally P f,\n      \u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op),\n      \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op),\n      \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)].TFAE", "state_after": "case tfae_1_to_4\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\n\u22a2 sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\ntfae_1_to_4 :\n  sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\n\u22a2 [sourceAffineLocally P f,\n      \u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op),\n      \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op),\n      \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)].TFAE"}, {"tactic": "tfae_have 4 \u2192 3", "annotated_tactic": ["tfae_have 4 \u2192 3", []], "state_before": "P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\ntfae_1_to_4 :\n  sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\n\u22a2 [sourceAffineLocally P f,\n      \u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op),\n      \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op),\n      \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)].TFAE", "state_after": "case tfae_4_to_3\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\ntfae_1_to_4 :\n  sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\n\u22a2 (\u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)) \u2192\n    \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\ntfae_1_to_4 :\n  sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\ntfae_4_to_3 :\n  (\u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)) \u2192\n    \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\n\u22a2 [sourceAffineLocally P f,\n      \u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op),\n      \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op),\n      \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)].TFAE"}, {"tactic": "tfae_have 3 \u2192 2", "annotated_tactic": ["tfae_have 3 \u2192 2", []], "state_before": "P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\ntfae_1_to_4 :\n  sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\ntfae_4_to_3 :\n  (\u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)) \u2192\n    \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\n\u22a2 [sourceAffineLocally P f,\n      \u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op),\n      \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op),\n      \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)].TFAE", "state_after": "case tfae_3_to_2\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\ntfae_1_to_4 :\n  sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\ntfae_4_to_3 :\n  (\u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)) \u2192\n    \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\n\u22a2 (\u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)) \u2192\n    \u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\ntfae_1_to_4 :\n  sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\ntfae_4_to_3 :\n  (\u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)) \u2192\n    \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\ntfae_3_to_2 :\n  (\u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)) \u2192\n    \u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\n\u22a2 [sourceAffineLocally P f,\n      \u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op),\n      \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op),\n      \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)].TFAE"}, {"tactic": "tfae_have 2 \u2192 1", "annotated_tactic": ["tfae_have 2 \u2192 1", []], "state_before": "P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\ntfae_1_to_4 :\n  sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\ntfae_4_to_3 :\n  (\u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)) \u2192\n    \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\ntfae_3_to_2 :\n  (\u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)) \u2192\n    \u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\n\u22a2 [sourceAffineLocally P f,\n      \u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op),\n      \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op),\n      \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)].TFAE", "state_after": "case tfae_2_to_1\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\ntfae_1_to_4 :\n  sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\ntfae_4_to_3 :\n  (\u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)) \u2192\n    \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\ntfae_3_to_2 :\n  (\u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)) \u2192\n    \u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\n\u22a2 (\u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)) \u2192\n    sourceAffineLocally P f\n\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\ntfae_1_to_4 :\n  sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\ntfae_4_to_3 :\n  (\u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)) \u2192\n    \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\ntfae_3_to_2 :\n  (\u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)) \u2192\n    \u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\ntfae_2_to_1 :\n  (\u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)) \u2192\n    sourceAffineLocally P f\n\u22a2 [sourceAffineLocally P f,\n      \u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op),\n      \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op),\n      \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)].TFAE"}, {"tactic": "tfae_finish", "annotated_tactic": ["tfae_finish", []], "state_before": "P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\ntfae_1_to_4 :\n  sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\ntfae_4_to_3 :\n  (\u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)) \u2192\n    \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\ntfae_3_to_2 :\n  (\u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)) \u2192\n    \u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\ntfae_2_to_1 :\n  (\u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)) \u2192\n    sourceAffineLocally P f\n\u22a2 [sourceAffineLocally P f,\n      \u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op),\n      \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op),\n      \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)].TFAE", "state_after": "no goals"}, {"tactic": "intro H U g _ hg", "annotated_tactic": ["intro H U g _ hg", []], "state_before": "case tfae_1_to_4\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\n\u22a2 sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)", "state_after": "case tfae_1_to_4\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d\u00b9 : IsAffine Y\nf : X \u27f6 Y\nH : sourceAffineLocally P f\nU : Scheme\ng : U \u27f6 X\ninst\u271d : IsAffine U\nhg : IsOpenImmersion g\n\u22a2 P (Scheme.\u0393.map (g \u226b f).op)"}, {"tactic": "specialize H \u27e8\u27e8_, hg.base_open.isOpen_range\u27e9, isAffineOpen_opensRange g\u27e9", "annotated_tactic": ["specialize H \u27e8\u27e8_, hg.base_open.isOpen_range\u27e9, <a>isAffineOpen_opensRange</a> g\u27e9", [{"full_name": "AlgebraicGeometry.isAffineOpen_opensRange", "def_path": "Mathlib/AlgebraicGeometry/AffineScheme.lean", "def_pos": [195, 9], "def_end_pos": [195, 32]}]], "state_before": "case tfae_1_to_4\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d\u00b9 : IsAffine Y\nf : X \u27f6 Y\nH : sourceAffineLocally P f\nU : Scheme\ng : U \u27f6 X\ninst\u271d : IsAffine U\nhg : IsOpenImmersion g\n\u22a2 P (Scheme.\u0393.map (g \u226b f).op)", "state_after": "case tfae_1_to_4\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d\u00b9 : IsAffine Y\nf : X \u27f6 Y\nU : Scheme\ng : U \u27f6 X\ninst\u271d : IsAffine U\nhg : IsOpenImmersion g\nH : P (Scheme.\u0393.map (X.ofRestrict \u22ef \u226b f).op)\n\u22a2 P (Scheme.\u0393.map (g \u226b f).op)"}, {"tactic": "rw [\u2190 hP.respectsIso.cancel_right_isIso _ (Scheme.\u0393.map (IsOpenImmersion.isoOfRangeEq g\n  (X.ofRestrict (Opens.openEmbedding \u27e8_, hg.base_open.isOpen_range\u27e9))\n  Subtype.range_coe.symm).hom.op),\n  \u2190 Scheme.\u0393.map_comp, \u2190 op_comp, IsOpenImmersion.isoOfRangeEq_hom_fac_assoc] at H", "annotated_tactic": ["rw [\u2190 hP.respectsIso.cancel_right_isIso _ (Scheme.\u0393.map (<a>IsOpenImmersion.isoOfRangeEq</a> g\n      (X.ofRestrict (<a>Opens.openEmbedding</a> \u27e8_, hg.base_open.isOpen_range\u27e9))\n      Subtype.range_coe.symm).hom.op),\n      \u2190 Scheme.\u0393.map_comp, \u2190 <a>op_comp</a>, <a>IsOpenImmersion.isoOfRangeEq_hom_fac_assoc</a>] at H", [{"full_name": "AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq", "def_path": "Mathlib/AlgebraicGeometry/OpenImmersion.lean", "def_pos": [502, 5], "def_end_pos": [502, 17]}, {"full_name": "TopologicalSpace.Opens.openEmbedding", "def_path": "Mathlib/Topology/Category/TopCat/Opens.lean", "def_pos": [138, 9], "def_end_pos": [138, 22]}, {"full_name": "CategoryTheory.op_comp", "def_path": "Mathlib/CategoryTheory/Opposites.lean", "def_pos": [79, 9], "def_end_pos": [79, 16]}, {"full_name": "AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq_hom_fac_assoc", "def_path": "Mathlib/AlgebraicGeometry/OpenImmersion.lean", "def_pos": [509, 9], "def_end_pos": [509, 16]}]], "state_before": "case tfae_1_to_4\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d\u00b9 : IsAffine Y\nf : X \u27f6 Y\nU : Scheme\ng : U \u27f6 X\ninst\u271d : IsAffine U\nhg : IsOpenImmersion g\nH : P (Scheme.\u0393.map (X.ofRestrict \u22ef \u226b f).op)\n\u22a2 P (Scheme.\u0393.map (g \u226b f).op)", "state_after": "case tfae_1_to_4\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d\u00b9 : IsAffine Y\nf : X \u27f6 Y\nU : Scheme\ng : U \u27f6 X\ninst\u271d : IsAffine U\nhg : IsOpenImmersion g\nH : P (Scheme.\u0393.map (g \u226b f).op)\n\u22a2 P (Scheme.\u0393.map (g \u226b f).op)"}, {"tactic": "exact H", "annotated_tactic": ["exact H", []], "state_before": "case tfae_1_to_4\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d\u00b9 : IsAffine Y\nf : X \u27f6 Y\nU : Scheme\ng : U \u27f6 X\ninst\u271d : IsAffine U\nhg : IsOpenImmersion g\nH : P (Scheme.\u0393.map (g \u226b f).op)\n\u22a2 P (Scheme.\u0393.map (g \u226b f).op)", "state_after": "no goals"}, {"tactic": "intro H \ud835\udcb0 _ i", "annotated_tactic": ["intro H \ud835\udcb0 _ i", []], "state_before": "case tfae_4_to_3\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\ntfae_1_to_4 :\n  sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\n\u22a2 (\u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)) \u2192\n    \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)", "state_after": "case tfae_4_to_3\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d\u00b9 : IsAffine Y\nf : X \u27f6 Y\ntfae_1_to_4 :\n  sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\nH : \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\n\ud835\udcb0 : X.OpenCover\ninst\u271d : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)\ni : \ud835\udcb0.J\n\u22a2 P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)"}, {"tactic": "apply H", "annotated_tactic": ["apply H", []], "state_before": "case tfae_4_to_3\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d\u00b9 : IsAffine Y\nf : X \u27f6 Y\ntfae_1_to_4 :\n  sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\nH : \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\n\ud835\udcb0 : X.OpenCover\ninst\u271d : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)\ni : \ud835\udcb0.J\n\u22a2 P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)", "state_after": "no goals"}, {"tactic": "intro H", "annotated_tactic": ["intro H", []], "state_before": "case tfae_3_to_2\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\ntfae_1_to_4 :\n  sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\ntfae_4_to_3 :\n  (\u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)) \u2192\n    \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\n\u22a2 (\u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)) \u2192\n    \u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)", "state_after": "case tfae_3_to_2\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\ntfae_1_to_4 :\n  sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\ntfae_4_to_3 :\n  (\u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)) \u2192\n    \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\nH : \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\n\u22a2 \u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)"}, {"tactic": "exact \u27e8X.affineCover, inferInstance, H _\u27e9", "annotated_tactic": ["exact \u27e8X.affineCover, <a>inferInstance</a>, H _\u27e9", [{"full_name": "inferInstance", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [99, 8], "def_end_pos": [99, 21]}]], "state_before": "case tfae_3_to_2\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\ntfae_1_to_4 :\n  sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\ntfae_4_to_3 :\n  (\u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)) \u2192\n    \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\nH : \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\n\u22a2 \u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)", "state_after": "no goals"}, {"tactic": "rintro \u27e8\ud835\udcb0, _, h\ud835\udcb0\u27e9", "annotated_tactic": ["rintro \u27e8\ud835\udcb0, _, h\ud835\udcb0\u27e9", []], "state_before": "case tfae_2_to_1\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\ntfae_1_to_4 :\n  sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\ntfae_4_to_3 :\n  (\u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)) \u2192\n    \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\ntfae_3_to_2 :\n  (\u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)) \u2192\n    \u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\n\u22a2 (\u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)) \u2192\n    sourceAffineLocally P f", "state_after": "case tfae_2_to_1.intro.intro\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\ntfae_1_to_4 :\n  sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\ntfae_4_to_3 :\n  (\u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)) \u2192\n    \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\ntfae_3_to_2 :\n  (\u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)) \u2192\n    \u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\n\ud835\udcb0 : X.OpenCover\nw\u271d : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)\nh\ud835\udcb0 : \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\n\u22a2 sourceAffineLocally P f"}, {"tactic": "exact sourceAffineLocally_of_source_openCover hP f \ud835\udcb0 h\ud835\udcb0", "annotated_tactic": ["exact <a>sourceAffineLocally_of_source_openCover</a> hP f \ud835\udcb0 h\ud835\udcb0", [{"full_name": "RingHom.PropertyIsLocal.sourceAffineLocally_of_source_openCover", "def_path": "Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean", "def_pos": [289, 9], "def_end_pos": [289, 48]}]], "state_before": "case tfae_2_to_1.intro.intro\nP : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop\nhP : PropertyIsLocal P\nX Y : Scheme\ninst\u271d : IsAffine Y\nf : X \u27f6 Y\ntfae_1_to_4 :\n  sourceAffineLocally P f \u2192\n    \u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)\ntfae_4_to_3 :\n  (\u2200 {U : Scheme} (g : U \u27f6 X) [inst : IsAffine U] [inst : IsOpenImmersion g], P (Scheme.\u0393.map (g \u226b f).op)) \u2192\n    \u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\ntfae_3_to_2 :\n  (\u2200 (\ud835\udcb0 : X.OpenCover) [inst : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)) \u2192\n    \u2203 \ud835\udcb0, \u2203 (_ : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)), \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\n\ud835\udcb0 : X.OpenCover\nw\u271d : \u2200 (i : \ud835\udcb0.J), IsAffine (\ud835\udcb0.obj i)\nh\ud835\udcb0 : \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)\n\u22a2 sourceAffineLocally P f", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Star/Module.lean", "full_name": "IsSelfAdjoint.coe_selfAdjointPart_apply", "start": [156, 1], "end": [158, 90], "traced_tactics": [{"tactic": "rw [selfAdjointPart_apply_coe, hx.star_eq, smul_add, invOf_two_smul_add_invOf_two_smul]", "annotated_tactic": ["rw [<a>selfAdjointPart_apply_coe</a>, hx.star_eq, <a>smul_add</a>, <a>invOf_two_smul_add_invOf_two_smul</a>]", [{"full_name": "selfAdjointPart_apply_coe", "def_path": "Mathlib/Algebra/Star/Module.lean", "def_pos": [120, 3], "def_end_pos": [120, 8]}, {"full_name": "smul_add", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [145, 9], "def_end_pos": [145, 17]}, {"full_name": "invOf_two_smul_add_invOf_two_smul", "def_path": "Mathlib/Algebra/Module/Defs.lean", "def_pos": [114, 9], "def_end_pos": [114, 42]}]], "state_before": "R : Type u_1\nA : Type u_2\ninst\u271d\u2077 : Semiring R\ninst\u271d\u2076 : StarMul R\ninst\u271d\u2075 : TrivialStar R\ninst\u271d\u2074 : AddCommGroup A\ninst\u271d\u00b3 : Module R A\ninst\u271d\u00b2 : StarAddMonoid A\ninst\u271d\u00b9 : StarModule R A\ninst\u271d : Invertible 2\nx : A\nhx : IsSelfAdjoint x\n\u22a2 \u2191((selfAdjointPart R) x) = x", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Perfection.lean", "full_name": "Perfection.coeff_iterate_frobenius'", "start": [147, 1], "end": [149, 98], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Module/Submodule/Pointwise.lean", "full_name": "Submodule.set_smul_le_of_le_le", "start": [394, 1], "end": [396, 75], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean", "full_name": "InnerProductGeometry.cos_angle_add_of_inner_eq_zero", "start": [132, 1], "end": [139, 51], "traced_tactics": [{"tactic": "rw [angle_add_eq_arccos_of_inner_eq_zero h,\n  Real.cos_arccos (le_trans (by norm_num) (div_nonneg (norm_nonneg _) (norm_nonneg _)))\n    (div_le_one_of_le _ (norm_nonneg _))]", "annotated_tactic": ["rw [<a>angle_add_eq_arccos_of_inner_eq_zero</a> h,\n    <a>Real.cos_arccos</a> (<a>le_trans</a> (by norm_num) (<a>div_nonneg</a> (<a>norm_nonneg</a> _) (<a>norm_nonneg</a> _)))\n      (<a>div_le_one_of_le</a> _ (<a>norm_nonneg</a> _))]", [{"full_name": "InnerProductGeometry.angle_add_eq_arccos_of_inner_eq_zero", "def_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean", "def_pos": [69, 9], "def_end_pos": [69, 45]}, {"full_name": "Real.cos_arccos", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean", "def_pos": [361, 9], "def_end_pos": [361, 19]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "div_nonneg", "def_path": "Mathlib/Algebra/Order/Field/Defs.lean", "def_pos": [83, 7], "def_end_pos": [83, 17]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [469, 30], "def_end_pos": [469, 41]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [469, 30], "def_end_pos": [469, 41]}, {"full_name": "div_le_one_of_le", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [165, 9], "def_end_pos": [165, 25]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [469, 30], "def_end_pos": [469, 41]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : \u27eax, y\u27eb_\u211d = 0\n\u22a2 Real.cos (angle x (x + y)) = \u2016x\u2016 / \u2016x + y\u2016", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : \u27eax, y\u27eb_\u211d = 0\n\u22a2 \u2016x\u2016 \u2264 \u2016x + y\u2016"}, {"tactic": "rw [mul_self_le_mul_self_iff (norm_nonneg _) (norm_nonneg _),\n  norm_add_sq_eq_norm_sq_add_norm_sq_real h]", "annotated_tactic": ["rw [<a>mul_self_le_mul_self_iff</a> (<a>norm_nonneg</a> _) (<a>norm_nonneg</a> _),\n    <a>norm_add_sq_eq_norm_sq_add_norm_sq_real</a> h]", [{"full_name": "mul_self_le_mul_self_iff", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [1005, 9], "def_end_pos": [1005, 33]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [469, 30], "def_end_pos": [469, 41]}, {"full_name": "norm_nonneg", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [469, 30], "def_end_pos": [469, 41]}, {"full_name": "norm_add_sq_eq_norm_sq_add_norm_sq_real", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [1472, 9], "def_end_pos": [1472, 48]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : \u27eax, y\u27eb_\u211d = 0\n\u22a2 \u2016x\u2016 \u2264 \u2016x + y\u2016", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : \u27eax, y\u27eb_\u211d = 0\n\u22a2 \u2016x\u2016 * \u2016x\u2016 \u2264 \u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016"}, {"tactic": "exact le_add_of_nonneg_right (mul_self_nonneg _)", "annotated_tactic": ["exact <a>le_add_of_nonneg_right</a> (<a>mul_self_nonneg</a> _)", [{"full_name": "le_add_of_nonneg_right", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [391, 15], "def_end_pos": [391, 37]}, {"full_name": "mul_self_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [1176, 7], "def_end_pos": [1176, 22]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : \u27eax, y\u27eb_\u211d = 0\n\u22a2 \u2016x\u2016 * \u2016x\u2016 \u2264 \u2016x\u2016 * \u2016x\u2016 + \u2016y\u2016 * \u2016y\u2016", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : \u27eax, y\u27eb_\u211d = 0\n\u22a2 -1 \u2264 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/MetricSpace/Lipschitz.lean", "full_name": "continuousAt_of_locally_lipschitz", "start": [371, 1], "end": [378, 7], "traced_tactics": [{"tactic": "refine tendsto_iff_dist_tendsto_zero.2 (squeeze_zero' (eventually_of_forall fun _ => dist_nonneg)\n  (mem_of_superset (ball_mem_nhds _ hr) h) ?_)", "annotated_tactic": ["refine <a>tendsto_iff_dist_tendsto_zero</a>.2 (<a>squeeze_zero'</a> (<a>eventually_of_forall</a> fun _ => <a>dist_nonneg</a>)\n    (<a>mem_of_superset</a> (<a>ball_mem_nhds</a> _ hr) h) ?_)", [{"full_name": "tendsto_iff_dist_tendsto_zero", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [1437, 9], "def_end_pos": [1437, 38]}, {"full_name": "squeeze_zero'", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Lemmas.lean", "def_pos": [48, 7], "def_end_pos": [48, 20]}, {"full_name": "Filter.eventually_of_forall", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1129, 9], "def_end_pos": [1129, 29]}, {"full_name": "dist_nonneg", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [259, 9], "def_end_pos": [259, 20]}, {"full_name": "Filter.mem_of_superset", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [151, 9], "def_end_pos": [151, 24]}, {"full_name": "Metric.ball_mem_nhds", "def_path": "Mathlib/Topology/MetricSpace/Pseudo/Defs.lean", "def_pos": [1016, 9], "def_end_pos": [1016, 22]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\nr : \u211d\nhr : 0 < r\nK : \u211d\nh : \u2200 (y : \u03b1), dist y x < r \u2192 dist (f y) (f x) \u2264 K * dist y x\n\u22a2 ContinuousAt f x", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\nr : \u211d\nhr : 0 < r\nK : \u211d\nh : \u2200 (y : \u03b1), dist y x < r \u2192 dist (f y) (f x) \u2264 K * dist y x\n\u22a2 Tendsto (fun a => K * dist a x) (\ud835\udcdd x) (\ud835\udcdd 0)"}, {"tactic": "refine (continuous_const.mul (continuous_id.dist continuous_const)).tendsto' _ _ ?_", "annotated_tactic": ["refine (continuous_const.mul (continuous_id.dist <a>continuous_const</a>)).<a>tendsto'</a> _ _ ?_", [{"full_name": "continuous_const", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1675, 9], "def_end_pos": [1675, 25]}, {"full_name": "Continuous.tendsto'", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1654, 9], "def_end_pos": [1654, 28]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\nr : \u211d\nhr : 0 < r\nK : \u211d\nh : \u2200 (y : \u03b1), dist y x < r \u2192 dist (f y) (f x) \u2264 K * dist y x\n\u22a2 Tendsto (fun a => K * dist a x) (\ud835\udcdd x) (\ud835\udcdd 0)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\nr : \u211d\nhr : 0 < r\nK : \u211d\nh : \u2200 (y : \u03b1), dist y x < r \u2192 dist (f y) (f x) \u2264 K * dist y x\n\u22a2 K * dist (id x) x = 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d\u00b9 : PseudoMetricSpace \u03b1\ninst\u271d : PseudoMetricSpace \u03b2\nf : \u03b1 \u2192 \u03b2\nx : \u03b1\nr : \u211d\nhr : 0 < r\nK : \u211d\nh : \u2200 (y : \u03b1), dist y x < r \u2192 dist (f y) (f x) \u2264 K * dist y x\n\u22a2 K * dist (id x) x = 0", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/Bornology.lean", "full_name": "orderBornology_isBounded", "start": [36, 1], "end": [37, 56], "traced_tactics": [{"tactic": "simp [IsBounded, IsCobounded, -isCobounded_compl_iff]", "annotated_tactic": ["simp [<a>IsBounded</a>, <a>IsCobounded</a>, -<a>isCobounded_compl_iff</a>]", [{"full_name": "Bornology.IsBounded", "def_path": "Mathlib/Topology/Bornology/Basic.lean", "def_pos": [128, 5], "def_end_pos": [128, 14]}, {"full_name": "Bornology.IsCobounded", "def_path": "Mathlib/Topology/Bornology/Basic.lean", "def_pos": [123, 5], "def_end_pos": [123, 16]}, {"full_name": "Bornology.isCobounded_compl_iff", "def_path": "Mathlib/Topology/Bornology/Basic.lean", "def_pos": [148, 9], "def_end_pos": [148, 30]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : Bornology \u03b1\ns t : Set \u03b1\ninst\u271d\u00b9 : Lattice \u03b1\ninst\u271d : Nonempty \u03b1\n\u22a2 IsBounded s \u2194 BddBelow s \u2227 BddAbove s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/ModelTheory/Semantics.lean", "full_name": "FirstOrder.Language.LHom.realize_onFormula", "start": [699, 1], "end": [701, 31], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "full_name": "MeasureTheory.OuterMeasure.coe_mkMetric", "start": [469, 1], "end": [471, 69], "traced_tactics": [{"tactic": "rw [\u2190 Measure.mkMetric_toOuterMeasure, Measure.coe_toOuterMeasure]", "annotated_tactic": ["rw [\u2190 <a>Measure.mkMetric_toOuterMeasure</a>, <a>Measure.coe_toOuterMeasure</a>]", [{"full_name": "MeasureTheory.Measure.mkMetric_toOuterMeasure", "def_path": "Mathlib/MeasureTheory/Measure/Hausdorff.lean", "def_pos": [462, 9], "def_end_pos": [462, 32]}, {"full_name": "MeasureTheory.Measure.coe_toOuterMeasure", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [153, 17], "def_end_pos": [153, 43]}]], "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 \u21d1(mkMetric m) = \u21d1(Measure.mkMetric m)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Lie/Subalgebra.lean", "full_name": "LieSubalgebra.incl_range", "start": [366, 1], "end": [368, 42], "traced_tactics": [{"tactic": "rw [\u2190 coe_to_submodule_eq_iff]", "annotated_tactic": ["rw [\u2190 <a>coe_to_submodule_eq_iff</a>]", [{"full_name": "LieSubalgebra.coe_to_submodule_eq_iff", 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"commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Function/SimpleFunc.lean", "full_name": "MeasureTheory.SimpleFunc.piecewise_same", "start": [261, 1], "end": [263, 42], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Abelian/Pseudoelements.lean", "full_name": "CategoryTheory.Abelian.Pseudoelement.zero_apply", "start": [275, 1], "end": [278, 9], "traced_tactics": [{"tactic": "rw [pseudoZero_def, pseudoApply_mk']", "annotated_tactic": ["rw [<a>pseudoZero_def</a>, <a>pseudoApply_mk'</a>]", [{"full_name": "CategoryTheory.Abelian.Pseudoelement.pseudoZero_def", "def_path": "Mathlib/CategoryTheory/Abelian/Pseudoelements.lean", "def_pos": [248, 9], "def_end_pos": [248, 23]}, {"full_name": "CategoryTheory.Abelian.Pseudoelement.pseudoApply_mk'", "def_path": "Mathlib/CategoryTheory/Abelian/Pseudoelements.lean", "def_pos": [187, 9], "def_end_pos": [187, 24]}]], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nP Q : C\na : Pseudoelement P\na' : Over P\n\u22a2 pseudoApply 0 \u27e6a'\u27e7 = 0", "state_after": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nP Q : C\na : Pseudoelement P\na' : Over P\n\u22a2 \u27e6Over.mk (a'.hom \u226b 0)\u27e7 = \u27e6Over.mk 0\u27e7"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u\ninst\u271d\u00b9 : Category.{v, u} C\ninst\u271d : Abelian C\nP Q : C\na : Pseudoelement P\na' : Over P\n\u22a2 \u27e6Over.mk (a'.hom \u226b 0)\u27e7 = \u27e6Over.mk 0\u27e7", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/DifferentialObject.lean", "full_name": "CategoryTheory.DifferentialObject.isoApp_trans", "start": [154, 1], "end": [155, 50], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/SymmDiff.lean", "full_name": "symmDiff_top'", "start": [343, 1], "end": [343, 57], "traced_tactics": [{"tactic": "simp [symmDiff]", "annotated_tactic": ["simp [<a>symmDiff</a>]", [{"full_name": "symmDiff", "def_path": "Mathlib/Order/SymmDiff.lean", "def_pos": [60, 5], "def_end_pos": [60, 13]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\n\u03c0 : \u03b9 \u2192 Type u_4\ninst\u271d : CoheytingAlgebra \u03b1\na : \u03b1\n\u22a2 a \u2206 \u22a4 = \uffe2a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/JacobsonIdeal.lean", "full_name": "Ideal.jacobson_eq_top_iff", "start": [79, 1], "end": [85, 64], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Group/Defs.lean", "full_name": "mul_inv_lt_inv_mul_iff", "start": [411, 1], "end": [413, 26], "traced_tactics": [{"tactic": "rw [\u2190 mul_lt_mul_iff_left d, \u2190 mul_lt_mul_iff_right b, mul_inv_cancel_left, mul_assoc,\n  inv_mul_cancel_right]", "annotated_tactic": ["rw [\u2190 <a>mul_lt_mul_iff_left</a> d, \u2190 <a>mul_lt_mul_iff_right</a> b, <a>mul_inv_cancel_left</a>, <a>mul_assoc</a>,\n    <a>inv_mul_cancel_right</a>]", [{"full_name": "mul_lt_mul_iff_left", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [105, 9], "def_end_pos": [105, 28]}, {"full_name": "mul_lt_mul_iff_right", "def_path": "Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean", "def_pos": [113, 9], "def_end_pos": [113, 29]}, {"full_name": "mul_inv_cancel_left", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1256, 9], "def_end_pos": [1256, 28]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}, {"full_name": "inv_mul_cancel_right", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1268, 9], "def_end_pos": [1268, 29]}]], "state_before": "\u03b1 : Type u\ninst\u271d\u00b3 : Group \u03b1\ninst\u271d\u00b2 : LT \u03b1\ninst\u271d\u00b9 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x * x_1) fun x x_1 => x < x_1\ninst\u271d : CovariantClass \u03b1 \u03b1 (swap fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b c d : \u03b1\n\u22a2 a * b\u207b\u00b9 < d\u207b\u00b9 * c \u2194 d * a < c * b", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Order/Lattice.lean", "full_name": "Filter.Tendsto.finset_inf_nhds_apply", "start": [177, 1], "end": [180, 38], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Shift/Pullback.lean", "full_name": "CategoryTheory.pullbackShiftFunctorZero_hom_app", "start": [70, 1], "end": [75, 6], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "C : Type u_1\ninst\u271d\u00b3 : Category.{?u.8027, u_1} C\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\n\u03c6 : A \u2192+ B\ninst\u271d : HasShift C B\nX : PullbackShift C \u03c6\na\u2081 a\u2082 a\u2083 : A\nh : a\u2081 + a\u2082 = a\u2083\nb\u2081 b\u2082 b\u2083 : B\nh\u2081 : b\u2081 = \u03c6 a\u2081\nh\u2082 : b\u2082 = \u03c6 a\u2082\nh\u2083 : b\u2083 = \u03c6 a\u2083\n\u22a2 0 = \u03c6 0", "state_after": "no goals"}, {"tactic": "rw [\u2190 cancel_epi ((shiftFunctorZero _ A).inv.app X), Iso.inv_hom_id_app,\n  pullbackShiftFunctorZero_inv_app, assoc, Iso.inv_hom_id_app_assoc, Iso.inv_hom_id_app]", "annotated_tactic": ["rw [\u2190 <a>cancel_epi</a> ((<a>shiftFunctorZero</a> _ A).inv.app X), <a>Iso.inv_hom_id_app</a>,\n    <a>pullbackShiftFunctorZero_inv_app</a>, <a>assoc</a>, <a>Iso.inv_hom_id_app_assoc</a>, <a>Iso.inv_hom_id_app</a>]", [{"full_name": "CategoryTheory.cancel_epi", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 19]}, {"full_name": "CategoryTheory.shiftFunctorZero", "def_path": "Mathlib/CategoryTheory/Shift/Basic.lean", "def_pos": [192, 5], "def_end_pos": [192, 21]}, {"full_name": "CategoryTheory.Iso.inv_hom_id_app", "def_path": "Mathlib/CategoryTheory/NatIso.lean", "def_pos": [71, 9], "def_end_pos": [71, 23]}, {"full_name": "CategoryTheory.pullbackShiftFunctorZero_inv_app", "def_path": "Mathlib/CategoryTheory/Shift/Pullback.lean", "def_pos": [62, 7], "def_end_pos": [62, 39]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.Iso.inv_hom_id_app_assoc", "def_path": "Mathlib/CategoryTheory/NatIso.lean", "def_pos": [70, 3], "def_end_pos": [70, 25]}, {"full_name": "CategoryTheory.Iso.inv_hom_id_app", "def_path": "Mathlib/CategoryTheory/NatIso.lean", "def_pos": [71, 9], "def_end_pos": [71, 23]}]], "state_before": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_4, u_1} C\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\n\u03c6 : A \u2192+ B\ninst\u271d : HasShift C B\nX : PullbackShift C \u03c6\na\u2081 a\u2082 a\u2083 : A\nh : a\u2081 + a\u2082 = a\u2083\nb\u2081 b\u2082 b\u2083 : B\nh\u2081 : b\u2081 = \u03c6 a\u2081\nh\u2082 : b\u2082 = \u03c6 a\u2082\nh\u2083 : b\u2083 = \u03c6 a\u2083\n\u22a2 (shiftFunctorZero (PullbackShift C \u03c6) A).hom.app X =\n    (pullbackShiftIso C \u03c6 0 0 \u22ef).hom.app X \u226b (shiftFunctorZero C B).hom.app X", "state_after": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_4, u_1} C\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\n\u03c6 : A \u2192+ B\ninst\u271d : HasShift C B\nX : PullbackShift C \u03c6\na\u2081 a\u2082 a\u2083 : A\nh : a\u2081 + a\u2082 = a\u2083\nb\u2081 b\u2082 b\u2083 : B\nh\u2081 : b\u2081 = \u03c6 a\u2081\nh\u2082 : b\u2082 = \u03c6 a\u2082\nh\u2083 : b\u2083 = \u03c6 a\u2083\n\u22a2 \ud835\udfd9 ((\ud835\udfed (PullbackShift C \u03c6)).obj X) = \ud835\udfd9 ((\ud835\udfed C).obj X)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "C : Type u_1\ninst\u271d\u00b3 : Category.{u_4, u_1} C\nA : Type u_2\nB : Type u_3\ninst\u271d\u00b2 : AddMonoid A\ninst\u271d\u00b9 : AddMonoid B\n\u03c6 : A \u2192+ B\ninst\u271d : HasShift C B\nX : PullbackShift C \u03c6\na\u2081 a\u2082 a\u2083 : A\nh : a\u2081 + a\u2082 = a\u2083\nb\u2081 b\u2082 b\u2083 : B\nh\u2081 : b\u2081 = \u03c6 a\u2081\nh\u2082 : b\u2082 = \u03c6 a\u2082\nh\u2083 : b\u2083 = \u03c6 a\u2083\n\u22a2 \ud835\udfd9 ((\ud835\udfed (PullbackShift C \u03c6)).obj X) = \ud835\udfd9 ((\ud835\udfed C).obj X)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Comp.lean", "full_name": "DifferentiableWithinAt.comp'", "start": [113, 1], "end": [116, 59], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Filtration.lean", "full_name": "Ideal.Filtration.Stable.exists_pow_smul_eq_of_ge", "start": [227, 1], "end": [232, 42], "traced_tactics": [{"tactic": "obtain \u27e8n\u2080, hn\u2080\u27e9 := h.exists_pow_smul_eq", "annotated_tactic": ["obtain \u27e8n\u2080, hn\u2080\u27e9 := h.exists_pow_smul_eq", []], "state_before": "R M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : I.Filtration M\nh : F.Stable\n\u22a2 \u2203 n\u2080, \u2200 n \u2265 n\u2080, F.N n = I ^ (n - n\u2080) \u2022 F.N n\u2080", "state_after": "case intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : I.Filtration M\nh : F.Stable\nn\u2080 : \u2115\nhn\u2080 : \u2200 (k : \u2115), F.N (n\u2080 + k) = I ^ k \u2022 F.N n\u2080\n\u22a2 \u2203 n\u2080, \u2200 n \u2265 n\u2080, F.N n = I ^ (n - n\u2080) \u2022 F.N n\u2080"}, {"tactic": "use n\u2080", "annotated_tactic": ["use n\u2080", []], "state_before": "case intro\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : I.Filtration M\nh : F.Stable\nn\u2080 : \u2115\nhn\u2080 : \u2200 (k : \u2115), F.N (n\u2080 + k) = I ^ k \u2022 F.N n\u2080\n\u22a2 \u2203 n\u2080, \u2200 n \u2265 n\u2080, F.N n = I ^ (n - n\u2080) \u2022 F.N n\u2080", "state_after": "case h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : I.Filtration M\nh : F.Stable\nn\u2080 : \u2115\nhn\u2080 : \u2200 (k : \u2115), F.N (n\u2080 + k) = I ^ k \u2022 F.N n\u2080\n\u22a2 \u2200 n \u2265 n\u2080, F.N n = I ^ (n - n\u2080) \u2022 F.N n\u2080"}, {"tactic": "intro n hn", "annotated_tactic": ["intro n hn", []], "state_before": "case h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : I.Filtration M\nh : F.Stable\nn\u2080 : \u2115\nhn\u2080 : \u2200 (k : \u2115), F.N (n\u2080 + k) = I ^ k \u2022 F.N n\u2080\n\u22a2 \u2200 n \u2265 n\u2080, F.N n = I ^ (n - n\u2080) \u2022 F.N n\u2080", "state_after": "case h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : I.Filtration M\nh : F.Stable\nn\u2080 : \u2115\nhn\u2080 : \u2200 (k : \u2115), F.N (n\u2080 + k) = I ^ k \u2022 F.N n\u2080\nn : \u2115\nhn : n \u2265 n\u2080\n\u22a2 F.N n = I ^ (n - n\u2080) \u2022 F.N n\u2080"}, {"tactic": "convert hn\u2080 (n - n\u2080)", "annotated_tactic": ["convert hn\u2080 (n - n\u2080)", []], "state_before": "case h\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : I.Filtration M\nh : F.Stable\nn\u2080 : \u2115\nhn\u2080 : \u2200 (k : \u2115), F.N (n\u2080 + k) = I ^ k \u2022 F.N n\u2080\nn : \u2115\nhn : n \u2265 n\u2080\n\u22a2 F.N n = I ^ (n - n\u2080) \u2022 F.N n\u2080", "state_after": "case h.e'_2.h.e'_8\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : I.Filtration M\nh : F.Stable\nn\u2080 : \u2115\nhn\u2080 : \u2200 (k : \u2115), F.N (n\u2080 + k) = I ^ k \u2022 F.N n\u2080\nn : \u2115\nhn : n \u2265 n\u2080\n\u22a2 n = n\u2080 + (n - n\u2080)"}, {"tactic": "rw [add_comm, tsub_add_cancel_of_le hn]", "annotated_tactic": ["rw [<a>add_comm</a>, <a>tsub_add_cancel_of_le</a> hn]", [{"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [331, 3], "def_end_pos": [331, 14]}, {"full_name": "tsub_add_cancel_of_le", "def_path": "Mathlib/Algebra/Order/Sub/Canonical.lean", "def_pos": [31, 9], "def_end_pos": [31, 30]}]], "state_before": "case h.e'_2.h.e'_8\nR M : Type u\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nI : Ideal R\nF F' : I.Filtration M\nh : F.Stable\nn\u2080 : \u2115\nhn\u2080 : \u2200 (k : \u2115), F.N (n\u2080 + k) = I ^ k \u2022 F.N n\u2080\nn : \u2115\nhn : n \u2265 n\u2080\n\u22a2 n = n\u2080 + (n - n\u2080)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Localization/LocalizerMorphism.lean", "full_name": "CategoryTheory.LocalizerMorphism.isEquivalence", "start": [139, 1], "end": [144, 25], "traced_tactics": [{"tactic": "rw [\u03a6.isEquivalence_iff L\u2081 L\u2082 G W\u2081.Q W\u2082.Q (\u03a6.localizedFunctor W\u2081.Q W\u2082.Q)]", "annotated_tactic": ["rw [\u03a6.isEquivalence_iff L\u2081 L\u2082 G W\u2081.Q W\u2082.Q (\u03a6.localizedFunctor W\u2081.Q W\u2082.Q)]", []], "state_before": "C\u2081 : Type u\u2081\nC\u2082 : Type u\u2082\nC\u2083 : Type u\u2083\nD\u2081 : Type u\u2084\nD\u2082 : Type u\u2085\nD\u2083 : Type u\u2086\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u2077 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d\u2076 : Category.{v\u2083, u\u2083} C\u2083\ninst\u271d\u2075 : Category.{v\u2084, u\u2084} D\u2081\ninst\u271d\u2074 : Category.{v\u2085, u\u2085} D\u2082\ninst\u271d\u00b3 : Category.{v\u2086, u\u2085} D\u2082\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nG : D\u2081 \u2964 D\u2082\nh : \u03a6.IsLocalizedEquivalence\ninst\u271d : CatCommSq \u03a6.functor L\u2081 L\u2082 G\n\u22a2 G.IsEquivalence", "state_after": "C\u2081 : Type u\u2081\nC\u2082 : Type u\u2082\nC\u2083 : Type u\u2083\nD\u2081 : Type u\u2084\nD\u2082 : Type u\u2085\nD\u2083 : Type u\u2086\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u2077 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d\u2076 : Category.{v\u2083, u\u2083} C\u2083\ninst\u271d\u2075 : Category.{v\u2084, u\u2084} D\u2081\ninst\u271d\u2074 : Category.{v\u2085, u\u2085} D\u2082\ninst\u271d\u00b3 : Category.{v\u2086, u\u2085} D\u2082\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\nL\u2081 : C\u2081 \u2964 D\u2081\ninst\u271d\u00b2 : L\u2081.IsLocalization W\u2081\nL\u2082 : C\u2082 \u2964 D\u2082\ninst\u271d\u00b9 : L\u2082.IsLocalization W\u2082\nG : D\u2081 \u2964 D\u2082\nh : \u03a6.IsLocalizedEquivalence\ninst\u271d : CatCommSq \u03a6.functor L\u2081 L\u2082 G\n\u22a2 (\u03a6.localizedFunctor W\u2081.Q W\u2082.Q).IsEquivalence"}, {"tactic": "exact h.isEquivalence", "annotated_tactic": ["exact h.isEquivalence", []], "state_before": "C\u2081 : Type u\u2081\nC\u2082 : Type u\u2082\nC\u2083 : Type u\u2083\nD\u2081 : Type u\u2084\nD\u2082 : Type u\u2085\nD\u2083 : Type u\u2086\ninst\u271d\u2078 : Category.{v\u2081, u\u2081} C\u2081\ninst\u271d\u2077 : Category.{v\u2082, u\u2082} C\u2082\ninst\u271d\u2076 : Category.{v\u2083, u\u2083} C\u2083\ninst\u271d\u2075 : Category.{v\u2084, u\u2084} D\u2081\ninst\u271d\u2074 : Category.{v\u2085, u\u2085} D\u2082\ninst\u271d\u00b3 : Category.{v\u2086, u\u2085} D\u2082\nW\u2081 : MorphismProperty C\u2081\nW\u2082 : MorphismProperty C\u2082\nW\u2083 : MorphismProperty C\u2083\n\u03a6 : LocalizerMorphism W\u2081 W\u2082\nL\u2081 : 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"29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Equiv/Set.lean", "full_name": "Equiv.setOf_apply_symm_eq_image_setOf", "start": [143, 1], "end": [145, 50], "traced_tactics": [{"tactic": "rw [Equiv.image_eq_preimage, preimage_setOf_eq]", "annotated_tactic": ["rw [<a>Equiv.image_eq_preimage</a>, <a>preimage_setOf_eq</a>]", [{"full_name": "Equiv.image_eq_preimage", "def_path": "Mathlib/Logic/Equiv/Set.lean", "def_pos": [41, 19], "def_end_pos": [41, 36]}, {"full_name": "Set.preimage_setOf_eq", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [104, 9], "def_end_pos": [104, 26]}]], "state_before": "\u03b1\u271d : Sort u\n\u03b2\u271d : Sort v\n\u03b3 : Sort w\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ne : \u03b1 \u2243 \u03b2\np : \u03b1 \u2192 Prop\n\u22a2 {b | p (e.symm b)} = \u21d1e '' {a | p a}", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": 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(fun x => decide (x < l)) (Ico n m) = Ico n l", "state_after": "no goals"}, {"tactic": "rw [eq_nil_of_le hln, filter_lt_of_le_bot hln]", "annotated_tactic": ["rw [<a>eq_nil_of_le</a> hln, <a>filter_lt_of_le_bot</a> hln]", [{"full_name": "List.Ico.eq_nil_of_le", "def_path": "Mathlib/Data/List/Intervals.lean", "def_pos": [72, 9], "def_end_pos": [72, 21]}, {"full_name": "List.Ico.filter_lt_of_le_bot", "def_path": "Mathlib/Data/List/Intervals.lean", "def_pos": [162, 9], "def_end_pos": [162, 28]}]], "state_before": "case inr\nn m l : \u2115\nhlm : l \u2264 m\nhln : l \u2264 n\n\u22a2 filter (fun x => decide (x < l)) (Ico n m) = Ico n l", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Quaternion.lean", "full_name": "QuaternionAlgebra.neg_mk", "start": [259, 1], "end": [260, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Constructions.lean", "full_name": "isOpen_setOf_disjoint_nhds_nhds", "start": [624, 1], "end": [629, 76], "traced_tactics": [{"tactic": "simp only [isOpen_iff_mem_nhds, Prod.forall, mem_setOf_eq]", "annotated_tactic": ["simp only [<a>isOpen_iff_mem_nhds</a>, <a>Prod.forall</a>, <a>mem_setOf_eq</a>]", [{"full_name": "isOpen_iff_mem_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [1202, 9], "def_end_pos": [1202, 28]}, {"full_name": "Prod.forall", "def_path": "Mathlib/Data/Prod/Basic.lean", "def_pos": [32, 9], "def_end_pos": [32, 17]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [79, 29], "def_end_pos": [79, 41]}]], "state_before": "X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : TopologicalSpace Z\ninst\u271d\u00b2 : TopologicalSpace W\ninst\u271d\u00b9 : TopologicalSpace \u03b5\ninst\u271d : TopologicalSpace \u03b6\n\u22a2 IsOpen {p | Disjoint (\ud835\udcdd p.1) (\ud835\udcdd p.2)}", "state_after": "X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : TopologicalSpace Z\ninst\u271d\u00b2 : TopologicalSpace W\ninst\u271d\u00b9 : TopologicalSpace \u03b5\ninst\u271d : TopologicalSpace \u03b6\n\u22a2 \u2200 (a b : X), Disjoint (\ud835\udcdd a) (\ud835\udcdd b) \u2192 {p | Disjoint (\ud835\udcdd p.1) (\ud835\udcdd p.2)} \u2208 \ud835\udcdd (a, b)"}, {"tactic": "intro x y h", "annotated_tactic": ["intro x y h", []], "state_before": "X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : TopologicalSpace Z\ninst\u271d\u00b2 : TopologicalSpace W\ninst\u271d\u00b9 : TopologicalSpace \u03b5\ninst\u271d : TopologicalSpace \u03b6\n\u22a2 \u2200 (a b : X), Disjoint (\ud835\udcdd a) (\ud835\udcdd b) \u2192 {p | Disjoint (\ud835\udcdd p.1) (\ud835\udcdd p.2)} \u2208 \ud835\udcdd (a, b)", "state_after": "X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : TopologicalSpace Z\ninst\u271d\u00b2 : TopologicalSpace W\ninst\u271d\u00b9 : TopologicalSpace \u03b5\ninst\u271d : TopologicalSpace \u03b6\nx y : X\nh : Disjoint (\ud835\udcdd x) (\ud835\udcdd y)\n\u22a2 {p | Disjoint (\ud835\udcdd p.1) (\ud835\udcdd p.2)} \u2208 \ud835\udcdd (x, y)"}, {"tactic": "obtain \u27e8U, hU, V, hV, hd\u27e9 := ((nhds_basis_opens x).disjoint_iff (nhds_basis_opens y)).mp h", "annotated_tactic": ["obtain \u27e8U, hU, V, hV, hd\u27e9 := ((<a>nhds_basis_opens</a> x).<a>disjoint_iff</a> (<a>nhds_basis_opens</a> y)).<a>mp</a> h", [{"full_name": "nhds_basis_opens", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [808, 9], "def_end_pos": [808, 25]}, {"full_name": "Filter.HasBasis.disjoint_iff", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [654, 9], "def_end_pos": [654, 30]}, {"full_name": "nhds_basis_opens", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [808, 9], "def_end_pos": [808, 25]}, {"full_name": "Iff.mp", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [118, 3], "def_end_pos": [118, 5]}]], "state_before": "X : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : TopologicalSpace Z\ninst\u271d\u00b2 : TopologicalSpace W\ninst\u271d\u00b9 : TopologicalSpace \u03b5\ninst\u271d : TopologicalSpace \u03b6\nx y : X\nh : Disjoint (\ud835\udcdd x) (\ud835\udcdd y)\n\u22a2 {p | Disjoint (\ud835\udcdd p.1) (\ud835\udcdd p.2)} \u2208 \ud835\udcdd (x, y)", "state_after": "case intro.intro.intro.intro\nX : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : TopologicalSpace Z\ninst\u271d\u00b2 : TopologicalSpace W\ninst\u271d\u00b9 : TopologicalSpace \u03b5\ninst\u271d : TopologicalSpace \u03b6\nx y : X\nh : Disjoint (\ud835\udcdd x) (\ud835\udcdd y)\nU : Set X\nhU : x \u2208 U \u2227 IsOpen U\nV : Set X\nhV : y \u2208 V \u2227 IsOpen V\nhd : Disjoint U V\n\u22a2 {p | Disjoint (\ud835\udcdd p.1) (\ud835\udcdd p.2)} \u2208 \ud835\udcdd (x, y)"}, {"tactic": "exact mem_nhds_prod_iff'.mpr \u27e8U, V, hU.2, hU.1, hV.2, hV.1, fun \u27e8x', y'\u27e9 \u27e8hx', hy'\u27e9 =>\n  disjoint_of_disjoint_of_mem hd (hU.2.mem_nhds hx') (hV.2.mem_nhds hy')\u27e9", "annotated_tactic": ["exact mem_nhds_prod_iff'.mpr \u27e8U, V, hU.2, hU.1, hV.2, hV.1, fun \u27e8x', y'\u27e9 \u27e8hx', hy'\u27e9 =>\n    <a>disjoint_of_disjoint_of_mem</a> hd (hU.2.<a>mem_nhds</a> hx') (hV.2.<a>mem_nhds</a> hy')\u27e9", [{"full_name": "Filter.disjoint_of_disjoint_of_mem", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [741, 9], "def_end_pos": [741, 36]}, {"full_name": "IsOpen.mem_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [870, 9], "def_end_pos": [870, 24]}, {"full_name": "IsOpen.mem_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [870, 9], "def_end_pos": [870, 24]}]], "state_before": "case intro.intro.intro.intro\nX : Type u\nY : Type v\nZ : Type u_1\nW : Type u_2\n\u03b5 : Type u_3\n\u03b6 : Type u_4\ninst\u271d\u2075 : TopologicalSpace X\ninst\u271d\u2074 : TopologicalSpace Y\ninst\u271d\u00b3 : TopologicalSpace Z\ninst\u271d\u00b2 : TopologicalSpace W\ninst\u271d\u00b9 : TopologicalSpace \u03b5\ninst\u271d : TopologicalSpace \u03b6\nx y : X\nh : Disjoint (\ud835\udcdd x) (\ud835\udcdd y)\nU : Set X\nhU : x \u2208 U \u2227 IsOpen U\nV : Set X\nhV : y \u2208 V \u2227 IsOpen V\nhd : Disjoint U V\n\u22a2 {p | Disjoint (\ud835\udcdd p.1) (\ud835\udcdd p.2)} \u2208 \ud835\udcdd (x, y)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Finset/Lattice.lean", "full_name": "Finset.min_insert", "start": [1480, 1], "end": [1481, 19], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Logic/Nontrivial/Defs.lean", "full_name": "Function.Surjective.nontrivial", "start": [109, 11], "end": [117, 25], "traced_tactics": [{"tactic": "rcases exists_pair_ne \u03b2 with \u27e8x, y, h\u27e9", "annotated_tactic": ["rcases <a>exists_pair_ne</a> \u03b2 with \u27e8x, y, h\u27e9", [{"full_name": "exists_pair_ne", "def_path": "Mathlib/Logic/Nontrivial/Defs.lean", "def_pos": [40, 9], "def_end_pos": [40, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Nontrivial \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\n\u22a2 Nontrivial \u03b1", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Nontrivial \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\nx y : \u03b2\nh : x \u2260 y\n\u22a2 Nontrivial \u03b1"}, {"tactic": "rcases hf x with \u27e8x', hx'\u27e9", "annotated_tactic": ["rcases hf x with \u27e8x', hx'\u27e9", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Nontrivial \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\nx y : \u03b2\nh : x \u2260 y\n\u22a2 Nontrivial \u03b1", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Nontrivial \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\nx y : \u03b2\nh : x \u2260 y\nx' : \u03b1\nhx' : f x' = x\n\u22a2 Nontrivial \u03b1"}, {"tactic": "rcases hf y with \u27e8y', hy'\u27e9", "annotated_tactic": ["rcases hf y with \u27e8y', hy'\u27e9", []], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Nontrivial \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\nx y : \u03b2\nh : x \u2260 y\nx' : \u03b1\nhx' : f x' = x\n\u22a2 Nontrivial \u03b1", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Nontrivial \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\nx y : \u03b2\nh : x \u2260 y\nx' : \u03b1\nhx' : f x' = x\ny' : \u03b1\nhy' : f y' = y\n\u22a2 Nontrivial \u03b1"}, {"tactic": "have : x' \u2260 y' := by\n  refine fun H \u21a6 h ?_\n  rw [\u2190 hx', \u2190 hy', H]", "annotated_tactic": ["have : x' \u2260 y' := by\n    refine fun H \u21a6 h ?_\n    rw [\u2190 hx', \u2190 hy', H]", []], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Nontrivial \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\nx y : \u03b2\nh : x \u2260 y\nx' : \u03b1\nhx' : f x' = x\ny' : \u03b1\nhy' : f y' = y\n\u22a2 Nontrivial \u03b1", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Nontrivial \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\nx y : \u03b2\nh : x \u2260 y\nx' : \u03b1\nhx' : f x' = x\ny' : \u03b1\nhy' : f y' = y\nthis : x' \u2260 y'\n\u22a2 Nontrivial \u03b1"}, {"tactic": "exact \u27e8\u27e8x', y', this\u27e9\u27e9", "annotated_tactic": ["exact \u27e8\u27e8x', y', this\u27e9\u27e9", []], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Nontrivial \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\nx y : \u03b2\nh : x \u2260 y\nx' : \u03b1\nhx' : f x' = x\ny' : \u03b1\nhy' : f y' = y\nthis : x' \u2260 y'\n\u22a2 Nontrivial \u03b1", "state_after": "no goals"}, {"tactic": "refine fun H \u21a6 h ?_", "annotated_tactic": ["refine fun H \u21a6 h ?_", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Nontrivial \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\nx y : \u03b2\nh : x \u2260 y\nx' : \u03b1\nhx' : f x' = x\ny' : \u03b1\nhy' : f y' = y\n\u22a2 x' \u2260 y'", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Nontrivial \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\nx y : \u03b2\nh : x \u2260 y\nx' : \u03b1\nhx' : f x' = x\ny' : \u03b1\nhy' : f y' = y\nH : x' = y'\n\u22a2 x = y"}, {"tactic": "rw [\u2190 hx', \u2190 hy', H]", "annotated_tactic": ["rw [\u2190 hx', \u2190 hy', H]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Nontrivial \u03b2\nf : \u03b1 \u2192 \u03b2\nhf : Surjective f\nx y : \u03b2\nh : x \u2260 y\nx' : \u03b1\nhx' : f x' = x\ny' : \u03b1\nhy' : f y' = y\nH : x' = y'\n\u22a2 x = y", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Homotopy/Product.lean", "full_name": "Path.Homotopic.comp_prod_eq_prod_comp", "start": [207, 1], "end": [212, 83], "traced_tactics": [{"tactic": "apply Quotient.inductionOn\u2082 (motive := _) q\u2081 q\u2082", "annotated_tactic": ["apply <a>Quotient.inductionOn\u2082</a> (motive := _) q\u2081 q\u2082", [{"full_name": "Quotient.inductionOn\u2082", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1717, 19], "def_end_pos": [1717, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\na\u2081 a\u2082 a\u2083 : \u03b1\nb\u2081 b\u2082 b\u2083 : \u03b2\np\u2081 p\u2081' : Path a\u2081 a\u2082\np\u2082 p\u2082' : Path b\u2081 b\u2082\nq\u2081 : Homotopic.Quotient a\u2081 a\u2082\nq\u2082 : Homotopic.Quotient b\u2081 b\u2082\nr\u2081 : Homotopic.Quotient a\u2082 a\u2083\nr\u2082 : Homotopic.Quotient b\u2082 b\u2083\n\u22a2 prod q\u2081 q\u2082 \u2b1d prod r\u2081 r\u2082 = prod (q\u2081 \u2b1d r\u2081) (q\u2082 \u2b1d r\u2082)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\na\u2081 a\u2082 a\u2083 : \u03b1\nb\u2081 b\u2082 b\u2083 : \u03b2\np\u2081 p\u2081' : Path a\u2081 a\u2082\np\u2082 p\u2082' : Path b\u2081 b\u2082\nq\u2081 : Homotopic.Quotient a\u2081 a\u2082\nq\u2082 : Homotopic.Quotient b\u2081 b\u2082\nr\u2081 : Homotopic.Quotient a\u2082 a\u2083\nr\u2082 : Homotopic.Quotient b\u2082 b\u2083\n\u22a2 \u2200 (a : Path a\u2081 a\u2082) (b : Path b\u2081 b\u2082), prod \u27e6a\u27e7 \u27e6b\u27e7 \u2b1d prod r\u2081 r\u2082 = prod (\u27e6a\u27e7 \u2b1d r\u2081) (\u27e6b\u27e7 \u2b1d r\u2082)"}, {"tactic": "intro a b", "annotated_tactic": ["intro a b", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\na\u2081 a\u2082 a\u2083 : \u03b1\nb\u2081 b\u2082 b\u2083 : \u03b2\np\u2081 p\u2081' : Path a\u2081 a\u2082\np\u2082 p\u2082' : Path b\u2081 b\u2082\nq\u2081 : Homotopic.Quotient a\u2081 a\u2082\nq\u2082 : Homotopic.Quotient b\u2081 b\u2082\nr\u2081 : Homotopic.Quotient a\u2082 a\u2083\nr\u2082 : Homotopic.Quotient b\u2082 b\u2083\n\u22a2 \u2200 (a : Path a\u2081 a\u2082) (b : Path b\u2081 b\u2082), prod \u27e6a\u27e7 \u27e6b\u27e7 \u2b1d prod r\u2081 r\u2082 = prod (\u27e6a\u27e7 \u2b1d r\u2081) (\u27e6b\u27e7 \u2b1d r\u2082)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\na\u2081 a\u2082 a\u2083 : \u03b1\nb\u2081 b\u2082 b\u2083 : \u03b2\np\u2081 p\u2081' : Path a\u2081 a\u2082\np\u2082 p\u2082' : Path b\u2081 b\u2082\nq\u2081 : Homotopic.Quotient a\u2081 a\u2082\nq\u2082 : Homotopic.Quotient b\u2081 b\u2082\nr\u2081 : Homotopic.Quotient a\u2082 a\u2083\nr\u2082 : Homotopic.Quotient b\u2082 b\u2083\na : Path a\u2081 a\u2082\nb : Path b\u2081 b\u2082\n\u22a2 prod \u27e6a\u27e7 \u27e6b\u27e7 \u2b1d prod r\u2081 r\u2082 = prod (\u27e6a\u27e7 \u2b1d r\u2081) (\u27e6b\u27e7 \u2b1d r\u2082)"}, {"tactic": "apply Quotient.inductionOn\u2082 (motive := _) r\u2081 r\u2082", "annotated_tactic": ["apply <a>Quotient.inductionOn\u2082</a> (motive := _) r\u2081 r\u2082", [{"full_name": "Quotient.inductionOn\u2082", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [1717, 19], "def_end_pos": [1717, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\na\u2081 a\u2082 a\u2083 : \u03b1\nb\u2081 b\u2082 b\u2083 : \u03b2\np\u2081 p\u2081' : Path a\u2081 a\u2082\np\u2082 p\u2082' : Path b\u2081 b\u2082\nq\u2081 : Homotopic.Quotient a\u2081 a\u2082\nq\u2082 : Homotopic.Quotient b\u2081 b\u2082\nr\u2081 : Homotopic.Quotient a\u2082 a\u2083\nr\u2082 : Homotopic.Quotient b\u2082 b\u2083\na : Path a\u2081 a\u2082\nb : Path b\u2081 b\u2082\n\u22a2 prod \u27e6a\u27e7 \u27e6b\u27e7 \u2b1d prod r\u2081 r\u2082 = prod (\u27e6a\u27e7 \u2b1d r\u2081) (\u27e6b\u27e7 \u2b1d r\u2082)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\na\u2081 a\u2082 a\u2083 : \u03b1\nb\u2081 b\u2082 b\u2083 : \u03b2\np\u2081 p\u2081' : Path a\u2081 a\u2082\np\u2082 p\u2082' : Path b\u2081 b\u2082\nq\u2081 : Homotopic.Quotient a\u2081 a\u2082\nq\u2082 : Homotopic.Quotient b\u2081 b\u2082\nr\u2081 : Homotopic.Quotient a\u2082 a\u2083\nr\u2082 : Homotopic.Quotient b\u2082 b\u2083\na : Path a\u2081 a\u2082\nb : Path b\u2081 b\u2082\n\u22a2 \u2200 (a_1 : Path a\u2082 a\u2083) (b_1 : Path b\u2082 b\u2083), prod \u27e6a\u27e7 \u27e6b\u27e7 \u2b1d prod \u27e6a_1\u27e7 \u27e6b_1\u27e7 = prod (\u27e6a\u27e7 \u2b1d \u27e6a_1\u27e7) (\u27e6b\u27e7 \u2b1d \u27e6b_1\u27e7)"}, {"tactic": "intros", "annotated_tactic": ["intros", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\na\u2081 a\u2082 a\u2083 : \u03b1\nb\u2081 b\u2082 b\u2083 : \u03b2\np\u2081 p\u2081' : Path a\u2081 a\u2082\np\u2082 p\u2082' : Path b\u2081 b\u2082\nq\u2081 : Homotopic.Quotient a\u2081 a\u2082\nq\u2082 : Homotopic.Quotient b\u2081 b\u2082\nr\u2081 : Homotopic.Quotient a\u2082 a\u2083\nr\u2082 : Homotopic.Quotient b\u2082 b\u2083\na : Path a\u2081 a\u2082\nb : Path b\u2081 b\u2082\n\u22a2 \u2200 (a_1 : Path a\u2082 a\u2083) (b_1 : Path b\u2082 b\u2083), prod \u27e6a\u27e7 \u27e6b\u27e7 \u2b1d prod \u27e6a_1\u27e7 \u27e6b_1\u27e7 = prod (\u27e6a\u27e7 \u2b1d \u27e6a_1\u27e7) (\u27e6b\u27e7 \u2b1d \u27e6b_1\u27e7)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\na\u2081 a\u2082 a\u2083 : \u03b1\nb\u2081 b\u2082 b\u2083 : \u03b2\np\u2081 p\u2081' : Path a\u2081 a\u2082\np\u2082 p\u2082' : Path b\u2081 b\u2082\nq\u2081 : Homotopic.Quotient a\u2081 a\u2082\nq\u2082 : Homotopic.Quotient b\u2081 b\u2082\nr\u2081 : Homotopic.Quotient a\u2082 a\u2083\nr\u2082 : Homotopic.Quotient b\u2082 b\u2083\na : Path a\u2081 a\u2082\nb : Path b\u2081 b\u2082\na\u271d : Path a\u2082 a\u2083\nb\u271d : Path b\u2082 b\u2083\n\u22a2 prod \u27e6a\u27e7 \u27e6b\u27e7 \u2b1d prod \u27e6a\u271d\u27e7 \u27e6b\u271d\u27e7 = prod (\u27e6a\u27e7 \u2b1d \u27e6a\u271d\u27e7) (\u27e6b\u27e7 \u2b1d \u27e6b\u271d\u27e7)"}, {"tactic": "simp only [prod_lift, \u2190 Path.Homotopic.comp_lift, Path.trans_prod_eq_prod_trans]", "annotated_tactic": ["simp only [<a>prod_lift</a>, \u2190 <a>Path.Homotopic.comp_lift</a>, <a>Path.trans_prod_eq_prod_trans</a>]", [{"full_name": "Path.Homotopic.prod_lift", "def_path": "Mathlib/Topology/Homotopy/Product.lean", "def_pos": [199, 9], "def_end_pos": [199, 18]}, {"full_name": "Path.Homotopic.comp_lift", "def_path": "Mathlib/Topology/Homotopy/Path.lean", "def_pos": [320, 9], "def_end_pos": [320, 18]}, {"full_name": "Path.trans_prod_eq_prod_trans", "def_path": "Mathlib/Topology/Connected/PathConnected.lean", "def_pos": [552, 9], "def_end_pos": [552, 33]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\na\u2081 a\u2082 a\u2083 : \u03b1\nb\u2081 b\u2082 b\u2083 : \u03b2\np\u2081 p\u2081' : Path a\u2081 a\u2082\np\u2082 p\u2082' : Path b\u2081 b\u2082\nq\u2081 : Homotopic.Quotient a\u2081 a\u2082\nq\u2082 : Homotopic.Quotient b\u2081 b\u2082\nr\u2081 : Homotopic.Quotient a\u2082 a\u2083\nr\u2082 : Homotopic.Quotient b\u2082 b\u2083\na : Path a\u2081 a\u2082\nb : Path b\u2081 b\u2082\na\u271d : Path a\u2082 a\u2083\nb\u271d : Path b\u2082 b\u2083\n\u22a2 prod \u27e6a\u27e7 \u27e6b\u27e7 \u2b1d prod \u27e6a\u271d\u27e7 \u27e6b\u271d\u27e7 = prod (\u27e6a\u27e7 \u2b1d \u27e6a\u271d\u27e7) (\u27e6b\u27e7 \u2b1d \u27e6b\u271d\u27e7)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean", "full_name": "InnerProductGeometry.sin_eq_one_iff_angle_eq_pi_div_two", "start": [371, 1], "end": [374, 7], "traced_tactics": [{"tactic": "refine \u27e8fun h => ?_, fun h => by rw [h, sin_pi_div_two]\u27e9", "annotated_tactic": ["refine \u27e8fun h => ?_, fun h => by rw [h, <a>sin_pi_div_two</a>]\u27e9", [{"full_name": "Real.sin_pi_div_two", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "def_pos": [494, 9], "def_end_pos": [494, 23]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\n\u22a2 sin (angle x y) = 1 \u2194 angle x y = \u03c0 / 2", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : sin (angle x y) = 1\n\u22a2 angle x y = \u03c0 / 2"}, {"tactic": "rw [\u2190 cos_eq_zero_iff_angle_eq_pi_div_two, \u2190 abs_eq_zero, abs_cos_eq_sqrt_one_sub_sin_sq, h]", "annotated_tactic": ["rw [\u2190 <a>cos_eq_zero_iff_angle_eq_pi_div_two</a>, \u2190 <a>abs_eq_zero</a>, <a>abs_cos_eq_sqrt_one_sub_sin_sq</a>, h]", [{"full_name": "InnerProductGeometry.cos_eq_zero_iff_angle_eq_pi_div_two", "def_path": "Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean", "def_pos": [352, 9], "def_end_pos": [352, 44]}, {"full_name": "abs_eq_zero", "def_path": "Mathlib/Algebra/Order/Group/Abs.lean", "def_pos": [300, 3], "def_end_pos": [300, 14]}, {"full_name": "Real.abs_cos_eq_sqrt_one_sub_sin_sq", "def_path": "Mathlib/Data/Complex/Exponential.lean", "def_pos": [1006, 9], "def_end_pos": [1006, 39]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : sin (angle x y) = 1\n\u22a2 angle x y = \u03c0 / 2", "state_after": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : sin (angle x y) = 1\n\u22a2 \u221a(1 - 1 ^ 2) = 0"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : sin (angle x y) = 1\n\u22a2 \u221a(1 - 1 ^ 2) = 0", "state_after": "no goals"}, {"tactic": "rw [h, sin_pi_div_two]", "annotated_tactic": ["rw [h, <a>sin_pi_div_two</a>]", [{"full_name": "Real.sin_pi_div_two", "def_path": "Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean", "def_pos": [494, 9], "def_end_pos": [494, 23]}]], "state_before": "V : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup V\ninst\u271d : InnerProductSpace \u211d V\nx y : V\nh : angle x y = \u03c0 / 2\n\u22a2 sin (angle x y) = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/LocallyConstant/Basic.lean", "full_name": "IsLocallyConstant.tfae", "start": [42, 11], "end": [63, 14], "traced_tactics": [{"tactic": "tfae_have 1 \u2192 4", "annotated_tactic": ["tfae_have 1 \u2192 4", []], "state_before": "X : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\n\u22a2 [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x].TFAE", "state_after": "case tfae_1_to_4\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\n\u22a2 IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\n\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\n\u22a2 [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x].TFAE"}, {"tactic": "tfae_have 4 \u2192 3", "annotated_tactic": ["tfae_have 4 \u2192 3", []], "state_before": "X : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\n\u22a2 [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x].TFAE", "state_after": "case tfae_4_to_3\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\n\u22a2 (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\n\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\n\u22a2 [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x].TFAE"}, {"tactic": "tfae_have 3 \u2192 2", "annotated_tactic": ["tfae_have 3 \u2192 2", []], "state_before": "X : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\n\u22a2 [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x].TFAE", "state_after": "case tfae_3_to_2\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\n\u22a2 (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\n\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\n\u22a2 [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x].TFAE"}, {"tactic": "tfae_have 2 \u2192 5", "annotated_tactic": ["tfae_have 2 \u2192 5", []], "state_before": "X : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\n\u22a2 [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x].TFAE", "state_after": "case tfae_2_to_5\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\n\u22a2 (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x\n\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 : (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x\n\u22a2 [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x].TFAE"}, {"tactic": "tfae_have 5 \u2192 1", "annotated_tactic": ["tfae_have 5 \u2192 1", []], "state_before": "X : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 : (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x\n\u22a2 [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x].TFAE", "state_after": "case tfae_5_to_1\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 : (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x\n\u22a2 (\u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x) \u2192 IsLocallyConstant f\n\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 : (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x\ntfae_5_to_1 : (\u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x) \u2192 IsLocallyConstant f\n\u22a2 [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x].TFAE"}, {"tactic": "tfae_finish", "annotated_tactic": ["tfae_finish", []], "state_before": "X : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 : (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x\ntfae_5_to_1 : (\u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x) \u2192 IsLocallyConstant f\n\u22a2 [IsLocallyConstant f, \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x, \u2200 (x : X), IsOpen {x' | f x' = f x},\n      \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y}), \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x].TFAE", "state_after": "no goals"}, {"tactic": "exact fun h y => h {y}", "annotated_tactic": ["exact fun h y => h {y}", []], "state_before": "case tfae_1_to_4\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\n\u22a2 IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})", "state_after": "no goals"}, {"tactic": "exact fun h x => h (f x)", "annotated_tactic": ["exact fun h x => h (f x)", []], "state_before": "case tfae_4_to_3\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\n\u22a2 (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}", "state_after": "no goals"}, {"tactic": "exact fun h x => IsOpen.mem_nhds (h x) rfl", "annotated_tactic": ["exact fun h x => <a>IsOpen.mem_nhds</a> (h x) <a>rfl</a>", [{"full_name": "IsOpen.mem_nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [870, 9], "def_end_pos": [870, 24]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}]], "state_before": "case tfae_3_to_2\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\n\u22a2 (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x", "state_after": "no goals"}, {"tactic": "intro h x", "annotated_tactic": ["intro h x", []], "state_before": "case tfae_2_to_5\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\n\u22a2 (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x", "state_after": "case tfae_2_to_5\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\nh : \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\nx : X\n\u22a2 \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x"}, {"tactic": "rcases mem_nhds_iff.1 (h x) with \u27e8U, eq, hU, hx\u27e9", "annotated_tactic": ["rcases <a>mem_nhds_iff</a>.1 (h x) with \u27e8U, eq, hU, hx\u27e9", [{"full_name": "mem_nhds_iff", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [840, 9], "def_end_pos": [840, 21]}]], "state_before": "case tfae_2_to_5\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\nh : \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\nx : X\n\u22a2 \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x", "state_after": "case tfae_2_to_5.intro.intro.intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\nh : \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\nx : X\nU : Set X\neq : U \u2286 {x_1 | (fun x' => f x' = f x) x_1}\nhU : IsOpen U\nhx : x \u2208 U\n\u22a2 \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x"}, {"tactic": "exact \u27e8U, hU, hx, eq\u27e9", "annotated_tactic": ["exact \u27e8U, hU, hx, eq\u27e9", []], "state_before": "case tfae_2_to_5.intro.intro.intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\nh : \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\nx : X\nU : Set X\neq : U \u2286 {x_1 | (fun x' => f x' = f x) x_1}\nhU : IsOpen U\nhx : x \u2208 U\n\u22a2 \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x", "state_after": "no goals"}, {"tactic": "intro h s", "annotated_tactic": ["intro h s", []], "state_before": "case tfae_5_to_1\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 : (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x\n\u22a2 (\u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x) \u2192 IsLocallyConstant f", "state_after": "case tfae_5_to_1\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 : (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x\nh : \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x\ns : Set Y\n\u22a2 IsOpen (f \u207b\u00b9' s)"}, {"tactic": "refine isOpen_iff_forall_mem_open.2 fun x hx \u21a6 ?_", "annotated_tactic": ["refine <a>isOpen_iff_forall_mem_open</a>.2 fun x hx \u21a6 ?_", [{"full_name": "isOpen_iff_forall_mem_open", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 35]}]], "state_before": "case tfae_5_to_1\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 : (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x\nh : \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x\ns : Set Y\n\u22a2 IsOpen (f \u207b\u00b9' s)", "state_after": "case tfae_5_to_1\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 : (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x\nh : \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x\ns : Set Y\nx : X\nhx : x \u2208 f \u207b\u00b9' s\n\u22a2 \u2203 t \u2286 f \u207b\u00b9' s, IsOpen t \u2227 x \u2208 t"}, {"tactic": "rcases h x with \u27e8U, hU, hxU, eq\u27e9", "annotated_tactic": ["rcases h x with \u27e8U, hU, hxU, eq\u27e9", []], "state_before": "case tfae_5_to_1\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 : (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x\nh : \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x\ns : Set Y\nx : X\nhx : x \u2208 f \u207b\u00b9' s\n\u22a2 \u2203 t \u2286 f \u207b\u00b9' s, IsOpen t \u2227 x \u2208 t", "state_after": "case tfae_5_to_1.intro.intro.intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 : (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x\nh : \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x\ns : Set Y\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nU : Set X\nhU : IsOpen U\nhxU : x \u2208 U\neq : \u2200 x' \u2208 U, f x' = f x\n\u22a2 \u2203 t \u2286 f \u207b\u00b9' s, IsOpen t \u2227 x \u2208 t"}, {"tactic": "exact \u27e8U, fun x' hx' => mem_preimage.2 <| (eq x' hx').symm \u25b8 hx, hU, hxU\u27e9", "annotated_tactic": ["exact \u27e8U, fun x' hx' => <a>mem_preimage</a>.2 <| (eq x' hx').<a>symm</a> \u25b8 hx, hU, hxU\u27e9", [{"full_name": "Set.mem_preimage", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [120, 9], "def_end_pos": [120, 21]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}]], "state_before": "case tfae_5_to_1.intro.intro.intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\n\u03b1 : Type u_4\ninst\u271d : TopologicalSpace X\nf : X \u2192 Y\ntfae_1_to_4 : IsLocallyConstant f \u2192 \u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})\ntfae_4_to_3 : (\u2200 (y : Y), IsOpen (f \u207b\u00b9' {y})) \u2192 \u2200 (x : X), IsOpen {x' | f x' = f x}\ntfae_3_to_2 : (\u2200 (x : X), IsOpen {x' | f x' = f x}) \u2192 \u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x\ntfae_2_to_5 : (\u2200 (x : X), \u2200\u1da0 (x' : X) in \ud835\udcdd x, f x' = f x) \u2192 \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x\nh : \u2200 (x : X), \u2203 U, IsOpen U \u2227 x \u2208 U \u2227 \u2200 x' \u2208 U, f x' = f x\ns : Set Y\nx : X\nhx : x \u2208 f \u207b\u00b9' s\nU : Set X\nhU : IsOpen U\nhxU : x \u2208 U\neq : \u2200 x' \u2208 U, f x' = f x\n\u22a2 \u2203 t \u2286 f \u207b\u00b9' s, IsOpen t \u2227 x \u2208 t", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean", "full_name": "Monotone.measurable", "start": [671, 11], "end": [674, 83], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/NumberTheory/DiophantineApproximation.lean", "full_name": "Real.continued_fraction_convergent_eq_convergent", "start": [380, 1], "end": [386, 14], "traced_tactics": [{"tactic": "induction' n with n ih generalizing \u03be", "annotated_tactic": ["induction' n with n ih generalizing \u03be", []], "state_before": "\u03be : \u211d\nn : \u2115\n\u22a2 (GeneralizedContinuedFraction.of \u03be).convergents n = \u2191(\u03be.convergent n)", "state_after": "case zero\n\u03be : \u211d\n\u22a2 (GeneralizedContinuedFraction.of \u03be).convergents 0 = \u2191(\u03be.convergent 0)\n\ncase succ\nn : \u2115\nih : \u2200 (\u03be : \u211d), (GeneralizedContinuedFraction.of \u03be).convergents n = \u2191(\u03be.convergent n)\n\u03be : \u211d\n\u22a2 (GeneralizedContinuedFraction.of \u03be).convergents (n + 1) = \u2191(\u03be.convergent (n + 1))"}, {"tactic": "simp only [Nat.zero_eq, zeroth_convergent_eq_h, of_h_eq_floor, convergent_zero,\n  Rat.cast_intCast]", "annotated_tactic": ["simp only [<a>Nat.zero_eq</a>, <a>zeroth_convergent_eq_h</a>, <a>of_h_eq_floor</a>, <a>convergent_zero</a>,\n      <a>Rat.cast_intCast</a>]", [{"full_name": "Nat.zero_eq", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [106, 17], "def_end_pos": [106, 24]}, {"full_name": "GeneralizedContinuedFraction.zeroth_convergent_eq_h", "def_path": "Mathlib/Algebra/ContinuedFractions/Translations.lean", "def_pos": [146, 9], "def_end_pos": [146, 31]}, {"full_name": "GeneralizedContinuedFraction.of_h_eq_floor", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean", "def_pos": [170, 9], "def_end_pos": [170, 22]}, {"full_name": "Real.convergent_zero", "def_path": "Mathlib/NumberTheory/DiophantineApproximation.lean", "def_pos": [341, 9], "def_end_pos": [341, 24]}, {"full_name": "Rat.cast_intCast", "def_path": "Mathlib/Data/Rat/Cast/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 21]}]], "state_before": "case zero\n\u03be : \u211d\n\u22a2 (GeneralizedContinuedFraction.of \u03be).convergents 0 = \u2191(\u03be.convergent 0)", "state_after": "no goals"}, {"tactic": "rw [convergents_succ, ih (fract \u03be)\u207b\u00b9, convergent_succ, one_div]", "annotated_tactic": ["rw [<a>convergents_succ</a>, ih (<a>fract</a> \u03be)\u207b\u00b9, <a>convergent_succ</a>, <a>one_div</a>]", [{"full_name": "GeneralizedContinuedFraction.convergents_succ", "def_path": "Mathlib/Algebra/ContinuedFractions/Computation/ApproximationCorollaries.lean", "def_pos": [80, 9], "def_end_pos": [80, 25]}, {"full_name": "Int.fract", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [651, 5], "def_end_pos": [651, 10]}, {"full_name": "Real.convergent_succ", "def_path": "Mathlib/NumberTheory/DiophantineApproximation.lean", "def_pos": [347, 9], "def_end_pos": [347, 24]}, {"full_name": "one_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [468, 9], "def_end_pos": [468, 16]}]], "state_before": "case succ\nn : \u2115\nih : \u2200 (\u03be : \u211d), (GeneralizedContinuedFraction.of \u03be).convergents n = \u2191(\u03be.convergent n)\n\u03be : \u211d\n\u22a2 (GeneralizedContinuedFraction.of \u03be).convergents (n + 1) = \u2191(\u03be.convergent (n + 1))", "state_after": "case succ\nn : \u2115\nih : \u2200 (\u03be : \u211d), (GeneralizedContinuedFraction.of \u03be).convergents n = \u2191(\u03be.convergent n)\n\u03be : \u211d\n\u22a2 \u2191\u230a\u03be\u230b + (\u2191((fract \u03be)\u207b\u00b9.convergent n))\u207b\u00b9 = \u2191(\u2191\u230a\u03be\u230b + ((fract \u03be)\u207b\u00b9.convergent n)\u207b\u00b9)"}, {"tactic": "norm_cast", "annotated_tactic": ["norm_cast", []], "state_before": "case succ\nn : \u2115\nih : \u2200 (\u03be : \u211d), (GeneralizedContinuedFraction.of \u03be).convergents n = \u2191(\u03be.convergent n)\n\u03be : \u211d\n\u22a2 \u2191\u230a\u03be\u230b + (\u2191((fract \u03be)\u207b\u00b9.convergent n))\u207b\u00b9 = \u2191(\u2191\u230a\u03be\u230b + ((fract \u03be)\u207b\u00b9.convergent n)\u207b\u00b9)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Analysis/Calculus/FDeriv/Add.lean", "full_name": "Differentiable.sum", "start": [392, 1], "end": [393, 99], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/CompleteLattice.lean", "full_name": "Prod.snd_iInf", "start": [1846, 1], "end": [1847, 39], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Geometry/Euclidean/MongePoint.lean", "full_name": "Affine.Simplex.sum_mongePointVSubFaceCentroidWeightsWithCircumcenter", "start": [187, 1], "end": [192, 35], "traced_tactics": [{"tactic": "rw [mongePointVSubFaceCentroidWeightsWithCircumcenter_eq_sub h]", "annotated_tactic": ["rw [<a>mongePointVSubFaceCentroidWeightsWithCircumcenter_eq_sub</a> h]", [{"full_name": "Affine.Simplex.mongePointVSubFaceCentroidWeightsWithCircumcenter_eq_sub", "def_path": "Mathlib/Geometry/Euclidean/MongePoint.lean", "def_pos": [169, 9], "def_end_pos": [169, 65]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ni\u2081 i\u2082 : Fin (n + 3)\nh : i\u2081 \u2260 i\u2082\n\u22a2 \u2211 i : PointsWithCircumcenterIndex (n + 2), mongePointVSubFaceCentroidWeightsWithCircumcenter i\u2081 i\u2082 i = 0", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ni\u2081 i\u2082 : Fin (n + 3)\nh : i\u2081 \u2260 i\u2082\n\u22a2 \u2211 i : PointsWithCircumcenterIndex (n + 2),\n      (mongePointWeightsWithCircumcenter n - centroidWeightsWithCircumcenter {i\u2081, i\u2082}\u1d9c) i =\n    0"}, {"tactic": "simp_rw [Pi.sub_apply, sum_sub_distrib, sum_mongePointWeightsWithCircumcenter]", "annotated_tactic": ["simp_rw [<a>Pi.sub_apply</a>, <a>sum_sub_distrib</a>, <a>sum_mongePointWeightsWithCircumcenter</a>]", [{"full_name": "Pi.sub_apply", "def_path": "Mathlib/Algebra/Group/Pi/Basic.lean", "def_pos": [195, 3], "def_end_pos": [195, 14]}, {"full_name": "Finset.sum_sub_distrib", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [2122, 3], "def_end_pos": [2122, 14]}, {"full_name": "Affine.Simplex.sum_mongePointWeightsWithCircumcenter", "def_path": "Mathlib/Geometry/Euclidean/MongePoint.lean", "def_pos": [118, 9], "def_end_pos": [118, 46]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ni\u2081 i\u2082 : Fin (n + 3)\nh : i\u2081 \u2260 i\u2082\n\u22a2 \u2211 i : PointsWithCircumcenterIndex (n + 2),\n      (mongePointWeightsWithCircumcenter n - centroidWeightsWithCircumcenter {i\u2081, i\u2082}\u1d9c) i =\n    0", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ni\u2081 i\u2082 : Fin (n + 3)\nh : i\u2081 \u2260 i\u2082\n\u22a2 1 - \u2211 x : PointsWithCircumcenterIndex (n + 2), centroidWeightsWithCircumcenter {i\u2081, i\u2082}\u1d9c x = 0"}, {"tactic": "rw [sum_centroidWeightsWithCircumcenter, sub_self]", "annotated_tactic": ["rw [<a>sum_centroidWeightsWithCircumcenter</a>, <a>sub_self</a>]", [{"full_name": "Affine.Simplex.sum_centroidWeightsWithCircumcenter", "def_path": "Mathlib/Geometry/Euclidean/Circumcenter.lean", "def_pos": [615, 9], "def_end_pos": [615, 44]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [1003, 30], "def_end_pos": [1003, 38]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ni\u2081 i\u2082 : Fin (n + 3)\nh : i\u2081 \u2260 i\u2082\n\u22a2 1 - \u2211 x : PointsWithCircumcenterIndex (n + 2), centroidWeightsWithCircumcenter {i\u2081, i\u2082}\u1d9c x = 0", "state_after": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ni\u2081 i\u2082 : Fin (n + 3)\nh : i\u2081 \u2260 i\u2082\n\u22a2 {i\u2081, i\u2082}\u1d9c.Nonempty"}, {"tactic": "simp [\u2190 card_pos, card_compl, h]", "annotated_tactic": ["simp [\u2190 <a>card_pos</a>, <a>card_compl</a>, h]", [{"full_name": "Finset.card_pos", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [74, 15], "def_end_pos": [74, 23]}, {"full_name": "Finset.card_compl", "def_path": "Mathlib/Data/Fintype/Card.lean", "def_pos": [290, 9], "def_end_pos": [290, 26]}]], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u00b3 : NormedAddCommGroup V\ninst\u271d\u00b2 : InnerProductSpace \u211d V\ninst\u271d\u00b9 : MetricSpace P\ninst\u271d : NormedAddTorsor V P\nn : \u2115\ni\u2081 i\u2082 : Fin (n + 3)\nh : i\u2081 \u2260 i\u2082\n\u22a2 {i\u2081, i\u2082}\u1d9c.Nonempty", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Filter/Partial.lean", "full_name": "Filter.rcomap_sets", "start": [110, 1], "end": [112, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/ConditionallyCompleteLattice/Basic.lean", "full_name": "le_ciSup", "start": [802, 1], "end": [803, 32], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/DFinsupp/Order.lean", "full_name": "DFinsupp.coe_sup", "start": [107, 1], "end": [108, 84], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "full_name": "Finset.prod_involution", "start": [1791, 1], "end": [1828, 67], "traced_tactics": [{"tactic": "haveI := Classical.decEq \u03b1", "annotated_tactic": ["haveI := <a>Classical.decEq</a> \u03b1", [{"full_name": "Classical.decEq", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1020, 19], "def_end_pos": [1020, 24]}]], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\n\u22a2 \u2200 (g : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b1),\n    (\u2200 (a : \u03b1) (ha : a \u2208 s), f a * f (g a ha) = 1) \u2192\n      (\u2200 (a : \u03b1) (ha : a \u2208 s), f a \u2260 1 \u2192 g a ha \u2260 a) \u2192\n        \u2200 (g_mem : \u2200 (a : \u03b1) (ha : a \u2208 s), g a ha \u2208 s), (\u2200 (a : \u03b1) (ha : a \u2208 s), g (g a ha) \u22ef = a) \u2192 \u220f x \u2208 s, f x = 1", "state_after": "\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nthis : DecidableEq \u03b1\n\u22a2 \u2200 (g : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b1),\n    (\u2200 (a : \u03b1) (ha : a \u2208 s), f a * f (g a ha) = 1) \u2192\n      (\u2200 (a : \u03b1) (ha : a \u2208 s), f a \u2260 1 \u2192 g a ha \u2260 a) \u2192\n        \u2200 (g_mem : \u2200 (a : \u03b1) (ha : a \u2208 s), g a ha \u2208 s), (\u2200 (a : \u03b1) (ha : a \u2208 s), g (g a ha) \u22ef = a) \u2192 \u220f x \u2208 s, f x = 1"}, {"tactic": "haveI := Classical.decEq \u03b2", "annotated_tactic": ["haveI := <a>Classical.decEq</a> \u03b2", [{"full_name": "Classical.decEq", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [1020, 19], "def_end_pos": [1020, 24]}]], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nthis : DecidableEq \u03b1\n\u22a2 \u2200 (g : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b1),\n    (\u2200 (a : \u03b1) (ha : a \u2208 s), f a * f (g a ha) = 1) \u2192\n      (\u2200 (a : \u03b1) (ha : a \u2208 s), f a \u2260 1 \u2192 g a ha \u2260 a) \u2192\n        \u2200 (g_mem : \u2200 (a : \u03b1) (ha : a \u2208 s), g a ha \u2208 s), (\u2200 (a : \u03b1) (ha : a \u2208 s), g (g a ha) \u22ef = a) \u2192 \u220f x \u2208 s, f x = 1", "state_after": "\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nthis\u271d : DecidableEq \u03b1\nthis : DecidableEq \u03b2\n\u22a2 \u2200 (g : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b1),\n    (\u2200 (a : \u03b1) (ha : a \u2208 s), f a * f (g a ha) = 1) \u2192\n      (\u2200 (a : \u03b1) (ha : a \u2208 s), f a \u2260 1 \u2192 g a ha \u2260 a) \u2192\n        \u2200 (g_mem : \u2200 (a : \u03b1) (ha : a \u2208 s), g a ha \u2208 s), (\u2200 (a : \u03b1) (ha : a \u2208 s), g (g a ha) \u22ef = a) \u2192 \u220f x \u2208 s, f x = 1"}, {"tactic": "exact\n  Finset.strongInductionOn s fun s ih g h g_ne g_mem g_inv =>\n    s.eq_empty_or_nonempty.elim (fun hs => hs.symm \u25b8 rfl) fun \u27e8x, hx\u27e9 =>\n      have hmem : \u2200 y \u2208 (s.erase x).erase (g x hx), y \u2208 s := fun y hy =>\n        mem_of_mem_erase (mem_of_mem_erase hy)\n      have g_inj : \u2200 {x hx y hy}, g x hx = g y hy \u2192 x = y := fun {x hx y hy} h => by\n        rw [\u2190 g_inv x hx, \u2190 g_inv y hy]; simp [h]\n      have ih' : (\u220f y \u2208 erase (erase s x) (g x hx), f y) = (1 : \u03b2) :=\n        ih ((s.erase x).erase (g x hx))\n          \u27e8Subset.trans (erase_subset _ _) (erase_subset _ _), fun h =>\n            not_mem_erase (g x hx) (s.erase x) (h (g_mem x hx))\u27e9\n          (fun y hy => g y (hmem y hy)) (fun y hy => h y (hmem y hy))\n          (fun y hy => g_ne y (hmem y hy))\n          (fun y hy =>\n            mem_erase.2\n              \u27e8fun h : g y _ = g x hx => by simp [g_inj h] at hy,\n                mem_erase.2\n                  \u27e8fun h : g y _ = x => by\n                    have : y = g x hx := g_inv y (hmem y hy) \u25b8 by simp [h]\n                    simp [this] at hy, g_mem y (hmem y hy)\u27e9\u27e9)\n          fun y hy => g_inv y (hmem y hy)\n      if hx1 : f x = 1 then\n        ih' \u25b8\n          Eq.symm\n            (prod_subset hmem fun y hy hy\u2081 =>\n              have : y = x \u2228 y = g x hx := by\n                simpa [hy, -not_and, mem_erase, not_and_or, or_comm] using hy\u2081\n              this.elim (fun hy => hy.symm \u25b8 hx1) fun hy =>\n                h x hx \u25b8 hy \u25b8 hx1.symm \u25b8 (one_mul _).symm)\n      else by\n        rw [\u2190 insert_erase hx, prod_insert (not_mem_erase _ _), \u2190\n          insert_erase (mem_erase.2 \u27e8g_ne x hx hx1, g_mem x hx\u27e9),\n          prod_insert (not_mem_erase _ _), ih', mul_one, h x hx]", "annotated_tactic": ["exact\n    <a>Finset.strongInductionOn</a> s fun s ih g h g_ne g_mem g_inv =>\n      s.eq_empty_or_nonempty.elim (fun hs => hs.symm \u25b8 <a>rfl</a>) fun \u27e8x, hx\u27e9 =>\n        have hmem : \u2200 y \u2208 (s.erase x).<a>erase</a> (g x hx), y \u2208 s := fun y hy =>\n          <a>mem_of_mem_erase</a> (<a>mem_of_mem_erase</a> hy)\n        have g_inj : \u2200 {x hx y hy}, g x hx = g y hy \u2192 x = y := fun {x hx y hy} h => by\n          rw [\u2190 g_inv x hx, \u2190 g_inv y hy]; simp [h]\n        have ih' : (\u220f y \u2208 <a>erase</a> (<a>erase</a> s x) (g x hx), f y) = (1 : \u03b2) :=\n          ih ((s.erase x).<a>erase</a> (g x hx))\n            \u27e8<a>Subset.trans</a> (<a>erase_subset</a> _ _) (<a>erase_subset</a> _ _), fun h =>\n              <a>not_mem_erase</a> (g x hx) (s.erase x) (h (g_mem x hx))\u27e9\n            (fun y hy => g y (hmem y hy)) (fun y hy => h y (hmem y hy))\n            (fun y hy => g_ne y (hmem y hy))\n            (fun y hy =>\n              <a>mem_erase</a>.2\n                \u27e8fun h : g y _ = g x hx => by simp [g_inj h] at hy,\n                  <a>mem_erase</a>.2\n                    \u27e8fun h : g y _ = x => by\n                      have : y = g x hx := g_inv y (hmem y hy) \u25b8 by simp [h]\n                      simp [this] at hy, g_mem y (hmem y hy)\u27e9\u27e9)\n            fun y hy => g_inv y (hmem y hy)\n        if hx1 : f x = 1 then\n          ih' \u25b8\n            <a>Eq.symm</a>\n              (<a>prod_subset</a> hmem fun y hy hy\u2081 =>\n                have : y = x \u2228 y = g x hx := by\n                  simpa [hy, -<a>not_and</a>, <a>mem_erase</a>, <a>not_and_or</a>, <a>or_comm</a>] using hy\u2081\n                this.elim (fun hy => hy.symm \u25b8 hx1) fun hy =>\n                  h x hx \u25b8 hy \u25b8 hx1.symm \u25b8 (<a>one_mul</a> _).<a>symm</a>)\n        else by\n          rw [\u2190 <a>insert_erase</a> hx, <a>prod_insert</a> (<a>not_mem_erase</a> _ _), \u2190\n            <a>insert_erase</a> (<a>mem_erase</a>.2 \u27e8g_ne x hx hx1, g_mem x hx\u27e9),\n            <a>prod_insert</a> (<a>not_mem_erase</a> _ _), ih', <a>mul_one</a>, h x hx]", [{"full_name": "Finset.strongInductionOn", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [845, 5], "def_end_pos": [845, 22]}, {"full_name": "rfl", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [294, 22], "def_end_pos": [294, 25]}, {"full_name": "Finset.erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1885, 5], "def_end_pos": [1885, 10]}, {"full_name": "Finset.mem_of_mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1927, 9], "def_end_pos": [1927, 25]}, {"full_name": "Finset.mem_of_mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1927, 9], "def_end_pos": [1927, 25]}, {"full_name": "Finset.erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1885, 5], "def_end_pos": [1885, 10]}, {"full_name": "Finset.erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1885, 5], "def_end_pos": [1885, 10]}, {"full_name": "Finset.erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1885, 5], "def_end_pos": [1885, 10]}, {"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": [1993, 9], "def_end_pos": [1993, 21]}, {"full_name": "Finset.erase_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1993, 9], "def_end_pos": [1993, 21]}, {"full_name": "Finset.not_mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1899, 9], "def_end_pos": [1899, 22]}, {"full_name": "Finset.mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1895, 9], "def_end_pos": [1895, 18]}, {"full_name": "Finset.mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1895, 9], "def_end_pos": [1895, 18]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}, {"full_name": "Finset.prod_subset", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [595, 9], "def_end_pos": [595, 20]}, {"full_name": "not_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [116, 17], "def_end_pos": [116, 24]}, {"full_name": "Finset.mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1895, 9], "def_end_pos": [1895, 18]}, {"full_name": "not_and_or", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [481, 9], "def_end_pos": [481, 19]}, {"full_name": "or_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [823, 9], "def_end_pos": [823, 16]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [477, 9], "def_end_pos": [477, 16]}, {"full_name": "Eq.symm", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [323, 9], "def_end_pos": [323, 16]}, {"full_name": "Finset.insert_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1974, 17], "def_end_pos": [1974, 29]}, {"full_name": "Finset.prod_insert", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [351, 9], "def_end_pos": [351, 20]}, {"full_name": "Finset.not_mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1899, 9], "def_end_pos": [1899, 22]}, {"full_name": "Finset.insert_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1974, 17], "def_end_pos": [1974, 29]}, {"full_name": "Finset.mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1895, 9], "def_end_pos": [1895, 18]}, {"full_name": "Finset.prod_insert", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [351, 9], "def_end_pos": [351, 20]}, {"full_name": "Finset.not_mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1899, 9], "def_end_pos": [1899, 22]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns\u271d s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\ns : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nthis\u271d : DecidableEq \u03b1\nthis : DecidableEq \u03b2\n\u22a2 \u2200 (g : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b1),\n    (\u2200 (a : \u03b1) (ha : a \u2208 s), f a * f (g a ha) = 1) \u2192\n      (\u2200 (a : \u03b1) (ha : a \u2208 s), f a \u2260 1 \u2192 g a ha \u2260 a) \u2192\n        \u2200 (g_mem : \u2200 (a : \u03b1) (ha : a \u2208 s), g a ha \u2208 s), (\u2200 (a : \u03b1) (ha : a \u2208 s), g (g a ha) \u22ef = a) \u2192 \u220f x \u2208 s, f x = 1", "state_after": "no goals"}, {"tactic": "rw [\u2190 g_inv x hx, \u2190 g_inv y hy]", "annotated_tactic": ["rw [\u2190 g_inv x hx, \u2190 g_inv y hy]", []], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns\u271d\u00b9 s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nthis\u271d : DecidableEq \u03b1\nthis : DecidableEq \u03b2\ns : Finset \u03b1\nih :\n  \u2200 t \u2282 s,\n    \u2200 (g : (a : \u03b1) \u2192 a \u2208 t \u2192 \u03b1),\n      (\u2200 (a : \u03b1) (ha : a \u2208 t), f a * f (g a ha) = 1) \u2192\n        (\u2200 (a : \u03b1) (ha : a \u2208 t), f a \u2260 1 \u2192 g a ha \u2260 a) \u2192\n          \u2200 (g_mem : \u2200 (a : \u03b1) (ha : a \u2208 t), g a ha \u2208 t), (\u2200 (a : \u03b1) (ha : a \u2208 t), g (g a ha) \u22ef = a) \u2192 \u220f x \u2208 t, f x = 1\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b1\nh\u271d : \u2200 (a : \u03b1) (ha : a \u2208 s), f a * f (g a ha) = 1\ng_ne : \u2200 (a : \u03b1) (ha : a \u2208 s), f a \u2260 1 \u2192 g a ha \u2260 a\ng_mem : \u2200 (a : \u03b1) (ha : a \u2208 s), g a ha \u2208 s\ng_inv : \u2200 (a : \u03b1) (ha : a \u2208 s), g (g a ha) \u22ef = a\nx\u271d\u00b9 : s.Nonempty\nx\u271d : \u03b1\nhx\u271d : x\u271d \u2208 s\nhmem : \u2200 y \u2208 (s.erase x\u271d).erase (g x\u271d hx\u271d), y \u2208 s\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nh : g x hx = g y hy\n\u22a2 x = y", "state_after": "\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns\u271d\u00b9 s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nthis\u271d : DecidableEq \u03b1\nthis : DecidableEq \u03b2\ns : Finset \u03b1\nih :\n  \u2200 t \u2282 s,\n    \u2200 (g : (a : \u03b1) \u2192 a \u2208 t \u2192 \u03b1),\n      (\u2200 (a : \u03b1) (ha : a \u2208 t), f a * f (g a ha) = 1) \u2192\n        (\u2200 (a : \u03b1) (ha : a \u2208 t), f a \u2260 1 \u2192 g a ha \u2260 a) \u2192\n          \u2200 (g_mem : \u2200 (a : \u03b1) (ha : a \u2208 t), g a ha \u2208 t), (\u2200 (a : \u03b1) (ha : a \u2208 t), g (g a ha) \u22ef = a) \u2192 \u220f x \u2208 t, f x = 1\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b1\nh\u271d : \u2200 (a : \u03b1) (ha : a \u2208 s), f a * f (g a ha) = 1\ng_ne : \u2200 (a : \u03b1) (ha : a \u2208 s), f a \u2260 1 \u2192 g a ha \u2260 a\ng_mem : \u2200 (a : \u03b1) (ha : a \u2208 s), g a ha \u2208 s\ng_inv : \u2200 (a : \u03b1) (ha : a \u2208 s), g (g a ha) \u22ef = a\nx\u271d\u00b9 : s.Nonempty\nx\u271d : \u03b1\nhx\u271d : x\u271d \u2208 s\nhmem : \u2200 y \u2208 (s.erase x\u271d).erase (g x\u271d hx\u271d), y \u2208 s\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nh : g x hx = g y hy\n\u22a2 g (g x hx) \u22ef = g (g y hy) \u22ef"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns\u271d\u00b9 s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nthis\u271d : DecidableEq \u03b1\nthis : DecidableEq \u03b2\ns : Finset \u03b1\nih :\n  \u2200 t \u2282 s,\n    \u2200 (g : (a : \u03b1) \u2192 a \u2208 t \u2192 \u03b1),\n      (\u2200 (a : \u03b1) (ha : a \u2208 t), f a * f (g a ha) = 1) \u2192\n        (\u2200 (a : \u03b1) (ha : a \u2208 t), f a \u2260 1 \u2192 g a ha \u2260 a) \u2192\n          \u2200 (g_mem : \u2200 (a : \u03b1) (ha : a \u2208 t), g a ha \u2208 t), (\u2200 (a : \u03b1) (ha : a \u2208 t), g (g a ha) \u22ef = a) \u2192 \u220f x \u2208 t, f x = 1\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b1\nh\u271d : \u2200 (a : \u03b1) (ha : a \u2208 s), f a * f (g a ha) = 1\ng_ne : \u2200 (a : \u03b1) (ha : a \u2208 s), f a \u2260 1 \u2192 g a ha \u2260 a\ng_mem : \u2200 (a : \u03b1) (ha : a \u2208 s), g a ha \u2208 s\ng_inv : \u2200 (a : \u03b1) (ha : a \u2208 s), g (g a ha) \u22ef = a\nx\u271d\u00b9 : s.Nonempty\nx\u271d : \u03b1\nhx\u271d : x\u271d \u2208 s\nhmem : \u2200 y \u2208 (s.erase x\u271d).erase (g x\u271d hx\u271d), y \u2208 s\nx : \u03b1\nhx : x \u2208 s\ny : \u03b1\nhy : y \u2208 s\nh : g x hx = g y hy\n\u22a2 g (g x hx) \u22ef = g (g y hy) \u22ef", "state_after": "no goals"}, {"tactic": "simp [g_inj h] at hy", "annotated_tactic": ["simp [g_inj h] at hy", []], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns\u271d\u00b9 s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nthis\u271d : DecidableEq \u03b1\nthis : DecidableEq \u03b2\ns : Finset \u03b1\nih :\n  \u2200 t \u2282 s,\n    \u2200 (g : (a : \u03b1) \u2192 a \u2208 t \u2192 \u03b1),\n      (\u2200 (a : \u03b1) (ha : a \u2208 t), f a * f (g a ha) = 1) \u2192\n        (\u2200 (a : \u03b1) (ha : a \u2208 t), f a \u2260 1 \u2192 g a ha \u2260 a) \u2192\n          \u2200 (g_mem : \u2200 (a : \u03b1) (ha : a \u2208 t), g a ha \u2208 t), (\u2200 (a : \u03b1) (ha : a \u2208 t), g (g a ha) \u22ef = a) \u2192 \u220f x \u2208 t, f x = 1\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b1\nh\u271d : \u2200 (a : \u03b1) (ha : a \u2208 s), f a * f (g a ha) = 1\ng_ne : \u2200 (a : \u03b1) (ha : a \u2208 s), f a \u2260 1 \u2192 g a ha \u2260 a\ng_mem : \u2200 (a : \u03b1) (ha : a \u2208 s), g a ha \u2208 s\ng_inv : \u2200 (a : \u03b1) (ha : a \u2208 s), g (g a ha) \u22ef = a\nx\u271d : s.Nonempty\nx : \u03b1\nhx : x \u2208 s\nhmem : \u2200 y \u2208 (s.erase x).erase (g x hx), y \u2208 s\ng_inj : \u2200 {x : \u03b1} {hx : x \u2208 s} {y : \u03b1} {hy : y \u2208 s}, g x hx = g y hy \u2192 x = y\ny : \u03b1\nhy : y \u2208 (s.erase x).erase (g x hx)\nh : g y \u22ef = g x hx\n\u22a2 False", "state_after": "no goals"}, {"tactic": "have : y = g x hx := g_inv y (hmem y hy) \u25b8 by simp [h]", "annotated_tactic": ["have : y = g x hx := g_inv y (hmem y hy) \u25b8 by simp [h]", []], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns\u271d\u00b9 s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nthis\u271d : DecidableEq \u03b1\nthis : DecidableEq \u03b2\ns : Finset \u03b1\nih :\n  \u2200 t \u2282 s,\n    \u2200 (g : (a : \u03b1) \u2192 a \u2208 t \u2192 \u03b1),\n      (\u2200 (a : \u03b1) (ha : a \u2208 t), f a * f (g a ha) = 1) \u2192\n        (\u2200 (a : \u03b1) (ha : a \u2208 t), f a \u2260 1 \u2192 g a ha \u2260 a) \u2192\n          \u2200 (g_mem : \u2200 (a : \u03b1) (ha : a \u2208 t), g a ha \u2208 t), (\u2200 (a : \u03b1) (ha : a \u2208 t), g (g a ha) \u22ef = a) \u2192 \u220f x \u2208 t, f x = 1\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b1\nh\u271d : \u2200 (a : \u03b1) (ha : a \u2208 s), f a * f (g a ha) = 1\ng_ne : \u2200 (a : \u03b1) (ha : a \u2208 s), f a \u2260 1 \u2192 g a ha \u2260 a\ng_mem : \u2200 (a : \u03b1) (ha : a \u2208 s), g a ha \u2208 s\ng_inv : \u2200 (a : \u03b1) (ha : a \u2208 s), g (g a ha) \u22ef = a\nx\u271d : s.Nonempty\nx : \u03b1\nhx : x \u2208 s\nhmem : \u2200 y \u2208 (s.erase x).erase (g x hx), y \u2208 s\ng_inj : \u2200 {x : \u03b1} {hx : x \u2208 s} {y : \u03b1} {hy : y \u2208 s}, g x hx = g y hy \u2192 x = y\ny : \u03b1\nhy : y \u2208 (s.erase x).erase (g x hx)\nh : g y \u22ef = x\n\u22a2 False", "state_after": "\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns\u271d\u00b9 s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nthis\u271d\u00b9 : DecidableEq \u03b1\nthis\u271d : DecidableEq \u03b2\ns : Finset \u03b1\nih :\n  \u2200 t \u2282 s,\n    \u2200 (g : (a : \u03b1) \u2192 a \u2208 t \u2192 \u03b1),\n      (\u2200 (a : \u03b1) (ha : a \u2208 t), f a * f (g a ha) = 1) \u2192\n        (\u2200 (a : \u03b1) (ha : a \u2208 t), f a \u2260 1 \u2192 g a ha \u2260 a) \u2192\n          \u2200 (g_mem : \u2200 (a : \u03b1) (ha : a \u2208 t), g a ha \u2208 t), (\u2200 (a : \u03b1) (ha : a \u2208 t), g (g a ha) \u22ef = a) \u2192 \u220f x \u2208 t, f x = 1\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b1\nh\u271d : \u2200 (a : \u03b1) (ha : a \u2208 s), f a * f (g a ha) = 1\ng_ne : \u2200 (a : \u03b1) (ha : a \u2208 s), f a \u2260 1 \u2192 g a ha \u2260 a\ng_mem : \u2200 (a : \u03b1) (ha : a \u2208 s), g a ha \u2208 s\ng_inv : \u2200 (a : \u03b1) (ha : a \u2208 s), g (g a ha) \u22ef = a\nx\u271d : s.Nonempty\nx : \u03b1\nhx : x \u2208 s\nhmem : \u2200 y \u2208 (s.erase x).erase (g x hx), y \u2208 s\ng_inj : \u2200 {x : \u03b1} {hx : x \u2208 s} {y : \u03b1} {hy : y \u2208 s}, g x hx = g y hy \u2192 x = y\ny : \u03b1\nhy : y \u2208 (s.erase x).erase (g x hx)\nh : g y \u22ef = x\nthis : y = g x hx\n\u22a2 False"}, {"tactic": "simp [this] at hy", "annotated_tactic": ["simp [this] at hy", []], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns\u271d\u00b9 s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nthis\u271d\u00b9 : DecidableEq \u03b1\nthis\u271d : DecidableEq \u03b2\ns : Finset \u03b1\nih :\n  \u2200 t \u2282 s,\n    \u2200 (g : (a : \u03b1) \u2192 a \u2208 t \u2192 \u03b1),\n      (\u2200 (a : \u03b1) (ha : a \u2208 t), f a * f (g a ha) = 1) \u2192\n        (\u2200 (a : \u03b1) (ha : a \u2208 t), f a \u2260 1 \u2192 g a ha \u2260 a) \u2192\n          \u2200 (g_mem : \u2200 (a : \u03b1) (ha : a \u2208 t), g a ha \u2208 t), (\u2200 (a : \u03b1) (ha : a \u2208 t), g (g a ha) \u22ef = a) \u2192 \u220f x \u2208 t, f x = 1\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b1\nh\u271d : \u2200 (a : \u03b1) (ha : a \u2208 s), f a * f (g a ha) = 1\ng_ne : \u2200 (a : \u03b1) (ha : a \u2208 s), f a \u2260 1 \u2192 g a ha \u2260 a\ng_mem : \u2200 (a : \u03b1) (ha : a \u2208 s), g a ha \u2208 s\ng_inv : \u2200 (a : \u03b1) (ha : a \u2208 s), g (g a ha) \u22ef = a\nx\u271d : s.Nonempty\nx : \u03b1\nhx : x \u2208 s\nhmem : \u2200 y \u2208 (s.erase x).erase (g x hx), y \u2208 s\ng_inj : \u2200 {x : \u03b1} {hx : x \u2208 s} {y : \u03b1} {hy : y \u2208 s}, g x hx = g y hy \u2192 x = y\ny : \u03b1\nhy : y \u2208 (s.erase x).erase (g x hx)\nh : g y \u22ef = x\nthis : y = g x hx\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns\u271d\u00b9 s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nthis\u271d : DecidableEq \u03b1\nthis : DecidableEq \u03b2\ns : Finset \u03b1\nih :\n  \u2200 t \u2282 s,\n    \u2200 (g : (a : \u03b1) \u2192 a \u2208 t \u2192 \u03b1),\n      (\u2200 (a : \u03b1) (ha : a \u2208 t), f a * f (g a ha) = 1) \u2192\n        (\u2200 (a : \u03b1) (ha : a \u2208 t), f a \u2260 1 \u2192 g a ha \u2260 a) \u2192\n          \u2200 (g_mem : \u2200 (a : \u03b1) (ha : a \u2208 t), g a ha \u2208 t), (\u2200 (a : \u03b1) (ha : a \u2208 t), g (g a ha) \u22ef = a) \u2192 \u220f x \u2208 t, f x = 1\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b1\nh\u271d : \u2200 (a : \u03b1) (ha : a \u2208 s), f a * f (g a ha) = 1\ng_ne : \u2200 (a : \u03b1) (ha : a \u2208 s), f a \u2260 1 \u2192 g a ha \u2260 a\ng_mem : \u2200 (a : \u03b1) (ha : a \u2208 s), g a ha \u2208 s\ng_inv : \u2200 (a : \u03b1) (ha : a \u2208 s), g (g a ha) \u22ef = a\nx\u271d : s.Nonempty\nx : \u03b1\nhx : x \u2208 s\nhmem : \u2200 y \u2208 (s.erase x).erase (g x hx), y \u2208 s\ng_inj : \u2200 {x : \u03b1} {hx : x \u2208 s} {y : \u03b1} {hy : y \u2208 s}, g x hx = g y hy \u2192 x = y\ny : \u03b1\nhy : y \u2208 (s.erase x).erase (g x hx)\nh : g y \u22ef = x\n\u22a2 g (g y \u22ef) \u22ef = g x hx", "state_after": "no goals"}, {"tactic": "simpa [hy, -not_and, mem_erase, not_and_or, or_comm] using hy\u2081", "annotated_tactic": ["simpa [hy, -<a>not_and</a>, <a>mem_erase</a>, <a>not_and_or</a>, <a>or_comm</a>] using hy\u2081", [{"full_name": "not_and", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [116, 17], "def_end_pos": [116, 24]}, {"full_name": "Finset.mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1895, 9], "def_end_pos": [1895, 18]}, {"full_name": "not_and_or", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [481, 9], "def_end_pos": [481, 19]}, {"full_name": "or_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [823, 9], "def_end_pos": [823, 16]}]], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns\u271d\u00b9 s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nthis\u271d : DecidableEq \u03b1\nthis : DecidableEq \u03b2\ns : Finset \u03b1\nih :\n  \u2200 t \u2282 s,\n    \u2200 (g : (a : \u03b1) \u2192 a \u2208 t \u2192 \u03b1),\n      (\u2200 (a : \u03b1) (ha : a \u2208 t), f a * f (g a ha) = 1) \u2192\n        (\u2200 (a : \u03b1) (ha : a \u2208 t), f a \u2260 1 \u2192 g a ha \u2260 a) \u2192\n          \u2200 (g_mem : \u2200 (a : \u03b1) (ha : a \u2208 t), g a ha \u2208 t), (\u2200 (a : \u03b1) (ha : a \u2208 t), g (g a ha) \u22ef = a) \u2192 \u220f x \u2208 t, f x = 1\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b1\nh : \u2200 (a : \u03b1) (ha : a \u2208 s), f a * f (g a ha) = 1\ng_ne : \u2200 (a : \u03b1) (ha : a \u2208 s), f a \u2260 1 \u2192 g a ha \u2260 a\ng_mem : \u2200 (a : \u03b1) (ha : a \u2208 s), g a ha \u2208 s\ng_inv : \u2200 (a : \u03b1) (ha : a \u2208 s), g (g a ha) \u22ef = a\nx\u271d : s.Nonempty\nx : \u03b1\nhx : x \u2208 s\nhmem : \u2200 y \u2208 (s.erase x).erase (g x hx), y \u2208 s\ng_inj : \u2200 {x : \u03b1} {hx : x \u2208 s} {y : \u03b1} {hy : y \u2208 s}, g x hx = g y hy \u2192 x = y\nih' : \u220f y \u2208 (s.erase x).erase (g x hx), f y = 1\nhx1 : f x = 1\ny : \u03b1\nhy : y \u2208 s\nhy\u2081 : y \u2209 (s.erase x).erase (g x hx)\n\u22a2 y = x \u2228 y = g x hx", "state_after": "no goals"}, {"tactic": "rw [\u2190 insert_erase hx, prod_insert (not_mem_erase _ _), \u2190\n  insert_erase (mem_erase.2 \u27e8g_ne x hx hx1, g_mem x hx\u27e9),\n  prod_insert (not_mem_erase _ _), ih', mul_one, h x hx]", "annotated_tactic": ["rw [\u2190 <a>insert_erase</a> hx, <a>prod_insert</a> (<a>not_mem_erase</a> _ _), \u2190\n            <a>insert_erase</a> (<a>mem_erase</a>.2 \u27e8g_ne x hx hx1, g_mem x hx\u27e9),\n            <a>prod_insert</a> (<a>not_mem_erase</a> _ _), ih', <a>mul_one</a>, h x hx]", [{"full_name": "Finset.insert_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1974, 17], "def_end_pos": [1974, 29]}, {"full_name": "Finset.prod_insert", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [351, 9], "def_end_pos": [351, 20]}, {"full_name": "Finset.not_mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1899, 9], "def_end_pos": [1899, 22]}, {"full_name": "Finset.insert_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1974, 17], "def_end_pos": [1974, 29]}, {"full_name": "Finset.mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1895, 9], "def_end_pos": [1895, 18]}, {"full_name": "Finset.prod_insert", "def_path": "Mathlib/Algebra/BigOperators/Group/Finset.lean", "def_pos": [351, 9], "def_end_pos": [351, 20]}, {"full_name": "Finset.not_mem_erase", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1899, 9], "def_end_pos": [1899, 22]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [483, 9], "def_end_pos": [483, 16]}]], "state_before": "\u03b9 : Type u_1\n\u03ba : Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_4\n\u03b3 : Type u_5\ns\u271d\u00b9 s\u2081 s\u2082 : Finset \u03b1\na : \u03b1\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : CommMonoid \u03b2\ns\u271d : Finset \u03b1\nf : \u03b1 \u2192 \u03b2\nthis\u271d : DecidableEq \u03b1\nthis : DecidableEq \u03b2\ns : Finset \u03b1\nih :\n  \u2200 t \u2282 s,\n    \u2200 (g : (a : \u03b1) \u2192 a \u2208 t \u2192 \u03b1),\n      (\u2200 (a : \u03b1) (ha : a \u2208 t), f a * f (g a ha) = 1) \u2192\n        (\u2200 (a : \u03b1) (ha : a \u2208 t), f a \u2260 1 \u2192 g a ha \u2260 a) \u2192\n          \u2200 (g_mem : \u2200 (a : \u03b1) (ha : a \u2208 t), g a ha \u2208 t), (\u2200 (a : \u03b1) (ha : a \u2208 t), g (g a ha) \u22ef = a) \u2192 \u220f x \u2208 t, f x = 1\ng : (a : \u03b1) \u2192 a \u2208 s \u2192 \u03b1\nh : \u2200 (a : \u03b1) (ha : a \u2208 s), f a * f (g a ha) = 1\ng_ne : \u2200 (a : \u03b1) (ha : a \u2208 s), f a \u2260 1 \u2192 g a ha \u2260 a\ng_mem : \u2200 (a : \u03b1) (ha : a \u2208 s), g a ha \u2208 s\ng_inv : \u2200 (a : \u03b1) (ha : a \u2208 s), g (g a ha) \u22ef = a\nx\u271d : s.Nonempty\nx : \u03b1\nhx : x \u2208 s\nhmem : \u2200 y \u2208 (s.erase x).erase (g x hx), y \u2208 s\ng_inj : \u2200 {x : \u03b1} {hx : x \u2208 s} {y : \u03b1} {hy : y \u2208 s}, g x hx = g y hy \u2192 x = y\nih' : \u220f y \u2208 (s.erase x).erase (g x hx), f y = 1\nhx1 : \u00acf x = 1\n\u22a2 \u220f x \u2208 s, f x = 1", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/RingTheory/Filtration.lean", "full_name": "Ideal.Filtration.Stable.inter_left", "start": [423, 1], "end": [425, 24], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/CategoryTheory/Abelian/RightDerived.lean", "full_name": "CategoryTheory.InjectiveResolution.isoRightDerivedObj_hom_naturality", "start": [120, 1], "end": [133, 6], "traced_tactics": [{"tactic": "dsimp [isoRightDerivedObj, Functor.rightDerived]", "annotated_tactic": ["dsimp [<a>isoRightDerivedObj</a>, <a>Functor.rightDerived</a>]", [{"full_name": "CategoryTheory.InjectiveResolution.isoRightDerivedObj", "def_path": "Mathlib/CategoryTheory/Abelian/RightDerived.lean", "def_pos": [111, 19], "def_end_pos": [111, 57]}, {"full_name": "CategoryTheory.Functor.rightDerived", "def_path": "Mathlib/CategoryTheory/Abelian/RightDerived.lean", "def_pos": [106, 19], "def_end_pos": [106, 39]}]], "state_before": "C : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} D\ninst\u271d\u00b3 : Abelian C\ninst\u271d\u00b2 : HasInjectiveResolutions C\ninst\u271d\u00b9 : Abelian D\nX Y : C\nf : X \u27f6 Y\nI : InjectiveResolution X\nJ : InjectiveResolution Y\n\u03c6 : I.cocomplex \u27f6 J.cocomplex\ncomm : I.\u03b9.f 0 \u226b \u03c6.f 0 = f \u226b J.\u03b9.f 0\nF : C \u2964 D\ninst\u271d : F.Additive\nn : \u2115\n\u22a2 (F.rightDerived n).map f \u226b (J.isoRightDerivedObj F n).hom =\n    (I.isoRightDerivedObj F n).hom \u226b\n      (F.mapHomologicalComplex (ComplexShape.up \u2115) \u22d9 HomologicalComplex.homologyFunctor D (ComplexShape.up \u2115) n).map \u03c6", "state_after": "C : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} D\ninst\u271d\u00b3 : Abelian C\ninst\u271d\u00b2 : HasInjectiveResolutions C\ninst\u271d\u00b9 : Abelian D\nX Y : C\nf : X \u27f6 Y\nI : InjectiveResolution X\nJ : InjectiveResolution Y\n\u03c6 : I.cocomplex \u27f6 J.cocomplex\ncomm : I.\u03b9.f 0 \u226b \u03c6.f 0 = f \u226b J.\u03b9.f 0\nF : C \u2964 D\ninst\u271d : F.Additive\nn : \u2115\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.up \u2115) n).map (F.rightDerivedToHomotopyCategory.map f) \u226b\n      (HomotopyCategory.homologyFunctor D (ComplexShape.up \u2115) n).map (J.isoRightDerivedToHomotopyCategoryObj F).hom \u226b\n        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up \u2115) n).hom.app\n          ((F.mapHomologicalComplex (ComplexShape.up \u2115)).obj J.cocomplex) =\n    ((HomotopyCategory.homologyFunctor D (ComplexShape.up \u2115) n).map (I.isoRightDerivedToHomotopyCategoryObj F).hom \u226b\n        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up \u2115) n).hom.app\n          ((F.mapHomologicalComplex (ComplexShape.up \u2115)).obj I.cocomplex)) \u226b\n      HomologicalComplex.homologyMap ((F.mapHomologicalComplex (ComplexShape.up \u2115)).map \u03c6) n"}, {"tactic": "rw [assoc, \u2190 Functor.map_comp_assoc,\n  InjectiveResolution.isoRightDerivedToHomotopyCategoryObj_hom_naturality f I J \u03c6 comm F,\n  Functor.map_comp, assoc]", "annotated_tactic": ["rw [<a>assoc</a>, \u2190 <a>Functor.map_comp_assoc</a>,\n    <a>InjectiveResolution.isoRightDerivedToHomotopyCategoryObj_hom_naturality</a> f I J \u03c6 comm F,\n    <a>Functor.map_comp</a>, <a>assoc</a>]", [{"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}, {"full_name": "CategoryTheory.Functor.map_comp_assoc", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [64, 7], "def_end_pos": [64, 29]}, {"full_name": "CategoryTheory.InjectiveResolution.isoRightDerivedToHomotopyCategoryObj_hom_naturality", "def_path": "Mathlib/CategoryTheory/Abelian/RightDerived.lean", "def_pos": [79, 7], "def_end_pos": [79, 78]}, {"full_name": "CategoryTheory.Functor.map_comp", "def_path": "Mathlib/CategoryTheory/Functor/Basic.lean", "def_pos": [43, 3], "def_end_pos": [43, 11]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [181, 3], "def_end_pos": [181, 8]}]], "state_before": "C : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} D\ninst\u271d\u00b3 : Abelian C\ninst\u271d\u00b2 : HasInjectiveResolutions C\ninst\u271d\u00b9 : Abelian D\nX Y : C\nf : X \u27f6 Y\nI : InjectiveResolution X\nJ : InjectiveResolution Y\n\u03c6 : I.cocomplex \u27f6 J.cocomplex\ncomm : I.\u03b9.f 0 \u226b \u03c6.f 0 = f \u226b J.\u03b9.f 0\nF : C \u2964 D\ninst\u271d : F.Additive\nn : \u2115\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.up \u2115) n).map (F.rightDerivedToHomotopyCategory.map f) \u226b\n      (HomotopyCategory.homologyFunctor D (ComplexShape.up \u2115) n).map (J.isoRightDerivedToHomotopyCategoryObj F).hom \u226b\n        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up \u2115) n).hom.app\n          ((F.mapHomologicalComplex (ComplexShape.up \u2115)).obj J.cocomplex) =\n    ((HomotopyCategory.homologyFunctor D (ComplexShape.up \u2115) n).map (I.isoRightDerivedToHomotopyCategoryObj F).hom \u226b\n        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up \u2115) n).hom.app\n          ((F.mapHomologicalComplex (ComplexShape.up \u2115)).obj I.cocomplex)) \u226b\n      HomologicalComplex.homologyMap ((F.mapHomologicalComplex (ComplexShape.up \u2115)).map \u03c6) n", "state_after": "C : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} D\ninst\u271d\u00b3 : Abelian C\ninst\u271d\u00b2 : HasInjectiveResolutions C\ninst\u271d\u00b9 : Abelian D\nX Y : C\nf : X \u27f6 Y\nI : InjectiveResolution X\nJ : InjectiveResolution Y\n\u03c6 : I.cocomplex \u27f6 J.cocomplex\ncomm : I.\u03b9.f 0 \u226b \u03c6.f 0 = f \u226b J.\u03b9.f 0\nF : C \u2964 D\ninst\u271d : F.Additive\nn : \u2115\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.up \u2115) n).map (I.isoRightDerivedToHomotopyCategoryObj F).hom \u226b\n      (HomotopyCategory.homologyFunctor D (ComplexShape.up \u2115) n).map\n          ((F.mapHomologicalComplex (ComplexShape.up \u2115) \u22d9 HomotopyCategory.quotient D (ComplexShape.up \u2115)).map \u03c6) \u226b\n        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up \u2115) n).hom.app\n          ((F.mapHomologicalComplex (ComplexShape.up \u2115)).obj J.cocomplex) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.up \u2115) n).map (I.isoRightDerivedToHomotopyCategoryObj F).hom \u226b\n      (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up \u2115) n).hom.app\n          ((F.mapHomologicalComplex (ComplexShape.up \u2115)).obj I.cocomplex) \u226b\n        HomologicalComplex.homologyMap ((F.mapHomologicalComplex (ComplexShape.up \u2115)).map \u03c6) n"}, {"tactic": "erw [(HomotopyCategory.homologyFunctorFactors D (ComplexShape.up \u2115) n).hom.naturality]", "annotated_tactic": ["erw [(<a>HomotopyCategory.homologyFunctorFactors</a> D (<a>ComplexShape.up</a> \u2115) n).hom.naturality]", [{"full_name": "HomotopyCategory.homologyFunctorFactors", "def_path": "Mathlib/Algebra/Homology/HomotopyCategory.lean", "def_pos": [236, 19], "def_end_pos": [236, 41]}, {"full_name": "ComplexShape.up", "def_path": "Mathlib/Algebra/Homology/ComplexShape.lean", "def_pos": [196, 5], "def_end_pos": [196, 7]}]], "state_before": "C : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} D\ninst\u271d\u00b3 : Abelian C\ninst\u271d\u00b2 : HasInjectiveResolutions C\ninst\u271d\u00b9 : Abelian D\nX Y : C\nf : X \u27f6 Y\nI : InjectiveResolution X\nJ : InjectiveResolution Y\n\u03c6 : I.cocomplex \u27f6 J.cocomplex\ncomm : I.\u03b9.f 0 \u226b \u03c6.f 0 = f \u226b J.\u03b9.f 0\nF : C \u2964 D\ninst\u271d : F.Additive\nn : \u2115\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.up \u2115) n).map (I.isoRightDerivedToHomotopyCategoryObj F).hom \u226b\n      (HomotopyCategory.homologyFunctor D (ComplexShape.up \u2115) n).map\n          ((F.mapHomologicalComplex (ComplexShape.up \u2115) \u22d9 HomotopyCategory.quotient D (ComplexShape.up \u2115)).map \u03c6) \u226b\n        (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up \u2115) n).hom.app\n          ((F.mapHomologicalComplex (ComplexShape.up \u2115)).obj J.cocomplex) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.up \u2115) n).map (I.isoRightDerivedToHomotopyCategoryObj F).hom \u226b\n      (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up \u2115) n).hom.app\n          ((F.mapHomologicalComplex (ComplexShape.up \u2115)).obj I.cocomplex) \u226b\n        HomologicalComplex.homologyMap ((F.mapHomologicalComplex (ComplexShape.up \u2115)).map \u03c6) n", "state_after": "C : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} D\ninst\u271d\u00b3 : Abelian C\ninst\u271d\u00b2 : HasInjectiveResolutions C\ninst\u271d\u00b9 : Abelian D\nX Y : C\nf : X \u27f6 Y\nI : InjectiveResolution X\nJ : InjectiveResolution Y\n\u03c6 : I.cocomplex \u27f6 J.cocomplex\ncomm : I.\u03b9.f 0 \u226b \u03c6.f 0 = f \u226b J.\u03b9.f 0\nF : C \u2964 D\ninst\u271d : F.Additive\nn : \u2115\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.up \u2115) n).map (I.isoRightDerivedToHomotopyCategoryObj F).hom \u226b\n      (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up \u2115) n).hom.app\n          ((F.mapHomologicalComplex (ComplexShape.up \u2115)).obj I.cocomplex) \u226b\n        (HomologicalComplex.homologyFunctor D (ComplexShape.up \u2115) n).map\n          ((F.mapHomologicalComplex (ComplexShape.up \u2115)).map \u03c6) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.up \u2115) n).map (I.isoRightDerivedToHomotopyCategoryObj F).hom \u226b\n      (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up \u2115) n).hom.app\n          ((F.mapHomologicalComplex (ComplexShape.up \u2115)).obj I.cocomplex) \u226b\n        HomologicalComplex.homologyMap ((F.mapHomologicalComplex (ComplexShape.up \u2115)).map \u03c6) n"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "C : Type u\ninst\u271d\u2075 : Category.{v, u} C\nD : Type u_1\ninst\u271d\u2074 : Category.{u_2, u_1} D\ninst\u271d\u00b3 : Abelian C\ninst\u271d\u00b2 : HasInjectiveResolutions C\ninst\u271d\u00b9 : Abelian D\nX Y : C\nf : X \u27f6 Y\nI : InjectiveResolution X\nJ : InjectiveResolution Y\n\u03c6 : I.cocomplex \u27f6 J.cocomplex\ncomm : I.\u03b9.f 0 \u226b \u03c6.f 0 = f \u226b J.\u03b9.f 0\nF : C \u2964 D\ninst\u271d : F.Additive\nn : \u2115\n\u22a2 (HomotopyCategory.homologyFunctor D (ComplexShape.up \u2115) n).map (I.isoRightDerivedToHomotopyCategoryObj F).hom \u226b\n      (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up \u2115) n).hom.app\n          ((F.mapHomologicalComplex (ComplexShape.up \u2115)).obj I.cocomplex) \u226b\n        (HomologicalComplex.homologyFunctor D (ComplexShape.up \u2115) n).map\n          ((F.mapHomologicalComplex (ComplexShape.up \u2115)).map \u03c6) =\n    (HomotopyCategory.homologyFunctor D (ComplexShape.up \u2115) n).map (I.isoRightDerivedToHomotopyCategoryObj F).hom \u226b\n      (HomotopyCategory.homologyFunctorFactors D (ComplexShape.up \u2115) n).hom.app\n          ((F.mapHomologicalComplex (ComplexShape.up \u2115)).obj I.cocomplex) \u226b\n        HomologicalComplex.homologyMap ((F.mapHomologicalComplex (ComplexShape.up \u2115)).map \u03c6) n", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Factorial/Basic.lean", "full_name": "Nat.succ_descFactorial", "start": [355, 1], "end": [359, 101], "traced_tactics": [{"tactic": "rw [Nat.sub_zero, descFactorial_zero, descFactorial_zero]", "annotated_tactic": ["rw [<a>Nat.sub_zero</a>, <a>descFactorial_zero</a>, <a>descFactorial_zero</a>]", [{"full_name": "Nat.sub_zero", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [268, 27], "def_end_pos": [268, 35]}, {"full_name": "Nat.descFactorial_zero", "def_path": "Mathlib/Data/Nat/Factorial/Basic.lean", "def_pos": [331, 9], "def_end_pos": [331, 27]}, {"full_name": "Nat.descFactorial_zero", "def_path": "Mathlib/Data/Nat/Factorial/Basic.lean", "def_pos": [331, 9], "def_end_pos": [331, 27]}]], "state_before": "n : \u2115\n\u22a2 (n + 1 - 0) * (n + 1).descFactorial 0 = (n + 1) * n.descFactorial 0", "state_after": "no goals"}, {"tactic": "rw [descFactorial, succ_descFactorial _ k, descFactorial_succ, succ_sub_succ, Nat.mul_left_comm]", "annotated_tactic": ["rw [<a>descFactorial</a>, succ_descFactorial _ k, <a>descFactorial_succ</a>, <a>succ_sub_succ</a>, <a>Nat.mul_left_comm</a>]", [{"full_name": "Nat.descFactorial", "def_path": "Mathlib/Data/Nat/Factorial/Basic.lean", "def_pos": [325, 5], "def_end_pos": [325, 18]}, {"full_name": "Nat.descFactorial_succ", "def_path": "Mathlib/Data/Nat/Factorial/Basic.lean", "def_pos": [336, 9], "def_end_pos": [336, 27]}, {"full_name": "Nat.succ_sub_succ", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [306, 9], "def_end_pos": [306, 22]}, {"full_name": "Nat.mul_left_comm", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [250, 19], "def_end_pos": [250, 32]}]], "state_before": "n k : \u2115\n\u22a2 (n + 1 - (k + 1)) * (n + 1).descFactorial (k + 1) = (n + 1) * n.descFactorial (k + 1)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Order/Atoms.lean", "full_name": "sSup_atoms_le_eq", "start": [460, 1], "end": [462, 98], "traced_tactics": [{"tactic": "rcases eq_sSup_atoms b with \u27e8s, rfl, hs\u27e9", "annotated_tactic": ["rcases <a>eq_sSup_atoms</a> b with \u27e8s, rfl, hs\u27e9", [{"full_name": "IsAtomistic.eq_sSup_atoms", "def_path": "Mathlib/Order/Atoms.lean", "def_pos": [411, 3], "def_end_pos": [411, 16]}]], "state_before": "\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsAtomistic \u03b1\nb : \u03b1\n\u22a2 sSup {a | IsAtom a \u2227 a \u2264 b} = b", "state_after": "case intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsAtomistic \u03b1\ns : Set \u03b1\nhs : \u2200 a \u2208 s, IsAtom a\n\u22a2 sSup {a | IsAtom a \u2227 a \u2264 sSup s} = sSup s"}, {"tactic": "exact le_antisymm (sSup_le fun _ => And.right) (sSup_le_sSup fun a ha => \u27e8hs a ha, le_sSup ha\u27e9)", "annotated_tactic": ["exact <a>le_antisymm</a> (<a>sSup_le</a> fun _ => <a>And.right</a>) (<a>sSup_le_sSup</a> fun a ha => \u27e8hs a ha, <a>le_sSup</a> ha\u27e9)", [{"full_name": "le_antisymm", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [188, 9], "def_end_pos": [188, 20]}, {"full_name": "sSup_le", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [78, 9], "def_end_pos": [78, 16]}, {"full_name": "And.right", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [524, 3], "def_end_pos": [524, 8]}, {"full_name": "sSup_le_sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [97, 9], "def_end_pos": [97, 21]}, {"full_name": "le_sSup", "def_path": "Mathlib/Order/CompleteLattice.lean", "def_pos": [74, 9], "def_end_pos": [74, 16]}]], "state_before": "case intro.intro\n\u03b9 : Sort u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d\u00b9 : CompleteLattice \u03b1\ninst\u271d : IsAtomistic \u03b1\ns : Set \u03b1\nhs : \u2200 a \u2208 s, IsAtom a\n\u22a2 sSup {a | IsAtom a \u2227 a \u2264 sSup s} = sSup s", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": ".lake/packages/batteries/Batteries/Data/String/Lemmas.lean", "full_name": "Substring.Valid.valid", "start": [951, 1], "end": [952, 79], "traced_tactics": [{"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "l m r : List Char\n\u22a2 l ++ (m ++ r) =\n      { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n            stopPos := { byteIdx := utf8Len l + utf8Len m } }.str.data \u2227\n    { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n            stopPos := { byteIdx := utf8Len l + utf8Len m } }.startPos.byteIdx =\n      utf8Len l", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "l m r : List Char\n\u22a2 l ++ m ++ r =\n      { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n            stopPos := { byteIdx := utf8Len l + utf8Len m } }.str.data \u2227\n    { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n            stopPos := { byteIdx := utf8Len l + utf8Len m } }.stopPos.byteIdx =\n      utf8Len (l ++ m)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "full_name": "MeasureTheory.lowerCrossingTime_lt_upperCrossingTime", "start": [252, 1], "end": [259, 53], "traced_tactics": [{"tactic": "refine lt_of_le_of_ne lowerCrossingTime_le_upperCrossingTime_succ fun h =>\n  not_le.2 hab <| le_trans (stoppedValue_upperCrossingTime hn) ?_", "annotated_tactic": ["refine <a>lt_of_le_of_ne</a> <a>lowerCrossingTime_le_upperCrossingTime_succ</a> fun h =>\n    <a>not_le</a>.2 hab <| <a>le_trans</a> (<a>stoppedValue_upperCrossingTime</a> hn) ?_", [{"full_name": "lt_of_le_of_ne", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [198, 9], "def_end_pos": [198, 23]}, {"full_name": "MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [206, 9], "def_end_pos": [206, 52]}, {"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [375, 9], "def_end_pos": [375, 15]}, {"full_name": "le_trans", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [56, 9], "def_end_pos": [56, 17]}, {"full_name": "MeasureTheory.stoppedValue_upperCrossingTime", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [236, 9], "def_end_pos": [236, 39]}]], "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\nhab : a < b\nhn : upperCrossingTime a b f N (n + 1) \u03c9 \u2260 N\n\u22a2 lowerCrossingTime a b f N n \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\nhab : a < b\nhn : upperCrossingTime a b f N (n + 1) \u03c9 \u2260 N\nh : lowerCrossingTime a b f N n \u03c9 = upperCrossingTime a b f N (n + 1) \u03c9\n\u22a2 stoppedValue f (upperCrossingTime a b f N (n + 1)) \u03c9 \u2264 a"}, {"tactic": "simp only [stoppedValue]", "annotated_tactic": ["simp only [<a>stoppedValue</a>]", [{"full_name": "MeasureTheory.stoppedValue", "def_path": "Mathlib/Probability/Process/Stopping.lean", "def_pos": [762, 5], "def_end_pos": [762, 17]}]], "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\nhab : a < b\nhn : upperCrossingTime a b f N (n + 1) \u03c9 \u2260 N\nh : lowerCrossingTime a b f N n \u03c9 = upperCrossingTime a b f N (n + 1) \u03c9\n\u22a2 stoppedValue f (upperCrossingTime a b f N (n + 1)) \u03c9 \u2264 a", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\nhab : a < b\nhn : upperCrossingTime a b f N (n + 1) \u03c9 \u2260 N\nh : lowerCrossingTime a b f N n \u03c9 = upperCrossingTime a b f N (n + 1) \u03c9\n\u22a2 f (upperCrossingTime a b f N (n + 1) \u03c9) \u03c9 \u2264 a"}, {"tactic": "rw [\u2190 h]", "annotated_tactic": ["rw [\u2190 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\nhab : a < b\nhn : upperCrossingTime a b f N (n + 1) \u03c9 \u2260 N\nh : lowerCrossingTime a b f N n \u03c9 = upperCrossingTime a b f N (n + 1) \u03c9\n\u22a2 f (upperCrossingTime a b f N (n + 1) \u03c9) \u03c9 \u2264 a", "state_after": "\u03a9 : Type u_1\n\u03b9 : Type u_2\nm0 : MeasurableSpace \u03a9\n\u03bc : Measure \u03a9\na b : \u211d\nf : \u2115 \u2192 \u03a9 \u2192 \u211d\nN n m : \u2115\n\u03c9 : \u03a9\nhab : a < b\nhn : upperCrossingTime a b f N (n + 1) \u03c9 \u2260 N\nh : lowerCrossingTime a b f N n \u03c9 = upperCrossingTime a b f N (n + 1) \u03c9\n\u22a2 f (lowerCrossingTime a b f N n \u03c9) \u03c9 \u2264 a"}, {"tactic": "exact stoppedValue_lowerCrossingTime (h.symm \u25b8 hn)", "annotated_tactic": ["exact <a>stoppedValue_lowerCrossingTime</a> (h.symm \u25b8 hn)", [{"full_name": "MeasureTheory.stoppedValue_lowerCrossingTime", "def_path": "Mathlib/Probability/Martingale/Upcrossing.lean", "def_pos": [230, 9], "def_end_pos": [230, 39]}]], "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\nhab : a < b\nhn : upperCrossingTime a b f N (n + 1) \u03c9 \u2260 N\nh : lowerCrossingTime a b f N n \u03c9 = upperCrossingTime a b f N (n + 1) \u03c9\n\u22a2 f (lowerCrossingTime a b f N n \u03c9) \u03c9 \u2264 a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/LocallyFinite.lean", "full_name": "locallyFinite_iff_smallSets", "start": [59, 1], "end": [63, 71], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/KummerExtension.lean", "full_name": "X_pow_mul_sub_C_irreducible", "start": [185, 1], "end": [195, 57], "traced_tactics": [{"tactic": "have hm' : m \u2260 0 := by\n  rintro rfl\n  rw [pow_zero, \u2190 C.map_one, \u2190 map_sub] at hm\n  exact not_irreducible_C _ hm", "annotated_tactic": ["have hm' : m \u2260 0 := by\n    rintro rfl\n    rw [<a>pow_zero</a>, \u2190 C.map_one, \u2190 <a>map_sub</a>] at hm\n    exact <a>not_irreducible_C</a> _ hm", [{"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [651, 9], "def_end_pos": [651, 17]}, {"full_name": "map_sub", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [461, 3], "def_end_pos": [461, 14]}, {"full_name": "Polynomial.not_irreducible_C", "def_path": "Mathlib/Algebra/Polynomial/FieldDivision.lean", "def_pos": [598, 9], "def_end_pos": [598, 26]}]], "state_before": "K : Type u\ninst\u271d : Field K\nn m : \u2115\na : K\nhm : Irreducible (X ^ m - C a)\nhn :\n  \u2200 (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n    minpoly K x = X ^ m - C a \u2192 Irreducible (X ^ n - C (AdjoinSimple.gen K x))\n\u22a2 Irreducible (X ^ (n * m) - C a)", "state_after": "K : Type u\ninst\u271d : Field K\nn m : \u2115\na : K\nhm : Irreducible (X ^ m - C a)\nhn :\n  \u2200 (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n    minpoly K x = X ^ m - C a \u2192 Irreducible (X ^ n - C (AdjoinSimple.gen K x))\nhm' : m \u2260 0\n\u22a2 Irreducible (X ^ (n * m) - C a)"}, {"tactic": "simpa [pow_mul] using irreducible_comp (monic_X_pow_sub_C a hm') (monic_X_pow n) hm\n  (by simpa only [Polynomial.map_pow, map_X] using hn)", "annotated_tactic": ["simpa [<a>pow_mul</a>] using <a>irreducible_comp</a> (<a>monic_X_pow_sub_C</a> a hm') (<a>monic_X_pow</a> n) hm\n    (by simpa only [<a>Polynomial.map_pow</a>, <a>map_X</a>] using hn)", [{"full_name": "pow_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [713, 32], "def_end_pos": [713, 39]}, {"full_name": "Polynomial.irreducible_comp", "def_path": "Mathlib/FieldTheory/Adjoin.lean", "def_pos": [1283, 9], "def_end_pos": [1283, 43]}, {"full_name": "Polynomial.monic_X_pow_sub_C", "def_path": "Mathlib/Algebra/Polynomial/Monic.lean", "def_pos": [399, 9], "def_end_pos": [399, 26]}, {"full_name": "Polynomial.monic_X_pow", "def_path": "Mathlib/Algebra/Polynomial/Degree/Definitions.lean", "def_pos": [890, 9], "def_end_pos": [890, 20]}, {"full_name": "Polynomial.map_pow", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [936, 19], "def_end_pos": [936, 26]}, {"full_name": "Polynomial.map_X", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [723, 9], "def_end_pos": [723, 14]}]], "state_before": "K : Type u\ninst\u271d : Field K\nn m : \u2115\na : K\nhm : Irreducible (X ^ m - C a)\nhn :\n  \u2200 (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n    minpoly K x = X ^ m - C a \u2192 Irreducible (X ^ n - C (AdjoinSimple.gen K x))\nhm' : m \u2260 0\n\u22a2 Irreducible (X ^ (n * m) - C a)", "state_after": "no goals"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "K : Type u\ninst\u271d : Field K\nn m : \u2115\na : K\nhm : Irreducible (X ^ m - C a)\nhn :\n  \u2200 (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n    minpoly K x = X ^ m - C a \u2192 Irreducible (X ^ n - C (AdjoinSimple.gen K x))\n\u22a2 m \u2260 0", "state_after": "K : Type u\ninst\u271d : Field K\nn : \u2115\na : K\nhm : Irreducible (X ^ 0 - C a)\nhn :\n  \u2200 (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n    minpoly K x = X ^ 0 - C a \u2192 Irreducible (X ^ n - C (AdjoinSimple.gen K x))\n\u22a2 False"}, {"tactic": "rw [pow_zero, \u2190 C.map_one, \u2190 map_sub] at hm", "annotated_tactic": ["rw [<a>pow_zero</a>, \u2190 C.map_one, \u2190 <a>map_sub</a>] at hm", [{"full_name": "pow_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [651, 9], "def_end_pos": [651, 17]}, {"full_name": "map_sub", "def_path": "Mathlib/Algebra/Group/Hom/Defs.lean", "def_pos": [461, 3], "def_end_pos": [461, 14]}]], "state_before": "K : Type u\ninst\u271d : Field K\nn : \u2115\na : K\nhm : Irreducible (X ^ 0 - C a)\nhn :\n  \u2200 (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n    minpoly K x = X ^ 0 - C a \u2192 Irreducible (X ^ n - C (AdjoinSimple.gen K x))\n\u22a2 False", "state_after": "K : Type u\ninst\u271d : Field K\nn : \u2115\na : K\nhm : Irreducible (C (1 - a))\nhn :\n  \u2200 (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n    minpoly K x = X ^ 0 - C a \u2192 Irreducible (X ^ n - C (AdjoinSimple.gen K x))\n\u22a2 False"}, {"tactic": "exact not_irreducible_C _ hm", "annotated_tactic": ["exact <a>not_irreducible_C</a> _ hm", [{"full_name": "Polynomial.not_irreducible_C", "def_path": "Mathlib/Algebra/Polynomial/FieldDivision.lean", "def_pos": [598, 9], "def_end_pos": [598, 26]}]], "state_before": "K : Type u\ninst\u271d : Field K\nn : \u2115\na : K\nhm : Irreducible (C (1 - a))\nhn :\n  \u2200 (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n    minpoly K x = X ^ 0 - C a \u2192 Irreducible (X ^ n - C (AdjoinSimple.gen K x))\n\u22a2 False", "state_after": "no goals"}, {"tactic": "simpa only [Polynomial.map_pow, map_X] using hn", "annotated_tactic": ["simpa only [<a>Polynomial.map_pow</a>, <a>map_X</a>] using hn", [{"full_name": "Polynomial.map_pow", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [936, 19], "def_end_pos": [936, 26]}, {"full_name": "Polynomial.map_X", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [723, 9], "def_end_pos": [723, 14]}]], "state_before": "K : Type u\ninst\u271d : Field K\nn m : \u2115\na : K\nhm : Irreducible (X ^ m - C a)\nhn :\n  \u2200 (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n    minpoly K x = X ^ m - C a \u2192 Irreducible (X ^ n - C (AdjoinSimple.gen K x))\nhm' : m \u2260 0\n\u22a2 \u2200 (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n    minpoly K x = X ^ m - C a \u2192 Irreducible (Polynomial.map (algebraMap K \u21a5K\u27eex\u27ef) (X ^ n) - C (AdjoinSimple.gen K x))", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/PiSystem.lean", "full_name": "isPiSystem_Ioc_mem", "start": [190, 1], "end": [192, 92], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/List/Indexes.lean", "full_name": "List.findIdx_le_length", "start": [279, 1], "end": [282, 54], "traced_tactics": [{"tactic": "by_cases e : \u2203 x \u2208 xs, p x", "annotated_tactic": ["by_cases e : \u2203 x \u2208 xs, p x", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Bool\nxs : List \u03b1\n\u22a2 findIdx p xs \u2264 xs.length", "state_after": "case pos\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Bool\nxs : List \u03b1\ne : \u2203 x \u2208 xs, p x = true\n\u22a2 findIdx p xs \u2264 xs.length\n\ncase neg\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Bool\nxs : List \u03b1\ne : \u00ac\u2203 x \u2208 xs, p x = true\n\u22a2 findIdx p xs \u2264 xs.length"}, {"tactic": "exact (findIdx_lt_length_of_exists e).le", "annotated_tactic": ["exact (<a>findIdx_lt_length_of_exists</a> e).<a>le</a>", [{"full_name": "List.findIdx_lt_length_of_exists", "def_path": ".lake/packages/batteries/Batteries/Data/List/Lemmas.lean", "def_pos": [691, 9], "def_end_pos": [691, 36]}, {"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [137, 7], "def_end_pos": [137, 15]}]], "state_before": "case pos\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Bool\nxs : List \u03b1\ne : \u2203 x \u2208 xs, p x = true\n\u22a2 findIdx p xs \u2264 xs.length", "state_after": "no goals"}, {"tactic": "push_neg at e", "annotated_tactic": ["push_neg at e", []], "state_before": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Bool\nxs : List \u03b1\ne : \u00ac\u2203 x \u2208 xs, p x = true\n\u22a2 findIdx p xs \u2264 xs.length", "state_after": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Bool\nxs : List \u03b1\ne : \u2200 x \u2208 xs, p x \u2260 true\n\u22a2 findIdx p xs \u2264 xs.length"}, {"tactic": "exact (findIdx_eq_length.mpr e).le", "annotated_tactic": ["exact (findIdx_eq_length.mpr e).<a>le</a>", [{"full_name": "Eq.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [154, 7], "def_end_pos": [154, 12]}]], "state_before": "case neg\n\u03b1 : Type u\n\u03b2 : Type v\np : \u03b1 \u2192 Bool\nxs : List \u03b1\ne : \u2200 x \u2208 xs, p x \u2260 true\n\u22a2 findIdx p xs \u2264 xs.length", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Eval.lean", "full_name": "Polynomial.eval\u2082_ofFinsupp", "start": [170, 1], "end": [173, 6], "traced_tactics": [{"tactic": "simp only [eval\u2082_eq_sum, sum, toFinsupp_sum, support, coeff]", "annotated_tactic": ["simp only [<a>eval\u2082_eq_sum</a>, <a>sum</a>, <a>toFinsupp_sum</a>, <a>support</a>, <a>coeff</a>]", [{"full_name": "Polynomial.eval\u2082_eq_sum", "def_path": "Mathlib/Algebra/Polynomial/Eval.lean", "def_pos": [48, 9], "def_end_pos": [48, 21]}, {"full_name": "Polynomial.sum", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [971, 5], "def_end_pos": [971, 8]}, {"full_name": "Polynomial.toFinsupp_sum", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [396, 9], "def_end_pos": [396, 22]}, {"full_name": "Polynomial.support", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [406, 5], "def_end_pos": [406, 12]}, {"full_name": "Polynomial.coeff", "def_path": "Mathlib/Algebra/Polynomial/Basic.lean", "def_pos": [663, 5], "def_end_pos": [663, 10]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np\u271d q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf\u271d : R \u2192+* S\nx\u271d : S\ninst\u271d : Semiring T\nf : R \u2192+* S\nx : S\np : R[\u2115]\n\u22a2 eval\u2082 f x { toFinsupp := p } = (liftNC \u2191f \u21d1((powersHom S) x)) p", "state_after": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np\u271d q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf\u271d : R \u2192+* S\nx\u271d : S\ninst\u271d : Semiring T\nf : R \u2192+* S\nx : S\np : R[\u2115]\n\u22a2 \u2211 x_1 \u2208 p.support, f (p x_1) * x ^ x_1 = (liftNC \u2191f \u21d1((powersHom S) x)) p"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n : \u2115\ninst\u271d\u00b2 : Semiring R\np\u271d q r : R[X]\ninst\u271d\u00b9 : Semiring S\nf\u271d : R \u2192+* S\nx\u271d : S\ninst\u271d : Semiring T\nf : R \u2192+* S\nx : S\np : R[\u2115]\n\u22a2 \u2211 x_1 \u2208 p.support, f (p x_1) * x ^ x_1 = (liftNC \u2191f \u21d1((powersHom S) x)) p", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Group/Subgroup/Pointwise.lean", "full_name": "Subgroup.smul_opposite_image_mul_preimage'", "start": [279, 1], "end": [281, 38], "traced_tactics": [{"tactic": "simp [preimage_preimage, mul_assoc]", "annotated_tactic": ["simp [<a>preimage_preimage</a>, <a>mul_assoc</a>]", [{"full_name": "Set.preimage_preimage", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [162, 9], "def_end_pos": [162, 26]}, {"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [289, 9], "def_end_pos": [289, 18]}]], "state_before": "\u03b1 : Type u_1\nG : Type u_2\nA : Type u_3\nS : Type u_4\ninst\u271d\u00b9 : Group G\ninst\u271d : AddGroup A\ns\u271d : Set G\ng : G\nh : G\u1d50\u1d52\u1d56\ns : Set G\n\u22a2 (fun y => h \u2022 y) '' ((fun x => g * x) \u207b\u00b9' s) = (fun x => g * x) \u207b\u00b9' ((fun y => h \u2022 y) '' s)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/LinearAlgebra/Matrix/Basis.lean", "full_name": "Basis.toMatrix_reindex'", "start": [245, 1], "end": [250, 35], "traced_tactics": [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03ba : Type u_3\n\u03ba' : Type u_4\nR : Type u_5\nM : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : Module R M\nR\u2082 : Type u_7\nM\u2082 : Type u_8\ninst\u271d\u2079 : CommRing R\u2082\ninst\u271d\u2078 : AddCommGroup M\u2082\ninst\u271d\u2077 : Module R\u2082 M\u2082\ne\u271d : Basis \u03b9 R M\nv\u271d : \u03b9' \u2192 M\ni : \u03b9\nj : \u03b9'\nN : Type u_9\ninst\u271d\u2076 : AddCommMonoid N\ninst\u271d\u2075 : Module R N\nb\u271d : Basis \u03b9 R M\nb' : Basis \u03b9' R M\nc : Basis \u03ba R N\nc' : Basis \u03ba' R N\nf : M \u2192\u2097[R] N\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : Finite \u03ba\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : DecidableEq \u03b9'\nb : Basis \u03b9 R M\nv : \u03b9' \u2192 M\ne : \u03b9 \u2243 \u03b9'\n\u22a2 (b.reindex e).toMatrix v = (reindexAlgEquiv R e) (b.toMatrix (v \u2218 \u21d1e))", "state_after": "case a\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03ba : Type u_3\n\u03ba' : Type u_4\nR : Type u_5\nM : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : Module R M\nR\u2082 : Type u_7\nM\u2082 : Type u_8\ninst\u271d\u2079 : CommRing R\u2082\ninst\u271d\u2078 : AddCommGroup M\u2082\ninst\u271d\u2077 : Module R\u2082 M\u2082\ne\u271d : Basis \u03b9 R M\nv\u271d : \u03b9' \u2192 M\ni : \u03b9\nj : \u03b9'\nN : Type u_9\ninst\u271d\u2076 : AddCommMonoid N\ninst\u271d\u2075 : Module R N\nb\u271d : Basis \u03b9 R M\nb' : Basis \u03b9' R M\nc : Basis \u03ba R N\nc' : Basis \u03ba' R N\nf : M \u2192\u2097[R] N\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : Finite \u03ba\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : DecidableEq \u03b9'\nb : Basis \u03b9 R M\nv : \u03b9' \u2192 M\ne : \u03b9 \u2243 \u03b9'\ni\u271d j\u271d : \u03b9'\n\u22a2 (b.reindex e).toMatrix v i\u271d j\u271d = (reindexAlgEquiv R e) (b.toMatrix (v \u2218 \u21d1e)) i\u271d j\u271d"}, {"tactic": "simp only [Basis.toMatrix_apply, Basis.repr_reindex, Matrix.reindexAlgEquiv_apply,\n  Matrix.reindex_apply, Matrix.submatrix_apply, Function.comp_apply, e.apply_symm_apply,\n  Finsupp.mapDomain_equiv_apply]", "annotated_tactic": ["simp only [<a>Basis.toMatrix_apply</a>, <a>Basis.repr_reindex</a>, <a>Matrix.reindexAlgEquiv_apply</a>,\n    <a>Matrix.reindex_apply</a>, <a>Matrix.submatrix_apply</a>, <a>Function.comp_apply</a>, e.apply_symm_apply,\n    <a>Finsupp.mapDomain_equiv_apply</a>]", [{"full_name": "Basis.toMatrix_apply", "def_path": "Mathlib/LinearAlgebra/Matrix/Basis.lean", "def_pos": [58, 9], "def_end_pos": [58, 23]}, {"full_name": "Basis.repr_reindex", "def_path": "Mathlib/LinearAlgebra/Basis.lean", "def_pos": [433, 9], "def_end_pos": [433, 21]}, {"full_name": "Matrix.reindexAlgEquiv_apply", "def_path": "Mathlib/LinearAlgebra/Matrix/Reindex.lean", "def_pos": [131, 9], "def_end_pos": [131, 30]}, {"full_name": "Matrix.reindex_apply", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [2777, 9], "def_end_pos": [2777, 22]}, {"full_name": "Matrix.submatrix_apply", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [2608, 9], "def_end_pos": [2608, 24]}, {"full_name": "Function.comp_apply", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [35, 17], "def_end_pos": [35, 36]}, {"full_name": "Finsupp.mapDomain_equiv_apply", "def_path": "Mathlib/Data/Finsupp/Basic.lean", "def_pos": [502, 9], "def_end_pos": [502, 30]}]], "state_before": "case a\n\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03ba : Type u_3\n\u03ba' : Type u_4\nR : Type u_5\nM : Type u_6\ninst\u271d\u00b9\u00b2 : CommSemiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : Module R M\nR\u2082 : Type u_7\nM\u2082 : Type u_8\ninst\u271d\u2079 : CommRing R\u2082\ninst\u271d\u2078 : AddCommGroup M\u2082\ninst\u271d\u2077 : Module R\u2082 M\u2082\ne\u271d : Basis \u03b9 R M\nv\u271d : \u03b9' \u2192 M\ni : \u03b9\nj : \u03b9'\nN : Type u_9\ninst\u271d\u2076 : AddCommMonoid N\ninst\u271d\u2075 : Module R N\nb\u271d : Basis \u03b9 R M\nb' : Basis \u03b9' R M\nc : Basis \u03ba R N\nc' : Basis \u03ba' R N\nf : M \u2192\u2097[R] N\ninst\u271d\u2074 : Fintype \u03b9'\ninst\u271d\u00b3 : Finite \u03ba\ninst\u271d\u00b2 : Fintype \u03b9\ninst\u271d\u00b9 : DecidableEq \u03b9\ninst\u271d : DecidableEq \u03b9'\nb : Basis \u03b9 R M\nv : \u03b9' \u2192 M\ne : \u03b9 \u2243 \u03b9'\ni\u271d j\u271d : \u03b9'\n\u22a2 (b.reindex e).toMatrix v i\u271d j\u271d = (reindexAlgEquiv R e) (b.toMatrix (v \u2218 \u21d1e)) i\u271d j\u271d", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "full_name": "MvQPF.has_good_supp_iff", "start": [180, 1], "end": [203, 40], "traced_tactics": [{"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "n : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\n\u22a2 (\u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u) \u2194\n    \u2203 a f,\n      abs \u27e8a, f\u27e9 = x \u2227\n        \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ", "state_after": "case mp\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\n\u22a2 (\u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u) \u2192\n    \u2203 a f,\n      abs \u27e8a, f\u27e9 = x \u2227\n        \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\n\ncase mpr\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\n\u22a2 (\u2203 a f,\n      abs \u27e8a, f\u27e9 = x \u2227\n        \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ) \u2192\n    \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u"}, {"tactic": "rintro \u27e8a, f, xeq, h\u27e9 p", "annotated_tactic": ["rintro \u27e8a, f, xeq, h\u27e9 p", []], "state_before": "case mpr\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\n\u22a2 (\u2203 a f,\n      abs \u27e8a, f\u27e9 = x \u2227\n        \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ) \u2192\n    \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u", "state_after": "case mpr.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\n\u22a2 LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u"}, {"tactic": "rw [liftP_iff]", "annotated_tactic": ["rw [<a>liftP_iff</a>]", [{"full_name": "MvQPF.liftP_iff", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 18]}]], "state_before": "case mpr.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\n\u22a2 LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u", "state_after": "case mpr.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\n\u22a2 (\u2203 a f, x = abs \u27e8a, f\u27e9 \u2227 \u2200 (i : Fin2 n) (j : (P F).B a i), p i (f i j)) \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "case mpr.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\n\u22a2 (\u2203 a f, x = abs \u27e8a, f\u27e9 \u2227 \u2200 (i : Fin2 n) (j : (P F).B a i), p i (f i j)) \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u", "state_after": "case mpr.intro.intro.intro.mp\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\n\u22a2 (\u2203 a f, x = abs \u27e8a, f\u27e9 \u2227 \u2200 (i : Fin2 n) (j : (P F).B a i), p i (f i j)) \u2192 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\n\ncase mpr.intro.intro.intro.mpr\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\n\u22a2 (\u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u) \u2192 \u2203 a f, x = abs \u27e8a, f\u27e9 \u2227 \u2200 (i : Fin2 n) (j : (P F).B a i), p i (f i j)"}, {"tactic": "intro h'", "annotated_tactic": ["intro h'", []], "state_before": "case mpr.intro.intro.intro.mpr\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\n\u22a2 (\u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u) \u2192 \u2203 a f, x = abs \u27e8a, f\u27e9 \u2227 \u2200 (i : Fin2 n) (j : (P F).B a i), p i (f i j)", "state_after": "case mpr.intro.intro.intro.mpr\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\nh' : \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\n\u22a2 \u2203 a f, x = abs \u27e8a, f\u27e9 \u2227 \u2200 (i : Fin2 n) (j : (P F).B a i), p i (f i j)"}, {"tactic": "refine \u27e8a, f, xeq.symm, ?_\u27e9", "annotated_tactic": ["refine \u27e8a, f, xeq.symm, ?_\u27e9", []], "state_before": "case mpr.intro.intro.intro.mpr\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\nh' : \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\n\u22a2 \u2203 a f, x = abs \u27e8a, f\u27e9 \u2227 \u2200 (i : Fin2 n) (j : (P F).B a i), p i (f i j)", "state_after": "case mpr.intro.intro.intro.mpr\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\nh' : \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\n\u22a2 \u2200 (i : Fin2 n) (j : (P F).B a i), p i (f i j)"}, {"tactic": "intro j y", "annotated_tactic": ["intro j y", []], "state_before": "case mpr.intro.intro.intro.mpr\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\nh' : \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\n\u22a2 \u2200 (i : Fin2 n) (j : (P F).B a i), p i (f i j)", "state_after": "case mpr.intro.intro.intro.mpr\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\nh' : \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\nj : Fin2 n\ny : (P F).B a j\n\u22a2 p j (f j y)"}, {"tactic": "apply h'", "annotated_tactic": ["apply h'", []], "state_before": "case mpr.intro.intro.intro.mpr\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\nh' : \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\nj : Fin2 n\ny : (P F).B a j\n\u22a2 p j (f j y)", "state_after": "case mpr.intro.intro.intro.mpr.a\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\nh' : \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\nj : Fin2 n\ny : (P F).B a j\n\u22a2 f j y \u2208 supp x j"}, {"tactic": "rw [mem_supp]", "annotated_tactic": ["rw [<a>mem_supp</a>]", [{"full_name": "MvQPF.mem_supp", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [160, 9], "def_end_pos": [160, 17]}]], "state_before": "case mpr.intro.intro.intro.mpr.a\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\nh' : \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\nj : Fin2 n\ny : (P F).B a j\n\u22a2 f j y \u2208 supp x j", "state_after": "case mpr.intro.intro.intro.mpr.a\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\nh' : \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\nj : Fin2 n\ny : (P F).B a j\n\u22a2 \u2200 (a : (P F).A) (f_1 : (P F).B a \u27f9 \u03b1), abs \u27e8a, f_1\u27e9 = x \u2192 f j y \u2208 f_1 j '' univ"}, {"tactic": "intro a' f' xeq'", "annotated_tactic": ["intro a' f' xeq'", []], "state_before": "case mpr.intro.intro.intro.mpr.a\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\nh' : \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\nj : Fin2 n\ny : (P F).B a j\n\u22a2 \u2200 (a : (P F).A) (f_1 : (P F).B a \u27f9 \u03b1), abs \u27e8a, f_1\u27e9 = x \u2192 f j y \u2208 f_1 j '' univ", "state_after": "case mpr.intro.intro.intro.mpr.a\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\nh' : \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\nj : Fin2 n\ny : (P F).B a j\na' : (P F).A\nf' : (P F).B a' \u27f9 \u03b1\nxeq' : abs \u27e8a', f'\u27e9 = x\n\u22a2 f j y \u2208 f' j '' univ"}, {"tactic": "apply h _ a' f' xeq'", "annotated_tactic": ["apply h _ a' f' xeq'", []], "state_before": "case mpr.intro.intro.intro.mpr.a\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\nh' : \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\nj : Fin2 n\ny : (P F).B a j\na' : (P F).A\nf' : (P F).B a' \u27f9 \u03b1\nxeq' : abs \u27e8a', f'\u27e9 = x\n\u22a2 f j y \u2208 f' j '' univ", "state_after": "case mpr.intro.intro.intro.mpr.a.a\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\nh' : \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\nj : Fin2 n\ny : (P F).B a j\na' : (P F).A\nf' : (P F).B a' \u27f9 \u03b1\nxeq' : abs \u27e8a', f'\u27e9 = x\n\u22a2 f j y \u2208 f j '' univ"}, {"tactic": "apply mem_image_of_mem _ (mem_univ _)", "annotated_tactic": ["apply <a>mem_image_of_mem</a> _ (<a>mem_univ</a> _)", [{"full_name": "Set.mem_image_of_mem", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [132, 9], "def_end_pos": [132, 25]}, {"full_name": "Set.mem_univ", "def_path": "Mathlib/Data/Set/Defs.lean", "def_pos": [82, 29], "def_end_pos": [82, 37]}]], "state_before": "case mpr.intro.intro.intro.mpr.a.a\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\nh' : \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\nj : Fin2 n\ny : (P F).B a j\na' : (P F).A\nf' : (P F).B a' \u27f9 \u03b1\nxeq' : abs \u27e8a', f'\u27e9 = x\n\u22a2 f j y \u2208 f j '' univ", "state_after": "no goals"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case mp\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\n\u22a2 (\u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u) \u2192\n    \u2203 a f,\n      abs \u27e8a, f\u27e9 = x \u2227\n        \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ", "state_after": "case mp\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\n\u22a2 \u2203 a f,\n    abs \u27e8a, f\u27e9 = x \u2227 \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ"}, {"tactic": "have : LiftP (supp x) x := by rw [h]; introv; exact id", "annotated_tactic": ["have : <a>LiftP</a> (<a>supp</a> x) x := by rw [h]; introv; exact <a>id</a>", [{"full_name": "MvFunctor.LiftP", "def_path": "Mathlib/Control/Functor/Multivariate.lean", "def_pos": [45, 5], "def_end_pos": [45, 10]}, {"full_name": "MvFunctor.supp", "def_path": "Mathlib/Control/Functor/Multivariate.lean", "def_pos": [58, 5], "def_end_pos": [58, 9]}, {"full_name": "id", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "case mp\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\n\u22a2 \u2203 a f,\n    abs \u27e8a, f\u27e9 = x \u2227 \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ", "state_after": "case mp\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\nthis : LiftP (supp x) x\n\u22a2 \u2203 a f,\n    abs \u27e8a, f\u27e9 = x \u2227 \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ"}, {"tactic": "rw [liftP_iff] at this", "annotated_tactic": ["rw [<a>liftP_iff</a>] at this", [{"full_name": "MvQPF.liftP_iff", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [122, 9], "def_end_pos": [122, 18]}]], "state_before": "case mp\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\nthis : LiftP (supp x) x\n\u22a2 \u2203 a f,\n    abs \u27e8a, f\u27e9 = x \u2227 \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ", "state_after": "case mp\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\nthis : \u2203 a f, x = abs \u27e8a, f\u27e9 \u2227 \u2200 (i : Fin2 n) (j : (P F).B a i), supp x i (f i j)\n\u22a2 \u2203 a f,\n    abs \u27e8a, f\u27e9 = x \u2227 \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ"}, {"tactic": "rcases this with \u27e8a, f, xeq, h'\u27e9", "annotated_tactic": ["rcases this with \u27e8a, f, xeq, h'\u27e9", []], "state_before": "case mp\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\nthis : \u2203 a f, x = abs \u27e8a, f\u27e9 \u2227 \u2200 (i : Fin2 n) (j : (P F).B a i), supp x i (f i j)\n\u22a2 \u2203 a f,\n    abs \u27e8a, f\u27e9 = x \u2227 \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ", "state_after": "case mp.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : x = abs \u27e8a, f\u27e9\nh' : \u2200 (i : Fin2 n) (j : (P F).B a i), supp x i (f i j)\n\u22a2 \u2203 a f,\n    abs \u27e8a, f\u27e9 = x \u2227 \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ"}, {"tactic": "refine \u27e8a, f, xeq.symm, ?_\u27e9", "annotated_tactic": ["refine \u27e8a, f, xeq.symm, ?_\u27e9", []], "state_before": "case mp.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : x = abs \u27e8a, f\u27e9\nh' : \u2200 (i : Fin2 n) (j : (P F).B a i), supp x i (f i j)\n\u22a2 \u2203 a f,\n    abs \u27e8a, f\u27e9 = x \u2227 \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ", "state_after": "case mp.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : x = abs \u27e8a, f\u27e9\nh' : \u2200 (i : Fin2 n) (j : (P F).B a i), supp x i (f i j)\n\u22a2 \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ"}, {"tactic": "intro a' f' h''", "annotated_tactic": ["intro a' f' h''", []], "state_before": "case mp.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : x = abs \u27e8a, f\u27e9\nh' : \u2200 (i : Fin2 n) (j : (P F).B a i), supp x i (f i j)\n\u22a2 \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ", "state_after": "case mp.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : x = abs \u27e8a, f\u27e9\nh' : \u2200 (i : Fin2 n) (j : (P F).B a i), supp x i (f i j)\na' : Fin2 n\nf' : (P F).A\nh'' : (P F).B f' \u27f9 \u03b1\n\u22a2 abs \u27e8f', h''\u27e9 = x \u2192 f a' '' univ \u2286 h'' a' '' univ"}, {"tactic": "rintro hu u \u27e8j, _h\u2082, hfi\u27e9", "annotated_tactic": ["rintro hu u \u27e8j, _h\u2082, hfi\u27e9", []], "state_before": "case mp.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : x = abs \u27e8a, f\u27e9\nh' : \u2200 (i : Fin2 n) (j : (P F).B a i), supp x i (f i j)\na' : Fin2 n\nf' : (P F).A\nh'' : (P F).B f' \u27f9 \u03b1\n\u22a2 abs \u27e8f', h''\u27e9 = x \u2192 f a' '' univ \u2286 h'' a' '' univ", "state_after": "case mp.intro.intro.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : x = abs \u27e8a, f\u27e9\nh' : \u2200 (i : Fin2 n) (j : (P F).B a i), supp x i (f i j)\na' : Fin2 n\nf' : (P F).A\nh'' : (P F).B f' \u27f9 \u03b1\nhu : abs \u27e8f', h''\u27e9 = x\nu : \u03b1 a'\nj : (P F).B a a'\n_h\u2082 : j \u2208 univ\nhfi : f a' j = u\n\u22a2 u \u2208 h'' a' '' univ"}, {"tactic": "have hh : u \u2208 supp x a' := by rw [\u2190 hfi]; apply h'", "annotated_tactic": ["have hh : u \u2208 <a>supp</a> x a' := by rw [\u2190 hfi]; apply h'", [{"full_name": "MvFunctor.supp", "def_path": "Mathlib/Control/Functor/Multivariate.lean", "def_pos": [58, 5], "def_end_pos": [58, 9]}]], "state_before": "case mp.intro.intro.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : x = abs \u27e8a, f\u27e9\nh' : \u2200 (i : Fin2 n) (j : (P F).B a i), supp x i (f i j)\na' : Fin2 n\nf' : (P F).A\nh'' : (P F).B f' \u27f9 \u03b1\nhu : abs \u27e8f', h''\u27e9 = x\nu : \u03b1 a'\nj : (P F).B a a'\n_h\u2082 : j \u2208 univ\nhfi : f a' j = u\n\u22a2 u \u2208 h'' a' '' univ", "state_after": "case mp.intro.intro.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : x = abs \u27e8a, f\u27e9\nh' : \u2200 (i : Fin2 n) (j : (P F).B a i), supp x i (f i j)\na' : Fin2 n\nf' : (P F).A\nh'' : (P F).B f' \u27f9 \u03b1\nhu : abs \u27e8f', h''\u27e9 = x\nu : \u03b1 a'\nj : (P F).B a a'\n_h\u2082 : j \u2208 univ\nhfi : f a' j = u\nhh : u \u2208 supp x a'\n\u22a2 u \u2208 h'' a' '' univ"}, {"tactic": "exact (mem_supp x _ u).mp hh _ _ hu", "annotated_tactic": ["exact (<a>mem_supp</a> x _ u).<a>mp</a> hh _ _ hu", [{"full_name": "MvQPF.mem_supp", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [160, 9], "def_end_pos": [160, 17]}, {"full_name": "Iff.mp", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [118, 3], "def_end_pos": [118, 5]}]], "state_before": "case mp.intro.intro.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : x = abs \u27e8a, f\u27e9\nh' : \u2200 (i : Fin2 n) (j : (P F).B a i), supp x i (f i j)\na' : Fin2 n\nf' : (P F).A\nh'' : (P F).B f' \u27f9 \u03b1\nhu : abs \u27e8f', h''\u27e9 = x\nu : \u03b1 a'\nj : (P F).B a a'\n_h\u2082 : j \u2208 univ\nhfi : f a' j = u\nhh : u \u2208 supp x a'\n\u22a2 u \u2208 h'' a' '' univ", "state_after": "no goals"}, {"tactic": "rw [h]", "annotated_tactic": ["rw [h]", []], "state_before": "n : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\n\u22a2 LiftP (supp x) x", "state_after": "n : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\n\u22a2 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, supp x i u"}, {"tactic": "introv", "annotated_tactic": ["introv", []], "state_before": "n : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\n\u22a2 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, supp x i u", "state_after": "n : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\ni : Fin2 n\nu : \u03b1 i\n\u22a2 u \u2208 supp x i \u2192 supp x i u"}, {"tactic": "exact id", "annotated_tactic": ["exact <a>id</a>", [{"full_name": "id", "def_path": ".lake/packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [33, 15], "def_end_pos": [33, 17]}]], "state_before": "n : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\ni : Fin2 n\nu : \u03b1 i\n\u22a2 u \u2208 supp x i \u2192 supp x i u", "state_after": "no goals"}, {"tactic": "rw [\u2190 hfi]", "annotated_tactic": ["rw [\u2190 hfi]", []], "state_before": "n : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : x = abs \u27e8a, f\u27e9\nh' : \u2200 (i : Fin2 n) (j : (P F).B a i), supp x i (f i j)\na' : Fin2 n\nf' : (P F).A\nh'' : (P F).B f' \u27f9 \u03b1\nhu : abs \u27e8f', h''\u27e9 = x\nu : \u03b1 a'\nj : (P F).B a a'\n_h\u2082 : j \u2208 univ\nhfi : f a' j = u\n\u22a2 u \u2208 supp x a'", "state_after": "n : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : x = abs \u27e8a, f\u27e9\nh' : \u2200 (i : Fin2 n) (j : (P F).B a i), supp x i (f i j)\na' : Fin2 n\nf' : (P F).A\nh'' : (P F).B f' \u27f9 \u03b1\nhu : abs \u27e8f', h''\u27e9 = x\nu : \u03b1 a'\nj : (P F).B a a'\n_h\u2082 : j \u2208 univ\nhfi : f a' j = u\n\u22a2 f a' j \u2208 supp x a'"}, {"tactic": "apply h'", "annotated_tactic": ["apply h'", []], "state_before": "n : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\nh : \u2200 (p : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop), LiftP p x \u2194 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : x = abs \u27e8a, f\u27e9\nh' : \u2200 (i : Fin2 n) (j : (P F).B a i), supp x i (f i j)\na' : Fin2 n\nf' : (P F).A\nh'' : (P F).B f' \u27f9 \u03b1\nhu : abs \u27e8f', h''\u27e9 = x\nu : \u03b1 a'\nj : (P F).B a a'\n_h\u2082 : j \u2208 univ\nhfi : f a' j = u\n\u22a2 f a' j \u2208 supp x a'", "state_after": "no goals"}, {"tactic": "rintro \u27e8a', f', xeq', h'\u27e9 i u usuppx", "annotated_tactic": ["rintro \u27e8a', f', xeq', h'\u27e9 i u usuppx", []], "state_before": "case mpr.intro.intro.intro.mp\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\n\u22a2 (\u2203 a f, x = abs \u27e8a, f\u27e9 \u2227 \u2200 (i : Fin2 n) (j : (P F).B a i), p i (f i j)) \u2192 \u2200 (i : Fin2 n), \u2200 u \u2208 supp x i, p i u", "state_after": "case mpr.intro.intro.intro.mp.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\na' : (P F).A\nf' : (P F).B a' \u27f9 \u03b1\nxeq' : x = abs \u27e8a', f'\u27e9\nh' : \u2200 (i : Fin2 n) (j : (P F).B a' i), p i (f' i j)\ni : Fin2 n\nu : \u03b1 i\nusuppx : u \u2208 supp x i\n\u22a2 p i u"}, {"tactic": "rcases (mem_supp x _ u).mp (@usuppx) a' f' xeq'.symm with \u27e8i, _, f'ieq\u27e9", "annotated_tactic": ["rcases (<a>mem_supp</a> x _ u).<a>mp</a> (@usuppx) a' f' xeq'.symm with \u27e8i, _, f'ieq\u27e9", [{"full_name": "MvQPF.mem_supp", "def_path": "Mathlib/Data/QPF/Multivariate/Basic.lean", "def_pos": [160, 9], "def_end_pos": [160, 17]}, {"full_name": "Iff.mp", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [118, 3], "def_end_pos": [118, 5]}]], "state_before": "case mpr.intro.intro.intro.mp.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\na' : (P F).A\nf' : (P F).B a' \u27f9 \u03b1\nxeq' : x = abs \u27e8a', f'\u27e9\nh' : \u2200 (i : Fin2 n) (j : (P F).B a' i), p i (f' i j)\ni : Fin2 n\nu : \u03b1 i\nusuppx : u \u2208 supp x i\n\u22a2 p i u", "state_after": "case mpr.intro.intro.intro.mp.intro.intro.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\na' : (P F).A\nf' : (P F).B a' \u27f9 \u03b1\nxeq' : x = abs \u27e8a', f'\u27e9\nh' : \u2200 (i : Fin2 n) (j : (P F).B a' i), p i (f' i j)\ni\u271d : Fin2 n\nu : \u03b1 i\u271d\nusuppx : u \u2208 supp x i\u271d\ni : (P F).B a' i\u271d\nleft\u271d : i \u2208 univ\nf'ieq : f' i\u271d i = u\n\u22a2 p i\u271d u"}, {"tactic": "rw [\u2190 f'ieq]", "annotated_tactic": ["rw [\u2190 f'ieq]", []], "state_before": "case mpr.intro.intro.intro.mp.intro.intro.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\na' : (P F).A\nf' : (P F).B a' \u27f9 \u03b1\nxeq' : x = abs \u27e8a', f'\u27e9\nh' : \u2200 (i : Fin2 n) (j : (P F).B a' i), p i (f' i j)\ni\u271d : Fin2 n\nu : \u03b1 i\u271d\nusuppx : u \u2208 supp x i\u271d\ni : (P F).B a' i\u271d\nleft\u271d : i \u2208 univ\nf'ieq : f' i\u271d i = u\n\u22a2 p i\u271d u", "state_after": "case mpr.intro.intro.intro.mp.intro.intro.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\na' : (P F).A\nf' : (P F).B a' \u27f9 \u03b1\nxeq' : x = abs \u27e8a', f'\u27e9\nh' : \u2200 (i : Fin2 n) (j : (P F).B a' i), p i (f' i j)\ni\u271d : Fin2 n\nu : \u03b1 i\u271d\nusuppx : u \u2208 supp x i\u271d\ni : (P F).B a' i\u271d\nleft\u271d : i \u2208 univ\nf'ieq : f' i\u271d i = u\n\u22a2 p i\u271d (f' i\u271d i)"}, {"tactic": "apply h'", "annotated_tactic": ["apply h'", []], "state_before": "case mpr.intro.intro.intro.mp.intro.intro.intro.intro.intro\nn : \u2115\nF : TypeVec.{u} n \u2192 Type u_1\nq : MvQPF F\n\u03b1 : TypeVec.{u} n\nx : F \u03b1\na : (P F).A\nf : (P F).B a \u27f9 \u03b1\nxeq : abs \u27e8a, f\u27e9 = x\nh : \u2200 (i : Fin2 n) (a' : (P F).A) (f' : (P F).B a' \u27f9 \u03b1), abs \u27e8a', f'\u27e9 = x \u2192 f i '' univ \u2286 f' i '' univ\np : (i : Fin2 n) \u2192 \u03b1 i \u2192 Prop\na' : (P F).A\nf' : (P F).B a' \u27f9 \u03b1\nxeq' : x = abs \u27e8a', f'\u27e9\nh' : \u2200 (i : Fin2 n) (j : (P F).B a' i), p i (f' i j)\ni\u271d : Fin2 n\nu : \u03b1 i\u271d\nusuppx : u \u2208 supp x i\u271d\ni : (P F).B a' i\u271d\nleft\u271d : i \u2208 univ\nf'ieq : f' i\u271d i = u\n\u22a2 p i\u271d (f' i\u271d i)", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Data/Nat/Bits.lean", "full_name": "Nat.bit1_eq_bit1", "start": [365, 1], "end": [366, 43], "traced_tactics": [{"tactic": "subst h", "annotated_tactic": ["subst h", []], "state_before": "m\u271d n\u271d m n : \u2115\nh : m = n\n\u22a2 bit1 m = bit1 n", "state_after": "m\u271d n m : \u2115\n\u22a2 bit1 m = bit1 m"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "m\u271d n m : \u2115\n\u22a2 bit1 m = bit1 m", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/GroupTheory/HNNExtension.lean", "full_name": "HNNExtension.NormalWord.unitsSMul_cancels_iff", "start": [395, 1], "end": [410, 28], "traced_tactics": [{"tactic": "by_cases h : Cancels u w", "annotated_tactic": ["by_cases h : <a>Cancels</a> u w", [{"full_name": "HNNExtension.NormalWord.Cancels", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [352, 5], "def_end_pos": [352, 12]}]], "state_before": "G : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : \u21a5A \u2243* \u21a5B\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\nu : \u2124\u02e3\nw : NormalWord d\n\u22a2 Cancels (-u) (unitsSMul \u03c6 u w) \u2194 \u00acCancels u w", "state_after": "case pos\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : \u21a5A \u2243* \u21a5B\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\nu : \u2124\u02e3\nw : NormalWord d\nh : Cancels u w\n\u22a2 Cancels (-u) (unitsSMul \u03c6 u w) \u2194 \u00acCancels u w\n\ncase neg\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : \u21a5A \u2243* \u21a5B\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\nu : \u2124\u02e3\nw : NormalWord d\nh : \u00acCancels u w\n\u22a2 Cancels (-u) (unitsSMul \u03c6 u w) \u2194 \u00acCancels u w"}, {"tactic": "simp only [unitsSMul, h, dite_true, not_true_eq_false, iff_false]", "annotated_tactic": ["simp only [<a>unitsSMul</a>, h, <a>dite_true</a>, <a>not_true_eq_false</a>, <a>iff_false</a>]", [{"full_name": "HNNExtension.NormalWord.unitsSMul", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [363, 19], "def_end_pos": [363, 28]}, {"full_name": "dite_true", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [93, 17], "def_end_pos": [93, 26]}, {"full_name": "not_true_eq_false", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [135, 17], "def_end_pos": [135, 34]}, {"full_name": "iff_false", "def_path": ".lake/packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [129, 17], "def_end_pos": [129, 26]}]], "state_before": "case pos\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : \u21a5A \u2243* \u21a5B\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\nu : \u2124\u02e3\nw : NormalWord d\nh : Cancels u w\n\u22a2 Cancels (-u) (unitsSMul \u03c6 u w) \u2194 \u00acCancels u w", "state_after": "case pos\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : \u21a5A \u2243* \u21a5B\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\nu : \u2124\u02e3\nw : NormalWord d\nh : Cancels u w\n\u22a2 \u00acCancels (-u) (unitsSMulWithCancel \u03c6 u w \u22ef)"}, {"tactic": "induction w using consRecOn with\n| ofGroup => simp [Cancels, unitsSMulWithCancel]\n| cons g u' w h1 h2 _ =>\n  intro hc\n  apply not_cancels_of_cons_hyp _ _ h2\n  simp only [Cancels, cons_head, cons_toList, List.head?_cons,\n    Option.map_some', Option.some.injEq] at h\n  cases h.2\n  simpa [Cancels, unitsSMulWithCancel,\n    Subgroup.mul_mem_cancel_left] using hc", "annotated_tactic": ["induction w using <a>consRecOn</a> with\n    | <a>ofGroup</a> => simp [<a>Cancels</a>, <a>unitsSMulWithCancel</a>]\n    | <a>cons</a> g u' w h1 h2 _ =>\n      intro hc\n      apply <a>not_cancels_of_cons_hyp</a> _ _ h2\n      simp only [<a>Cancels</a>, <a>cons_head</a>, <a>cons_toList</a>, <a>List.head?_cons</a>,\n        <a>Option.map_some'</a>, Option.some.injEq] at h\n      cases h.2\n      simpa [<a>Cancels</a>, <a>unitsSMulWithCancel</a>,\n        <a>Subgroup.mul_mem_cancel_left</a>] using hc", [{"full_name": "HNNExtension.NormalWord.consRecOn", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [286, 5], "def_end_pos": [286, 14]}, {"full_name": "HNNExtension.NormalWord.ofGroup", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [240, 5], "def_end_pos": [240, 12]}, {"full_name": "HNNExtension.NormalWord.Cancels", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [352, 5], "def_end_pos": [352, 12]}, {"full_name": "HNNExtension.NormalWord.unitsSMulWithCancel", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [356, 5], "def_end_pos": [356, 24]}, {"full_name": "HNNExtension.NormalWord.cons", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [268, 5], "def_end_pos": [268, 9]}, {"full_name": "HNNExtension.NormalWord.not_cancels_of_cons_hyp", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [385, 9], "def_end_pos": [385, 32]}, {"full_name": "HNNExtension.NormalWord.Cancels", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [352, 5], "def_end_pos": [352, 12]}, {"full_name": "HNNExtension.NormalWord.cons_head", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [267, 3], "def_end_pos": [267, 8]}, {"full_name": "HNNExtension.NormalWord.cons_toList", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [267, 3], "def_end_pos": [267, 8]}, {"full_name": "List.head?_cons", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [301, 17], "def_end_pos": [301, 27]}, {"full_name": "Option.map_some'", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Option/Basic.lean", "def_pos": [115, 17], "def_end_pos": [115, 26]}, {"full_name": "HNNExtension.NormalWord.Cancels", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [352, 5], "def_end_pos": [352, 12]}, {"full_name": "HNNExtension.NormalWord.unitsSMulWithCancel", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [356, 5], "def_end_pos": [356, 24]}, {"full_name": "Subgroup.mul_mem_cancel_left", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [612, 19], "def_end_pos": [612, 38]}]], "state_before": "case pos\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : \u21a5A \u2243* \u21a5B\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\nu : \u2124\u02e3\nw : NormalWord d\nh : Cancels u w\n\u22a2 \u00acCancels (-u) (unitsSMulWithCancel \u03c6 u w \u22ef)", "state_after": "no goals"}, {"tactic": "simp [Cancels, unitsSMulWithCancel]", "annotated_tactic": ["simp [<a>Cancels</a>, <a>unitsSMulWithCancel</a>]", [{"full_name": "HNNExtension.NormalWord.Cancels", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [352, 5], "def_end_pos": [352, 12]}, {"full_name": "HNNExtension.NormalWord.unitsSMulWithCancel", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [356, 5], "def_end_pos": [356, 24]}]], "state_before": "case pos.ofGroup\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : \u21a5A \u2243* \u21a5B\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\nu : \u2124\u02e3\ng\u271d : G\nh : Cancels u (ofGroup g\u271d)\n\u22a2 \u00acCancels (-u) (unitsSMulWithCancel \u03c6 u (ofGroup g\u271d) \u22ef)", "state_after": "no goals"}, {"tactic": "intro hc", "annotated_tactic": ["intro hc", []], "state_before": "case pos.cons\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : \u21a5A \u2243* \u21a5B\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\nu : \u2124\u02e3\ng : G\nu' : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 d.set u'\nh2 : \u2200 u'_1 \u2208 Option.map Prod.fst w.toList.head?, w.head \u2208 toSubgroup A B u' \u2192 u' = u'_1\na\u271d : \u2200 (h : Cancels u w), \u00acCancels (-u) (unitsSMulWithCancel \u03c6 u w \u22ef)\nh : Cancels u (cons g u' w h1 h2)\n\u22a2 \u00acCancels (-u) (unitsSMulWithCancel \u03c6 u (cons g u' w h1 h2) \u22ef)", "state_after": "case pos.cons\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : \u21a5A \u2243* \u21a5B\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\nu : \u2124\u02e3\ng : G\nu' : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 d.set u'\nh2 : \u2200 u'_1 \u2208 Option.map Prod.fst w.toList.head?, w.head \u2208 toSubgroup A B u' \u2192 u' = u'_1\na\u271d : \u2200 (h : Cancels u w), \u00acCancels (-u) (unitsSMulWithCancel \u03c6 u w \u22ef)\nh : Cancels u (cons g u' w h1 h2)\nhc : Cancels (-u) (unitsSMulWithCancel \u03c6 u (cons g u' w h1 h2) \u22ef)\n\u22a2 False"}, {"tactic": "apply not_cancels_of_cons_hyp _ _ h2", "annotated_tactic": ["apply <a>not_cancels_of_cons_hyp</a> _ _ h2", [{"full_name": "HNNExtension.NormalWord.not_cancels_of_cons_hyp", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [385, 9], "def_end_pos": [385, 32]}]], "state_before": "case pos.cons\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : \u21a5A \u2243* \u21a5B\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\nu : \u2124\u02e3\ng : G\nu' : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 d.set u'\nh2 : \u2200 u'_1 \u2208 Option.map Prod.fst w.toList.head?, w.head \u2208 toSubgroup A B u' \u2192 u' = u'_1\na\u271d : \u2200 (h : Cancels u w), \u00acCancels (-u) (unitsSMulWithCancel \u03c6 u w \u22ef)\nh : Cancels u (cons g u' w h1 h2)\nhc : Cancels (-u) (unitsSMulWithCancel \u03c6 u (cons g u' w h1 h2) \u22ef)\n\u22a2 False", "state_after": "case pos.cons\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : \u21a5A \u2243* \u21a5B\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\nu : \u2124\u02e3\ng : G\nu' : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 d.set u'\nh2 : \u2200 u'_1 \u2208 Option.map Prod.fst w.toList.head?, w.head \u2208 toSubgroup A B u' \u2192 u' = u'_1\na\u271d : \u2200 (h : Cancels u w), \u00acCancels (-u) (unitsSMulWithCancel \u03c6 u w \u22ef)\nh : Cancels u (cons g u' w h1 h2)\nhc : Cancels (-u) (unitsSMulWithCancel \u03c6 u (cons g u' w h1 h2) \u22ef)\n\u22a2 Cancels u' w"}, {"tactic": "simp only [Cancels, cons_head, cons_toList, List.head?_cons,\n  Option.map_some', Option.some.injEq] at h", "annotated_tactic": ["simp only [<a>Cancels</a>, <a>cons_head</a>, <a>cons_toList</a>, <a>List.head?_cons</a>,\n        <a>Option.map_some'</a>, Option.some.injEq] at h", [{"full_name": "HNNExtension.NormalWord.Cancels", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [352, 5], "def_end_pos": [352, 12]}, {"full_name": "HNNExtension.NormalWord.cons_head", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [267, 3], "def_end_pos": [267, 8]}, {"full_name": "HNNExtension.NormalWord.cons_toList", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [267, 3], "def_end_pos": [267, 8]}, {"full_name": "List.head?_cons", "def_path": ".lake/packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [301, 17], "def_end_pos": [301, 27]}, {"full_name": "Option.map_some'", "def_path": ".lake/packages/lean4/src/lean/Init/Data/Option/Basic.lean", "def_pos": [115, 17], "def_end_pos": [115, 26]}]], "state_before": "case pos.cons\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : \u21a5A \u2243* \u21a5B\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\nu : \u2124\u02e3\ng : G\nu' : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 d.set u'\nh2 : \u2200 u'_1 \u2208 Option.map Prod.fst w.toList.head?, w.head \u2208 toSubgroup A B u' \u2192 u' = u'_1\na\u271d : \u2200 (h : Cancels u w), \u00acCancels (-u) (unitsSMulWithCancel \u03c6 u w \u22ef)\nh : Cancels u (cons g u' w h1 h2)\nhc : Cancels (-u) (unitsSMulWithCancel \u03c6 u (cons g u' w h1 h2) \u22ef)\n\u22a2 Cancels u' w", "state_after": "case pos.cons\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : \u21a5A \u2243* \u21a5B\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\nu : \u2124\u02e3\ng : G\nu' : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 d.set u'\nh2 : \u2200 u'_1 \u2208 Option.map Prod.fst w.toList.head?, w.head \u2208 toSubgroup A B u' \u2192 u' = u'_1\na\u271d : \u2200 (h : Cancels u w), \u00acCancels (-u) (unitsSMulWithCancel \u03c6 u w \u22ef)\nh\u271d : Cancels u (cons g u' w h1 h2)\nhc : Cancels (-u) (unitsSMulWithCancel \u03c6 u (cons g u' w h1 h2) \u22ef)\nh : g \u2208 toSubgroup A B u \u2227 u' = -u\n\u22a2 Cancels u' w"}, {"tactic": "cases h.2", "annotated_tactic": ["cases h.2", []], "state_before": "case pos.cons\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : \u21a5A \u2243* \u21a5B\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\nu : \u2124\u02e3\ng : G\nu' : \u2124\u02e3\nw : NormalWord d\nh1 : w.head \u2208 d.set u'\nh2 : \u2200 u'_1 \u2208 Option.map Prod.fst w.toList.head?, w.head \u2208 toSubgroup A B u' \u2192 u' = u'_1\na\u271d : \u2200 (h : Cancels u w), \u00acCancels (-u) (unitsSMulWithCancel \u03c6 u w \u22ef)\nh\u271d : Cancels u (cons g u' w h1 h2)\nhc : Cancels (-u) (unitsSMulWithCancel \u03c6 u (cons g u' w h1 h2) \u22ef)\nh : g \u2208 toSubgroup A B u \u2227 u' = -u\n\u22a2 Cancels u' w", "state_after": "case pos.cons.refl\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : \u21a5A \u2243* \u21a5B\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\nu : \u2124\u02e3\ng : G\nw : NormalWord d\na\u271d : \u2200 (h : Cancels u w), \u00acCancels (-u) (unitsSMulWithCancel \u03c6 u w \u22ef)\nh1 : w.head \u2208 d.set (-u)\nh2 : \u2200 u' \u2208 Option.map Prod.fst w.toList.head?, w.head \u2208 toSubgroup A B (-u) \u2192 -u = u'\nh\u271d : Cancels u (cons g (-u) w h1 h2)\nhc : Cancels (-u) (unitsSMulWithCancel \u03c6 u (cons g (-u) w h1 h2) \u22ef)\nh : g \u2208 toSubgroup A B u \u2227 -u = -u\n\u22a2 Cancels (-u) w"}, {"tactic": "simpa [Cancels, unitsSMulWithCancel,\n  Subgroup.mul_mem_cancel_left] using hc", "annotated_tactic": ["simpa [<a>Cancels</a>, <a>unitsSMulWithCancel</a>,\n        <a>Subgroup.mul_mem_cancel_left</a>] using hc", [{"full_name": "HNNExtension.NormalWord.Cancels", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [352, 5], "def_end_pos": [352, 12]}, {"full_name": "HNNExtension.NormalWord.unitsSMulWithCancel", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [356, 5], "def_end_pos": [356, 24]}, {"full_name": "Subgroup.mul_mem_cancel_left", "def_path": "Mathlib/Algebra/Group/Subgroup/Basic.lean", "def_pos": [612, 19], "def_end_pos": [612, 38]}]], "state_before": "case pos.cons.refl\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : \u21a5A \u2243* \u21a5B\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\nu : \u2124\u02e3\ng : G\nw : NormalWord d\na\u271d : \u2200 (h : Cancels u w), \u00acCancels (-u) (unitsSMulWithCancel \u03c6 u w \u22ef)\nh1 : w.head \u2208 d.set (-u)\nh2 : \u2200 u' \u2208 Option.map Prod.fst w.toList.head?, w.head \u2208 toSubgroup A B (-u) \u2192 -u = u'\nh\u271d : Cancels u (cons g (-u) w h1 h2)\nhc : Cancels (-u) (unitsSMulWithCancel \u03c6 u (cons g (-u) w h1 h2) \u22ef)\nh : g \u2208 toSubgroup A B u \u2227 -u = -u\n\u22a2 Cancels (-u) w", "state_after": "no goals"}, {"tactic": "simp only [unitsSMul, dif_neg h]", "annotated_tactic": ["simp only [<a>unitsSMul</a>, <a>dif_neg</a> h]", [{"full_name": "HNNExtension.NormalWord.unitsSMul", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [363, 19], "def_end_pos": [363, 28]}, {"full_name": "dif_neg", "def_path": ".lake/packages/lean4/src/lean/Init/Core.lean", "def_pos": [954, 9], "def_end_pos": [954, 16]}]], "state_before": "case neg\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : \u21a5A \u2243* \u21a5B\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\nu : \u2124\u02e3\nw : NormalWord d\nh : \u00acCancels u w\n\u22a2 Cancels (-u) (unitsSMul \u03c6 u w) \u2194 \u00acCancels u w", "state_after": "case neg\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : \u21a5A \u2243* \u21a5B\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\nu : \u2124\u02e3\nw : NormalWord d\nh : \u00acCancels u w\n\u22a2 Cancels (-u) (cons (\u2191(unitsSMulGroup \u03c6 d u w.head).1) u ((\u2191(unitsSMulGroup \u03c6 d u w.head).2 * w.head\u207b\u00b9) \u2022 w) \u22ef \u22ef) \u2194\n    \u00acCancels u w"}, {"tactic": "simpa [Cancels] using h", "annotated_tactic": ["simpa [<a>Cancels</a>] using h", [{"full_name": "HNNExtension.NormalWord.Cancels", "def_path": "Mathlib/GroupTheory/HNNExtension.lean", "def_pos": [352, 5], "def_end_pos": [352, 12]}]], "state_before": "case neg\nG : Type u_1\ninst\u271d\u00b3 : Group G\nA B : Subgroup G\n\u03c6 : \u21a5A \u2243* \u21a5B\nH : Type u_2\ninst\u271d\u00b2 : Group H\nM : Type u_3\ninst\u271d\u00b9 : Monoid M\nd : TransversalPair G A B\ninst\u271d : DecidableEq G\nu : \u2124\u02e3\nw : NormalWord d\nh : \u00acCancels u w\n\u22a2 Cancels (-u) (cons (\u2191(unitsSMulGroup \u03c6 d u w.head).1) u ((\u2191(unitsSMulGroup \u03c6 d u w.head).2 * w.head\u207b\u00b9) \u2022 w) \u22ef \u22ef) \u2194\n    \u00acCancels u w", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Order/Hom/Ring.lean", "full_name": "OrderRingIso.coe_toRingEquiv", "start": [394, 1], "end": [395, 6], "traced_tactics": []}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Algebra/Polynomial/Monic.lean", "full_name": "Polynomial.monic_multiset_prod_of_monic", "start": [278, 1], "end": [284, 100], "traced_tactics": [{"tactic": "revert ht", "annotated_tactic": ["revert ht", []], "state_before": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : CommSemiring R\np : R[X]\nt : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\nht : \u2200 i \u2208 t, (f i).Monic\n\u22a2 (Multiset.map f t).prod.Monic", "state_after": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : CommSemiring R\np : R[X]\nt : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\n\u22a2 (\u2200 i \u2208 t, (f i).Monic) \u2192 (Multiset.map f t).prod.Monic"}, {"tactic": "refine t.induction_on ?_ ?_", "annotated_tactic": ["refine t.induction_on ?_ ?_", []], "state_before": "R : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : CommSemiring R\np : R[X]\nt : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\n\u22a2 (\u2200 i \u2208 t, (f i).Monic) \u2192 (Multiset.map f t).prod.Monic", "state_after": "case refine_1\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : CommSemiring R\np : R[X]\nt : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\n\u22a2 (\u2200 i \u2208 0, (f i).Monic) \u2192 (Multiset.map f 0).prod.Monic\n\ncase refine_2\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : CommSemiring R\np : R[X]\nt : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\n\u22a2 \u2200 (a : \u03b9) (s : Multiset \u03b9),\n    ((\u2200 i \u2208 s, (f i).Monic) \u2192 (Multiset.map f s).prod.Monic) \u2192\n      (\u2200 i \u2208 a ::\u2098 s, (f i).Monic) \u2192 (Multiset.map f (a ::\u2098 s)).prod.Monic"}, {"tactic": "intro a t ih ht", "annotated_tactic": ["intro a t ih ht", []], "state_before": "case refine_2\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : CommSemiring R\np : R[X]\nt : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\n\u22a2 \u2200 (a : \u03b9) (s : Multiset \u03b9),\n    ((\u2200 i \u2208 s, (f i).Monic) \u2192 (Multiset.map f s).prod.Monic) \u2192\n      (\u2200 i \u2208 a ::\u2098 s, (f i).Monic) \u2192 (Multiset.map f (a ::\u2098 s)).prod.Monic", "state_after": "case refine_2\nR : Type u\nS : Type v\na\u271d b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : CommSemiring R\np : R[X]\nt\u271d : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\na : \u03b9\nt : Multiset \u03b9\nih : (\u2200 i \u2208 t, (f i).Monic) \u2192 (Multiset.map f t).prod.Monic\nht : \u2200 i \u2208 a ::\u2098 t, (f i).Monic\n\u22a2 (Multiset.map f (a ::\u2098 t)).prod.Monic"}, {"tactic": "rw [Multiset.map_cons, Multiset.prod_cons]", "annotated_tactic": ["rw [<a>Multiset.map_cons</a>, <a>Multiset.prod_cons</a>]", [{"full_name": "Multiset.map_cons", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [1225, 9], "def_end_pos": [1225, 17]}, {"full_name": "Multiset.prod_cons", "def_path": "Mathlib/Algebra/BigOperators/Group/Multiset.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}]], "state_before": "case refine_2\nR : Type u\nS : Type v\na\u271d b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : CommSemiring R\np : R[X]\nt\u271d : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\na : \u03b9\nt : Multiset \u03b9\nih : (\u2200 i \u2208 t, (f i).Monic) \u2192 (Multiset.map f t).prod.Monic\nht : \u2200 i \u2208 a ::\u2098 t, (f i).Monic\n\u22a2 (Multiset.map f (a ::\u2098 t)).prod.Monic", "state_after": "case refine_2\nR : Type u\nS : Type v\na\u271d b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : CommSemiring R\np : R[X]\nt\u271d : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\na : \u03b9\nt : Multiset \u03b9\nih : (\u2200 i \u2208 t, (f i).Monic) \u2192 (Multiset.map f t).prod.Monic\nht : \u2200 i \u2208 a ::\u2098 t, (f i).Monic\n\u22a2 (f a * (Multiset.map f t).prod).Monic"}, {"tactic": "exact (ht _ (Multiset.mem_cons_self _ _)).mul (ih fun _ hi => ht _ (Multiset.mem_cons_of_mem hi))", "annotated_tactic": ["exact (ht _ (<a>Multiset.mem_cons_self</a> _ _)).<a>mul</a> (ih fun _ hi => ht _ (<a>Multiset.mem_cons_of_mem</a> hi))", [{"full_name": "Multiset.mem_cons_self", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [258, 9], "def_end_pos": [258, 22]}, {"full_name": "Polynomial.Monic.mul", "def_path": "Mathlib/Algebra/Polynomial/Monic.lean", "def_pos": [117, 9], "def_end_pos": [117, 18]}, {"full_name": "Multiset.mem_cons_of_mem", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [253, 9], "def_end_pos": [253, 24]}]], "state_before": "case refine_2\nR : Type u\nS : Type v\na\u271d b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : CommSemiring R\np : R[X]\nt\u271d : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\na : \u03b9\nt : Multiset \u03b9\nih : (\u2200 i \u2208 t, (f i).Monic) \u2192 (Multiset.map f t).prod.Monic\nht : \u2200 i \u2208 a ::\u2098 t, (f i).Monic\n\u22a2 (f a * (Multiset.map f t).prod).Monic", "state_after": "no goals"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case refine_1\nR : Type u\nS : Type v\na b : R\nm n : \u2115\n\u03b9 : Type y\ninst\u271d : CommSemiring R\np : R[X]\nt : Multiset \u03b9\nf : \u03b9 \u2192 R[X]\n\u22a2 (\u2200 i \u2208 0, (f i).Monic) \u2192 (Multiset.map f 0).prod.Monic", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "full_name": "intervalIntegral.integral_congr", "start": [909, 1], "end": [913, 71], "traced_tactics": [{"tactic": "rcases le_total a b with hab | hab <;>\n  simpa [hab, integral_of_le, integral_of_ge] using\n    setIntegral_congr measurableSet_Ioc (h.mono Ioc_subset_Icc_self)", "annotated_tactic": ["rcases <a>le_total</a> a b with hab | hab <;>\n    simpa [hab, <a>integral_of_le</a>, <a>integral_of_ge</a>] using\n      <a>setIntegral_congr</a> <a>measurableSet_Ioc</a> (h.mono <a>Ioc_subset_Icc_self</a>)", [{"full_name": "le_total", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [297, 9], "def_end_pos": [297, 17]}, {"full_name": "intervalIntegral.integral_of_le", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [487, 9], "def_end_pos": [487, 23]}, {"full_name": "intervalIntegral.integral_of_ge", "def_path": "Mathlib/MeasureTheory/Integral/IntervalIntegral.lean", "def_pos": [500, 9], "def_end_pos": [500, 23]}, {"full_name": "MeasureTheory.setIntegral_congr", "def_path": "Mathlib/MeasureTheory/Integral/SetIntegral.lean", "def_pos": [95, 9], "def_end_pos": [95, 26]}, {"full_name": "measurableSet_Ioc", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean", "def_pos": [197, 9], "def_end_pos": [197, 26]}, {"full_name": "Set.Ioc_subset_Icc_self", "def_path": "Mathlib/Order/Interval/Set/Basic.lean", "def_pos": [529, 9], "def_end_pos": [529, 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\u271d b\u271d c d : \u211d\nf g : \u211d \u2192 E\n\u03bc : Measure \u211d\na b : \u211d\nh : EqOn f g [[a, b]]\n\u22a2 \u222b (x : \u211d) in a..b, f x \u2202\u03bc = \u222b (x : \u211d) in a..b, g x \u2202\u03bc", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/Topology/Instances/ENNReal.lean", "full_name": "Real.diam_Ioc", "start": [1581, 1], "end": [1582, 61], "traced_tactics": [{"tactic": "simp [Metric.diam, ENNReal.toReal_ofReal (sub_nonneg.2 h)]", "annotated_tactic": ["simp [<a>Metric.diam</a>, <a>ENNReal.toReal_ofReal</a> (<a>sub_nonneg</a>.2 h)]", [{"full_name": "Metric.diam", "def_path": "Mathlib/Topology/MetricSpace/Bounded.lean", "def_pos": [388, 19], "def_end_pos": [388, 23]}, {"full_name": "ENNReal.toReal_ofReal", "def_path": "Mathlib/Data/ENNReal/Basic.lean", "def_pos": [217, 9], "def_end_pos": [217, 22]}, {"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : PseudoEMetricSpace \u03b1\na b : \u211d\nh : a \u2264 b\n\u22a2 Metric.diam (Ioc a b) = b - a", "state_after": "no goals"}]}, {"url": "https://github.com/leanprover-community/mathlib4", "commit": "29dcec074de168ac2bf835a77ef68bbe069194c5", "file_path": "Mathlib/FieldTheory/SeparableDegree.lean", "full_name": "Polynomial.natSepDegree_eq_of_isAlgClosed", "start": [342, 1], "end": [344, 62], "traced_tactics": []}]