url
stringclasses 3
values | commit
stringclasses 3
values | file_path
stringlengths 20
79
| full_name
stringlengths 3
115
| start
sequence | end
sequence | traced_tactics
stringlengths 2
997k
|
|---|---|---|---|---|---|---|
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Set/Pointwise/BigOperators.lean | Set.list_prod_subset_list_prod | [
108,
1
] | [
114,
56
] | [{"tactic": "induction' t with h tl ih", "annotated_tactic": ["induction' t with h tl ih", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nt : List \u03b9\nf\u2081 f\u2082 : \u03b9 \u2192 Set \u03b1\nhf : \u2200 (i : \u03b9), i \u2208 t \u2192 f\u2081 i \u2286 f\u2082 i\n\u22a2 List.prod (List.map f\u2081 t) \u2286 List.prod (List.map f\u2082 t)", "state_after": "case nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nt : List \u03b9\nf\u2081 f\u2082 : \u03b9 \u2192 Set \u03b1\nhf\u271d : \u2200 (i : \u03b9), i \u2208 t \u2192 f\u2081 i \u2286 f\u2082 i\nhf : \u2200 (i : \u03b9), i \u2208 [] \u2192 f\u2081 i \u2286 f\u2082 i\n\u22a2 List.prod (List.map f\u2081 []) \u2286 List.prod (List.map f\u2082 [])\n\ncase cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nt : List \u03b9\nf\u2081 f\u2082 : \u03b9 \u2192 Set \u03b1\nhf\u271d : \u2200 (i : \u03b9), i \u2208 t \u2192 f\u2081 i \u2286 f\u2082 i\nh : \u03b9\ntl : List \u03b9\nih : (\u2200 (i : \u03b9), i \u2208 tl \u2192 f\u2081 i \u2286 f\u2082 i) \u2192 List.prod (List.map f\u2081 tl) \u2286 List.prod (List.map f\u2082 tl)\nhf : \u2200 (i : \u03b9), i \u2208 h :: tl \u2192 f\u2081 i \u2286 f\u2082 i\n\u22a2 List.prod (List.map f\u2081 (h :: tl)) \u2286 List.prod (List.map f\u2082 (h :: tl))"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nt : List \u03b9\nf\u2081 f\u2082 : \u03b9 \u2192 Set \u03b1\nhf\u271d : \u2200 (i : \u03b9), i \u2208 t \u2192 f\u2081 i \u2286 f\u2082 i\nhf : \u2200 (i : \u03b9), i \u2208 [] \u2192 f\u2081 i \u2286 f\u2082 i\n\u22a2 List.prod (List.map f\u2081 []) \u2286 List.prod (List.map f\u2082 [])", "state_after": "no goals"}, {"tactic": "simp_rw [List.map_cons, List.prod_cons]", "annotated_tactic": ["simp_rw [<a>List.map_cons</a>, <a>List.prod_cons</a>]", [{"full_name": "List.map_cons", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [89, 17], "def_end_pos": [89, 25]}, {"full_name": "List.prod_cons", "def_path": "Mathlib/Data/List/BigOperators/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 18]}]], "state_before": "case cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nt : List \u03b9\nf\u2081 f\u2082 : \u03b9 \u2192 Set \u03b1\nhf\u271d : \u2200 (i : \u03b9), i \u2208 t \u2192 f\u2081 i \u2286 f\u2082 i\nh : \u03b9\ntl : List \u03b9\nih : (\u2200 (i : \u03b9), i \u2208 tl \u2192 f\u2081 i \u2286 f\u2082 i) \u2192 List.prod (List.map f\u2081 tl) \u2286 List.prod (List.map f\u2082 tl)\nhf : \u2200 (i : \u03b9), i \u2208 h :: tl \u2192 f\u2081 i \u2286 f\u2082 i\n\u22a2 List.prod (List.map f\u2081 (h :: tl)) \u2286 List.prod (List.map f\u2082 (h :: tl))", "state_after": "case cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nt : List \u03b9\nf\u2081 f\u2082 : \u03b9 \u2192 Set \u03b1\nhf\u271d : \u2200 (i : \u03b9), i \u2208 t \u2192 f\u2081 i \u2286 f\u2082 i\nh : \u03b9\ntl : List \u03b9\nih : (\u2200 (i : \u03b9), i \u2208 tl \u2192 f\u2081 i \u2286 f\u2082 i) \u2192 List.prod (List.map f\u2081 tl) \u2286 List.prod (List.map f\u2082 tl)\nhf : \u2200 (i : \u03b9), i \u2208 h :: tl \u2192 f\u2081 i \u2286 f\u2082 i\n\u22a2 f\u2081 h * List.prod (List.map f\u2081 tl) \u2286 f\u2082 h * List.prod (List.map f\u2082 tl)"}, {"tactic": "exact mul_subset_mul (hf h <| List.mem_cons_self _ _)\n (ih fun i hi \u21a6 hf i <| List.mem_cons_of_mem _ hi)", "annotated_tactic": ["exact <a>mul_subset_mul</a> (hf h <| <a>List.mem_cons_self</a> _ _)\n (ih fun i hi \u21a6 hf i <| <a>List.mem_cons_of_mem</a> _ hi)", [{"full_name": "Set.mul_subset_mul", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [416, 9], "def_end_pos": [416, 23]}, {"full_name": "List.mem_cons_self", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [66, 9], "def_end_pos": [66, 22]}, {"full_name": "List.mem_cons_of_mem", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [68, 9], "def_end_pos": [68, 24]}]], "state_before": "case cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\nF : Type u_4\ninst\u271d\u00b2 : CommMonoid \u03b1\ninst\u271d\u00b9 : CommMonoid \u03b2\ninst\u271d : MonoidHomClass F \u03b1 \u03b2\nt : List \u03b9\nf\u2081 f\u2082 : \u03b9 \u2192 Set \u03b1\nhf\u271d : \u2200 (i : \u03b9), i \u2208 t \u2192 f\u2081 i \u2286 f\u2082 i\nh : \u03b9\ntl : List \u03b9\nih : (\u2200 (i : \u03b9), i \u2208 tl \u2192 f\u2081 i \u2286 f\u2082 i) \u2192 List.prod (List.map f\u2081 tl) \u2286 List.prod (List.map f\u2082 tl)\nhf : \u2200 (i : \u03b9), i \u2208 h :: tl \u2192 f\u2081 i \u2286 f\u2082 i\n\u22a2 f\u2081 h * List.prod (List.map f\u2081 tl) \u2286 f\u2082 h * List.prod (List.map f\u2082 tl)", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | HasDerivWithinAt.ccos | [
185,
1
] | [
187,
60
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Algebra/Algebra/Basic.lean | algebraMap.coe_pow | [
165,
1
] | [
166,
31
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Algebra/Opposites.lean | MulOpposite.unop_smul | [
323,
1
] | [
324,
6
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Logic/Embedding/Basic.lean | subtypeOrLeftEmbedding_apply_left | [
481,
1
] | [
484,
13
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | Finset.sum_centroidWeightsIndicator_eq_one_of_nonempty | [
920,
1
] | [
923,
53
] | [{"tactic": "rw [sum_centroidWeightsIndicator]", "annotated_tactic": ["rw [<a>sum_centroidWeightsIndicator</a>]", [{"full_name": "Finset.sum_centroidWeightsIndicator", "def_path": "Mathlib/LinearAlgebra/AffineSpace/Combination.lean", "def_pos": [904, 9], "def_end_pos": [904, 37]}]], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2075 : DivisionRing k\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module k V\ninst\u271d\u00b2 : AffineSpace V P\n\u03b9 : Type u_4\ns : Finset \u03b9\n\u03b9\u2082 : Type u_5\ns\u2082 : Finset \u03b9\u2082\ninst\u271d\u00b9 : CharZero k\ninst\u271d : Fintype \u03b9\nh : Finset.Nonempty s\n\u22a2 \u2211 i : \u03b9, centroidWeightsIndicator k s i = 1", "state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2075 : DivisionRing k\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module k V\ninst\u271d\u00b2 : AffineSpace V P\n\u03b9 : Type u_4\ns : Finset \u03b9\n\u03b9\u2082 : Type u_5\ns\u2082 : Finset \u03b9\u2082\ninst\u271d\u00b9 : CharZero k\ninst\u271d : Fintype \u03b9\nh : Finset.Nonempty s\n\u22a2 \u2211 i in s, centroidWeights k s i = 1"}, {"tactic": "exact s.sum_centroidWeights_eq_one_of_nonempty k h", "annotated_tactic": ["exact s.sum_centroidWeights_eq_one_of_nonempty k h", []], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u2075 : DivisionRing k\ninst\u271d\u2074 : AddCommGroup V\ninst\u271d\u00b3 : Module k V\ninst\u271d\u00b2 : AffineSpace V P\n\u03b9 : Type u_4\ns : Finset \u03b9\n\u03b9\u2082 : Type u_5\ns\u2082 : Finset \u03b9\u2082\ninst\u271d\u00b9 : CharZero k\ninst\u271d : Fintype \u03b9\nh : Finset.Nonempty s\n\u22a2 \u2211 i in s, centroidWeights k s i = 1", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Set/Prod.lean | Set.offDiag_eq_sep_prod | [
596,
1
] | [
597,
30
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Real/ENNReal.lean | ENNReal.range_coe' | [
160,
1
] | [
160,
61
] | [] |
https://github.com/leanprover/std4 | 869c615eb10130c0637a7bc038e2b80253559913 | lake-packages/std/Std/Data/Sum/Lemmas.lean | Sum.liftRel_subrelation_lex | [
200,
1
] | [
200,
85
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Order/Heyting/Basic.lean | PUnit.sup_eq | [
1356,
1
] | [
1357,
6
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/MeasureTheory/Function/Jacobian.lean | exists_partition_approximatesLinearOn_of_hasFDerivWithinAt | [
254,
1
] | [
269,
94
] | [{"tactic": "rcases exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt f s f' hf' r rpos with\n \u27e8t, A, t_closed, st, t_approx, ht\u27e9", "annotated_tactic": ["rcases <a>exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt</a> f s f' hf' r rpos with\n \u27e8t, A, t_closed, st, t_approx, ht\u27e9", [{"full_name": "exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt", "def_path": "Mathlib/MeasureTheory/Function/Jacobian.lean", "def_pos": [111, 9], "def_end_pos": [111, 70]}]], "state_before": "E : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\n\u22a2 \u2203 t A,\n Pairwise (Disjoint on t) \u2227\n (\u2200 (n : \u2115), MeasurableSet (t n)) \u2227\n s \u2286 \u22c3 n, t n \u2227\n (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227\n (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y)", "state_after": "case intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 n, t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2203 t A,\n Pairwise (Disjoint on t) \u2227\n (\u2200 (n : \u2115), MeasurableSet (t n)) \u2227\n s \u2286 \u22c3 n, t n \u2227\n (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227\n (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y)"}, {"tactic": "refine'\n \u27e8disjointed t, A, disjoint_disjointed _,\n MeasurableSet.disjointed fun n => (t_closed n).measurableSet, _, _, ht\u27e9", "annotated_tactic": ["refine'\n \u27e8<a>disjointed</a> t, A, <a>disjoint_disjointed</a> _,\n <a>MeasurableSet.disjointed</a> fun n => (t_closed n).<a>measurableSet</a>, _, _, ht\u27e9", [{"full_name": "disjointed", "def_path": "Mathlib/Order/Disjointed.lean", "def_pos": [49, 5], "def_end_pos": [49, 15]}, {"full_name": "disjoint_disjointed", "def_path": "Mathlib/Order/Disjointed.lean", "def_pos": [74, 9], "def_end_pos": [74, 28]}, {"full_name": "MeasurableSet.disjointed", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [236, 19], "def_end_pos": [236, 43]}, {"full_name": "IsClosed.measurableSet", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean", "def_pos": [340, 9], "def_end_pos": [340, 31]}]], "state_before": "case intro.intro.intro.intro.intro\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 n, t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2203 t A,\n Pairwise (Disjoint on t) \u2227\n (\u2200 (n : \u2115), MeasurableSet (t n)) \u2227\n s \u2286 \u22c3 n, t n \u2227\n (\u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))) \u2227\n (Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y)", "state_after": "case intro.intro.intro.intro.intro.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 n, t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 s \u2286 \u22c3 n, disjointed t n\n\ncase intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 n, t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 disjointed t n) (r (A n))"}, {"tactic": "rw [iUnion_disjointed]", "annotated_tactic": ["rw [<a>iUnion_disjointed</a>]", [{"full_name": "iUnion_disjointed", "def_path": "Mathlib/Order/Disjointed.lean", "def_pos": [165, 9], "def_end_pos": [165, 26]}]], "state_before": "case intro.intro.intro.intro.intro.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 n, t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 s \u2286 \u22c3 n, disjointed t n", "state_after": "case intro.intro.intro.intro.intro.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 n, t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 s \u2286 \u22c3 n, t n"}, {"tactic": "exact st", "annotated_tactic": ["exact st", []], "state_before": "case intro.intro.intro.intro.intro.refine'_1\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 n, t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 s \u2286 \u22c3 n, t n", "state_after": "no goals"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "case intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 n, t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\n\u22a2 \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 disjointed t n) (r (A n))", "state_after": "case intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 n, t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\nn : \u2115\n\u22a2 ApproximatesLinearOn f (A n) (s \u2229 disjointed t n) (r (A n))"}, {"tactic": "exact (t_approx n).mono_set (inter_subset_inter_right _ (disjointed_subset _ _))", "annotated_tactic": ["exact (t_approx n).<a>mono_set</a> (<a>inter_subset_inter_right</a> _ (<a>disjointed_subset</a> _ _))", [{"full_name": "ApproximatesLinearOn.mono_set", "def_path": "Mathlib/Analysis/Calculus/Inverse.lean", "def_pos": [142, 9], "def_end_pos": [142, 17]}, {"full_name": "Set.inter_subset_inter_right", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [1032, 9], "def_end_pos": [1032, 33]}, {"full_name": "disjointed_subset", "def_path": "Mathlib/Order/Disjointed.lean", "def_pos": [161, 9], "def_end_pos": [161, 26]}]], "state_before": "case intro.intro.intro.intro.intro.refine'_2\nE : Type u_1\nF : Type u_2\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedSpace \u211d E\ninst\u271d\u2076 : FiniteDimensional \u211d E\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \u211d F\ns\u271d : Set E\nf\u271d : E \u2192 E\nf'\u271d : E \u2192 E \u2192L[\u211d] E\ninst\u271d\u00b3 : MeasurableSpace E\ninst\u271d\u00b2 : BorelSpace E\n\u03bc : Measure E\ninst\u271d\u00b9 : IsAddHaarMeasure \u03bc\ninst\u271d : SecondCountableTopology F\nf : E \u2192 F\ns : Set E\nf' : E \u2192 E \u2192L[\u211d] F\nhf' : \u2200 (x : E), x \u2208 s \u2192 HasFDerivWithinAt f (f' x) s x\nr : (E \u2192L[\u211d] F) \u2192 \u211d\u22650\nrpos : \u2200 (A : E \u2192L[\u211d] F), r A \u2260 0\nt : \u2115 \u2192 Set E\nA : \u2115 \u2192 E \u2192L[\u211d] F\nt_closed : \u2200 (n : \u2115), IsClosed (t n)\nst : s \u2286 \u22c3 n, t n\nt_approx : \u2200 (n : \u2115), ApproximatesLinearOn f (A n) (s \u2229 t n) (r (A n))\nht : Set.Nonempty s \u2192 \u2200 (n : \u2115), \u2203 y, y \u2208 s \u2227 A n = f' y\nn : \u2115\n\u22a2 ApproximatesLinearOn f (A n) (s \u2229 disjointed t n) (r (A n))", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/ModelTheory/Encoding.lean | FirstOrder.Language.BoundedFormula.listEncode_sigma_injective | [
306,
1
] | [
308,
43
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Algebra/GroupPower/Order.lean | Left.one_le_pow_of_le | [
113,
1
] | [
117,
57
] | [{"tactic": "rw [pow_succ]", "annotated_tactic": ["rw [<a>pow_succ</a>]", [{"full_name": "pow_succ", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [645, 9], "def_end_pos": [645, 17]}]], "state_before": "\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : M\nhx : 1 \u2264 x\nn : \u2115\n\u22a2 1 \u2264 x ^ (n + 1)", "state_after": "\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : M\nhx : 1 \u2264 x\nn : \u2115\n\u22a2 1 \u2264 x * x ^ n"}, {"tactic": "exact Left.one_le_mul hx <| Left.one_le_pow_of_le hx", "annotated_tactic": ["exact <a>Left.one_le_mul</a> hx <| Left.one_le_pow_of_le hx", [{"full_name": "Left.one_le_mul", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [769, 9], "def_end_pos": [769, 24]}]], "state_before": "\u03b2 : Type u_1\nA : Type u_2\nG : Type u_3\nM : Type u_4\nR : Type u_5\ninst\u271d\u00b2 : Monoid M\ninst\u271d\u00b9 : Preorder M\ninst\u271d : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x \u2264 x_1\nx : M\nhx : 1 \u2264 x\nn : \u2115\n\u22a2 1 \u2264 x * x ^ n", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Topology/Sequences.lean | SeqCompact.lebesgue_number_lemma_of_metric | [
407,
8
] | [
410,
55
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Fin/Basic.lean | Fin.succAbove_pred | [
1424,
1
] | [
1427,
54
] | [{"tactic": "rw [succAbove_above, succ_pred]", "annotated_tactic": ["rw [<a>succAbove_above</a>, <a>succ_pred</a>]", [{"full_name": "Fin.succAbove_above", "def_path": "Mathlib/Data/Fin/Basic.lean", "def_pos": [1356, 9], "def_end_pos": [1356, 24]}, {"full_name": "Fin.succ_pred", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [485, 17], "def_end_pos": [485, 26]}]], "state_before": "n m : \u2115\nx y : Fin (n + 1)\nh : x < y\nhy : optParam (y \u2260 0) (_ : y \u2260 0)\n\u22a2 succAbove x (pred y hy) = y", "state_after": "case h\nn m : \u2115\nx y : Fin (n + 1)\nh : x < y\nhy : optParam (y \u2260 0) (_ : y \u2260 0)\n\u22a2 x \u2264 castSucc (pred y hy)"}, {"tactic": "simpa [le_iff_val_le_val] using Nat.le_pred_of_lt h", "annotated_tactic": ["simpa [<a>le_iff_val_le_val</a>] using <a>Nat.le_pred_of_lt</a> h", [{"full_name": "Fin.le_iff_val_le_val", "def_path": "Mathlib/Data/Fin/Basic.lean", "def_pos": [219, 9], "def_end_pos": [219, 26]}, {"full_name": "Nat.le_pred_of_lt", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [268, 9], "def_end_pos": [268, 22]}]], "state_before": "case h\nn m : \u2115\nx y : Fin (n + 1)\nh : x < y\nhy : optParam (y \u2260 0) (_ : y \u2260 0)\n\u22a2 x \u2264 castSucc (pred y hy)", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/DFinsupp/Order.lean | DFinsupp.bot_eq_zero | [
155,
11
] | [
156,
6
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Analysis/Convex/Between.lean | Wbtw.right_mem_affineSpan_of_left_ne | [
696,
1
] | [
699,
42
] | [{"tactic": "rcases h.right_mem_image_Ici_of_left_ne hne with \u27e8r, \u27e8-, rfl\u27e9\u27e9", "annotated_tactic": ["rcases h.right_mem_image_Ici_of_left_ne hne with \u27e8r, \u27e8-, rfl\u27e9\u27e9", []], "state_before": "R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\nhne : x \u2260 y\n\u22a2 z \u2208 affineSpan R {x, y}", "state_after": "case intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nhne : x \u2260 y\nr : R\nh : Wbtw R x y (\u2191(lineMap x y) r)\n\u22a2 \u2191(lineMap x y) r \u2208 affineSpan R {x, y}"}, {"tactic": "exact lineMap_mem_affineSpan_pair _ _ _", "annotated_tactic": ["exact <a>lineMap_mem_affineSpan_pair</a> _ _ _", [{"full_name": "AffineMap.lineMap_mem_affineSpan_pair", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [1323, 9], "def_end_pos": [1323, 46]}]], "state_before": "case intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst\u271d\u00b3 : LinearOrderedField R\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module R V\ninst\u271d : AddTorsor V P\nx y : P\nhne : x \u2260 y\nr : R\nh : Wbtw R x y (\u2191(lineMap x y) r)\n\u22a2 \u2191(lineMap x y) r \u2208 affineSpan R {x, y}", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/LinearAlgebra/Matrix/Nondegenerate.lean | Matrix.nondegenerate_of_det_ne_zero | [
50,
1
] | [
63,
18
] | [{"tactic": "intro v hv", "annotated_tactic": ["intro v hv", []], "state_before": "m : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\n\u22a2 Nondegenerate M", "state_after": "m : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\nhv : \u2200 (w : m \u2192 A), v \u2b1d\u1d65 mulVec M w = 0\n\u22a2 v = 0"}, {"tactic": "ext i", "annotated_tactic": ["ext i", []], "state_before": "m : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\nhv : \u2200 (w : m \u2192 A), v \u2b1d\u1d65 mulVec M w = 0\n\u22a2 v = 0", "state_after": "case h\nm : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\nhv : \u2200 (w : m \u2192 A), v \u2b1d\u1d65 mulVec M w = 0\ni : m\n\u22a2 v i = OfNat.ofNat 0 i"}, {"tactic": "specialize hv (M.cramer (Pi.single i 1))", "annotated_tactic": ["specialize hv (M.cramer (<a>Pi.single</a> i 1))", [{"full_name": "Pi.single", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [231, 3], "def_end_pos": [231, 14]}]], "state_before": "case h\nm : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\nhv : \u2200 (w : m \u2192 A), v \u2b1d\u1d65 mulVec M w = 0\ni : m\n\u22a2 v i = OfNat.ofNat 0 i", "state_after": "case h\nm : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\ni : m\nhv : v \u2b1d\u1d65 mulVec M (\u2191(cramer M) (Pi.single i 1)) = 0\n\u22a2 v i = OfNat.ofNat 0 i"}, {"tactic": "refine' (mul_eq_zero.mp _).resolve_right hM", "annotated_tactic": ["refine' (mul_eq_zero.mp _).<a>resolve_right</a> hM", [{"full_name": "Or.resolve_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [272, 9], "def_end_pos": [272, 25]}]], "state_before": "case h\nm : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\ni : m\nhv : v \u2b1d\u1d65 mulVec M (\u2191(cramer M) (Pi.single i 1)) = 0\n\u22a2 v i = OfNat.ofNat 0 i", "state_after": "case h\nm : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\ni : m\nhv : v \u2b1d\u1d65 mulVec M (\u2191(cramer M) (Pi.single i 1)) = 0\n\u22a2 v i * det M = 0"}, {"tactic": "convert hv", "annotated_tactic": ["convert hv", []], "state_before": "case h\nm : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\ni : m\nhv : v \u2b1d\u1d65 mulVec M (\u2191(cramer M) (Pi.single i 1)) = 0\n\u22a2 v i * det M = 0", "state_after": "case h.e'_2\nm : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\ni : m\nhv : v \u2b1d\u1d65 mulVec M (\u2191(cramer M) (Pi.single i 1)) = 0\n\u22a2 v i * det M = v \u2b1d\u1d65 mulVec M (\u2191(cramer M) (Pi.single i 1))"}, {"tactic": "simp only [mulVec_cramer M (Pi.single i 1), dotProduct, Pi.smul_apply, smul_eq_mul]", "annotated_tactic": ["simp only [<a>mulVec_cramer</a> M (<a>Pi.single</a> i 1), <a>dotProduct</a>, <a>Pi.smul_apply</a>, <a>smul_eq_mul</a>]", [{"full_name": "Matrix.mulVec_cramer", "def_path": "Mathlib/LinearAlgebra/Matrix/Adjugate.lean", "def_pos": [312, 9], "def_end_pos": [312, 22]}, {"full_name": "Pi.single", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [231, 3], "def_end_pos": [231, 14]}, {"full_name": "Matrix.dotProduct", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [705, 5], "def_end_pos": [705, 15]}, {"full_name": "Pi.smul_apply", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [116, 60], "def_end_pos": [116, 70]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}]], "state_before": "case h.e'_2\nm : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\ni : m\nhv : v \u2b1d\u1d65 mulVec M (\u2191(cramer M) (Pi.single i 1)) = 0\n\u22a2 v i * det M = v \u2b1d\u1d65 mulVec M (\u2191(cramer M) (Pi.single i 1))", "state_after": "case h.e'_2\nm : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\ni : m\nhv : v \u2b1d\u1d65 mulVec M (\u2191(cramer M) (Pi.single i 1)) = 0\n\u22a2 v i * det M = Finset.sum Finset.univ fun x => v x * (det M * Pi.single i 1 x)"}, {"tactic": "rw [Finset.sum_eq_single i, Pi.single_eq_same, mul_one]", "annotated_tactic": ["rw [<a>Finset.sum_eq_single</a> i, <a>Pi.single_eq_same</a>, <a>mul_one</a>]", [{"full_name": "Finset.sum_eq_single", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [798, 3], "def_end_pos": [798, 14]}, {"full_name": "Pi.single_eq_same", "def_path": "Mathlib/Data/Pi/Algebra.lean", "def_pos": [237, 3], "def_end_pos": [237, 14]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "case h.e'_2\nm : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\ni : m\nhv : v \u2b1d\u1d65 mulVec M (\u2191(cramer M) (Pi.single i 1)) = 0\n\u22a2 v i * det M = Finset.sum Finset.univ fun x => v x * (det M * Pi.single i 1 x)", "state_after": "case h.e'_2.h\u2080\nm : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\ni : m\nhv : v \u2b1d\u1d65 mulVec M (\u2191(cramer M) (Pi.single i 1)) = 0\n\u22a2 \u2200 (b : m), b \u2208 Finset.univ \u2192 b \u2260 i \u2192 v b * (det M * Pi.single i 1 b) = 0\n\ncase h.e'_2.h\u2081\nm : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\ni : m\nhv : v \u2b1d\u1d65 mulVec M (\u2191(cramer M) (Pi.single i 1)) = 0\n\u22a2 \u00aci \u2208 Finset.univ \u2192 v i * (det M * Pi.single i 1 i) = 0"}, {"tactic": "intro j _ hj", "annotated_tactic": ["intro j _ hj", []], "state_before": "case h.e'_2.h\u2080\nm : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\ni : m\nhv : v \u2b1d\u1d65 mulVec M (\u2191(cramer M) (Pi.single i 1)) = 0\n\u22a2 \u2200 (b : m), b \u2208 Finset.univ \u2192 b \u2260 i \u2192 v b * (det M * Pi.single i 1 b) = 0", "state_after": "case h.e'_2.h\u2080\nm : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\ni : m\nhv : v \u2b1d\u1d65 mulVec M (\u2191(cramer M) (Pi.single i 1)) = 0\nj : m\na\u271d : j \u2208 Finset.univ\nhj : j \u2260 i\n\u22a2 v j * (det M * Pi.single i 1 j) = 0"}, {"tactic": "simp [hj]", "annotated_tactic": ["simp [hj]", []], "state_before": "case h.e'_2.h\u2080\nm : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\ni : m\nhv : v \u2b1d\u1d65 mulVec M (\u2191(cramer M) (Pi.single i 1)) = 0\nj : m\na\u271d : j \u2208 Finset.univ\nhj : j \u2260 i\n\u22a2 v j * (det M * Pi.single i 1 j) = 0", "state_after": "no goals"}, {"tactic": "intros", "annotated_tactic": ["intros", []], "state_before": "case h.e'_2.h\u2081\nm : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\ni : m\nhv : v \u2b1d\u1d65 mulVec M (\u2191(cramer M) (Pi.single i 1)) = 0\n\u22a2 \u00aci \u2208 Finset.univ \u2192 v i * (det M * Pi.single i 1 i) = 0", "state_after": "case h.e'_2.h\u2081\nm : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\ni : m\nhv : v \u2b1d\u1d65 mulVec M (\u2191(cramer M) (Pi.single i 1)) = 0\na\u271d : \u00aci \u2208 Finset.univ\n\u22a2 v i * (det M * Pi.single i 1 i) = 0"}, {"tactic": "have := Finset.mem_univ i", "annotated_tactic": ["have := <a>Finset.mem_univ</a> i", [{"full_name": "Finset.mem_univ", "def_path": "Mathlib/Data/Fintype/Basic.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "case h.e'_2.h\u2081\nm : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\ni : m\nhv : v \u2b1d\u1d65 mulVec M (\u2191(cramer M) (Pi.single i 1)) = 0\na\u271d : \u00aci \u2208 Finset.univ\n\u22a2 v i * (det M * Pi.single i 1 i) = 0", "state_after": "case h.e'_2.h\u2081\nm : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\ni : m\nhv : v \u2b1d\u1d65 mulVec M (\u2191(cramer M) (Pi.single i 1)) = 0\na\u271d : \u00aci \u2208 Finset.univ\nthis : i \u2208 Finset.univ\n\u22a2 v i * (det M * Pi.single i 1 i) = 0"}, {"tactic": "contradiction", "annotated_tactic": ["contradiction", []], "state_before": "case h.e'_2.h\u2081\nm : Type u_1\nR : Type u_2\nA : Type u_3\ninst\u271d\u2074 : Fintype m\ninst\u271d\u00b3 : CommRing R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : IsDomain A\ninst\u271d : DecidableEq m\nM : Matrix m m A\nhM : det M \u2260 0\nv : m \u2192 A\ni : m\nhv : v \u2b1d\u1d65 mulVec M (\u2191(cramer M) (Pi.single i 1)) = 0\na\u271d : \u00aci \u2208 Finset.univ\nthis : i \u2208 Finset.univ\n\u22a2 v i * (det M * Pi.single i 1 i) = 0", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/LinearAlgebra/Multilinear/Basic.lean | MultilinearMap.map_neg | [
1132,
1
] | [
1135,
88
] | [{"tactic": "rw [\u2190 MultilinearMap.map_add, add_left_neg, f.map_coord_zero i (update_same i 0 m)]", "annotated_tactic": ["rw [\u2190 <a>MultilinearMap.map_add</a>, <a>add_left_neg</a>, f.map_coord_zero i (<a>update_same</a> i 0 m)]", [{"full_name": "MultilinearMap.map_add", "def_path": "Mathlib/LinearAlgebra/Multilinear/Basic.lean", "def_pos": [161, 19], "def_end_pos": [161, 26]}, {"full_name": "add_left_neg", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1118, 3], "def_end_pos": [1118, 14]}, {"full_name": "Function.update_same", "def_path": "Mathlib/Logic/Function/Basic.lean", "def_pos": [555, 9], "def_end_pos": [555, 20]}]], "state_before": "R : Type uR\nS : Type uS\n\u03b9 : Type u\u03b9\nn : \u2115\nM : Fin (Nat.succ n) \u2192 Type v\nM\u2081 : \u03b9 \u2192 Type v\u2081\nM\u2082 : Type v\u2082\nM\u2083 : Type v\u2083\nM' : Type v'\ninst\u271d\u2075 : Semiring R\ninst\u271d\u2074 : (i : \u03b9) \u2192 AddCommGroup (M\u2081 i)\ninst\u271d\u00b3 : AddCommGroup M\u2082\ninst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (M\u2081 i)\ninst\u271d\u00b9 : Module R M\u2082\nf : MultilinearMap R M\u2081 M\u2082\ninst\u271d : DecidableEq \u03b9\nm : (i : \u03b9) \u2192 M\u2081 i\ni : \u03b9\nx : M\u2081 i\n\u22a2 \u2191f (update m i (-x)) + \u2191f (update m i x) = 0", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/RingTheory/Ideal/Quotient.lean | Ideal.Quotient.maximal_of_isField | [
234,
1
] | [
243,
77
] | [{"tactic": "apply Ideal.isMaximal_iff.2", "annotated_tactic": ["apply <a>Ideal.isMaximal_iff</a>.2", [{"full_name": "Ideal.isMaximal_iff", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [302, 9], "def_end_pos": [302, 22]}]], "state_before": "R : Type u\ninst\u271d : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx y : R\nI : Ideal R\nhqf : IsField (R \u29f8 I)\n\u22a2 IsMaximal I", "state_after": "R : Type u\ninst\u271d : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx y : R\nI : Ideal R\nhqf : IsField (R \u29f8 I)\n\u22a2 \u00ac1 \u2208 I \u2227 \u2200 (J : Ideal R) (x : R), I \u2264 J \u2192 \u00acx \u2208 I \u2192 x \u2208 J \u2192 1 \u2208 J"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "R : Type u\ninst\u271d : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx y : R\nI : Ideal R\nhqf : IsField (R \u29f8 I)\n\u22a2 \u00ac1 \u2208 I \u2227 \u2200 (J : Ideal R) (x : R), I \u2264 J \u2192 \u00acx \u2208 I \u2192 x \u2208 J \u2192 1 \u2208 J", "state_after": "case left\nR : Type u\ninst\u271d : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx y : R\nI : Ideal R\nhqf : IsField (R \u29f8 I)\n\u22a2 \u00ac1 \u2208 I\n\ncase right\nR : Type u\ninst\u271d : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx y : R\nI : Ideal R\nhqf : IsField (R \u29f8 I)\n\u22a2 \u2200 (J : Ideal R) (x : R), I \u2264 J \u2192 \u00acx \u2208 I \u2192 x \u2208 J \u2192 1 \u2208 J"}, {"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "case left\nR : Type u\ninst\u271d : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx y : R\nI : Ideal R\nhqf : IsField (R \u29f8 I)\n\u22a2 \u00ac1 \u2208 I", "state_after": "case left\nR : Type u\ninst\u271d : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx y : R\nI : Ideal R\nhqf : IsField (R \u29f8 I)\nh : 1 \u2208 I\n\u22a2 False"}, {"tactic": "rcases hqf.exists_pair_ne with \u27e8\u27e8x\u27e9, \u27e8y\u27e9, hxy\u27e9", "annotated_tactic": ["rcases hqf.exists_pair_ne with \u27e8\u27e8x\u27e9, \u27e8y\u27e9, hxy\u27e9", []], "state_before": "case left\nR : Type u\ninst\u271d : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx y : R\nI : Ideal R\nhqf : IsField (R \u29f8 I)\nh : 1 \u2208 I\n\u22a2 False", "state_after": "case left.intro.mk.intro.mk\nR : Type u\ninst\u271d : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx\u271d y\u271d : R\nI : Ideal R\nhqf : IsField (R \u29f8 I)\nh : 1 \u2208 I\nw\u271d\u00b9 : R \u29f8 I\nx : R\nw\u271d : R \u29f8 I\ny : R\nhxy : Quot.mk Setoid.r x \u2260 Quot.mk Setoid.r y\n\u22a2 False"}, {"tactic": "exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) \u25b8 I.mul_mem_left _ h))", "annotated_tactic": ["exact hxy (<a>Ideal.Quotient.eq</a>.2 (<a>mul_one</a> (x - y) \u25b8 I.mul_mem_left _ h))", [{"full_name": "Ideal.Quotient.eq", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [121, 19], "def_end_pos": [121, 21]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}]], "state_before": "case left.intro.mk.intro.mk\nR : Type u\ninst\u271d : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx\u271d y\u271d : R\nI : Ideal R\nhqf : IsField (R \u29f8 I)\nh : 1 \u2208 I\nw\u271d\u00b9 : R \u29f8 I\nx : R\nw\u271d : R \u29f8 I\ny : R\nhxy : Quot.mk Setoid.r x \u2260 Quot.mk Setoid.r y\n\u22a2 False", "state_after": "no goals"}, {"tactic": "intro J x hIJ hxnI hxJ", "annotated_tactic": ["intro J x hIJ hxnI hxJ", []], "state_before": "case right\nR : Type u\ninst\u271d : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx y : R\nI : Ideal R\nhqf : IsField (R \u29f8 I)\n\u22a2 \u2200 (J : Ideal R) (x : R), I \u2264 J \u2192 \u00acx \u2208 I \u2192 x \u2208 J \u2192 1 \u2208 J", "state_after": "case right\nR : Type u\ninst\u271d : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx\u271d y : R\nI : Ideal R\nhqf : IsField (R \u29f8 I)\nJ : Ideal R\nx : R\nhIJ : I \u2264 J\nhxnI : \u00acx \u2208 I\nhxJ : x \u2208 J\n\u22a2 1 \u2208 J"}, {"tactic": "rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with \u27e8\u27e8y\u27e9, hy\u27e9", "annotated_tactic": ["rcases hqf.mul_inv_cancel (<a>mt</a> <a>Ideal.Quotient.eq_zero_iff_mem</a>.1 hxnI) with \u27e8\u27e8y\u27e9, hy\u27e9", [{"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}, {"full_name": "Ideal.Quotient.eq_zero_iff_mem", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [129, 9], "def_end_pos": [129, 24]}]], "state_before": "case right\nR : Type u\ninst\u271d : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx\u271d y : R\nI : Ideal R\nhqf : IsField (R \u29f8 I)\nJ : Ideal R\nx : R\nhIJ : I \u2264 J\nhxnI : \u00acx \u2208 I\nhxJ : x \u2208 J\n\u22a2 1 \u2208 J", "state_after": "case right.intro.mk\nR : Type u\ninst\u271d : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx\u271d y\u271d : R\nI : Ideal R\nhqf : IsField (R \u29f8 I)\nJ : Ideal R\nx : R\nhIJ : I \u2264 J\nhxnI : \u00acx \u2208 I\nhxJ : x \u2208 J\nw\u271d : R \u29f8 I\ny : R\nhy : \u2191(mk I) x * Quot.mk Setoid.r y = 1\n\u22a2 1 \u2208 J"}, {"tactic": "rw [\u2190 zero_add (1 : R), \u2190 sub_self (x * y), sub_add]", "annotated_tactic": ["rw [\u2190 <a>zero_add</a> (1 : R), \u2190 <a>sub_self</a> (x * y), <a>sub_add</a>]", [{"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}, {"full_name": "sub_self", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [734, 30], "def_end_pos": [734, 38]}, {"full_name": "sub_add", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [531, 3], "def_end_pos": [531, 14]}]], "state_before": "case right.intro.mk\nR : Type u\ninst\u271d : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx\u271d y\u271d : R\nI : Ideal R\nhqf : IsField (R \u29f8 I)\nJ : Ideal R\nx : R\nhIJ : I \u2264 J\nhxnI : \u00acx \u2208 I\nhxJ : x \u2208 J\nw\u271d : R \u29f8 I\ny : R\nhy : \u2191(mk I) x * Quot.mk Setoid.r y = 1\n\u22a2 1 \u2208 J", "state_after": "case right.intro.mk\nR : Type u\ninst\u271d : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx\u271d y\u271d : R\nI : Ideal R\nhqf : IsField (R \u29f8 I)\nJ : Ideal R\nx : R\nhIJ : I \u2264 J\nhxnI : \u00acx \u2208 I\nhxJ : x \u2208 J\nw\u271d : R \u29f8 I\ny : R\nhy : \u2191(mk I) x * Quot.mk Setoid.r y = 1\n\u22a2 x * y - (x * y - 1) \u2208 J"}, {"tactic": "refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))", "annotated_tactic": ["refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (<a>Ideal.Quotient.eq</a>.1 hy))", [{"full_name": "Ideal.Quotient.eq", "def_path": "Mathlib/RingTheory/Ideal/Quotient.lean", "def_pos": [121, 19], "def_end_pos": [121, 21]}]], "state_before": "case right.intro.mk\nR : Type u\ninst\u271d : CommRing R\nI\u271d : Ideal R\na b : R\nS : Type v\nx\u271d y\u271d : R\nI : Ideal R\nhqf : IsField (R \u29f8 I)\nJ : Ideal R\nx : R\nhIJ : I \u2264 J\nhxnI : \u00acx \u2208 I\nhxJ : x \u2208 J\nw\u271d : R \u29f8 I\ny : R\nhy : \u2191(mk I) x * Quot.mk Setoid.r y = 1\n\u22a2 x * y - (x * y - 1) \u2208 J", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Algebra/FreeAlgebra.lean | FreeAlgebra.algebraMap_eq_zero_iff | [
498,
1
] | [
499,
68
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Multiset/Basic.lean | Multiset.count_cons_of_ne | [
2364,
1
] | [
2365,
27
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Probability/Process/Stopping.lean | MeasureTheory.IsStoppingTime.measurable | [
579,
11
] | [
582,
90
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Order/Ideal.lean | IsCoatom.isMaximal | [
241,
1
] | [
242,
81
] | [{"tactic": "simp [hI.2 _ hJ]", "annotated_tactic": ["simp [hI.2 _ hJ]", []], "state_before": "P : Type u_1\ninst\u271d\u00b2 : LE P\ninst\u271d\u00b9 : IsDirected P fun x x_1 => x \u2264 x_1\ninst\u271d : Nonempty P\nI : Ideal P\nhI : IsCoatom I\nsrc\u271d : IsProper I := IsCoatom.isProper hI\nx\u271d : Ideal P\nhJ : I < x\u271d\n\u22a2 \u2191x\u271d = univ", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Topology/DenseEmbedding.lean | DenseInducing.continuous | [
54,
11
] | [
55,
27
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Order/Filter/Bases.lean | Filter.hasBasis_generate | [
241,
1
] | [
243,
84
] | [{"tactic": "simp only [mem_generate_iff, exists_prop, and_assoc, and_left_comm]", "annotated_tactic": ["simp only [<a>mem_generate_iff</a>, <a>exists_prop</a>, <a>and_assoc</a>, <a>and_left_comm</a>]", [{"full_name": "Filter.mem_generate_iff", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [363, 9], "def_end_pos": [363, 25]}, {"full_name": "exists_prop", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [485, 17], "def_end_pos": [485, 28]}, {"full_name": "and_assoc", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [177, 9], "def_end_pos": [177, 18]}, {"full_name": "and_left_comm", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [180, 9], "def_end_pos": [180, 22]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns\u271d : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\ns : Set (Set \u03b1)\nU : Set \u03b1\n\u22a2 U \u2208 generate s \u2194 \u2203 i, (Set.Finite i \u2227 i \u2286 s) \u2227 \u22c2\u2080 i \u2286 U", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/CategoryTheory/Monoidal/Center.lean | CategoryTheory.Center.rightUnitor_hom_f | [
284,
1
] | [
285,
6
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Algebra/Squarefree.lean | squarefree_one | [
46,
1
] | [
47,
24
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/List/Basic.lean | List.map_injective_iff | [
1774,
1
] | [
1784,
33
] | [{"tactic": "constructor <;> intro h x y hxy", "annotated_tactic": ["constructor <;> intro h x y hxy", []], "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 : \u03b1 \u2192 \u03b2\n\u22a2 Injective (map f) \u2194 Injective f", "state_after": "case mp\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nf : \u03b1 \u2192 \u03b2\nh : Injective (map f)\nx y : \u03b1\nhxy : f x = f y\n\u22a2 x = y\n\ncase mpr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nf : \u03b1 \u2192 \u03b2\nh : Injective f\nx y : List \u03b1\nhxy : map f x = map f y\n\u22a2 x = y"}, {"tactic": "suffices [x] = [y] by simpa using this", "annotated_tactic": ["suffices [x] = [y] by simpa using this", []], "state_before": "case mp\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nf : \u03b1 \u2192 \u03b2\nh : Injective (map f)\nx y : \u03b1\nhxy : f x = f y\n\u22a2 x = y", "state_after": "case mp\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nf : \u03b1 \u2192 \u03b2\nh : Injective (map f)\nx y : \u03b1\nhxy : f x = f y\n\u22a2 [x] = [y]"}, {"tactic": "apply h", "annotated_tactic": ["apply h", []], "state_before": "case mp\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nf : \u03b1 \u2192 \u03b2\nh : Injective (map f)\nx y : \u03b1\nhxy : f x = f y\n\u22a2 [x] = [y]", "state_after": "case mp.a\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nf : \u03b1 \u2192 \u03b2\nh : Injective (map f)\nx y : \u03b1\nhxy : f x = f y\n\u22a2 map f [x] = map f [y]"}, {"tactic": "simp [hxy]", "annotated_tactic": ["simp [hxy]", []], "state_before": "case mp.a\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nf : \u03b1 \u2192 \u03b2\nh : Injective (map f)\nx y : \u03b1\nhxy : f x = f y\n\u22a2 map f [x] = map f [y]", "state_after": "no goals"}, {"tactic": "simpa using this", "annotated_tactic": ["simpa using this", []], "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 : \u03b1 \u2192 \u03b2\nh : Injective (map f)\nx y : \u03b1\nhxy : f x = f y\nthis : [x] = [y]\n\u22a2 x = y", "state_after": "no goals"}, {"tactic": "induction' y with yh yt y_ih generalizing x", "annotated_tactic": ["induction' y with yh yt y_ih generalizing x", []], "state_before": "case mpr\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nf : \u03b1 \u2192 \u03b2\nh : Injective f\nx y : List \u03b1\nhxy : map f x = map f y\n\u22a2 x = y", "state_after": "case mpr.nil\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nf : \u03b1 \u2192 \u03b2\nh : Injective f\nx\u271d y : List \u03b1\nhxy\u271d : map f x\u271d = map f y\nx : List \u03b1\nhxy : map f x = map f []\n\u22a2 x = []\n\ncase mpr.cons\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nf : \u03b1 \u2192 \u03b2\nh : Injective f\nx\u271d y : List \u03b1\nhxy\u271d : map f x\u271d = map f y\nyh : \u03b1\nyt : List \u03b1\ny_ih : \u2200 \u2983x : List \u03b1\u2984, map f x = map f yt \u2192 x = yt\nx : List \u03b1\nhxy : map f x = map f (yh :: yt)\n\u22a2 x = yh :: yt"}, {"tactic": "cases x", "annotated_tactic": ["cases x", []], "state_before": "case mpr.cons\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nf : \u03b1 \u2192 \u03b2\nh : Injective f\nx\u271d y : List \u03b1\nhxy\u271d : map f x\u271d = map f y\nyh : \u03b1\nyt : List \u03b1\ny_ih : \u2200 \u2983x : List \u03b1\u2984, map f x = map f yt \u2192 x = yt\nx : List \u03b1\nhxy : map f x = map f (yh :: yt)\n\u22a2 x = yh :: yt", "state_after": "case mpr.cons.nil\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nf : \u03b1 \u2192 \u03b2\nh : Injective f\nx y : List \u03b1\nhxy\u271d : map f x = map f y\nyh : \u03b1\nyt : List \u03b1\ny_ih : \u2200 \u2983x : List \u03b1\u2984, map f x = map f yt \u2192 x = yt\nhxy : map f [] = map f (yh :: yt)\n\u22a2 [] = yh :: yt\n\ncase mpr.cons.cons\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nf : \u03b1 \u2192 \u03b2\nh : Injective f\nx y : List \u03b1\nhxy\u271d : map f x = map f y\nyh : \u03b1\nyt : List \u03b1\ny_ih : \u2200 \u2983x : List \u03b1\u2984, map f x = map f yt \u2192 x = yt\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nhxy : map f (head\u271d :: tail\u271d) = map f (yh :: yt)\n\u22a2 head\u271d :: tail\u271d = yh :: yt"}, {"tactic": "simpa using hxy", "annotated_tactic": ["simpa using hxy", []], "state_before": "case mpr.nil\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nf : \u03b1 \u2192 \u03b2\nh : Injective f\nx\u271d y : List \u03b1\nhxy\u271d : map f x\u271d = map f y\nx : List \u03b1\nhxy : map f x = map f []\n\u22a2 x = []", "state_after": "no goals"}, {"tactic": "simp at hxy", "annotated_tactic": ["simp at hxy", []], "state_before": "case mpr.cons.nil\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nf : \u03b1 \u2192 \u03b2\nh : Injective f\nx y : List \u03b1\nhxy\u271d : map f x = map f y\nyh : \u03b1\nyt : List \u03b1\ny_ih : \u2200 \u2983x : List \u03b1\u2984, map f x = map f yt \u2192 x = yt\nhxy : map f [] = map f (yh :: yt)\n\u22a2 [] = yh :: yt", "state_after": "no goals"}, {"tactic": "simp only [map, cons.injEq] at hxy", "annotated_tactic": ["simp only [<a>map</a>, cons.injEq] at hxy", [{"full_name": "List.map", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [151, 19], "def_end_pos": [151, 22]}]], "state_before": "case mpr.cons.cons\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nf : \u03b1 \u2192 \u03b2\nh : Injective f\nx y : List \u03b1\nhxy\u271d : map f x = map f y\nyh : \u03b1\nyt : List \u03b1\ny_ih : \u2200 \u2983x : List \u03b1\u2984, map f x = map f yt \u2192 x = yt\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nhxy : map f (head\u271d :: tail\u271d) = map f (yh :: yt)\n\u22a2 head\u271d :: tail\u271d = yh :: yt", "state_after": "case mpr.cons.cons\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nf : \u03b1 \u2192 \u03b2\nh : Injective f\nx y : List \u03b1\nhxy\u271d : map f x = map f y\nyh : \u03b1\nyt : List \u03b1\ny_ih : \u2200 \u2983x : List \u03b1\u2984, map f x = map f yt \u2192 x = yt\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nhxy : f head\u271d = f yh \u2227 map f tail\u271d = map f yt\n\u22a2 head\u271d :: tail\u271d = yh :: yt"}, {"tactic": "simp [y_ih hxy.2, h hxy.1]", "annotated_tactic": ["simp [y_ih hxy.2, h hxy.1]", []], "state_before": "case mpr.cons.cons\n\u03b9 : Type u_1\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\nl\u2081 l\u2082 : List \u03b1\nf : \u03b1 \u2192 \u03b2\nh : Injective f\nx y : List \u03b1\nhxy\u271d : map f x = map f y\nyh : \u03b1\nyt : List \u03b1\ny_ih : \u2200 \u2983x : List \u03b1\u2984, map f x = map f yt \u2192 x = yt\nhead\u271d : \u03b1\ntail\u271d : List \u03b1\nhxy : f head\u271d = f yh \u2227 map f tail\u271d = map f yt\n\u22a2 head\u271d :: tail\u271d = yh :: yt", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Analysis/NormedSpace/Star/Spectrum.lean | unitary.spectrum_subset_circle | [
31,
1
] | [
41,
61
] | [{"tactic": "nontriviality E", "annotated_tactic": ["nontriviality E", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u22a2 spectrum \ud835\udd5c \u2191u \u2286 Metric.sphere 0 1", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\n\u22a2 spectrum \ud835\udd5c \u2191u \u2286 Metric.sphere 0 1"}, {"tactic": "refine' fun k hk => mem_sphere_zero_iff_norm.mpr (le_antisymm _ _)", "annotated_tactic": ["refine' fun k hk => mem_sphere_zero_iff_norm.mpr (<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": "\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\n\u22a2 spectrum \ud835\udd5c \u2191u \u2286 Metric.sphere 0 1", "state_after": "case refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\nk : \ud835\udd5c\nhk : k \u2208 spectrum \ud835\udd5c \u2191u\n\u22a2 \u2016k\u2016 \u2264 1\n\ncase refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\nk : \ud835\udd5c\nhk : k \u2208 spectrum \ud835\udd5c \u2191u\n\u22a2 1 \u2264 \u2016k\u2016"}, {"tactic": "simpa only [CstarRing.norm_coe_unitary u] using norm_le_norm_of_mem hk", "annotated_tactic": ["simpa only [<a>CstarRing.norm_coe_unitary</a> u] using <a>norm_le_norm_of_mem</a> hk", [{"full_name": "CstarRing.norm_coe_unitary", "def_path": "Mathlib/Analysis/NormedSpace/Star/Basic.lean", "def_pos": [215, 9], "def_end_pos": [215, 25]}, {"full_name": "spectrum.norm_le_norm_of_mem", "def_path": "Mathlib/Analysis/NormedSpace/Spectrum.lean", "def_pos": [120, 9], "def_end_pos": [120, 28]}]], "state_before": "case refine'_1\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\nk : \ud835\udd5c\nhk : k \u2208 spectrum \ud835\udd5c \u2191u\n\u22a2 \u2016k\u2016 \u2264 1", "state_after": "no goals"}, {"tactic": "rw [\u2190 unitary.val_toUnits_apply u] at hk", "annotated_tactic": ["rw [\u2190 <a>unitary.val_toUnits_apply</a> u] at hk", [{"full_name": "unitary.val_toUnits_apply", "def_path": "Mathlib/Algebra/Star/Unitary.lean", "def_pos": [130, 3], "def_end_pos": [130, 8]}]], "state_before": "case refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\nk : \ud835\udd5c\nhk : k \u2208 spectrum \ud835\udd5c \u2191u\n\u22a2 1 \u2264 \u2016k\u2016", "state_after": "case refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\nk : \ud835\udd5c\nhk : k \u2208 spectrum \ud835\udd5c \u2191(\u2191toUnits u)\n\u22a2 1 \u2264 \u2016k\u2016"}, {"tactic": "have hnk := ne_zero_of_mem_of_unit hk", "annotated_tactic": ["have hnk := <a>ne_zero_of_mem_of_unit</a> hk", [{"full_name": "spectrum.ne_zero_of_mem_of_unit", "def_path": "Mathlib/Algebra/Algebra/Spectrum.lean", "def_pos": [199, 9], "def_end_pos": [199, 31]}]], "state_before": "case refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\nk : \ud835\udd5c\nhk : k \u2208 spectrum \ud835\udd5c \u2191(\u2191toUnits u)\n\u22a2 1 \u2264 \u2016k\u2016", "state_after": "case refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\nk : \ud835\udd5c\nhk : k \u2208 spectrum \ud835\udd5c \u2191(\u2191toUnits u)\nhnk : k \u2260 0\n\u22a2 1 \u2264 \u2016k\u2016"}, {"tactic": "rw [\u2190 inv_inv (unitary.toUnits u), \u2190 spectrum.map_inv, Set.mem_inv] at hk", "annotated_tactic": ["rw [\u2190 <a>inv_inv</a> (<a>unitary.toUnits</a> u), \u2190 <a>spectrum.map_inv</a>, <a>Set.mem_inv</a>] at hk", [{"full_name": "inv_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [800, 9], "def_end_pos": [800, 16]}, {"full_name": "unitary.toUnits", "def_path": "Mathlib/Algebra/Star/Unitary.lean", "def_pos": [131, 5], "def_end_pos": [131, 12]}, {"full_name": "spectrum.map_inv", "def_path": "Mathlib/Algebra/Algebra/Spectrum.lean", "def_pos": [374, 19], "def_end_pos": [374, 26]}, {"full_name": "Set.mem_inv", "def_path": "Mathlib/Data/Set/Pointwise/Basic.lean", "def_pos": [180, 9], "def_end_pos": [180, 16]}]], "state_before": "case refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\nk : \ud835\udd5c\nhk : k \u2208 spectrum \ud835\udd5c \u2191(\u2191toUnits u)\nhnk : k \u2260 0\n\u22a2 1 \u2264 \u2016k\u2016", "state_after": "case refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\nk : \ud835\udd5c\nhk : k\u207b\u00b9 \u2208 spectrum \ud835\udd5c \u2191(\u2191toUnits u)\u207b\u00b9\nhnk : k \u2260 0\n\u22a2 1 \u2264 \u2016k\u2016"}, {"tactic": "have : \u2016k\u2016\u207b\u00b9 \u2264 \u2016(\u2191(unitary.toUnits u)\u207b\u00b9 : E)\u2016 :=\n by simpa only [norm_inv] using norm_le_norm_of_mem hk", "annotated_tactic": ["have : \u2016k\u2016\u207b\u00b9 \u2264 \u2016(\u2191(<a>unitary.toUnits</a> u)\u207b\u00b9 : E)\u2016 :=\n by simpa only [<a>norm_inv</a>] using <a>norm_le_norm_of_mem</a> hk", [{"full_name": "unitary.toUnits", "def_path": "Mathlib/Algebra/Star/Unitary.lean", "def_pos": [131, 5], "def_end_pos": [131, 12]}, {"full_name": "norm_inv", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [572, 9], "def_end_pos": [572, 17]}, {"full_name": "spectrum.norm_le_norm_of_mem", "def_path": "Mathlib/Analysis/NormedSpace/Spectrum.lean", "def_pos": [120, 9], "def_end_pos": [120, 28]}]], "state_before": "case refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\nk : \ud835\udd5c\nhk : k\u207b\u00b9 \u2208 spectrum \ud835\udd5c \u2191(\u2191toUnits u)\u207b\u00b9\nhnk : k \u2260 0\n\u22a2 1 \u2264 \u2016k\u2016", "state_after": "case refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\nk : \ud835\udd5c\nhk : k\u207b\u00b9 \u2208 spectrum \ud835\udd5c \u2191(\u2191toUnits u)\u207b\u00b9\nhnk : k \u2260 0\nthis : \u2016k\u2016\u207b\u00b9 \u2264 \u2016\u2191(\u2191toUnits u)\u207b\u00b9\u2016\n\u22a2 1 \u2264 \u2016k\u2016"}, {"tactic": "simpa using inv_le_of_inv_le (norm_pos_iff.mpr hnk) this", "annotated_tactic": ["simpa using <a>inv_le_of_inv_le</a> (norm_pos_iff.mpr hnk) this", [{"full_name": "inv_le_of_inv_le", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [266, 9], "def_end_pos": [266, 25]}]], "state_before": "case refine'_2\n\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\nk : \ud835\udd5c\nhk : k\u207b\u00b9 \u2208 spectrum \ud835\udd5c \u2191(\u2191toUnits u)\u207b\u00b9\nhnk : k \u2260 0\nthis : \u2016k\u2016\u207b\u00b9 \u2264 \u2016\u2191(\u2191toUnits u)\u207b\u00b9\u2016\n\u22a2 1 \u2264 \u2016k\u2016", "state_after": "no goals"}, {"tactic": "simpa only [norm_inv] using norm_le_norm_of_mem hk", "annotated_tactic": ["simpa only [<a>norm_inv</a>] using <a>norm_le_norm_of_mem</a> hk", [{"full_name": "norm_inv", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [572, 9], "def_end_pos": [572, 17]}, {"full_name": "spectrum.norm_le_norm_of_mem", "def_path": "Mathlib/Analysis/NormedSpace/Spectrum.lean", "def_pos": [120, 9], "def_end_pos": [120, 28]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u2075 : NormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u2074 : NormedRing E\ninst\u271d\u00b3 : StarRing E\ninst\u271d\u00b2 : CstarRing E\ninst\u271d\u00b9 : NormedAlgebra \ud835\udd5c E\ninst\u271d : CompleteSpace E\nu : { x // x \u2208 unitary E }\n\u271d : Nontrivial E\nk : \ud835\udd5c\nhk : k\u207b\u00b9 \u2208 spectrum \ud835\udd5c \u2191(\u2191toUnits u)\u207b\u00b9\nhnk : k \u2260 0\n\u22a2 \u2016k\u2016\u207b\u00b9 \u2264 \u2016\u2191(\u2191toUnits u)\u207b\u00b9\u2016", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Topology/Maps.lean | quotientMap_iff_closed | [
282,
1
] | [
285,
49
] | [{"tactic": "simp only [isOpen_compl_iff, preimage_compl]", "annotated_tactic": ["simp only [<a>isOpen_compl_iff</a>, <a>preimage_compl</a>]", [{"full_name": "isOpen_compl_iff", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [205, 17], "def_end_pos": [205, 33]}, {"full_name": "Set.preimage_compl", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [97, 9], "def_end_pos": [97, 23]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : TopologicalSpace \u03b2\nf : \u03b1 \u2192 \u03b2\n\u22a2 (\u2200 (x : Set \u03b2), IsOpen x\u1d9c \u2194 IsOpen (f \u207b\u00b9' x\u1d9c)) \u2194 \u2200 (s : Set \u03b2), IsClosed s \u2194 IsClosed (f \u207b\u00b9' s)", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean | BoxIntegral.TaggedPrepartition.IsSubordinate.infPrepartition | [
268,
1
] | [
270,
26
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Algebra/GroupPower/Lemmas.lean | SemiconjBy.cast_int_mul_right | [
1038,
1
] | [
1039,
48
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Sigma/Basic.lean | Function.Injective.of_sigma_map | [
134,
1
] | [
137,
73
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Topology/UniformSpace/UniformConvergence.lean | tendstoUniformlyOn_iff_tendstoUniformlyOnFilter | [
107,
1
] | [
112,
7
] | [{"tactic": "simp only [TendstoUniformlyOn, TendstoUniformlyOnFilter]", "annotated_tactic": ["simp only [<a>TendstoUniformlyOn</a>, <a>TendstoUniformlyOnFilter</a>]", [{"full_name": "TendstoUniformlyOn", "def_path": "Mathlib/Topology/UniformSpace/UniformConvergence.lean", "def_pos": [103, 5], "def_end_pos": [103, 23]}, {"full_name": "TendstoUniformlyOnFilter", "def_path": "Mathlib/Topology/UniformSpace/UniformConvergence.lean", "def_pos": [85, 5], "def_end_pos": [85, 29]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\n\u22a2 TendstoUniformlyOn F f p s \u2194 TendstoUniformlyOnFilter F f p (\ud835\udcdf s)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\n\u22a2 (\u2200 (u : Set (\u03b2 \u00d7 \u03b2)), u \u2208 \ud835\udce4 \u03b2 \u2192 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b1), x \u2208 s \u2192 (f x, F n x) \u2208 u) \u2194\n \u2200 (u : Set (\u03b2 \u00d7 \u03b2)), u \u2208 \ud835\udce4 \u03b2 \u2192 \u2200\u1da0 (n : \u03b9 \u00d7 \u03b1) in p \u00d7\u02e2 \ud835\udcdf s, (f n.2, F n.1 n.2) \u2208 u"}, {"tactic": "apply forall\u2082_congr", "annotated_tactic": ["apply <a>forall\u2082_congr</a>", [{"full_name": "forall\u2082_congr", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [383, 9], "def_end_pos": [383, 22]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\n\u22a2 (\u2200 (u : Set (\u03b2 \u00d7 \u03b2)), u \u2208 \ud835\udce4 \u03b2 \u2192 \u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b1), x \u2208 s \u2192 (f x, F n x) \u2208 u) \u2194\n \u2200 (u : Set (\u03b2 \u00d7 \u03b2)), u \u2208 \ud835\udce4 \u03b2 \u2192 \u2200\u1da0 (n : \u03b9 \u00d7 \u03b1) in p \u00d7\u02e2 \ud835\udcdf s, (f n.2, F n.1 n.2) \u2208 u", "state_after": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\n\u22a2 \u2200 (a : Set (\u03b2 \u00d7 \u03b2)),\n a \u2208 \ud835\udce4 \u03b2 \u2192\n ((\u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b1), x \u2208 s \u2192 (f x, F n x) \u2208 a) \u2194 \u2200\u1da0 (n : \u03b9 \u00d7 \u03b1) in p \u00d7\u02e2 \ud835\udcdf s, (f n.2, F n.1 n.2) \u2208 a)"}, {"tactic": "simp_rw [eventually_prod_principal_iff]", "annotated_tactic": ["simp_rw [<a>eventually_prod_principal_iff</a>]", [{"full_name": "Filter.eventually_prod_principal_iff", "def_path": "Mathlib/Order/Filter/Prod.lean", "def_pos": [103, 9], "def_end_pos": [103, 38]}]], "state_before": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\n\u22a2 \u2200 (a : Set (\u03b2 \u00d7 \u03b2)),\n a \u2208 \ud835\udce4 \u03b2 \u2192\n ((\u2200\u1da0 (n : \u03b9) in p, \u2200 (x : \u03b1), x \u2208 s \u2192 (f x, F n x) \u2208 a) \u2194 \u2200\u1da0 (n : \u03b9 \u00d7 \u03b1) in p \u00d7\u02e2 \ud835\udcdf s, (f n.2, F n.1 n.2) \u2208 a)", "state_after": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\n\u22a2 \u2200 (a : Set (\u03b2 \u00d7 \u03b2)), a \u2208 \ud835\udce4 \u03b2 \u2192 True"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Type x\ninst\u271d : UniformSpace \u03b2\nF : \u03b9 \u2192 \u03b1 \u2192 \u03b2\nf : \u03b1 \u2192 \u03b2\ns s' : Set \u03b1\nx : \u03b1\np : Filter \u03b9\np' : Filter \u03b1\ng : \u03b9 \u2192 \u03b1\n\u22a2 \u2200 (a : Set (\u03b2 \u00d7 \u03b2)), a \u2208 \ud835\udce4 \u03b2 \u2192 True", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/List/Perm.lean | List.singleton_perm | [
211,
1
] | [
212,
26
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | ProjectiveSpectrum.zeroLocus_vanishingIdeal_eq_closure | [
360,
1
] | [
368,
46
] | [{"tactic": "apply Set.Subset.antisymm", "annotated_tactic": ["apply <a>Set.Subset.antisymm</a>", [{"full_name": "Set.Subset.antisymm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [370, 9], "def_end_pos": [370, 24]}]], "state_before": "R : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\n\u22a2 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t) = closure t", "state_after": "case h\u2081\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\n\u22a2 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t) \u2286 closure t\n\ncase h\u2082\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\n\u22a2 closure t \u2286 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t)"}, {"tactic": "rintro x hx t' \u27e8ht', ht\u27e9", "annotated_tactic": ["rintro x hx t' \u27e8ht', ht\u27e9", []], "state_before": "case h\u2081\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\n\u22a2 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t) \u2286 closure t", "state_after": "case h\u2081.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\nx : ProjectiveSpectrum \ud835\udc9c\nhx : x \u2208 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t)\nt' : Set (ProjectiveSpectrum \ud835\udc9c)\nht' : IsClosed t'\nht : t \u2286 t'\n\u22a2 x \u2208 t'"}, {"tactic": "obtain \u27e8fs, rfl\u27e9 : \u2203 s, t' = zeroLocus \ud835\udc9c s := by rwa [isClosed_iff_zeroLocus] at ht'", "annotated_tactic": ["obtain \u27e8fs, rfl\u27e9 : \u2203 s, t' = <a>zeroLocus</a> \ud835\udc9c s := by rwa [<a>isClosed_iff_zeroLocus</a>] at ht'", [{"full_name": "ProjectiveSpectrum.zeroLocus", "def_path": "Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean", "def_pos": [71, 5], "def_end_pos": [71, 14]}, {"full_name": "ProjectiveSpectrum.isClosed_iff_zeroLocus", "def_path": "Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean", "def_pos": [351, 9], "def_end_pos": [351, 31]}]], "state_before": "case h\u2081.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\nx : ProjectiveSpectrum \ud835\udc9c\nhx : x \u2208 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t)\nt' : Set (ProjectiveSpectrum \ud835\udc9c)\nht' : IsClosed t'\nht : t \u2286 t'\n\u22a2 x \u2208 t'", "state_after": "case h\u2081.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\nx : ProjectiveSpectrum \ud835\udc9c\nhx : x \u2208 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t)\nfs : Set A\nht' : IsClosed (zeroLocus \ud835\udc9c fs)\nht : t \u2286 zeroLocus \ud835\udc9c fs\n\u22a2 x \u2208 zeroLocus \ud835\udc9c fs"}, {"tactic": "rw [subset_zeroLocus_iff_subset_vanishingIdeal] at ht", "annotated_tactic": ["rw [<a>subset_zeroLocus_iff_subset_vanishingIdeal</a>] at ht", [{"full_name": "ProjectiveSpectrum.subset_zeroLocus_iff_subset_vanishingIdeal", "def_path": "Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean", "def_pos": [153, 9], "def_end_pos": [153, 51]}]], "state_before": "case h\u2081.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\nx : ProjectiveSpectrum \ud835\udc9c\nhx : x \u2208 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t)\nfs : Set A\nht' : IsClosed (zeroLocus \ud835\udc9c fs)\nht : t \u2286 zeroLocus \ud835\udc9c fs\n\u22a2 x \u2208 zeroLocus \ud835\udc9c fs", "state_after": "case h\u2081.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\nx : ProjectiveSpectrum \ud835\udc9c\nhx : x \u2208 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t)\nfs : Set A\nht' : IsClosed (zeroLocus \ud835\udc9c fs)\nht : fs \u2286 \u2191(vanishingIdeal t)\n\u22a2 x \u2208 zeroLocus \ud835\udc9c fs"}, {"tactic": "exact Set.Subset.trans ht hx", "annotated_tactic": ["exact <a>Set.Subset.trans</a> ht hx", [{"full_name": "Set.Subset.trans", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [362, 9], "def_end_pos": [362, 21]}]], "state_before": "case h\u2081.intro.intro\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\nx : ProjectiveSpectrum \ud835\udc9c\nhx : x \u2208 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t)\nfs : Set A\nht' : IsClosed (zeroLocus \ud835\udc9c fs)\nht : fs \u2286 \u2191(vanishingIdeal t)\n\u22a2 x \u2208 zeroLocus \ud835\udc9c fs", "state_after": "no goals"}, {"tactic": "rwa [isClosed_iff_zeroLocus] at ht'", "annotated_tactic": ["rwa [<a>isClosed_iff_zeroLocus</a>] at ht'", [{"full_name": "ProjectiveSpectrum.isClosed_iff_zeroLocus", "def_path": "Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean", "def_pos": [351, 9], "def_end_pos": [351, 31]}]], "state_before": "R : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\nx : ProjectiveSpectrum \ud835\udc9c\nhx : x \u2208 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t)\nt' : Set (ProjectiveSpectrum \ud835\udc9c)\nht' : IsClosed t'\nht : t \u2286 t'\n\u22a2 \u2203 s, t' = zeroLocus \ud835\udc9c s", "state_after": "no goals"}, {"tactic": "rw [(isClosed_zeroLocus _ _).closure_subset_iff]", "annotated_tactic": ["rw [(<a>isClosed_zeroLocus</a> _ _).<a>closure_subset_iff</a>]", [{"full_name": "ProjectiveSpectrum.isClosed_zeroLocus", "def_path": "Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean", "def_pos": [355, 9], "def_end_pos": [355, 27]}, {"full_name": "IsClosed.closure_subset_iff", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [465, 9], "def_end_pos": [465, 36]}]], "state_before": "case h\u2082\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\n\u22a2 closure t \u2286 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t)", "state_after": "case h\u2082\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\n\u22a2 t \u2286 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t)"}, {"tactic": "exact subset_zeroLocus_vanishingIdeal \ud835\udc9c t", "annotated_tactic": ["exact <a>subset_zeroLocus_vanishingIdeal</a> \ud835\udc9c t", [{"full_name": "ProjectiveSpectrum.subset_zeroLocus_vanishingIdeal", "def_path": "Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean", "def_pos": [172, 9], "def_end_pos": [172, 40]}]], "state_before": "case h\u2082\nR : Type u_1\nA : Type u_2\ninst\u271d\u00b3 : CommSemiring R\ninst\u271d\u00b2 : CommRing A\ninst\u271d\u00b9 : Algebra R A\n\ud835\udc9c : \u2115 \u2192 Submodule R A\ninst\u271d : GradedAlgebra \ud835\udc9c\nt : Set (ProjectiveSpectrum \ud835\udc9c)\n\u22a2 t \u2286 zeroLocus \ud835\udc9c \u2191(vanishingIdeal t)", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/RingTheory/Ideal/Norm.lean | Ideal.irreducible_of_irreducible_absNorm | [
408,
1
] | [
416,
58
] | [{"tactic": "simpa only [Ideal.isUnit_iff, Nat.isUnit_iff, absNorm_eq_one_iff] using h", "annotated_tactic": ["simpa only [<a>Ideal.isUnit_iff</a>, <a>Nat.isUnit_iff</a>, <a>absNorm_eq_one_iff</a>] using h", [{"full_name": "Ideal.isUnit_iff", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [1336, 9], "def_end_pos": [1336, 19]}, {"full_name": "Nat.isUnit_iff", "def_path": "Mathlib/Data/Nat/Units.lean", "def_pos": [25, 19], "def_end_pos": [25, 29]}, {"full_name": "Ideal.absNorm_eq_one_iff", "def_path": "Mathlib/RingTheory/Ideal/Norm.lean", "def_pos": [274, 9], "def_end_pos": [274, 27]}]], "state_before": "S : Type u_1\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : IsDomain S\ninst\u271d\u00b3 : Infinite S\ninst\u271d\u00b2 : IsDedekindDomain S\ninst\u271d\u00b9 : Module.Free \u2124 S\ninst\u271d : Module.Finite \u2124 S\nI : Ideal S\nhI : Irreducible (\u2191absNorm I)\nh : IsUnit I\n\u22a2 IsUnit (\u2191absNorm I)", "state_after": "no goals"}, {"tactic": "rintro a b rfl", "annotated_tactic": ["rintro a b rfl", []], "state_before": "S : Type u_1\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : IsDomain S\ninst\u271d\u00b3 : Infinite S\ninst\u271d\u00b2 : IsDedekindDomain S\ninst\u271d\u00b9 : Module.Free \u2124 S\ninst\u271d : Module.Finite \u2124 S\nI : Ideal S\nhI : Irreducible (\u2191absNorm I)\n\u22a2 \u2200 (a b : Ideal S), I = a * b \u2192 IsUnit a \u2228 IsUnit b", "state_after": "S : Type u_1\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : IsDomain S\ninst\u271d\u00b3 : Infinite S\ninst\u271d\u00b2 : IsDedekindDomain S\ninst\u271d\u00b9 : Module.Free \u2124 S\ninst\u271d : Module.Finite \u2124 S\na b : Ideal S\nhI : Irreducible (\u2191absNorm (a * b))\n\u22a2 IsUnit a \u2228 IsUnit b"}, {"tactic": "simpa only [Ideal.isUnit_iff, Nat.isUnit_iff, absNorm_eq_one_iff] using\n hI.isUnit_or_isUnit (_root_.map_mul absNorm a b)", "annotated_tactic": ["simpa only [<a>Ideal.isUnit_iff</a>, <a>Nat.isUnit_iff</a>, <a>absNorm_eq_one_iff</a>] using\n hI.isUnit_or_isUnit (<a>_root_.map_mul</a> <a>absNorm</a> a b)", [{"full_name": "Ideal.isUnit_iff", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [1336, 9], "def_end_pos": [1336, 19]}, {"full_name": "Nat.isUnit_iff", "def_path": "Mathlib/Data/Nat/Units.lean", "def_pos": [25, 19], "def_end_pos": [25, 29]}, {"full_name": "Ideal.absNorm_eq_one_iff", "def_path": "Mathlib/RingTheory/Ideal/Norm.lean", "def_pos": [274, 9], "def_end_pos": [274, 27]}, {"full_name": "map_mul", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [299, 9], "def_end_pos": [299, 16]}, {"full_name": "Ideal.absNorm", "def_path": "Mathlib/RingTheory/Ideal/Norm.lean", "def_pos": [250, 19], "def_end_pos": [250, 32]}]], "state_before": "S : Type u_1\ninst\u271d\u2075 : CommRing S\ninst\u271d\u2074 : IsDomain S\ninst\u271d\u00b3 : Infinite S\ninst\u271d\u00b2 : IsDedekindDomain S\ninst\u271d\u00b9 : Module.Free \u2124 S\ninst\u271d : Module.Finite \u2124 S\na b : Ideal S\nhI : Irreducible (\u2191absNorm (a * b))\n\u22a2 IsUnit a \u2228 IsUnit b", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Polynomial/Degree/TrailingDegree.lean | Polynomial.coeff_mul_natTrailingDegree_add_natTrailingDegree | [
382,
1
] | [
396,
52
] | [{"tactic": "rw [coeff_mul]", "annotated_tactic": ["rw [<a>coeff_mul</a>]", [{"full_name": "Polynomial.coeff_mul", "def_path": "Mathlib/Data/Polynomial/Coeff.lean", "def_pos": [116, 9], "def_end_pos": [116, 18]}]], "state_before": "R : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 coeff (p * q) (natTrailingDegree p + natTrailingDegree q) = trailingCoeff p * trailingCoeff q", "state_after": "R : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 \u2211 x in Nat.antidiagonal (natTrailingDegree p + natTrailingDegree q), coeff p x.1 * coeff q x.2 =\n trailingCoeff p * trailingCoeff q"}, {"tactic": "refine'\n Finset.sum_eq_single (p.natTrailingDegree, q.natTrailingDegree) _ fun h =>\n (h (Nat.mem_antidiagonal.mpr rfl)).elim", "annotated_tactic": ["refine'\n <a>Finset.sum_eq_single</a> (p.natTrailingDegree, q.natTrailingDegree) _ fun h =>\n (h (Nat.mem_antidiagonal.mpr <a>rfl</a>)).<a>elim</a>", [{"full_name": "Finset.sum_eq_single", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [798, 3], "def_end_pos": [798, 14]}, {"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "R : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 \u2211 x in Nat.antidiagonal (natTrailingDegree p + natTrailingDegree q), coeff p x.1 * coeff q x.2 =\n trailingCoeff p * trailingCoeff q", "state_after": "R : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 \u2200 (b : \u2115 \u00d7 \u2115),\n b \u2208 Nat.antidiagonal (natTrailingDegree p + natTrailingDegree q) \u2192\n b \u2260 (natTrailingDegree p, natTrailingDegree q) \u2192 coeff p b.1 * coeff q b.2 = 0"}, {"tactic": "rintro \u27e8i, j\u27e9 h\u2081 h\u2082", "annotated_tactic": ["rintro \u27e8i, j\u27e9 h\u2081 h\u2082", []], "state_before": "R : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\n\u22a2 \u2200 (b : \u2115 \u00d7 \u2115),\n b \u2208 Nat.antidiagonal (natTrailingDegree p + natTrailingDegree q) \u2192\n b \u2260 (natTrailingDegree p, natTrailingDegree q) \u2192 coeff p b.1 * coeff q b.2 = 0", "state_after": "case mk\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\ni j : \u2115\nh\u2081 : (i, j) \u2208 Nat.antidiagonal (natTrailingDegree p + natTrailingDegree q)\nh\u2082 : (i, j) \u2260 (natTrailingDegree p, natTrailingDegree q)\n\u22a2 coeff p (i, j).1 * coeff q (i, j).2 = 0"}, {"tactic": "rw [Nat.mem_antidiagonal] at h\u2081", "annotated_tactic": ["rw [<a>Nat.mem_antidiagonal</a>] at h\u2081", [{"full_name": "Finset.Nat.mem_antidiagonal", "def_path": "Mathlib/Data/Finset/NatAntidiagonal.lean", "def_pos": [37, 9], "def_end_pos": [37, 25]}]], "state_before": "case mk\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\ni j : \u2115\nh\u2081 : (i, j) \u2208 Nat.antidiagonal (natTrailingDegree p + natTrailingDegree q)\nh\u2082 : (i, j) \u2260 (natTrailingDegree p, natTrailingDegree q)\n\u22a2 coeff p (i, j).1 * coeff q (i, j).2 = 0", "state_after": "case mk\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\ni j : \u2115\nh\u2081 : (i, j).1 + (i, j).2 = natTrailingDegree p + natTrailingDegree q\nh\u2082 : (i, j) \u2260 (natTrailingDegree p, natTrailingDegree q)\n\u22a2 coeff p (i, j).1 * coeff q (i, j).2 = 0"}, {"tactic": "by_cases hi : i < p.natTrailingDegree", "annotated_tactic": ["by_cases hi : i < p.natTrailingDegree", []], "state_before": "case mk\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\ni j : \u2115\nh\u2081 : (i, j).1 + (i, j).2 = natTrailingDegree p + natTrailingDegree q\nh\u2082 : (i, j) \u2260 (natTrailingDegree p, natTrailingDegree q)\n\u22a2 coeff p (i, j).1 * coeff q (i, j).2 = 0", "state_after": "case pos\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\ni j : \u2115\nh\u2081 : (i, j).1 + (i, j).2 = natTrailingDegree p + natTrailingDegree q\nh\u2082 : (i, j) \u2260 (natTrailingDegree p, natTrailingDegree q)\nhi : i < natTrailingDegree p\n\u22a2 coeff p (i, j).1 * coeff q (i, j).2 = 0\n\ncase neg\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\ni j : \u2115\nh\u2081 : (i, j).1 + (i, j).2 = natTrailingDegree p + natTrailingDegree q\nh\u2082 : (i, j) \u2260 (natTrailingDegree p, natTrailingDegree q)\nhi : \u00aci < natTrailingDegree p\n\u22a2 coeff p (i, j).1 * coeff q (i, j).2 = 0"}, {"tactic": "by_cases hj : j < q.natTrailingDegree", "annotated_tactic": ["by_cases hj : j < q.natTrailingDegree", []], "state_before": "case neg\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\ni j : \u2115\nh\u2081 : (i, j).1 + (i, j).2 = natTrailingDegree p + natTrailingDegree q\nh\u2082 : (i, j) \u2260 (natTrailingDegree p, natTrailingDegree q)\nhi : \u00aci < natTrailingDegree p\n\u22a2 coeff p (i, j).1 * coeff q (i, j).2 = 0", "state_after": "case pos\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\ni j : \u2115\nh\u2081 : (i, j).1 + (i, j).2 = natTrailingDegree p + natTrailingDegree q\nh\u2082 : (i, j) \u2260 (natTrailingDegree p, natTrailingDegree q)\nhi : \u00aci < natTrailingDegree p\nhj : j < natTrailingDegree q\n\u22a2 coeff p (i, j).1 * coeff q (i, j).2 = 0\n\ncase neg\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\ni j : \u2115\nh\u2081 : (i, j).1 + (i, j).2 = natTrailingDegree p + natTrailingDegree q\nh\u2082 : (i, j) \u2260 (natTrailingDegree p, natTrailingDegree q)\nhi : \u00aci < natTrailingDegree p\nhj : \u00acj < natTrailingDegree q\n\u22a2 coeff p (i, j).1 * coeff q (i, j).2 = 0"}, {"tactic": "rw [not_lt] at hi hj", "annotated_tactic": ["rw [<a>not_lt</a>] at hi hj", [{"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}]], "state_before": "case neg\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\ni j : \u2115\nh\u2081 : (i, j).1 + (i, j).2 = natTrailingDegree p + natTrailingDegree q\nh\u2082 : (i, j) \u2260 (natTrailingDegree p, natTrailingDegree q)\nhi : \u00aci < natTrailingDegree p\nhj : \u00acj < natTrailingDegree q\n\u22a2 coeff p (i, j).1 * coeff q (i, j).2 = 0", "state_after": "case neg\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\ni j : \u2115\nh\u2081 : (i, j).1 + (i, j).2 = natTrailingDegree p + natTrailingDegree q\nh\u2082 : (i, j) \u2260 (natTrailingDegree p, natTrailingDegree q)\nhi : natTrailingDegree p \u2264 i\nhj : natTrailingDegree q \u2264 j\n\u22a2 coeff p (i, j).1 * coeff q (i, j).2 = 0"}, {"tactic": "refine' (h\u2082 (Prod.ext_iff.mpr _).symm).elim", "annotated_tactic": ["refine' (h\u2082 (Prod.ext_iff.mpr _).<a>symm</a>).<a>elim</a>", [{"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case neg\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\ni j : \u2115\nh\u2081 : (i, j).1 + (i, j).2 = natTrailingDegree p + natTrailingDegree q\nh\u2082 : (i, j) \u2260 (natTrailingDegree p, natTrailingDegree q)\nhi : natTrailingDegree p \u2264 i\nhj : natTrailingDegree q \u2264 j\n\u22a2 coeff p (i, j).1 * coeff q (i, j).2 = 0", "state_after": "case neg\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\ni j : \u2115\nh\u2081 : (i, j).1 + (i, j).2 = natTrailingDegree p + natTrailingDegree q\nh\u2082 : (i, j) \u2260 (natTrailingDegree p, natTrailingDegree q)\nhi : natTrailingDegree p \u2264 i\nhj : natTrailingDegree q \u2264 j\n\u22a2 (natTrailingDegree p, natTrailingDegree q).1 = (i, j).1 \u2227 (natTrailingDegree p, natTrailingDegree q).2 = (i, j).2"}, {"tactic": "exact (add_eq_add_iff_eq_and_eq hi hj).mp h\u2081.symm", "annotated_tactic": ["exact (<a>add_eq_add_iff_eq_and_eq</a> hi hj).<a>mp</a> h\u2081.symm", [{"full_name": "add_eq_add_iff_eq_and_eq", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [319, 3], "def_end_pos": [319, 14]}, {"full_name": "Iff.mp", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [90, 3], "def_end_pos": [90, 5]}]], "state_before": "case neg\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\ni j : \u2115\nh\u2081 : (i, j).1 + (i, j).2 = natTrailingDegree p + natTrailingDegree q\nh\u2082 : (i, j) \u2260 (natTrailingDegree p, natTrailingDegree q)\nhi : natTrailingDegree p \u2264 i\nhj : natTrailingDegree q \u2264 j\n\u22a2 (natTrailingDegree p, natTrailingDegree q).1 = (i, j).1 \u2227 (natTrailingDegree p, natTrailingDegree q).2 = (i, j).2", "state_after": "no goals"}, {"tactic": "rw [coeff_eq_zero_of_lt_natTrailingDegree hi, zero_mul]", "annotated_tactic": ["rw [<a>coeff_eq_zero_of_lt_natTrailingDegree</a> hi, <a>zero_mul</a>]", [{"full_name": "Polynomial.coeff_eq_zero_of_lt_natTrailingDegree", "def_path": "Mathlib/Data/Polynomial/Degree/TrailingDegree.lean", "def_pos": [268, 9], "def_end_pos": [268, 46]}, {"full_name": "MulZeroClass.zero_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [36, 3], "def_end_pos": [36, 11]}]], "state_before": "case pos\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\ni j : \u2115\nh\u2081 : (i, j).1 + (i, j).2 = natTrailingDegree p + natTrailingDegree q\nh\u2082 : (i, j) \u2260 (natTrailingDegree p, natTrailingDegree q)\nhi : i < natTrailingDegree p\n\u22a2 coeff p (i, j).1 * coeff q (i, j).2 = 0", "state_after": "no goals"}, {"tactic": "rw [coeff_eq_zero_of_lt_natTrailingDegree hj, mul_zero]", "annotated_tactic": ["rw [<a>coeff_eq_zero_of_lt_natTrailingDegree</a> hj, <a>mul_zero</a>]", [{"full_name": "Polynomial.coeff_eq_zero_of_lt_natTrailingDegree", "def_path": "Mathlib/Data/Polynomial/Degree/TrailingDegree.lean", "def_pos": [268, 9], "def_end_pos": [268, 46]}, {"full_name": "MulZeroClass.mul_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Defs.lean", "def_pos": [38, 3], "def_end_pos": [38, 11]}]], "state_before": "case pos\nR : Type u\nS : Type v\na b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\ni j : \u2115\nh\u2081 : (i, j).1 + (i, j).2 = natTrailingDegree p + natTrailingDegree q\nh\u2082 : (i, j) \u2260 (natTrailingDegree p, natTrailingDegree q)\nhi : \u00aci < natTrailingDegree p\nhj : j < natTrailingDegree q\n\u22a2 coeff p (i, j).1 * coeff q (i, j).2 = 0", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | RightDerivMeasurableAux.le_of_mem_A | [
522,
1
] | [
527,
78
] | [{"tactic": "rcases hx with \u27e8r', r'mem, hr'\u27e9", "annotated_tactic": ["rcases hx with \u27e8r', r'mem, hr'\u27e9", []], "state_before": "F : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nr \u03b5 : \u211d\nL : F\nx : \u211d\nhx : x \u2208 A f L r \u03b5\ny z : \u211d\nhy : y \u2208 Icc x (x + r / 2)\nhz : z \u2208 Icc x (x + r / 2)\n\u22a2 \u2016f z - f y - (z - y) \u2022 L\u2016 \u2264 \u03b5 * r", "state_after": "case intro.intro\nF : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nr \u03b5 : \u211d\nL : F\nx y z : \u211d\nhy : y \u2208 Icc x (x + r / 2)\nhz : z \u2208 Icc x (x + r / 2)\nr' : \u211d\nr'mem : r' \u2208 Ioc (r / 2) r\nhr' : \u2200 (y : \u211d), y \u2208 Icc x (x + r') \u2192 \u2200 (z : \u211d), z \u2208 Icc x (x + r') \u2192 \u2016f z - f y - (z - y) \u2022 L\u2016 \u2264 \u03b5 * r\n\u22a2 \u2016f z - f y - (z - y) \u2022 L\u2016 \u2264 \u03b5 * r"}, {"tactic": "have A : x + r / 2 \u2264 x + r' := by linarith [r'mem.1]", "annotated_tactic": ["have A : x + r / 2 \u2264 x + r' := by linarith [r'mem.1]", []], "state_before": "case intro.intro\nF : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nr \u03b5 : \u211d\nL : F\nx y z : \u211d\nhy : y \u2208 Icc x (x + r / 2)\nhz : z \u2208 Icc x (x + r / 2)\nr' : \u211d\nr'mem : r' \u2208 Ioc (r / 2) r\nhr' : \u2200 (y : \u211d), y \u2208 Icc x (x + r') \u2192 \u2200 (z : \u211d), z \u2208 Icc x (x + r') \u2192 \u2016f z - f y - (z - y) \u2022 L\u2016 \u2264 \u03b5 * r\n\u22a2 \u2016f z - f y - (z - y) \u2022 L\u2016 \u2264 \u03b5 * r", "state_after": "case intro.intro\nF : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nr \u03b5 : \u211d\nL : F\nx y z : \u211d\nhy : y \u2208 Icc x (x + r / 2)\nhz : z \u2208 Icc x (x + r / 2)\nr' : \u211d\nr'mem : r' \u2208 Ioc (r / 2) r\nhr' : \u2200 (y : \u211d), y \u2208 Icc x (x + r') \u2192 \u2200 (z : \u211d), z \u2208 Icc x (x + r') \u2192 \u2016f z - f y - (z - y) \u2022 L\u2016 \u2264 \u03b5 * r\nA : x + r / 2 \u2264 x + r'\n\u22a2 \u2016f z - f y - (z - y) \u2022 L\u2016 \u2264 \u03b5 * r"}, {"tactic": "exact hr' _ ((Icc_subset_Icc le_rfl A) hy) _ ((Icc_subset_Icc le_rfl A) hz)", "annotated_tactic": ["exact hr' _ ((<a>Icc_subset_Icc</a> <a>le_rfl</a> A) hy) _ ((<a>Icc_subset_Icc</a> <a>le_rfl</a> A) hz)", [{"full_name": "Set.Icc_subset_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [455, 9], "def_end_pos": [455, 23]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}, {"full_name": "Set.Icc_subset_Icc", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [455, 9], "def_end_pos": [455, 23]}, {"full_name": "le_rfl", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [170, 9], "def_end_pos": [170, 15]}]], "state_before": "case intro.intro\nF : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nr \u03b5 : \u211d\nL : F\nx y z : \u211d\nhy : y \u2208 Icc x (x + r / 2)\nhz : z \u2208 Icc x (x + r / 2)\nr' : \u211d\nr'mem : r' \u2208 Ioc (r / 2) r\nhr' : \u2200 (y : \u211d), y \u2208 Icc x (x + r') \u2192 \u2200 (z : \u211d), z \u2208 Icc x (x + r') \u2192 \u2016f z - f y - (z - y) \u2022 L\u2016 \u2264 \u03b5 * r\nA : x + r / 2 \u2264 x + r'\n\u22a2 \u2016f z - f y - (z - y) \u2022 L\u2016 \u2264 \u03b5 * r", "state_after": "no goals"}, {"tactic": "linarith [r'mem.1]", "annotated_tactic": ["linarith [r'mem.1]", []], "state_before": "F : Type u_1\ninst\u271d\u00b9 : NormedAddCommGroup F\ninst\u271d : NormedSpace \u211d F\nf : \u211d \u2192 F\nK : Set F\nr \u03b5 : \u211d\nL : F\nx y z : \u211d\nhy : y \u2208 Icc x (x + r / 2)\nhz : z \u2208 Icc x (x + r / 2)\nr' : \u211d\nr'mem : r' \u2208 Ioc (r / 2) r\nhr' : \u2200 (y : \u211d), y \u2208 Icc x (x + r') \u2192 \u2200 (z : \u211d), z \u2208 Icc x (x + r') \u2192 \u2016f z - f y - (z - y) \u2022 L\u2016 \u2264 \u03b5 * r\n\u22a2 x + r / 2 \u2264 x + r'", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/AlgebraicGeometry/StructureSheaf.lean | AlgebraicGeometry.StructureSheaf.stalkAlgebra_map | [
938,
1
] | [
940,
6
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/RingTheory/Valuation/ValuationSubring.lean | Valuation.mem_valuationSubring_iff | [
437,
1
] | [
437,
90
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Geometry/Manifold/MFDeriv.lean | tangentMapWithin_comp_at | [
1028,
1
] | [
1035,
6
] | [{"tactic": "simp only [tangentMapWithin, mfld_simps]", "annotated_tactic": ["simp only [<a>tangentMapWithin</a>, mfld_simps]", [{"full_name": "tangentMapWithin", "def_path": "Mathlib/Geometry/Manifold/MFDeriv.lean", "def_pos": [304, 5], "def_end_pos": [304, 21]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\np : TangentBundle I M\nhg : MDifferentiableWithinAt I' I'' g u (f p.proj)\nhf : MDifferentiableWithinAt I I' f s p.proj\nh : s \u2286 f \u207b\u00b9' u\nhps : UniqueMDiffWithinAt I s p.proj\n\u22a2 tangentMapWithin I I'' (g \u2218 f) s p = tangentMapWithin I' I'' g u (tangentMapWithin I I' f s p)", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\np : TangentBundle I M\nhg : MDifferentiableWithinAt I' I'' g u (f p.proj)\nhf : MDifferentiableWithinAt I I' f s p.proj\nh : s \u2286 f \u207b\u00b9' u\nhps : UniqueMDiffWithinAt I s p.proj\n\u22a2 \u2191(mfderivWithin I I'' (g \u2218 f) s p.proj) p.snd =\n \u2191(mfderivWithin I' I'' g u (f p.proj)) (\u2191(mfderivWithin I I' f s p.proj) p.snd)"}, {"tactic": "rw [mfderivWithin_comp p.1 hg hf h hps]", "annotated_tactic": ["rw [<a>mfderivWithin_comp</a> p.1 hg hf h hps]", [{"full_name": "mfderivWithin_comp", "def_path": "Mathlib/Geometry/Manifold/MFDeriv.lean", "def_pos": [999, 9], "def_end_pos": [999, 27]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\np : TangentBundle I M\nhg : MDifferentiableWithinAt I' I'' g u (f p.proj)\nhf : MDifferentiableWithinAt I I' f s p.proj\nh : s \u2286 f \u207b\u00b9' u\nhps : UniqueMDiffWithinAt I s p.proj\n\u22a2 \u2191(mfderivWithin I I'' (g \u2218 f) s p.proj) p.snd =\n \u2191(mfderivWithin I' I'' g u (f p.proj)) (\u2191(mfderivWithin I I' f s p.proj) p.snd)", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\np : TangentBundle I M\nhg : MDifferentiableWithinAt I' I'' g u (f p.proj)\nhf : MDifferentiableWithinAt I I' f s p.proj\nh : s \u2286 f \u207b\u00b9' u\nhps : UniqueMDiffWithinAt I s p.proj\n\u22a2 \u2191(ContinuousLinearMap.comp (mfderivWithin I' I'' g u (f p.proj)) (mfderivWithin I I' f s p.proj)) p.snd =\n \u2191(mfderivWithin I' I'' g u (f p.proj)) (\u2191(mfderivWithin I I' f s p.proj) p.snd)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c\nE : Type u_2\ninst\u271d\u00b9\u2074 : NormedAddCommGroup E\ninst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E\nH : Type u_3\ninst\u271d\u00b9\u00b2 : TopologicalSpace H\nI : ModelWithCorners \ud835\udd5c E H\nM : Type u_4\ninst\u271d\u00b9\u00b9 : TopologicalSpace M\ninst\u271d\u00b9\u2070 : ChartedSpace H M\nE' : Type u_5\ninst\u271d\u2079 : NormedAddCommGroup E'\ninst\u271d\u2078 : NormedSpace \ud835\udd5c E'\nH' : Type u_6\ninst\u271d\u2077 : TopologicalSpace H'\nI' : ModelWithCorners \ud835\udd5c E' H'\nM' : Type u_7\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : ChartedSpace H' M'\nE'' : Type u_8\ninst\u271d\u2074 : NormedAddCommGroup E''\ninst\u271d\u00b3 : NormedSpace \ud835\udd5c E''\nH'' : Type u_9\ninst\u271d\u00b2 : TopologicalSpace H''\nI'' : ModelWithCorners \ud835\udd5c E'' H''\nM'' : Type u_10\ninst\u271d\u00b9 : TopologicalSpace M''\ninst\u271d : ChartedSpace H'' M''\nf f\u2080 f\u2081 : M \u2192 M'\nx : M\ns t : Set M\ng : M' \u2192 M''\nu : Set M'\nIs : SmoothManifoldWithCorners I M\nI's : SmoothManifoldWithCorners I' M'\nI''s : SmoothManifoldWithCorners I'' M''\nf' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x)\ng' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x))\np : TangentBundle I M\nhg : MDifferentiableWithinAt I' I'' g u (f p.proj)\nhf : MDifferentiableWithinAt I I' f s p.proj\nh : s \u2286 f \u207b\u00b9' u\nhps : UniqueMDiffWithinAt I s p.proj\n\u22a2 \u2191(ContinuousLinearMap.comp (mfderivWithin I' I'' g u (f p.proj)) (mfderivWithin I I' f s p.proj)) p.snd =\n \u2191(mfderivWithin I' I'' g u (f p.proj)) (\u2191(mfderivWithin I I' f s p.proj) p.snd)", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Set/Prod.lean | Set.mem_pi | [
670,
1
] | [
671,
10
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Set/Sigma.lean | Set.sigma_univ_range_eq | [
197,
1
] | [
199,
25
] | [{"tactic": "simp [range]", "annotated_tactic": ["simp [<a>range</a>]", [{"full_name": "Set.range", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [668, 5], "def_end_pos": [668, 10]}]], "state_before": "\u03b9 : Type u_1\n\u03b9' : Type u_2\n\u03b1 : \u03b9 \u2192 Type u_3\n\u03b2 : \u03b9 \u2192 Type u_4\ns s\u2081 s\u2082 : Set \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i)\nu : Set ((i : \u03b9) \u00d7 \u03b1 i)\nx : (i : \u03b9) \u00d7 \u03b1 i\ni j : \u03b9\na : \u03b1 i\nf : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 i\n\u22a2 \u2200 (x : (i : \u03b9) \u00d7 \u03b2 i),\n (x \u2208 Set.Sigma univ fun i => range (f i)) \u2194 x \u2208 range fun x => { fst := x.fst, snd := f x.fst x.snd }", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | span_gramSchmidtNormed_range | [
329,
1
] | [
331,
67
] | [{"tactic": "simpa only [image_univ.symm] using span_gramSchmidtNormed f univ", "annotated_tactic": ["simpa only [image_univ.symm] using <a>span_gramSchmidtNormed</a> f <a>univ</a>", [{"full_name": "span_gramSchmidtNormed", "def_path": "Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean", "def_pos": [316, 9], "def_end_pos": [316, 31]}, {"full_name": "Set.univ", "def_path": "Mathlib/Init/Set.lean", "def_pos": [90, 5], "def_end_pos": [90, 9]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\ninst\u271d\u2075 : IsROrC \ud835\udd5c\ninst\u271d\u2074 : NormedAddCommGroup E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E\n\u03b9 : Type u_3\ninst\u271d\u00b2 : LinearOrder \u03b9\ninst\u271d\u00b9 : LocallyFiniteOrderBot \u03b9\ninst\u271d : IsWellOrder \u03b9 fun x x_1 => x < x_1\nf : \u03b9 \u2192 E\n\u22a2 span \ud835\udd5c (Set.range (gramSchmidtNormed \ud835\udd5c f)) = span \ud835\udd5c (Set.range (gramSchmidt \ud835\udd5c f))", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Int/ModEq.lean | Dvd.dvd.zero_modEq_int | [
90,
1
] | [
91,
24
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/GroupTheory/Subgroup/Finite.lean | Subgroup.eq_bot_of_card_eq | [
162,
1
] | [
163,
35
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Algebra/Group/WithOne/Defs.lean | WithOne.one_ne_coe | [
151,
1
] | [
152,
18
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/CategoryTheory/Iso.lean | CategoryTheory.Iso.trans_assoc | [
181,
1
] | [
183,
45
] | [{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX Y Z Z' : C\n\u03b1 : X \u2245 Y\n\u03b2 : Y \u2245 Z\n\u03b3 : Z \u2245 Z'\n\u22a2 (\u03b1 \u226a\u226b \u03b2) \u226a\u226b \u03b3 = \u03b1 \u226a\u226b \u03b2 \u226a\u226b \u03b3", "state_after": "case w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z Z' : C\n\u03b1 : X \u2245 Y\n\u03b2 : Y \u2245 Z\n\u03b3 : Z \u2245 Z'\n\u22a2 ((\u03b1 \u226a\u226b \u03b2) \u226a\u226b \u03b3).hom = (\u03b1 \u226a\u226b \u03b2 \u226a\u226b \u03b3).hom"}, {"tactic": "simp only [trans_hom, Category.assoc]", "annotated_tactic": ["simp only [<a>trans_hom</a>, <a>Category.assoc</a>]", [{"full_name": "CategoryTheory.Iso.trans_hom", "def_path": "Mathlib/CategoryTheory/Iso.lean", "def_pos": [152, 10], "def_end_pos": [152, 15]}, {"full_name": "CategoryTheory.Category.assoc", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [159, 3], "def_end_pos": [159, 8]}]], "state_before": "case w\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y Z Z' : C\n\u03b1 : X \u2245 Y\n\u03b2 : Y \u2245 Z\n\u03b3 : Z \u2245 Z'\n\u22a2 ((\u03b1 \u226a\u226b \u03b2) \u226a\u226b \u03b3).hom = (\u03b1 \u226a\u226b \u03b2 \u226a\u226b \u03b3).hom", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Matrix/Kronecker.lean | Matrix.det_kroneckerTMul | [
562,
1
] | [
570,
36
] | [{"tactic": "refine' (det_kroneckerMapBilinear (TensorProduct.mk R \u03b1 \u03b2) tmul_mul_tmul _ _).trans _", "annotated_tactic": ["refine' (<a>det_kroneckerMapBilinear</a> (<a>TensorProduct.mk</a> R \u03b1 \u03b2) <a>tmul_mul_tmul</a> _ _).<a>trans</a> _", [{"full_name": "Matrix.det_kroneckerMapBilinear", "def_path": "Mathlib/Data/Matrix/Kronecker.lean", "def_pos": [236, 9], "def_end_pos": [236, 33]}, {"full_name": "TensorProduct.mk", "def_path": "Mathlib/LinearAlgebra/TensorProduct.lean", "def_pos": [387, 5], "def_end_pos": [387, 7]}, {"full_name": "Algebra.TensorProduct.tmul_mul_tmul", "def_path": "Mathlib/RingTheory/TensorProduct.lean", "def_pos": [233, 9], "def_end_pos": [233, 22]}, {"full_name": "Eq.trans", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [322, 9], "def_end_pos": [322, 17]}]], "state_before": "R : Type u_1\n\u03b1 : Type u_2\n\u03b1' : Type u_3\n\u03b2 : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\nl : Type u_8\nm : Type u_9\nn : Type u_10\np : Type u_11\nq : Type u_12\nr : Type u_13\nl' : Type u_14\nm' : Type u_15\nn' : Type u_16\np' : Type u_17\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing \u03b1\ninst\u271d\u2076 : CommRing \u03b2\ninst\u271d\u2075 : Algebra R \u03b1\ninst\u271d\u2074 : Algebra R \u03b2\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nA : Matrix m m \u03b1\nB : Matrix n n \u03b2\n\u22a2 det (kroneckerMap (tmul R) A B) = (det A ^ Fintype.card n) \u2297\u209c[R] (det B ^ Fintype.card m)", "state_after": "R : Type u_1\n\u03b1 : Type u_2\n\u03b1' : Type u_3\n\u03b2 : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\nl : Type u_8\nm : Type u_9\nn : Type u_10\np : Type u_11\nq : Type u_12\nr : Type u_13\nl' : Type u_14\nm' : Type u_15\nn' : Type u_16\np' : Type u_17\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing \u03b1\ninst\u271d\u2076 : CommRing \u03b2\ninst\u271d\u2075 : Algebra R \u03b1\ninst\u271d\u2074 : Algebra R \u03b2\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nA : Matrix m m \u03b1\nB : Matrix n n \u03b2\n\u22a2 det (map A fun a => \u2191(\u2191(mk R \u03b1 \u03b2) a) 1) ^ Fintype.card n * det (map B fun b => \u2191(\u2191(mk R \u03b1 \u03b2) 1) b) ^ Fintype.card m =\n (det A ^ Fintype.card n) \u2297\u209c[R] (det B ^ Fintype.card m)"}, {"tactic": "simp (config := { eta := false }) only [mk_apply, \u2190 includeLeft_apply (S := R),\n \u2190 includeRight_apply]", "annotated_tactic": ["simp (config := { eta := <a>false</a> }) only [<a>mk_apply</a>, \u2190 <a>includeLeft_apply</a> (S := R),\n \u2190 <a>includeRight_apply</a>]", [{"full_name": "Bool.false", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [547, 5], "def_end_pos": [547, 10]}, {"full_name": "TensorProduct.mk_apply", "def_path": "Mathlib/LinearAlgebra/TensorProduct.lean", "def_pos": [395, 9], "def_end_pos": [395, 17]}, {"full_name": "Algebra.TensorProduct.includeLeft_apply", "def_path": "Mathlib/RingTheory/TensorProduct.lean", "def_pos": [399, 9], "def_end_pos": [399, 26]}, {"full_name": "Algebra.TensorProduct.includeRight_apply", "def_path": "Mathlib/RingTheory/TensorProduct.lean", "def_pos": [415, 9], "def_end_pos": [415, 27]}]], "state_before": "R : Type u_1\n\u03b1 : Type u_2\n\u03b1' : Type u_3\n\u03b2 : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\nl : Type u_8\nm : Type u_9\nn : Type u_10\np : Type u_11\nq : Type u_12\nr : Type u_13\nl' : Type u_14\nm' : Type u_15\nn' : Type u_16\np' : Type u_17\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing \u03b1\ninst\u271d\u2076 : CommRing \u03b2\ninst\u271d\u2075 : Algebra R \u03b1\ninst\u271d\u2074 : Algebra R \u03b2\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nA : Matrix m m \u03b1\nB : Matrix n n \u03b2\n\u22a2 det (map A fun a => \u2191(\u2191(mk R \u03b1 \u03b2) a) 1) ^ Fintype.card n * det (map B fun b => \u2191(\u2191(mk R \u03b1 \u03b2) 1) b) ^ Fintype.card m =\n (det A ^ Fintype.card n) \u2297\u209c[R] (det B ^ Fintype.card m)", "state_after": "R : Type u_1\n\u03b1 : Type u_2\n\u03b1' : Type u_3\n\u03b2 : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\nl : Type u_8\nm : Type u_9\nn : Type u_10\np : Type u_11\nq : Type u_12\nr : Type u_13\nl' : Type u_14\nm' : Type u_15\nn' : Type u_16\np' : Type u_17\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing \u03b1\ninst\u271d\u2076 : CommRing \u03b2\ninst\u271d\u2075 : Algebra R \u03b1\ninst\u271d\u2074 : Algebra R \u03b2\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nA : Matrix m m \u03b1\nB : Matrix n n \u03b2\n\u22a2 det (map A fun a => \u2191includeLeft a) ^ Fintype.card n * det (map B fun b => \u2191includeRight b) ^ Fintype.card m =\n (det A ^ Fintype.card n) \u2297\u209c[R] (det B ^ Fintype.card m)"}, {"tactic": "simp only [\u2190 AlgHom.mapMatrix_apply, \u2190 AlgHom.map_det]", "annotated_tactic": ["simp only [\u2190 <a>AlgHom.mapMatrix_apply</a>, \u2190 <a>AlgHom.map_det</a>]", [{"full_name": "AlgHom.mapMatrix_apply", "def_path": "Mathlib/Data/Matrix/Basic.lean", "def_pos": [1586, 3], "def_end_pos": [1586, 8]}, {"full_name": "AlgHom.map_det", "def_path": "Mathlib/LinearAlgebra/Matrix/Determinant.lean", "def_pos": [337, 9], "def_end_pos": [337, 30]}]], "state_before": "R : Type u_1\n\u03b1 : Type u_2\n\u03b1' : Type u_3\n\u03b2 : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\nl : Type u_8\nm : Type u_9\nn : Type u_10\np : Type u_11\nq : Type u_12\nr : Type u_13\nl' : Type u_14\nm' : Type u_15\nn' : Type u_16\np' : Type u_17\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing \u03b1\ninst\u271d\u2076 : CommRing \u03b2\ninst\u271d\u2075 : Algebra R \u03b1\ninst\u271d\u2074 : Algebra R \u03b2\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nA : Matrix m m \u03b1\nB : Matrix n n \u03b2\n\u22a2 det (map A fun a => \u2191includeLeft a) ^ Fintype.card n * det (map B fun b => \u2191includeRight b) ^ Fintype.card m =\n (det A ^ Fintype.card n) \u2297\u209c[R] (det B ^ Fintype.card m)", "state_after": "R : Type u_1\n\u03b1 : Type u_2\n\u03b1' : Type u_3\n\u03b2 : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\nl : Type u_8\nm : Type u_9\nn : Type u_10\np : Type u_11\nq : Type u_12\nr : Type u_13\nl' : Type u_14\nm' : Type u_15\nn' : Type u_16\np' : Type u_17\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing \u03b1\ninst\u271d\u2076 : CommRing \u03b2\ninst\u271d\u2075 : Algebra R \u03b1\ninst\u271d\u2074 : Algebra R \u03b2\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nA : Matrix m m \u03b1\nB : Matrix n n \u03b2\n\u22a2 \u2191includeLeft (det A) ^ Fintype.card n * \u2191includeRight (det B) ^ Fintype.card m =\n (det A ^ Fintype.card n) \u2297\u209c[R] (det B ^ Fintype.card m)"}, {"tactic": "simp only [includeLeft_apply, includeRight_apply, tmul_pow, tmul_mul_tmul, one_pow,\n _root_.mul_one, _root_.one_mul]", "annotated_tactic": ["simp only [<a>includeLeft_apply</a>, <a>includeRight_apply</a>, <a>tmul_pow</a>, <a>tmul_mul_tmul</a>, <a>one_pow</a>,\n <a>_root_.mul_one</a>, <a>_root_.one_mul</a>]", [{"full_name": "Algebra.TensorProduct.includeLeft_apply", "def_path": "Mathlib/RingTheory/TensorProduct.lean", "def_pos": [399, 9], "def_end_pos": [399, 26]}, {"full_name": "Algebra.TensorProduct.includeRight_apply", "def_path": "Mathlib/RingTheory/TensorProduct.lean", "def_pos": [415, 9], "def_end_pos": [415, 27]}, {"full_name": "Algebra.TensorProduct.tmul_pow", "def_path": "Mathlib/RingTheory/TensorProduct.lean", "def_pos": [347, 9], "def_end_pos": [347, 17]}, {"full_name": "Algebra.TensorProduct.tmul_mul_tmul", "def_path": "Mathlib/RingTheory/TensorProduct.lean", "def_pos": [233, 9], "def_end_pos": [233, 22]}, {"full_name": "one_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [90, 9], "def_end_pos": [90, 16]}, {"full_name": "mul_one", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [470, 9], "def_end_pos": [470, 16]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}]], "state_before": "R : Type u_1\n\u03b1 : Type u_2\n\u03b1' : Type u_3\n\u03b2 : Type u_4\n\u03b2' : Type u_5\n\u03b3 : Type u_6\n\u03b3' : Type u_7\nl : Type u_8\nm : Type u_9\nn : Type u_10\np : Type u_11\nq : Type u_12\nr : Type u_13\nl' : Type u_14\nm' : Type u_15\nn' : Type u_16\np' : Type u_17\ninst\u271d\u2078 : CommRing R\ninst\u271d\u2077 : CommRing \u03b1\ninst\u271d\u2076 : CommRing \u03b2\ninst\u271d\u2075 : Algebra R \u03b1\ninst\u271d\u2074 : Algebra R \u03b2\ninst\u271d\u00b3 : Fintype m\ninst\u271d\u00b2 : Fintype n\ninst\u271d\u00b9 : DecidableEq m\ninst\u271d : DecidableEq n\nA : Matrix m m \u03b1\nB : Matrix n n \u03b2\n\u22a2 \u2191includeLeft (det A) ^ Fintype.card n * \u2191includeRight (det B) ^ Fintype.card m =\n (det A ^ Fintype.card n) \u2297\u209c[R] (det B ^ Fintype.card m)", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Analysis/Normed/Group/SemiNormedGroupCat/Kernels.lean | SemiNormedGroupCat.explicitCokernelIso_hom_desc | [
394,
1
] | [
399,
40
] | [{"tactic": "ext1", "annotated_tactic": ["ext1", []], "state_before": "X Y Z : SemiNormedGroupCat\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 : SemiNormedGroupCat\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": "SemiNormedGroupCat.explicitCokernelDesc", "def_path": "Mathlib/Analysis/Normed/Group/SemiNormedGroupCat/Kernels.lean", "def_pos": [222, 5], "def_end_pos": [222, 25]}, {"full_name": "SemiNormedGroupCat.explicitCokernel\u03c0", "def_path": "Mathlib/Analysis/Normed/Group/SemiNormedGroupCat/Kernels.lean", "def_pos": [230, 5], "def_end_pos": [230, 22]}, {"full_name": "SemiNormedGroupCat.explicitCokernelIso", "def_path": "Mathlib/Analysis/Normed/Group/SemiNormedGroupCat/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": [652, 5], "def_end_pos": [652, 29]}]], "state_before": "case h\nX Y Z : SemiNormedGroupCat\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"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Polynomial/EraseLead.lean | Polynomial.eraseLead_add_of_natDegree_lt_right | [
170,
1
] | [
178,
57
] | [{"tactic": "ext n", "annotated_tactic": ["ext n", []], "state_before": "R : Type u_1\ninst\u271d : Semiring R\nf p q : R[X]\npq : natDegree p < natDegree q\n\u22a2 eraseLead (p + q) = p + eraseLead q", "state_after": "case a\nR : Type u_1\ninst\u271d : Semiring R\nf p q : R[X]\npq : natDegree p < natDegree q\nn : \u2115\n\u22a2 coeff (eraseLead (p + q)) n = coeff (p + eraseLead q) n"}, {"tactic": "by_cases nd : n = q.natDegree", "annotated_tactic": ["by_cases nd : n = q.natDegree", []], "state_before": "case a\nR : Type u_1\ninst\u271d : Semiring R\nf p q : R[X]\npq : natDegree p < natDegree q\nn : \u2115\n\u22a2 coeff (eraseLead (p + q)) n = coeff (p + eraseLead q) n", "state_after": "case pos\nR : Type u_1\ninst\u271d : Semiring R\nf p q : R[X]\npq : natDegree p < natDegree q\nn : \u2115\nnd : n = natDegree q\n\u22a2 coeff (eraseLead (p + q)) n = coeff (p + eraseLead q) n\n\ncase neg\nR : Type u_1\ninst\u271d : Semiring R\nf p q : R[X]\npq : natDegree p < natDegree q\nn : \u2115\nnd : \u00acn = natDegree q\n\u22a2 coeff (eraseLead (p + q)) n = coeff (p + eraseLead q) n"}, {"tactic": "rw [nd, eraseLead_coeff, if_pos (natDegree_add_eq_right_of_natDegree_lt pq).symm]", "annotated_tactic": ["rw [nd, <a>eraseLead_coeff</a>, <a>if_pos</a> (<a>natDegree_add_eq_right_of_natDegree_lt</a> pq).<a>symm</a>]", [{"full_name": "Polynomial.eraseLead_coeff", "def_path": "Mathlib/Data/Polynomial/EraseLead.lean", "def_pos": [46, 9], "def_end_pos": [46, 24]}, {"full_name": "if_pos", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [790, 9], "def_end_pos": [790, 15]}, {"full_name": "Polynomial.natDegree_add_eq_right_of_natDegree_lt", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [715, 9], "def_end_pos": [715, 47]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case pos\nR : Type u_1\ninst\u271d : Semiring R\nf p q : R[X]\npq : natDegree p < natDegree q\nn : \u2115\nnd : n = natDegree q\n\u22a2 coeff (eraseLead (p + q)) n = coeff (p + eraseLead q) n", "state_after": "case pos\nR : Type u_1\ninst\u271d : Semiring R\nf p q : R[X]\npq : natDegree p < natDegree q\nn : \u2115\nnd : n = natDegree q\n\u22a2 0 = coeff (p + eraseLead q) (natDegree q)"}, {"tactic": "simpa using (coeff_eq_zero_of_natDegree_lt pq).symm", "annotated_tactic": ["simpa using (<a>coeff_eq_zero_of_natDegree_lt</a> pq).<a>symm</a>", [{"full_name": "Polynomial.coeff_eq_zero_of_natDegree_lt", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [349, 9], "def_end_pos": [349, 38]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case pos\nR : Type u_1\ninst\u271d : Semiring R\nf p q : R[X]\npq : natDegree p < natDegree q\nn : \u2115\nnd : n = natDegree q\n\u22a2 0 = coeff (p + eraseLead q) (natDegree q)", "state_after": "no goals"}, {"tactic": "rw [eraseLead_coeff, coeff_add, coeff_add, eraseLead_coeff, if_neg, if_neg nd]", "annotated_tactic": ["rw [<a>eraseLead_coeff</a>, <a>coeff_add</a>, <a>coeff_add</a>, <a>eraseLead_coeff</a>, <a>if_neg</a>, <a>if_neg</a> nd]", [{"full_name": "Polynomial.eraseLead_coeff", "def_path": "Mathlib/Data/Polynomial/EraseLead.lean", "def_pos": [46, 9], "def_end_pos": [46, 24]}, {"full_name": "Polynomial.coeff_add", "def_path": "Mathlib/Data/Polynomial/Coeff.lean", "def_pos": [41, 9], "def_end_pos": [41, 18]}, {"full_name": "Polynomial.coeff_add", "def_path": "Mathlib/Data/Polynomial/Coeff.lean", "def_pos": [41, 9], "def_end_pos": [41, 18]}, {"full_name": "Polynomial.eraseLead_coeff", "def_path": "Mathlib/Data/Polynomial/EraseLead.lean", "def_pos": [46, 9], "def_end_pos": [46, 24]}, {"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}, {"full_name": "if_neg", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [795, 9], "def_end_pos": [795, 15]}]], "state_before": "case neg\nR : Type u_1\ninst\u271d : Semiring R\nf p q : R[X]\npq : natDegree p < natDegree q\nn : \u2115\nnd : \u00acn = natDegree q\n\u22a2 coeff (eraseLead (p + q)) n = coeff (p + eraseLead q) n", "state_after": "case neg.hnc\nR : Type u_1\ninst\u271d : Semiring R\nf p q : R[X]\npq : natDegree p < natDegree q\nn : \u2115\nnd : \u00acn = natDegree q\n\u22a2 \u00acn = natDegree (p + q)"}, {"tactic": "rintro rfl", "annotated_tactic": ["rintro rfl", []], "state_before": "case neg.hnc\nR : Type u_1\ninst\u271d : Semiring R\nf p q : R[X]\npq : natDegree p < natDegree q\nn : \u2115\nnd : \u00acn = natDegree q\n\u22a2 \u00acn = natDegree (p + q)", "state_after": "case neg.hnc\nR : Type u_1\ninst\u271d : Semiring R\nf p q : R[X]\npq : natDegree p < natDegree q\nnd : \u00acnatDegree (p + q) = natDegree q\n\u22a2 False"}, {"tactic": "exact nd (natDegree_add_eq_right_of_natDegree_lt pq)", "annotated_tactic": ["exact nd (<a>natDegree_add_eq_right_of_natDegree_lt</a> pq)", [{"full_name": "Polynomial.natDegree_add_eq_right_of_natDegree_lt", "def_path": "Mathlib/Data/Polynomial/Degree/Definitions.lean", "def_pos": [715, 9], "def_end_pos": [715, 47]}]], "state_before": "case neg.hnc\nR : Type u_1\ninst\u271d : Semiring R\nf p q : R[X]\npq : natDegree p < natDegree q\nnd : \u00acnatDegree (p + q) = natDegree q\n\u22a2 False", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Topology/UniformSpace/Completion.lean | CauchyFilter.uniformEmbedding_pureCauchy | [
170,
1
] | [
172,
72
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/RingTheory/UniqueFactorizationDomain.lean | irreducible_iff_prime_of_exists_unique_irreducible_factors | [
375,
1
] | [
412,
23
] | [{"tactic": "simp [ha0]", "annotated_tactic": ["simp [ha0]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : a * b = 0\nha0 : a = 0\n\u22a2 p \u2223 a \u2228 p \u2223 b", "state_after": "no goals"}, {"tactic": "simp [hb0]", "annotated_tactic": ["simp [hb0]", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : a * b = 0\nhb0 : b = 0\n\u22a2 p \u2223 a \u2228 p \u2223 b", "state_after": "no goals"}, {"tactic": "have hx0 : x \u2260 0 := fun hx0 => by simp_all", "annotated_tactic": ["have hx0 : x \u2260 0 := fun hx0 => by simp_all", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\n\u22a2 p \u2223 a \u2228 p \u2223 b", "state_after": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\n\u22a2 p \u2223 a \u2228 p \u2223 b"}, {"tactic": "have ha0 : a \u2260 0 := left_ne_zero_of_mul hab0", "annotated_tactic": ["have ha0 : a \u2260 0 := <a>left_ne_zero_of_mul</a> hab0", [{"full_name": "left_ne_zero_of_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [51, 9], "def_end_pos": [51, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\n\u22a2 p \u2223 a \u2228 p \u2223 b", "state_after": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\n\u22a2 p \u2223 a \u2228 p \u2223 b"}, {"tactic": "have hb0 : b \u2260 0 := right_ne_zero_of_mul hab0", "annotated_tactic": ["have hb0 : b \u2260 0 := <a>right_ne_zero_of_mul</a> hab0", [{"full_name": "right_ne_zero_of_mul", "def_path": "Mathlib/Algebra/GroupWithZero/Basic.lean", "def_pos": [55, 9], "def_end_pos": [55, 29]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\n\u22a2 p \u2223 a \u2228 p \u2223 b", "state_after": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\n\u22a2 p \u2223 a \u2228 p \u2223 b"}, {"tactic": "cases' eif x hx0 with fx hfx", "annotated_tactic": ["cases' eif x hx0 with fx hfx", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\n\u22a2 p \u2223 a \u2228 p \u2223 b", "state_after": "case intro\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\n\u22a2 p \u2223 a \u2228 p \u2223 b"}, {"tactic": "cases' eif a ha0 with fa hfa", "annotated_tactic": ["cases' eif a ha0 with fa hfa", []], "state_before": "case intro\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\n\u22a2 p \u2223 a \u2228 p \u2223 b", "state_after": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\n\u22a2 p \u2223 a \u2228 p \u2223 b"}, {"tactic": "cases' eif b hb0 with fb hfb", "annotated_tactic": ["cases' eif b hb0 with fb hfb", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\n\u22a2 p \u2223 a \u2228 p \u2223 b", "state_after": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\n\u22a2 p \u2223 a \u2228 p \u2223 b"}, {"tactic": "exact\n let \u27e8q, hqf, hq\u27e9 := Multiset.exists_mem_of_rel_of_mem h (Multiset.mem_cons_self p _)\n (Multiset.mem_add.1 hqf).elim\n (fun hqa =>\n Or.inl <| hq.dvd_iff_dvd_left.2 <| hfa.2.dvd_iff_dvd_right.1 (Multiset.dvd_prod hqa))\n fun hqb =>\n Or.inr <| hq.dvd_iff_dvd_left.2 <| hfb.2.dvd_iff_dvd_right.1 (Multiset.dvd_prod hqb)", "annotated_tactic": ["exact\n let \u27e8q, hqf, hq\u27e9 := <a>Multiset.exists_mem_of_rel_of_mem</a> h (<a>Multiset.mem_cons_self</a> p _)\n (<a>Multiset.mem_add</a>.1 hqf).<a>elim</a>\n (fun hqa =>\n <a>Or.inl</a> <| hq.dvd_iff_dvd_left.2 <| hfa.2.<a>dvd_iff_dvd_right</a>.1 (<a>Multiset.dvd_prod</a> hqa))\n fun hqb =>\n <a>Or.inr</a> <| hq.dvd_iff_dvd_left.2 <| hfb.2.<a>dvd_iff_dvd_right</a>.1 (<a>Multiset.dvd_prod</a> hqb)", [{"full_name": "Multiset.exists_mem_of_rel_of_mem", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [2790, 9], "def_end_pos": [2790, 33]}, {"full_name": "Multiset.mem_cons_self", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [244, 9], "def_end_pos": [244, 22]}, {"full_name": "Multiset.mem_add", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [683, 9], "def_end_pos": [683, 16]}, {"full_name": "Or.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [533, 9], "def_end_pos": [533, 16]}, {"full_name": "Or.inl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [517, 5], "def_end_pos": [517, 8]}, {"full_name": "Associated.dvd_iff_dvd_right", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [579, 9], "def_end_pos": [579, 37]}, {"full_name": "Multiset.dvd_prod", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Lemmas.lean", "def_pos": [18, 9], "def_end_pos": [18, 17]}, {"full_name": "Or.inr", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [519, 5], "def_end_pos": [519, 8]}, {"full_name": "Associated.dvd_iff_dvd_right", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [579, 9], "def_end_pos": [579, 37]}, {"full_name": "Multiset.dvd_prod", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Lemmas.lean", "def_pos": [18, 9], "def_end_pos": [18, 17]}]], "state_before": "case intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\nh : Multiset.Rel Associated (p ::\u2098 fx) (fa + fb)\n\u22a2 p \u2223 a \u2228 p \u2223 b", "state_after": "no goals"}, {"tactic": "simp_all", "annotated_tactic": ["simp_all", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x = 0\n\u22a2 False", "state_after": "no goals"}, {"tactic": "apply uif", "annotated_tactic": ["apply uif", []], "state_before": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\n\u22a2 Multiset.Rel Associated (p ::\u2098 fx) (fa + fb)", "state_after": "case a\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\n\u22a2 \u2200 (x : \u03b1), x \u2208 p ::\u2098 fx \u2192 Irreducible x\n\ncase a\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\n\u22a2 \u2200 (x : \u03b1), x \u2208 fa + fb \u2192 Irreducible x\n\ncase a\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\n\u22a2 Multiset.prod (p ::\u2098 fx) ~\u1d64 Multiset.prod (fa + fb)"}, {"tactic": "calc\n Multiset.prod (p ::\u2098 fx) ~\u1d64 a * b := by\n rw [hx, Multiset.prod_cons]; exact hfx.2.mul_left _\n _ ~\u1d64 fa.prod * fb.prod := (hfa.2.symm.mul_mul hfb.2.symm)\n _ = _ := by rw [Multiset.prod_add]", "annotated_tactic": ["calc\n <a>Multiset.prod</a> (p ::\u2098 fx) ~\u1d64 a * b := by\n rw [hx, <a>Multiset.prod_cons</a>]; exact hfx.2.<a>mul_left</a> _\n _ ~\u1d64 fa.prod * fb.prod := (hfa.2.symm.mul_mul hfb.2.<a>symm</a>)\n _ = _ := by rw [<a>Multiset.prod_add</a>]", [{"full_name": "Multiset.prod", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "def_pos": [44, 5], "def_end_pos": [44, 9]}, {"full_name": "Multiset.prod_cons", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 18]}, {"full_name": "Associated.mul_left", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [528, 9], "def_end_pos": [528, 28]}, {"full_name": "Associated.symm", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [398, 19], "def_end_pos": [398, 23]}, {"full_name": "Multiset.prod_add", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "def_pos": [115, 9], "def_end_pos": [115, 17]}]], "state_before": "case a\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\n\u22a2 Multiset.prod (p ::\u2098 fx) ~\u1d64 Multiset.prod (fa + fb)", "state_after": "no goals"}, {"tactic": "exact fun i hi => (Multiset.mem_cons.1 hi).elim (fun hip => hip.symm \u25b8 hpi) (hfx.1 _)", "annotated_tactic": ["exact fun i hi => (<a>Multiset.mem_cons</a>.1 hi).<a>elim</a> (fun hip => hip.symm \u25b8 hpi) (hfx.1 _)", [{"full_name": "Multiset.mem_cons", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [235, 9], "def_end_pos": [235, 17]}, {"full_name": "Or.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [533, 9], "def_end_pos": [533, 16]}]], "state_before": "case a\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\n\u22a2 \u2200 (x : \u03b1), x \u2208 p ::\u2098 fx \u2192 Irreducible x", "state_after": "no goals"}, {"tactic": "exact fun i hi => (Multiset.mem_add.1 hi).elim (hfa.1 _) (hfb.1 _)", "annotated_tactic": ["exact fun i hi => (<a>Multiset.mem_add</a>.1 hi).<a>elim</a> (hfa.1 _) (hfb.1 _)", [{"full_name": "Multiset.mem_add", "def_path": "Mathlib/Data/Multiset/Basic.lean", "def_pos": [683, 9], "def_end_pos": [683, 16]}, {"full_name": "Or.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [533, 9], "def_end_pos": [533, 16]}]], "state_before": "case a\n\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\n\u22a2 \u2200 (x : \u03b1), x \u2208 fa + fb \u2192 Irreducible x", "state_after": "no goals"}, {"tactic": "rw [hx, Multiset.prod_cons]", "annotated_tactic": ["rw [hx, <a>Multiset.prod_cons</a>]", [{"full_name": "Multiset.prod_cons", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\n\u22a2 Multiset.prod (p ::\u2098 fx) ~\u1d64 a * b", "state_after": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\n\u22a2 p * Multiset.prod fx ~\u1d64 p * x"}, {"tactic": "exact hfx.2.mul_left _", "annotated_tactic": ["exact hfx.2.<a>mul_left</a> _", [{"full_name": "Associated.mul_left", "def_path": "Mathlib/Algebra/Associated.lean", "def_pos": [528, 9], "def_end_pos": [528, 28]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\n\u22a2 p * Multiset.prod fx ~\u1d64 p * x", "state_after": "no goals"}, {"tactic": "rw [Multiset.prod_add]", "annotated_tactic": ["rw [<a>Multiset.prod_add</a>]", [{"full_name": "Multiset.prod_add", "def_path": "Mathlib/Algebra/BigOperators/Multiset/Basic.lean", "def_pos": [115, 9], "def_end_pos": [115, 17]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : CancelCommMonoidWithZero \u03b1\neif : \u2200 (a : \u03b1), a \u2260 0 \u2192 \u2203 f, (\u2200 (b : \u03b1), b \u2208 f \u2192 Irreducible b) \u2227 Multiset.prod f ~\u1d64 a\nuif :\n \u2200 (f g : Multiset \u03b1),\n (\u2200 (x : \u03b1), x \u2208 f \u2192 Irreducible x) \u2192\n (\u2200 (x : \u03b1), x \u2208 g \u2192 Irreducible x) \u2192 Multiset.prod f ~\u1d64 Multiset.prod g \u2192 Multiset.Rel Associated f g\np : \u03b1\nthis : DecidableEq \u03b1 := Classical.decEq \u03b1\nhpi : Irreducible p\na b : \u03b1\nx\u271d : p \u2223 a * b\nx : \u03b1\nhx : a * b = p * x\nhab0 : \u00aca * b = 0\nhx0 : x \u2260 0\nha0 : a \u2260 0\nhb0 : b \u2260 0\nfx : Multiset \u03b1\nhfx : (\u2200 (b : \u03b1), b \u2208 fx \u2192 Irreducible b) \u2227 Multiset.prod fx ~\u1d64 x\nfa : Multiset \u03b1\nhfa : (\u2200 (b : \u03b1), b \u2208 fa \u2192 Irreducible b) \u2227 Multiset.prod fa ~\u1d64 a\nfb : Multiset \u03b1\nhfb : (\u2200 (b : \u03b1), b \u2208 fb \u2192 Irreducible b) \u2227 Multiset.prod fb ~\u1d64 b\n\u22a2 Multiset.prod fa * Multiset.prod fb = Multiset.prod (fa + fb)", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Finset/LocallyFinite.lean | Finset.Ico_inter_Ico_consecutive | [
620,
1
] | [
621,
46
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Topology/Order/NhdsSet.lean | Iio_mem_nhdsSet_Ico | [
115,
1
] | [
116,
47
] | [{"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b2 : LinearOrder \u03b1\ninst\u271d\u00b9 : TopologicalSpace \u03b1\ninst\u271d : OrderClosedTopology \u03b1\na b c d : \u03b1\nh : b \u2264 c\n\u22a2 Iio c \u2208 \ud835\udcdd\u02e2 (Iio b)", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.mem_ae_map_iff | [
2460,
1
] | [
2462,
80
] | [{"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/Measure/MeasureSpaceDef.lean", "def_pos": [384, 9], "def_end_pos": [384, 19]}, {"full_name": "MeasureTheory.Measure.map_apply_of_aemeasurable", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1228, 9], "def_end_pos": [1228, 34]}, {"full_name": "Set.preimage_compl", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [97, 9], "def_end_pos": [97, 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\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"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Analysis/Convex/StrictConvexBetween.lean | Wbtw.dist_le_max_dist | [
41,
1
] | [
46,
35
] | [{"tactic": "by_cases hp\u2081 : p\u2082 = p\u2081", "annotated_tactic": ["by_cases hp\u2081 : p\u2082 = p\u2081", []], "state_before": "V : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : PseudoMetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\ninst\u271d : StrictConvexSpace \u211d V\np p\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\n\u22a2 dist p\u2082 p \u2264 max (dist p\u2081 p) (dist p\u2083 p)", "state_after": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : PseudoMetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\ninst\u271d : StrictConvexSpace \u211d V\np p\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\nhp\u2081 : p\u2082 = p\u2081\n\u22a2 dist p\u2082 p \u2264 max (dist p\u2081 p) (dist p\u2083 p)\n\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : PseudoMetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\ninst\u271d : StrictConvexSpace \u211d V\np p\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\nhp\u2081 : \u00acp\u2082 = p\u2081\n\u22a2 dist p\u2082 p \u2264 max (dist p\u2081 p) (dist p\u2083 p)"}, {"tactic": "by_cases hp\u2083 : p\u2082 = p\u2083", "annotated_tactic": ["by_cases hp\u2083 : p\u2082 = p\u2083", []], "state_before": "case neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : PseudoMetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\ninst\u271d : StrictConvexSpace \u211d V\np p\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\nhp\u2081 : \u00acp\u2082 = p\u2081\n\u22a2 dist p\u2082 p \u2264 max (dist p\u2081 p) (dist p\u2083 p)", "state_after": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : PseudoMetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\ninst\u271d : StrictConvexSpace \u211d V\np p\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\nhp\u2081 : \u00acp\u2082 = p\u2081\nhp\u2083 : p\u2082 = p\u2083\n\u22a2 dist p\u2082 p \u2264 max (dist p\u2081 p) (dist p\u2083 p)\n\ncase neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : PseudoMetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\ninst\u271d : StrictConvexSpace \u211d V\np p\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\nhp\u2081 : \u00acp\u2082 = p\u2081\nhp\u2083 : \u00acp\u2082 = p\u2083\n\u22a2 dist p\u2082 p \u2264 max (dist p\u2081 p) (dist p\u2083 p)"}, {"tactic": "have hs : Sbtw \u211d p\u2081 p\u2082 p\u2083 := \u27e8h, hp\u2081, hp\u2083\u27e9", "annotated_tactic": ["have hs : <a>Sbtw</a> \u211d p\u2081 p\u2082 p\u2083 := \u27e8h, hp\u2081, hp\u2083\u27e9", [{"full_name": "Sbtw", "def_path": "Mathlib/Analysis/Convex/Between.lean", "def_pos": [149, 5], "def_end_pos": [149, 9]}]], "state_before": "case neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : PseudoMetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\ninst\u271d : StrictConvexSpace \u211d V\np p\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\nhp\u2081 : \u00acp\u2082 = p\u2081\nhp\u2083 : \u00acp\u2082 = p\u2083\n\u22a2 dist p\u2082 p \u2264 max (dist p\u2081 p) (dist p\u2083 p)", "state_after": "case neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : PseudoMetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\ninst\u271d : StrictConvexSpace \u211d V\np p\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\nhp\u2081 : \u00acp\u2082 = p\u2081\nhp\u2083 : \u00acp\u2082 = p\u2083\nhs : Sbtw \u211d p\u2081 p\u2082 p\u2083\n\u22a2 dist p\u2082 p \u2264 max (dist p\u2081 p) (dist p\u2083 p)"}, {"tactic": "exact (hs.dist_lt_max_dist _).le", "annotated_tactic": ["exact (hs.dist_lt_max_dist _).<a>le</a>", [{"full_name": "LT.lt.le", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [142, 7], "def_end_pos": [142, 15]}]], "state_before": "case neg\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : PseudoMetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\ninst\u271d : StrictConvexSpace \u211d V\np p\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\nhp\u2081 : \u00acp\u2082 = p\u2081\nhp\u2083 : \u00acp\u2082 = p\u2083\nhs : Sbtw \u211d p\u2081 p\u2082 p\u2083\n\u22a2 dist p\u2082 p \u2264 max (dist p\u2081 p) (dist p\u2083 p)", "state_after": "no goals"}, {"tactic": "simp [hp\u2081]", "annotated_tactic": ["simp [hp\u2081]", []], "state_before": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : PseudoMetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\ninst\u271d : StrictConvexSpace \u211d V\np p\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\nhp\u2081 : p\u2082 = p\u2081\n\u22a2 dist p\u2082 p \u2264 max (dist p\u2081 p) (dist p\u2083 p)", "state_after": "no goals"}, {"tactic": "simp [hp\u2083]", "annotated_tactic": ["simp [hp\u2083]", []], "state_before": "case pos\nV : Type u_1\nP : Type u_2\ninst\u271d\u2074 : NormedAddCommGroup V\ninst\u271d\u00b3 : NormedSpace \u211d V\ninst\u271d\u00b2 : PseudoMetricSpace P\ninst\u271d\u00b9 : NormedAddTorsor V P\ninst\u271d : StrictConvexSpace \u211d V\np p\u2081 p\u2082 p\u2083 : P\nh : Wbtw \u211d p\u2081 p\u2082 p\u2083\nhp\u2081 : \u00acp\u2082 = p\u2081\nhp\u2083 : p\u2082 = p\u2083\n\u22a2 dist p\u2082 p \u2264 max (dist p\u2081 p) (dist p\u2083 p)", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Analysis/NormedSpace/OperatorNorm.lean | ContinuousLinearMap.op_nnnorm_prod | [
630,
1
] | [
631,
34
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/LinearAlgebra/LinearIndependent.lean | eq_of_linearIndependent_of_span_subtype | [
1013,
1
] | [
1025,
29
] | [{"tactic": "let f : t \u21aa s :=\n \u27e8fun x => \u27e8x.1, h x.2\u27e9, fun a b hab => Subtype.coe_injective (Subtype.mk.inj hab)\u27e9", "annotated_tactic": ["let f : t \u21aa s :=\n \u27e8fun x => \u27e8x.1, h x.2\u27e9, fun a b hab => <a>Subtype.coe_injective</a> (Subtype.mk.inj hab)\u27e9", [{"full_name": "Subtype.coe_injective", "def_path": "Mathlib/Data/Subtype.lean", "def_pos": [119, 9], "def_end_pos": [119, 22]}]], "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\n\u22a2 s = t", "state_after": "\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n inj' :=\n (_ :\n \u2200 (a b : \u2191t),\n (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n a = b) }\n\u22a2 s = t"}, {"tactic": "have h_surj : Surjective f := by\n apply surjective_of_linearIndependent_of_span hs f _\n convert hst <;> simp [comp]", "annotated_tactic": ["have h_surj : <a>Surjective</a> f := by\n apply <a>surjective_of_linearIndependent_of_span</a> hs f _\n convert hst <;> simp [<a>comp</a>]", [{"full_name": "Function.Surjective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [119, 5], "def_end_pos": [119, 15]}, {"full_name": "surjective_of_linearIndependent_of_span", "def_path": "Mathlib/LinearAlgebra/LinearIndependent.lean", "def_pos": [992, 9], "def_end_pos": [992, 48]}, {"full_name": "Function.comp", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}]], "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n inj' :=\n (_ :\n \u2200 (a b : \u2191t),\n (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n a = b) }\n\u22a2 s = t", "state_after": "\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n inj' :=\n (_ :\n \u2200 (a b : \u2191t),\n (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n a = b) }\nh_surj : Surjective \u2191f\n\u22a2 s = t"}, {"tactic": "apply surjective_of_linearIndependent_of_span hs f _", "annotated_tactic": ["apply <a>surjective_of_linearIndependent_of_span</a> hs f _", [{"full_name": "surjective_of_linearIndependent_of_span", "def_path": "Mathlib/LinearAlgebra/LinearIndependent.lean", "def_pos": [992, 9], "def_end_pos": [992, 48]}]], "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n inj' :=\n (_ :\n \u2200 (a b : \u2191t),\n (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n a = b) }\n\u22a2 Surjective \u2191f", "state_after": "\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n inj' :=\n (_ :\n \u2200 (a b : \u2191t),\n (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n a = b) }\n\u22a2 (range fun x => \u2191x) \u2286 \u2191(span R (range ((fun x => \u2191x) \u2218 \u2191f)))"}, {"tactic": "convert hst <;> simp [comp]", "annotated_tactic": ["convert hst <;> simp [<a>comp</a>]", [{"full_name": "Function.comp", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [52, 15], "def_end_pos": [52, 28]}]], "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n inj' :=\n (_ :\n \u2200 (a b : \u2191t),\n (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n a = b) }\n\u22a2 (range fun x => \u2191x) \u2286 \u2191(span R (range ((fun x => \u2191x) \u2218 \u2191f)))", "state_after": "no goals"}, {"tactic": "apply Subset.antisymm _ h", "annotated_tactic": ["apply <a>Subset.antisymm</a> _ h", [{"full_name": "Set.Subset.antisymm", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [370, 9], "def_end_pos": [370, 24]}]], "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n inj' :=\n (_ :\n \u2200 (a b : \u2191t),\n (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n a = b) }\nh_surj : Surjective \u2191f\n\u22a2 s = t", "state_after": "\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n inj' :=\n (_ :\n \u2200 (a b : \u2191t),\n (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n a = b) }\nh_surj : Surjective \u2191f\n\u22a2 s \u2286 t"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n inj' :=\n (_ :\n \u2200 (a b : \u2191t),\n (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n a = b) }\nh_surj : Surjective \u2191f\n\u22a2 s \u2286 t", "state_after": "\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n inj' :=\n (_ :\n \u2200 (a b : \u2191t),\n (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n a = b) }\nh_surj : Surjective \u2191f\nx : M\nhx : x \u2208 s\n\u22a2 x \u2208 t"}, {"tactic": "rcases h_surj \u27e8x, hx\u27e9 with \u27e8y, hy\u27e9", "annotated_tactic": ["rcases h_surj \u27e8x, hx\u27e9 with \u27e8y, hy\u27e9", []], "state_before": "\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n inj' :=\n (_ :\n \u2200 (a b : \u2191t),\n (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n a = b) }\nh_surj : Surjective \u2191f\nx : M\nhx : x \u2208 s\n\u22a2 x \u2208 t", "state_after": "case intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y\u271d : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n inj' :=\n (_ :\n \u2200 (a b : \u2191t),\n (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n a = b) }\nh_surj : Surjective \u2191f\nx : M\nhx : x \u2208 s\ny : \u2191t\nhy : \u2191f y = { val := x, property := hx }\n\u22a2 x \u2208 t"}, {"tactic": "convert y.mem", "annotated_tactic": ["convert y.mem", []], "state_before": "case intro\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y\u271d : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n inj' :=\n (_ :\n \u2200 (a b : \u2191t),\n (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n a = b) }\nh_surj : Surjective \u2191f\nx : M\nhx : x \u2208 s\ny : \u2191t\nhy : \u2191f y = { val := x, property := hx }\n\u22a2 x \u2208 t", "state_after": "case h.e'_4\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y\u271d : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n inj' :=\n (_ :\n \u2200 (a b : \u2191t),\n (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n a = b) }\nh_surj : Surjective \u2191f\nx : M\nhx : x \u2208 s\ny : \u2191t\nhy : \u2191f y = { val := x, property := hx }\n\u22a2 x = \u2191y"}, {"tactic": "rw [\u2190 Subtype.mk.inj hy]", "annotated_tactic": ["rw [\u2190 Subtype.mk.inj hy]", []], "state_before": "case h.e'_4\n\u03b9 : Type u'\n\u03b9' : Type u_1\nR : Type u_2\nK : Type u_3\nM : Type u_4\nM' : Type u_5\nM'' : Type u_6\nV : Type u\nV' : Type u_7\nv : \u03b9 \u2192 M\ninst\u271d\u2077 : Ring R\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M'\ninst\u271d\u2074 : AddCommGroup M''\ninst\u271d\u00b3 : Module R M\ninst\u271d\u00b2 : Module R M'\ninst\u271d\u00b9 : Module R M''\na b : R\nx\u271d y\u271d : M\ninst\u271d : Nontrivial R\ns t : Set M\nhs : LinearIndependent R fun x => \u2191x\nh : t \u2286 s\nhst : s \u2286 \u2191(span R t)\nf : \u2191t \u21aa \u2191s :=\n { toFun := fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) },\n inj' :=\n (_ :\n \u2200 (a b : \u2191t),\n (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) a = (fun x => { val := \u2191x, property := (_ : \u2191x \u2208 s) }) b \u2192\n a = b) }\nh_surj : Surjective \u2191f\nx : M\nhx : x \u2208 s\ny : \u2191t\nhy : \u2191f y = { val := x, property := hx }\n\u22a2 x = \u2191y", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Topology/Algebra/GroupWithZero.lean | ContinuousWithinAt.inv₀ | [
121,
8
] | [
123,
13
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Finset/Sigma.lean | Finset.pairwiseDisjoint_map_sigmaMk | [
75,
1
] | [
81,
43
] | [{"tactic": "intro i _ j _ hij", "annotated_tactic": ["intro i _ j _ hij", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : Type u_3\ns s\u2081 s\u2082 : Finset \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Finset (\u03b1 i)\n\u22a2 Set.PairwiseDisjoint \u2191s fun i => map (Embedding.sigmaMk i) (t i)", "state_after": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : Type u_3\ns s\u2081 s\u2082 : Finset \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Finset (\u03b1 i)\ni : \u03b9\na\u271d\u00b9 : i \u2208 \u2191s\nj : \u03b9\na\u271d : j \u2208 \u2191s\nhij : i \u2260 j\n\u22a2 (_root_.Disjoint on fun i => map (Embedding.sigmaMk i) (t i)) i j"}, {"tactic": "rw [Function.onFun, disjoint_left]", "annotated_tactic": ["rw [<a>Function.onFun</a>, <a>disjoint_left</a>]", [{"full_name": "Function.onFun", "def_path": "Mathlib/Init/Function.lean", "def_pos": [49, 5], "def_end_pos": [49, 10]}, {"full_name": "Finset.disjoint_left", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [939, 9], "def_end_pos": [939, 22]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : Type u_3\ns s\u2081 s\u2082 : Finset \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Finset (\u03b1 i)\ni : \u03b9\na\u271d\u00b9 : i \u2208 \u2191s\nj : \u03b9\na\u271d : j \u2208 \u2191s\nhij : i \u2260 j\n\u22a2 (_root_.Disjoint on fun i => map (Embedding.sigmaMk i) (t i)) i j", "state_after": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : Type u_3\ns s\u2081 s\u2082 : Finset \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Finset (\u03b1 i)\ni : \u03b9\na\u271d\u00b9 : i \u2208 \u2191s\nj : \u03b9\na\u271d : j \u2208 \u2191s\nhij : i \u2260 j\n\u22a2 \u2200 \u2983a : (x : \u03b9) \u00d7 \u03b1 x\u2984, a \u2208 map (Embedding.sigmaMk i) (t i) \u2192 \u00aca \u2208 map (Embedding.sigmaMk j) (t j)"}, {"tactic": "simp_rw [mem_map, Function.Embedding.sigmaMk_apply]", "annotated_tactic": ["simp_rw [<a>mem_map</a>, <a>Function.Embedding.sigmaMk_apply</a>]", [{"full_name": "Finset.mem_map", "def_path": "Mathlib/Data/Finset/Image.lean", "def_pos": [68, 9], "def_end_pos": [68, 16]}, {"full_name": "Function.Embedding.sigmaMk_apply", "def_path": "Mathlib/Logic/Embedding/Basic.lean", "def_pos": [320, 9], "def_end_pos": [320, 14]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : Type u_3\ns s\u2081 s\u2082 : Finset \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Finset (\u03b1 i)\ni : \u03b9\na\u271d\u00b9 : i \u2208 \u2191s\nj : \u03b9\na\u271d : j \u2208 \u2191s\nhij : i \u2260 j\n\u22a2 \u2200 \u2983a : (x : \u03b9) \u00d7 \u03b1 x\u2984, a \u2208 map (Embedding.sigmaMk i) (t i) \u2192 \u00aca \u2208 map (Embedding.sigmaMk j) (t j)", "state_after": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : Type u_3\ns s\u2081 s\u2082 : Finset \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Finset (\u03b1 i)\ni : \u03b9\na\u271d\u00b9 : i \u2208 \u2191s\nj : \u03b9\na\u271d : j \u2208 \u2191s\nhij : i \u2260 j\n\u22a2 \u2200 \u2983a : (x : \u03b9) \u00d7 \u03b1 x\u2984,\n (\u2203 a_1, a_1 \u2208 t i \u2227 { fst := i, snd := a_1 } = a) \u2192 \u00ac\u2203 a_2, a_2 \u2208 t j \u2227 { fst := j, snd := a_2 } = a"}, {"tactic": "rintro _ \u27e8y, _, rfl\u27e9 \u27e8z, _, hz'\u27e9", "annotated_tactic": ["rintro _ \u27e8y, _, rfl\u27e9 \u27e8z, _, hz'\u27e9", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : Type u_3\ns s\u2081 s\u2082 : Finset \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Finset (\u03b1 i)\ni : \u03b9\na\u271d\u00b9 : i \u2208 \u2191s\nj : \u03b9\na\u271d : j \u2208 \u2191s\nhij : i \u2260 j\n\u22a2 \u2200 \u2983a : (x : \u03b9) \u00d7 \u03b1 x\u2984,\n (\u2203 a_1, a_1 \u2208 t i \u2227 { fst := i, snd := a_1 } = a) \u2192 \u00ac\u2203 a_2, a_2 \u2208 t j \u2227 { fst := j, snd := a_2 } = a", "state_after": "case intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : Type u_3\ns s\u2081 s\u2082 : Finset \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Finset (\u03b1 i)\ni : \u03b9\na\u271d\u00b9 : i \u2208 \u2191s\nj : \u03b9\na\u271d : j \u2208 \u2191s\nhij : i \u2260 j\ny : \u03b1 i\nleft\u271d\u00b9 : y \u2208 t i\nz : \u03b1 j\nleft\u271d : z \u2208 t j\nhz' : { fst := j, snd := z } = { fst := i, snd := y }\n\u22a2 False"}, {"tactic": "exact hij (congr_arg Sigma.fst hz'.symm)", "annotated_tactic": ["exact hij (<a>congr_arg</a> <a>Sigma.fst</a> hz'.symm)", [{"full_name": "congr_arg", "def_path": "Mathlib/Init/Logic.lean", "def_pos": [43, 7], "def_end_pos": [43, 16]}, {"full_name": "Sigma.fst", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [142, 3], "def_end_pos": [142, 6]}]], "state_before": "case intro.intro.intro.intro\n\u03b9 : Type u_1\n\u03b1 : \u03b9 \u2192 Type u_2\n\u03b2 : Type u_3\ns s\u2081 s\u2082 : Finset \u03b9\nt t\u2081 t\u2082 : (i : \u03b9) \u2192 Finset (\u03b1 i)\ni : \u03b9\na\u271d\u00b9 : i \u2208 \u2191s\nj : \u03b9\na\u271d : j \u2208 \u2191s\nhij : i \u2260 j\ny : \u03b1 i\nleft\u271d\u00b9 : y \u2208 t i\nz : \u03b1 j\nleft\u271d : z \u2208 t j\nhz' : { fst := j, snd := z } = { fst := i, snd := y }\n\u22a2 False", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Algebra/Invertible/Basic.lean | Ring.inverse_invertible | [
170,
1
] | [
171,
41
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | Complex.hasDerivAt_sinh | [
112,
1
] | [
113,
39
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/RingTheory/Ideal/Basic.lean | Ring.isField_iff_isSimpleOrder_ideal | [
782,
1
] | [
790,
37
] | [{"tactic": "cases subsingleton_or_nontrivial R", "annotated_tactic": ["cases <a>subsingleton_or_nontrivial</a> R", [{"full_name": "subsingleton_or_nontrivial", "def_path": "Mathlib/Logic/Nontrivial/Defs.lean", "def_pos": [94, 9], "def_end_pos": [94, 35]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommSemiring R\n\u22a2 IsField R \u2194 IsSimpleOrder (Ideal R)", "state_after": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommSemiring R\nh\u271d : Subsingleton R\n\u22a2 IsField R \u2194 IsSimpleOrder (Ideal R)\n\ncase inr\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommSemiring R\nh\u271d : Nontrivial R\n\u22a2 IsField R \u2194 IsSimpleOrder (Ideal R)"}, {"tactic": "rw [\u2190 not_iff_not, Ring.not_isField_iff_exists_ideal_bot_lt_and_lt_top, \u2190 not_iff_not]", "annotated_tactic": ["rw [\u2190 <a>not_iff_not</a>, <a>Ring.not_isField_iff_exists_ideal_bot_lt_and_lt_top</a>, \u2190 <a>not_iff_not</a>]", [{"full_name": "not_iff_not", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [439, 9], "def_end_pos": [439, 20]}, {"full_name": "Ring.not_isField_iff_exists_ideal_bot_lt_and_lt_top", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [753, 9], "def_end_pos": [753, 55]}, {"full_name": "not_iff_not", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [439, 9], "def_end_pos": [439, 20]}]], "state_before": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommSemiring R\nh\u271d : Nontrivial R\n\u22a2 IsField R \u2194 IsSimpleOrder (Ideal R)", "state_after": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommSemiring R\nh\u271d : Nontrivial R\n\u22a2 (\u00ac\u2203 I, \u22a5 < I \u2227 I < \u22a4) \u2194 \u00ac\u00acIsSimpleOrder (Ideal R)"}, {"tactic": "push_neg", "annotated_tactic": ["push_neg", []], "state_before": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommSemiring R\nh\u271d : Nontrivial R\n\u22a2 (\u00ac\u2203 I, \u22a5 < I \u2227 I < \u22a4) \u2194 \u00ac\u00acIsSimpleOrder (Ideal R)", "state_after": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommSemiring R\nh\u271d : Nontrivial R\n\u22a2 (\u2200 (I : Ideal R), \u22a5 < I \u2192 \u00acI < \u22a4) \u2194 IsSimpleOrder (Ideal R)"}, {"tactic": "simp_rw [lt_top_iff_ne_top, bot_lt_iff_ne_bot, \u2190 or_iff_not_imp_left, not_ne_iff]", "annotated_tactic": ["simp_rw [<a>lt_top_iff_ne_top</a>, <a>bot_lt_iff_ne_bot</a>, \u2190 <a>or_iff_not_imp_left</a>, <a>not_ne_iff</a>]", [{"full_name": "lt_top_iff_ne_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [173, 9], "def_end_pos": [173, 26]}, {"full_name": "bot_lt_iff_ne_bot", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [371, 9], "def_end_pos": [371, 26]}, {"full_name": "or_iff_not_imp_left", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [360, 9], "def_end_pos": [360, 28]}, {"full_name": "not_ne_iff", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [253, 9], "def_end_pos": [253, 19]}]], "state_before": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommSemiring R\nh\u271d : Nontrivial R\n\u22a2 (\u2200 (I : Ideal R), \u22a5 < I \u2192 \u00acI < \u22a4) \u2194 IsSimpleOrder (Ideal R)", "state_after": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommSemiring R\nh\u271d : Nontrivial R\n\u22a2 (\u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4) \u2194 IsSimpleOrder (Ideal R)"}, {"tactic": "exact \u27e8fun h => \u27e8h\u27e9, fun h => h.2\u27e9", "annotated_tactic": ["exact \u27e8fun h => \u27e8h\u27e9, fun h => h.2\u27e9", []], "state_before": "case inr\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommSemiring R\nh\u271d : Nontrivial R\n\u22a2 (\u2200 (I : Ideal R), I = \u22a5 \u2228 I = \u22a4) \u2194 IsSimpleOrder (Ideal R)", "state_after": "no goals"}, {"tactic": "exact\n \u27e8fun h => (not_isField_of_subsingleton _ h).elim, fun h =>\n (false_of_nontrivial_of_subsingleton <| Ideal R).elim\u27e9", "annotated_tactic": ["exact\n \u27e8fun h => (<a>not_isField_of_subsingleton</a> _ h).<a>elim</a>, fun h =>\n (<a>false_of_nontrivial_of_subsingleton</a> <| <a>Ideal</a> R).<a>elim</a>\u27e9", [{"full_name": "not_isField_of_subsingleton", "def_path": "Mathlib/Algebra/Field/IsField.lean", "def_pos": [57, 9], "def_end_pos": [57, 36]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}, {"full_name": "false_of_nontrivial_of_subsingleton", "def_path": "Mathlib/Logic/Nontrivial/Defs.lean", "def_pos": [99, 9], "def_end_pos": [99, 44]}, {"full_name": "Ideal", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [43, 5], "def_end_pos": [43, 10]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case inl\n\u03b1 : Type u\n\u03b2 : Type v\nR : Type u_1\ninst\u271d : CommSemiring R\nh\u271d : Subsingleton R\n\u22a2 IsField R \u2194 IsSimpleOrder (Ideal R)", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Topology/Algebra/Module/Basic.lean | ContinuousLinearEquiv.coe_funUnique | [
2525,
1
] | [
2526,
6
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Polynomial/Reverse.lean | Polynomial.reflect_monomial | [
167,
1
] | [
168,
80
] | [{"tactic": "rw [\u2190 one_mul (X ^ n), \u2190 one_mul (X ^ revAt N n), \u2190 C_1, reflect_C_mul_X_pow]", "annotated_tactic": ["rw [\u2190 <a>one_mul</a> (<a>X</a> ^ n), \u2190 <a>one_mul</a> (<a>X</a> ^ <a>revAt</a> N n), \u2190 <a>C_1</a>, <a>reflect_C_mul_X_pow</a>]", [{"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "Polynomial.X", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [566, 5], "def_end_pos": [566, 6]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "Polynomial.X", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [566, 5], "def_end_pos": [566, 6]}, {"full_name": "Polynomial.revAt", "def_path": "Mathlib/Data/Polynomial/Reverse.lean", "def_pos": [59, 5], "def_end_pos": [59, 10]}, {"full_name": "Polynomial.C_1", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [518, 9], "def_end_pos": [518, 12]}, {"full_name": "Polynomial.reflect_C_mul_X_pow", "def_path": "Mathlib/Data/Polynomial/Reverse.lean", "def_pos": [147, 9], "def_end_pos": [147, 28]}]], "state_before": "R : Type u_1\ninst\u271d : Semiring R\nf : R[X]\nN n : \u2115\n\u22a2 reflect N (X ^ n) = X ^ \u2191(revAt N) n", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Algebra/Group/Basic.lean | mul_div_cancel'' | [
740,
1
] | [
741,
51
] | [{"tactic": "rw [div_eq_mul_inv, mul_inv_cancel_right a b]", "annotated_tactic": ["rw [<a>div_eq_mul_inv</a>, <a>mul_inv_cancel_right</a> a b]", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}, {"full_name": "mul_inv_cancel_right", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [1159, 9], "def_end_pos": [1159, 29]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nG : Type u_3\ninst\u271d : Group G\na\u271d b\u271d c d a b : G\n\u22a2 a * b / b = a", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Order/Compare.lean | Ordering.Compares.eq_lt | [
82,
1
] | [
85,
77
] | [{"tactic": "injection h", "annotated_tactic": ["injection h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b1\na b : \u03b1\nh\u271d : Compares eq a b\nh : eq = lt\n\u22a2 a < b", "state_after": "no goals"}, {"tactic": "injection h", "annotated_tactic": ["injection h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d : Preorder \u03b1\na b : \u03b1\nh\u271d : Compares gt a b\nh : gt = lt\n\u22a2 a < b", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Finset/Basic.lean | Finset.singleton_inter_of_mem | [
1706,
1
] | [
1707,
78
] | [{"tactic": "rw [insert_inter_of_mem H, empty_inter]", "annotated_tactic": ["rw [<a>insert_inter_of_mem</a> H, <a>empty_inter</a>]", [{"full_name": "Finset.insert_inter_of_mem", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1680, 9], "def_end_pos": [1680, 28]}, {"full_name": "Finset.empty_inter", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1670, 9], "def_end_pos": [1670, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ninst\u271d : DecidableEq \u03b1\ns\u271d s\u2081 s\u2082 t t\u2081 t\u2082 u v : Finset \u03b1\na\u271d b a : \u03b1\ns : Finset \u03b1\nH : a \u2208 s\n\u22a2 insert a \u2205 \u2229 s = insert a \u2205", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Computability/Partrec.lean | Computable.of_eq | [
280,
1
] | [
281,
26
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Logic/Function/Basic.lean | Function.partialInv_left | [
419,
1
] | [
420,
48
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | exists_stronglyMeasurable_limit_of_tendsto_ae | [
1685,
1
] | [
1697,
18
] | [{"tactic": "borelize \u03b2", "annotated_tactic": ["borelize \u03b2", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))"}, {"tactic": "obtain \u27e8g, _, hg\u27e9 :\n \u2203 (g : \u03b1 \u2192 \u03b2) (_ : Measurable g), \u2200\u1d50 x \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x)) :=\n measurable_limit_of_tendsto_metrizable_ae (fun n => (hf n).aemeasurable) h_ae_tendsto", "annotated_tactic": ["obtain \u27e8g, _, hg\u27e9 :\n \u2203 (g : \u03b1 \u2192 \u03b2) (_ : <a>Measurable</a> g), \u2200\u1d50 x \u2202\u03bc, <a>Tendsto</a> (fun n => f n x) <a>atTop</a> (\ud835\udcdd (g x)) :=\n <a>measurable_limit_of_tendsto_metrizable_ae</a> (fun n => (hf n).<a>aemeasurable</a>) h_ae_tendsto", [{"full_name": "Measurable", "def_path": "Mathlib/MeasureTheory/MeasurableSpace/Defs.lean", "def_pos": [535, 5], "def_end_pos": [535, 15]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "measurable_limit_of_tendsto_metrizable_ae", "def_path": "Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean", "def_pos": [146, 9], "def_end_pos": [146, 50]}, {"full_name": "MeasureTheory.AEStronglyMeasurable.aemeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1220, 19], "def_end_pos": [1220, 31]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nw\u271d : Measurable g\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))"}, {"tactic": "have Hg : AEStronglyMeasurable g \u03bc := aestronglyMeasurable_of_tendsto_ae _ hf hg", "annotated_tactic": ["have Hg : <a>AEStronglyMeasurable</a> g \u03bc := <a>aestronglyMeasurable_of_tendsto_ae</a> _ hf hg", [{"full_name": "MeasureTheory.AEStronglyMeasurable", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [93, 5], "def_end_pos": [93, 25]}, {"full_name": "aestronglyMeasurable_of_tendsto_ae", "def_path": "Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean", "def_pos": [1664, 9], "def_end_pos": [1664, 50]}]], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nw\u271d : Measurable g\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nw\u271d : Measurable g\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nHg : AEStronglyMeasurable g \u03bc\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))"}, {"tactic": "refine' \u27e8Hg.mk g, Hg.stronglyMeasurable_mk, _\u27e9", "annotated_tactic": ["refine' \u27e8Hg.mk g, Hg.stronglyMeasurable_mk, _\u27e9", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nw\u271d : Measurable g\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nHg : AEStronglyMeasurable g \u03bc\n\u22a2 \u2203 f_lim hf_lim_meas, \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (f_lim x))", "state_after": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nw\u271d : Measurable g\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nHg : AEStronglyMeasurable g \u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (AEStronglyMeasurable.mk g Hg x))"}, {"tactic": "filter_upwards [hg, Hg.ae_eq_mk] with x hx h'x", "annotated_tactic": ["filter_upwards [hg, Hg.ae_eq_mk] with x hx h'x", []], "state_before": "case intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nw\u271d : Measurable g\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nHg : AEStronglyMeasurable g \u03bc\n\u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (AEStronglyMeasurable.mk g Hg x))", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nw\u271d : Measurable g\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nHg : AEStronglyMeasurable g \u03bc\nx : \u03b1\nhx : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nh'x : g x = AEStronglyMeasurable.mk g Hg x\n\u22a2 Tendsto (fun n => f n x) atTop (\ud835\udcdd (AEStronglyMeasurable.mk g Hg x))"}, {"tactic": "rwa [h'x] at hx", "annotated_tactic": ["rwa [h'x] at hx", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Type u_4\ninst\u271d\u00b3 : Countable \u03b9\nm : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : TopologicalSpace \u03b3\nf\u271d g\u271d : \u03b1 \u2192 \u03b2\ninst\u271d : PseudoMetrizableSpace \u03b2\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nhf : \u2200 (n : \u2115), AEStronglyMeasurable (f n) \u03bc\nh_ae_tendsto : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u2203 l, Tendsto (fun n => f n x) atTop (\ud835\udcdd l)\nthis\u271d\u00b9 : MeasurableSpace \u03b2 := borel \u03b2\nthis\u271d : BorelSpace \u03b2\ng : \u03b1 \u2192 \u03b2\nw\u271d : Measurable g\nhg : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nHg : AEStronglyMeasurable g \u03bc\nx : \u03b1\nhx : Tendsto (fun n => f n x) atTop (\ud835\udcdd (g x))\nh'x : g x = AEStronglyMeasurable.mk g Hg x\n\u22a2 Tendsto (fun n => f n x) atTop (\ud835\udcdd (AEStronglyMeasurable.mk g Hg x))", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Algebra/Order/Ring/Lemmas.lean | mul_le_mul_right | [
214,
1
] | [
215,
64
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Order/JordanHolder.lean | CompositionSeries.append_succ_natAdd | [
535,
1
] | [
537,
48
] | [{"tactic": "rw [coe_append, append_succ_natAdd_aux _ _ i]", "annotated_tactic": ["rw [<a>coe_append</a>, <a>append_succ_natAdd_aux</a> _ _ i]", [{"full_name": "CompositionSeries.coe_append", "def_path": "Mathlib/Order/JordanHolder.lean", "def_pos": [511, 9], "def_end_pos": [511, 19]}, {"full_name": "CompositionSeries.append_succ_natAdd_aux", "def_path": "Mathlib/Order/JordanHolder.lean", "def_pos": [485, 9], "def_end_pos": [485, 31]}]], "state_before": "X : Type u\ninst\u271d\u00b9 : Lattice X\ninst\u271d : JordanHolderLattice X\ns\u2081 s\u2082 : CompositionSeries X\nh : top s\u2081 = bot s\u2082\ni : Fin s\u2082.length\n\u22a2 series (append s\u2081 s\u2082 h) (Fin.succ (Fin.natAdd s\u2081.length i)) = series s\u2082 (Fin.succ i)", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Order/Bounds/Basic.lean | IsGreatest.dual | [
157,
1
] | [
158,
4
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/RingTheory/AdjoinRoot.lean | AdjoinRoot.coe_injective' | [
417,
1
] | [
418,
19
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Order/Filter/Pointwise.lean | Filter.mul_eq_bot_iff | [
320,
1
] | [
321,
18
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | IsSelfAdjoint.conj_adjoint | [
290,
1
] | [
294,
45
] | [{"tactic": "rw [isSelfAdjoint_iff'] at hT \u22a2", "annotated_tactic": ["rw [<a>isSelfAdjoint_iff'</a>] at hT \u22a2", [{"full_name": "ContinuousLinearMap.isSelfAdjoint_iff'", "def_path": "Mathlib/Analysis/InnerProductSpace/Adjoint.lean", "def_pos": [223, 9], "def_end_pos": [223, 27]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nT : E \u2192L[\ud835\udd5c] E\nhT : IsSelfAdjoint T\nS : E \u2192L[\ud835\udd5c] F\n\u22a2 IsSelfAdjoint (comp S (comp T (\u2191adjoint S)))", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nT : E \u2192L[\ud835\udd5c] E\nhT : \u2191adjoint T = T\nS : E \u2192L[\ud835\udd5c] F\n\u22a2 \u2191adjoint (comp S (comp T (\u2191adjoint S))) = comp S (comp T (\u2191adjoint S))"}, {"tactic": "simp only [hT, adjoint_comp, adjoint_adjoint]", "annotated_tactic": ["simp only [hT, <a>adjoint_comp</a>, <a>adjoint_adjoint</a>]", [{"full_name": "ContinuousLinearMap.adjoint_comp", "def_path": "Mathlib/Analysis/InnerProductSpace/Adjoint.lean", "def_pos": [142, 9], "def_end_pos": [142, 21]}, {"full_name": "ContinuousLinearMap.adjoint_adjoint", "def_path": "Mathlib/Analysis/InnerProductSpace/Adjoint.lean", "def_pos": [135, 9], "def_end_pos": [135, 24]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nT : E \u2192L[\ud835\udd5c] E\nhT : \u2191adjoint T = T\nS : E \u2192L[\ud835\udd5c] F\n\u22a2 \u2191adjoint (comp S (comp T (\u2191adjoint S))) = comp S (comp T (\u2191adjoint S))", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nT : E \u2192L[\ud835\udd5c] E\nhT : \u2191adjoint T = T\nS : E \u2192L[\ud835\udd5c] F\n\u22a2 comp (comp S T) (\u2191adjoint S) = comp S (comp T (\u2191adjoint S))"}, {"tactic": "exact ContinuousLinearMap.comp_assoc _ _ _", "annotated_tactic": ["exact <a>ContinuousLinearMap.comp_assoc</a> _ _ _", [{"full_name": "ContinuousLinearMap.comp_assoc", "def_path": "Mathlib/Topology/Algebra/Module/Basic.lean", "def_pos": [848, 9], "def_end_pos": [848, 19]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2078 : IsROrC \ud835\udd5c\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedAddCommGroup F\ninst\u271d\u2075 : NormedAddCommGroup G\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b2 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b9 : CompleteSpace E\ninst\u271d : CompleteSpace F\nT : E \u2192L[\ud835\udd5c] E\nhT : \u2191adjoint T = T\nS : E \u2192L[\ud835\udd5c] F\n\u22a2 comp (comp S T) (\u2191adjoint S) = comp S (comp T (\u2191adjoint S))", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Sum/Order.lean | Sum.not_inl_le_inr | [
149,
1
] | [
150,
22
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Finset/Basic.lean | Finset.union_sdiff_distrib | [
2244,
1
] | [
2245,
12
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/GroupTheory/Nilpotent.lean | nilpotencyClass_quotient_center | [
619,
1
] | [
636,
73
] | [{"tactic": "generalize hn : Group.nilpotencyClass G = n", "annotated_tactic": ["generalize hn : <a>Group.nilpotencyClass</a> G = n", [{"full_name": "Group.nilpotencyClass", "def_path": "Mathlib/GroupTheory/Nilpotent.lean", "def_pos": [362, 19], "def_end_pos": [362, 40]}]], "state_before": "G : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\n\u22a2 nilpotencyClass (G \u29f8 center G) = nilpotencyClass G - 1", "state_after": "G : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nn : \u2115\nhn : nilpotencyClass G = n\n\u22a2 nilpotencyClass (G \u29f8 center G) = n - 1"}, {"tactic": "rcases n with (rfl | n)", "annotated_tactic": ["rcases n with (rfl | n)", []], "state_before": "G : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nn : \u2115\nhn : nilpotencyClass G = n\n\u22a2 nilpotencyClass (G \u29f8 center G) = n - 1", "state_after": "case zero\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nhn : nilpotencyClass G = Nat.zero\n\u22a2 nilpotencyClass (G \u29f8 center G) = Nat.zero - 1\n\ncase succ\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nn : \u2115\nhn : nilpotencyClass G = Nat.succ n\n\u22a2 nilpotencyClass (G \u29f8 center G) = Nat.succ n - 1"}, {"tactic": "simp [nilpotencyClass_zero_iff_subsingleton] at *", "annotated_tactic": ["simp [<a>nilpotencyClass_zero_iff_subsingleton</a>] at *", [{"full_name": "nilpotencyClass_zero_iff_subsingleton", "def_path": "Mathlib/GroupTheory/Nilpotent.lean", "def_pos": [611, 9], "def_end_pos": [611, 46]}]], "state_before": "case zero\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nhn : nilpotencyClass G = Nat.zero\n\u22a2 nilpotencyClass (G \u29f8 center G) = Nat.zero - 1", "state_after": "case zero\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nhn : Subsingleton G\n\u22a2 Subsingleton (G \u29f8 center G)"}, {"tactic": "exact Quotient.instSubsingletonQuotient (leftRel (center G))", "annotated_tactic": ["exact <a>Quotient.instSubsingletonQuotient</a> (<a>leftRel</a> (<a>center</a> G))", [{"full_name": "Quotient.instSubsingletonQuotient", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [229, 10], "def_end_pos": [229, 34]}, {"full_name": "QuotientGroup.leftRel", "def_path": "Mathlib/GroupTheory/Coset.lean", "def_pos": [313, 5], "def_end_pos": [313, 12]}, {"full_name": "Subgroup.center", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [2085, 5], "def_end_pos": [2085, 11]}]], "state_before": "case zero\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nhn : Subsingleton G\n\u22a2 Subsingleton (G \u29f8 center G)", "state_after": "no goals"}, {"tactic": "suffices Group.nilpotencyClass (G \u29f8 center G) = n by simpa", "annotated_tactic": ["suffices <a>Group.nilpotencyClass</a> (G \u29f8 <a>center</a> G) = n by simpa", [{"full_name": "Group.nilpotencyClass", "def_path": "Mathlib/GroupTheory/Nilpotent.lean", "def_pos": [362, 19], "def_end_pos": [362, 40]}, {"full_name": "Subgroup.center", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [2085, 5], "def_end_pos": [2085, 11]}]], "state_before": "case succ\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nn : \u2115\nhn : nilpotencyClass G = Nat.succ n\n\u22a2 nilpotencyClass (G \u29f8 center G) = Nat.succ n - 1", "state_after": "case succ\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nn : \u2115\nhn : nilpotencyClass G = Nat.succ n\n\u22a2 nilpotencyClass (G \u29f8 center G) = n"}, {"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 succ\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nn : \u2115\nhn : nilpotencyClass G = Nat.succ n\n\u22a2 nilpotencyClass (G \u29f8 center G) = n", "state_after": "case succ.a\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nn : \u2115\nhn : nilpotencyClass G = Nat.succ n\n\u22a2 nilpotencyClass (G \u29f8 center G) \u2264 n\n\ncase succ.a\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nn : \u2115\nhn : nilpotencyClass G = Nat.succ n\n\u22a2 n \u2264 nilpotencyClass (G \u29f8 center G)"}, {"tactic": "simpa", "annotated_tactic": ["simpa", []], "state_before": "G : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nn : \u2115\nhn : nilpotencyClass G = Nat.succ n\nthis : nilpotencyClass (G \u29f8 center G) = n\n\u22a2 nilpotencyClass (G \u29f8 center G) = Nat.succ n - 1", "state_after": "no goals"}, {"tactic": "apply upperCentralSeries_eq_top_iff_nilpotencyClass_le.mp", "annotated_tactic": ["apply upperCentralSeries_eq_top_iff_nilpotencyClass_le.mp", []], "state_before": "case succ.a\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nn : \u2115\nhn : nilpotencyClass G = Nat.succ n\n\u22a2 nilpotencyClass (G \u29f8 center G) \u2264 n", "state_after": "case succ.a\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nn : \u2115\nhn : nilpotencyClass G = Nat.succ n\n\u22a2 upperCentralSeries (G \u29f8 center G) n = \u22a4"}, {"tactic": "apply @comap_injective G _ _ _ (mk' (center G)) (surjective_quot_mk _)", "annotated_tactic": ["apply @<a>comap_injective</a> G _ _ _ (<a>mk'</a> (<a>center</a> G)) (<a>surjective_quot_mk</a> _)", [{"full_name": "Subgroup.comap_injective", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [3137, 9], "def_end_pos": [3137, 24]}, {"full_name": "QuotientGroup.mk'", "def_path": "Mathlib/GroupTheory/QuotientGroup.lean", "def_pos": [76, 5], "def_end_pos": [76, 8]}, {"full_name": "Subgroup.center", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [2085, 5], "def_end_pos": [2085, 11]}, {"full_name": "surjective_quot_mk", "def_path": "Mathlib/Data/Quot.lean", "def_pos": [345, 9], "def_end_pos": [345, 27]}]], "state_before": "case succ.a\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nn : \u2115\nhn : nilpotencyClass G = Nat.succ n\n\u22a2 upperCentralSeries (G \u29f8 center G) n = \u22a4", "state_after": "case succ.a.a\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nn : \u2115\nhn : nilpotencyClass G = Nat.succ n\n\u22a2 comap (mk' (center G)) (upperCentralSeries (G \u29f8 center G) n) = comap (mk' (center G)) \u22a4"}, {"tactic": "rw [comap_upperCentralSeries_quotient_center, comap_top, \u2190 hn]", "annotated_tactic": ["rw [<a>comap_upperCentralSeries_quotient_center</a>, <a>comap_top</a>, \u2190 hn]", [{"full_name": "comap_upperCentralSeries_quotient_center", "def_path": "Mathlib/GroupTheory/Nilpotent.lean", "def_pos": [592, 9], "def_end_pos": [592, 49]}, {"full_name": "Subgroup.comap_top", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [1603, 9], "def_end_pos": [1603, 18]}]], "state_before": "case succ.a.a\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nn : \u2115\nhn : nilpotencyClass G = Nat.succ n\n\u22a2 comap (mk' (center G)) (upperCentralSeries (G \u29f8 center G) n) = comap (mk' (center G)) \u22a4", "state_after": "case succ.a.a\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nn : \u2115\nhn : nilpotencyClass G = Nat.succ n\n\u22a2 upperCentralSeries G (nilpotencyClass G) = \u22a4"}, {"tactic": "exact upperCentralSeries_nilpotencyClass", "annotated_tactic": ["exact <a>upperCentralSeries_nilpotencyClass</a>", [{"full_name": "upperCentralSeries_nilpotencyClass", "def_path": "Mathlib/GroupTheory/Nilpotent.lean", "def_pos": [368, 9], "def_end_pos": [368, 43]}]], "state_before": "case succ.a.a\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nn : \u2115\nhn : nilpotencyClass G = Nat.succ n\n\u22a2 upperCentralSeries G (nilpotencyClass G) = \u22a4", "state_after": "no goals"}, {"tactic": "apply le_of_add_le_add_right", "annotated_tactic": ["apply <a>le_of_add_le_add_right</a>", [{"full_name": "le_of_add_le_add_right", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [74, 15], "def_end_pos": [74, 37]}]], "state_before": "case succ.a\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nn : \u2115\nhn : nilpotencyClass G = Nat.succ n\n\u22a2 n \u2264 nilpotencyClass (G \u29f8 center G)", "state_after": "case succ.a.bc\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nn : \u2115\nhn : nilpotencyClass G = Nat.succ n\n\u22a2 n + ?succ.a.a \u2264 nilpotencyClass (G \u29f8 center G) + ?succ.a.a\n\ncase succ.a.a\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nn : \u2115\nhn : nilpotencyClass G = Nat.succ n\n\u22a2 \u2115"}, {"tactic": "calc\n n + 1 = n.succ := rfl\n _ = Group.nilpotencyClass G := (symm hn)\n _ \u2264 Group.nilpotencyClass (G \u29f8 center G) + 1 :=\n nilpotencyClass_le_of_ker_le_center _ (le_of_eq (ker_mk' _)) _", "annotated_tactic": ["calc\n n + 1 = n.succ := <a>rfl</a>\n _ = <a>Group.nilpotencyClass</a> G := (<a>symm</a> hn)\n _ \u2264 <a>Group.nilpotencyClass</a> (G \u29f8 <a>center</a> G) + 1 :=\n <a>nilpotencyClass_le_of_ker_le_center</a> _ (<a>le_of_eq</a> (<a>ker_mk'</a> _)) _", [{"full_name": "rfl", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [281, 22], "def_end_pos": [281, 25]}, {"full_name": "Group.nilpotencyClass", "def_path": "Mathlib/GroupTheory/Nilpotent.lean", "def_pos": [362, 19], "def_end_pos": [362, 40]}, {"full_name": "symm", "def_path": "Mathlib/Init/Algebra/Classes.lean", "def_pos": [312, 9], "def_end_pos": [312, 13]}, {"full_name": "Group.nilpotencyClass", "def_path": "Mathlib/GroupTheory/Nilpotent.lean", "def_pos": [362, 19], "def_end_pos": [362, 40]}, {"full_name": "Subgroup.center", "def_path": "Mathlib/GroupTheory/Subgroup/Basic.lean", "def_pos": [2085, 5], "def_end_pos": [2085, 11]}, {"full_name": "nilpotencyClass_le_of_ker_le_center", "def_path": "Mathlib/GroupTheory/Nilpotent.lean", "def_pos": [525, 9], "def_end_pos": [525, 44]}, {"full_name": "le_of_eq", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "QuotientGroup.ker_mk'", "def_path": "Mathlib/GroupTheory/QuotientGroup.lean", "def_pos": [133, 9], "def_end_pos": [133, 16]}]], "state_before": "case succ.a.bc\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nn : \u2115\nhn : nilpotencyClass G = Nat.succ n\n\u22a2 n + ?succ.a.a \u2264 nilpotencyClass (G \u29f8 center G) + ?succ.a.a\n\ncase succ.a.a\nG : Type u_1\ninst\u271d\u00b9 : Group G\nH : Subgroup G\ninst\u271d : Normal H\nhH : Group.IsNilpotent G\nn : \u2115\nhn : nilpotencyClass G = Nat.succ n\n\u22a2 \u2115", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean | CategoryTheory.Limits.MultispanIndex.multispan_obj_left | [
266,
1
] | [
267,
6
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Geometry/Manifold/ContMDiff.lean | SmoothOn.comp' | [
985,
8
] | [
987,
14
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Algebra/IndicatorFunction.lean | Set.mulIndicator_le' | [
795,
1
] | [
796,
75
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/RingTheory/Ideal/Operations.lean | Ideal.mem_colon_singleton | [
399,
1
] | [
401,
84
] | [{"tactic": "simp only [\u2190 Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul]", "annotated_tactic": ["simp only [\u2190 <a>Ideal.submodule_span_eq</a>, <a>Submodule.mem_colon_singleton</a>, <a>smul_eq_mul</a>]", [{"full_name": "Ideal.submodule_span_eq", "def_path": "Mathlib/RingTheory/Ideal/Basic.lean", "def_pos": [114, 9], "def_end_pos": [114, 26]}, {"full_name": "Submodule.mem_colon_singleton", "def_path": "Mathlib/RingTheory/Ideal/Operations.lean", "def_pos": [390, 9], "def_end_pos": [390, 28]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}]], "state_before": "R : Type u\nM : Type v\nF : Type u_1\nG : Type u_2\ninst\u271d\u00b2 : CommRing R\ninst\u271d\u00b9 : AddCommGroup M\ninst\u271d : Module R M\nN N\u2081 N\u2082 P P\u2081 P\u2082 : Submodule R M\nI : Ideal R\nx r : R\n\u22a2 r \u2208 colon I (Ideal.span {x}) \u2194 r * x \u2208 I", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/LinearAlgebra/BilinearForm/TensorProduct.lean | BilinForm.tensorDistribEquiv_tmul | [
128,
1
] | [
132,
6
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Real/ENNReal.lean | ENNReal.toReal_sSup | [
2452,
1
] | [
2454,
83
] | [{"tactic": "simp only [ENNReal.toReal, toNNReal_sSup s hf, NNReal.coe_sSup, Set.image_image]", "annotated_tactic": ["simp only [<a>ENNReal.toReal</a>, <a>toNNReal_sSup</a> s hf, <a>NNReal.coe_sSup</a>, <a>Set.image_image</a>]", [{"full_name": "ENNReal.toReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [168, 15], "def_end_pos": [168, 21]}, {"full_name": "ENNReal.toNNReal_sSup", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [2431, 9], "def_end_pos": [2431, 22]}, {"full_name": "NNReal.coe_sSup", "def_path": "Mathlib/Data/Real/NNReal.lean", "def_pos": [490, 9], "def_end_pos": [490, 17]}, {"full_name": "Set.image_image", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [299, 9], "def_end_pos": [299, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\n\u03b9 : Sort u_3\nf g : \u03b9 \u2192 \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nhf : \u2200 (r : \u211d\u22650\u221e), r \u2208 s \u2192 r \u2260 \u22a4\n\u22a2 ENNReal.toReal (sSup s) = sSup (ENNReal.toReal '' s)", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Int/Basic.lean | Int.neg_nat_succ | [
183,
1
] | [
183,
75
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Order/Monotone/Basic.lean | monotone_fst | [
1185,
1
] | [
1185,
70
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Algebra/Symmetrized.lean | SymAlg.sym_injective | [
99,
1
] | [
100,
16
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Multiset/LocallyFinite.lean | Multiset.Ioo_self | [
109,
1
] | [
109,
81
] | [{"tactic": "rw [Ioo, Finset.Ioo_self, Finset.empty_val]", "annotated_tactic": ["rw [<a>Ioo</a>, <a>Finset.Ioo_self</a>, <a>Finset.empty_val</a>]", [{"full_name": "Multiset.Ioo", "def_path": "Mathlib/Order/LocallyFinite.lean", "def_pos": [546, 5], "def_end_pos": [546, 8]}, {"full_name": "Finset.Ioo_self", "def_path": "Mathlib/Data/Finset/LocallyFinite.lean", "def_pos": [303, 9], "def_end_pos": [303, 17]}, {"full_name": "Finset.empty_val", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [543, 9], "def_end_pos": [543, 18]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : Preorder \u03b1\ninst\u271d : LocallyFiniteOrder \u03b1\na b c : \u03b1\n\u22a2 Ioo a a = 0", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/Vector/Basic.lean | Vector.get_set_of_ne | [
628,
1
] | [
633,
18
] | [{"tactic": "cases v", "annotated_tactic": ["cases v", []], "state_before": "n : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nv : Vector \u03b1 n\ni j : Fin n\nh : i \u2260 j\na : \u03b1\n\u22a2 get (set v i a) j = get v j", "state_after": "case mk\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ni j : Fin n\nh : i \u2260 j\na : \u03b1\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 get (set { val := val\u271d, property := property\u271d } i a) j = get { val := val\u271d, property := property\u271d } j"}, {"tactic": "cases i", "annotated_tactic": ["cases i", []], "state_before": "case mk\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\ni j : Fin n\nh : i \u2260 j\na : \u03b1\nval\u271d : List \u03b1\nproperty\u271d : List.length val\u271d = n\n\u22a2 get (set { val := val\u271d, property := property\u271d } i a) j = get { val := val\u271d, property := property\u271d } j", "state_after": "case mk.mk\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nj : Fin n\na : \u03b1\nval\u271d\u00b9 : List \u03b1\nproperty\u271d : List.length val\u271d\u00b9 = n\nval\u271d : \u2115\nisLt\u271d : val\u271d < n\nh : { val := val\u271d, isLt := isLt\u271d } \u2260 j\n\u22a2 get (set { val := val\u271d\u00b9, property := property\u271d } { val := val\u271d, isLt := isLt\u271d } a) j =\n get { val := val\u271d\u00b9, property := property\u271d } j"}, {"tactic": "cases j", "annotated_tactic": ["cases j", []], "state_before": "case mk.mk\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\nj : Fin n\na : \u03b1\nval\u271d\u00b9 : List \u03b1\nproperty\u271d : List.length val\u271d\u00b9 = n\nval\u271d : \u2115\nisLt\u271d : val\u271d < n\nh : { val := val\u271d, isLt := isLt\u271d } \u2260 j\n\u22a2 get (set { val := val\u271d\u00b9, property := property\u271d } { val := val\u271d, isLt := isLt\u271d } a) j =\n get { val := val\u271d\u00b9, property := property\u271d } j", "state_after": "case mk.mk.mk\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nval\u271d\u00b2 : List \u03b1\nproperty\u271d : List.length val\u271d\u00b2 = n\nval\u271d\u00b9 : \u2115\nisLt\u271d\u00b9 : val\u271d\u00b9 < n\nval\u271d : \u2115\nisLt\u271d : val\u271d < n\nh : { val := val\u271d\u00b9, isLt := isLt\u271d\u00b9 } \u2260 { val := val\u271d, isLt := isLt\u271d }\n\u22a2 get (set { val := val\u271d\u00b2, property := property\u271d } { val := val\u271d\u00b9, isLt := isLt\u271d\u00b9 } a) { val := val\u271d, isLt := isLt\u271d } =\n get { val := val\u271d\u00b2, property := property\u271d } { val := val\u271d, isLt := isLt\u271d }"}, {"tactic": "simp only [set, get_eq_get, toList_mk, Fin.cast_mk, ne_eq]", "annotated_tactic": ["simp only [<a>set</a>, <a>get_eq_get</a>, <a>toList_mk</a>, <a>Fin.cast_mk</a>, <a>ne_eq</a>]", [{"full_name": "Vector.set", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [613, 5], "def_end_pos": [613, 8]}, {"full_name": "Vector.get_eq_get", "def_path": "Mathlib/Data/Vector/Basic.lean", "def_pos": [116, 9], "def_end_pos": [116, 19]}, {"full_name": "Vector.toList_mk", "def_path": "Mathlib/Data/Vector.lean", "def_pos": [245, 9], "def_end_pos": [245, 18]}, {"full_name": "Fin.cast_mk", "def_path": "lake-packages/std/Std/Data/Fin/Lemmas.lean", "def_pos": [301, 17], "def_end_pos": [301, 24]}, {"full_name": "ne_eq", "def_path": "lake-packages/lean4/src/lean/Init/SimpLemmas.lean", "def_pos": [76, 17], "def_end_pos": [76, 22]}]], "state_before": "case mk.mk.mk\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nval\u271d\u00b2 : List \u03b1\nproperty\u271d : List.length val\u271d\u00b2 = n\nval\u271d\u00b9 : \u2115\nisLt\u271d\u00b9 : val\u271d\u00b9 < n\nval\u271d : \u2115\nisLt\u271d : val\u271d < n\nh : { val := val\u271d\u00b9, isLt := isLt\u271d\u00b9 } \u2260 { val := val\u271d, isLt := isLt\u271d }\n\u22a2 get (set { val := val\u271d\u00b2, property := property\u271d } { val := val\u271d\u00b9, isLt := isLt\u271d\u00b9 } a) { val := val\u271d, isLt := isLt\u271d } =\n get { val := val\u271d\u00b2, property := property\u271d } { val := val\u271d, isLt := isLt\u271d }", "state_after": "case mk.mk.mk\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nval\u271d\u00b2 : List \u03b1\nproperty\u271d : List.length val\u271d\u00b2 = n\nval\u271d\u00b9 : \u2115\nisLt\u271d\u00b9 : val\u271d\u00b9 < n\nval\u271d : \u2115\nisLt\u271d : val\u271d < n\nh : { val := val\u271d\u00b9, isLt := isLt\u271d\u00b9 } \u2260 { val := val\u271d, isLt := isLt\u271d }\n\u22a2 List.get (List.set val\u271d\u00b2 val\u271d\u00b9 a)\n { val := val\u271d,\n isLt :=\n (_ :\n val\u271d <\n List.length\n (toList\n { val := List.set val\u271d\u00b2 val\u271d\u00b9 a,\n property :=\n (_ :\n List.length\n (List.set (\u2191{ val := val\u271d\u00b2, property := property\u271d }) (\u2191{ val := val\u271d\u00b9, isLt := isLt\u271d\u00b9 })\n a) =\n n) })) } =\n List.get val\u271d\u00b2 { val := val\u271d, isLt := (_ : val\u271d < List.length (toList { val := val\u271d\u00b2, property := property\u271d })) }"}, {"tactic": "rw [List.get_set_of_ne]", "annotated_tactic": ["rw [<a>List.get_set_of_ne</a>]", [{"full_name": "List.get_set_of_ne", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [1518, 9], "def_end_pos": [1518, 22]}]], "state_before": "case mk.mk.mk\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nval\u271d\u00b2 : List \u03b1\nproperty\u271d : List.length val\u271d\u00b2 = n\nval\u271d\u00b9 : \u2115\nisLt\u271d\u00b9 : val\u271d\u00b9 < n\nval\u271d : \u2115\nisLt\u271d : val\u271d < n\nh : { val := val\u271d\u00b9, isLt := isLt\u271d\u00b9 } \u2260 { val := val\u271d, isLt := isLt\u271d }\n\u22a2 List.get (List.set val\u271d\u00b2 val\u271d\u00b9 a)\n { val := val\u271d,\n isLt :=\n (_ :\n val\u271d <\n List.length\n (toList\n { val := List.set val\u271d\u00b2 val\u271d\u00b9 a,\n property :=\n (_ :\n List.length\n (List.set (\u2191{ val := val\u271d\u00b2, property := property\u271d }) (\u2191{ val := val\u271d\u00b9, isLt := isLt\u271d\u00b9 })\n a) =\n n) })) } =\n List.get val\u271d\u00b2 { val := val\u271d, isLt := (_ : val\u271d < List.length (toList { val := val\u271d\u00b2, property := property\u271d })) }", "state_after": "case mk.mk.mk.h\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nval\u271d\u00b2 : List \u03b1\nproperty\u271d : List.length val\u271d\u00b2 = n\nval\u271d\u00b9 : \u2115\nisLt\u271d\u00b9 : val\u271d\u00b9 < n\nval\u271d : \u2115\nisLt\u271d : val\u271d < n\nh : { val := val\u271d\u00b9, isLt := isLt\u271d\u00b9 } \u2260 { val := val\u271d, isLt := isLt\u271d }\n\u22a2 val\u271d\u00b9 \u2260 val\u271d"}, {"tactic": "simpa using h", "annotated_tactic": ["simpa using h", []], "state_before": "case mk.mk.mk.h\nn : \u2115\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na : \u03b1\nval\u271d\u00b2 : List \u03b1\nproperty\u271d : List.length val\u271d\u00b2 = n\nval\u271d\u00b9 : \u2115\nisLt\u271d\u00b9 : val\u271d\u00b9 < n\nval\u271d : \u2115\nisLt\u271d : val\u271d < n\nh : { val := val\u271d\u00b9, isLt := isLt\u271d\u00b9 } \u2260 { val := val\u271d, isLt := isLt\u271d }\n\u22a2 val\u271d\u00b9 \u2260 val\u271d", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Topology/Algebra/Module/Alternating.lean | ContinuousAlternatingMap.toAlternatingMap_injective | [
98,
1
] | [
100,
50
] | [{"tactic": "convert FunLike.ext'_iff.1 h", "annotated_tactic": ["convert <a>FunLike.ext'_iff</a>.1 h", [{"full_name": "FunLike.ext'_iff", "def_path": "Mathlib/Data/FunLike/Basic.lean", "def_pos": [173, 9], "def_end_pos": [173, 17]}]], "state_before": "R : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\n\u03b9 : Type u_6\ninst\u271d\u00b9\u00b2 : Semiring R\ninst\u271d\u00b9\u00b9 : AddCommMonoid M\ninst\u271d\u00b9\u2070 : Module R M\ninst\u271d\u2079 : TopologicalSpace M\ninst\u271d\u2078 : AddCommMonoid M'\ninst\u271d\u2077 : Module R M'\ninst\u271d\u2076 : TopologicalSpace M'\ninst\u271d\u2075 : AddCommMonoid N\ninst\u271d\u2074 : Module R N\ninst\u271d\u00b3 : TopologicalSpace N\ninst\u271d\u00b2 : AddCommMonoid N'\ninst\u271d\u00b9 : Module R N'\ninst\u271d : TopologicalSpace N'\nn : \u2115\nf\u271d g\u271d f g : M[\u039b^\u03b9]\u2192L[R]N\nh : toAlternatingMap f = toAlternatingMap g\n\u22a2 \u2191f = \u2191g", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/MeasureTheory/Integral/Lebesgue.lean | MeasureTheory.lintegral_sub_le' | [
938,
1
] | [
945,
46
] | [{"tactic": "rw [tsub_le_iff_right]", "annotated_tactic": ["rw [<a>tsub_le_iff_right</a>]", [{"full_name": "tsub_le_iff_right", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [65, 9], "def_end_pos": [65, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\n\u22a2 \u222b\u207b (x : \u03b1), g x \u2202\u03bc - \u222b\u207b (x : \u03b1), f x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), g 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 : AEMeasurable f\n\u22a2 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), g x - f x \u2202\u03bc + \u222b\u207b (x : \u03b1), f x \u2202\u03bc"}, {"tactic": "by_cases hfi : \u222b\u207b x, f x \u2202\u03bc = \u221e", "annotated_tactic": ["by_cases hfi : \u222b\u207b x, f x \u2202\u03bc = \u221e", []], "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 : AEMeasurable f\n\u22a2 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), g x - f x \u2202\u03bc + \u222b\u207b (x : \u03b1), f x \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), g x - f x \u2202\u03bc + \u222b\u207b (x : \u03b1), f x \u2202\u03bc\n\ncase neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhfi : \u00ac\u222b\u207b (x : \u03b1), f x \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), g x - f x \u2202\u03bc + \u222b\u207b (x : \u03b1), f x \u2202\u03bc"}, {"tactic": "rw [hfi, add_top]", "annotated_tactic": ["rw [hfi, <a>add_top</a>]", [{"full_name": "add_top", "def_path": "Mathlib/Algebra/Order/Monoid/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 16]}]], "state_before": "case pos\n\u03b1 : Type u_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 : AEMeasurable f\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), g x - f x \u2202\u03bc + \u222b\u207b (x : \u03b1), f x \u2202\u03bc", "state_after": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u22a4"}, {"tactic": "exact le_top", "annotated_tactic": ["exact <a>le_top</a>", [{"full_name": "le_top", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [98, 9], "def_end_pos": [98, 15]}]], "state_before": "case pos\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhfi : \u222b\u207b (x : \u03b1), f x \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u22a4", "state_after": "no goals"}, {"tactic": "rw [\u2190 lintegral_add_right' _ hf]", "annotated_tactic": ["rw [\u2190 <a>lintegral_add_right'</a> _ hf]", [{"full_name": "MeasureTheory.lintegral_add_right'", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [572, 9], "def_end_pos": [572, 29]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhfi : \u00ac\u222b\u207b (x : \u03b1), f x \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (x : \u03b1), g x - f x \u2202\u03bc + \u222b\u207b (x : \u03b1), f x \u2202\u03bc", "state_after": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhfi : \u00ac\u222b\u207b (x : \u03b1), f x \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), g a - f a + f a \u2202\u03bc"}, {"tactic": "exact lintegral_mono fun x => le_tsub_add", "annotated_tactic": ["exact <a>lintegral_mono</a> fun x => <a>le_tsub_add</a>", [{"full_name": "MeasureTheory.lintegral_mono", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [99, 9], "def_end_pos": [99, 23]}, {"full_name": "le_tsub_add", "def_path": "Mathlib/Algebra/Order/Sub/Defs.lean", "def_pos": [74, 9], "def_end_pos": [74, 20]}]], "state_before": "case neg\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\nm : MeasurableSpace \u03b1\n\u03bc \u03bd : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\u22650\u221e\nhf : AEMeasurable f\nhfi : \u00ac\u222b\u207b (x : \u03b1), f x \u2202\u03bc = \u22a4\n\u22a2 \u222b\u207b (x : \u03b1), g x \u2202\u03bc \u2264 \u222b\u207b (a : \u03b1), g a - f a + f a \u2202\u03bc", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Order/BooleanAlgebra.lean | inf_sdiff_inf | [
111,
1
] | [
111,
89
] | [{"tactic": "rw [inf_comm, inf_inf_sdiff]", "annotated_tactic": ["rw [<a>inf_comm</a>, <a>inf_inf_sdiff</a>]", [{"full_name": "inf_comm", "def_path": "Mathlib/Order/Lattice.lean", "def_pos": [500, 9], "def_end_pos": [500, 17]}, {"full_name": "inf_inf_sdiff", "def_path": "Mathlib/Order/BooleanAlgebra.lean", "def_pos": [102, 9], "def_end_pos": [102, 22]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type u_1\nw x\u271d y\u271d z : \u03b1\ninst\u271d : GeneralizedBooleanAlgebra \u03b1\nx y : \u03b1\n\u22a2 x \\ y \u2293 (x \u2293 y) = \u22a5", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Topology/LocalHomeomorph.lean | LocalHomeomorph.ext | [
328,
11
] | [
330,
52
] | [] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Data/List/Lemmas.lean | List.mapAccumr_eq_foldr | [
93,
1
] | [
100,
58
] | [{"tactic": "simp only [mapAccumr, foldr, mapAccumr_eq_foldr f as]", "annotated_tactic": ["simp only [<a>mapAccumr</a>, <a>foldr</a>, mapAccumr_eq_foldr f as]", [{"full_name": "List.mapAccumr", "def_path": "Mathlib/Init/Data/List/Lemmas.lean", "def_pos": [139, 5], "def_end_pos": [139, 14]}, {"full_name": "List.foldr", "def_path": "lake-packages/lean4/src/lean/Init/Data/List/Basic.lean", "def_pos": [514, 19], "def_end_pos": [514, 24]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03c3 : Type u_4\nf : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2\na : \u03b1\nas : List \u03b1\ns : \u03c3\n\u22a2 mapAccumr f (a :: as) s =\n foldr\n (fun a s =>\n let r := f a s.1;\n (r.1, r.2 :: s.2))\n (s, []) (a :: as)", "state_after": "no goals"}] |
https://github.com/leanprover-community/mathlib4 | 3ce43c18f614b76e161f911b75a3e1ef641620ff | Mathlib/Analysis/Convex/Basic.lean | Convex.smul_mem_of_zero_mem | [
505,
1
] | [
507,
62
] | [{"tactic": "simpa using hs.add_smul_mem zero_mem (by simpa using hx) ht", "annotated_tactic": ["simpa using hs.add_smul_mem zero_mem (by simpa using hx) ht", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\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\nhs : Convex \ud835\udd5c s\nx : E\nzero_mem : 0 \u2208 s\nhx : x \u2208 s\nt : \ud835\udd5c\nht : t \u2208 Icc 0 1\n\u22a2 t \u2022 x \u2208 s", "state_after": "no goals"}, {"tactic": "simpa using hx", "annotated_tactic": ["simpa using hx", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\n\u03b2 : Type u_4\ninst\u271d\u2074 : OrderedRing \ud835\udd5c\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\nhs : Convex \ud835\udd5c s\nx : E\nzero_mem : 0 \u2208 s\nhx : x \u2208 s\nt : \ud835\udd5c\nht : t \u2208 Icc 0 1\n\u22a2 0 + x \u2208 s", "state_after": "no goals"}] |
Subsets and Splits