url
stringclasses 3
values | commit
stringclasses 3
values | file_path
stringlengths 20
79
| full_name
stringlengths 3
115
| start
list | end
list | traced_tactics
stringlengths 2
997k
|
|---|---|---|---|---|---|---|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Field/IsField.lean
|
Semifield.toIsField
|
[
41,
1
] |
[
43,
64
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Order/Basic.lean
|
pi_Ioo_mem_nhds
|
[
1595,
1
] |
[
1597,
35
] |
[{"tactic": "refine' mem_of_superset (set_pi_mem_nhds Set.finite_univ fun i _ => _) (pi_univ_Ioo_subset a b)", "annotated_tactic": ["refine' <a>mem_of_superset</a> (<a>set_pi_mem_nhds</a> <a>Set.finite_univ</a> fun i _ => _) (<a>pi_univ_Ioo_subset</a> a b)", [{"full_name": "Filter.mem_of_superset", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [152, 9], "def_end_pos": [152, 24]}, {"full_name": "set_pi_mem_nhds", "def_path": "Mathlib/Topology/Constructions.lean", "def_pos": [1389, 9], "def_end_pos": [1389, 24]}, {"full_name": "Set.finite_univ", "def_path": "Mathlib/Data/Set/Finite.lean", "def_pos": [722, 9], "def_end_pos": [722, 20]}, {"full_name": "Set.pi_univ_Ioo_subset", "def_path": "Mathlib/Data/Set/Intervals/Pi.lean", "def_pos": [72, 9], "def_end_pos": [72, 27]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : LinearOrder \u03b1\ninst\u271d\u2075 : OrderTopology \u03b1\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : Finite \u03b9\ninst\u271d\u00b3 : (i : \u03b9) \u2192 LinearOrder (\u03c0 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), OrderTopology (\u03c0 i)\na b x : (i : \u03b9) \u2192 \u03c0 i\na' b' x' : \u03b9 \u2192 \u03b1\ninst\u271d : Nonempty \u03b9\nha : \u2200 (i : \u03b9), a i < x i\nhb : \u2200 (i : \u03b9), x i < b i\n\u22a2 Ioo a b \u2208 \ud835\udcdd x", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : LinearOrder \u03b1\ninst\u271d\u2075 : OrderTopology \u03b1\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : Finite \u03b9\ninst\u271d\u00b3 : (i : \u03b9) \u2192 LinearOrder (\u03c0 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), OrderTopology (\u03c0 i)\na b x : (i : \u03b9) \u2192 \u03c0 i\na' b' x' : \u03b9 \u2192 \u03b1\ninst\u271d : Nonempty \u03b9\nha : \u2200 (i : \u03b9), a i < x i\nhb : \u2200 (i : \u03b9), x i < b i\ni : \u03b9\nx\u271d : i \u2208 univ\n\u22a2 Ioo (a i) (b i) \u2208 \ud835\udcdd (x i)"}, {"tactic": "exact Ioo_mem_nhds (ha i) (hb i)", "annotated_tactic": ["exact <a>Ioo_mem_nhds</a> (ha i) (hb i)", [{"full_name": "Ioo_mem_nhds", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [375, 9], "def_end_pos": [375, 21]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\ninst\u271d\u2077 : TopologicalSpace \u03b1\ninst\u271d\u2076 : LinearOrder \u03b1\ninst\u271d\u2075 : OrderTopology \u03b1\n\u03b9 : Type u_1\n\u03c0 : \u03b9 \u2192 Type u_2\ninst\u271d\u2074 : Finite \u03b9\ninst\u271d\u00b3 : (i : \u03b9) \u2192 LinearOrder (\u03c0 i)\ninst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i)\ninst\u271d\u00b9 : \u2200 (i : \u03b9), OrderTopology (\u03c0 i)\na b x : (i : \u03b9) \u2192 \u03c0 i\na' b' x' : \u03b9 \u2192 \u03b1\ninst\u271d : Nonempty \u03b9\nha : \u2200 (i : \u03b9), a i < x i\nhb : \u2200 (i : \u03b9), x i < b i\ni : \u03b9\nx\u271d : i \u2208 univ\n\u22a2 Ioo (a i) (b i) \u2208 \ud835\udcdd (x i)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/BoxIntegral/Partition/Split.lean
|
BoxIntegral.Prepartition.iUnion_compl
|
[
366,
1
] |
[
367,
38
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/Normed/Group/Pointwise.lean
|
div_ball_one
|
[
214,
1
] |
[
214,
95
] |
[{"tactic": "simp [div_eq_mul_inv, mul_ball_one]", "annotated_tactic": ["simp [<a>div_eq_mul_inv</a>, <a>mul_ball_one</a>]", [{"full_name": "div_eq_mul_inv", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [977, 9], "def_end_pos": [977, 23]}, {"full_name": "mul_ball_one", "def_path": "Mathlib/Analysis/Normed/Group/Pointwise.lean", "def_pos": [204, 9], "def_end_pos": [204, 21]}]], "state_before": "E : Type u_1\ninst\u271d : SeminormedCommGroup E\n\u03b5 \u03b4 : \u211d\ns t : Set E\nx y : E\n\u22a2 s / ball 1 \u03b4 = thickening \u03b4 s", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Set/Basic.lean
|
Set.diff_diff
|
[
1939,
1
] |
[
1940,
19
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
MulLECancellable.injective_left
|
[
1632,
11
] |
[
1634,
95
] |
[{"tactic": "dsimp", "annotated_tactic": ["dsimp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Mul \u03b1\ni : IsSymmOp \u03b1 \u03b1 fun x x_1 => x * x_1\ninst\u271d : PartialOrder \u03b1\na : \u03b1\nha : MulLECancellable a\nb c : \u03b1\nh : (fun x => x * a) b = (fun x => x * a) c\n\u22a2 (fun x x_1 => x * x_1) a b = (fun x x_1 => x * x_1) a c", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Mul \u03b1\ni : IsSymmOp \u03b1 \u03b1 fun x x_1 => x * x_1\ninst\u271d : PartialOrder \u03b1\na : \u03b1\nha : MulLECancellable a\nb c : \u03b1\nh : (fun x => x * a) b = (fun x => x * a) c\n\u22a2 a * b = a * c"}, {"tactic": "rwa [i.symm_op a, i.symm_op a]", "annotated_tactic": ["rwa [i.symm_op a, i.symm_op a]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\ninst\u271d\u00b9 : Mul \u03b1\ni : IsSymmOp \u03b1 \u03b1 fun x x_1 => x * x_1\ninst\u271d : PartialOrder \u03b1\na : \u03b1\nha : MulLECancellable a\nb c : \u03b1\nh : (fun x => x * a) b = (fun x => x * a) c\n\u22a2 a * b = a * c", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/Analytic/Linear.lean
|
ContinuousLinearMap.analyticAt_bilinear
|
[
122,
11
] |
[
124,
46
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/CategoryTheory/Limits/Shapes/Images.lean
|
CategoryTheory.Limits.MonoFactorisation.ext
|
[
108,
1
] |
[
115,
25
] |
[{"tactic": "cases' F with _ Fm _ _ Ffac", "annotated_tactic": ["cases' F with _ Fm _ _ Ffac", []], "state_before": "C : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : X \u27f6 Y\nF F' : MonoFactorisation f\nhI : F.I = F'.I\nhm : F.m = eqToHom hI \u226b F'.m\n\u22a2 F = F'", "state_after": "case mk\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : X \u27f6 Y\nF' : MonoFactorisation f\nI\u271d : C\nFm : I\u271d \u27f6 Y\nm_mono\u271d : Mono Fm\ne\u271d : X \u27f6 I\u271d\nFfac : e\u271d \u226b Fm = f\nhI : (mk I\u271d Fm e\u271d).I = F'.I\nhm : (mk I\u271d Fm e\u271d).m = eqToHom hI \u226b F'.m\n\u22a2 mk I\u271d Fm e\u271d = F'"}, {"tactic": "cases' F' with _ Fm' _ _ Ffac'", "annotated_tactic": ["cases' F' with _ Fm' _ _ Ffac'", []], "state_before": "case mk\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : X \u27f6 Y\nF' : MonoFactorisation f\nI\u271d : C\nFm : I\u271d \u27f6 Y\nm_mono\u271d : Mono Fm\ne\u271d : X \u27f6 I\u271d\nFfac : e\u271d \u226b Fm = f\nhI : (mk I\u271d Fm e\u271d).I = F'.I\nhm : (mk I\u271d Fm e\u271d).m = eqToHom hI \u226b F'.m\n\u22a2 mk I\u271d Fm e\u271d = F'", "state_after": "case mk.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : X \u27f6 Y\nI\u271d\u00b9 : C\nFm : I\u271d\u00b9 \u27f6 Y\nm_mono\u271d\u00b9 : Mono Fm\ne\u271d\u00b9 : X \u27f6 I\u271d\u00b9\nFfac : e\u271d\u00b9 \u226b Fm = f\nI\u271d : C\nFm' : I\u271d \u27f6 Y\nm_mono\u271d : Mono Fm'\ne\u271d : X \u27f6 I\u271d\nFfac' : e\u271d \u226b Fm' = f\nhI : (mk I\u271d\u00b9 Fm e\u271d\u00b9).I = (mk I\u271d Fm' e\u271d).I\nhm : (mk I\u271d\u00b9 Fm e\u271d\u00b9).m = eqToHom hI \u226b (mk I\u271d Fm' e\u271d).m\n\u22a2 mk I\u271d\u00b9 Fm e\u271d\u00b9 = mk I\u271d Fm' e\u271d"}, {"tactic": "cases' hI", "annotated_tactic": ["cases' hI", []], "state_before": "case mk.mk\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : X \u27f6 Y\nI\u271d\u00b9 : C\nFm : I\u271d\u00b9 \u27f6 Y\nm_mono\u271d\u00b9 : Mono Fm\ne\u271d\u00b9 : X \u27f6 I\u271d\u00b9\nFfac : e\u271d\u00b9 \u226b Fm = f\nI\u271d : C\nFm' : I\u271d \u27f6 Y\nm_mono\u271d : Mono Fm'\ne\u271d : X \u27f6 I\u271d\nFfac' : e\u271d \u226b Fm' = f\nhI : (mk I\u271d\u00b9 Fm e\u271d\u00b9).I = (mk I\u271d Fm' e\u271d).I\nhm : (mk I\u271d\u00b9 Fm e\u271d\u00b9).m = eqToHom hI \u226b (mk I\u271d Fm' e\u271d).m\n\u22a2 mk I\u271d\u00b9 Fm e\u271d\u00b9 = mk I\u271d Fm' e\u271d", "state_after": "case mk.mk.refl\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : X \u27f6 Y\nI\u271d : C\nFm : I\u271d \u27f6 Y\nm_mono\u271d\u00b9 : Mono Fm\ne\u271d\u00b9 : X \u27f6 I\u271d\nFfac : e\u271d\u00b9 \u226b Fm = f\nFm' : I\u271d \u27f6 Y\nm_mono\u271d : Mono Fm'\ne\u271d : X \u27f6 I\u271d\nFfac' : e\u271d \u226b Fm' = f\nhm : (mk I\u271d Fm e\u271d\u00b9).m = eqToHom (_ : (mk I\u271d Fm e\u271d\u00b9).I = (mk I\u271d Fm e\u271d\u00b9).I) \u226b (mk I\u271d Fm' e\u271d).m\n\u22a2 mk I\u271d Fm e\u271d\u00b9 = mk I\u271d Fm' e\u271d"}, {"tactic": "simp at hm", "annotated_tactic": ["simp at hm", []], "state_before": "case mk.mk.refl\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : X \u27f6 Y\nI\u271d : C\nFm : I\u271d \u27f6 Y\nm_mono\u271d\u00b9 : Mono Fm\ne\u271d\u00b9 : X \u27f6 I\u271d\nFfac : e\u271d\u00b9 \u226b Fm = f\nFm' : I\u271d \u27f6 Y\nm_mono\u271d : Mono Fm'\ne\u271d : X \u27f6 I\u271d\nFfac' : e\u271d \u226b Fm' = f\nhm : (mk I\u271d Fm e\u271d\u00b9).m = eqToHom (_ : (mk I\u271d Fm e\u271d\u00b9).I = (mk I\u271d Fm e\u271d\u00b9).I) \u226b (mk I\u271d Fm' e\u271d).m\n\u22a2 mk I\u271d Fm e\u271d\u00b9 = mk I\u271d Fm' e\u271d", "state_after": "case mk.mk.refl\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : X \u27f6 Y\nI\u271d : C\nFm : I\u271d \u27f6 Y\nm_mono\u271d\u00b9 : Mono Fm\ne\u271d\u00b9 : X \u27f6 I\u271d\nFfac : e\u271d\u00b9 \u226b Fm = f\nFm' : I\u271d \u27f6 Y\nm_mono\u271d : Mono Fm'\ne\u271d : X \u27f6 I\u271d\nFfac' : e\u271d \u226b Fm' = f\nhm : Fm = Fm'\n\u22a2 mk I\u271d Fm e\u271d\u00b9 = mk I\u271d Fm' e\u271d"}, {"tactic": "congr", "annotated_tactic": ["congr", []], "state_before": "case mk.mk.refl\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : X \u27f6 Y\nI\u271d : C\nFm : I\u271d \u27f6 Y\nm_mono\u271d\u00b9 : Mono Fm\ne\u271d\u00b9 : X \u27f6 I\u271d\nFfac : e\u271d\u00b9 \u226b Fm = f\nFm' : I\u271d \u27f6 Y\nm_mono\u271d : Mono Fm'\ne\u271d : X \u27f6 I\u271d\nFfac' : e\u271d \u226b Fm' = f\nhm : Fm = Fm'\n\u22a2 mk I\u271d Fm e\u271d\u00b9 = mk I\u271d Fm' e\u271d", "state_after": "case mk.mk.refl.e_e\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : X \u27f6 Y\nI\u271d : C\nFm : I\u271d \u27f6 Y\nm_mono\u271d\u00b9 : Mono Fm\ne\u271d\u00b9 : X \u27f6 I\u271d\nFfac : e\u271d\u00b9 \u226b Fm = f\nFm' : I\u271d \u27f6 Y\nm_mono\u271d : Mono Fm'\ne\u271d : X \u27f6 I\u271d\nFfac' : e\u271d \u226b Fm' = f\nhm : Fm = Fm'\n\u22a2 e\u271d\u00b9 = e\u271d"}, {"tactic": "apply (cancel_mono Fm).1", "annotated_tactic": ["apply (<a>cancel_mono</a> Fm).1", [{"full_name": "CategoryTheory.cancel_mono", "def_path": "Mathlib/CategoryTheory/Category/Basic.lean", "def_pos": [292, 9], "def_end_pos": [292, 20]}]], "state_before": "case mk.mk.refl.e_e\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : X \u27f6 Y\nI\u271d : C\nFm : I\u271d \u27f6 Y\nm_mono\u271d\u00b9 : Mono Fm\ne\u271d\u00b9 : X \u27f6 I\u271d\nFfac : e\u271d\u00b9 \u226b Fm = f\nFm' : I\u271d \u27f6 Y\nm_mono\u271d : Mono Fm'\ne\u271d : X \u27f6 I\u271d\nFfac' : e\u271d \u226b Fm' = f\nhm : Fm = Fm'\n\u22a2 e\u271d\u00b9 = e\u271d", "state_after": "case mk.mk.refl.e_e\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : X \u27f6 Y\nI\u271d : C\nFm : I\u271d \u27f6 Y\nm_mono\u271d\u00b9 : Mono Fm\ne\u271d\u00b9 : X \u27f6 I\u271d\nFfac : e\u271d\u00b9 \u226b Fm = f\nFm' : I\u271d \u27f6 Y\nm_mono\u271d : Mono Fm'\ne\u271d : X \u27f6 I\u271d\nFfac' : e\u271d \u226b Fm' = f\nhm : Fm = Fm'\n\u22a2 e\u271d\u00b9 \u226b Fm = e\u271d \u226b Fm"}, {"tactic": "rw [Ffac, hm, Ffac']", "annotated_tactic": ["rw [Ffac, hm, Ffac']", []], "state_before": "case mk.mk.refl.e_e\nC : Type u\ninst\u271d : Category.{v, u} C\nX Y : C\nf : X \u27f6 Y\nI\u271d : C\nFm : I\u271d \u27f6 Y\nm_mono\u271d\u00b9 : Mono Fm\ne\u271d\u00b9 : X \u27f6 I\u271d\nFfac : e\u271d\u00b9 \u226b Fm = f\nFm' : I\u271d \u27f6 Y\nm_mono\u271d : Mono Fm'\ne\u271d : X \u27f6 I\u271d\nFfac' : e\u271d \u226b Fm' = f\nhm : Fm = Fm'\n\u22a2 e\u271d\u00b9 \u226b Fm = e\u271d \u226b Fm", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/LinearAlgebra/SymplecticGroup.lean
|
Matrix.isUnit_det_J
|
[
74,
1
] |
[
75,
63
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/List/Sublists.lean
|
List.sublists_cons_perm_append
|
[
419,
1
] |
[
423,
99
] |
[{"tactic": "rw [sublists'_cons]", "annotated_tactic": ["rw [<a>sublists'_cons</a>]", [{"full_name": "List.sublists'_cons", "def_path": "Mathlib/Data/List/Sublists.lean", "def_pos": [75, 9], "def_end_pos": [75, 23]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\n\u22a2 sublists' (a :: l) ~ sublists l ++ map (cons a) (sublists l)", "state_after": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\n\u22a2 sublists' l ++ map (cons a) (sublists' l) ~ sublists l ++ map (cons a) (sublists l)"}, {"tactic": "exact Perm.append (sublists_perm_sublists' _).symm (Perm.map _ (sublists_perm_sublists' _).symm)", "annotated_tactic": ["exact <a>Perm.append</a> (<a>sublists_perm_sublists'</a> _).<a>symm</a> (<a>Perm.map</a> _ (<a>sublists_perm_sublists'</a> _).<a>symm</a>)", [{"full_name": "List.Perm.append", "def_path": "Mathlib/Data/List/Perm.lean", "def_pos": [120, 9], "def_end_pos": [120, 20]}, {"full_name": "List.sublists_perm_sublists'", "def_path": "Mathlib/Data/List/Sublists.lean", "def_pos": [412, 9], "def_end_pos": [412, 32]}, {"full_name": "List.Perm.symm", "def_path": "Mathlib/Data/List/Perm.lean", "def_pos": [56, 19], "def_end_pos": [56, 28]}, {"full_name": "List.Perm.map", "def_path": "Mathlib/Data/List/Perm.lean", "def_pos": [254, 9], "def_end_pos": [254, 17]}, {"full_name": "List.sublists_perm_sublists'", "def_path": "Mathlib/Data/List/Sublists.lean", "def_pos": [412, 9], "def_end_pos": [412, 32]}, {"full_name": "List.Perm.symm", "def_path": "Mathlib/Data/List/Perm.lean", "def_pos": [56, 19], "def_end_pos": [56, 28]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\na : \u03b1\nl : List \u03b1\n\u22a2 sublists' l ++ map (cons a) (sublists' l) ~ sublists l ++ map (cons a) (sublists l)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/GroupTheory/Subsemigroup/Basic.lean
|
Subsemigroup.subset_closure
|
[
313,
1
] |
[
313,
86
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Maps.lean
|
Embedding.map_nhds_eq
|
[
230,
1
] |
[
232,
21
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/SetTheory/Game/Ordinal.lean
|
Ordinal.toPGame_le_iff
|
[
154,
1
] |
[
155,
75
] |
[{"tactic": "contrapose", "annotated_tactic": ["contrapose", []], "state_before": "a b : Ordinal.{u_1}\n\u22a2 toPGame a \u2264 toPGame b \u2192 a \u2264 b", "state_after": "a b : Ordinal.{u_1}\n\u22a2 \u00aca \u2264 b \u2192 \u00actoPGame a \u2264 toPGame b"}, {"tactic": "rw [not_le, PGame.not_le]", "annotated_tactic": ["rw [<a>not_le</a>, <a>PGame.not_le</a>]", [{"full_name": "not_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [373, 9], "def_end_pos": [373, 15]}, {"full_name": "SetTheory.PGame.not_le", "def_path": "Mathlib/SetTheory/Game/PGame.lean", "def_pos": [410, 19], "def_end_pos": [410, 25]}]], "state_before": "a b : Ordinal.{u_1}\n\u22a2 \u00aca \u2264 b \u2192 \u00actoPGame a \u2264 toPGame b", "state_after": "a b : Ordinal.{u_1}\n\u22a2 b < a \u2192 toPGame b \u29cf toPGame a"}, {"tactic": "exact toPGame_lf", "annotated_tactic": ["exact <a>toPGame_lf</a>", [{"full_name": "Ordinal.toPGame_lf", "def_path": "Mathlib/SetTheory/Game/Ordinal.lean", "def_pos": [130, 9], "def_end_pos": [130, 19]}]], "state_before": "a b : Ordinal.{u_1}\n\u22a2 b < a \u2192 toPGame b \u29cf toPGame a", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Pi/Algebra.lean
|
Pi.mulSingle_eq_of_ne
|
[
244,
1
] |
[
245,
30
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/MeasureTheory/Group/FundamentalDomain.lean
|
MeasureTheory.IsFundamentalDomain.smul_of_comm
|
[
181,
1
] |
[
185,
34
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Logic/Function/Basic.lean
|
Function.Surjective.forall
|
[
192,
11
] |
[
196,
14
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Finset/Pointwise.lean
|
Finset.smul_finset_singleton
|
[
1653,
1
] |
[
1654,
22
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Algebra/RestrictScalars.lean
|
RestrictScalars.ringEquiv_map_smul
|
[
210,
1
] |
[
213,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Algebra/Hom.lean
|
AlgHom.map_sum
|
[
269,
11
] |
[
271,
16
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Basic.lean
|
TopologicalSpace.ext
|
[
114,
11
] |
[
116,
43
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Combinatorics/Additive/Behrend.lean
|
Behrend.dValue_pos
|
[
406,
1
] |
[
426,
54
] |
[{"tactic": "have hN\u2080 : 0 < (N : \u211d) := cast_pos.2 (succ_pos'.trans_le hN\u2083)", "annotated_tactic": ["have hN\u2080 : 0 < (N : \u211d) := <a>cast_pos</a>.2 (succ_pos'.trans_le hN\u2083)", [{"full_name": "Nat.cast_pos", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\n\u22a2 0 < dValue N", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 0 < dValue N"}, {"tactic": "rw [dValue, floor_pos, \u2190 log_le_log zero_lt_one, log_one, log_div _ two_ne_zero, log_rpow hN\u2080,\n div_mul_eq_mul_div, one_mul, sub_nonneg, le_div_iff]", "annotated_tactic": ["rw [<a>dValue</a>, <a>floor_pos</a>, \u2190 <a>log_le_log</a> <a>zero_lt_one</a>, <a>log_one</a>, <a>log_div</a> _ <a>two_ne_zero</a>, <a>log_rpow</a> hN\u2080,\n <a>div_mul_eq_mul_div</a>, <a>one_mul</a>, <a>sub_nonneg</a>, <a>le_div_iff</a>]", [{"full_name": "Behrend.dValue", "def_path": "Mathlib/Combinatorics/Additive/Behrend.lean", "def_pos": [375, 19], "def_end_pos": [375, 25]}, {"full_name": "Nat.floor_pos", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [210, 9], "def_end_pos": [210, 18]}, {"full_name": "Real.log_le_log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [144, 9], "def_end_pos": [144, 19]}, {"full_name": "zero_lt_one", "def_path": "Mathlib/Algebra/Order/ZeroLEOne.lean", "def_pos": [39, 15], "def_end_pos": [39, 26]}, {"full_name": "Real.log_one", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [101, 9], "def_end_pos": [101, 16]}, {"full_name": "Real.log_div", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [133, 9], "def_end_pos": [133, 16]}, {"full_name": "two_ne_zero", "def_path": "Mathlib/Algebra/NeZero.lean", "def_pos": [62, 7], "def_end_pos": [62, 18]}, {"full_name": "Real.log_rpow", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [401, 9], "def_end_pos": [401, 17]}, {"full_name": "div_mul_eq_mul_div", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [552, 9], "def_end_pos": [552, 27]}, {"full_name": "one_mul", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [464, 9], "def_end_pos": [464, 16]}, {"full_name": "sub_nonneg", "def_path": "Mathlib/Algebra/Order/Group/Defs.lean", "def_pos": [720, 30], "def_end_pos": [720, 40]}, {"full_name": "le_div_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [132, 9], "def_end_pos": [132, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 0 < dValue N", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 log 2 * \u2191(nValue N) \u2264 log \u2191N\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 0 < \u2191(nValue N)\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 \u2191N ^ (1 / \u2191(nValue N)) \u2260 0\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 0 < \u2191N ^ (1 / \u2191(nValue N)) / 2"}, {"tactic": "have : (nValue N : \u211d) \u2264 2 * sqrt (log N) := by\n apply (ceil_lt_add_one <| sqrt_nonneg _).le.trans\n rw [two_mul, add_le_add_iff_left]\n apply le_sqrt_of_sq_le\n rw [one_pow, le_log_iff_exp_le hN\u2080]\n exact (exp_one_lt_d9.le.trans <| by norm_num).trans (cast_le.2 hN\u2083)", "annotated_tactic": ["have : (<a>nValue</a> N : \u211d) \u2264 2 * <a>sqrt</a> (<a>log</a> N) := by\n apply (<a>ceil_lt_add_one</a> <| <a>sqrt_nonneg</a> _).le.trans\n rw [<a>two_mul</a>, <a>add_le_add_iff_left</a>]\n apply <a>le_sqrt_of_sq_le</a>\n rw [<a>one_pow</a>, <a>le_log_iff_exp_le</a> hN\u2080]\n exact (exp_one_lt_d9.le.trans <| by norm_num).<a>trans</a> (<a>cast_le</a>.2 hN\u2083)", [{"full_name": "Behrend.nValue", "def_path": "Mathlib/Combinatorics/Additive/Behrend.lean", "def_pos": [370, 19], "def_end_pos": [370, 25]}, {"full_name": "Real.sqrt", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [166, 19], "def_end_pos": [166, 23]}, {"full_name": "Real.log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [43, 19], "def_end_pos": [43, 22]}, {"full_name": "Nat.ceil_lt_add_one", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [497, 9], "def_end_pos": [497, 24]}, {"full_name": "Real.sqrt_nonneg", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [194, 9], "def_end_pos": [194, 20]}, {"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}, {"full_name": "add_le_add_iff_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}, {"full_name": "Real.le_sqrt_of_sq_le", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [326, 9], "def_end_pos": [326, 25]}, {"full_name": "one_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [90, 9], "def_end_pos": [90, 16]}, {"full_name": "Real.le_log_iff_exp_le", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [171, 9], "def_end_pos": [171, 26]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "Nat.cast_le", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [91, 9], "def_end_pos": [91, 16]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 log 2 * \u2191(nValue N) \u2264 log \u2191N", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 log 2 * \u2191(nValue N) \u2264 log \u2191N"}, {"tactic": "apply (mul_le_mul_of_nonneg_left this <| log_nonneg one_le_two).trans _", "annotated_tactic": ["apply (<a>mul_le_mul_of_nonneg_left</a> this <| <a>log_nonneg</a> <a>one_le_two</a>).<a>trans</a> _", [{"full_name": "mul_le_mul_of_nonneg_left", "def_path": "Mathlib/Algebra/Order/Ring/Lemmas.lean", "def_pos": [152, 9], "def_end_pos": [152, 34]}, {"full_name": "Real.log_nonneg", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [209, 9], "def_end_pos": [209, 19]}, {"full_name": "one_le_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [50, 7], "def_end_pos": [50, 17]}, {"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 log 2 * \u2191(nValue N) \u2264 log \u2191N", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 log 2 * (2 * Real.sqrt (log \u2191N)) \u2264 log \u2191N"}, {"tactic": "rw [\u2190 mul_assoc, \u2190 le_div_iff (Real.sqrt_pos.2 <| log_pos <| one_lt_cast.2 _), div_sqrt]", "annotated_tactic": ["rw [\u2190 <a>mul_assoc</a>, \u2190 <a>le_div_iff</a> (<a>Real.sqrt_pos</a>.2 <| <a>log_pos</a> <| <a>one_lt_cast</a>.2 _), <a>div_sqrt</a>]", [{"full_name": "mul_assoc", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [264, 9], "def_end_pos": [264, 18]}, {"full_name": "le_div_iff", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [132, 9], "def_end_pos": [132, 19]}, {"full_name": "Real.sqrt_pos", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [350, 9], "def_end_pos": [350, 17]}, {"full_name": "Real.log_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [182, 9], "def_end_pos": [182, 16]}, {"full_name": "Nat.one_lt_cast", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [101, 9], "def_end_pos": [101, 20]}, {"full_name": "Real.div_sqrt", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [419, 9], "def_end_pos": [419, 17]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 log 2 * (2 * Real.sqrt (log \u2191N)) \u2264 log \u2191N", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 log 2 * 2 \u2264 Real.sqrt (log \u2191N)\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 1 < N"}, {"tactic": "exact hN\u2083.trans_lt' (by norm_num)", "annotated_tactic": ["exact hN\u2083.trans_lt' (by norm_num)", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 1 < N", "state_after": "no goals"}, {"tactic": "apply (ceil_lt_add_one <| sqrt_nonneg _).le.trans", "annotated_tactic": ["apply (<a>ceil_lt_add_one</a> <| <a>sqrt_nonneg</a> _).le.trans", [{"full_name": "Nat.ceil_lt_add_one", "def_path": "Mathlib/Algebra/Order/Floor.lean", "def_pos": [497, 9], "def_end_pos": [497, 24]}, {"full_name": "Real.sqrt_nonneg", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [194, 9], "def_end_pos": [194, 20]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 Real.sqrt (log \u2191N) + 1 \u2264 2 * Real.sqrt (log \u2191N)"}, {"tactic": "rw [two_mul, add_le_add_iff_left]", "annotated_tactic": ["rw [<a>two_mul</a>, <a>add_le_add_iff_left</a>]", [{"full_name": "two_mul", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [177, 9], "def_end_pos": [177, 16]}, {"full_name": "add_le_add_iff_left", "def_path": "Mathlib/Algebra/Order/Monoid/Lemmas.lean", "def_pos": [82, 3], "def_end_pos": [82, 14]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 Real.sqrt (log \u2191N) + 1 \u2264 2 * Real.sqrt (log \u2191N)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 1 \u2264 Real.sqrt (log \u2191N)"}, {"tactic": "apply le_sqrt_of_sq_le", "annotated_tactic": ["apply <a>le_sqrt_of_sq_le</a>", [{"full_name": "Real.le_sqrt_of_sq_le", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [326, 9], "def_end_pos": [326, 25]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 1 \u2264 Real.sqrt (log \u2191N)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 1 ^ 2 \u2264 log \u2191N"}, {"tactic": "rw [one_pow, le_log_iff_exp_le hN\u2080]", "annotated_tactic": ["rw [<a>one_pow</a>, <a>le_log_iff_exp_le</a> hN\u2080]", [{"full_name": "one_pow", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [90, 9], "def_end_pos": [90, 16]}, {"full_name": "Real.le_log_iff_exp_le", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [171, 9], "def_end_pos": [171, 26]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 1 ^ 2 \u2264 log \u2191N", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 rexp 1 \u2264 \u2191N"}, {"tactic": "exact (exp_one_lt_d9.le.trans <| by norm_num).trans (cast_le.2 hN\u2083)", "annotated_tactic": ["exact (exp_one_lt_d9.le.trans <| by norm_num).<a>trans</a> (<a>cast_le</a>.2 hN\u2083)", [{"full_name": "LE.le.trans", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [120, 7], "def_end_pos": [120, 18]}, {"full_name": "Nat.cast_le", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [91, 9], "def_end_pos": [91, 16]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 rexp 1 \u2264 \u2191N", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 2.7182818286 \u2264 \u21918", "state_after": "no goals"}, {"tactic": "apply log_two_mul_two_le_sqrt_log_eight.trans", "annotated_tactic": ["apply log_two_mul_two_le_sqrt_log_eight.trans", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 log 2 * 2 \u2264 Real.sqrt (log \u2191N)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 Real.sqrt (log 8) \u2264 Real.sqrt (log \u2191N)"}, {"tactic": "apply Real.sqrt_le_sqrt", "annotated_tactic": ["apply <a>Real.sqrt_le_sqrt</a>", [{"full_name": "Real.sqrt_le_sqrt", "def_path": "Mathlib/Data/Real/Sqrt.lean", "def_pos": [274, 9], "def_end_pos": [274, 21]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 Real.sqrt (log 8) \u2264 Real.sqrt (log \u2191N)", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 log 8 \u2264 log \u2191N"}, {"tactic": "rw [log_le_log _ hN\u2080]", "annotated_tactic": ["rw [<a>log_le_log</a> _ hN\u2080]", [{"full_name": "Real.log_le_log", "def_path": "Mathlib/Analysis/SpecialFunctions/Log/Basic.lean", "def_pos": [144, 9], "def_end_pos": [144, 19]}]], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 log 8 \u2264 log \u2191N", "state_after": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 8 \u2264 \u2191N\n\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 0 < 8"}, {"tactic": "exact_mod_cast hN\u2083", "annotated_tactic": ["exact_mod_cast hN\u2083", []], "state_before": "case h\n\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 8 \u2264 \u2191N", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 0 < 8", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\nthis : \u2191(nValue N) \u2264 2 * Real.sqrt (log \u2191N)\n\u22a2 1 < 8", "state_after": "no goals"}, {"tactic": "exact cast_pos.2 (nValue_pos <| hN\u2083.trans' <| by norm_num)", "annotated_tactic": ["exact <a>cast_pos</a>.2 (<a>nValue_pos</a> <| hN\u2083.trans' <| by norm_num)", [{"full_name": "Nat.cast_pos", "def_path": "Mathlib/Data/Nat/Cast/Order.lean", "def_pos": [72, 9], "def_end_pos": [72, 17]}, {"full_name": "Behrend.nValue_pos", "def_path": "Mathlib/Combinatorics/Additive/Behrend.lean", "def_pos": [379, 9], "def_end_pos": [379, 19]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 0 < \u2191(nValue N)", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 2 \u2264 8", "state_after": "no goals"}, {"tactic": "exact (rpow_pos_of_pos hN\u2080 _).ne'", "annotated_tactic": ["exact (<a>rpow_pos_of_pos</a> hN\u2080 _).<a>ne'</a>", [{"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}, {"full_name": "LT.lt.ne'", "def_path": "Mathlib/Order/Basic.lean", "def_pos": [328, 9], "def_end_pos": [328, 12]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 \u2191N ^ (1 / \u2191(nValue N)) \u2260 0", "state_after": "no goals"}, {"tactic": "exact div_pos (rpow_pos_of_pos hN\u2080 _) zero_lt_two", "annotated_tactic": ["exact <a>div_pos</a> (<a>rpow_pos_of_pos</a> hN\u2080 _) <a>zero_lt_two</a>", [{"full_name": "div_pos", "def_path": "Mathlib/Algebra/Order/Field/Basic.lean", "def_pos": [89, 9], "def_end_pos": [89, 16]}, {"full_name": "Real.rpow_pos_of_pos", "def_path": "Mathlib/Analysis/SpecialFunctions/Pow/Real.lean", "def_pos": [92, 9], "def_end_pos": [92, 24]}, {"full_name": "zero_lt_two", "def_path": "Mathlib/Algebra/Order/Monoid/NatCast.lean", "def_pos": [71, 15], "def_end_pos": [71, 26]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\nn d k N : \u2115\nx : Fin n \u2192 \u2115\nhN\u2083 : 8 \u2264 N\nhN\u2080 : 0 < \u2191N\n\u22a2 0 < \u2191N ^ (1 / \u2191(nValue N)) / 2", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Computability/Primrec.lean
|
Nat.Primrec'.of_prim
|
[
1522,
1
] |
[
1540,
93
] |
[{"tactic": "simp [encodek]", "annotated_tactic": ["simp [<a>encodek</a>]", [{"full_name": "Encodable.encodek", "def_path": "Mathlib/Logic/Encodable/Basic.lean", "def_pos": [53, 3], "def_end_pos": [53, 10]}]], "state_before": "n : \u2115\nf : Vector \u2115 n \u2192 \u2115\nthis : \u2200 (f : \u2115 \u2192 \u2115), Nat.Primrec f \u2192 Primrec' fun v => f (Vector.head v)\nhf : Primrec f\ni : Vector \u2115 n\n\u22a2 Nat.pred ((fun m => encode (Option.map f (decode m))) (encode i)) = f i", "state_after": "no goals"}, {"tactic": "induction hf", "annotated_tactic": ["induction hf", []], "state_before": "n : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\nhf : Nat.Primrec f\n\u22a2 Primrec' fun v => f (Vector.head v)", "state_after": "case zero\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun x => 0) (Vector.head v)\n\ncase succ\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => Nat.succ (Vector.head v)\n\ncase left\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).1) (Vector.head v)\n\ncase right\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)\n\ncase pair\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)\n\ncase comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)"}, {"tactic": "case zero => exact const 0", "annotated_tactic": ["case zero => exact <a>const</a> 0", [{"full_name": "Nat.Primrec'.const", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1399, 9], "def_end_pos": [1399, 14]}]], "state_before": "case zero\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun x => 0) (Vector.head v)\n\ncase succ\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => Nat.succ (Vector.head v)\n\ncase left\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).1) (Vector.head v)\n\ncase right\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)\n\ncase pair\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)\n\ncase comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)", "state_after": "case succ\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => Nat.succ (Vector.head v)\n\ncase left\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).1) (Vector.head v)\n\ncase right\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)\n\ncase pair\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)\n\ncase comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)"}, {"tactic": "case succ => exact succ", "annotated_tactic": ["case succ => exact <a>succ</a>", [{"full_name": "Nat.Primrec'.succ", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1363, 5], "def_end_pos": [1363, 9]}]], "state_before": "case succ\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => Nat.succ (Vector.head v)\n\ncase left\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).1) (Vector.head v)\n\ncase right\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)\n\ncase pair\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)\n\ncase comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)", "state_after": "case left\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).1) (Vector.head v)\n\ncase right\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)\n\ncase pair\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)\n\ncase comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)"}, {"tactic": "case left => exact unpair\u2081 head", "annotated_tactic": ["case left => exact <a>unpair\u2081</a> <a>head</a>", [{"full_name": "Nat.Primrec'.unpair\u2081", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1508, 9], "def_end_pos": [1508, 16]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}]], "state_before": "case left\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).1) (Vector.head v)\n\ncase right\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)\n\ncase pair\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)\n\ncase comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)", "state_after": "case right\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)\n\ncase pair\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)\n\ncase comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)"}, {"tactic": "case right => exact unpair\u2082 head", "annotated_tactic": ["case right => exact <a>unpair\u2082</a> <a>head</a>", [{"full_name": "Nat.Primrec'.unpair\u2082", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1515, 9], "def_end_pos": [1515, 16]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}]], "state_before": "case right\nn : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)\n\ncase pair\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)\n\ncase comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)", "state_after": "case pair\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)\n\ncase comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)"}, {"tactic": "case pair f g _ _ hf hg => exact natPair.comp\u2082 _ hf hg", "annotated_tactic": ["case pair f g _ _ hf hg => exact natPair.comp\u2082 _ hf hg", []], "state_before": "case pair\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f\u271d n) (g\u271d n)) (Vector.head v)\n\ncase comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)", "state_after": "case comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)"}, {"tactic": "case comp f g _ _ hf hg => exact hf.comp\u2081 _ hg", "annotated_tactic": ["case comp f g _ _ hf hg => exact hf.comp\u2081 _ hg", []], "state_before": "case comp\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f\u271d (g\u271d n)) (Vector.head v)\n\ncase prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)", "state_after": "case prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)"}, {"tactic": "case prec f g _ _ hf hg =>\n simpa using\n prec' (unpair\u2082 head) (hf.comp\u2081 _ (unpair\u2081 head))\n (hg.comp\u2081 _ <|\n natPair.comp\u2082 _ (unpair\u2081 <| tail <| tail head) (natPair.comp\u2082 _ head (tail head)))", "annotated_tactic": ["case prec f g _ _ hf hg =>\n simpa using\n <a>prec'</a> (<a>unpair\u2082</a> <a>head</a>) (hf.comp\u2081 _ (<a>unpair\u2081</a> <a>head</a>))\n (hg.comp\u2081 _ <|\n natPair.comp\u2082 _ (<a>unpair\u2081</a> <| <a>tail</a> <| <a>tail</a> <a>head</a>) (natPair.comp\u2082 _ <a>head</a> (<a>tail</a> <a>head</a>)))", [{"full_name": "Nat.Primrec'.prec'", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1443, 9], "def_end_pos": [1443, 14]}, {"full_name": "Nat.Primrec'.unpair\u2082", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1515, 9], "def_end_pos": [1515, 16]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}, {"full_name": "Nat.Primrec'.unpair\u2081", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1508, 9], "def_end_pos": [1508, 16]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}, {"full_name": "Nat.Primrec'.unpair\u2081", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1508, 9], "def_end_pos": [1508, 16]}, {"full_name": "Nat.Primrec'.tail", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1408, 9], "def_end_pos": [1408, 13]}, {"full_name": "Nat.Primrec'.tail", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1408, 9], "def_end_pos": [1408, 13]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}, {"full_name": "Nat.Primrec'.tail", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1408, 9], "def_end_pos": [1408, 13]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}]], "state_before": "case prec\nn : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf f\u271d g\u271d : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\u271d\na\u271d : Nat.Primrec g\u271d\na_ih\u271d\u00b9 : Primrec' fun v => f\u271d (Vector.head v)\na_ih\u271d : Primrec' fun v => g\u271d (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f\u271d z) (fun y IH => g\u271d (pair z (pair y IH))) n) (Vector.head v)", "state_after": "no goals"}, {"tactic": "exact const 0", "annotated_tactic": ["exact <a>const</a> 0", [{"full_name": "Nat.Primrec'.const", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1399, 9], "def_end_pos": [1399, 14]}]], "state_before": "n : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun x => 0) (Vector.head v)", "state_after": "no goals"}, {"tactic": "exact succ", "annotated_tactic": ["exact <a>succ</a>", [{"full_name": "Nat.Primrec'.succ", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1363, 5], "def_end_pos": [1363, 9]}]], "state_before": "n : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => Nat.succ (Vector.head v)", "state_after": "no goals"}, {"tactic": "exact unpair\u2081 head", "annotated_tactic": ["exact <a>unpair\u2081</a> <a>head</a>", [{"full_name": "Nat.Primrec'.unpair\u2081", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1508, 9], "def_end_pos": [1508, 16]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}]], "state_before": "n : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).1) (Vector.head v)", "state_after": "no goals"}, {"tactic": "exact unpair\u2082 head", "annotated_tactic": ["exact <a>unpair\u2082</a> <a>head</a>", [{"full_name": "Nat.Primrec'.unpair\u2082", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1515, 9], "def_end_pos": [1515, 16]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}]], "state_before": "n : \u2115\nf\u271d : Vector \u2115 n \u2192 \u2115\nf : \u2115 \u2192 \u2115\n\u22a2 Primrec' fun v => (fun n => (unpair n).2) (Vector.head v)", "state_after": "no goals"}, {"tactic": "exact natPair.comp\u2082 _ hf hg", "annotated_tactic": ["exact natPair.comp\u2082 _ hf hg", []], "state_before": "n : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf\u271d f g : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\na\u271d : Nat.Primrec g\nhf : Primrec' fun v => f (Vector.head v)\nhg : Primrec' fun v => g (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => pair (f n) (g n)) (Vector.head v)", "state_after": "no goals"}, {"tactic": "exact hf.comp\u2081 _ hg", "annotated_tactic": ["exact hf.comp\u2081 _ hg", []], "state_before": "n : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf\u271d f g : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\na\u271d : Nat.Primrec g\nhf : Primrec' fun v => f (Vector.head v)\nhg : Primrec' fun v => g (Vector.head v)\n\u22a2 Primrec' fun v => (fun n => f (g n)) (Vector.head v)", "state_after": "no goals"}, {"tactic": "simpa using\n prec' (unpair\u2082 head) (hf.comp\u2081 _ (unpair\u2081 head))\n (hg.comp\u2081 _ <|\n natPair.comp\u2082 _ (unpair\u2081 <| tail <| tail head) (natPair.comp\u2082 _ head (tail head)))", "annotated_tactic": ["simpa using\n <a>prec'</a> (<a>unpair\u2082</a> <a>head</a>) (hf.comp\u2081 _ (<a>unpair\u2081</a> <a>head</a>))\n (hg.comp\u2081 _ <|\n natPair.comp\u2082 _ (<a>unpair\u2081</a> <| <a>tail</a> <| <a>tail</a> <a>head</a>) (natPair.comp\u2082 _ <a>head</a> (<a>tail</a> <a>head</a>)))", [{"full_name": "Nat.Primrec'.prec'", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1443, 9], "def_end_pos": [1443, 14]}, {"full_name": "Nat.Primrec'.unpair\u2082", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1515, 9], "def_end_pos": [1515, 16]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}, {"full_name": "Nat.Primrec'.unpair\u2081", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1508, 9], "def_end_pos": [1508, 16]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}, {"full_name": "Nat.Primrec'.unpair\u2081", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1508, 9], "def_end_pos": [1508, 16]}, {"full_name": "Nat.Primrec'.tail", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1408, 9], "def_end_pos": [1408, 13]}, {"full_name": "Nat.Primrec'.tail", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1408, 9], "def_end_pos": [1408, 13]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}, {"full_name": "Nat.Primrec'.tail", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1408, 9], "def_end_pos": [1408, 13]}, {"full_name": "Nat.Primrec'.head", "def_path": "Mathlib/Computability/Primrec.lean", "def_pos": [1404, 9], "def_end_pos": [1404, 13]}]], "state_before": "n : \u2115\nf\u271d\u00b9 : Vector \u2115 n \u2192 \u2115\nf\u271d f g : \u2115 \u2192 \u2115\na\u271d\u00b9 : Nat.Primrec f\na\u271d : Nat.Primrec g\nhf : Primrec' fun v => f (Vector.head v)\nhg : Primrec' fun v => g (Vector.head v)\n\u22a2 Primrec' fun v => unpaired (fun z n => Nat.rec (f z) (fun y IH => g (pair z (pair y IH))) n) (Vector.head v)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiffAt.neg
|
[
1305,
1
] |
[
1306,
53
] |
[{"tactic": "rw [\u2190 contDiffWithinAt_univ] at *", "annotated_tactic": ["rw [\u2190 <a>contDiffWithinAt_univ</a>] at *", [{"full_name": "contDiffWithinAt_univ", "def_path": "Mathlib/Analysis/Calculus/ContDiffDef.lean", "def_pos": [1335, 9], "def_end_pos": [1335, 30]}]], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf\u271d f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nf : E \u2192 F\nhf : ContDiffAt \ud835\udd5c n f x\n\u22a2 ContDiffAt \ud835\udd5c n (fun x => -f x) x", "state_after": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf\u271d f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nf : E \u2192 F\nhf : ContDiffWithinAt \ud835\udd5c n f univ x\n\u22a2 ContDiffWithinAt \ud835\udd5c n (fun x => -f x) univ x"}, {"tactic": "exact hf.neg", "annotated_tactic": ["exact hf.neg", []], "state_before": "\ud835\udd5c : Type u_1\ninst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\nD : Type uD\ninst\u271d\u2079 : NormedAddCommGroup D\ninst\u271d\u2078 : NormedSpace \ud835\udd5c D\nE : Type uE\ninst\u271d\u2077 : NormedAddCommGroup E\ninst\u271d\u2076 : NormedSpace \ud835\udd5c E\nF : Type uF\ninst\u271d\u2075 : NormedAddCommGroup F\ninst\u271d\u2074 : NormedSpace \ud835\udd5c F\nG : Type uG\ninst\u271d\u00b3 : NormedAddCommGroup G\ninst\u271d\u00b2 : NormedSpace \ud835\udd5c G\nX : Type u_2\ninst\u271d\u00b9 : NormedAddCommGroup X\ninst\u271d : NormedSpace \ud835\udd5c X\ns s\u2081 t u : Set E\nf\u271d f\u2081 : E \u2192 F\ng : F \u2192 G\nx x\u2080 : E\nc : F\nb : E \u00d7 F \u2192 G\nm n : \u2115\u221e\np : E \u2192 FormalMultilinearSeries \ud835\udd5c E F\nf : E \u2192 F\nhf : ContDiffWithinAt \ud835\udd5c n f univ x\n\u22a2 ContDiffWithinAt \ud835\udd5c n (fun x => -f x) univ x", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/ContinuousFunction/ZeroAtInfty.lean
|
ZeroAtInftyContinuousMap.coe_toContinuousMap
|
[
99,
1
] |
[
100,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/LocallyConvex/Bounded.lean
|
Bornology.IsVonNBounded.image
|
[
121,
1
] |
[
137,
54
] |
[{"tactic": "let \u03c3' := RingEquiv.ofBijective \u03c3 \u27e8\u03c3.injective, \u03c3.surjective\u27e9", "annotated_tactic": ["let \u03c3' := <a>RingEquiv.ofBijective</a> \u03c3 \u27e8\u03c3.injective, \u03c3.surjective\u27e9", [{"full_name": "RingEquiv.ofBijective", "def_path": "Mathlib/Algebra/Ring/Equiv.lean", "def_pos": [459, 19], "def_end_pos": [459, 30]}]], "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u22a2 IsVonNBounded \ud835\udd5c\u2082 (\u2191f '' s)", "state_after": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u22a2 IsVonNBounded \ud835\udd5c\u2082 (\u2191f '' s)"}, {"tactic": "have \u03c3_iso : Isometry \u03c3 := AddMonoidHomClass.isometry_of_norm \u03c3 fun x => RingHomIsometric.is_iso", "annotated_tactic": ["have \u03c3_iso : <a>Isometry</a> \u03c3 := <a>AddMonoidHomClass.isometry_of_norm</a> \u03c3 fun x => <a>RingHomIsometric.is_iso</a>", [{"full_name": "Isometry", "def_path": "Mathlib/Topology/MetricSpace/Isometry.lean", "def_pos": [35, 5], "def_end_pos": [35, 13]}, {"full_name": "AddMonoidHomClass.isometry_of_norm", "def_path": "Mathlib/Analysis/Normed/Group/Basic.lean", "def_pos": [894, 12], "def_end_pos": [894, 23]}, {"full_name": "RingHomIsometric.is_iso", "def_path": "Mathlib/Analysis/Normed/Field/Basic.lean", "def_pos": [886, 3], "def_end_pos": [886, 9]}]], "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u22a2 IsVonNBounded \ud835\udd5c\u2082 (\u2191f '' s)", "state_after": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u22a2 IsVonNBounded \ud835\udd5c\u2082 (\u2191f '' s)"}, {"tactic": "have \u03c3'_symm_iso : Isometry \u03c3'.symm := \u03c3_iso.right_inv \u03c3'.right_inv", "annotated_tactic": ["have \u03c3'_symm_iso : <a>Isometry</a> \u03c3'.symm := \u03c3_iso.right_inv \u03c3'.right_inv", [{"full_name": "Isometry", "def_path": "Mathlib/Topology/MetricSpace/Isometry.lean", "def_pos": [35, 5], "def_end_pos": [35, 13]}]], "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u22a2 IsVonNBounded \ud835\udd5c\u2082 (\u2191f '' s)", "state_after": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\n\u22a2 IsVonNBounded \ud835\udd5c\u2082 (\u2191f '' s)"}, {"tactic": "have f_tendsto_zero := f.continuous.tendsto 0", "annotated_tactic": ["have f_tendsto_zero := f.continuous.tendsto 0", []], "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\n\u22a2 IsVonNBounded \ud835\udd5c\u2082 (\u2191f '' s)", "state_after": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd (\u2191f 0))\n\u22a2 IsVonNBounded \ud835\udd5c\u2082 (\u2191f '' s)"}, {"tactic": "rw [map_zero] at f_tendsto_zero", "annotated_tactic": ["rw [<a>map_zero</a>] at f_tendsto_zero", [{"full_name": "map_zero", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [201, 3], "def_end_pos": [201, 14]}]], "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd (\u2191f 0))\n\u22a2 IsVonNBounded \ud835\udd5c\u2082 (\u2191f '' s)", "state_after": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u22a2 IsVonNBounded \ud835\udd5c\u2082 (\u2191f '' s)"}, {"tactic": "intro V hV", "annotated_tactic": ["intro V hV", []], "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\n\u22a2 IsVonNBounded \ud835\udd5c\u2082 (\u2191f '' s)", "state_after": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\n\u22a2 Absorbs \ud835\udd5c\u2082 V (\u2191f '' s)"}, {"tactic": "rcases hs (f_tendsto_zero hV) with \u27e8r, hrpos, hr\u27e9", "annotated_tactic": ["rcases hs (f_tendsto_zero hV) with \u27e8r, hrpos, hr\u27e9", []], "state_before": "\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\n\u22a2 Absorbs \ud835\udd5c\u2082 V (\u2191f '' s)", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\n\u22a2 Absorbs \ud835\udd5c\u2082 V (\u2191f '' s)"}, {"tactic": "refine' \u27e8r, hrpos, fun a ha => _\u27e9", "annotated_tactic": ["refine' \u27e8r, hrpos, fun a ha => _\u27e9", []], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\n\u22a2 Absorbs \ud835\udd5c\u2082 V (\u2191f '' s)", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\na : \ud835\udd5c\u2082\nha : r \u2264 \u2016a\u2016\n\u22a2 \u2191f '' s \u2286 a \u2022 V"}, {"tactic": "rw [\u2190 \u03c3'.apply_symm_apply a]", "annotated_tactic": ["rw [\u2190 \u03c3'.apply_symm_apply a]", []], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\na : \ud835\udd5c\u2082\nha : r \u2264 \u2016a\u2016\n\u22a2 \u2191f '' s \u2286 a \u2022 V", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\na : \ud835\udd5c\u2082\nha : r \u2264 \u2016a\u2016\n\u22a2 \u2191f '' s \u2286 \u2191\u03c3' (\u2191(RingEquiv.symm \u03c3') a) \u2022 V"}, {"tactic": "have hanz : a \u2260 0 := norm_pos_iff.mp (hrpos.trans_le ha)", "annotated_tactic": ["have hanz : a \u2260 0 := norm_pos_iff.mp (hrpos.trans_le ha)", []], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\na : \ud835\udd5c\u2082\nha : r \u2264 \u2016a\u2016\n\u22a2 \u2191f '' s \u2286 \u2191\u03c3' (\u2191(RingEquiv.symm \u03c3') a) \u2022 V", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\na : \ud835\udd5c\u2082\nha : r \u2264 \u2016a\u2016\nhanz : a \u2260 0\n\u22a2 \u2191f '' s \u2286 \u2191\u03c3' (\u2191(RingEquiv.symm \u03c3') a) \u2022 V"}, {"tactic": "have : \u03c3'.symm a \u2260 0 := (map_ne_zero \u03c3'.symm.toRingHom).mpr hanz", "annotated_tactic": ["have : \u03c3'.symm a \u2260 0 := (<a>map_ne_zero</a> \u03c3'.symm.toRingHom).<a>mpr</a> hanz", [{"full_name": "map_ne_zero", "def_path": "Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean", "def_pos": [214, 9], "def_end_pos": [214, 20]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\na : \ud835\udd5c\u2082\nha : r \u2264 \u2016a\u2016\nhanz : a \u2260 0\n\u22a2 \u2191f '' s \u2286 \u2191\u03c3' (\u2191(RingEquiv.symm \u03c3') a) \u2022 V", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\na : \ud835\udd5c\u2082\nha : r \u2264 \u2016a\u2016\nhanz : a \u2260 0\nthis : \u2191(RingEquiv.symm \u03c3') a \u2260 0\n\u22a2 \u2191f '' s \u2286 \u2191\u03c3' (\u2191(RingEquiv.symm \u03c3') a) \u2022 V"}, {"tactic": "change _ \u2286 \u03c3 _ \u2022 _", "annotated_tactic": ["change _ \u2286 \u03c3 _ \u2022 _", []], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\na : \ud835\udd5c\u2082\nha : r \u2264 \u2016a\u2016\nhanz : a \u2260 0\nthis : \u2191(RingEquiv.symm \u03c3') a \u2260 0\n\u22a2 \u2191f '' s \u2286 \u2191\u03c3' (\u2191(RingEquiv.symm \u03c3') a) \u2022 V", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\na : \ud835\udd5c\u2082\nha : r \u2264 \u2016a\u2016\nhanz : a \u2260 0\nthis : \u2191(RingEquiv.symm \u03c3') a \u2260 0\n\u22a2 \u2191f '' s \u2286 \u2191\u03c3 (\u2191(RingEquiv.symm \u03c3') a) \u2022 V"}, {"tactic": "rw [Set.image_subset_iff, preimage_smul_set\u209b\u2097 _ _ _ f this.isUnit]", "annotated_tactic": ["rw [<a>Set.image_subset_iff</a>, <a>preimage_smul_set\u209b\u2097</a> _ _ _ f this.isUnit]", [{"full_name": "Set.image_subset_iff", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [497, 9], "def_end_pos": [497, 25]}, {"full_name": "preimage_smul_set\u209b\u2097", "def_path": "Mathlib/Algebra/Module/LinearMap.lean", "def_pos": [379, 9], "def_end_pos": [379, 35]}]], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\na : \ud835\udd5c\u2082\nha : r \u2264 \u2016a\u2016\nhanz : a \u2260 0\nthis : \u2191(RingEquiv.symm \u03c3') a \u2260 0\n\u22a2 \u2191f '' s \u2286 \u2191\u03c3 (\u2191(RingEquiv.symm \u03c3') a) \u2022 V", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\na : \ud835\udd5c\u2082\nha : r \u2264 \u2016a\u2016\nhanz : a \u2260 0\nthis : \u2191(RingEquiv.symm \u03c3') a \u2260 0\n\u22a2 s \u2286 \u2191(RingEquiv.symm \u03c3') a \u2022 \u2191f \u207b\u00b9' V"}, {"tactic": "refine' hr (\u03c3'.symm a) _", "annotated_tactic": ["refine' hr (\u03c3'.symm a) _", []], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\na : \ud835\udd5c\u2082\nha : r \u2264 \u2016a\u2016\nhanz : a \u2260 0\nthis : \u2191(RingEquiv.symm \u03c3') a \u2260 0\n\u22a2 s \u2286 \u2191(RingEquiv.symm \u03c3') a \u2022 \u2191f \u207b\u00b9' V", "state_after": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\na : \ud835\udd5c\u2082\nha : r \u2264 \u2016a\u2016\nhanz : a \u2260 0\nthis : \u2191(RingEquiv.symm \u03c3') a \u2260 0\n\u22a2 r \u2264 \u2016\u2191(RingEquiv.symm \u03c3') a\u2016"}, {"tactic": "rwa [\u03c3'_symm_iso.norm_map_of_map_zero (map_zero _)]", "annotated_tactic": ["rwa [\u03c3'_symm_iso.norm_map_of_map_zero (<a>map_zero</a> _)]", [{"full_name": "map_zero", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [201, 3], "def_end_pos": [201, 14]}]], "state_before": "case intro.intro\n\ud835\udd5c : Type u_1\n\ud835\udd5c' : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_5\n\u03b9 : Type u_6\n\ud835\udd5c\u2081 : Type u_7\n\ud835\udd5c\u2082 : Type u_8\ninst\u271d\u2079 : NormedDivisionRing \ud835\udd5c\u2081\ninst\u271d\u2078 : NormedDivisionRing \ud835\udd5c\u2082\ninst\u271d\u2077 : AddCommGroup E\ninst\u271d\u2076 : Module \ud835\udd5c\u2081 E\ninst\u271d\u2075 : AddCommGroup F\ninst\u271d\u2074 : Module \ud835\udd5c\u2082 F\ninst\u271d\u00b3 : TopologicalSpace E\ninst\u271d\u00b2 : TopologicalSpace F\n\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082\ninst\u271d\u00b9 : RingHomSurjective \u03c3\ninst\u271d : RingHomIsometric \u03c3\ns : Set E\nhs : IsVonNBounded \ud835\udd5c\u2081 s\nf : E \u2192SL[\u03c3] F\n\u03c3' : \ud835\udd5c\u2081 \u2243+* \ud835\udd5c\u2082 := RingEquiv.ofBijective \u03c3 (_ : Function.Injective \u2191\u03c3 \u2227 Function.Surjective \u2191\u03c3)\n\u03c3_iso : Isometry \u2191\u03c3\n\u03c3'_symm_iso : Isometry \u2191(RingEquiv.symm \u03c3')\nf_tendsto_zero : Tendsto (\u2191f) (\ud835\udcdd 0) (\ud835\udcdd 0)\nV : Set F\nhV : V \u2208 \ud835\udcdd 0\nr : \u211d\nhrpos : 0 < r\nhr : \u2200 (a : \ud835\udd5c\u2081), r \u2264 \u2016a\u2016 \u2192 s \u2286 a \u2022 \u2191f \u207b\u00b9' V\na : \ud835\udd5c\u2082\nha : r \u2264 \u2016a\u2016\nhanz : a \u2260 0\nthis : \u2191(RingEquiv.symm \u03c3') a \u2260 0\n\u22a2 r \u2264 \u2016\u2191(RingEquiv.symm \u03c3') a\u2016", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Order/Filter/Bases.lean
|
Filter.HasBasis.filter_eq
|
[
320,
1
] |
[
322,
43
] |
[{"tactic": "ext U", "annotated_tactic": ["ext U", []], "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 : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : HasBasis l p s\n\u22a2 IsBasis.filter (_ : IsBasis p s) = l", "state_after": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : HasBasis l p s\nU : Set \u03b1\n\u22a2 U \u2208 IsBasis.filter (_ : IsBasis p s) \u2194 U \u2208 l"}, {"tactic": "simp [h.mem_iff, IsBasis.mem_filter_iff]", "annotated_tactic": ["simp [h.mem_iff, <a>IsBasis.mem_filter_iff</a>]", [{"full_name": "Filter.IsBasis.mem_filter_iff", "def_path": "Mathlib/Order/Filter/Bases.lean", "def_pos": [216, 19], "def_end_pos": [216, 33]}]], "state_before": "case a\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b9 : Sort u_4\n\u03b9' : Sort u_5\nl l' : Filter \u03b1\np : \u03b9 \u2192 Prop\ns : \u03b9 \u2192 Set \u03b1\nt : Set \u03b1\ni : \u03b9\np' : \u03b9' \u2192 Prop\ns' : \u03b9' \u2192 Set \u03b1\ni' : \u03b9'\nh : HasBasis l p s\nU : Set \u03b1\n\u22a2 U \u2208 IsBasis.filter (_ : IsBasis p s) \u2194 U \u2208 l", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Order/Filter/Prod.lean
|
Filter.Eventually.diag_of_prod_right
|
[
202,
1
] |
[
206,
85
] |
[{"tactic": "intro h", "annotated_tactic": ["intro h", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ns : Set \u03b1\nt : Set \u03b2\nf\u271d : Filter \u03b1\ng\u271d : Filter \u03b2\nf : Filter \u03b1\ng : Filter \u03b3\np : \u03b1 \u00d7 \u03b3 \u00d7 \u03b3 \u2192 Prop\n\u22a2 (\u2200\u1da0 (x : \u03b1 \u00d7 \u03b3 \u00d7 \u03b3) in f \u00d7\u02e2 g \u00d7\u02e2 g, p x) \u2192 \u2200\u1da0 (x : \u03b1 \u00d7 \u03b3) in f \u00d7\u02e2 g, p (x.1, x.2, x.2)", "state_after": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ns : Set \u03b1\nt : Set \u03b2\nf\u271d : Filter \u03b1\ng\u271d : Filter \u03b2\nf : Filter \u03b1\ng : Filter \u03b3\np : \u03b1 \u00d7 \u03b3 \u00d7 \u03b3 \u2192 Prop\nh : \u2200\u1da0 (x : \u03b1 \u00d7 \u03b3 \u00d7 \u03b3) in f \u00d7\u02e2 g \u00d7\u02e2 g, p x\n\u22a2 \u2200\u1da0 (x : \u03b1 \u00d7 \u03b3) in f \u00d7\u02e2 g, p (x.1, x.2, x.2)"}, {"tactic": "obtain \u27e8t, ht, s, hs, hst\u27e9 := eventually_prod_iff.1 h", "annotated_tactic": ["obtain \u27e8t, ht, s, hs, hst\u27e9 := <a>eventually_prod_iff</a>.1 h", [{"full_name": "Filter.eventually_prod_iff", "def_path": "Mathlib/Order/Filter/Prod.lean", "def_pos": [129, 9], "def_end_pos": [129, 28]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ns : Set \u03b1\nt : Set \u03b2\nf\u271d : Filter \u03b1\ng\u271d : Filter \u03b2\nf : Filter \u03b1\ng : Filter \u03b3\np : \u03b1 \u00d7 \u03b3 \u00d7 \u03b3 \u2192 Prop\nh : \u2200\u1da0 (x : \u03b1 \u00d7 \u03b3 \u00d7 \u03b3) in f \u00d7\u02e2 g \u00d7\u02e2 g, p x\n\u22a2 \u2200\u1da0 (x : \u03b1 \u00d7 \u03b3) in f \u00d7\u02e2 g, p (x.1, x.2, x.2)", "state_after": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ns\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf\u271d : Filter \u03b1\ng\u271d : Filter \u03b2\nf : Filter \u03b1\ng : Filter \u03b3\np : \u03b1 \u00d7 \u03b3 \u00d7 \u03b3 \u2192 Prop\nh : \u2200\u1da0 (x : \u03b1 \u00d7 \u03b3 \u00d7 \u03b3) in f \u00d7\u02e2 g \u00d7\u02e2 g, p x\nt : \u03b1 \u2192 Prop\nht : \u2200\u1da0 (x : \u03b1) in f, t x\ns : \u03b3 \u00d7 \u03b3 \u2192 Prop\nhs : \u2200\u1da0 (y : \u03b3 \u00d7 \u03b3) in g \u00d7\u02e2 g, s y\nhst : \u2200 {x : \u03b1}, t x \u2192 \u2200 {y : \u03b3 \u00d7 \u03b3}, s y \u2192 p (x, y)\n\u22a2 \u2200\u1da0 (x : \u03b1 \u00d7 \u03b3) in f \u00d7\u02e2 g, p (x.1, x.2, x.2)"}, {"tactic": "refine' (ht.prod_mk hs.diag_of_prod).mono fun x hx => by simp only [hst hx.1 hx.2]", "annotated_tactic": ["refine' (ht.prod_mk hs.diag_of_prod).<a>mono</a> fun x hx => by simp only [hst hx.1 hx.2]", [{"full_name": "Filter.Eventually.mono", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1140, 9], "def_end_pos": [1140, 24]}]], "state_before": "case intro.intro.intro.intro\n\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ns\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf\u271d : Filter \u03b1\ng\u271d : Filter \u03b2\nf : Filter \u03b1\ng : Filter \u03b3\np : \u03b1 \u00d7 \u03b3 \u00d7 \u03b3 \u2192 Prop\nh : \u2200\u1da0 (x : \u03b1 \u00d7 \u03b3 \u00d7 \u03b3) in f \u00d7\u02e2 g \u00d7\u02e2 g, p x\nt : \u03b1 \u2192 Prop\nht : \u2200\u1da0 (x : \u03b1) in f, t x\ns : \u03b3 \u00d7 \u03b3 \u2192 Prop\nhs : \u2200\u1da0 (y : \u03b3 \u00d7 \u03b3) in g \u00d7\u02e2 g, s y\nhst : \u2200 {x : \u03b1}, t x \u2192 \u2200 {y : \u03b3 \u00d7 \u03b3}, s y \u2192 p (x, y)\n\u22a2 \u2200\u1da0 (x : \u03b1 \u00d7 \u03b3) in f \u00d7\u02e2 g, p (x.1, x.2, x.2)", "state_after": "no goals"}, {"tactic": "simp only [hst hx.1 hx.2]", "annotated_tactic": ["simp only [hst hx.1 hx.2]", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\n\u03b9 : Sort u_5\ns\u271d : Set \u03b1\nt\u271d : Set \u03b2\nf\u271d : Filter \u03b1\ng\u271d : Filter \u03b2\nf : Filter \u03b1\ng : Filter \u03b3\np : \u03b1 \u00d7 \u03b3 \u00d7 \u03b3 \u2192 Prop\nh : \u2200\u1da0 (x : \u03b1 \u00d7 \u03b3 \u00d7 \u03b3) in f \u00d7\u02e2 g \u00d7\u02e2 g, p x\nt : \u03b1 \u2192 Prop\nht : \u2200\u1da0 (x : \u03b1) in f, t x\ns : \u03b3 \u00d7 \u03b3 \u2192 Prop\nhs : \u2200\u1da0 (y : \u03b3 \u00d7 \u03b3) in g \u00d7\u02e2 g, s y\nhst : \u2200 {x : \u03b1}, t x \u2192 \u2200 {y : \u03b3 \u00d7 \u03b3}, s y \u2192 p (x, y)\nx : \u03b1 \u00d7 \u03b3\nhx : t x.1 \u2227 s (x.2, x.2)\n\u22a2 p (x.1, x.2, x.2)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/SpecialFunctions/Exponential.lean
|
hasFDerivAt_exp_zero
|
[
168,
1
] |
[
169,
41
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/GroupTheory/Perm/Support.lean
|
Equiv.Perm.card_support_eq_two
|
[
617,
1
] |
[
633,
32
] |
[{"tactic": "constructor <;> intro h", "annotated_tactic": ["constructor <;> intro h", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\n\u22a2 card (support f) = 2 \u2194 IsSwap f", "state_after": "case mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\n\u22a2 IsSwap f\n\ncase mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : IsSwap f\n\u22a2 card (support f) = 2"}, {"tactic": "obtain \u27e8x, t, hmem, hins, ht\u27e9 := card_eq_succ.1 h", "annotated_tactic": ["obtain \u27e8x, t, hmem, hins, ht\u27e9 := <a>card_eq_succ</a>.1 h", [{"full_name": "Finset.card_eq_succ", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [627, 9], "def_end_pos": [627, 21]}]], "state_before": "case mp\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\n\u22a2 IsSwap f", "state_after": "case mp.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx : \u03b1\nt : Finset \u03b1\nhmem : \u00acx \u2208 t\nhins : insert x t = support f\nht : card t = 1\n\u22a2 IsSwap f"}, {"tactic": "obtain \u27e8y, rfl\u27e9 := card_eq_one.1 ht", "annotated_tactic": ["obtain \u27e8y, rfl\u27e9 := <a>card_eq_one</a>.1 ht", [{"full_name": "Finset.card_eq_one", "def_path": "Mathlib/Data/Finset/Card.lean", "def_pos": [522, 9], "def_end_pos": [522, 20]}]], "state_before": "case mp.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx : \u03b1\nt : Finset \u03b1\nhmem : \u00acx \u2208 t\nhins : insert x t = support f\nht : card t = 1\n\u22a2 IsSwap f", "state_after": "case mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx \u2208 {y}\nhins : {x, y} = support f\nht : card {y} = 1\n\u22a2 IsSwap f"}, {"tactic": "rw [mem_singleton] at hmem", "annotated_tactic": ["rw [<a>mem_singleton</a>] at hmem", [{"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 22]}]], "state_before": "case mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx \u2208 {y}\nhins : {x, y} = support f\nht : card {y} = 1\n\u22a2 IsSwap f", "state_after": "case mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\n\u22a2 IsSwap f"}, {"tactic": "refine' \u27e8x, y, hmem, _\u27e9", "annotated_tactic": ["refine' \u27e8x, y, hmem, _\u27e9", []], "state_before": "case mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\n\u22a2 IsSwap f", "state_after": "case mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\n\u22a2 f = swap x y"}, {"tactic": "ext a", "annotated_tactic": ["ext a", []], "state_before": "case mp.intro.intro.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\n\u22a2 f = swap x y", "state_after": "case mp.intro.intro.intro.intro.intro.H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\na : \u03b1\n\u22a2 \u2191f a = \u2191(swap x y) a"}, {"tactic": "have key : \u2200 b, f b \u2260 b \u2194 _ := fun b => by rw [\u2190 mem_support, \u2190 hins, mem_insert, mem_singleton]", "annotated_tactic": ["have key : \u2200 b, f b \u2260 b \u2194 _ := fun b => by rw [\u2190 <a>mem_support</a>, \u2190 hins, <a>mem_insert</a>, <a>mem_singleton</a>]", [{"full_name": "Equiv.Perm.mem_support", "def_path": "Mathlib/GroupTheory/Perm/Support.lean", "def_pos": [290, 9], "def_end_pos": [290, 20]}, {"full_name": "Finset.mem_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1087, 9], "def_end_pos": [1087, 19]}, {"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 22]}]], "state_before": "case mp.intro.intro.intro.intro.intro.H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\na : \u03b1\n\u22a2 \u2191f a = \u2191(swap x y) a", "state_after": "case mp.intro.intro.intro.intro.intro.H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\na : \u03b1\nkey : \u2200 (b : \u03b1), \u2191f b \u2260 b \u2194 b = x \u2228 b = y\n\u22a2 \u2191f a = \u2191(swap x y) a"}, {"tactic": "by_cases ha : f a = a", "annotated_tactic": ["by_cases ha : f a = a", []], "state_before": "case mp.intro.intro.intro.intro.intro.H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\na : \u03b1\nkey : \u2200 (b : \u03b1), \u2191f b \u2260 b \u2194 b = x \u2228 b = y\n\u22a2 \u2191f a = \u2191(swap x y) a", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\na : \u03b1\nkey : \u2200 (b : \u03b1), \u2191f b \u2260 b \u2194 b = x \u2228 b = y\nha : \u2191f a = a\n\u22a2 \u2191f a = \u2191(swap x y) a\n\ncase neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\na : \u03b1\nkey : \u2200 (b : \u03b1), \u2191f b \u2260 b \u2194 b = x \u2228 b = y\nha : \u00ac\u2191f a = a\n\u22a2 \u2191f a = \u2191(swap x y) a"}, {"tactic": "rw [\u2190 mem_support, \u2190 hins, mem_insert, mem_singleton]", "annotated_tactic": ["rw [\u2190 <a>mem_support</a>, \u2190 hins, <a>mem_insert</a>, <a>mem_singleton</a>]", [{"full_name": "Equiv.Perm.mem_support", "def_path": "Mathlib/GroupTheory/Perm/Support.lean", "def_pos": [290, 9], "def_end_pos": [290, 20]}, {"full_name": "Finset.mem_insert", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1087, 9], "def_end_pos": [1087, 19]}, {"full_name": "Finset.mem_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [678, 9], "def_end_pos": [678, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\na b : \u03b1\n\u22a2 \u2191f b \u2260 b \u2194 ?m.196555 b", "state_after": "no goals"}, {"tactic": "have ha' := not_or.mp (mt (key a).mpr (not_not.mpr ha))", "annotated_tactic": ["have ha' := not_or.mp (<a>mt</a> (key a).<a>mpr</a> (not_not.mpr ha))", [{"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}, {"full_name": "Iff.mpr", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [92, 3], "def_end_pos": [92, 6]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\na : \u03b1\nkey : \u2200 (b : \u03b1), \u2191f b \u2260 b \u2194 b = x \u2228 b = y\nha : \u2191f a = a\n\u22a2 \u2191f a = \u2191(swap x y) a", "state_after": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\na : \u03b1\nkey : \u2200 (b : \u03b1), \u2191f b \u2260 b \u2194 b = x \u2228 b = y\nha : \u2191f a = a\nha' : \u00aca = x \u2227 \u00aca = y\n\u22a2 \u2191f a = \u2191(swap x y) a"}, {"tactic": "rw [ha, swap_apply_of_ne_of_ne ha'.1 ha'.2]", "annotated_tactic": ["rw [ha, <a>swap_apply_of_ne_of_ne</a> ha'.1 ha'.2]", [{"full_name": "Equiv.swap_apply_of_ne_of_ne", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1650, 9], "def_end_pos": [1650, 31]}]], "state_before": "case pos\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\na : \u03b1\nkey : \u2200 (b : \u03b1), \u2191f b \u2260 b \u2194 b = x \u2228 b = y\nha : \u2191f a = a\nha' : \u00aca = x \u2227 \u00aca = y\n\u22a2 \u2191f a = \u2191(swap x y) a", "state_after": "no goals"}, {"tactic": "have ha' := (key (f a)).mp (mt f.apply_eq_iff_eq.mp ha)", "annotated_tactic": ["have ha' := (key (f a)).<a>mp</a> (<a>mt</a> f.apply_eq_iff_eq.mp ha)", [{"full_name": "Iff.mp", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [90, 3], "def_end_pos": [90, 5]}, {"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}]], "state_before": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\na : \u03b1\nkey : \u2200 (b : \u03b1), \u2191f b \u2260 b \u2194 b = x \u2228 b = y\nha : \u00ac\u2191f a = a\n\u22a2 \u2191f a = \u2191(swap x y) a", "state_after": "case neg\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\na : \u03b1\nkey : \u2200 (b : \u03b1), \u2191f b \u2260 b \u2194 b = x \u2228 b = y\nha : \u00ac\u2191f a = a\nha' : \u2191f a = x \u2228 \u2191f a = y\n\u22a2 \u2191f a = \u2191(swap x y) a"}, {"tactic": "obtain rfl | rfl := (key a).mp ha", "annotated_tactic": ["obtain rfl | rfl := (key a).<a>mp</a> ha", [{"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\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx y : \u03b1\nhmem : \u00acx = y\nhins : {x, y} = support f\nht : card {y} = 1\na : \u03b1\nkey : \u2200 (b : \u03b1), \u2191f b \u2260 b \u2194 b = x \u2228 b = y\nha : \u00ac\u2191f a = a\nha' : \u2191f a = x \u2228 \u2191f a = y\n\u22a2 \u2191f a = \u2191(swap x y) a", "state_after": "case neg.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\ny : \u03b1\nht : card {y} = 1\na : \u03b1\nha : \u00ac\u2191f a = a\nhmem : \u00aca = y\nhins : {a, y} = support f\nkey : \u2200 (b : \u03b1), \u2191f b \u2260 b \u2194 b = a \u2228 b = y\nha' : \u2191f a = a \u2228 \u2191f a = y\n\u22a2 \u2191f a = \u2191(swap a y) a\n\ncase neg.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx a : \u03b1\nha : \u00ac\u2191f a = a\nhmem : \u00acx = a\nhins : {x, a} = support f\nht : card {a} = 1\nkey : \u2200 (b : \u03b1), \u2191f b \u2260 b \u2194 b = x \u2228 b = a\nha' : \u2191f a = x \u2228 \u2191f a = a\n\u22a2 \u2191f a = \u2191(swap x a) a"}, {"tactic": "rw [Or.resolve_left ha' ha, swap_apply_left]", "annotated_tactic": ["rw [<a>Or.resolve_left</a> ha' ha, <a>swap_apply_left</a>]", [{"full_name": "Or.resolve_left", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [268, 9], "def_end_pos": [268, 24]}, {"full_name": "Equiv.swap_apply_left", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1641, 9], "def_end_pos": [1641, 24]}]], "state_before": "case neg.inl\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\ny : \u03b1\nht : card {y} = 1\na : \u03b1\nha : \u00ac\u2191f a = a\nhmem : \u00aca = y\nhins : {a, y} = support f\nkey : \u2200 (b : \u03b1), \u2191f b \u2260 b \u2194 b = a \u2228 b = y\nha' : \u2191f a = a \u2228 \u2191f a = y\n\u22a2 \u2191f a = \u2191(swap a y) a", "state_after": "no goals"}, {"tactic": "rw [Or.resolve_right ha' ha, swap_apply_right]", "annotated_tactic": ["rw [<a>Or.resolve_right</a> ha' ha, <a>swap_apply_right</a>]", [{"full_name": "Or.resolve_right", "def_path": "lake-packages/std/Std/Logic.lean", "def_pos": [272, 9], "def_end_pos": [272, 25]}, {"full_name": "Equiv.swap_apply_right", "def_path": "Mathlib/Logic/Equiv/Basic.lean", "def_pos": [1646, 9], "def_end_pos": [1646, 25]}]], "state_before": "case neg.inr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : card (support f) = 2\nx a : \u03b1\nha : \u00ac\u2191f a = a\nhmem : \u00acx = a\nhins : {x, a} = support f\nht : card {a} = 1\nkey : \u2200 (b : \u03b1), \u2191f b \u2260 b \u2194 b = x \u2228 b = a\nha' : \u2191f a = x \u2228 \u2191f a = a\n\u22a2 \u2191f a = \u2191(swap x a) a", "state_after": "no goals"}, {"tactic": "obtain \u27e8x, y, hxy, rfl\u27e9 := h", "annotated_tactic": ["obtain \u27e8x, y, hxy, rfl\u27e9 := h", []], "state_before": "case mpr\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf\u271d g f : Perm \u03b1\nh : IsSwap f\n\u22a2 card (support f) = 2", "state_after": "case mpr.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nhxy : x \u2260 y\n\u22a2 card (support (swap x y)) = 2"}, {"tactic": "exact card_support_swap hxy", "annotated_tactic": ["exact <a>card_support_swap</a> hxy", [{"full_name": "Equiv.Perm.card_support_swap", "def_path": "Mathlib/GroupTheory/Perm/Support.lean", "def_pos": [611, 9], "def_end_pos": [611, 26]}]], "state_before": "case mpr.intro.intro.intro\n\u03b1 : Type u_1\ninst\u271d\u00b9 : DecidableEq \u03b1\ninst\u271d : Fintype \u03b1\nf g : Perm \u03b1\nx y : \u03b1\nhxy : x \u2260 y\n\u22a2 card (support (swap x y)) = 2", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/MetricSpace/DilationEquiv.lean
|
DilationEquiv.surjective
|
[
112,
11
] |
[
112,
75
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Set/Lattice.lean
|
Set.iUnion_iUnion_eq_left
|
[
863,
1
] |
[
865,
20
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/LinearAlgebra/Span.lean
|
Submodule.mem_span_singleton_self
|
[
405,
1
] |
[
406,
18
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Order/Filter/Basic.lean
|
Filter.frequently_sup
|
[
1408,
1
] |
[
1410,
60
] |
[{"tactic": "simp only [Filter.Frequently, eventually_sup, not_and_or]", "annotated_tactic": ["simp only [<a>Filter.Frequently</a>, <a>eventually_sup</a>, <a>not_and_or</a>]", [{"full_name": "Filter.Frequently", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1259, 15], "def_end_pos": [1259, 25]}, {"full_name": "Filter.eventually_sup", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1223, 9], "def_end_pos": [1223, 23]}, {"full_name": "not_and_or", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [473, 9], "def_end_pos": [473, 19]}]], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b4 : Type u_1\n\u03b9 : Sort x\np : \u03b1 \u2192 Prop\nf g : Filter \u03b1\n\u22a2 (\u2203\u1da0 (x : \u03b1) in f \u2294 g, p x) \u2194 (\u2203\u1da0 (x : \u03b1) in f, p x) \u2228 \u2203\u1da0 (x : \u03b1) in g, p x", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Order/Filter/Pointwise.lean
|
Filter.Tendsto.inv_inv
|
[
894,
11
] |
[
895,
55
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.refl_apply
|
[
269,
9
] |
[
269,
63
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.sup_le_lsub
|
[
1605,
1
] |
[
1606,
35
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/CategoryTheory/Elements.lean
|
CategoryTheory.CategoryOfElements.to_comma_map_right
|
[
149,
1
] |
[
150,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Logic/Embedding/Basic.lean
|
subtypeOrLeftEmbedding_apply_right
|
[
487,
1
] |
[
490,
13
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/UniformSpace/Completion.lean
|
UniformSpace.Completion.uniformContinuous_extension
|
[
541,
1
] |
[
542,
32
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_comp
|
[
1311,
1
] |
[
1313,
29
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
Measurable.ennnorm
|
[
2277,
1
] |
[
2278,
31
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/GroupTheory/FreeAbelianGroup.lean
|
FreeAbelianGroup.seq_neg
|
[
310,
1
] |
[
311,
31
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/InnerProductSpace/Adjoint.lean
|
ContinuousLinearMap.eq_adjoint_iff
|
[
172,
1
] |
[
175,
76
] |
[{"tactic": "refine' \u27e8fun h x y => by rw [h, adjoint_inner_left], fun h => _\u27e9", "annotated_tactic": ["refine' \u27e8fun h x y => by rw [h, <a>adjoint_inner_left</a>], fun h => _\u27e9", [{"full_name": "ContinuousLinearMap.adjoint_inner_left", "def_path": "Mathlib/Analysis/InnerProductSpace/Adjoint.lean", "def_pos": [124, 9], "def_end_pos": [124, 27]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\nB : F \u2192L[\ud835\udd5c] E\n\u22a2 A = \u2191adjoint B \u2194 \u2200 (x : E) (y : (fun x => F) x), inner (\u2191A x) y = inner x (\u2191B y)", "state_after": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\nB : F \u2192L[\ud835\udd5c] E\nh : \u2200 (x : E) (y : (fun x => F) x), inner (\u2191A x) y = inner x (\u2191B y)\n\u22a2 A = \u2191adjoint B"}, {"tactic": "ext x", "annotated_tactic": ["ext x", []], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\nB : F \u2192L[\ud835\udd5c] E\nh : \u2200 (x : E) (y : (fun x => F) x), inner (\u2191A x) y = inner x (\u2191B y)\n\u22a2 A = \u2191adjoint B", "state_after": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\nB : F \u2192L[\ud835\udd5c] E\nh : \u2200 (x : E) (y : (fun x => F) x), inner (\u2191A x) y = inner x (\u2191B y)\nx : E\n\u22a2 \u2191A x = \u2191(\u2191adjoint B) x"}, {"tactic": "exact ext_inner_right \ud835\udd5c fun y => by simp only [adjoint_inner_left, h x y]", "annotated_tactic": ["exact <a>ext_inner_right</a> \ud835\udd5c fun y => by simp only [<a>adjoint_inner_left</a>, h x y]", [{"full_name": "ext_inner_right", "def_path": "Mathlib/Analysis/InnerProductSpace/Basic.lean", "def_pos": [691, 9], "def_end_pos": [691, 24]}, {"full_name": "ContinuousLinearMap.adjoint_inner_left", "def_path": "Mathlib/Analysis/InnerProductSpace/Adjoint.lean", "def_pos": [124, 9], "def_end_pos": [124, 27]}]], "state_before": "case h\n\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\nB : F \u2192L[\ud835\udd5c] E\nh : \u2200 (x : E) (y : (fun x => F) x), inner (\u2191A x) y = inner x (\u2191B y)\nx : E\n\u22a2 \u2191A x = \u2191(\u2191adjoint B) x", "state_after": "no goals"}, {"tactic": "rw [h, adjoint_inner_left]", "annotated_tactic": ["rw [h, <a>adjoint_inner_left</a>]", [{"full_name": "ContinuousLinearMap.adjoint_inner_left", "def_path": "Mathlib/Analysis/InnerProductSpace/Adjoint.lean", "def_pos": [124, 9], "def_end_pos": [124, 27]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\nB : F \u2192L[\ud835\udd5c] E\nh : A = \u2191adjoint B\nx : E\ny : (fun x => F) x\n\u22a2 inner (\u2191A x) y = inner x (\u2191B y)", "state_after": "no goals"}, {"tactic": "simp only [adjoint_inner_left, h x y]", "annotated_tactic": ["simp only [<a>adjoint_inner_left</a>, h x y]", [{"full_name": "ContinuousLinearMap.adjoint_inner_left", "def_path": "Mathlib/Analysis/InnerProductSpace/Adjoint.lean", "def_pos": [124, 9], "def_end_pos": [124, 27]}]], "state_before": "\ud835\udd5c : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst\u271d\u2079 : IsROrC \ud835\udd5c\ninst\u271d\u2078 : NormedAddCommGroup E\ninst\u271d\u2077 : NormedAddCommGroup F\ninst\u271d\u2076 : NormedAddCommGroup G\ninst\u271d\u2075 : InnerProductSpace \ud835\udd5c E\ninst\u271d\u2074 : InnerProductSpace \ud835\udd5c F\ninst\u271d\u00b3 : InnerProductSpace \ud835\udd5c G\ninst\u271d\u00b2 : CompleteSpace E\ninst\u271d\u00b9 : CompleteSpace G\ninst\u271d : CompleteSpace F\nA : E \u2192L[\ud835\udd5c] F\nB : F \u2192L[\ud835\udd5c] E\nh : \u2200 (x : E) (y : (fun x => F) x), inner (\u2191A x) y = inner x (\u2191B y)\nx : E\ny : (fun x => F) x\n\u22a2 inner (\u2191A x) y = inner (\u2191(\u2191adjoint B) x) y", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
|
volume_regionBetween_eq_lintegral
|
[
553,
1
] |
[
577,
87
] |
[{"tactic": "have h\u2081 :\n (fun y => ENNReal.ofReal ((g - f) y)) =\u1d50[\u03bc.restrict s] fun y =>\n ENNReal.ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y) :=\n (hg.ae_eq_mk.sub hf.ae_eq_mk).fun_comp ENNReal.ofReal", "annotated_tactic": ["have h\u2081 :\n (fun y => <a>ENNReal.ofReal</a> ((g - f) y)) =\u1d50[\u03bc.restrict s] fun y =>\n <a>ENNReal.ofReal</a> ((<a>AEMeasurable.mk</a> g hg - <a>AEMeasurable.mk</a> f hf) y) :=\n (hg.ae_eq_mk.sub hf.ae_eq_mk).<a>fun_comp</a> <a>ENNReal.ofReal</a>", [{"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}, {"full_name": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [722, 5], "def_end_pos": [722, 7]}, {"full_name": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [722, 5], "def_end_pos": [722, 7]}, {"full_name": "Filter.EventuallyEq.fun_comp", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1520, 9], "def_end_pos": [1520, 30]}, {"full_name": "ENNReal.ofReal", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [172, 29], "def_end_pos": [172, 35]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc"}, {"tactic": "have h\u2082 :\n (\u03bc.restrict s).prod volume (regionBetween f g s) =\n (\u03bc.restrict s).prod volume\n (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) := by\n apply measure_congr\n apply EventuallyEq.rfl.inter\n exact\n ((quasiMeasurePreserving_fst.ae_eq_comp hf.ae_eq_mk).comp\u2082 _ EventuallyEq.rfl).inter\n (EventuallyEq.rfl.comp\u2082 _ <| quasiMeasurePreserving_fst.ae_eq_comp hg.ae_eq_mk)", "annotated_tactic": ["have h\u2082 :\n (\u03bc.restrict s).<a>prod</a> <a>volume</a> (<a>regionBetween</a> f g s) =\n (\u03bc.restrict s).<a>prod</a> <a>volume</a>\n (<a>regionBetween</a> (<a>AEMeasurable.mk</a> f hf) (<a>AEMeasurable.mk</a> g hg) s) := by\n apply <a>measure_congr</a>\n apply EventuallyEq.rfl.inter\n exact\n ((quasiMeasurePreserving_fst.ae_eq_comp hf.ae_eq_mk).<a>comp\u2082</a> _ <a>EventuallyEq.rfl</a>).<a>inter</a>\n (EventuallyEq.rfl.comp\u2082 _ <| quasiMeasurePreserving_fst.ae_eq_comp hg.ae_eq_mk)", [{"full_name": "MeasureTheory.Measure.prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [292, 27], "def_end_pos": [292, 31]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "regionBetween", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [466, 5], "def_end_pos": [466, 18]}, {"full_name": "MeasureTheory.Measure.prod", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [292, 27], "def_end_pos": [292, 31]}, {"full_name": "MeasureTheory.MeasureSpace.volume", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [663, 3], "def_end_pos": [663, 9]}, {"full_name": "regionBetween", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [466, 5], "def_end_pos": [466, 18]}, {"full_name": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [722, 5], "def_end_pos": [722, 7]}, {"full_name": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [722, 5], "def_end_pos": [722, 7]}, {"full_name": "MeasureTheory.measure_congr", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [608, 9], "def_end_pos": [608, 22]}, {"full_name": "Filter.EventuallyEq.comp\u2082", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1525, 9], "def_end_pos": [1525, 27]}, {"full_name": "Filter.EventuallyEq.rfl", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1493, 9], "def_end_pos": [1493, 25]}, {"full_name": "Filter.EventuallyEq.inter", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1580, 9], "def_end_pos": [1580, 27]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc"}, {"tactic": "rw [lintegral_congr_ae h\u2081, \u2190\n volume_regionBetween_eq_lintegral' hf.measurable_mk hg.measurable_mk hs]", "annotated_tactic": ["rw [<a>lintegral_congr_ae</a> h\u2081, \u2190\n <a>volume_regionBetween_eq_lintegral'</a> hf.measurable_mk hg.measurable_mk hs]", [{"full_name": "MeasureTheory.lintegral_congr_ae", "def_path": "Mathlib/MeasureTheory/Integral/Lebesgue.lean", "def_pos": [304, 9], "def_end_pos": [304, 27]}, {"full_name": "volume_regionBetween_eq_lintegral'", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [531, 9], "def_end_pos": [531, 43]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u222b\u207b (y : \u03b1) in s, ofReal ((g - f) y) \u2202\u03bc", "state_after": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) =\n \u2191\u2191(Measure.prod \u03bc volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)"}, {"tactic": "convert h\u2082 using 1", "annotated_tactic": ["convert h\u2082 using 1", []], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) =\n \u2191\u2191(Measure.prod \u03bc volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)", "state_after": "case h.e'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s)\n\ncase h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)"}, {"tactic": "apply measure_congr", "annotated_tactic": ["apply <a>measure_congr</a>", [{"full_name": "MeasureTheory.measure_congr", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [608, 9], "def_end_pos": [608, 22]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)", "state_after": "case H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 regionBetween f g s =\u1da0[ae (Measure.prod (Measure.restrict \u03bc s) volume)]\n regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s"}, {"tactic": "apply EventuallyEq.rfl.inter", "annotated_tactic": ["apply EventuallyEq.rfl.inter", []], "state_before": "case H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 regionBetween f g s =\u1da0[ae (Measure.prod (Measure.restrict \u03bc s) volume)]\n regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s", "state_after": "case H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 (fun p => Ioo (f p.1) (g p.1) p.2) =\u1da0[ae (Measure.prod (Measure.restrict \u03bc s) volume)] fun p =>\n Ioo (AEMeasurable.mk f hf p.1) (AEMeasurable.mk g hg p.1) p.2"}, {"tactic": "exact\n ((quasiMeasurePreserving_fst.ae_eq_comp hf.ae_eq_mk).comp\u2082 _ EventuallyEq.rfl).inter\n (EventuallyEq.rfl.comp\u2082 _ <| quasiMeasurePreserving_fst.ae_eq_comp hg.ae_eq_mk)", "annotated_tactic": ["exact\n ((quasiMeasurePreserving_fst.ae_eq_comp hf.ae_eq_mk).<a>comp\u2082</a> _ <a>EventuallyEq.rfl</a>).<a>inter</a>\n (EventuallyEq.rfl.comp\u2082 _ <| quasiMeasurePreserving_fst.ae_eq_comp hg.ae_eq_mk)", [{"full_name": "Filter.EventuallyEq.comp\u2082", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1525, 9], "def_end_pos": [1525, 27]}, {"full_name": "Filter.EventuallyEq.rfl", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1493, 9], "def_end_pos": [1493, 25]}, {"full_name": "Filter.EventuallyEq.inter", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1580, 9], "def_end_pos": [1580, 27]}]], "state_before": "case H\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\n\u22a2 (fun p => Ioo (f p.1) (g p.1) p.2) =\u1da0[ae (Measure.prod (Measure.restrict \u03bc s) volume)] fun p =>\n Ioo (AEMeasurable.mk f hf p.1) (AEMeasurable.mk g hg p.1) p.2", "state_after": "no goals"}, {"tactic": "rw [Measure.restrict_prod_eq_prod_univ]", "annotated_tactic": ["rw [<a>Measure.restrict_prod_eq_prod_univ</a>]", [{"full_name": "MeasureTheory.Measure.restrict_prod_eq_prod_univ", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [645, 9], "def_end_pos": [645, 35]}]], "state_before": "case h.e'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) = \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s)", "state_after": "case h.e'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) =\n \u2191\u2191(Measure.restrict (Measure.prod \u03bc volume) (s \u00d7\u02e2 univ)) (regionBetween f g s)"}, {"tactic": "exact (Measure.restrict_eq_self _ (regionBetween_subset f g s)).symm", "annotated_tactic": ["exact (<a>Measure.restrict_eq_self</a> _ (<a>regionBetween_subset</a> f g s)).<a>symm</a>", [{"full_name": "MeasureTheory.Measure.restrict_eq_self", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1587, 9], "def_end_pos": [1587, 25]}, {"full_name": "regionBetween_subset", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [470, 9], "def_end_pos": [470, 29]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case h.e'_2\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween f g s) =\n \u2191\u2191(Measure.restrict (Measure.prod \u03bc volume) (s \u00d7\u02e2 univ)) (regionBetween f g s)", "state_after": "no goals"}, {"tactic": "rw [Measure.restrict_prod_eq_prod_univ]", "annotated_tactic": ["rw [<a>Measure.restrict_prod_eq_prod_univ</a>]", [{"full_name": "MeasureTheory.Measure.restrict_prod_eq_prod_univ", "def_path": "Mathlib/MeasureTheory/Constructions/Prod/Basic.lean", "def_pos": [645, 9], "def_end_pos": [645, 35]}]], "state_before": "case h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)", "state_after": "case h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =\n \u2191\u2191(Measure.restrict (Measure.prod \u03bc volume) (s \u00d7\u02e2 univ))\n (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)"}, {"tactic": "exact\n (Measure.restrict_eq_self _\n (regionBetween_subset (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)).symm", "annotated_tactic": ["exact\n (<a>Measure.restrict_eq_self</a> _\n (<a>regionBetween_subset</a> (<a>AEMeasurable.mk</a> f hf) (<a>AEMeasurable.mk</a> g hg) s)).<a>symm</a>", [{"full_name": "MeasureTheory.Measure.restrict_eq_self", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpace.lean", "def_pos": [1587, 9], "def_end_pos": [1587, 25]}, {"full_name": "regionBetween_subset", "def_path": "Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean", "def_pos": [470, 9], "def_end_pos": [470, 29]}, {"full_name": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [722, 5], "def_end_pos": [722, 7]}, {"full_name": "AEMeasurable.mk", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [722, 5], "def_end_pos": [722, 7]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case h.e'_3\n\u03b1 : Type u_1\ninst\u271d\u00b9 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\nf g : \u03b1 \u2192 \u211d\ns : Set \u03b1\ninst\u271d : SigmaFinite \u03bc\nhf : AEMeasurable f\nhg : AEMeasurable g\nhs : MeasurableSet s\nh\u2081 :\n (fun y => ofReal ((g - f) y)) =\u1da0[ae (Measure.restrict \u03bc s)] fun y =>\n ofReal ((AEMeasurable.mk g hg - AEMeasurable.mk f hf) y)\nh\u2082 :\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween f g s) =\n \u2191\u2191(Measure.prod (Measure.restrict \u03bc s) volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)\n\u22a2 \u2191\u2191(Measure.prod \u03bc volume) (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s) =\n \u2191\u2191(Measure.restrict (Measure.prod \u03bc volume) (s \u00d7\u02e2 univ))\n (regionBetween (AEMeasurable.mk f hf) (AEMeasurable.mk g hg) s)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Nat/Cast/Defs.lean
|
Nat.cast_two
|
[
193,
1
] |
[
193,
71
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Set/Image.lean
|
Set.range_inr_inter_range_inl
|
[
926,
1
] |
[
927,
46
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/GroupPower/Ring.lean
|
Commute.sq_sub_sq
|
[
238,
11
] |
[
239,
42
] |
[{"tactic": "rw [sq, sq, h.mul_self_sub_mul_self_eq]", "annotated_tactic": ["rw [<a>sq</a>, <a>sq</a>, h.mul_self_sub_mul_self_eq]", [{"full_name": "sq", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [106, 7], "def_end_pos": [106, 9]}, {"full_name": "sq", "def_path": "Mathlib/Algebra/GroupPower/Basic.lean", "def_pos": [106, 7], "def_end_pos": [106, 9]}]], "state_before": "R : Type u_1\nS : Type u_2\nM : Type u_3\ninst\u271d : Ring R\na b : R\nh : Commute a b\n\u22a2 a ^ 2 - b ^ 2 = (a + b) * (a - b)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/LinearAlgebra/Lagrange.lean
|
Lagrange.basis_singleton
|
[
222,
1
] |
[
223,
51
] |
[{"tactic": "rw [Lagrange.basis, erase_singleton, prod_empty]", "annotated_tactic": ["rw [<a>Lagrange.basis</a>, <a>erase_singleton</a>, <a>prod_empty</a>]", [{"full_name": "Lagrange.basis", "def_path": "Mathlib/LinearAlgebra/Lagrange.lean", "def_pos": [212, 15], "def_end_pos": [212, 20]}, {"full_name": "Finset.erase_singleton", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [1902, 9], "def_end_pos": [1902, 24]}, {"full_name": "Finset.prod_empty", "def_path": "Mathlib/Algebra/BigOperators/Basic.lean", "def_pos": [299, 9], "def_end_pos": [299, 19]}]], "state_before": "F : Type u_1\ninst\u271d\u00b9 : Field F\n\u03b9 : Type u_2\ninst\u271d : DecidableEq \u03b9\ns : Finset \u03b9\nv : \u03b9 \u2192 F\ni\u271d j i : \u03b9\n\u22a2 Lagrange.basis {i} v i = 1", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Option/Basic.lean
|
Option.join_pmap_eq_pmap_join
|
[
269,
1
] |
[
271,
37
] |
[{"tactic": "rcases x with (_ | _ | x) <;> simp", "annotated_tactic": ["rcases x with (_ | _ | x) <;> simp", []], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\n\u03b4 : Type u_4\np : \u03b1 \u2192 Prop\nf\u271d : (a : \u03b1) \u2192 p a \u2192 \u03b2\nx\u271d : Option \u03b1\nf : (a : \u03b1) \u2192 p a \u2192 \u03b2\nx : Option (Option \u03b1)\nH : \u2200 (a : Option \u03b1), a \u2208 x \u2192 \u2200 (a_2 : \u03b1), a_2 \u2208 a \u2192 p a_2\n\u22a2 join (pmap (pmap f) x H) = pmap f (join x) (_ : \u2200 (a : \u03b1), a \u2208 join x \u2192 p a)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Num/Lemmas.lean
|
Num.cast_one
|
[
282,
1
] |
[
283,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Order/Cover.lean
|
Covby.ge_of_gt
|
[
451,
1
] |
[
452,
22
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/MetricSpace/Isometry.lean
|
IsometryEquiv.ext
|
[
370,
1
] |
[
371,
20
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/List/Indexes.lean
|
List.oldMapIdxCore_eq
|
[
46,
11
] |
[
51,
84
] |
[{"tactic": "induction' l with hd tl hl generalizing f n", "annotated_tactic": ["induction' l with hd tl hl generalizing f n", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\nl : List \u03b1\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 List.oldMapIdxCore f n l = List.oldMapIdx (fun i a => f (i + n) a) l", "state_after": "case nil\n\u03b1 : Type u\n\u03b2 : Type v\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nn\u271d : \u2115\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 List.oldMapIdxCore f n [] = List.oldMapIdx (fun i a => f (i + n) a) []\n\ncase cons\n\u03b1 : Type u\n\u03b2 : Type v\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nn\u271d : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (f : \u2115 \u2192 \u03b1 \u2192 \u03b2) (n : \u2115), List.oldMapIdxCore f n tl = List.oldMapIdx (fun i a => f (i + n) a) tl\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 List.oldMapIdxCore f n (hd :: tl) = List.oldMapIdx (fun i a => f (i + n) a) (hd :: tl)"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case nil\n\u03b1 : Type u\n\u03b2 : Type v\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nn\u271d : \u2115\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 List.oldMapIdxCore f n [] = List.oldMapIdx (fun i a => f (i + n) a) []", "state_after": "no goals"}, {"tactic": "rw [List.oldMapIdx]", "annotated_tactic": ["rw [<a>List.oldMapIdx</a>]", [{"full_name": "List.oldMapIdx", "def_path": "Mathlib/Data/List/Indexes.lean", "def_pos": [37, 15], "def_end_pos": [37, 24]}]], "state_before": "case cons\n\u03b1 : Type u\n\u03b2 : Type v\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nn\u271d : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (f : \u2115 \u2192 \u03b1 \u2192 \u03b2) (n : \u2115), List.oldMapIdxCore f n tl = List.oldMapIdx (fun i a => f (i + n) a) tl\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 List.oldMapIdxCore f n (hd :: tl) = List.oldMapIdx (fun i a => f (i + n) a) (hd :: tl)", "state_after": "case cons\n\u03b1 : Type u\n\u03b2 : Type v\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nn\u271d : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (f : \u2115 \u2192 \u03b1 \u2192 \u03b2) (n : \u2115), List.oldMapIdxCore f n tl = List.oldMapIdx (fun i a => f (i + n) a) tl\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 List.oldMapIdxCore f n (hd :: tl) = List.oldMapIdxCore (fun i a => f (i + n) a) 0 (hd :: tl)"}, {"tactic": "simp only [List.oldMapIdxCore, hl, add_left_comm, add_comm, add_zero, zero_add]", "annotated_tactic": ["simp only [<a>List.oldMapIdxCore</a>, hl, <a>add_left_comm</a>, <a>add_comm</a>, <a>add_zero</a>, <a>zero_add</a>]", [{"full_name": "List.oldMapIdxCore", "def_path": "Mathlib/Data/List/Indexes.lean", "def_pos": [31, 15], "def_end_pos": [31, 28]}, {"full_name": "add_left_comm", "def_path": "Mathlib/Algebra/Group/Basic.lean", "def_pos": [95, 3], "def_end_pos": [95, 14]}, {"full_name": "add_comm", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [301, 3], "def_end_pos": [301, 14]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}, {"full_name": "zero_add", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [463, 3], "def_end_pos": [463, 14]}]], "state_before": "case cons\n\u03b1 : Type u\n\u03b2 : Type v\nf\u271d : \u2115 \u2192 \u03b1 \u2192 \u03b2\nn\u271d : \u2115\nhd : \u03b1\ntl : List \u03b1\nhl : \u2200 (f : \u2115 \u2192 \u03b1 \u2192 \u03b2) (n : \u2115), List.oldMapIdxCore f n tl = List.oldMapIdx (fun i a => f (i + n) a) tl\nf : \u2115 \u2192 \u03b1 \u2192 \u03b2\nn : \u2115\n\u22a2 List.oldMapIdxCore f n (hd :: tl) = List.oldMapIdxCore (fun i a => f (i + n) a) 0 (hd :: tl)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Order/Heyting/Basic.lean
|
himp_compl_comm
|
[
813,
1
] |
[
813,
95
] |
[{"tactic": "simp_rw [\u2190 himp_bot, himp_left_comm]", "annotated_tactic": ["simp_rw [\u2190 <a>himp_bot</a>, <a>himp_left_comm</a>]", [{"full_name": "himp_bot", "def_path": "Mathlib/Order/Heyting/Basic.lean", "def_pos": [777, 9], "def_end_pos": [777, 17]}, {"full_name": "himp_left_comm", "def_path": "Mathlib/Order/Heyting/Basic.lean", "def_pos": [394, 9], "def_end_pos": [394, 23]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\n\u03b2 : Type u_3\ninst\u271d : HeytingAlgebra \u03b1\na\u271d b\u271d c a b : \u03b1\n\u22a2 a \u21e8 b\u1d9c = b \u21e8 a\u1d9c", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.tendsto_sin_pi_div_two
|
[
1064,
1
] |
[
1066,
7
] |
[{"tactic": "convert continuous_sin.continuousWithinAt.tendsto", "annotated_tactic": ["convert continuous_sin.continuousWithinAt.tendsto", []], "state_before": "\u22a2 Tendsto sin (\ud835\udcdd[Iio (\u03c0 / 2)] (\u03c0 / 2)) (\ud835\udcdd 1)", "state_after": "case h.e'_5.h.e'_3\n\n\u22a2 1 = sin (\u03c0 / 2)"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case h.e'_5.h.e'_3\n\n\u22a2 1 = sin (\u03c0 / 2)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Polynomial/Coeff.lean
|
Polynomial.C_mul'
|
[
167,
1
] |
[
169,
44
] |
[{"tactic": "ext", "annotated_tactic": ["ext", []], "state_before": "R : Type u\nS : Type v\na\u271d b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\na : R\nf : R[X]\n\u22a2 \u2191C a * f = a \u2022 f", "state_after": "case a\nR : Type u\nS : Type v\na\u271d b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\na : R\nf : R[X]\nn\u271d : \u2115\n\u22a2 coeff (\u2191C a * f) n\u271d = coeff (a \u2022 f) n\u271d"}, {"tactic": "rw [coeff_C_mul, coeff_smul, smul_eq_mul]", "annotated_tactic": ["rw [<a>coeff_C_mul</a>, <a>coeff_smul</a>, <a>smul_eq_mul</a>]", [{"full_name": "Polynomial.coeff_C_mul", "def_path": "Mathlib/Data/Polynomial/Coeff.lean", "def_pos": [161, 9], "def_end_pos": [161, 20]}, {"full_name": "Polynomial.coeff_smul", "def_path": "Mathlib/Data/Polynomial/Coeff.lean", "def_pos": [54, 9], "def_end_pos": [54, 19]}, {"full_name": "smul_eq_mul", "def_path": "Mathlib/GroupTheory/GroupAction/Defs.lean", "def_pos": [93, 9], "def_end_pos": [93, 20]}]], "state_before": "case a\nR : Type u\nS : Type v\na\u271d b : R\nn m : \u2115\ninst\u271d : Semiring R\np q r : R[X]\na : R\nf : R[X]\nn\u271d : \u2115\n\u22a2 coeff (\u2191C a * f) n\u271d = coeff (a \u2022 f) n\u271d", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/MeasureTheory/Measure/FiniteMeasure.lean
|
MeasureTheory.FiniteMeasure.tendsto_iff_forall_lintegral_tendsto
|
[
532,
1
] |
[
539,
26
] |
[{"tactic": "rw [tendsto_iff_forall_toWeakDualBCNN_tendsto]", "annotated_tactic": ["rw [<a>tendsto_iff_forall_toWeakDualBCNN_tendsto</a>]", [{"full_name": "MeasureTheory.FiniteMeasure.tendsto_iff_forall_toWeakDualBCNN_tendsto", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [487, 9], "def_end_pos": [487, 50]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 Tendsto \u03bcs F (\ud835\udcdd \u03bc) \u2194 \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))", "state_after": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 (\u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u2191(toWeakDualBCNN (\u03bcs i)) f) F (\ud835\udcdd (\u2191(toWeakDualBCNN \u03bc) f))) \u2194\n \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))"}, {"tactic": "simp_rw [toWeakDualBCNN_apply _ _, \u2190 testAgainstNN_coe_eq, ENNReal.tendsto_coe,\n ENNReal.toNNReal_coe]", "annotated_tactic": ["simp_rw [<a>toWeakDualBCNN_apply</a> _ _, \u2190 <a>testAgainstNN_coe_eq</a>, <a>ENNReal.tendsto_coe</a>,\n <a>ENNReal.toNNReal_coe</a>]", [{"full_name": "MeasureTheory.FiniteMeasure.toWeakDualBCNN_apply", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [443, 9], "def_end_pos": [443, 29]}, {"full_name": "MeasureTheory.FiniteMeasure.testAgainstNN_coe_eq", "def_path": "Mathlib/MeasureTheory/Measure/FiniteMeasure.lean", "def_pos": [326, 9], "def_end_pos": [326, 29]}, {"full_name": "ENNReal.tendsto_coe", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [71, 9], "def_end_pos": [71, 20]}, {"full_name": "ENNReal.toNNReal_coe", "def_path": "Mathlib/Data/Real/ENNReal.lean", "def_pos": [176, 9], "def_end_pos": [176, 21]}]], "state_before": "\u03a9 : Type u_1\ninst\u271d\u2076 : MeasurableSpace \u03a9\nR : Type u_2\ninst\u271d\u2075 : SMul R \u211d\u22650\ninst\u271d\u2074 : SMul R \u211d\u22650\u221e\ninst\u271d\u00b3 : IsScalarTower R \u211d\u22650 \u211d\u22650\u221e\ninst\u271d\u00b2 : IsScalarTower R \u211d\u22650\u221e \u211d\u22650\u221e\ninst\u271d\u00b9 : TopologicalSpace \u03a9\ninst\u271d : OpensMeasurableSpace \u03a9\n\u03b3 : Type u_3\nF : Filter \u03b3\n\u03bcs : \u03b3 \u2192 FiniteMeasure \u03a9\n\u03bc : FiniteMeasure \u03a9\n\u22a2 (\u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u2191(toWeakDualBCNN (\u03bcs i)) f) F (\ud835\udcdd (\u2191(toWeakDualBCNN \u03bc) f))) \u2194\n \u2200 (f : \u03a9 \u2192\u1d47 \u211d\u22650), Tendsto (fun i => \u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191(\u03bcs i)) F (\ud835\udcdd (\u222b\u207b (x : \u03a9), \u2191(\u2191f x) \u2202\u2191\u03bc))", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Order/Archimedean.lean
|
exists_lt_nsmul
|
[
51,
1
] |
[
54,
97
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/List/Perm.lean
|
List.Perm.take_inter
|
[
1152,
1
] |
[
1157,
22
] |
[{"tactic": "simp only [List.inter]", "annotated_tactic": ["simp only [<a>List.inter</a>]", [{"full_name": "List.inter", "def_path": "lake-packages/std/Std/Data/List/Basic.lean", "def_pos": [433, 25], "def_end_pos": [433, 30]}]], "state_before": "\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\n\u22a2 take n xs ~ List.inter ys (take n xs)", "state_after": "\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\n\u22a2 take n xs ~ List.filter (fun x => decide (x \u2208 take n xs)) ys"}, {"tactic": "conv_lhs => rw [Nodup.take_eq_filter_mem ((Perm.nodup_iff h).2 h')]", "annotated_tactic": ["conv_lhs => rw [<a>Nodup.take_eq_filter_mem</a> ((<a>Perm.nodup_iff</a> h).2 h')]", [{"full_name": "List.Nodup.take_eq_filter_mem", "def_path": "Mathlib/Data/List/Nodup.lean", "def_pos": [452, 9], "def_end_pos": [452, 33]}, {"full_name": "List.Perm.nodup_iff", "def_path": "Mathlib/Data/List/Perm.lean", "def_pos": [1061, 9], "def_end_pos": [1061, 23]}]], "state_before": "\u03b1\u271d : Type uu\n\u03b2 : Type vv\nl\u2081 l\u2082 : List \u03b1\u271d\n\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\nxs ys : List \u03b1\nn : \u2115\nh : xs ~ ys\nh' : Nodup ys\n\u22a2 take n xs ~ List.filter (fun x => decide (x \u2208 take n xs)) xs", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ico_subset_Ico_right
|
[
451,
1
] |
[
452,
26
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean
|
MeasureTheory.ae_const_le_iff_forall_lt_measure_zero
|
[
126,
1
] |
[
158,
22
] |
[{"tactic": "rw [ae_iff]", "annotated_tactic": ["rw [<a>ae_iff</a>]", [{"full_name": "MeasureTheory.ae_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [388, 9], "def_end_pos": [388, 15]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 (\u2200\u1d50 (x : \u03b1) \u2202\u03bc, c \u2264 f x) \u2194 \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 \u2191\u2191\u03bc {a | \u00acc \u2264 f a} = 0 \u2194 \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0"}, {"tactic": "push_neg", "annotated_tactic": ["push_neg", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 \u2191\u2191\u03bc {a | \u00acc \u2264 f a} = 0 \u2194 \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0 \u2194 \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0"}, {"tactic": "constructor", "annotated_tactic": ["constructor", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0 \u2194 \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0", "state_after": "case mp\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0 \u2192 \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\ncase mpr\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 (\u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0) \u2192 \u2191\u2191\u03bc {a | f a < c} = 0"}, {"tactic": "intro hc", "annotated_tactic": ["intro hc", []], "state_before": "case mpr\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 (\u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0) \u2192 \u2191\u2191\u03bc {a | f a < c} = 0", "state_after": "case mpr\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0"}, {"tactic": "by_cases h : \u2200 b, c \u2264 b", "annotated_tactic": ["by_cases h : \u2200 b, c \u2264 b", []], "state_before": "case mpr\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u2200 (b : \u03b2), c \u2264 b\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0"}, {"tactic": "by_cases H : \u00acIsLUB (Set.Iio c) c", "annotated_tactic": ["by_cases H : \u00ac<a>IsLUB</a> (<a>Set.Iio</a> c) c", [{"full_name": "IsLUB", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [76, 5], "def_end_pos": [76, 10]}, {"full_name": "Set.Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [54, 5], "def_end_pos": [54, 8]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00acIsLUB (Set.Iio c) c\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0\n\ncase neg\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00ac\u00acIsLUB (Set.Iio c) c\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0"}, {"tactic": "push_neg at H h", "annotated_tactic": ["push_neg at H h", []], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00ac\u00acIsLUB (Set.Iio c) c\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0", "state_after": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0"}, {"tactic": "obtain \u27e8u, _, u_lt, u_lim, -\u27e9 :\n \u2203 u : \u2115 \u2192 \u03b2,\n StrictMono u \u2227 (\u2200 n : \u2115, u n < c) \u2227 Tendsto u atTop (nhds c) \u2227 \u2200 n : \u2115, u n \u2208 Set.Iio c :=\n H.exists_seq_strictMono_tendsto_of_not_mem (lt_irrefl c) h", "annotated_tactic": ["obtain \u27e8u, _, u_lt, u_lim, -\u27e9 :\n \u2203 u : \u2115 \u2192 \u03b2,\n <a>StrictMono</a> u \u2227 (\u2200 n : \u2115, u n < c) \u2227 <a>Tendsto</a> u <a>atTop</a> (<a>nhds</a> c) \u2227 \u2200 n : \u2115, u n \u2208 <a>Set.Iio</a> c :=\n H.exists_seq_strictMono_tendsto_of_not_mem (<a>lt_irrefl</a> c) h", [{"full_name": "StrictMono", "def_path": "Mathlib/Order/Monotone/Basic.lean", "def_pos": [97, 5], "def_end_pos": [97, 15]}, {"full_name": "Filter.Tendsto", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [2939, 5], "def_end_pos": [2939, 12]}, {"full_name": "Filter.atTop", "def_path": "Mathlib/Order/Filter/AtTopBot.lean", "def_pos": [41, 5], "def_end_pos": [41, 10]}, {"full_name": "nhds", "def_path": "Mathlib/Topology/Basic.lean", "def_pos": [831, 17], "def_end_pos": [831, 21]}, {"full_name": "Set.Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [54, 5], "def_end_pos": [54, 8]}, {"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}]], "state_before": "case neg\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0", "state_after": "case neg.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0"}, {"tactic": "rw [h_Union, measure_iUnion_null_iff]", "annotated_tactic": ["rw [h_Union, <a>measure_iUnion_null_iff</a>]", [{"full_name": "MeasureTheory.measure_iUnion_null_iff", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [275, 9], "def_end_pos": [275, 32]}]], "state_before": "case neg.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nh_Union : {x | f x < c} = \u22c3 n, {x | f x \u2264 u n}\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0", "state_after": "case neg.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nh_Union : {x | f x < c} = \u22c3 n, {x | f x \u2264 u n}\n\u22a2 \u2200 (i : \u2115), \u2191\u2191\u03bc {x | f x \u2264 u i} = 0"}, {"tactic": "intro n", "annotated_tactic": ["intro n", []], "state_before": "case neg.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nh_Union : {x | f x < c} = \u22c3 n, {x | f x \u2264 u n}\n\u22a2 \u2200 (i : \u2115), \u2191\u2191\u03bc {x | f x \u2264 u i} = 0", "state_after": "case neg.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nh_Union : {x | f x < c} = \u22c3 n, {x | f x \u2264 u n}\nn : \u2115\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 u n} = 0"}, {"tactic": "exact hc _ (u_lt n)", "annotated_tactic": ["exact hc _ (u_lt n)", []], "state_before": "case neg.intro.intro.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nh_Union : {x | f x < c} = \u22c3 n, {x | f x \u2264 u n}\nn : \u2115\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 u n} = 0", "state_after": "no goals"}, {"tactic": "intro h b hb", "annotated_tactic": ["intro h b hb", []], "state_before": "case mp\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0 \u2192 \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0", "state_after": "case mp\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nh : \u2191\u2191\u03bc {a | f a < c} = 0\nb : \u03b2\nhb : b < c\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0"}, {"tactic": "exact measure_mono_null (fun y hy => (lt_of_le_of_lt hy hb : _)) h", "annotated_tactic": ["exact <a>measure_mono_null</a> (fun y hy => (<a>lt_of_le_of_lt</a> hy hb : _)) h", [{"full_name": "MeasureTheory.measure_mono_null", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [197, 9], "def_end_pos": [197, 26]}, {"full_name": "lt_of_le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [122, 9], "def_end_pos": [122, 23]}]], "state_before": "case mp\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nh : \u2191\u2191\u03bc {a | f a < c} = 0\nb : \u03b2\nhb : b < c\n\u22a2 \u2191\u2191\u03bc {x | f x \u2264 b} = 0", "state_after": "no goals"}, {"tactic": "have : {a : \u03b1 | f a < c} = \u2205 := by\n apply Set.eq_empty_iff_forall_not_mem.2 fun x hx => ?_\n exact (lt_irrefl _ (lt_of_lt_of_le hx (h (f x)))).elim", "annotated_tactic": ["have : {a : \u03b1 | f a < c} = \u2205 := by\n apply <a>Set.eq_empty_iff_forall_not_mem</a>.2 fun x hx => ?_\n exact (<a>lt_irrefl</a> _ (<a>lt_of_lt_of_le</a> hx (h (f x)))).<a>elim</a>", [{"full_name": "Set.eq_empty_iff_forall_not_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [582, 9], "def_end_pos": [582, 36]}, {"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u2200 (b : \u03b2), c \u2264 b\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u2200 (b : \u03b2), c \u2264 b\nthis : {a | f a < c} = \u2205\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0"}, {"tactic": "simp [this]", "annotated_tactic": ["simp [this]", []], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u2200 (b : \u03b2), c \u2264 b\nthis : {a | f a < c} = \u2205\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0", "state_after": "no goals"}, {"tactic": "apply Set.eq_empty_iff_forall_not_mem.2 fun x hx => ?_", "annotated_tactic": ["apply <a>Set.eq_empty_iff_forall_not_mem</a>.2 fun x hx => ?_", [{"full_name": "Set.eq_empty_iff_forall_not_mem", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [582, 9], "def_end_pos": [582, 36]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u2200 (b : \u03b2), c \u2264 b\n\u22a2 {a | f a < c} = \u2205", "state_after": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u2200 (b : \u03b2), c \u2264 b\nx : \u03b1\nhx : x \u2208 {a | f a < c}\n\u22a2 False"}, {"tactic": "exact (lt_irrefl _ (lt_of_lt_of_le hx (h (f x)))).elim", "annotated_tactic": ["exact (<a>lt_irrefl</a> _ (<a>lt_of_lt_of_le</a> hx (h (f x)))).<a>elim</a>", [{"full_name": "lt_irrefl", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [79, 9], "def_end_pos": [79, 18]}, {"full_name": "lt_of_lt_of_le", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [115, 9], "def_end_pos": [115, 23]}, {"full_name": "False.elim", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [223, 21], "def_end_pos": [223, 31]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u2200 (b : \u03b2), c \u2264 b\nx : \u03b1\nhx : x \u2208 {a | f a < c}\n\u22a2 False", "state_after": "no goals"}, {"tactic": "have : c \u2208 upperBounds (Set.Iio c) := fun y hy => le_of_lt hy", "annotated_tactic": ["have : c \u2208 <a>upperBounds</a> (<a>Set.Iio</a> c) := fun y hy => <a>le_of_lt</a> hy", [{"full_name": "upperBounds", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [46, 5], "def_end_pos": [46, 16]}, {"full_name": "Set.Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [54, 5], "def_end_pos": [54, 8]}, {"full_name": "le_of_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [110, 9], "def_end_pos": [110, 17]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00acIsLUB (Set.Iio c) c\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0", "state_after": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00acIsLUB (Set.Iio c) c\nthis : c \u2208 upperBounds (Set.Iio c)\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0"}, {"tactic": "obtain \u27e8b, b_up, bc\u27e9 : \u2203 b : \u03b2, b \u2208 upperBounds (Set.Iio c) \u2227 b < c := by\n simpa [IsLUB, IsLeast, this, lowerBounds] using H", "annotated_tactic": ["obtain \u27e8b, b_up, bc\u27e9 : \u2203 b : \u03b2, b \u2208 <a>upperBounds</a> (<a>Set.Iio</a> c) \u2227 b < c := by\n simpa [<a>IsLUB</a>, <a>IsLeast</a>, this, <a>lowerBounds</a>] using H", [{"full_name": "upperBounds", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [46, 5], "def_end_pos": [46, 16]}, {"full_name": "Set.Iio", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [54, 5], "def_end_pos": [54, 8]}, {"full_name": "IsLUB", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [76, 5], "def_end_pos": [76, 10]}, {"full_name": "IsLeast", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [66, 5], "def_end_pos": [66, 12]}, {"full_name": "lowerBounds", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [51, 5], "def_end_pos": [51, 16]}]], "state_before": "case pos\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00acIsLUB (Set.Iio c) c\nthis : c \u2208 upperBounds (Set.Iio c)\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0", "state_after": "case pos.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00acIsLUB (Set.Iio c) c\nthis : c \u2208 upperBounds (Set.Iio c)\nb : \u03b2\nb_up : b \u2208 upperBounds (Set.Iio c)\nbc : b < c\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0"}, {"tactic": "exact measure_mono_null (fun x hx => b_up hx) (hc b bc)", "annotated_tactic": ["exact <a>measure_mono_null</a> (fun x hx => b_up hx) (hc b bc)", [{"full_name": "MeasureTheory.measure_mono_null", "def_path": "Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean", "def_pos": [197, 9], "def_end_pos": [197, 26]}]], "state_before": "case pos.intro.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00acIsLUB (Set.Iio c) c\nthis : c \u2208 upperBounds (Set.Iio c)\nb : \u03b2\nb_up : b \u2208 upperBounds (Set.Iio c)\nbc : b < c\n\u22a2 \u2191\u2191\u03bc {a | f a < c} = 0", "state_after": "no goals"}, {"tactic": "simpa [IsLUB, IsLeast, this, lowerBounds] using H", "annotated_tactic": ["simpa [<a>IsLUB</a>, <a>IsLeast</a>, this, <a>lowerBounds</a>] using H", [{"full_name": "IsLUB", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [76, 5], "def_end_pos": [76, 10]}, {"full_name": "IsLeast", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [66, 5], "def_end_pos": [66, 12]}, {"full_name": "lowerBounds", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [51, 5], "def_end_pos": [51, 16]}]], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nh : \u00ac\u2200 (b : \u03b2), c \u2264 b\nH : \u00acIsLUB (Set.Iio c) c\nthis : c \u2208 upperBounds (Set.Iio c)\n\u22a2 \u2203 b, b \u2208 upperBounds (Set.Iio c) \u2227 b < c", "state_after": "no goals"}, {"tactic": "ext1 x", "annotated_tactic": ["ext1 x", []], "state_before": "\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\n\u22a2 {x | f x < c} = \u22c3 n, {x | f x \u2264 u n}", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\n\u22a2 x \u2208 {x | f x < c} \u2194 x \u2208 \u22c3 n, {x | f x \u2264 u n}"}, {"tactic": "simp_rw [Set.mem_iUnion, Set.mem_setOf_eq]", "annotated_tactic": ["simp_rw [<a>Set.mem_iUnion</a>, <a>Set.mem_setOf_eq</a>]", [{"full_name": "Set.mem_iUnion", "def_path": "Mathlib/Data/Set/Lattice.lean", "def_pos": [201, 9], "def_end_pos": [201, 19]}, {"full_name": "Set.mem_setOf_eq", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [256, 29], "def_end_pos": [256, 41]}]], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\n\u22a2 x \u2208 {x | f x < c} \u2194 x \u2208 \u22c3 n, {x | f x \u2264 u n}", "state_after": "case h\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\n\u22a2 f x < c \u2194 \u2203 i, f x \u2264 u i"}, {"tactic": "constructor <;> intro h", "annotated_tactic": ["constructor <;> intro h", []], "state_before": "case h\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\n\u22a2 f x < c \u2194 \u2203 i, f x \u2264 u i", "state_after": "case h.mp\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh\u271d : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\nh : f x < c\n\u22a2 \u2203 i, f x \u2264 u i\n\ncase h.mpr\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh\u271d : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\nh : \u2203 i, f x \u2264 u i\n\u22a2 f x < c"}, {"tactic": "obtain \u27e8n, hn\u27e9 := ((tendsto_order.1 u_lim).1 _ h).exists", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 := ((<a>tendsto_order</a>.1 u_lim).1 _ h).<a>exists</a>", [{"full_name": "tendsto_order", "def_path": "Mathlib/Topology/Order/Basic.lean", "def_pos": [919, 9], "def_end_pos": [919, 22]}, {"full_name": "Filter.Eventually.exists", "def_path": "Mathlib/Order/Filter/Basic.lean", "def_pos": [1308, 9], "def_end_pos": [1308, 26]}]], "state_before": "case h.mp\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh\u271d : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\nh : f x < c\n\u22a2 \u2203 i, f x \u2264 u i", "state_after": "case h.mp.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh\u271d : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\nh : f x < c\nn : \u2115\nhn : f x < u n\n\u22a2 \u2203 i, f x \u2264 u i"}, {"tactic": "exact \u27e8n, hn.le\u27e9", "annotated_tactic": ["exact \u27e8n, hn.le\u27e9", []], "state_before": "case h.mp.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh\u271d : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\nh : f x < c\nn : \u2115\nhn : f x < u n\n\u22a2 \u2203 i, f x \u2264 u i", "state_after": "no goals"}, {"tactic": "obtain \u27e8n, hn\u27e9 := h", "annotated_tactic": ["obtain \u27e8n, hn\u27e9 := h", []], "state_before": "case h.mpr\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh\u271d : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\nh : \u2203 i, f x \u2264 u i\n\u22a2 f x < c", "state_after": "case h.mpr.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\nn : \u2115\nhn : f x \u2264 u n\n\u22a2 f x < c"}, {"tactic": "exact hn.trans_lt (u_lt _)", "annotated_tactic": ["exact hn.trans_lt (u_lt _)", []], "state_before": "case h.mpr.intro\n\u03b1 : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace \u03b1\n\u03bc : Measure \u03b1\ns t : Set \u03b1\ninst\u271d\u2076 : NormedAddCommGroup E\ninst\u271d\u2075 : NormedSpace \u211d E\ninst\u271d\u2074 : CompleteSpace E\np : \u211d\u22650\u221e\n\u03b2 : Type u_3\ninst\u271d\u00b3 : LinearOrder \u03b2\ninst\u271d\u00b2 : TopologicalSpace \u03b2\ninst\u271d\u00b9 : OrderTopology \u03b2\ninst\u271d : FirstCountableTopology \u03b2\nf : \u03b1 \u2192 \u03b2\nc : \u03b2\nhc : \u2200 (b : \u03b2), b < c \u2192 \u2191\u2191\u03bc {x | f x \u2264 b} = 0\nH : IsLUB (Set.Iio c) c\nh : \u2203 b, b < c\nu : \u2115 \u2192 \u03b2\nleft\u271d : StrictMono u\nu_lt : \u2200 (n : \u2115), u n < c\nu_lim : Tendsto u atTop (\ud835\udcdd c)\nx : \u03b1\nn : \u2115\nhn : f x \u2264 u n\n\u22a2 f x < c", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Order/Monotone/Basic.lean
|
Subsingleton.strictAnti
|
[
537,
11
] |
[
538,
51
] |
[]
|
https://github.com/leanprover/std4
|
869c615eb10130c0637a7bc038e2b80253559913
|
lake-packages/std/Std/Control/ForInStep/Lemmas.lean
|
ForInStep.bindList_append
|
[
44,
9
] |
[
47,
43
] |
[{"tactic": "induction l\u2081 generalizing s <;> simp [*]", "annotated_tactic": ["induction l\u2081 generalizing s <;> simp [*]", []], "state_before": "m : Type u_1 \u2192 Type u_2\n\u03b1 : Type u_3\n\u03b2 : Type u_1\ninst\u271d\u00b9 : Monad m\ninst\u271d : LawfulMonad m\nf : \u03b1 \u2192 \u03b2 \u2192 m (ForInStep \u03b2)\ns : ForInStep \u03b2\nl\u2081 l\u2082 : List \u03b1\n\u22a2 bindList f (l\u2081 ++ l\u2082) s = do\n let x \u2190 bindList f l\u2081 s\n bindList f l\u2082 x", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.tsum_sub
|
[
975,
1
] |
[
978,
73
] |
[{"tactic": "simp only [\u2190 ENNReal.tsum_add, this]", "annotated_tactic": ["simp only [\u2190 <a>ENNReal.tsum_add</a>, this]", [{"full_name": "ENNReal.tsum_add", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [823, 19], "def_end_pos": [823, 27]}]], "state_before": "\u03b1 : Type u_1\n\u03b2 : Type u_2\n\u03b3 : Type u_3\na b c d : \u211d\u22650\u221e\nr p q : \u211d\u22650\nx y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e\ns : Set \u211d\u22650\u221e\nf\u271d g\u271d : \u03b1 \u2192 \u211d\u22650\u221e\nf g : \u2115 \u2192 \u211d\u22650\u221e\nh\u2081 : \u2211' (i : \u2115), g i \u2260 \u22a4\nh\u2082 : g \u2264 f\nthis : \u2200 (i : \u2115), f i - g i + g i = f i\n\u22a2 \u2211' (i : \u2115), (f i - g i) + \u2211' (i : \u2115), g i = \u2211' (i : \u2115), f i", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.mul_X_sub_int_cast_comp
|
[
1314,
1
] |
[
1316,
97
] |
[{"tactic": "rw [mul_sub, sub_comp, mul_X_comp, \u2190 Nat.cast_comm, nat_cast_mul_comp, Nat.cast_comm, mul_sub]", "annotated_tactic": ["rw [<a>mul_sub</a>, <a>sub_comp</a>, <a>mul_X_comp</a>, \u2190 <a>Nat.cast_comm</a>, <a>nat_cast_mul_comp</a>, <a>Nat.cast_comm</a>, <a>mul_sub</a>]", [{"full_name": "mul_sub", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [365, 7], "def_end_pos": [365, 14]}, {"full_name": "Polynomial.sub_comp", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [1300, 9], "def_end_pos": [1300, 17]}, {"full_name": "Polynomial.mul_X_comp", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [602, 9], "def_end_pos": [602, 19]}, {"full_name": "Nat.cast_comm", "def_path": "Mathlib/Data/Nat/Cast/Commute.lean", "def_pos": [33, 9], "def_end_pos": [33, 18]}, {"full_name": "Polynomial.nat_cast_mul_comp", "def_path": "Mathlib/Data/Polynomial/Eval.lean", "def_pos": [636, 9], "def_end_pos": [636, 26]}, {"full_name": "Nat.cast_comm", "def_path": "Mathlib/Data/Nat/Cast/Commute.lean", "def_pos": [33, 9], "def_end_pos": [33, 18]}, {"full_name": "mul_sub", "def_path": "Mathlib/Algebra/Ring/Defs.lean", "def_pos": [365, 7], "def_end_pos": [365, 14]}]], "state_before": "R : Type u\nS : Type v\nT : Type w\n\u03b9 : Type y\na b : R\nm n\u271d : \u2115\ninst\u271d : Ring R\np q r : R[X]\nn : \u2115\n\u22a2 comp (p * (X - \u2191n)) q = comp p q * (q - \u2191n)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Order/Monoid/Defs.lean
|
add_top
|
[
110,
1
] |
[
111,
41
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Hom/Units.lean
|
IsUnit.div_eq_of_eq_mul
|
[
426,
11
] |
[
427,
17
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/RepresentationTheory/Action.lean
|
Action.comp_hom
|
[
158,
1
] |
[
160,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/LinearAlgebra/Matrix/Transvection.lean
|
Matrix.TransvectionStruct.toMatrix_reindexEquiv_prod
|
[
306,
1
] |
[
312,
45
] |
[{"tactic": "induction' L with t L IH", "annotated_tactic": ["induction' L with t L IH", []], "state_before": "n : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nL : List (TransvectionStruct n R)\n\u22a2 List.prod (List.map (toMatrix \u2218 reindexEquiv e) L) = \u2191(reindexAlgEquiv R e) (List.prod (List.map toMatrix L))", "state_after": "case nil\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\n\u22a2 List.prod (List.map (toMatrix \u2218 reindexEquiv e) []) = \u2191(reindexAlgEquiv R e) (List.prod (List.map toMatrix []))\n\ncase cons\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map (toMatrix \u2218 reindexEquiv e) L) = \u2191(reindexAlgEquiv R e) (List.prod (List.map toMatrix L))\n\u22a2 List.prod (List.map (toMatrix \u2218 reindexEquiv e) (t :: L)) =\n \u2191(reindexAlgEquiv R e) (List.prod (List.map toMatrix (t :: L)))"}, {"tactic": "simp", "annotated_tactic": ["simp", []], "state_before": "case nil\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\n\u22a2 List.prod (List.map (toMatrix \u2218 reindexEquiv e) []) = \u2191(reindexAlgEquiv R e) (List.prod (List.map toMatrix []))", "state_after": "no goals"}, {"tactic": "simp only [toMatrix_reindexEquiv, IH, Function.comp_apply, List.prod_cons,\n reindexAlgEquiv_apply, List.map]", "annotated_tactic": ["simp only [<a>toMatrix_reindexEquiv</a>, IH, <a>Function.comp_apply</a>, <a>List.prod_cons</a>,\n <a>reindexAlgEquiv_apply</a>, <a>List.map</a>]", [{"full_name": "Matrix.TransvectionStruct.toMatrix_reindexEquiv", "def_path": "Mathlib/LinearAlgebra/Matrix/Transvection.lean", "def_pos": [296, 9], "def_end_pos": [296, 30]}, {"full_name": "Function.comp_apply", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [33, 17], "def_end_pos": [33, 36]}, {"full_name": "List.prod_cons", "def_path": "Mathlib/Data/List/BigOperators/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 18]}, {"full_name": "Matrix.reindexAlgEquiv_apply", "def_path": "Mathlib/LinearAlgebra/Matrix/Reindex.lean", "def_pos": [132, 9], "def_end_pos": [132, 30]}, {"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 cons\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map (toMatrix \u2218 reindexEquiv e) L) = \u2191(reindexAlgEquiv R e) (List.prod (List.map toMatrix L))\n\u22a2 List.prod (List.map (toMatrix \u2218 reindexEquiv e) (t :: L)) =\n \u2191(reindexAlgEquiv R e) (List.prod (List.map toMatrix (t :: L)))", "state_after": "case cons\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map (toMatrix \u2218 reindexEquiv e) L) = \u2191(reindexAlgEquiv R e) (List.prod (List.map toMatrix L))\n\u22a2 \u2191(reindex e e) (toMatrix t) * \u2191(reindex e e) (List.prod (List.map toMatrix L)) =\n \u2191(reindex e e) (toMatrix t * List.prod (List.map toMatrix L))"}, {"tactic": "exact (reindexAlgEquiv_mul _ _ _ _).symm", "annotated_tactic": ["exact (<a>reindexAlgEquiv_mul</a> _ _ _ _).<a>symm</a>", [{"full_name": "Matrix.reindexAlgEquiv_mul", "def_path": "Mathlib/LinearAlgebra/Matrix/Reindex.lean", "def_pos": [147, 9], "def_end_pos": [147, 28]}, {"full_name": "Eq.symm", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [310, 9], "def_end_pos": [310, 16]}]], "state_before": "case cons\nn : Type u_1\np : Type u_2\nR : Type u\u2082\n\ud835\udd5c : Type u_3\ninst\u271d\u2075 : Field \ud835\udd5c\ninst\u271d\u2074 : DecidableEq n\ninst\u271d\u00b3 : DecidableEq p\ninst\u271d\u00b2 : CommRing R\ni j : n\ninst\u271d\u00b9 : Fintype n\ninst\u271d : Fintype p\ne : n \u2243 p\nt : TransvectionStruct n R\nL : List (TransvectionStruct n R)\nIH : List.prod (List.map (toMatrix \u2218 reindexEquiv e) L) = \u2191(reindexAlgEquiv R e) (List.prod (List.map toMatrix L))\n\u22a2 \u2191(reindex e e) (toMatrix t) * \u2191(reindex e e) (List.prod (List.map toMatrix L)) =\n \u2191(reindex e e) (toMatrix t * List.prod (List.map toMatrix L))", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
Summable.congr
|
[
110,
1
] |
[
111,
29
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/ModelTheory/FinitelyGenerated.lean
|
FirstOrder.Language.Substructure.CG.of_map_embedding
|
[
162,
1
] |
[
173,
28
] |
[{"tactic": "rcases hs with \u27e8t, h1, h2\u27e9", "annotated_tactic": ["rcases hs with \u27e8t, h1, h2\u27e9", []], "state_before": "L : Language\nM : Type u_1\ninst\u271d\u00b9 : Structure L M\nN : Type u_2\ninst\u271d : Structure L N\nf : M \u21aa[L] N\ns : Substructure L M\nhs : CG (Substructure.map (Embedding.toHom f) s)\n\u22a2 CG s", "state_after": "case intro.intro\nL : Language\nM : Type u_1\ninst\u271d\u00b9 : Structure L M\nN : Type u_2\ninst\u271d : Structure L N\nf : M \u21aa[L] N\ns : Substructure L M\nt : Set N\nh1 : Set.Countable t\nh2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s\n\u22a2 CG s"}, {"tactic": "rw [cg_def]", "annotated_tactic": ["rw [<a>cg_def</a>]", [{"full_name": "FirstOrder.Language.Substructure.cg_def", "def_path": "Mathlib/ModelTheory/FinitelyGenerated.lean", "def_pos": [107, 9], "def_end_pos": [107, 15]}]], "state_before": "case intro.intro\nL : Language\nM : Type u_1\ninst\u271d\u00b9 : Structure L M\nN : Type u_2\ninst\u271d : Structure L N\nf : M \u21aa[L] N\ns : Substructure L M\nt : Set N\nh1 : Set.Countable t\nh2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s\n\u22a2 CG s", "state_after": "case intro.intro\nL : Language\nM : Type u_1\ninst\u271d\u00b9 : Structure L M\nN : Type u_2\ninst\u271d : Structure L N\nf : M \u21aa[L] N\ns : Substructure L M\nt : Set N\nh1 : Set.Countable t\nh2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s\n\u22a2 \u2203 S, Set.Countable S \u2227 LowerAdjoint.toFun (closure L) S = s"}, {"tactic": "refine' \u27e8f \u207b\u00b9' t, h1.preimage f.injective, _\u27e9", "annotated_tactic": ["refine' \u27e8f \u207b\u00b9' t, h1.preimage f.injective, _\u27e9", []], "state_before": "case intro.intro\nL : Language\nM : Type u_1\ninst\u271d\u00b9 : Structure L M\nN : Type u_2\ninst\u271d : Structure L N\nf : M \u21aa[L] N\ns : Substructure L M\nt : Set N\nh1 : Set.Countable t\nh2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s\n\u22a2 \u2203 S, Set.Countable S \u2227 LowerAdjoint.toFun (closure L) S = s", "state_after": "case intro.intro\nL : Language\nM : Type u_1\ninst\u271d\u00b9 : Structure L M\nN : Type u_2\ninst\u271d : Structure L N\nf : M \u21aa[L] N\ns : Substructure L M\nt : Set N\nh1 : Set.Countable t\nh2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s\n\u22a2 LowerAdjoint.toFun (closure L) (\u2191f \u207b\u00b9' t) = s"}, {"tactic": "have hf : Function.Injective f.toHom := f.injective", "annotated_tactic": ["have hf : <a>Function.Injective</a> f.toHom := f.injective", [{"full_name": "Function.Injective", "def_path": "Mathlib/Init/Function.lean", "def_pos": [109, 5], "def_end_pos": [109, 14]}]], "state_before": "case intro.intro\nL : Language\nM : Type u_1\ninst\u271d\u00b9 : Structure L M\nN : Type u_2\ninst\u271d : Structure L N\nf : M \u21aa[L] N\ns : Substructure L M\nt : Set N\nh1 : Set.Countable t\nh2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s\n\u22a2 LowerAdjoint.toFun (closure L) (\u2191f \u207b\u00b9' t) = s", "state_after": "case intro.intro\nL : Language\nM : Type u_1\ninst\u271d\u00b9 : Structure L M\nN : Type u_2\ninst\u271d : Structure L N\nf : M \u21aa[L] N\ns : Substructure L M\nt : Set N\nh1 : Set.Countable t\nh2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s\nhf : Function.Injective \u2191(Embedding.toHom f)\n\u22a2 LowerAdjoint.toFun (closure L) (\u2191f \u207b\u00b9' t) = s"}, {"tactic": "refine' map_injective_of_injective hf _", "annotated_tactic": ["refine' <a>map_injective_of_injective</a> hf _", [{"full_name": "FirstOrder.Language.Substructure.map_injective_of_injective", "def_path": "Mathlib/ModelTheory/Substructures.lean", "def_pos": [572, 9], "def_end_pos": [572, 35]}]], "state_before": "case intro.intro\nL : Language\nM : Type u_1\ninst\u271d\u00b9 : Structure L M\nN : Type u_2\ninst\u271d : Structure L N\nf : M \u21aa[L] N\ns : Substructure L M\nt : Set N\nh1 : Set.Countable t\nh2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s\nhf : Function.Injective \u2191(Embedding.toHom f)\n\u22a2 LowerAdjoint.toFun (closure L) (\u2191f \u207b\u00b9' t) = s", "state_after": "case intro.intro\nL : Language\nM : Type u_1\ninst\u271d\u00b9 : Structure L M\nN : Type u_2\ninst\u271d : Structure L N\nf : M \u21aa[L] N\ns : Substructure L M\nt : Set N\nh1 : Set.Countable t\nh2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s\nhf : Function.Injective \u2191(Embedding.toHom f)\n\u22a2 Substructure.map (Embedding.toHom f) (LowerAdjoint.toFun (closure L) (\u2191f \u207b\u00b9' t)) =\n Substructure.map (Embedding.toHom f) s"}, {"tactic": "rw [\u2190 h2, map_closure, Embedding.coe_toHom, image_preimage_eq_of_subset]", "annotated_tactic": ["rw [\u2190 h2, <a>map_closure</a>, <a>Embedding.coe_toHom</a>, <a>image_preimage_eq_of_subset</a>]", [{"full_name": "FirstOrder.Language.Substructure.map_closure", "def_path": "Mathlib/ModelTheory/Substructures.lean", "def_pos": [544, 9], "def_end_pos": [544, 20]}, {"full_name": "FirstOrder.Language.Embedding.coe_toHom", "def_path": "Mathlib/ModelTheory/Basic.lean", "def_pos": [641, 9], "def_end_pos": [641, 18]}, {"full_name": "Set.image_preimage_eq_of_subset", "def_path": "Mathlib/Data/Set/Image.lean", "def_pos": [802, 9], "def_end_pos": [802, 36]}]], "state_before": "case intro.intro\nL : Language\nM : Type u_1\ninst\u271d\u00b9 : Structure L M\nN : Type u_2\ninst\u271d : Structure L N\nf : M \u21aa[L] N\ns : Substructure L M\nt : Set N\nh1 : Set.Countable t\nh2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s\nhf : Function.Injective \u2191(Embedding.toHom f)\n\u22a2 Substructure.map (Embedding.toHom f) (LowerAdjoint.toFun (closure L) (\u2191f \u207b\u00b9' t)) =\n Substructure.map (Embedding.toHom f) s", "state_after": "case intro.intro\nL : Language\nM : Type u_1\ninst\u271d\u00b9 : Structure L M\nN : Type u_2\ninst\u271d : Structure L N\nf : M \u21aa[L] N\ns : Substructure L M\nt : Set N\nh1 : Set.Countable t\nh2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s\nhf : Function.Injective \u2191(Embedding.toHom f)\n\u22a2 t \u2286 range \u2191f"}, {"tactic": "intro x hx", "annotated_tactic": ["intro x hx", []], "state_before": "case intro.intro\nL : Language\nM : Type u_1\ninst\u271d\u00b9 : Structure L M\nN : Type u_2\ninst\u271d : Structure L N\nf : M \u21aa[L] N\ns : Substructure L M\nt : Set N\nh1 : Set.Countable t\nh2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s\nhf : Function.Injective \u2191(Embedding.toHom f)\n\u22a2 t \u2286 range \u2191f", "state_after": "case intro.intro\nL : Language\nM : Type u_1\ninst\u271d\u00b9 : Structure L M\nN : Type u_2\ninst\u271d : Structure L N\nf : M \u21aa[L] N\ns : Substructure L M\nt : Set N\nh1 : Set.Countable t\nh2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s\nhf : Function.Injective \u2191(Embedding.toHom f)\nx : N\nhx : x \u2208 t\n\u22a2 x \u2208 range \u2191f"}, {"tactic": "have h' := subset_closure (L := L) hx", "annotated_tactic": ["have h' := <a>subset_closure</a> (L := L) hx", [{"full_name": "FirstOrder.Language.Substructure.subset_closure", "def_path": "Mathlib/ModelTheory/Substructures.lean", "def_pos": [266, 9], "def_end_pos": [266, 23]}]], "state_before": "case intro.intro\nL : Language\nM : Type u_1\ninst\u271d\u00b9 : Structure L M\nN : Type u_2\ninst\u271d : Structure L N\nf : M \u21aa[L] N\ns : Substructure L M\nt : Set N\nh1 : Set.Countable t\nh2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s\nhf : Function.Injective \u2191(Embedding.toHom f)\nx : N\nhx : x \u2208 t\n\u22a2 x \u2208 range \u2191f", "state_after": "case intro.intro\nL : Language\nM : Type u_1\ninst\u271d\u00b9 : Structure L M\nN : Type u_2\ninst\u271d : Structure L N\nf : M \u21aa[L] N\ns : Substructure L M\nt : Set N\nh1 : Set.Countable t\nh2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s\nhf : Function.Injective \u2191(Embedding.toHom f)\nx : N\nhx : x \u2208 t\nh' : x \u2208 \u2191(LowerAdjoint.toFun (closure L) t)\n\u22a2 x \u2208 range \u2191f"}, {"tactic": "rw [h2] at h'", "annotated_tactic": ["rw [h2] at h'", []], "state_before": "case intro.intro\nL : Language\nM : Type u_1\ninst\u271d\u00b9 : Structure L M\nN : Type u_2\ninst\u271d : Structure L N\nf : M \u21aa[L] N\ns : Substructure L M\nt : Set N\nh1 : Set.Countable t\nh2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s\nhf : Function.Injective \u2191(Embedding.toHom f)\nx : N\nhx : x \u2208 t\nh' : x \u2208 \u2191(LowerAdjoint.toFun (closure L) t)\n\u22a2 x \u2208 range \u2191f", "state_after": "case intro.intro\nL : Language\nM : Type u_1\ninst\u271d\u00b9 : Structure L M\nN : Type u_2\ninst\u271d : Structure L N\nf : M \u21aa[L] N\ns : Substructure L M\nt : Set N\nh1 : Set.Countable t\nh2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s\nhf : Function.Injective \u2191(Embedding.toHom f)\nx : N\nhx : x \u2208 t\nh' : x \u2208 \u2191(Substructure.map (Embedding.toHom f) s)\n\u22a2 x \u2208 range \u2191f"}, {"tactic": "exact Hom.map_le_range h'", "annotated_tactic": ["exact <a>Hom.map_le_range</a> h'", [{"full_name": "FirstOrder.Language.Hom.map_le_range", "def_path": "Mathlib/ModelTheory/Substructures.lean", "def_pos": [864, 9], "def_end_pos": [864, 21]}]], "state_before": "case intro.intro\nL : Language\nM : Type u_1\ninst\u271d\u00b9 : Structure L M\nN : Type u_2\ninst\u271d : Structure L N\nf : M \u21aa[L] N\ns : Substructure L M\nt : Set N\nh1 : Set.Countable t\nh2 : LowerAdjoint.toFun (closure L) t = Substructure.map (Embedding.toHom f) s\nhf : Function.Injective \u2191(Embedding.toHom f)\nx : N\nhx : x \u2208 t\nh' : x \u2208 \u2191(Substructure.map (Embedding.toHom f) s)\n\u22a2 x \u2208 range \u2191f", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Ioc_subset_Ioi_self
|
[
416,
1
] |
[
417,
53
] |
[{"tactic": "simpa [\u2190 coe_subset] using Set.Ioc_subset_Ioi_self", "annotated_tactic": ["simpa [\u2190 <a>coe_subset</a>] using <a>Set.Ioc_subset_Ioi_self</a>", [{"full_name": "Finset.coe_subset", "def_path": "Mathlib/Data/Finset/Basic.lean", "def_pos": [376, 9], "def_end_pos": [376, 19]}, {"full_name": "Set.Ioc_subset_Ioi_self", "def_path": "Mathlib/Data/Set/Intervals/Basic.lean", "def_pos": [526, 9], "def_end_pos": [526, 28]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : LocallyFiniteOrder \u03b1\na a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1\ninst\u271d : LocallyFiniteOrderTop \u03b1\n\u22a2 Ioc a b \u2286 Ioi a", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Geometry/Manifold/ContMDiff.lean
|
Smooth.smoothOn
|
[
767,
1
] |
[
768,
27
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Polynomial/Coeff.lean
|
Polynomial.coeff_X_pow_mul'
|
[
270,
1
] |
[
272,
48
] |
[{"tactic": "rw [(commute_X_pow p n).eq, coeff_mul_X_pow']", "annotated_tactic": ["rw [(<a>commute_X_pow</a> p n).<a>eq</a>, <a>coeff_mul_X_pow'</a>]", [{"full_name": "Polynomial.commute_X_pow", "def_path": "Mathlib/Data/Polynomial/Basic.lean", "def_pos": [634, 9], "def_end_pos": [634, 22]}, {"full_name": "Commute.eq", "def_path": "Mathlib/Algebra/Group/Commute/Defs.lean", "def_pos": [47, 19], "def_end_pos": [47, 21]}, {"full_name": "Polynomial.coeff_mul_X_pow'", "def_path": "Mathlib/Data/Polynomial/Coeff.lean", "def_pos": [261, 9], "def_end_pos": [261, 25]}]], "state_before": "R : Type u\nS : Type v\na b : R\nn\u271d m : \u2115\ninst\u271d : Semiring R\np\u271d q r p : R[X]\nn d : \u2115\n\u22a2 coeff (X ^ n * p) d = if n \u2264 d then coeff p (d - n) else 0", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/SetTheory/Ordinal/Exponential.lean
|
Ordinal.log_zero_left
|
[
278,
1
] |
[
279,
33
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/Module/Basic.lean
|
map_inv_int_cast_smul
|
[
482,
1
] |
[
488,
35
] |
[{"tactic": "obtain \u27e8n, rfl | rfl\u27e9 := z.eq_nat_or_neg", "annotated_tactic": ["obtain \u27e8n, rfl | rfl\u27e9 := z.eq_nat_or_neg", []], "state_before": "\u03b1 : Type u_1\nR\u271d : Type u_2\nk : Type u_3\nS\u271d : Type u_4\nM : Type u_5\nM\u2082 : Type u_6\nM\u2083 : Type u_7\n\u03b9 : Type u_8\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M\u2082\nF : Type u_9\ninst\u271d\u2074 : AddMonoidHomClass F M M\u2082\nf : F\nR : Type u_10\nS : Type u_11\ninst\u271d\u00b3 : DivisionRing R\ninst\u271d\u00b2 : DivisionRing S\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module S M\u2082\nz : \u2124\nx : M\n\u22a2 \u2191f ((\u2191z)\u207b\u00b9 \u2022 x) = (\u2191z)\u207b\u00b9 \u2022 \u2191f x", "state_after": "case intro.inl\n\u03b1 : Type u_1\nR\u271d : Type u_2\nk : Type u_3\nS\u271d : Type u_4\nM : Type u_5\nM\u2082 : Type u_6\nM\u2083 : Type u_7\n\u03b9 : Type u_8\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M\u2082\nF : Type u_9\ninst\u271d\u2074 : AddMonoidHomClass F M M\u2082\nf : F\nR : Type u_10\nS : Type u_11\ninst\u271d\u00b3 : DivisionRing R\ninst\u271d\u00b2 : DivisionRing S\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module S M\u2082\nx : M\nn : \u2115\n\u22a2 \u2191f ((\u2191\u2191n)\u207b\u00b9 \u2022 x) = (\u2191\u2191n)\u207b\u00b9 \u2022 \u2191f x\n\ncase intro.inr\n\u03b1 : Type u_1\nR\u271d : Type u_2\nk : Type u_3\nS\u271d : Type u_4\nM : Type u_5\nM\u2082 : Type u_6\nM\u2083 : Type u_7\n\u03b9 : Type u_8\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M\u2082\nF : Type u_9\ninst\u271d\u2074 : AddMonoidHomClass F M M\u2082\nf : F\nR : Type u_10\nS : Type u_11\ninst\u271d\u00b3 : DivisionRing R\ninst\u271d\u00b2 : DivisionRing S\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module S M\u2082\nx : M\nn : \u2115\n\u22a2 \u2191f ((\u2191(-\u2191n))\u207b\u00b9 \u2022 x) = (\u2191(-\u2191n))\u207b\u00b9 \u2022 \u2191f x"}, {"tactic": "rw [Int.cast_Nat_cast, Int.cast_Nat_cast, map_inv_nat_cast_smul _ R S]", "annotated_tactic": ["rw [<a>Int.cast_Nat_cast</a>, <a>Int.cast_Nat_cast</a>, <a>map_inv_nat_cast_smul</a> _ R S]", [{"full_name": "Int.cast_Nat_cast", "def_path": "Mathlib/Data/Int/Basic.lean", "def_pos": [79, 7], "def_end_pos": [79, 20]}, {"full_name": "Int.cast_Nat_cast", "def_path": "Mathlib/Data/Int/Basic.lean", "def_pos": [79, 7], "def_end_pos": [79, 20]}, {"full_name": "map_inv_nat_cast_smul", "def_path": "Mathlib/Algebra/Module/Basic.lean", "def_pos": [464, 9], "def_end_pos": [464, 30]}]], "state_before": "case intro.inl\n\u03b1 : Type u_1\nR\u271d : Type u_2\nk : Type u_3\nS\u271d : Type u_4\nM : Type u_5\nM\u2082 : Type u_6\nM\u2083 : Type u_7\n\u03b9 : Type u_8\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M\u2082\nF : Type u_9\ninst\u271d\u2074 : AddMonoidHomClass F M M\u2082\nf : F\nR : Type u_10\nS : Type u_11\ninst\u271d\u00b3 : DivisionRing R\ninst\u271d\u00b2 : DivisionRing S\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module S M\u2082\nx : M\nn : \u2115\n\u22a2 \u2191f ((\u2191\u2191n)\u207b\u00b9 \u2022 x) = (\u2191\u2191n)\u207b\u00b9 \u2022 \u2191f x", "state_after": "no goals"}, {"tactic": "simp_rw [Int.cast_neg, Int.cast_Nat_cast, inv_neg, neg_smul, map_neg,\n map_inv_nat_cast_smul _ R S]", "annotated_tactic": ["simp_rw [<a>Int.cast_neg</a>, <a>Int.cast_Nat_cast</a>, <a>inv_neg</a>, <a>neg_smul</a>, <a>map_neg</a>,\n <a>map_inv_nat_cast_smul</a> _ R S]", [{"full_name": "Int.cast_neg", "def_path": "Mathlib/Data/Int/Cast/Basic.lean", "def_pos": [83, 9], "def_end_pos": [83, 17]}, {"full_name": "Int.cast_Nat_cast", "def_path": "Mathlib/Data/Int/Basic.lean", "def_pos": [79, 7], "def_end_pos": [79, 20]}, {"full_name": "inv_neg", "def_path": "Mathlib/Algebra/Field/Basic.lean", "def_pos": [130, 9], "def_end_pos": [130, 16]}, {"full_name": "neg_smul", "def_path": "Mathlib/Algebra/Module/Basic.lean", "def_pos": [278, 9], "def_end_pos": [278, 17]}, {"full_name": "map_neg", "def_path": "Mathlib/Algebra/Hom/Group/Defs.lean", "def_pos": [413, 3], "def_end_pos": [413, 14]}, {"full_name": "map_inv_nat_cast_smul", "def_path": "Mathlib/Algebra/Module/Basic.lean", "def_pos": [464, 9], "def_end_pos": [464, 30]}]], "state_before": "case intro.inr\n\u03b1 : Type u_1\nR\u271d : Type u_2\nk : Type u_3\nS\u271d : Type u_4\nM : Type u_5\nM\u2082 : Type u_6\nM\u2083 : Type u_7\n\u03b9 : Type u_8\ninst\u271d\u2076 : AddCommGroup M\ninst\u271d\u2075 : AddCommGroup M\u2082\nF : Type u_9\ninst\u271d\u2074 : AddMonoidHomClass F M M\u2082\nf : F\nR : Type u_10\nS : Type u_11\ninst\u271d\u00b3 : DivisionRing R\ninst\u271d\u00b2 : DivisionRing S\ninst\u271d\u00b9 : Module R M\ninst\u271d : Module S M\u2082\nx : M\nn : \u2115\n\u22a2 \u2191f ((\u2191(-\u2191n))\u207b\u00b9 \u2022 x) = (\u2191(-\u2191n))\u207b\u00b9 \u2022 \u2191f x", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Order/Filter/Prod.lean
|
Filter.prod_assoc
|
[
300,
1
] |
[
303,
32
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Nat/Count.lean
|
Nat.count_succ_eq_count_iff
|
[
110,
1
] |
[
111,
44
] |
[{"tactic": "by_cases h : p n <;> simp [h, count_succ]", "annotated_tactic": ["by_cases h : p n <;> simp [h, <a>count_succ</a>]", [{"full_name": "Nat.count_succ", "def_path": "Mathlib/Data/Nat/Count.lean", "def_pos": [65, 9], "def_end_pos": [65, 19]}]], "state_before": "p : \u2115 \u2192 Prop\ninst\u271d : DecidablePred p\nn : \u2115\n\u22a2 count p (n + 1) = count p n \u2194 \u00acp n", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Finset/Basic.lean
|
Finset.ssubset_iff_subset_ne
|
[
420,
1
] |
[
421,
28
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean
|
Set.mem_ordConnectedComponent_ordConnectedProj
|
[
114,
1
] |
[
116,
85
] |
[]
|
https://github.com/leanprover/std4
|
869c615eb10130c0637a7bc038e2b80253559913
|
lake-packages/std/Std/Data/List/Lemmas.lean
|
List.mem_diff_of_mem
|
[
1709,
1
] |
[
1713,
94
] |
[{"tactic": "rw [diff_cons]", "annotated_tactic": ["rw [<a>diff_cons</a>]", [{"full_name": "List.diff_cons", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1668, 17], "def_end_pos": [1668, 26]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b1\nl\u2082 : List \u03b1\nh\u2081 : a \u2208 l\u2081\nh\u2082 : \u00aca \u2208 b :: l\u2082\n\u22a2 a \u2208 List.diff l\u2081 (b :: l\u2082)", "state_after": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b1\nl\u2082 : List \u03b1\nh\u2081 : a \u2208 l\u2081\nh\u2082 : \u00aca \u2208 b :: l\u2082\n\u22a2 a \u2208 List.diff (List.erase l\u2081 b) l\u2082"}, {"tactic": "exact mem_diff_of_mem ((mem_erase_of_ne <| ne_of_not_mem_cons h\u2082).2 h\u2081) (mt (.tail _) h\u2082)", "annotated_tactic": ["exact mem_diff_of_mem ((<a>mem_erase_of_ne</a> <| <a>ne_of_not_mem_cons</a> h\u2082).2 h\u2081) (<a>mt</a> (.tail _) h\u2082)", [{"full_name": "List.mem_erase_of_ne", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [1208, 17], "def_end_pos": [1208, 32]}, {"full_name": "List.ne_of_not_mem_cons", "def_path": "lake-packages/std/Std/Data/List/Lemmas.lean", "def_pos": [96, 9], "def_end_pos": [96, 27]}, {"full_name": "mt", "def_path": "lake-packages/lean4/src/lean/Init/Core.lean", "def_pos": [516, 9], "def_end_pos": [516, 11]}]], "state_before": "\u03b1 : Type u_1\ninst\u271d : DecidableEq \u03b1\na : \u03b1\nl\u2081 : List \u03b1\nb : \u03b1\nl\u2082 : List \u03b1\nh\u2081 : a \u2208 l\u2081\nh\u2082 : \u00aca \u2208 b :: l\u2082\n\u22a2 a \u2208 List.diff (List.erase l\u2081 b) l\u2082", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean
|
Matrix.SpecialLinearGroup.row_ne_zero
|
[
166,
1
] |
[
167,
63
] |
[{"tactic": "simp [h]", "annotated_tactic": ["simp [h]", []], "state_before": "n : Type u\ninst\u271d\u00b3 : DecidableEq n\ninst\u271d\u00b2 : Fintype n\nR : Type v\ninst\u271d\u00b9 : CommRing R\nA B : SpecialLinearGroup n R\ninst\u271d : Nontrivial R\ng : SpecialLinearGroup n R\ni : n\nh : \u2191g i = 0\n\u22a2 \u2200 (j : n), \u2191g i j = 0", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Order/Bounds/Basic.lean
|
BddAbove.image2
|
[
1389,
1
] |
[
1391,
52
] |
[{"tactic": "rintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9", "annotated_tactic": ["rintro \u27e8a, ha\u27e9 \u27e8b, hb\u27e9", []], "state_before": "\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na : \u03b1\nb : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Monotone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Monotone (f a)\n\u22a2 BddAbove s \u2192 BddAbove t \u2192 BddAbove (Set.image2 f s t)", "state_after": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Monotone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Monotone (f a)\na : \u03b1\nha : a \u2208 upperBounds s\nb : \u03b2\nhb : b \u2208 upperBounds t\n\u22a2 BddAbove (Set.image2 f s t)"}, {"tactic": "exact \u27e8f a b, mem_upperBounds_image2 h\u2080 h\u2081 ha hb\u27e9", "annotated_tactic": ["exact \u27e8f a b, <a>mem_upperBounds_image2</a> h\u2080 h\u2081 ha hb\u27e9", [{"full_name": "mem_upperBounds_image2", "def_path": "Mathlib/Order/Bounds/Basic.lean", "def_pos": [1366, 9], "def_end_pos": [1366, 31]}]], "state_before": "case intro.intro\n\u03b1 : Type u\n\u03b2 : Type v\n\u03b3 : Type w\n\u03b9 : Sort x\ninst\u271d\u00b2 : Preorder \u03b1\ninst\u271d\u00b9 : Preorder \u03b2\ninst\u271d : Preorder \u03b3\nf : \u03b1 \u2192 \u03b2 \u2192 \u03b3\ns : Set \u03b1\nt : Set \u03b2\na\u271d : \u03b1\nb\u271d : \u03b2\nh\u2080 : \u2200 (b : \u03b2), Monotone (swap f b)\nh\u2081 : \u2200 (a : \u03b1), Monotone (f a)\na : \u03b1\nha : a \u2208 upperBounds s\nb : \u03b2\nhb : b \u2208 upperBounds t\n\u22a2 BddAbove (Set.image2 f s t)", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Cardinal.add_eq_left_iff
|
[
785,
1
] |
[
802,
24
] |
[{"tactic": "rw [max_le_iff]", "annotated_tactic": ["rw [<a>max_le_iff</a>]", [{"full_name": "max_le_iff", "def_path": "Mathlib/Order/MinMax.lean", "def_pos": [48, 9], "def_end_pos": [48, 19]}]], "state_before": "a b : Cardinal.{u_1}\n\u22a2 a + b = a \u2194 max \u2135\u2080 b \u2264 a \u2228 b = 0", "state_after": "a b : Cardinal.{u_1}\n\u22a2 a + b = a \u2194 \u2135\u2080 \u2264 a \u2227 b \u2264 a \u2228 b = 0"}, {"tactic": "refine' \u27e8fun h => _, _\u27e9", "annotated_tactic": ["refine' \u27e8fun h => _, _\u27e9", []], "state_before": "a b : Cardinal.{u_1}\n\u22a2 a + b = a \u2194 \u2135\u2080 \u2264 a \u2227 b \u2264 a \u2228 b = 0", "state_after": "case refine'_1\na b : Cardinal.{u_1}\nh : a + b = a\n\u22a2 \u2135\u2080 \u2264 a \u2227 b \u2264 a \u2228 b = 0\n\ncase refine'_2\na b : Cardinal.{u_1}\n\u22a2 \u2135\u2080 \u2264 a \u2227 b \u2264 a \u2228 b = 0 \u2192 a + b = a"}, {"tactic": "cases' le_or_lt \u2135\u2080 a with ha ha", "annotated_tactic": ["cases' <a>le_or_lt</a> \u2135\u2080 a with ha ha", [{"full_name": "le_or_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [340, 9], "def_end_pos": [340, 17]}]], "state_before": "case refine'_1\na b : Cardinal.{u_1}\nh : a + b = a\n\u22a2 \u2135\u2080 \u2264 a \u2227 b \u2264 a \u2228 b = 0", "state_after": "case refine'_1.inl\na b : Cardinal.{u_1}\nh : a + b = a\nha : \u2135\u2080 \u2264 a\n\u22a2 \u2135\u2080 \u2264 a \u2227 b \u2264 a \u2228 b = 0\n\ncase refine'_1.inr\na b : Cardinal.{u_1}\nh : a + b = a\nha : a < \u2135\u2080\n\u22a2 \u2135\u2080 \u2264 a \u2227 b \u2264 a \u2228 b = 0"}, {"tactic": "right", "annotated_tactic": ["right", []], "state_before": "case refine'_1.inr\na b : Cardinal.{u_1}\nh : a + b = a\nha : a < \u2135\u2080\n\u22a2 \u2135\u2080 \u2264 a \u2227 b \u2264 a \u2228 b = 0", "state_after": "case refine'_1.inr.h\na b : Cardinal.{u_1}\nh : a + b = a\nha : a < \u2135\u2080\n\u22a2 b = 0"}, {"tactic": "rw [\u2190 h, add_lt_aleph0_iff, lt_aleph0, lt_aleph0] at ha", "annotated_tactic": ["rw [\u2190 h, <a>add_lt_aleph0_iff</a>, <a>lt_aleph0</a>, <a>lt_aleph0</a>] at ha", [{"full_name": "Cardinal.add_lt_aleph0_iff", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1537, 9], "def_end_pos": [1537, 26]}, {"full_name": "Cardinal.lt_aleph0", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1429, 9], "def_end_pos": [1429, 18]}, {"full_name": "Cardinal.lt_aleph0", "def_path": "Mathlib/SetTheory/Cardinal/Basic.lean", "def_pos": [1429, 9], "def_end_pos": [1429, 18]}]], "state_before": "case refine'_1.inr.h\na b : Cardinal.{u_1}\nh : a + b = a\nha : a < \u2135\u2080\n\u22a2 b = 0", "state_after": "case refine'_1.inr.h\na b : Cardinal.{u_1}\nh : a + b = a\nha : (\u2203 n, a = \u2191n) \u2227 \u2203 n, b = \u2191n\n\u22a2 b = 0"}, {"tactic": "rcases ha with \u27e8\u27e8n, rfl\u27e9, \u27e8m, rfl\u27e9\u27e9", "annotated_tactic": ["rcases ha with \u27e8\u27e8n, rfl\u27e9, \u27e8m, rfl\u27e9\u27e9", []], "state_before": "case refine'_1.inr.h\na b : Cardinal.{u_1}\nh : a + b = a\nha : (\u2203 n, a = \u2191n) \u2227 \u2203 n, b = \u2191n\n\u22a2 b = 0", "state_after": "case refine'_1.inr.h.intro.intro.intro\nn m : \u2115\nh : \u2191n + \u2191m = \u2191n\n\u22a2 \u2191m = 0"}, {"tactic": "norm_cast at h \u22a2", "annotated_tactic": ["norm_cast at h \u22a2", []], "state_before": "case refine'_1.inr.h.intro.intro.intro\nn m : \u2115\nh : \u2191n + \u2191m = \u2191n\n\u22a2 \u2191m = 0", "state_after": "case refine'_1.inr.h.intro.intro.intro\nn m : \u2115\nh : n + m = n\n\u22a2 m = 0"}, {"tactic": "rw [\u2190 add_right_inj, h, add_zero]", "annotated_tactic": ["rw [\u2190 <a>add_right_inj</a>, h, <a>add_zero</a>]", [{"full_name": "add_right_inj", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [188, 3], "def_end_pos": [188, 14]}, {"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "case refine'_1.inr.h.intro.intro.intro\nn m : \u2115\nh : n + m = n\n\u22a2 m = 0", "state_after": "no goals"}, {"tactic": "left", "annotated_tactic": ["left", []], "state_before": "case refine'_1.inl\na b : Cardinal.{u_1}\nh : a + b = a\nha : \u2135\u2080 \u2264 a\n\u22a2 \u2135\u2080 \u2264 a \u2227 b \u2264 a \u2228 b = 0", "state_after": "case refine'_1.inl.h\na b : Cardinal.{u_1}\nh : a + b = a\nha : \u2135\u2080 \u2264 a\n\u22a2 \u2135\u2080 \u2264 a \u2227 b \u2264 a"}, {"tactic": "use ha", "annotated_tactic": ["use ha", []], "state_before": "case refine'_1.inl.h\na b : Cardinal.{u_1}\nh : a + b = a\nha : \u2135\u2080 \u2264 a\n\u22a2 \u2135\u2080 \u2264 a \u2227 b \u2264 a", "state_after": "case right\na b : Cardinal.{u_1}\nh : a + b = a\nha : \u2135\u2080 \u2264 a\n\u22a2 b \u2264 a"}, {"tactic": "rw [\u2190 not_lt]", "annotated_tactic": ["rw [\u2190 <a>not_lt</a>]", [{"full_name": "not_lt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [368, 9], "def_end_pos": [368, 15]}]], "state_before": "case right\na b : Cardinal.{u_1}\nh : a + b = a\nha : \u2135\u2080 \u2264 a\n\u22a2 b \u2264 a", "state_after": "case right\na b : Cardinal.{u_1}\nh : a + b = a\nha : \u2135\u2080 \u2264 a\n\u22a2 \u00aca < b"}, {"tactic": "apply fun hb => ne_of_gt _ h", "annotated_tactic": ["apply fun hb => <a>ne_of_gt</a> _ h", [{"full_name": "ne_of_gt", "def_path": "Mathlib/Init/Order/Defs.lean", "def_pos": [104, 9], "def_end_pos": [104, 17]}]], "state_before": "case right\na b : Cardinal.{u_1}\nh : a + b = a\nha : \u2135\u2080 \u2264 a\n\u22a2 \u00aca < b", "state_after": "a b : Cardinal.{u_1}\nh : a + b = a\nha : \u2135\u2080 \u2264 a\n\u22a2 a < b \u2192 a < a + b"}, {"tactic": "intro hb", "annotated_tactic": ["intro hb", []], "state_before": "a b : Cardinal.{u_1}\nh : a + b = a\nha : \u2135\u2080 \u2264 a\n\u22a2 a < b \u2192 a < a + b", "state_after": "a b : Cardinal.{u_1}\nh : a + b = a\nha : \u2135\u2080 \u2264 a\nhb : a < b\n\u22a2 a < a + b"}, {"tactic": "exact hb.trans_le (self_le_add_left b a)", "annotated_tactic": ["exact hb.trans_le (<a>self_le_add_left</a> b a)", [{"full_name": "self_le_add_left", "def_path": "Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean", "def_pos": [173, 3], "def_end_pos": [173, 14]}]], "state_before": "a b : Cardinal.{u_1}\nh : a + b = a\nha : \u2135\u2080 \u2264 a\nhb : a < b\n\u22a2 a < a + b", "state_after": "no goals"}, {"tactic": "rintro (\u27e8h1, h2\u27e9 | h3)", "annotated_tactic": ["rintro (\u27e8h1, h2\u27e9 | h3)", []], "state_before": "case refine'_2\na b : Cardinal.{u_1}\n\u22a2 \u2135\u2080 \u2264 a \u2227 b \u2264 a \u2228 b = 0 \u2192 a + b = a", "state_after": "case refine'_2.inl.intro\na b : Cardinal.{u_1}\nh1 : \u2135\u2080 \u2264 a\nh2 : b \u2264 a\n\u22a2 a + b = a\n\ncase refine'_2.inr\na b : Cardinal.{u_1}\nh3 : b = 0\n\u22a2 a + b = a"}, {"tactic": "rw [add_eq_max h1, max_eq_left h2]", "annotated_tactic": ["rw [<a>add_eq_max</a> h1, <a>max_eq_left</a> h2]", [{"full_name": "Cardinal.add_eq_max", "def_path": "Mathlib/SetTheory/Cardinal/Ordinal.lean", "def_pos": [726, 9], "def_end_pos": [726, 19]}, {"full_name": "max_eq_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [133, 9], "def_end_pos": [133, 20]}]], "state_before": "case refine'_2.inl.intro\na b : Cardinal.{u_1}\nh1 : \u2135\u2080 \u2264 a\nh2 : b \u2264 a\n\u22a2 a + b = a", "state_after": "no goals"}, {"tactic": "rw [h3, add_zero]", "annotated_tactic": ["rw [h3, <a>add_zero</a>]", [{"full_name": "add_zero", "def_path": "Mathlib/Algebra/Group/Defs.lean", "def_pos": [469, 3], "def_end_pos": [469, 14]}]], "state_before": "case refine'_2.inr\na b : Cardinal.{u_1}\nh3 : b = 0\n\u22a2 a + b = a", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/LinearAlgebra/AdicCompletion.lean
|
adicCompletion.ext
|
[
254,
1
] |
[
255,
25
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.tendsto_nat_nhds_top
|
[
178,
1
] |
[
180,
96
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/LinearAlgebra/Basic.lean
|
LinearMap.subtype_comp_restrict
|
[
227,
1
] |
[
229,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/CategoryTheory/Monoidal/Opposite.lean
|
CategoryTheory.mop_id_unmop
|
[
146,
1
] |
[
147,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/Int/ModEq.lean
|
Int.ModEq.of_div
|
[
241,
1
] |
[
242,
76
] |
[{"tactic": "convert h.mul_left' <;> rwa [Int.mul_ediv_cancel']", "annotated_tactic": ["convert h.mul_left' <;> rwa [<a>Int.mul_ediv_cancel'</a>]", [{"full_name": "Int.mul_ediv_cancel'", "def_path": "lake-packages/std/Std/Data/Int/DivMod.lean", "def_pos": [710, 19], "def_end_pos": [710, 35]}]], "state_before": "m n a b c d : \u2124\nh : a / c \u2261 b / c [ZMOD m / c]\nha\u271d\u00b9 : c \u2223 a\nha\u271d : c \u2223 b\nha : c \u2223 m\n\u22a2 a \u2261 b [ZMOD m]", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.span_singleton_mul_left_injective
|
[
594,
1
] |
[
596,
38
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
bot_lt_affineSpan
|
[
1183,
1
] |
[
1185,
34
] |
[{"tactic": "rw [bot_lt_iff_ne_bot, nonempty_iff_ne_empty]", "annotated_tactic": ["rw [<a>bot_lt_iff_ne_bot</a>, <a>nonempty_iff_ne_empty</a>]", [{"full_name": "bot_lt_iff_ne_bot", "def_path": "Mathlib/Order/BoundedOrder.lean", "def_pos": [371, 9], "def_end_pos": [371, 26]}, {"full_name": "Set.nonempty_iff_ne_empty", "def_path": "Mathlib/Data/Set/Basic.lean", "def_pos": [610, 9], "def_end_pos": [610, 30]}]], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\n\u22a2 \u22a5 < affineSpan k s \u2194 Set.Nonempty s", "state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\n\u22a2 affineSpan k s \u2260 \u22a5 \u2194 s \u2260 \u2205"}, {"tactic": "exact (affineSpan_eq_bot _).not", "annotated_tactic": ["exact (<a>affineSpan_eq_bot</a> _).<a>not</a>", [{"full_name": "affineSpan_eq_bot", "def_path": "Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean", "def_pos": [1177, 9], "def_end_pos": [1177, 26]}, {"full_name": "Iff.not", "def_path": "Mathlib/Logic/Basic.lean", "def_pos": [274, 7], "def_end_pos": [274, 14]}]], "state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst\u271d\u00b3 : Ring k\ninst\u271d\u00b2 : AddCommGroup V\ninst\u271d\u00b9 : Module k V\ninst\u271d : AffineSpace V P\n\u03b9 : Type u_4\ns : Set P\n\u22a2 affineSpan k s \u2260 \u22a5 \u2194 s \u2260 \u2205", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/CategoryTheory/Preadditive/Opposite.lean
|
CategoryTheory.unop_add
|
[
36,
1
] |
[
37,
6
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Order/Filter/Pointwise.lean
|
Filter.NeBot.div
|
[
473,
1
] |
[
474,
13
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Topology/MetricSpace/PiNat.lean
|
PiCountable.dist_summable
|
[
833,
1
] |
[
837,
56
] |
[{"tactic": "refine summable_of_nonneg_of_le (fun i => ?_) (fun i => min_le_left _ _)\n summable_geometric_two_encode", "annotated_tactic": ["refine <a>summable_of_nonneg_of_le</a> (fun i => ?_) (fun i => <a>min_le_left</a> _ _)\n <a>summable_geometric_two_encode</a>", [{"full_name": "summable_of_nonneg_of_le", "def_path": "Mathlib/Topology/Instances/ENNReal.lean", "def_pos": [1297, 9], "def_end_pos": [1297, 33]}, {"full_name": "min_le_left", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [33, 9], "def_end_pos": [33, 20]}, {"full_name": "summable_geometric_two_encode", "def_path": "Mathlib/Analysis/SpecificLimits/Basic.lean", "def_pos": [232, 9], "def_end_pos": [232, 38]}]], "state_before": "E : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\nx y : (i : \u03b9) \u2192 F i\n\u22a2 Summable fun i => min ((1 / 2) ^ encode i) (dist (x i) (y i))", "state_after": "E : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\nx y : (i : \u03b9) \u2192 F i\ni : \u03b9\n\u22a2 0 \u2264 min ((1 / 2) ^ encode i) (dist (x i) (y i))"}, {"tactic": "exact le_min (pow_nonneg (by norm_num) _) dist_nonneg", "annotated_tactic": ["exact <a>le_min</a> (<a>pow_nonneg</a> (by norm_num) _) <a>dist_nonneg</a>", [{"full_name": "le_min", "def_path": "Mathlib/Init/Order/LinearOrder.lean", "def_pos": [47, 9], "def_end_pos": [47, 15]}, {"full_name": "pow_nonneg", "def_path": "Mathlib/Algebra/Order/Ring/Defs.lean", "def_pos": [244, 9], "def_end_pos": [244, 19]}, {"full_name": "dist_nonneg", "def_path": "Mathlib/Topology/MetricSpace/Basic.lean", "def_pos": [270, 9], "def_end_pos": [270, 20]}]], "state_before": "E : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\nx y : (i : \u03b9) \u2192 F i\ni : \u03b9\n\u22a2 0 \u2264 min ((1 / 2) ^ encode i) (dist (x i) (y i))", "state_after": "no goals"}, {"tactic": "norm_num", "annotated_tactic": ["norm_num", []], "state_before": "E : \u2115 \u2192 Type u_1\n\u03b9 : Type u_2\ninst\u271d\u00b9 : Encodable \u03b9\nF : \u03b9 \u2192 Type u_3\ninst\u271d : (i : \u03b9) \u2192 MetricSpace (F i)\nx y : (i : \u03b9) \u2192 F i\ni : \u03b9\n\u22a2 0 \u2264 1 / 2", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/LinearAlgebra/Span.lean
|
Submodule.span_int_eq_addSubgroup_closure
|
[
208,
1
] |
[
213,
86
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.tendsto_mul_self_atTop
|
[
931,
1
] |
[
932,
40
] |
[]
|
https://github.com/leanprover/std4
|
869c615eb10130c0637a7bc038e2b80253559913
|
lake-packages/std/Std/Data/List/Lemmas.lean
|
List.infix_append
|
[
1737,
1
] |
[
1737,
83
] |
[]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Algebra/DualQuaternion.lean
|
Quaternion.fst_re_dualNumberEquiv_symm
|
[
110,
1
] |
[
112,
6
] |
[]
|
https://github.com/leanprover/std4
|
869c615eb10130c0637a7bc038e2b80253559913
|
lake-packages/std/Std/Data/Nat/Lemmas.lean
|
Nat.pow_mul'
|
[
829,
11
] |
[
830,
34
] |
[{"tactic": "rw [\u2190Nat.pow_mul, Nat.mul_comm]", "annotated_tactic": ["rw [\u2190<a>Nat.pow_mul</a>, <a>Nat.mul_comm</a>]", [{"full_name": "Nat.pow_mul", "def_path": "lake-packages/std/Std/Data/Nat/Lemmas.lean", "def_pos": [824, 19], "def_end_pos": [824, 26]}, {"full_name": "Nat.mul_comm", "def_path": "lake-packages/lean4/src/lean/Init/Data/Nat/Basic.lean", "def_pos": [174, 19], "def_end_pos": [174, 27]}]], "state_before": "a m n : Nat\n\u22a2 a ^ (m * n) = (a ^ n) ^ m", "state_after": "no goals"}]
|
https://github.com/leanprover-community/mathlib4
|
3ce43c18f614b76e161f911b75a3e1ef641620ff
|
Mathlib/Data/List/BigOperators/Lemmas.lean
|
MulOpposite.op_list_prod
|
[
142,
1
] |
[
146,
89
] |
[{"tactic": "intro l", "annotated_tactic": ["intro l", []], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : Monoid M\n\u22a2 \u2200 (l : List M), op (prod l) = prod (reverse (map op l))", "state_after": "\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : Monoid M\nl : List M\n\u22a2 op (prod l) = prod (reverse (map op l))"}, {"tactic": "induction l with\n| nil => rfl\n| cons x xs ih =>\nrw [List.prod_cons, List.map_cons, List.reverse_cons', List.prod_concat, op_mul, ih]", "annotated_tactic": ["induction l with\n | <a>nil</a> => rfl\n | <a>cons</a> x xs ih =>\n rw [<a>List.prod_cons</a>, <a>List.map_cons</a>, <a>List.reverse_cons'</a>, <a>List.prod_concat</a>, <a>op_mul</a>, ih]", [{"full_name": "List.nil", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2184, 5], "def_end_pos": [2184, 8]}, {"full_name": "List.cons", "def_path": "lake-packages/lean4/src/lean/Init/Prelude.lean", "def_pos": [2187, 5], "def_end_pos": [2187, 9]}, {"full_name": "List.prod_cons", "def_path": "Mathlib/Data/List/BigOperators/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 18]}, {"full_name": "List.map_cons", "def_path": "lake-packages/std/Std/Data/List/Init/Lemmas.lean", "def_pos": [89, 17], "def_end_pos": [89, 25]}, {"full_name": "List.reverse_cons'", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [562, 9], "def_end_pos": [562, 22]}, {"full_name": "List.prod_concat", "def_path": "Mathlib/Data/List/BigOperators/Basic.lean", "def_pos": [58, 9], "def_end_pos": [58, 20]}, {"full_name": "MulOpposite.op_mul", "def_path": "Mathlib/Algebra/Opposites.lean", "def_pos": [283, 9], "def_end_pos": [283, 15]}]], "state_before": "\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : Monoid M\nl : List M\n\u22a2 op (prod l) = prod (reverse (map op l))", "state_after": "no goals"}, {"tactic": "rfl", "annotated_tactic": ["rfl", []], "state_before": "case nil\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : Monoid M\n\u22a2 op (prod []) = prod (reverse (map op []))", "state_after": "no goals"}, {"tactic": "rw [List.prod_cons, List.map_cons, List.reverse_cons', List.prod_concat, op_mul, ih]", "annotated_tactic": ["rw [<a>List.prod_cons</a>, <a>List.map_cons</a>, <a>List.reverse_cons'</a>, <a>List.prod_concat</a>, <a>op_mul</a>, ih]", [{"full_name": "List.prod_cons", "def_path": "Mathlib/Data/List/BigOperators/Basic.lean", "def_pos": [41, 9], "def_end_pos": [41, 18]}, {"full_name": "List.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.reverse_cons'", "def_path": "Mathlib/Data/List/Basic.lean", "def_pos": [562, 9], "def_end_pos": [562, 22]}, {"full_name": "List.prod_concat", "def_path": "Mathlib/Data/List/BigOperators/Basic.lean", "def_pos": [58, 9], "def_end_pos": [58, 20]}, {"full_name": "MulOpposite.op_mul", "def_path": "Mathlib/Algebra/Opposites.lean", "def_pos": [283, 9], "def_end_pos": [283, 15]}]], "state_before": "case cons\n\u03b9 : Type u_1\n\u03b1 : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\nM\u2080 : Type u_6\nG : Type u_7\nR : Type u_8\ninst\u271d : Monoid M\nx : M\nxs : List M\nih : op (prod xs) = prod (reverse (map op xs))\n\u22a2 op (prod (x :: xs)) = prod (reverse (map op (x :: xs)))", "state_after": "no goals"}]
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Subsets and Splits
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