file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/FieldTheory/Tower.lean
|
rank_mul_rank
|
[
{
"state_after": "no goals",
"state_before": "F✝ : Type u\nK✝ : Type v\nA✝ : Type w\ninst✝²¹ : CommRing F✝\ninst✝²⁰ : Ring K✝\ninst✝¹⁹ : AddCommGroup A✝\ninst✝¹⁸ : Algebra F✝ K✝\ninst✝¹⁷ : Module K✝ A✝\ninst✝¹⁶ : Module F✝ A✝\ninst✝¹⁵ : IsScalarTower F✝ K✝ A✝\ninst✝¹⁴ : StrongRankCondition F✝\ninst✝¹³ : StrongRankCondition K✝\ninst✝¹² : Module.Free F✝ K✝\ninst✝¹¹ : Module.Free K✝ A✝\nF : Type u\nK A : Type v\ninst✝¹⁰ : CommRing F\ninst✝⁹ : Ring K\ninst✝⁸ : AddCommGroup A\ninst✝⁷ : Algebra F K\ninst✝⁶ : Module K A\ninst✝⁵ : Module F A\ninst✝⁴ : IsScalarTower F K A\ninst✝³ : StrongRankCondition F\ninst✝² : StrongRankCondition K\ninst✝¹ : Module.Free F K\ninst✝ : Module.Free K A\n⊢ Module.rank F K * Module.rank K A = Module.rank F A",
"tactic": "convert lift_rank_mul_lift_rank F K A <;> rw [lift_id]"
}
] |
[
76,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieHom.range_subset_idealRange
|
[] |
[
938,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
937,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.map_map
|
[] |
[
469,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
467,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.AEFinStronglyMeasurable.const_smul
|
[] |
[
1900,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1897,
11
] |
Mathlib/RingTheory/OreLocalization/Basic.lean
|
OreLocalization.mul_zero
|
[
{
"state_after": "case c\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr : R\ns : { x // x ∈ S }\n⊢ r /ₒ s * 0 = 0",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nx : OreLocalization R S\n⊢ x * 0 = 0",
"tactic": "induction' x using OreLocalization.ind with r s"
},
{
"state_after": "case c\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr : R\ns : { x // x ∈ S }\n⊢ r * 0 /ₒ (1 * 1) = 0 /ₒ 1",
"state_before": "case c\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr : R\ns : { x // x ∈ S }\n⊢ r /ₒ s * 0 = 0",
"tactic": "rw [OreLocalization.zero_def, oreDiv_mul_char r 0 s 1 0 1 (by simp)]"
},
{
"state_after": "no goals",
"state_before": "case c\nR : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr : R\ns : { x // x ∈ S }\n⊢ r * 0 /ₒ (1 * 1) = 0 /ₒ 1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\nS : Submonoid R\ninst✝ : OreSet S\nr : R\ns : { x // x ∈ S }\n⊢ 0 * ↑1 = ↑s * 0",
"tactic": "simp"
}
] |
[
700,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
698,
11
] |
Mathlib/Data/Real/EReal.lean
|
EReal.mul_top_of_pos
|
[] |
[
925,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
922,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
zsmul_int_int
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.285397\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\na b : ℤ\n⊢ a • b = a * b",
"tactic": "simp"
}
] |
[
618,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
618,
1
] |
Mathlib/Topology/VectorBundle/Basic.lean
|
Trivialization.linearEquivAt_symm_apply
|
[] |
[
212,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/Topology/Order/Hom/Basic.lean
|
ContinuousOrderHom.coe_id
|
[] |
[
149,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean
|
span_gramSchmidtNormed
|
[
{
"state_after": "𝕜 : Type u_3\nE : Type u_2\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\ns : Set ι\ni : ι\nhi : i ∈ s\n⊢ gramSchmidt 𝕜 f i ∈ span 𝕜 (gramSchmidtNormed 𝕜 f '' ↑{i})",
"state_before": "𝕜 : Type u_3\nE : Type u_2\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\ns : Set ι\n⊢ span 𝕜 (gramSchmidtNormed 𝕜 f '' s) = span 𝕜 (gramSchmidt 𝕜 f '' s)",
"tactic": "refine' span_eq_span\n (Set.image_subset_iff.2 fun i hi => smul_mem _ _ <| subset_span <| mem_image_of_mem _ hi)\n (Set.image_subset_iff.2 fun i hi =>\n span_mono (image_subset _ <| singleton_subset_set_iff.2 hi) _)"
},
{
"state_after": "𝕜 : Type u_3\nE : Type u_2\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\ns : Set ι\ni : ι\nhi : i ∈ s\n⊢ gramSchmidt 𝕜 f i ∈ span 𝕜 {gramSchmidtNormed 𝕜 f i}",
"state_before": "𝕜 : Type u_3\nE : Type u_2\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\ns : Set ι\ni : ι\nhi : i ∈ s\n⊢ gramSchmidt 𝕜 f i ∈ span 𝕜 (gramSchmidtNormed 𝕜 f '' ↑{i})",
"tactic": "simp only [coe_singleton, Set.image_singleton]"
},
{
"state_after": "case pos\n𝕜 : Type u_3\nE : Type u_2\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\ns : Set ι\ni : ι\nhi : i ∈ s\nh : gramSchmidt 𝕜 f i = 0\n⊢ gramSchmidt 𝕜 f i ∈ span 𝕜 {gramSchmidtNormed 𝕜 f i}\n\ncase neg\n𝕜 : Type u_3\nE : Type u_2\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\ns : Set ι\ni : ι\nhi : i ∈ s\nh : ¬gramSchmidt 𝕜 f i = 0\n⊢ gramSchmidt 𝕜 f i ∈ span 𝕜 {gramSchmidtNormed 𝕜 f i}",
"state_before": "𝕜 : Type u_3\nE : Type u_2\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\ns : Set ι\ni : ι\nhi : i ∈ s\n⊢ gramSchmidt 𝕜 f i ∈ span 𝕜 {gramSchmidtNormed 𝕜 f i}",
"tactic": "by_cases h : gramSchmidt 𝕜 f i = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\n𝕜 : Type u_3\nE : Type u_2\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\ns : Set ι\ni : ι\nhi : i ∈ s\nh : gramSchmidt 𝕜 f i = 0\n⊢ gramSchmidt 𝕜 f i ∈ span 𝕜 {gramSchmidtNormed 𝕜 f i}",
"tactic": "simp [h]"
},
{
"state_after": "case neg\n𝕜 : Type u_3\nE : Type u_2\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\ns : Set ι\ni : ι\nhi : i ∈ s\nh : ¬gramSchmidt 𝕜 f i = 0\n⊢ ↑‖gramSchmidt 𝕜 f i‖ ≠ 0",
"state_before": "case neg\n𝕜 : Type u_3\nE : Type u_2\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\ns : Set ι\ni : ι\nhi : i ∈ s\nh : ¬gramSchmidt 𝕜 f i = 0\n⊢ gramSchmidt 𝕜 f i ∈ span 𝕜 {gramSchmidtNormed 𝕜 f i}",
"tactic": "refine' mem_span_singleton.2 ⟨‖gramSchmidt 𝕜 f i‖, smul_inv_smul₀ _ _⟩"
},
{
"state_after": "no goals",
"state_before": "case neg\n𝕜 : Type u_3\nE : Type u_2\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : IsWellOrder ι fun x x_1 => x < x_1\nf : ι → E\ns : Set ι\ni : ι\nhi : i ∈ s\nh : ¬gramSchmidt 𝕜 f i = 0\n⊢ ↑‖gramSchmidt 𝕜 f i‖ ≠ 0",
"tactic": "exact_mod_cast norm_ne_zero_iff.2 h"
}
] |
[
329,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
319,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.trans_refl
|
[] |
[
860,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
859,
1
] |
Mathlib/CategoryTheory/Limits/HasLimits.lean
|
CategoryTheory.Limits.colimit.map_post
|
[
{
"state_after": "case w\nJ : Type u₁\ninst✝⁵ : Category J\nK : Type u₂\ninst✝⁴ : Category K\nC : Type u\ninst✝³ : Category C\nF : J ⥤ C\ninst✝² : HasColimitsOfShape J C\nG : J ⥤ C\nα : F ⟶ G\nD : Type u'\ninst✝¹ : Category D\ninst✝ : HasColimitsOfShape J D\nH : C ⥤ D\nj✝ : J\n⊢ ι (F ⋙ H) j✝ ≫ post F H ≫ H.map (colim.map α) = ι (F ⋙ H) j✝ ≫ colim.map (whiskerRight α H) ≫ post G H",
"state_before": "J : Type u₁\ninst✝⁵ : Category J\nK : Type u₂\ninst✝⁴ : Category K\nC : Type u\ninst✝³ : Category C\nF : J ⥤ C\ninst✝² : HasColimitsOfShape J C\nG : J ⥤ C\nα : F ⟶ G\nD : Type u'\ninst✝¹ : Category D\ninst✝ : HasColimitsOfShape J D\nH : C ⥤ D\n⊢ post F H ≫ H.map (colim.map α) = colim.map (whiskerRight α H) ≫ post G H",
"tactic": "ext"
},
{
"state_after": "case w\nJ : Type u₁\ninst✝⁵ : Category J\nK : Type u₂\ninst✝⁴ : Category K\nC : Type u\ninst✝³ : Category C\nF : J ⥤ C\ninst✝² : HasColimitsOfShape J C\nG : J ⥤ C\nα : F ⟶ G\nD : Type u'\ninst✝¹ : Category D\ninst✝ : HasColimitsOfShape J D\nH : C ⥤ D\nj✝ : J\n⊢ H.map (α.app j✝) ≫ H.map (ι G j✝) = ι (F ⋙ H) j✝ ≫ colim.map (whiskerRight α H) ≫ post G H",
"state_before": "case w\nJ : Type u₁\ninst✝⁵ : Category J\nK : Type u₂\ninst✝⁴ : Category K\nC : Type u\ninst✝³ : Category C\nF : J ⥤ C\ninst✝² : HasColimitsOfShape J C\nG : J ⥤ C\nα : F ⟶ G\nD : Type u'\ninst✝¹ : Category D\ninst✝ : HasColimitsOfShape J D\nH : C ⥤ D\nj✝ : J\n⊢ ι (F ⋙ H) j✝ ≫ post F H ≫ H.map (colim.map α) = ι (F ⋙ H) j✝ ≫ colim.map (whiskerRight α H) ≫ post G H",
"tactic": "rw [← assoc, colimit.ι_post, ← H.map_comp, colimit.ι_map, H.map_comp]"
},
{
"state_after": "case w\nJ : Type u₁\ninst✝⁵ : Category J\nK : Type u₂\ninst✝⁴ : Category K\nC : Type u\ninst✝³ : Category C\nF : J ⥤ C\ninst✝² : HasColimitsOfShape J C\nG : J ⥤ C\nα : F ⟶ G\nD : Type u'\ninst✝¹ : Category D\ninst✝ : HasColimitsOfShape J D\nH : C ⥤ D\nj✝ : J\n⊢ H.map (α.app j✝) ≫ H.map (ι G j✝) = (whiskerRight α H).app j✝ ≫ H.map (ι G j✝)",
"state_before": "case w\nJ : Type u₁\ninst✝⁵ : Category J\nK : Type u₂\ninst✝⁴ : Category K\nC : Type u\ninst✝³ : Category C\nF : J ⥤ C\ninst✝² : HasColimitsOfShape J C\nG : J ⥤ C\nα : F ⟶ G\nD : Type u'\ninst✝¹ : Category D\ninst✝ : HasColimitsOfShape J D\nH : C ⥤ D\nj✝ : J\n⊢ H.map (α.app j✝) ≫ H.map (ι G j✝) = ι (F ⋙ H) j✝ ≫ colim.map (whiskerRight α H) ≫ post G H",
"tactic": "rw [← assoc, colimit.ι_map, assoc, colimit.ι_post]"
},
{
"state_after": "no goals",
"state_before": "case w\nJ : Type u₁\ninst✝⁵ : Category J\nK : Type u₂\ninst✝⁴ : Category K\nC : Type u\ninst✝³ : Category C\nF : J ⥤ C\ninst✝² : HasColimitsOfShape J C\nG : J ⥤ C\nα : F ⟶ G\nD : Type u'\ninst✝¹ : Category D\ninst✝ : HasColimitsOfShape J D\nH : C ⥤ D\nj✝ : J\n⊢ H.map (α.app j✝) ≫ H.map (ι G j✝) = (whiskerRight α H).app j✝ ≫ H.map (ι G j✝)",
"tactic": "rfl"
}
] |
[
1141,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1131,
1
] |
Mathlib/Order/WithBot.lean
|
WithTop.recTopCoe_coe
|
[] |
[
630,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
628,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_eq_zero_iff
|
[] |
[
883,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
882,
1
] |
Mathlib/CategoryTheory/Yoneda.lean
|
CategoryTheory.yonedaEquiv_apply
|
[] |
[
388,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
1
] |
Mathlib/Data/Polynomial/AlgebraMap.lean
|
Polynomial.algHom_ext'
|
[] |
[
92,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Images.lean
|
CategoryTheory.Limits.image.map_comp
|
[
{
"state_after": "case e_self\nC : Type u\ninst✝⁶ : Category C\nf g : Arrow C\ninst✝⁵ : HasImage f.hom\ninst✝⁴ : HasImage g.hom\nsq : f ⟶ g\ninst✝³ : HasImageMap sq\nh : Arrow C\ninst✝² : HasImage h.hom\nsq' : g ⟶ h\ninst✝¹ : HasImageMap sq'\ninst✝ : HasImageMap (sq ≫ sq')\n⊢ HasImageMap.imageMap (sq ≫ sq') = imageMapComp sq sq'",
"state_before": "C : Type u\ninst✝⁶ : Category C\nf g : Arrow C\ninst✝⁵ : HasImage f.hom\ninst✝⁴ : HasImage g.hom\nsq : f ⟶ g\ninst✝³ : HasImageMap sq\nh : Arrow C\ninst✝² : HasImage h.hom\nsq' : g ⟶ h\ninst✝¹ : HasImageMap sq'\ninst✝ : HasImageMap (sq ≫ sq')\n⊢ (HasImageMap.imageMap (sq ≫ sq')).map = (imageMapComp sq sq').map",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_self\nC : Type u\ninst✝⁶ : Category C\nf g : Arrow C\ninst✝⁵ : HasImage f.hom\ninst✝⁴ : HasImage g.hom\nsq : f ⟶ g\ninst✝³ : HasImageMap sq\nh : Arrow C\ninst✝² : HasImage h.hom\nsq' : g ⟶ h\ninst✝¹ : HasImageMap sq'\ninst✝ : HasImageMap (sq ≫ sq')\n⊢ HasImageMap.imageMap (sq ≫ sq') = imageMapComp sq sq'",
"tactic": "simp only [eq_iff_true_of_subsingleton]"
}
] |
[
826,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
823,
1
] |
Mathlib/GroupTheory/IsFreeGroup.lean
|
IsFreeGroup.of_eq_freeGroup_of
|
[] |
[
84,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Data/Fin/Interval.lean
|
Fin.map_valEmbedding_Ioc
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\na b : Fin n\n⊢ map valEmbedding (Ioc a b) = Ioc ↑a ↑b",
"tactic": "simp [Ioc_eq_finset_subtype, Finset.fin, Finset.map_map, Ioc_filter_lt_of_lt_right]"
}
] |
[
70,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.mem_centralizer_iff
|
[] |
[
2281,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2280,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
ofMul_list_prod
|
[
{
"state_after": "ι : Type ?u.987998\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : Monoid α\ns : List α\n⊢ List.prod s = List.sum s",
"state_before": "ι : Type ?u.987998\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : Monoid α\ns : List α\n⊢ ↑ofMul (List.prod s) = List.sum (List.map (↑ofMul) s)",
"tactic": "simp [ofMul]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.987998\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : Monoid α\ns : List α\n⊢ List.prod s = List.sum s",
"tactic": "rfl"
}
] |
[
2292,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2292,
1
] |
Mathlib/GroupTheory/Nilpotent.lean
|
upperCentralSeriesStep_eq_comap_center
|
[
{
"state_after": "case h\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nx✝ : G\n⊢ x✝ ∈ upperCentralSeriesStep H ↔ x✝ ∈ comap (mk' H) (center (G ⧸ H))",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\n⊢ upperCentralSeriesStep H = comap (mk' H) (center (G ⧸ H))",
"tactic": "ext"
},
{
"state_after": "case h\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nx✝ : G\n⊢ x✝ ∈ upperCentralSeriesStep H ↔ ∀ (x : G), ↑x * ↑(mk' H) x✝ = ↑(mk' H) x✝ * ↑x",
"state_before": "case h\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nx✝ : G\n⊢ x✝ ∈ upperCentralSeriesStep H ↔ x✝ ∈ comap (mk' H) (center (G ⧸ H))",
"tactic": "rw [mem_comap, mem_center_iff, forall_mk]"
},
{
"state_after": "case h.h\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nx✝ : G\n⊢ ∀ (a : G), x✝ * a * x✝⁻¹ * a⁻¹ ∈ H ↔ ↑a * ↑(mk' H) x✝ = ↑(mk' H) x✝ * ↑a",
"state_before": "case h\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nx✝ : G\n⊢ x✝ ∈ upperCentralSeriesStep H ↔ ∀ (x : G), ↑x * ↑(mk' H) x✝ = ↑(mk' H) x✝ * ↑x",
"tactic": "apply forall_congr'"
},
{
"state_after": "case h.h\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nx✝ y : G\n⊢ x✝ * y * x✝⁻¹ * y⁻¹ ∈ H ↔ ↑y * ↑(mk' H) x✝ = ↑(mk' H) x✝ * ↑y",
"state_before": "case h.h\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nx✝ : G\n⊢ ∀ (a : G), x✝ * a * x✝⁻¹ * a⁻¹ ∈ H ↔ ↑a * ↑(mk' H) x✝ = ↑(mk' H) x✝ * ↑a",
"tactic": "intro y"
},
{
"state_after": "no goals",
"state_before": "case h.h\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Normal H\nx✝ y : G\n⊢ x✝ * y * x✝⁻¹ * y⁻¹ ∈ H ↔ ↑y * ↑(mk' H) x✝ = ↑(mk' H) x✝ * ↑y",
"tactic": "rw [coe_mk', ← QuotientGroup.mk_mul, ← QuotientGroup.mk_mul, eq_comm, eq_iff_div_mem,\n div_eq_mul_inv, mul_inv_rev, mul_assoc]"
}
] |
[
122,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/CategoryTheory/Abelian/Basic.lean
|
CategoryTheory.Abelian.mono_inl_of_factor_thru_epi_mono_factorization
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁴ : Category C\ninst✝³ : Abelian C\ninst✝² : HasPushouts C\nW X Y Z : C\nf✝ : X ⟶ Y\ng✝ : X ⟶ Z\nf : X ⟶ Y\ng : X ⟶ Z\ng₁ : X ⟶ W\ninst✝¹ : Epi g₁\ng₂ : W ⟶ Z\ninst✝ : Mono g₂\nhg : g₁ ≫ g₂ = g\nf' : W ⟶ Y\nhf : g₁ ≫ f' = f\nt : PushoutCocone f g\nht : IsColimit t\n⊢ Mono (PushoutCocone.inl t)",
"tactic": "apply mono_inl_of_isColimit _ _ (PushoutCocone.isColimitOfFactors _ _ _ _ _ hf hg t ht)"
}
] |
[
780,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
777,
1
] |
Mathlib/CategoryTheory/IsConnected.lean
|
CategoryTheory.nat_trans_from_is_connected
|
[
{
"state_after": "J : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u₂\ninst✝¹ : Category C\ninst✝ : IsPreconnected J\nX Y : C\nα : (Functor.const J).obj X ⟶ (Functor.const J).obj Y\nx✝¹ x✝ : J\nf : x✝¹ ⟶ x✝\nthis : ((Functor.const J).obj X).map f ≫ α.app x✝ = α.app x✝¹ ≫ ((Functor.const J).obj Y).map f\n⊢ (fun j => α.app j) x✝¹ = (fun j => α.app j) x✝",
"state_before": "J : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u₂\ninst✝¹ : Category C\ninst✝ : IsPreconnected J\nX Y : C\nα : (Functor.const J).obj X ⟶ (Functor.const J).obj Y\nx✝¹ x✝ : J\nf : x✝¹ ⟶ x✝\n⊢ (fun j => α.app j) x✝¹ = (fun j => α.app j) x✝",
"tactic": "have := α.naturality f"
},
{
"state_after": "J : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u₂\ninst✝¹ : Category C\ninst✝ : IsPreconnected J\nX Y : C\nα : (Functor.const J).obj X ⟶ (Functor.const J).obj Y\nx✝¹ x✝ : J\nf : x✝¹ ⟶ x✝\nthis : α.app x✝ = α.app x✝¹\n⊢ (fun j => α.app j) x✝¹ = (fun j => α.app j) x✝",
"state_before": "J : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u₂\ninst✝¹ : Category C\ninst✝ : IsPreconnected J\nX Y : C\nα : (Functor.const J).obj X ⟶ (Functor.const J).obj Y\nx✝¹ x✝ : J\nf : x✝¹ ⟶ x✝\nthis : ((Functor.const J).obj X).map f ≫ α.app x✝ = α.app x✝¹ ≫ ((Functor.const J).obj Y).map f\n⊢ (fun j => α.app j) x✝¹ = (fun j => α.app j) x✝",
"tactic": "erw [id_comp, comp_id] at this"
},
{
"state_after": "no goals",
"state_before": "J : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u₂\ninst✝¹ : Category C\ninst✝ : IsPreconnected J\nX Y : C\nα : (Functor.const J).obj X ⟶ (Functor.const J).obj Y\nx✝¹ x✝ : J\nf : x✝¹ ⟶ x✝\nthis : α.app x✝ = α.app x✝¹\n⊢ (fun j => α.app j) x✝¹ = (fun j => α.app j) x✝",
"tactic": "exact this.symm"
}
] |
[
371,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
1
] |
Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean
|
ZMod.χ₄_eq_neg_one_pow
|
[
{
"state_after": "n : ℕ\nhn : n % 2 = 1\n⊢ (if n % 2 = 0 then 0 else if n % 4 = 1 then 1 else -1) = (-1) ^ (n / 2)",
"state_before": "n : ℕ\nhn : n % 2 = 1\n⊢ ↑χ₄ ↑n = (-1) ^ (n / 2)",
"tactic": "rw [χ₄_nat_eq_if_mod_four]"
},
{
"state_after": "n : ℕ\nhn : n % 2 = 1\n⊢ (if n % 4 = 1 then 1 else -1) = (-1) ^ (n / 2)",
"state_before": "n : ℕ\nhn : n % 2 = 1\n⊢ (if n % 2 = 0 then 0 else if n % 4 = 1 then 1 else -1) = (-1) ^ (n / 2)",
"tactic": "simp only [hn, Nat.one_ne_zero, if_false]"
},
{
"state_after": "n : ℕ\nhn : n % 2 = 1\n⊢ (if n % 4 = 1 then 1 else -1) = (-1) ^ ((4 * (n / 4) + n % 4) / 2)",
"state_before": "n : ℕ\nhn : n % 2 = 1\n⊢ (if n % 4 = 1 then 1 else -1) = (-1) ^ (n / 2)",
"tactic": "nth_rw 3 [← Nat.div_add_mod n 4]"
},
{
"state_after": "n : ℕ\nhn : n % 2 = 1\n⊢ (if n % 4 = 1 then 1 else -1) = (-1) ^ ((2 * 2 * (n / 4) + n % 4) / 2)",
"state_before": "n : ℕ\nhn : n % 2 = 1\n⊢ (if n % 4 = 1 then 1 else -1) = (-1) ^ ((4 * (n / 4) + n % 4) / 2)",
"tactic": "nth_rw 3 [(by norm_num : 4 = 2 * 2)]"
},
{
"state_after": "n : ℕ\nhn : n % 2 = 1\n⊢ (if n % 4 = 1 then 1 else -1) = (-1) ^ (n % 4 / 2)",
"state_before": "n : ℕ\nhn : n % 2 = 1\n⊢ (if n % 4 = 1 then 1 else -1) = (-1) ^ ((2 * 2 * (n / 4) + n % 4) / 2)",
"tactic": "rw [mul_assoc, add_comm, Nat.add_mul_div_left _ _ (by norm_num : 0 < 2), pow_add, pow_mul,\n neg_one_sq, one_pow, mul_one]"
},
{
"state_after": "n : ℕ\nhn : n % 2 = 1\nhelp : ∀ (m : ℕ), m < 4 → m % 2 = 1 → (if m = 1 then 1 else -1) = (-1) ^ (m / 2)\n⊢ (if n % 4 = 1 then 1 else -1) = (-1) ^ (n % 4 / 2)",
"state_before": "n : ℕ\nhn : n % 2 = 1\n⊢ (if n % 4 = 1 then 1 else -1) = (-1) ^ (n % 4 / 2)",
"tactic": "have help : ∀ m : ℕ, m < 4 → m % 2 = 1 → ite (m = 1) (1 : ℤ) (-1) = (-1) ^ (m / 2) := by decide"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nhn : n % 2 = 1\nhelp : ∀ (m : ℕ), m < 4 → m % 2 = 1 → (if m = 1 then 1 else -1) = (-1) ^ (m / 2)\n⊢ (if n % 4 = 1 then 1 else -1) = (-1) ^ (n % 4 / 2)",
"tactic": "exact\n help (n % 4) (Nat.mod_lt n (by norm_num))\n ((Nat.mod_mod_of_dvd n (by norm_num : 2 ∣ 4)).trans hn)"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nhn : n % 2 = 1\n⊢ 4 = 2 * 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nhn : n % 2 = 1\n⊢ 0 < 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nhn : n % 2 = 1\n⊢ ∀ (m : ℕ), m < 4 → m % 2 = 1 → (if m = 1 then 1 else -1) = (-1) ^ (m / 2)",
"tactic": "decide"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nhn : n % 2 = 1\nhelp : ∀ (m : ℕ), m < 4 → m % 2 = 1 → (if m = 1 then 1 else -1) = (-1) ^ (m / 2)\n⊢ 4 > 0",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nhn : n % 2 = 1\nhelp : ∀ (m : ℕ), m < 4 → m % 2 = 1 → (if m = 1 then 1 else -1) = (-1) ^ (m / 2)\n⊢ 2 ∣ 4",
"tactic": "norm_num"
}
] |
[
94,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean
|
SimpleGraph.adjMatrix_dotProduct
|
[
{
"state_after": "no goals",
"state_before": "V : Type u_2\nα : Type u_1\nβ : Type ?u.32376\nG : SimpleGraph V\ninst✝² : DecidableRel G.Adj\ninst✝¹ : Fintype V\ninst✝ : NonAssocSemiring α\nv : V\nvec : V → α\n⊢ adjMatrix α G v ⬝ᵥ vec = ∑ u in neighborFinset G v, vec u",
"tactic": "simp [neighborFinset_eq_filter, dotProduct, sum_filter]"
}
] |
[
204,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/CategoryTheory/Category/TwoP.lean
|
pointedToTwoPSnd_comp_swap
|
[] |
[
142,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
|
NonUnitalSubsemiring.prod_mono_right
|
[] |
[
795,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
793,
1
] |
Mathlib/Algebra/EuclideanDomain/Defs.lean
|
EuclideanDomain.mod_add_div
|
[] |
[
129,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Analysis/InnerProductSpace/Calculus.lean
|
HasFDerivAt.inner
|
[] |
[
112,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Data/Fin/Interval.lean
|
Fin.card_Ici
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\na b : Fin n\n⊢ card (Ici a) = n - ↑a",
"tactic": "cases n with\n| zero => exact Fin.elim0 a\n| succ =>\n rw [← card_map, map_valEmbedding_Ici, Nat.card_Icc, Nat.succ_sub_one]\n assumption"
},
{
"state_after": "no goals",
"state_before": "case zero\na b : Fin Nat.zero\n⊢ card (Ici a) = Nat.zero - ↑a",
"tactic": "exact Fin.elim0 a"
},
{
"state_after": "case succ.b\nn✝ : ℕ\na b : Fin (Nat.succ n✝)\n⊢ Fin (Nat.succ n✝)",
"state_before": "case succ\nn✝ : ℕ\na b : Fin (Nat.succ n✝)\n⊢ card (Ici a) = Nat.succ n✝ - ↑a",
"tactic": "rw [← card_map, map_valEmbedding_Ici, Nat.card_Icc, Nat.succ_sub_one]"
},
{
"state_after": "no goals",
"state_before": "case succ.b\nn✝ : ℕ\na b : Fin (Nat.succ n✝)\n⊢ Fin (Nat.succ n✝)",
"tactic": "assumption"
}
] |
[
180,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Topology/Order/Basic.lean
|
isOpen_Iio
|
[] |
[
298,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
297,
1
] |
Mathlib/Data/Fintype/Perm.lean
|
card_perms_of_finset
|
[
{
"state_after": "case mk.mk\nα : Type u_1\nβ : Type ?u.52511\nγ : Type ?u.52514\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nval✝ : Multiset α\nl : List α\nhs : Multiset.Nodup (Quot.mk Setoid.r l)\n⊢ card (permsOfFinset { val := Quot.mk Setoid.r l, nodup := hs }) = (card { val := Quot.mk Setoid.r l, nodup := hs })!",
"state_before": "α : Type u_1\nβ : Type ?u.52511\nγ : Type ?u.52514\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\n⊢ ∀ (s : Finset α), card (permsOfFinset s) = (card s)!",
"tactic": "rintro ⟨⟨l⟩, hs⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.mk\nα : Type u_1\nβ : Type ?u.52511\nγ : Type ?u.52514\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nval✝ : Multiset α\nl : List α\nhs : Multiset.Nodup (Quot.mk Setoid.r l)\n⊢ card (permsOfFinset { val := Quot.mk Setoid.r l, nodup := hs }) = (card { val := Quot.mk Setoid.r l, nodup := hs })!",
"tactic": "exact length_permsOfList l"
}
] |
[
149,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
|
DiscreteValuationRing.iff_pid_with_one_nonzero_prime
|
[
{
"state_after": "case mp\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ DiscreteValuationRing R → IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ IsPrime P\n\ncase mpr\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ (IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ IsPrime P) → DiscreteValuationRing R",
"state_before": "R✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ DiscreteValuationRing R ↔ IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ IsPrime P",
"tactic": "constructor"
},
{
"state_after": "case mp\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\n⊢ IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ IsPrime P",
"state_before": "case mp\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ DiscreteValuationRing R → IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ IsPrime P",
"tactic": "intro RDVR"
},
{
"state_after": "case mp.mk\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\n⊢ IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ IsPrime P",
"state_before": "case mp\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\n⊢ IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ IsPrime P",
"tactic": "rcases id RDVR with ⟨Rlocal⟩"
},
{
"state_after": "case mp.mk.left\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\n⊢ IsPrincipalIdealRing R\n\ncase mp.mk.right\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\n⊢ ∃! P, P ≠ ⊥ ∧ IsPrime P",
"state_before": "case mp.mk\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\n⊢ IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ IsPrime P",
"tactic": "constructor"
},
{
"state_after": "case mp.mk.right\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\n⊢ ∃! P, P ≠ ⊥ ∧ IsPrime P",
"state_before": "case mp.mk.left\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\n⊢ IsPrincipalIdealRing R\n\ncase mp.mk.right\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\n⊢ ∃! P, P ≠ ⊥ ∧ IsPrime P",
"tactic": "assumption"
},
{
"state_after": "case mp.mk.right\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\n⊢ (fun P => P ≠ ⊥ ∧ IsPrime P) (maximalIdeal R) ∧ ∀ (y : Ideal R), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = maximalIdeal R",
"state_before": "case mp.mk.right\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\n⊢ ∃! P, P ≠ ⊥ ∧ IsPrime P",
"tactic": "use LocalRing.maximalIdeal R"
},
{
"state_after": "case mp.mk.right.left\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\n⊢ (fun P => P ≠ ⊥ ∧ IsPrime P) (maximalIdeal R)\n\ncase mp.mk.right.right\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\n⊢ ∀ (y : Ideal R), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = maximalIdeal R",
"state_before": "case mp.mk.right\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\n⊢ (fun P => P ≠ ⊥ ∧ IsPrime P) (maximalIdeal R) ∧ ∀ (y : Ideal R), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = maximalIdeal R",
"tactic": "constructor"
},
{
"state_after": "case mp.mk.right.right\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\n⊢ ∀ (y : Ideal R), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = maximalIdeal R",
"state_before": "case mp.mk.right.left\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\n⊢ (fun P => P ≠ ⊥ ∧ IsPrime P) (maximalIdeal R)\n\ncase mp.mk.right.right\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\n⊢ ∀ (y : Ideal R), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = maximalIdeal R",
"tactic": "exact ⟨Rlocal, inferInstance⟩"
},
{
"state_after": "case mp.mk.right.right.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\nQ : Ideal R\nhQ1 : Q ≠ ⊥\nhQ2 : IsPrime Q\n⊢ Q = maximalIdeal R",
"state_before": "case mp.mk.right.right\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\n⊢ ∀ (y : Ideal R), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = maximalIdeal R",
"tactic": "rintro Q ⟨hQ1, hQ2⟩"
},
{
"state_after": "case mp.mk.right.right.intro.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\nq : R\nhQ1 : Submodule.span R {q} ≠ ⊥\nhQ2 : IsPrime (Submodule.span R {q})\n⊢ Submodule.span R {q} = maximalIdeal R",
"state_before": "case mp.mk.right.right.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\nQ : Ideal R\nhQ1 : Q ≠ ⊥\nhQ2 : IsPrime Q\n⊢ Q = maximalIdeal R",
"tactic": "obtain ⟨q, rfl⟩ := (IsPrincipalIdealRing.principal Q).1"
},
{
"state_after": "case mp.mk.right.right.intro.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\nq : R\nhQ1 : Submodule.span R {q} ≠ ⊥\nhQ2 : IsPrime (Submodule.span R {q})\nhq : q ≠ 0\n⊢ Submodule.span R {q} = maximalIdeal R",
"state_before": "case mp.mk.right.right.intro.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\nq : R\nhQ1 : Submodule.span R {q} ≠ ⊥\nhQ2 : IsPrime (Submodule.span R {q})\n⊢ Submodule.span R {q} = maximalIdeal R",
"tactic": "have hq : q ≠ 0 := by\n rintro rfl\n apply hQ1\n simp"
},
{
"state_after": "case mp.mk.right.right.intro.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\nq : R\nhQ1 : Submodule.span R {q} ≠ ⊥\nhQ2 : Prime q\nhq : q ≠ 0\n⊢ Submodule.span R {q} = maximalIdeal R",
"state_before": "case mp.mk.right.right.intro.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\nq : R\nhQ1 : Submodule.span R {q} ≠ ⊥\nhQ2 : IsPrime (Submodule.span R {q})\nhq : q ≠ 0\n⊢ Submodule.span R {q} = maximalIdeal R",
"tactic": "erw [span_singleton_prime hq] at hQ2"
},
{
"state_after": "case mp.mk.right.right.intro.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\nq : R\nhQ1 : Submodule.span R {q} ≠ ⊥\nhq : q ≠ 0\nhQ2 : Irreducible q\n⊢ Submodule.span R {q} = maximalIdeal R",
"state_before": "case mp.mk.right.right.intro.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\nq : R\nhQ1 : Submodule.span R {q} ≠ ⊥\nhQ2 : Prime q\nhq : q ≠ 0\n⊢ Submodule.span R {q} = maximalIdeal R",
"tactic": "replace hQ2 := hQ2.irreducible"
},
{
"state_after": "case mp.mk.right.right.intro.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\nq : R\nhQ1 : Submodule.span R {q} ≠ ⊥\nhq : q ≠ 0\nhQ2✝ : Irreducible q\nhQ2 : maximalIdeal R = span {q}\n⊢ Submodule.span R {q} = maximalIdeal R",
"state_before": "case mp.mk.right.right.intro.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\nq : R\nhQ1 : Submodule.span R {q} ≠ ⊥\nhq : q ≠ 0\nhQ2 : Irreducible q\n⊢ Submodule.span R {q} = maximalIdeal R",
"tactic": "rw [irreducible_iff_uniformizer] at hQ2"
},
{
"state_after": "no goals",
"state_before": "case mp.mk.right.right.intro.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\nq : R\nhQ1 : Submodule.span R {q} ≠ ⊥\nhq : q ≠ 0\nhQ2✝ : Irreducible q\nhQ2 : maximalIdeal R = span {q}\n⊢ Submodule.span R {q} = maximalIdeal R",
"tactic": "exact hQ2.symm"
},
{
"state_after": "R✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\nhQ1 : Submodule.span R {0} ≠ ⊥\nhQ2 : IsPrime (Submodule.span R {0})\n⊢ False",
"state_before": "R✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\nq : R\nhQ1 : Submodule.span R {q} ≠ ⊥\nhQ2 : IsPrime (Submodule.span R {q})\n⊢ q ≠ 0",
"tactic": "rintro rfl"
},
{
"state_after": "R✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\nhQ1 : Submodule.span R {0} ≠ ⊥\nhQ2 : IsPrime (Submodule.span R {0})\n⊢ Submodule.span R {0} = ⊥",
"state_before": "R✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\nhQ1 : Submodule.span R {0} ≠ ⊥\nhQ2 : IsPrime (Submodule.span R {0})\n⊢ False",
"tactic": "apply hQ1"
},
{
"state_after": "no goals",
"state_before": "R✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRDVR : DiscreteValuationRing R\ntoIsPrincipalIdealRing✝ : IsPrincipalIdealRing R\ntoLocalRing✝ : LocalRing R\nRlocal : maximalIdeal R ≠ ⊥\nhQ1 : Submodule.span R {0} ≠ ⊥\nhQ2 : IsPrime (Submodule.span R {0})\n⊢ Submodule.span R {0} = ⊥",
"tactic": "simp"
},
{
"state_after": "case mpr.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRPID : IsPrincipalIdealRing R\nPunique : ∃! P, P ≠ ⊥ ∧ IsPrime P\n⊢ DiscreteValuationRing R",
"state_before": "case mpr\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ (IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ IsPrime P) → DiscreteValuationRing R",
"tactic": "rintro ⟨RPID, Punique⟩"
},
{
"state_after": "case mpr.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRPID : IsPrincipalIdealRing R\nPunique : ∃! P, P ≠ ⊥ ∧ IsPrime P\nthis : LocalRing R\n⊢ DiscreteValuationRing R",
"state_before": "case mpr.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRPID : IsPrincipalIdealRing R\nPunique : ∃! P, P ≠ ⊥ ∧ IsPrime P\n⊢ DiscreteValuationRing R",
"tactic": "haveI : LocalRing R := LocalRing.of_unique_nonzero_prime Punique"
},
{
"state_after": "case mpr.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRPID : IsPrincipalIdealRing R\nPunique : ∃! P, P ≠ ⊥ ∧ IsPrime P\nthis : LocalRing R\n⊢ maximalIdeal R ≠ ⊥",
"state_before": "case mpr.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRPID : IsPrincipalIdealRing R\nPunique : ∃! P, P ≠ ⊥ ∧ IsPrime P\nthis : LocalRing R\n⊢ DiscreteValuationRing R",
"tactic": "refine' { not_a_field' := _ }"
},
{
"state_after": "case mpr.intro.intro.intro.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRPID : IsPrincipalIdealRing R\nthis : LocalRing R\nP : Ideal R\nright✝ : ∀ (y : Ideal R), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = P\nhP1 : P ≠ ⊥\nhP2 : IsPrime P\n⊢ maximalIdeal R ≠ ⊥",
"state_before": "case mpr.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRPID : IsPrincipalIdealRing R\nPunique : ∃! P, P ≠ ⊥ ∧ IsPrime P\nthis : LocalRing R\n⊢ maximalIdeal R ≠ ⊥",
"tactic": "rcases Punique with ⟨P, ⟨hP1, hP2⟩, _⟩"
},
{
"state_after": "case mpr.intro.intro.intro.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRPID : IsPrincipalIdealRing R\nthis : LocalRing R\nP : Ideal R\nright✝ : ∀ (y : Ideal R), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = P\nhP1 : P ≠ ⊥\nhP2 : IsPrime P\nhPM : P ≤ maximalIdeal R\n⊢ maximalIdeal R ≠ ⊥",
"state_before": "case mpr.intro.intro.intro.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRPID : IsPrincipalIdealRing R\nthis : LocalRing R\nP : Ideal R\nright✝ : ∀ (y : Ideal R), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = P\nhP1 : P ≠ ⊥\nhP2 : IsPrime P\n⊢ maximalIdeal R ≠ ⊥",
"tactic": "have hPM : P ≤ maximalIdeal R := le_maximalIdeal hP2.1"
},
{
"state_after": "case mpr.intro.intro.intro.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRPID : IsPrincipalIdealRing R\nthis : LocalRing R\nP : Ideal R\nright✝ : ∀ (y : Ideal R), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = P\nhP1 : P ≠ ⊥\nhP2 : IsPrime P\nhPM : P ≤ maximalIdeal R\nh : maximalIdeal R = ⊥\n⊢ False",
"state_before": "case mpr.intro.intro.intro.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRPID : IsPrincipalIdealRing R\nthis : LocalRing R\nP : Ideal R\nright✝ : ∀ (y : Ideal R), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = P\nhP1 : P ≠ ⊥\nhP2 : IsPrime P\nhPM : P ≤ maximalIdeal R\n⊢ maximalIdeal R ≠ ⊥",
"tactic": "intro h"
},
{
"state_after": "case mpr.intro.intro.intro.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRPID : IsPrincipalIdealRing R\nthis : LocalRing R\nP : Ideal R\nright✝ : ∀ (y : Ideal R), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = P\nhP1 : P ≠ ⊥\nhP2 : IsPrime P\nhPM : P = ⊥\nh : maximalIdeal R = ⊥\n⊢ False",
"state_before": "case mpr.intro.intro.intro.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRPID : IsPrincipalIdealRing R\nthis : LocalRing R\nP : Ideal R\nright✝ : ∀ (y : Ideal R), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = P\nhP1 : P ≠ ⊥\nhP2 : IsPrime P\nhPM : P ≤ maximalIdeal R\nh : maximalIdeal R = ⊥\n⊢ False",
"tactic": "rw [h, le_bot_iff] at hPM"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro.intro\nR✝ : Type u\ninst✝⁴ : CommRing R✝\ninst✝³ : IsDomain R✝\ninst✝² : DiscreteValuationRing R✝\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRPID : IsPrincipalIdealRing R\nthis : LocalRing R\nP : Ideal R\nright✝ : ∀ (y : Ideal R), (fun P => P ≠ ⊥ ∧ IsPrime P) y → y = P\nhP1 : P ≠ ⊥\nhP2 : IsPrime P\nhPM : P = ⊥\nh : maximalIdeal R = ⊥\n⊢ False",
"tactic": "exact hP1 hPM"
}
] |
[
149,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
HasDerivWithinAt.csin
|
[] |
[
219,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Probability/CondCount.lean
|
ProbabilityTheory.condCount_isProbabilityMeasure
|
[
{
"state_after": "case h0\nΩ : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns : Set Ω\nhs : Set.Finite s\nhs' : Set.Nonempty s\n⊢ ↑↑Measure.count s ≠ 0\n\ncase ht\nΩ : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns : Set Ω\nhs : Set.Finite s\nhs' : Set.Nonempty s\n⊢ ↑↑Measure.count s ≠ ⊤",
"state_before": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns : Set Ω\nhs : Set.Finite s\nhs' : Set.Nonempty s\n⊢ ↑↑(condCount s) Set.univ = 1",
"tactic": "rw [condCount, cond_apply _ hs.measurableSet, Set.inter_univ, ENNReal.inv_mul_cancel]"
},
{
"state_after": "no goals",
"state_before": "case h0\nΩ : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns : Set Ω\nhs : Set.Finite s\nhs' : Set.Nonempty s\n⊢ ↑↑Measure.count s ≠ 0",
"tactic": "exact fun h => hs'.ne_empty <| Measure.empty_of_count_eq_zero h"
},
{
"state_after": "no goals",
"state_before": "case ht\nΩ : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns : Set Ω\nhs : Set.Finite s\nhs' : Set.Nonempty s\n⊢ ↑↑Measure.count s ≠ ⊤",
"tactic": "exact (Measure.count_apply_lt_top.2 hs).ne"
}
] |
[
88,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
sum_range_diag_flip
|
[
{
"state_after": "α✝ : Type ?u.61826\ninst✝⁴ : LinearOrderedField α✝\ninst✝³ : Archimedean α✝\nβ : Type ?u.61890\ninst✝² : Ring β\nabv : β → α✝\ninst✝¹ : IsAbsoluteValue abv\nα : Type u_1\ninst✝ : AddCommMonoid α\nn : ℕ\nf : ℕ → ℕ → α\n⊢ ∑ x in Finset.sigma (range n) fun m => range (m + 1), f x.snd (x.fst - x.snd) =\n ∑ x in Finset.sigma (range n) fun m => range (n - m), f x.fst x.snd",
"state_before": "α✝ : Type ?u.61826\ninst✝⁴ : LinearOrderedField α✝\ninst✝³ : Archimedean α✝\nβ : Type ?u.61890\ninst✝² : Ring β\nabv : β → α✝\ninst✝¹ : IsAbsoluteValue abv\nα : Type u_1\ninst✝ : AddCommMonoid α\nn : ℕ\nf : ℕ → ℕ → α\n⊢ ∑ m in range n, ∑ k in range (m + 1), f k (m - k) = ∑ m in range n, ∑ k in range (n - m), f m k",
"tactic": "rw [sum_sigma', sum_sigma']"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.61826\ninst✝⁴ : LinearOrderedField α✝\ninst✝³ : Archimedean α✝\nβ : Type ?u.61890\ninst✝² : Ring β\nabv : β → α✝\ninst✝¹ : IsAbsoluteValue abv\nα : Type u_1\ninst✝ : AddCommMonoid α\nn : ℕ\nf : ℕ → ℕ → α\n⊢ ∑ x in Finset.sigma (range n) fun m => range (m + 1), f x.snd (x.fst - x.snd) =\n ∑ x in Finset.sigma (range n) fun m => range (n - m), f x.fst x.snd",
"tactic": "exact\n sum_bij (fun a _ => ⟨a.2, a.1 - a.2⟩)\n (fun a ha =>\n have h₁ : a.1 < n := mem_range.1 (mem_sigma.1 ha).1\n have h₂ : a.2 < Nat.succ a.1 := mem_range.1 (mem_sigma.1 ha).2\n mem_sigma.2\n ⟨mem_range.2 (lt_of_lt_of_le h₂ h₁),\n mem_range.2 ((tsub_lt_tsub_iff_right (Nat.le_of_lt_succ h₂)).2 h₁)⟩)\n (fun _ _ => rfl)\n (fun ⟨a₁, a₂⟩ ⟨b₁, b₂⟩ ha hb h =>\n have ha : a₁ < n ∧ a₂ ≤ a₁ :=\n ⟨mem_range.1 (mem_sigma.1 ha).1, Nat.le_of_lt_succ (mem_range.1 (mem_sigma.1 ha).2)⟩\n have hb : b₁ < n ∧ b₂ ≤ b₁ :=\n ⟨mem_range.1 (mem_sigma.1 hb).1, Nat.le_of_lt_succ (mem_range.1 (mem_sigma.1 hb).2)⟩\n have h : a₂ = b₂ ∧ _ := by simpa using h\n have h' : a₁ = b₁ - b₂ + a₂ := (tsub_eq_iff_eq_add_of_le ha.2).1 (eq_of_heq h.2)\n Sigma.mk.inj_iff.2 ⟨tsub_add_cancel_of_le hb.2 ▸ h'.symm ▸ h.1 ▸ rfl, heq_of_eq h.1⟩)\n fun ⟨a₁, a₂⟩ ha =>\n have ha : a₁ < n ∧ a₂ < n - a₁ :=\n ⟨mem_range.1 (mem_sigma.1 ha).1, mem_range.1 (mem_sigma.1 ha).2⟩\n ⟨⟨a₂ + a₁, a₁⟩,\n ⟨mem_sigma.2\n ⟨mem_range.2 (lt_tsub_iff_right.1 ha.2),\n mem_range.2 (Nat.lt_succ_of_le (Nat.le_add_left _ _))⟩,\n Sigma.mk.inj_iff.2 ⟨rfl, heq_of_eq (add_tsub_cancel_right _ _).symm⟩⟩⟩"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.61826\ninst✝⁴ : LinearOrderedField α✝\ninst✝³ : Archimedean α✝\nβ : Type ?u.61890\ninst✝² : Ring β\nabv : β → α✝\ninst✝¹ : IsAbsoluteValue abv\nα : Type u_1\ninst✝ : AddCommMonoid α\nn : ℕ\nf : ℕ → ℕ → α\nx✝¹ x✝ : (_ : ℕ) × ℕ\na₁ a₂ : ℕ\nha✝ : { fst := a₁, snd := a₂ } ∈ Finset.sigma (range n) fun m => range (m + 1)\nb₁ b₂ : ℕ\nhb✝ : { fst := b₁, snd := b₂ } ∈ Finset.sigma (range n) fun m => range (m + 1)\nh :\n (fun a x => { fst := a.snd, snd := a.fst - a.snd }) { fst := a₁, snd := a₂ } ha✝ =\n (fun a x => { fst := a.snd, snd := a.fst - a.snd }) { fst := b₁, snd := b₂ } hb✝\nha : a₁ < n ∧ a₂ ≤ a₁\nhb : b₁ < n ∧ b₂ ≤ b₁\n⊢ a₂ = b₂ ∧ HEq (a₁ - a₂) (b₁ - b₂)",
"tactic": "simpa using h"
}
] |
[
216,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/CategoryTheory/Subobject/FactorThru.lean
|
CategoryTheory.Subobject.factorThru_ofLE
|
[
{
"state_after": "case h\nC : Type u₁\ninst✝¹ : Category C\nX Y✝ Z✝ : C\nD : Type u₂\ninst✝ : Category D\nY Z : C\nP Q : Subobject Y\nf : Z ⟶ Y\nh : P ≤ Q\nw : Factors P f\n⊢ factorThru Q f (_ : Factors Q f) ≫ arrow Q = (factorThru P f w ≫ ofLE P Q h) ≫ arrow Q",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nX Y✝ Z✝ : C\nD : Type u₂\ninst✝ : Category D\nY Z : C\nP Q : Subobject Y\nf : Z ⟶ Y\nh : P ≤ Q\nw : Factors P f\n⊢ factorThru Q f (_ : Factors Q f) = factorThru P f w ≫ ofLE P Q h",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nC : Type u₁\ninst✝¹ : Category C\nX Y✝ Z✝ : C\nD : Type u₂\ninst✝ : Category D\nY Z : C\nP Q : Subobject Y\nf : Z ⟶ Y\nh : P ≤ Q\nw : Factors P f\n⊢ factorThru Q f (_ : Factors Q f) ≫ arrow Q = (factorThru P f w ≫ ofLE P Q h) ≫ arrow Q",
"tactic": "simp"
}
] |
[
178,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
AddMonoidAlgebra.single_mul_apply_of_not_exists_add
|
[] |
[
1652,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1650,
1
] |
Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean
|
MeasureTheory.withDensityᵥ_toReal
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.66433\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type ?u.66452\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhfi : Integrable fun x => ENNReal.toReal (f x)\n⊢ (withDensityᵥ μ fun x => ENNReal.toReal (f x)) = toSignedMeasure (withDensity μ f)",
"state_before": "α : Type u_1\nβ : Type ?u.66433\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type ?u.66452\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\n⊢ (withDensityᵥ μ fun x => ENNReal.toReal (f x)) = toSignedMeasure (withDensity μ f)",
"tactic": "have hfi := integrable_toReal_of_lintegral_ne_top hfm hf"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.66433\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type ?u.66452\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhfi : Integrable fun x => ENNReal.toReal (f x)\nthis : IsFiniteMeasure (withDensity μ fun a => f a)\n⊢ (withDensityᵥ μ fun x => ENNReal.toReal (f x)) = toSignedMeasure (withDensity μ f)",
"state_before": "α : Type u_1\nβ : Type ?u.66433\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type ?u.66452\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhfi : Integrable fun x => ENNReal.toReal (f x)\n⊢ (withDensityᵥ μ fun x => ENNReal.toReal (f x)) = toSignedMeasure (withDensity μ f)",
"tactic": "haveI := isFiniteMeasure_withDensity hf"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.66433\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type ?u.66452\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhfi : Integrable fun x => ENNReal.toReal (f x)\nthis : IsFiniteMeasure (withDensity μ fun a => f a)\ni : Set α\n⊢ MeasurableSet i → ↑(withDensityᵥ μ fun x => ENNReal.toReal (f x)) i = ↑(toSignedMeasure (withDensity μ f)) i",
"state_before": "α : Type u_1\nβ : Type ?u.66433\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type ?u.66452\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhfi : Integrable fun x => ENNReal.toReal (f x)\nthis : IsFiniteMeasure (withDensity μ fun a => f a)\n⊢ (withDensityᵥ μ fun x => ENNReal.toReal (f x)) = toSignedMeasure (withDensity μ f)",
"tactic": "ext i"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.66433\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type ?u.66452\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhfi : Integrable fun x => ENNReal.toReal (f x)\nthis : IsFiniteMeasure (withDensity μ fun a => f a)\ni : Set α\nhi : MeasurableSet i\n⊢ ↑(withDensityᵥ μ fun x => ENNReal.toReal (f x)) i = ↑(toSignedMeasure (withDensity μ f)) i",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.66433\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type ?u.66452\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhfi : Integrable fun x => ENNReal.toReal (f x)\nthis : IsFiniteMeasure (withDensity μ fun a => f a)\ni : Set α\n⊢ MeasurableSet i → ↑(withDensityᵥ μ fun x => ENNReal.toReal (f x)) i = ↑(toSignedMeasure (withDensity μ f)) i",
"tactic": "intro hi"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.66433\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type ?u.66452\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhfi : Integrable fun x => ENNReal.toReal (f x)\nthis : IsFiniteMeasure (withDensity μ fun a => f a)\ni : Set α\nhi : MeasurableSet i\n⊢ ∀ᵐ (x : α) ∂restrict μ i, f x < ⊤",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.66433\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type ?u.66452\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhfi : Integrable fun x => ENNReal.toReal (f x)\nthis : IsFiniteMeasure (withDensity μ fun a => f a)\ni : Set α\nhi : MeasurableSet i\n⊢ ↑(withDensityᵥ μ fun x => ENNReal.toReal (f x)) i = ↑(toSignedMeasure (withDensity μ f)) i",
"tactic": "rw [withDensityᵥ_apply hfi hi, toSignedMeasure_apply_measurable hi, withDensity_apply _ hi,\n integral_toReal hfm.restrict]"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.66433\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type ?u.66452\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhfi : Integrable fun x => ENNReal.toReal (f x)\nthis : IsFiniteMeasure (withDensity μ fun a => f a)\ni : Set α\nhi : MeasurableSet i\n⊢ (∫⁻ (x : α) in i, f x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.66433\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type ?u.66452\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhfi : Integrable fun x => ENNReal.toReal (f x)\nthis : IsFiniteMeasure (withDensity μ fun a => f a)\ni : Set α\nhi : MeasurableSet i\n⊢ ∀ᵐ (x : α) ∂restrict μ i, f x < ⊤",
"tactic": "refine' ae_lt_top' hfm.restrict (ne_top_of_le_ne_top hf _)"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.66433\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type ?u.66452\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhfi : Integrable fun x => ENNReal.toReal (f x)\nthis : IsFiniteMeasure (withDensity μ fun a => f a)\ni : Set α\nhi : MeasurableSet i\n⊢ (∫⁻ (x : α) in i, f x ∂μ) ≤ ∫⁻ (x : α) in Set.univ, f x ∂μ",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.66433\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type ?u.66452\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhfi : Integrable fun x => ENNReal.toReal (f x)\nthis : IsFiniteMeasure (withDensity μ fun a => f a)\ni : Set α\nhi : MeasurableSet i\n⊢ (∫⁻ (x : α) in i, f x ∂μ) ≤ ∫⁻ (x : α), f x ∂μ",
"tactic": "conv_rhs => rw [← set_lintegral_univ]"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.66433\nm : MeasurableSpace α\nμ ν : Measure α\nE : Type ?u.66452\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf✝ g : α → E\nf : α → ℝ≥0∞\nhfm : AEMeasurable f\nhf : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nhfi : Integrable fun x => ENNReal.toReal (f x)\nthis : IsFiniteMeasure (withDensity μ fun a => f a)\ni : Set α\nhi : MeasurableSet i\n⊢ (∫⁻ (x : α) in i, f x ∂μ) ≤ ∫⁻ (x : α) in Set.univ, f x ∂μ",
"tactic": "exact lintegral_mono_set (Set.subset_univ _)"
}
] |
[
179,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Data/Set/Intervals/OrdConnected.lean
|
Set.ordConnected_iff
|
[] |
[
57,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Algebra/DirectSum/Decomposition.lean
|
DirectSum.decompose_symm_sub
|
[] |
[
225,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.IsEquiv.trans
|
[] |
[
383,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
382,
1
] |
Mathlib/Analysis/Calculus/IteratedDeriv.lean
|
contDiffOn_of_continuousOn_differentiableOn_deriv
|
[
{
"state_after": "case Hcont\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type ?u.34703\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn✝ : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx : 𝕜\nn : ℕ∞\nHcont : ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x => iteratedDerivWithin m f s x) s\nHdiff : ∀ (m : ℕ), ↑m < n → DifferentiableOn 𝕜 (fun x => iteratedDerivWithin m f s x) s\n⊢ ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x => iteratedFDerivWithin 𝕜 m f s x) s\n\ncase Hdiff\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type ?u.34703\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn✝ : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx : 𝕜\nn : ℕ∞\nHcont : ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x => iteratedDerivWithin m f s x) s\nHdiff : ∀ (m : ℕ), ↑m < n → DifferentiableOn 𝕜 (fun x => iteratedDerivWithin m f s x) s\n⊢ ∀ (m : ℕ), ↑m < n → DifferentiableOn 𝕜 (fun x => iteratedFDerivWithin 𝕜 m f s x) s",
"state_before": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type ?u.34703\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn✝ : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx : 𝕜\nn : ℕ∞\nHcont : ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x => iteratedDerivWithin m f s x) s\nHdiff : ∀ (m : ℕ), ↑m < n → DifferentiableOn 𝕜 (fun x => iteratedDerivWithin m f s x) s\n⊢ ContDiffOn 𝕜 n f s",
"tactic": "apply contDiffOn_of_continuousOn_differentiableOn"
},
{
"state_after": "no goals",
"state_before": "case Hcont\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type ?u.34703\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn✝ : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx : 𝕜\nn : ℕ∞\nHcont : ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x => iteratedDerivWithin m f s x) s\nHdiff : ∀ (m : ℕ), ↑m < n → DifferentiableOn 𝕜 (fun x => iteratedDerivWithin m f s x) s\n⊢ ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x => iteratedFDerivWithin 𝕜 m f s x) s",
"tactic": "simpa only [iteratedFDerivWithin_eq_equiv_comp, LinearIsometryEquiv.comp_continuousOn_iff]"
},
{
"state_after": "no goals",
"state_before": "case Hdiff\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type ?u.34703\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn✝ : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx : 𝕜\nn : ℕ∞\nHcont : ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x => iteratedDerivWithin m f s x) s\nHdiff : ∀ (m : ℕ), ↑m < n → DifferentiableOn 𝕜 (fun x => iteratedDerivWithin m f s x) s\n⊢ ∀ (m : ℕ), ↑m < n → DifferentiableOn 𝕜 (fun x => iteratedFDerivWithin 𝕜 m f s x) s",
"tactic": "simpa only [iteratedFDerivWithin_eq_equiv_comp, LinearIsometryEquiv.comp_differentiableOn_iff]"
}
] |
[
139,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.powerset_insert
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\n⊢ t ∈ 𝒫 insert a s ↔ t ∈ 𝒫 s ∪ insert a '' 𝒫 s",
"state_before": "α : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\n⊢ 𝒫 insert a s = 𝒫 s ∪ insert a '' 𝒫 s",
"tactic": "ext t"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\n⊢ t ⊆ insert a s ↔ t ⊆ s ∨ ∃ x, x ⊆ s ∧ insert a x = t",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\n⊢ t ∈ 𝒫 insert a s ↔ t ∈ 𝒫 s ∪ insert a '' 𝒫 s",
"tactic": "simp_rw [mem_union, mem_image, mem_powerset_iff]"
},
{
"state_after": "case h.mp\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\n⊢ t ⊆ insert a s → t ⊆ s ∨ ∃ x, x ⊆ s ∧ insert a x = t\n\ncase h.mpr\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\n⊢ (t ⊆ s ∨ ∃ x, x ⊆ s ∧ insert a x = t) → t ⊆ insert a s",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\n⊢ t ⊆ insert a s ↔ t ⊆ s ∨ ∃ x, x ⊆ s ∧ insert a x = t",
"tactic": "constructor"
},
{
"state_after": "case h.mp\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\nh : t ⊆ insert a s\n⊢ t ⊆ s ∨ ∃ x, x ⊆ s ∧ insert a x = t",
"state_before": "case h.mp\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\n⊢ t ⊆ insert a s → t ⊆ s ∨ ∃ x, x ⊆ s ∧ insert a x = t",
"tactic": "intro h"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\nh : t ⊆ insert a s\nhs : a ∈ t\n⊢ t ⊆ s ∨ ∃ x, x ⊆ s ∧ insert a x = t\n\ncase neg\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\nh : t ⊆ insert a s\nhs : ¬a ∈ t\n⊢ t ⊆ s ∨ ∃ x, x ⊆ s ∧ insert a x = t",
"state_before": "case h.mp\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\nh : t ⊆ insert a s\n⊢ t ⊆ s ∨ ∃ x, x ⊆ s ∧ insert a x = t",
"tactic": "by_cases hs : a ∈ t"
},
{
"state_after": "case pos.h\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\nh : t ⊆ insert a s\nhs : a ∈ t\n⊢ ∃ x, x ⊆ s ∧ insert a x = t",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\nh : t ⊆ insert a s\nhs : a ∈ t\n⊢ t ⊆ s ∨ ∃ x, x ⊆ s ∧ insert a x = t",
"tactic": "right"
},
{
"state_after": "case pos.h.refine'_1\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\nh : t ⊆ insert a s\nhs : a ∈ t\n⊢ t \\ {a} ⊆ s\n\ncase pos.h.refine'_2\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\nh : t ⊆ insert a s\nhs : a ∈ t\n⊢ insert a (t \\ {a}) = t",
"state_before": "case pos.h\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\nh : t ⊆ insert a s\nhs : a ∈ t\n⊢ ∃ x, x ⊆ s ∧ insert a x = t",
"tactic": "refine' ⟨t \\ {a}, _, _⟩"
},
{
"state_after": "case pos.h.refine'_1\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\nh : t ⊆ insert a s\nhs : a ∈ t\n⊢ t ⊆ insert a s",
"state_before": "case pos.h.refine'_1\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\nh : t ⊆ insert a s\nhs : a ∈ t\n⊢ t \\ {a} ⊆ s",
"tactic": "rw [diff_singleton_subset_iff]"
},
{
"state_after": "no goals",
"state_before": "case pos.h.refine'_1\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\nh : t ⊆ insert a s\nhs : a ∈ t\n⊢ t ⊆ insert a s",
"tactic": "assumption"
},
{
"state_after": "no goals",
"state_before": "case pos.h.refine'_2\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\nh : t ⊆ insert a s\nhs : a ∈ t\n⊢ insert a (t \\ {a}) = t",
"tactic": "rw [insert_diff_singleton, insert_eq_of_mem hs]"
},
{
"state_after": "case neg.h\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\nh : t ⊆ insert a s\nhs : ¬a ∈ t\n⊢ t ⊆ s",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\nh : t ⊆ insert a s\nhs : ¬a ∈ t\n⊢ t ⊆ s ∨ ∃ x, x ⊆ s ∧ insert a x = t",
"tactic": "left"
},
{
"state_after": "no goals",
"state_before": "case neg.h\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\nh : t ⊆ insert a s\nhs : ¬a ∈ t\n⊢ t ⊆ s",
"tactic": "exact (subset_insert_iff_of_not_mem hs).mp h"
},
{
"state_after": "case h.mpr.inl\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\nh : t ⊆ s\n⊢ t ⊆ insert a s\n\ncase h.mpr.inr.intro.intro\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\ns' : Set α\nh₁ : s' ⊆ s\n⊢ insert a s' ⊆ insert a s",
"state_before": "case h.mpr\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\n⊢ (t ⊆ s ∨ ∃ x, x ⊆ s ∧ insert a x = t) → t ⊆ insert a s",
"tactic": "rintro (h | ⟨s', h₁, rfl⟩)"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.inl\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\nt : Set α\nh : t ⊆ s\n⊢ t ⊆ insert a s",
"tactic": "exact subset_trans h (subset_insert a s)"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.inr.intro.intro\nα : Type u_1\nβ : Type ?u.61243\nγ : Type ?u.61246\nι : Sort ?u.61249\nι' : Sort ?u.61252\ns : Set α\na : α\ns' : Set α\nh₁ : s' ⊆ s\n⊢ insert a s' ⊆ insert a s",
"tactic": "exact insert_subset_insert h₁"
}
] |
[
631,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
616,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
Subsemigroup.comap_inf
|
[] |
[
344,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.pushout.mapLift_comp
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁵ : Category C\nD : Type u₂\ninst✝⁴ : Category D\nW X✝ Y✝ Z X Y S T S' : C\nf : T ⟶ X\ng : T ⟶ Y\ni : S ⟶ T\ni' : S' ⟶ S\ninst✝³ : HasPushout f g\ninst✝² : HasPushout (i ≫ f) (i ≫ g)\ninst✝¹ : HasPushout (i' ≫ i ≫ f) (i' ≫ i ≫ g)\ninst✝ : HasPushout ((i' ≫ i) ≫ f) ((i' ≫ i) ≫ g)\n⊢ mapLift f g (i' ≫ i) =\n (congrHom (_ : (i' ≫ i) ≫ f = i' ≫ i ≫ f) (_ : (i' ≫ i) ≫ g = i' ≫ i ≫ g)).hom ≫\n mapLift (i ≫ f) (i ≫ g) i' ≫ mapLift f g i",
"tactic": "aesop_cat"
}
] |
[
1429,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1423,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
lcm_dvd_lcm_mul_left
|
[] |
[
849,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
848,
1
] |
Mathlib/LinearAlgebra/Ray.lean
|
sameRay_neg_swap
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁴ : StrictOrderedCommRing R\nM : Type u_2\nN : Type ?u.144746\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nx y : M\n⊢ SameRay R (-x) y ↔ SameRay R x (-y)",
"tactic": "rw [← sameRay_neg_iff, neg_neg]"
}
] |
[
406,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
406,
1
] |
Mathlib/Analysis/Convex/Gauge.lean
|
gauge_nonneg
|
[] |
[
133,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/Order/InitialSeg.lean
|
PrincipalSeg.init_iff
|
[] |
[
285,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
Mathlib/SetTheory/Ordinal/Topology.lean
|
Ordinal.mem_closed_iff_bsup
|
[
{
"state_after": "no goals",
"state_before": "s : Set Ordinal\na : Ordinal\nhs : IsClosed s\n⊢ a ∈ s ↔ ∃ o _ho f, (∀ (i : Ordinal) (hi : i < o), f i hi ∈ s) ∧ bsup o f = a",
"tactic": "rw [← mem_closure_iff_bsup, hs.closure_eq]"
}
] |
[
152,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.sep_eq_of_subset
|
[] |
[
1425,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1424,
1
] |
Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
|
Real.volume_le_diam
|
[
{
"state_after": "case pos\nι : Type ?u.374130\ninst✝ : Fintype ι\ns : Set ℝ\nhs : Metric.Bounded s\n⊢ ↑↑volume s ≤ EMetric.diam s\n\ncase neg\nι : Type ?u.374130\ninst✝ : Fintype ι\ns : Set ℝ\nhs : ¬Metric.Bounded s\n⊢ ↑↑volume s ≤ EMetric.diam s",
"state_before": "ι : Type ?u.374130\ninst✝ : Fintype ι\ns : Set ℝ\n⊢ ↑↑volume s ≤ EMetric.diam s",
"tactic": "by_cases hs : Metric.Bounded s"
},
{
"state_after": "case pos\nι : Type ?u.374130\ninst✝ : Fintype ι\ns : Set ℝ\nhs : Metric.Bounded s\n⊢ ↑↑volume s ≤ ↑↑volume (Icc (sInf s) (sSup s))",
"state_before": "case pos\nι : Type ?u.374130\ninst✝ : Fintype ι\ns : Set ℝ\nhs : Metric.Bounded s\n⊢ ↑↑volume s ≤ EMetric.diam s",
"tactic": "rw [Real.ediam_eq hs, ← volume_Icc]"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type ?u.374130\ninst✝ : Fintype ι\ns : Set ℝ\nhs : Metric.Bounded s\n⊢ ↑↑volume s ≤ ↑↑volume (Icc (sInf s) (sSup s))",
"tactic": "exact volume.mono (Real.subset_Icc_sInf_sSup_of_bounded hs)"
},
{
"state_after": "case neg\nι : Type ?u.374130\ninst✝ : Fintype ι\ns : Set ℝ\nhs : ¬Metric.Bounded s\n⊢ ↑↑volume s ≤ ⊤",
"state_before": "case neg\nι : Type ?u.374130\ninst✝ : Fintype ι\ns : Set ℝ\nhs : ¬Metric.Bounded s\n⊢ ↑↑volume s ≤ EMetric.diam s",
"tactic": "rw [Metric.ediam_of_unbounded hs]"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type ?u.374130\ninst✝ : Fintype ι\ns : Set ℝ\nhs : ¬Metric.Bounded s\n⊢ ↑↑volume s ≤ ⊤",
"tactic": "exact le_top"
}
] |
[
200,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/GroupTheory/IsFreeGroup.lean
|
IsFreeGroup.unique_lift
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : IsFreeGroup G\nH : Type u_2\ninst✝ : Group H\nf : Generators G → H\n⊢ ∃! F, ∀ (a : Generators G), ↑F (of a) = f a",
"tactic": "simpa only [Function.funext_iff] using lift.symm.bijective.existsUnique f"
}
] |
[
130,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/Order/OrderIsoNat.lean
|
RelEmbedding.coe_natLt
|
[] |
[
46,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
1
] |
Mathlib/Data/Nat/Dist.lean
|
Nat.dist_comm
|
[
{
"state_after": "no goals",
"state_before": "n m : ℕ\n⊢ dist n m = dist m n",
"tactic": "simp [dist.def, add_comm]"
}
] |
[
31,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
31,
1
] |
Mathlib/Algebra/Quaternion.lean
|
QuaternionAlgebra.one_im
|
[] |
[
205,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
9
] |
Mathlib/Data/Int/GCD.lean
|
Int.lcm_assoc
|
[
{
"state_after": "i j k : ℤ\n⊢ Nat.lcm (Nat.lcm (natAbs i) (natAbs j)) (natAbs k) = Nat.lcm (natAbs i) (Nat.lcm (natAbs j) (natAbs k))",
"state_before": "i j k : ℤ\n⊢ lcm (↑(lcm i j)) k = lcm i ↑(lcm j k)",
"tactic": "rw [Int.lcm, Int.lcm, Int.lcm, Int.lcm, natAbs_ofNat, natAbs_ofNat]"
},
{
"state_after": "no goals",
"state_before": "i j k : ℤ\n⊢ Nat.lcm (Nat.lcm (natAbs i) (natAbs j)) (natAbs k) = Nat.lcm (natAbs i) (Nat.lcm (natAbs j) (natAbs k))",
"tactic": "apply Nat.lcm_assoc"
}
] |
[
450,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
448,
1
] |
Mathlib/NumberTheory/Divisors.lean
|
Nat.prod_divisors_prime_pow
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\nα : Type u_1\ninst✝ : CommMonoid α\nk p : ℕ\nf : ℕ → α\nh : Prime p\n⊢ ∏ x in divisors (p ^ k), f x = ∏ x in range (k + 1), f (p ^ x)",
"tactic": "simp [h, divisors_prime_pow]"
}
] |
[
464,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
462,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ico_bot
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.57096\ninst✝¹ : Preorder α\ninst✝ : OrderBot α\na : α\n⊢ Ico ⊥ a = Iio a",
"tactic": "simp [← Ici_inter_Iio]"
}
] |
[
1031,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1031,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.findMin'_dual
|
[] |
[
582,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
580,
1
] |
Mathlib/Order/Closure.lean
|
ClosureOperator.closure_le_mk₃_iff
|
[] |
[
146,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Topology/Bases.lean
|
TopologicalSpace.IsTopologicalBasis.open_eq_sUnion
|
[] |
[
197,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.length_append
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w : V\np : Walk G u v\nq : Walk G v w\n⊢ length (append p q) = length p + length q",
"tactic": "induction p with\n| nil => simp\n| cons _ _ ih => simp [ih, add_comm, add_left_comm, add_assoc]"
},
{
"state_after": "no goals",
"state_before": "case nil\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w u✝ : V\nq : Walk G u✝ w\n⊢ length (append nil q) = length nil + length q",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w u✝ v✝ w✝ : V\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : ∀ (q : Walk G w✝ w), length (append p✝ q) = length p✝ + length q\nq : Walk G w✝ w\n⊢ length (append (cons h✝ p✝) q) = length (cons h✝ p✝) + length q",
"tactic": "simp [ih, add_comm, add_left_comm, add_assoc]"
}
] |
[
405,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
401,
1
] |
Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
|
Real.volume_pi_Ioo_toReal
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\ninst✝ : Fintype ι\na b : ι → ℝ\nh : a ≤ b\n⊢ ENNReal.toReal (↑↑volume (Set.pi univ fun i => Ioo (a i) (b i))) = ∏ i : ι, (b i - a i)",
"tactic": "simp only [volume_pi_Ioo, ENNReal.toReal_prod, ENNReal.toReal_ofReal (sub_nonneg.2 (h _))]"
}
] |
[
234,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
232,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Ordered.lean
|
map_lt_lineMap_iff_slope_lt_slope_right
|
[] |
[
266,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
263,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.ite_le_natDegree_coeff
|
[
{
"state_after": "case inl\nR : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nI : Decidable (n < 1 + natDegree p)\nh : n < 1 + natDegree p\n⊢ coeff p n = coeff p n\n\ncase inr\nR : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nI : Decidable (n < 1 + natDegree p)\nh : ¬n < 1 + natDegree p\n⊢ 0 = coeff p n",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nI : Decidable (n < 1 + natDegree p)\n⊢ (if n < 1 + natDegree p then coeff p n else 0) = coeff p n",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nI : Decidable (n < 1 + natDegree p)\nh : n < 1 + natDegree p\n⊢ coeff p n = coeff p n",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case inr\nR : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn : ℕ\nI : Decidable (n < 1 + natDegree p)\nh : ¬n < 1 + natDegree p\n⊢ 0 = coeff p n",
"tactic": "exact (coeff_eq_zero_of_natDegree_lt (not_le.1 fun w => h (Nat.lt_one_add_iff.2 w))).symm"
}
] |
[
383,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
379,
1
] |
Mathlib/Topology/Bases.lean
|
TopologicalSpace.eq_generateFrom_countableBasis
|
[] |
[
644,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
642,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.pullbackSymmetry_hom_of_mono_eq
|
[
{
"state_after": "case h₀\nC : Type u\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nW X Y Z : C\nf : X ⟶ Y\ninst✝ : Mono f\n⊢ (pullbackSymmetry f f).hom ≫ pullback.fst = 𝟙 (pullback f f) ≫ pullback.fst\n\ncase h₁\nC : Type u\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nW X Y Z : C\nf : X ⟶ Y\ninst✝ : Mono f\n⊢ (pullbackSymmetry f f).hom ≫ pullback.snd = 𝟙 (pullback f f) ≫ pullback.snd",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nW X Y Z : C\nf : X ⟶ Y\ninst✝ : Mono f\n⊢ (pullbackSymmetry f f).hom = 𝟙 (pullback f f)",
"tactic": "ext"
},
{
"state_after": "case h₁\nC : Type u\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nW X Y Z : C\nf : X ⟶ Y\ninst✝ : Mono f\n⊢ (pullbackSymmetry f f).hom ≫ pullback.snd = 𝟙 (pullback f f) ≫ pullback.snd",
"state_before": "case h₀\nC : Type u\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nW X Y Z : C\nf : X ⟶ Y\ninst✝ : Mono f\n⊢ (pullbackSymmetry f f).hom ≫ pullback.fst = 𝟙 (pullback f f) ≫ pullback.fst\n\ncase h₁\nC : Type u\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nW X Y Z : C\nf : X ⟶ Y\ninst✝ : Mono f\n⊢ (pullbackSymmetry f f).hom ≫ pullback.snd = 𝟙 (pullback f f) ≫ pullback.snd",
"tactic": "simp [fst_eq_snd_of_mono_eq]"
},
{
"state_after": "no goals",
"state_before": "case h₁\nC : Type u\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nW X Y Z : C\nf : X ⟶ Y\ninst✝ : Mono f\n⊢ (pullbackSymmetry f f).hom ≫ pullback.snd = 𝟙 (pullback f f) ≫ pullback.snd",
"tactic": "simp [fst_eq_snd_of_mono_eq]"
}
] |
[
1938,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1937,
1
] |
Mathlib/Algebra/GroupPower/Ring.lean
|
Commute.sq_eq_sq_iff_eq_or_eq_neg
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.123904\nM : Type ?u.123907\ninst✝¹ : Ring R\na b : R\ninst✝ : NoZeroDivisors R\nh : Commute a b\n⊢ a ^ 2 = b ^ 2 ↔ a = b ∨ a = -b",
"tactic": "rw [← sub_eq_zero, h.sq_sub_sq, mul_eq_zero, add_eq_zero_iff_eq_neg, sub_eq_zero, or_comm]"
}
] |
[
258,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
11
] |
Mathlib/Data/Finset/Image.lean
|
Finset.image_const
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.117445\ninst✝ : DecidableEq β\nf g : α → β\ns✝ : Finset α\nt : Finset β\na : α\nb✝ c : β\ns : Finset α\nh : Finset.Nonempty s\nb b' : β\n⊢ b' ∈ image (fun x => b) s ↔ b' ∈ {b}",
"tactic": "simp only [mem_image, exists_prop, exists_and_right, h.bex, true_and_iff, mem_singleton,\n eq_comm]"
}
] |
[
617,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
614,
1
] |
Mathlib/CategoryTheory/Sites/Sheafification.lean
|
CategoryTheory.GrothendieckTopology.toSheafification_app
|
[] |
[
530,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
529,
1
] |
Mathlib/Logic/Basic.lean
|
apply_ite₂
|
[] |
[
1234,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1232,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.Nonempty.right
|
[] |
[
516,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
515,
1
] |
Mathlib/Logic/Equiv/Option.lean
|
Equiv.removeNone_optionCongr
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.8105\ne✝ : Option α ≃ Option β\ne : α ≃ β\nx : α\n⊢ ↑(optionCongr e) (some x) = some (↑e x)",
"tactic": "simp [EquivFunctor.map]"
}
] |
[
163,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/Analysis/NormedSpace/Star/Multiplier.lean
|
DoubleCentralizer.toProdMulOpposite_injective
|
[] |
[
340,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
336,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.memℒp_re_im_iff
|
[
{
"state_after": "α : Type u_1\nE : Type ?u.5158781\nF : Type ?u.5158784\nG : Type ?u.5158787\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedAddCommGroup G\ng : E → F\nc : ℝ≥0\n𝕜 : Type ?u.5387273\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\nK : Type u_2\ninst✝ : IsROrC K\nf : α → K\n⊢ Memℒp (fun x => ↑IsROrC.re (f x)) p ∧ Memℒp (fun x => ↑IsROrC.im (f x)) p → Memℒp f p",
"state_before": "α : Type u_1\nE : Type ?u.5158781\nF : Type ?u.5158784\nG : Type ?u.5158787\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedAddCommGroup G\ng : E → F\nc : ℝ≥0\n𝕜 : Type ?u.5387273\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\nK : Type u_2\ninst✝ : IsROrC K\nf : α → K\n⊢ Memℒp (fun x => ↑IsROrC.re (f x)) p ∧ Memℒp (fun x => ↑IsROrC.im (f x)) p ↔ Memℒp f p",
"tactic": "refine' ⟨_, fun hf => ⟨hf.re, hf.im⟩⟩"
},
{
"state_after": "case intro\nα : Type u_1\nE : Type ?u.5158781\nF : Type ?u.5158784\nG : Type ?u.5158787\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedAddCommGroup G\ng : E → F\nc : ℝ≥0\n𝕜 : Type ?u.5387273\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\nK : Type u_2\ninst✝ : IsROrC K\nf : α → K\nhre : Memℒp (fun x => ↑IsROrC.re (f x)) p\nhim : Memℒp (fun x => ↑IsROrC.im (f x)) p\n⊢ Memℒp f p",
"state_before": "α : Type u_1\nE : Type ?u.5158781\nF : Type ?u.5158784\nG : Type ?u.5158787\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedAddCommGroup G\ng : E → F\nc : ℝ≥0\n𝕜 : Type ?u.5387273\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\nK : Type u_2\ninst✝ : IsROrC K\nf : α → K\n⊢ Memℒp (fun x => ↑IsROrC.re (f x)) p ∧ Memℒp (fun x => ↑IsROrC.im (f x)) p → Memℒp f p",
"tactic": "rintro ⟨hre, him⟩"
},
{
"state_after": "case h.e'_5\nα : Type u_1\nE : Type ?u.5158781\nF : Type ?u.5158784\nG : Type ?u.5158787\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedAddCommGroup G\ng : E → F\nc : ℝ≥0\n𝕜 : Type ?u.5387273\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\nK : Type u_2\ninst✝ : IsROrC K\nf : α → K\nhre : Memℒp (fun x => ↑IsROrC.re (f x)) p\nhim : Memℒp (fun x => ↑IsROrC.im (f x)) p\n⊢ f = (fun x => ↑(↑IsROrC.re (f x))) + fun x => IsROrC.I * ↑(↑IsROrC.im (f x))",
"state_before": "case intro\nα : Type u_1\nE : Type ?u.5158781\nF : Type ?u.5158784\nG : Type ?u.5158787\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedAddCommGroup G\ng : E → F\nc : ℝ≥0\n𝕜 : Type ?u.5387273\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\nK : Type u_2\ninst✝ : IsROrC K\nf : α → K\nhre : Memℒp (fun x => ↑IsROrC.re (f x)) p\nhim : Memℒp (fun x => ↑IsROrC.im (f x)) p\n⊢ Memℒp f p",
"tactic": "convert MeasureTheory.Memℒp.add (E := K) hre.ofReal (him.ofReal.const_mul IsROrC.I)"
},
{
"state_after": "case h.e'_5.h\nα : Type u_1\nE : Type ?u.5158781\nF : Type ?u.5158784\nG : Type ?u.5158787\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedAddCommGroup G\ng : E → F\nc : ℝ≥0\n𝕜 : Type ?u.5387273\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\nK : Type u_2\ninst✝ : IsROrC K\nf : α → K\nhre : Memℒp (fun x => ↑IsROrC.re (f x)) p\nhim : Memℒp (fun x => ↑IsROrC.im (f x)) p\nx : α\n⊢ f x = ((fun x => ↑(↑IsROrC.re (f x))) + fun x => IsROrC.I * ↑(↑IsROrC.im (f x))) x",
"state_before": "case h.e'_5\nα : Type u_1\nE : Type ?u.5158781\nF : Type ?u.5158784\nG : Type ?u.5158787\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedAddCommGroup G\ng : E → F\nc : ℝ≥0\n𝕜 : Type ?u.5387273\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\nK : Type u_2\ninst✝ : IsROrC K\nf : α → K\nhre : Memℒp (fun x => ↑IsROrC.re (f x)) p\nhim : Memℒp (fun x => ↑IsROrC.im (f x)) p\n⊢ f = (fun x => ↑(↑IsROrC.re (f x))) + fun x => IsROrC.I * ↑(↑IsROrC.im (f x))",
"tactic": "ext1 x"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h\nα : Type u_1\nE : Type ?u.5158781\nF : Type ?u.5158784\nG : Type ?u.5158787\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : MeasureTheory.Measure α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedAddCommGroup G\ng : E → F\nc : ℝ≥0\n𝕜 : Type ?u.5387273\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace 𝕜 F\nK : Type u_2\ninst✝ : IsROrC K\nf : α → K\nhre : Memℒp (fun x => ↑IsROrC.re (f x)) p\nhim : Memℒp (fun x => ↑IsROrC.im (f x)) p\nx : α\n⊢ f x = ((fun x => ↑(↑IsROrC.re (f x))) + fun x => IsROrC.I * ↑(↑IsROrC.im (f x))) x",
"tactic": "rw [Pi.add_apply, mul_comm, IsROrC.re_add_im]"
}
] |
[
981,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
975,
1
] |
Mathlib/LinearAlgebra/Multilinear/FiniteDimensional.lean
|
MultilinearMap.free_and_finite
|
[
{
"state_after": "case intro\nι : Type u_2\nR : Type u_1\nM₂ : Type u_3\nM₁ : ι → Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : Module R M₂\ninst✝⁵ : Module.Finite R M₂\ninst✝⁴ : Module.Free R M₂\ninst✝³ : (i : ι) → AddCommGroup (M₁ i)\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : ∀ (i : ι), Module.Finite R (M₁ i)\ninst✝ : ∀ (i : ι), Module.Free R (M₁ i)\nval✝ : Fintype ι\n⊢ Module.Free R (MultilinearMap R M₁ M₂) ∧ Module.Finite R (MultilinearMap R M₁ M₂)",
"state_before": "ι : Type u_2\nR : Type u_1\nM₂ : Type u_3\nM₁ : ι → Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : Module R M₂\ninst✝⁵ : Module.Finite R M₂\ninst✝⁴ : Module.Free R M₂\ninst✝³ : (i : ι) → AddCommGroup (M₁ i)\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : ∀ (i : ι), Module.Finite R (M₁ i)\ninst✝ : ∀ (i : ι), Module.Free R (M₁ i)\n⊢ Module.Free R (MultilinearMap R M₁ M₂) ∧ Module.Finite R (MultilinearMap R M₁ M₂)",
"tactic": "cases nonempty_fintype ι"
},
{
"state_after": "case intro\nι : Type u_2\nR : Type u_1\nM₂ : Type u_3\nM₁ : ι → Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : Module R M₂\ninst✝⁵ : Module.Finite R M₂\ninst✝⁴ : Module.Free R M₂\ninst✝³ : (i : ι) → AddCommGroup (M₁ i)\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : ∀ (i : ι), Module.Finite R (M₁ i)\ninst✝ : ∀ (i : ι), Module.Free R (M₁ i)\nval✝ : Fintype ι\nthis :\n ∀ [inst : (i : Fin (Fintype.card ι)) → AddCommGroup (M₁ (↑(Fintype.equivFin ι).symm i))]\n [inst_1 : (i : Fin (Fintype.card ι)) → Module R (M₁ (↑(Fintype.equivFin ι).symm i))]\n [inst_2 : ∀ (i : Fin (Fintype.card ι)), Module.Finite R (M₁ (↑(Fintype.equivFin ι).symm i))]\n [inst_3 : ∀ (i : Fin (Fintype.card ι)), Module.Free R (M₁ (↑(Fintype.equivFin ι).symm i))],\n Module.Free R (MultilinearMap R (fun x => M₁ (↑(Fintype.equivFin ι).symm x)) M₂) ∧\n Module.Finite R (MultilinearMap R (fun x => M₁ (↑(Fintype.equivFin ι).symm x)) M₂)\n⊢ Module.Free R (MultilinearMap R M₁ M₂) ∧ Module.Finite R (MultilinearMap R M₁ M₂)",
"state_before": "case intro\nι : Type u_2\nR : Type u_1\nM₂ : Type u_3\nM₁ : ι → Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : Module R M₂\ninst✝⁵ : Module.Finite R M₂\ninst✝⁴ : Module.Free R M₂\ninst✝³ : (i : ι) → AddCommGroup (M₁ i)\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : ∀ (i : ι), Module.Finite R (M₁ i)\ninst✝ : ∀ (i : ι), Module.Free R (M₁ i)\nval✝ : Fintype ι\n⊢ Module.Free R (MultilinearMap R M₁ M₂) ∧ Module.Finite R (MultilinearMap R M₁ M₂)",
"tactic": "have := @free_and_finite_fin R M₂ _ _ _ _ _ (Fintype.card ι)\n (fun x => M₁ ((Fintype.equivFin ι).symm x))"
},
{
"state_after": "case intro.intro\nι : Type u_2\nR : Type u_1\nM₂ : Type u_3\nM₁ : ι → Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : Module R M₂\ninst✝⁵ : Module.Finite R M₂\ninst✝⁴ : Module.Free R M₂\ninst✝³ : (i : ι) → AddCommGroup (M₁ i)\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : ∀ (i : ι), Module.Finite R (M₁ i)\ninst✝ : ∀ (i : ι), Module.Free R (M₁ i)\nval✝ : Fintype ι\nthis :\n ∀ [inst : (i : Fin (Fintype.card ι)) → AddCommGroup (M₁ (↑(Fintype.equivFin ι).symm i))]\n [inst_1 : (i : Fin (Fintype.card ι)) → Module R (M₁ (↑(Fintype.equivFin ι).symm i))]\n [inst_2 : ∀ (i : Fin (Fintype.card ι)), Module.Finite R (M₁ (↑(Fintype.equivFin ι).symm i))]\n [inst_3 : ∀ (i : Fin (Fintype.card ι)), Module.Free R (M₁ (↑(Fintype.equivFin ι).symm i))],\n Module.Free R (MultilinearMap R (fun x => M₁ (↑(Fintype.equivFin ι).symm x)) M₂) ∧\n Module.Finite R (MultilinearMap R (fun x => M₁ (↑(Fintype.equivFin ι).symm x)) M₂)\nl : Module.Free R (MultilinearMap R (fun x => M₁ (↑(Fintype.equivFin ι).symm x)) M₂)\nr : Module.Finite R (MultilinearMap R (fun x => M₁ (↑(Fintype.equivFin ι).symm x)) M₂)\n⊢ Module.Free R (MultilinearMap R M₁ M₂) ∧ Module.Finite R (MultilinearMap R M₁ M₂)",
"state_before": "case intro\nι : Type u_2\nR : Type u_1\nM₂ : Type u_3\nM₁ : ι → Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : Module R M₂\ninst✝⁵ : Module.Finite R M₂\ninst✝⁴ : Module.Free R M₂\ninst✝³ : (i : ι) → AddCommGroup (M₁ i)\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : ∀ (i : ι), Module.Finite R (M₁ i)\ninst✝ : ∀ (i : ι), Module.Free R (M₁ i)\nval✝ : Fintype ι\nthis :\n ∀ [inst : (i : Fin (Fintype.card ι)) → AddCommGroup (M₁ (↑(Fintype.equivFin ι).symm i))]\n [inst_1 : (i : Fin (Fintype.card ι)) → Module R (M₁ (↑(Fintype.equivFin ι).symm i))]\n [inst_2 : ∀ (i : Fin (Fintype.card ι)), Module.Finite R (M₁ (↑(Fintype.equivFin ι).symm i))]\n [inst_3 : ∀ (i : Fin (Fintype.card ι)), Module.Free R (M₁ (↑(Fintype.equivFin ι).symm i))],\n Module.Free R (MultilinearMap R (fun x => M₁ (↑(Fintype.equivFin ι).symm x)) M₂) ∧\n Module.Finite R (MultilinearMap R (fun x => M₁ (↑(Fintype.equivFin ι).symm x)) M₂)\n⊢ Module.Free R (MultilinearMap R M₁ M₂) ∧ Module.Finite R (MultilinearMap R M₁ M₂)",
"tactic": "cases' this with l r"
},
{
"state_after": "case intro.intro\nι : Type u_2\nR : Type u_1\nM₂ : Type u_3\nM₁ : ι → Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : Module R M₂\ninst✝⁵ : Module.Finite R M₂\ninst✝⁴ : Module.Free R M₂\ninst✝³ : (i : ι) → AddCommGroup (M₁ i)\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : ∀ (i : ι), Module.Finite R (M₁ i)\ninst✝ : ∀ (i : ι), Module.Free R (M₁ i)\nval✝ : Fintype ι\nthis :\n ∀ [inst : (i : Fin (Fintype.card ι)) → AddCommGroup (M₁ (↑(Fintype.equivFin ι).symm i))]\n [inst_1 : (i : Fin (Fintype.card ι)) → Module R (M₁ (↑(Fintype.equivFin ι).symm i))]\n [inst_2 : ∀ (i : Fin (Fintype.card ι)), Module.Finite R (M₁ (↑(Fintype.equivFin ι).symm i))]\n [inst_3 : ∀ (i : Fin (Fintype.card ι)), Module.Free R (M₁ (↑(Fintype.equivFin ι).symm i))],\n Module.Free R (MultilinearMap R (fun x => M₁ (↑(Fintype.equivFin ι).symm x)) M₂) ∧\n Module.Finite R (MultilinearMap R (fun x => M₁ (↑(Fintype.equivFin ι).symm x)) M₂)\nl : Module.Free R (MultilinearMap R (fun x => M₁ (↑(Fintype.equivFin ι).symm x)) M₂)\nr : Module.Finite R (MultilinearMap R (fun x => M₁ (↑(Fintype.equivFin ι).symm x)) M₂)\ne : MultilinearMap R M₁ M₂ ≃ₗ[R] MultilinearMap R (fun i => M₁ (↑(Fintype.equivFin ι).symm i)) M₂\n⊢ Module.Free R (MultilinearMap R M₁ M₂) ∧ Module.Finite R (MultilinearMap R M₁ M₂)",
"state_before": "case intro.intro\nι : Type u_2\nR : Type u_1\nM₂ : Type u_3\nM₁ : ι → Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : Module R M₂\ninst✝⁵ : Module.Finite R M₂\ninst✝⁴ : Module.Free R M₂\ninst✝³ : (i : ι) → AddCommGroup (M₁ i)\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : ∀ (i : ι), Module.Finite R (M₁ i)\ninst✝ : ∀ (i : ι), Module.Free R (M₁ i)\nval✝ : Fintype ι\nthis :\n ∀ [inst : (i : Fin (Fintype.card ι)) → AddCommGroup (M₁ (↑(Fintype.equivFin ι).symm i))]\n [inst_1 : (i : Fin (Fintype.card ι)) → Module R (M₁ (↑(Fintype.equivFin ι).symm i))]\n [inst_2 : ∀ (i : Fin (Fintype.card ι)), Module.Finite R (M₁ (↑(Fintype.equivFin ι).symm i))]\n [inst_3 : ∀ (i : Fin (Fintype.card ι)), Module.Free R (M₁ (↑(Fintype.equivFin ι).symm i))],\n Module.Free R (MultilinearMap R (fun x => M₁ (↑(Fintype.equivFin ι).symm x)) M₂) ∧\n Module.Finite R (MultilinearMap R (fun x => M₁ (↑(Fintype.equivFin ι).symm x)) M₂)\nl : Module.Free R (MultilinearMap R (fun x => M₁ (↑(Fintype.equivFin ι).symm x)) M₂)\nr : Module.Finite R (MultilinearMap R (fun x => M₁ (↑(Fintype.equivFin ι).symm x)) M₂)\n⊢ Module.Free R (MultilinearMap R M₁ M₂) ∧ Module.Finite R (MultilinearMap R M₁ M₂)",
"tactic": "have e := domDomCongrLinearEquiv' R M₁ M₂ (Fintype.equivFin ι)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nι : Type u_2\nR : Type u_1\nM₂ : Type u_3\nM₁ : ι → Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : Module R M₂\ninst✝⁵ : Module.Finite R M₂\ninst✝⁴ : Module.Free R M₂\ninst✝³ : (i : ι) → AddCommGroup (M₁ i)\ninst✝² : (i : ι) → Module R (M₁ i)\ninst✝¹ : ∀ (i : ι), Module.Finite R (M₁ i)\ninst✝ : ∀ (i : ι), Module.Free R (M₁ i)\nval✝ : Fintype ι\nthis :\n ∀ [inst : (i : Fin (Fintype.card ι)) → AddCommGroup (M₁ (↑(Fintype.equivFin ι).symm i))]\n [inst_1 : (i : Fin (Fintype.card ι)) → Module R (M₁ (↑(Fintype.equivFin ι).symm i))]\n [inst_2 : ∀ (i : Fin (Fintype.card ι)), Module.Finite R (M₁ (↑(Fintype.equivFin ι).symm i))]\n [inst_3 : ∀ (i : Fin (Fintype.card ι)), Module.Free R (M₁ (↑(Fintype.equivFin ι).symm i))],\n Module.Free R (MultilinearMap R (fun x => M₁ (↑(Fintype.equivFin ι).symm x)) M₂) ∧\n Module.Finite R (MultilinearMap R (fun x => M₁ (↑(Fintype.equivFin ι).symm x)) M₂)\nl : Module.Free R (MultilinearMap R (fun x => M₁ (↑(Fintype.equivFin ι).symm x)) M₂)\nr : Module.Finite R (MultilinearMap R (fun x => M₁ (↑(Fintype.equivFin ι).symm x)) M₂)\ne : MultilinearMap R M₁ M₂ ≃ₗ[R] MultilinearMap R (fun i => M₁ (↑(Fintype.equivFin ι).symm i)) M₂\n⊢ Module.Free R (MultilinearMap R M₁ M₂) ∧ Module.Finite R (MultilinearMap R M₁ M₂)",
"tactic": "exact ⟨Module.Free.of_equiv e.symm, Module.Finite.equiv e.symm⟩"
}
] |
[
68,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
9
] |
Mathlib/Combinatorics/SetFamily/Compression/Down.lean
|
Down.erase_mem_compression_of_mem_compression
|
[
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\n⊢ s ∈ 𝒜 ∧ erase s a ∈ 𝒜 ∨ ¬s ∈ 𝒜 ∧ insert a s ∈ 𝒜 →\n erase s a ∈ 𝒜 ∧ erase s a ∈ 𝒜 ∨ ¬erase s a ∈ 𝒜 ∧ insert a (erase s a) ∈ 𝒜",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\n⊢ s ∈ 𝓓 a 𝒜 → erase s a ∈ 𝓓 a 𝒜",
"tactic": "simp_rw [mem_compression, erase_idem]"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\nh : ¬s ∈ 𝒜 ∧ insert a s ∈ 𝒜\n⊢ ¬erase s a ∈ 𝒜 ∧ insert a (erase s a) ∈ 𝒜",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\n⊢ s ∈ 𝒜 ∧ erase s a ∈ 𝒜 ∨ ¬s ∈ 𝒜 ∧ insert a s ∈ 𝒜 →\n erase s a ∈ 𝒜 ∧ erase s a ∈ 𝒜 ∨ ¬erase s a ∈ 𝒜 ∧ insert a (erase s a) ∈ 𝒜",
"tactic": "refine' Or.imp (fun h => ⟨h.2, h.2⟩) fun h => _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\nh : ¬s ∈ 𝒜 ∧ insert a s ∈ 𝒜\n⊢ ¬erase s a ∈ 𝒜 ∧ insert a (erase s a) ∈ 𝒜",
"tactic": "rwa [erase_eq_of_not_mem (insert_ne_self.1 <| ne_of_mem_of_not_mem h.2 h.1)]"
}
] |
[
187,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.eventually_congr
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.142230\nι : Sort x\nf : Filter α\np q : α → Prop\nh : ∀ᶠ (x : α) in f, p x ↔ q x\nhq : ∀ᶠ (x : α) in f, q x\n⊢ ∀ᶠ (x : α) in f, q x ↔ p x",
"tactic": "simpa only [Iff.comm] using h"
}
] |
[
1153,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1151,
1
] |
Mathlib/Topology/Algebra/Order/MonotoneContinuity.lean
|
continuousWithinAt_right_of_monotoneOn_of_closure_image_mem_nhdsWithin
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : DenselyOrdered β\nf : α → β\ns : Set α\na : α\nh_mono : MonotoneOn f s\nhs : s ∈ 𝓝[Ici a] a\nhfs : closure (f '' s) ∈ 𝓝[Ici (f a)] f a\nb : β\nhb : b > f a\n⊢ ∃ c, c ∈ s ∧ f c ∈ Ioo (f a) b",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : DenselyOrdered β\nf : α → β\ns : Set α\na : α\nh_mono : MonotoneOn f s\nhs : s ∈ 𝓝[Ici a] a\nhfs : closure (f '' s) ∈ 𝓝[Ici (f a)] f a\n⊢ ContinuousWithinAt f (Ici a) a",
"tactic": "refine' continuousWithinAt_right_of_monotoneOn_of_exists_between h_mono hs fun b hb => _"
},
{
"state_after": "case intro.intro.intro\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : DenselyOrdered β\nf : α → β\ns : Set α\na : α\nh_mono : MonotoneOn f s\nhs : s ∈ 𝓝[Ici a] a\nhfs : closure (f '' s) ∈ 𝓝[Ici (f a)] f a\nb : β\nhb : b > f a\nb' : β\nhb' : Ico (f a) b' ⊆ closure (f '' s)\nhab' : f a < b'\nhbb' : b' ≤ b\n⊢ ∃ c, c ∈ s ∧ f c ∈ Ioo (f a) b",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : DenselyOrdered β\nf : α → β\ns : Set α\na : α\nh_mono : MonotoneOn f s\nhs : s ∈ 𝓝[Ici a] a\nhfs : closure (f '' s) ∈ 𝓝[Ici (f a)] f a\nb : β\nhb : b > f a\n⊢ ∃ c, c ∈ s ∧ f c ∈ Ioo (f a) b",
"tactic": "rcases(mem_nhdsWithin_Ici_iff_exists_mem_Ioc_Ico_subset hb).1 hfs with ⟨b', ⟨hab', hbb'⟩, hb'⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : DenselyOrdered β\nf : α → β\ns : Set α\na : α\nh_mono : MonotoneOn f s\nhs : s ∈ 𝓝[Ici a] a\nhfs : closure (f '' s) ∈ 𝓝[Ici (f a)] f a\nb : β\nhb : b > f a\nb' : β\nhb' : Ico (f a) b' ⊆ closure (f '' s)\nhab' : f a < b'\nhbb' : b' ≤ b\nc' : β\nhc' : f a < c' ∧ c' < b'\n⊢ ∃ c, c ∈ s ∧ f c ∈ Ioo (f a) b",
"state_before": "case intro.intro.intro\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : DenselyOrdered β\nf : α → β\ns : Set α\na : α\nh_mono : MonotoneOn f s\nhs : s ∈ 𝓝[Ici a] a\nhfs : closure (f '' s) ∈ 𝓝[Ici (f a)] f a\nb : β\nhb : b > f a\nb' : β\nhb' : Ico (f a) b' ⊆ closure (f '' s)\nhab' : f a < b'\nhbb' : b' ≤ b\n⊢ ∃ c, c ∈ s ∧ f c ∈ Ioo (f a) b",
"tactic": "rcases exists_between hab' with ⟨c', hc'⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : DenselyOrdered β\nf : α → β\ns : Set α\na : α\nh_mono : MonotoneOn f s\nhs : s ∈ 𝓝[Ici a] a\nhfs : closure (f '' s) ∈ 𝓝[Ici (f a)] f a\nb : β\nhb : b > f a\nb' : β\nhb' : Ico (f a) b' ⊆ closure (f '' s)\nhab' : f a < b'\nhbb' : b' ≤ b\nc' : β\nhc' : f a < c' ∧ c' < b'\nc : α\nhcs : c ∈ s\nhc : f c ∈ Ioo (f a) b'\n⊢ ∃ c, c ∈ s ∧ f c ∈ Ioo (f a) b",
"state_before": "case intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : DenselyOrdered β\nf : α → β\ns : Set α\na : α\nh_mono : MonotoneOn f s\nhs : s ∈ 𝓝[Ici a] a\nhfs : closure (f '' s) ∈ 𝓝[Ici (f a)] f a\nb : β\nhb : b > f a\nb' : β\nhb' : Ico (f a) b' ⊆ closure (f '' s)\nhab' : f a < b'\nhbb' : b' ≤ b\nc' : β\nhc' : f a < c' ∧ c' < b'\n⊢ ∃ c, c ∈ s ∧ f c ∈ Ioo (f a) b",
"tactic": "rcases mem_closure_iff.1 (hb' ⟨hc'.1.le, hc'.2⟩) (Ioo (f a) b') isOpen_Ioo hc' with\n ⟨_, hc, ⟨c, hcs, rfl⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_2\nβ : Type u_1\ninst✝⁶ : LinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : LinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : DenselyOrdered β\nf : α → β\ns : Set α\na : α\nh_mono : MonotoneOn f s\nhs : s ∈ 𝓝[Ici a] a\nhfs : closure (f '' s) ∈ 𝓝[Ici (f a)] f a\nb : β\nhb : b > f a\nb' : β\nhb' : Ico (f a) b' ⊆ closure (f '' s)\nhab' : f a < b'\nhbb' : b' ≤ b\nc' : β\nhc' : f a < c' ∧ c' < b'\nc : α\nhcs : c ∈ s\nhc : f c ∈ Ioo (f a) b'\n⊢ ∃ c, c ∈ s ∧ f c ∈ Ioo (f a) b",
"tactic": "exact ⟨c, hcs, hc.1, hc.2.trans_le hbb'⟩"
}
] |
[
93,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
|
collinear_singleton
|
[
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type ?u.247366\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : P\n⊢ Module.rank k { x // x ∈ ⊥ } ≤ 1",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type ?u.247366\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : P\n⊢ Collinear k {p}",
"tactic": "rw [collinear_iff_rank_le_one, vectorSpan_singleton]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type ?u.247366\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : P\n⊢ Module.rank k { x // x ∈ ⊥ } ≤ 1",
"tactic": "simp"
}
] |
[
373,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
371,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Quaternion.im_idem
|
[] |
[
817,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
817,
9
] |
Mathlib/Data/Polynomial/Mirror.lean
|
Polynomial.natTrailingDegree_mul_mirror
|
[
{
"state_after": "case pos\nR : Type u_1\ninst✝¹ : Semiring R\np q : R[X]\ninst✝ : NoZeroDivisors R\nhp : p = 0\n⊢ natTrailingDegree (p * mirror p) = 2 * natTrailingDegree p\n\ncase neg\nR : Type u_1\ninst✝¹ : Semiring R\np q : R[X]\ninst✝ : NoZeroDivisors R\nhp : ¬p = 0\n⊢ natTrailingDegree (p * mirror p) = 2 * natTrailingDegree p",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\np q : R[X]\ninst✝ : NoZeroDivisors R\n⊢ natTrailingDegree (p * mirror p) = 2 * natTrailingDegree p",
"tactic": "by_cases hp : p = 0"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u_1\ninst✝¹ : Semiring R\np q : R[X]\ninst✝ : NoZeroDivisors R\nhp : ¬p = 0\n⊢ natTrailingDegree (p * mirror p) = 2 * natTrailingDegree p",
"tactic": "rw [natTrailingDegree_mul hp (mt mirror_eq_zero.mp hp), mirror_natTrailingDegree, two_mul]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_1\ninst✝¹ : Semiring R\np q : R[X]\ninst✝ : NoZeroDivisors R\nhp : p = 0\n⊢ natTrailingDegree (p * mirror p) = 2 * natTrailingDegree p",
"tactic": "rw [hp, MulZeroClass.zero_mul, natTrailingDegree_zero, MulZeroClass.mul_zero]"
}
] |
[
187,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
UpperSet.coe_sInf
|
[] |
[
549,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
548,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.mem_neighborFinset
|
[] |
[
1355,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1354,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.map_map_symm
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nP' : Type u_3\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\nloc' : IsLocalization S P'\nP'' : Type ?u.569528\ninst✝¹ : CommRing P''\ninst✝ : Algebra R P''\nloc'' : IsLocalization S P''\nI J : FractionalIdeal S P\ng✝ : P →ₐ[R] P'\ng : P ≃ₐ[R] P'\n⊢ map (↑(AlgEquiv.symm g)) (map (↑g) I) = I",
"tactic": "rw [← map_comp, g.symm_comp, map_id]"
}
] |
[
791,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
790,
1
] |
Mathlib/Logic/Hydra.lean
|
Relation.not_cutExpand_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nr : α → α → Prop\ninst✝ : IsIrrefl α r\ns : Multiset α\n⊢ ¬CutExpand r s 0",
"tactic": "classical\nrw [cutExpand_iff]\nrintro ⟨_, _, _, ⟨⟩, _⟩"
},
{
"state_after": "α : Type u_1\nr : α → α → Prop\ninst✝ : IsIrrefl α r\ns : Multiset α\n⊢ ¬∃ t a, (∀ (a' : α), a' ∈ t → r a' a) ∧ a ∈ 0 ∧ s = erase 0 a + t",
"state_before": "α : Type u_1\nr : α → α → Prop\ninst✝ : IsIrrefl α r\ns : Multiset α\n⊢ ¬CutExpand r s 0",
"tactic": "rw [cutExpand_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nr : α → α → Prop\ninst✝ : IsIrrefl α r\ns : Multiset α\n⊢ ¬∃ t a, (∀ (a' : α), a' ∈ t → r a' a) ∧ a ∈ 0 ∧ s = erase 0 a + t",
"tactic": "rintro ⟨_, _, _, ⟨⟩, _⟩"
}
] |
[
107,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
Ideal.bot_lt_of_maximal
|
[
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : ¬IsField R\nI : Ideal R\nIbot : ⊥ < I\nItop : I < ⊤\n⊢ ⊥ < M",
"state_before": "α : Type u\nβ : Type v\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : ¬IsField R\n⊢ ⊥ < M",
"tactic": "rcases Ring.not_isField_iff_exists_ideal_bot_lt_and_lt_top.1 non_field with ⟨I, Ibot, Itop⟩"
},
{
"state_after": "case intro.intro.left\nα : Type u\nβ : Type v\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : ¬IsField R\nI : Ideal R\nIbot : ⊥ < I\nItop : I < ⊤\n⊢ ↑⊥ ⊆ ↑M\n\ncase intro.intro.right\nα : Type u\nβ : Type v\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : ¬IsField R\nI : Ideal R\nIbot : ⊥ < I\nItop : I < ⊤\n⊢ ¬↑M ⊆ ↑⊥",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : ¬IsField R\nI : Ideal R\nIbot : ⊥ < I\nItop : I < ⊤\n⊢ ⊥ < M",
"tactic": "constructor"
},
{
"state_after": "case intro.intro.right\nα : Type u\nβ : Type v\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : ¬IsField R\nI : Ideal R\nIbot : ⊥ < I\nItop : I < ⊤\nmle : ↑M ⊆ ↑⊥\n⊢ False",
"state_before": "case intro.intro.right\nα : Type u\nβ : Type v\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : ¬IsField R\nI : Ideal R\nIbot : ⊥ < I\nItop : I < ⊤\n⊢ ¬↑M ⊆ ↑⊥",
"tactic": "intro mle"
},
{
"state_after": "case intro.intro.right\nα : Type u\nβ : Type v\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : ¬IsField R\nI : Ideal R\nIbot : ⊥ < I\nItop : I < ⊤\nmle : ↑M ⊆ ↑⊥\n⊢ ⊤ < ⊤",
"state_before": "case intro.intro.right\nα : Type u\nβ : Type v\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : ¬IsField R\nI : Ideal R\nIbot : ⊥ < I\nItop : I < ⊤\nmle : ↑M ⊆ ↑⊥\n⊢ False",
"tactic": "apply lt_irrefl (⊤ : Ideal R)"
},
{
"state_after": "case intro.intro.right\nα : Type u\nβ : Type v\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : ¬IsField R\nI : Ideal R\nIbot : ⊥ < I\nItop : I < ⊤\nmle : ↑M ⊆ ↑⊥\nthis : M = ⊥\n⊢ ⊤ < ⊤",
"state_before": "case intro.intro.right\nα : Type u\nβ : Type v\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : ¬IsField R\nI : Ideal R\nIbot : ⊥ < I\nItop : I < ⊤\nmle : ↑M ⊆ ↑⊥\n⊢ ⊤ < ⊤",
"tactic": "have : M = ⊥ := eq_bot_iff.mpr mle"
},
{
"state_after": "case intro.intro.right\nα : Type u\nβ : Type v\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : ¬IsField R\nI : Ideal R\nIbot : M < I\nItop : I < ⊤\nmle : ↑M ⊆ ↑⊥\nthis : M = ⊥\n⊢ ⊤ < ⊤",
"state_before": "case intro.intro.right\nα : Type u\nβ : Type v\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : ¬IsField R\nI : Ideal R\nIbot : ⊥ < I\nItop : I < ⊤\nmle : ↑M ⊆ ↑⊥\nthis : M = ⊥\n⊢ ⊤ < ⊤",
"tactic": "rw [←this] at Ibot"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.right\nα : Type u\nβ : Type v\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : ¬IsField R\nI : Ideal R\nIbot : M < I\nItop : I < ⊤\nmle : ↑M ⊆ ↑⊥\nthis : M = ⊥\n⊢ ⊤ < ⊤",
"tactic": "rwa [hm.1.2 I Ibot] at Itop"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.left\nα : Type u\nβ : Type v\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nM : Ideal R\nhm : IsMaximal M\nnon_field : ¬IsField R\nI : Ideal R\nIbot : ⊥ < I\nItop : I < ⊤\n⊢ ↑⊥ ⊆ ↑M",
"tactic": "simp"
}
] |
[
816,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
809,
1
] |
Mathlib/Data/Option/Basic.lean
|
Option.none_orElse'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.15823\nγ : Type ?u.15826\nδ : Type ?u.15829\nx : Option α\n⊢ (Option.orElse none fun x_1 => x) = x",
"tactic": "cases x <;> rfl"
}
] |
[
277,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/Algebra/Periodic.lean
|
Function.Antiperiodic.const_inv_smul₀
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf g : α → β\nc c₁ c₂ x : α\ninst✝³ : AddCommMonoid α\ninst✝² : Neg β\ninst✝¹ : DivisionSemiring γ\ninst✝ : Module γ α\nh : Antiperiodic f c\na : γ\nha : a ≠ 0\n⊢ Antiperiodic (fun x => f (a⁻¹ • x)) (a • c)",
"tactic": "simpa only [inv_inv] using h.const_smul₀ (inv_ne_zero ha)"
}
] |
[
480,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
478,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Finset.compl_singleton
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.18666\nγ : Type ?u.18669\ninst✝¹ : Fintype α\ns t : Finset α\ninst✝ : DecidableEq α\na✝ a : α\n⊢ {a}ᶜ = erase univ a",
"tactic": "rw [compl_eq_univ_sdiff, sdiff_singleton_eq_erase]"
}
] |
[
249,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
248,
1
] |
Mathlib/Order/LocallyFinite.lean
|
Set.finite_Ico
|
[] |
[
653,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
652,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Perm.diff_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nl t₁ t₂ : List α\nh : t₁ ~ t₂\n⊢ List.diff l t₁ = List.diff l t₂",
"tactic": "induction h generalizing l <;>\n first |simp [*, Perm.erase, erase_comm]"
},
{
"state_after": "no goals",
"state_before": "case trans\nα : Type uu\nβ : Type vv\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nt₁ t₂ l₁✝ l₂✝ l₃✝ : List α\na✝¹ : l₁✝ ~ l₂✝\na✝ : l₂✝ ~ l₃✝\na_ih✝¹ : ∀ (l : List α), List.diff l l₁✝ = List.diff l l₂✝\na_ih✝ : ∀ (l : List α), List.diff l l₂✝ = List.diff l l₃✝\nl : List α\n⊢ List.diff l l₁✝ = List.diff l l₃✝",
"tactic": "simp [*, Perm.erase, erase_comm]"
}
] |
[
802,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
800,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Finset.image_univ_of_surjective
|
[] |
[
259,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Order/PFilter.lean
|
Order.PFilter.antitone_principal
|
[] |
[
132,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Data/Set/Pairwise/Lattice.lean
|
Set.pairwiseDisjoint_sUnion
|
[] |
[
65,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.destruct_map
|
[
{
"state_after": "case bisim\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns : WSeq α\n⊢ Computation.IsBisimulation fun c1 c2 =>\n ∃ s, c1 = destruct (map f s) ∧ c2 = Computation.map (Option.map (Prod.map f (map f))) (destruct s)\n\ncase r\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns : WSeq α\n⊢ ∃ s_1,\n destruct (map f s) = destruct (map f s_1) ∧\n Computation.map (Option.map (Prod.map f (map f))) (destruct s) =\n Computation.map (Option.map (Prod.map f (map f))) (destruct s_1)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns : WSeq α\n⊢ destruct (map f s) = Computation.map (Option.map (Prod.map f (map f))) (destruct s)",
"tactic": "apply\n Computation.eq_of_bisim fun c1 c2 =>\n ∃ s,\n c1 = destruct (map f s) ∧\n c2 = Computation.map (Option.map (Prod.map f (map f))) (destruct s)"
},
{
"state_after": "case bisim\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns : WSeq α\nc1 c2 : Computation (Option (β × WSeq β))\nh : ∃ s, c1 = destruct (map f s) ∧ c2 = Computation.map (Option.map (Prod.map f (map f))) (destruct s)\n⊢ Computation.BisimO\n (fun c1 c2 => ∃ s, c1 = destruct (map f s) ∧ c2 = Computation.map (Option.map (Prod.map f (map f))) (destruct s))\n (Computation.destruct c1) (Computation.destruct c2)",
"state_before": "case bisim\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns : WSeq α\n⊢ Computation.IsBisimulation fun c1 c2 =>\n ∃ s, c1 = destruct (map f s) ∧ c2 = Computation.map (Option.map (Prod.map f (map f))) (destruct s)",
"tactic": "intro c1 c2 h"
},
{
"state_after": "case bisim.intro\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns✝ : WSeq α\nc1 c2 : Computation (Option (β × WSeq β))\ns : WSeq α\nh : c1 = destruct (map f s) ∧ c2 = Computation.map (Option.map (Prod.map f (map f))) (destruct s)\n⊢ Computation.BisimO\n (fun c1 c2 => ∃ s, c1 = destruct (map f s) ∧ c2 = Computation.map (Option.map (Prod.map f (map f))) (destruct s))\n (Computation.destruct c1) (Computation.destruct c2)",
"state_before": "case bisim\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns : WSeq α\nc1 c2 : Computation (Option (β × WSeq β))\nh : ∃ s, c1 = destruct (map f s) ∧ c2 = Computation.map (Option.map (Prod.map f (map f))) (destruct s)\n⊢ Computation.BisimO\n (fun c1 c2 => ∃ s, c1 = destruct (map f s) ∧ c2 = Computation.map (Option.map (Prod.map f (map f))) (destruct s))\n (Computation.destruct c1) (Computation.destruct c2)",
"tactic": "cases' h with s h"
},
{
"state_after": "case bisim.intro\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns✝ : WSeq α\nc1 c2 : Computation (Option (β × WSeq β))\ns : WSeq α\nh : c1 = destruct (map f s) ∧ c2 = Computation.map (Option.map (Prod.map f (map f))) (destruct s)\n⊢ Computation.BisimO\n (fun c1 c2 => ∃ s, c1 = destruct (map f s) ∧ c2 = Computation.map (Option.map (Prod.map f (map f))) (destruct s))\n (Computation.destruct (destruct (map f s)))\n (Computation.destruct (Computation.map (Option.map (Prod.map f (map f))) (destruct s)))",
"state_before": "case bisim.intro\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns✝ : WSeq α\nc1 c2 : Computation (Option (β × WSeq β))\ns : WSeq α\nh : c1 = destruct (map f s) ∧ c2 = Computation.map (Option.map (Prod.map f (map f))) (destruct s)\n⊢ Computation.BisimO\n (fun c1 c2 => ∃ s, c1 = destruct (map f s) ∧ c2 = Computation.map (Option.map (Prod.map f (map f))) (destruct s))\n (Computation.destruct c1) (Computation.destruct c2)",
"tactic": "rw [h.left, h.right]"
},
{
"state_after": "case bisim.intro.h3\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns✝¹ : WSeq α\nc1 c2 : Computation (Option (β × WSeq β))\ns✝ : WSeq α\nh✝ : c1 = destruct (map f s✝) ∧ c2 = Computation.map (Option.map (Prod.map f (map f))) (destruct s✝)\ns : WSeq α\nh : c1 = destruct (map f (think s)) ∧ c2 = Computation.map (Option.map (Prod.map f (map f))) (destruct (think s))\n⊢ ∃ s_1,\n destruct (map f s) = destruct (map f s_1) ∧\n Computation.map (Option.map (Prod.map f (map f))) (destruct s) =\n Computation.map (Option.map (Prod.map f (map f))) (destruct s_1)",
"state_before": "case bisim.intro\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns✝ : WSeq α\nc1 c2 : Computation (Option (β × WSeq β))\ns : WSeq α\nh : c1 = destruct (map f s) ∧ c2 = Computation.map (Option.map (Prod.map f (map f))) (destruct s)\n⊢ Computation.BisimO\n (fun c1 c2 => ∃ s, c1 = destruct (map f s) ∧ c2 = Computation.map (Option.map (Prod.map f (map f))) (destruct s))\n (Computation.destruct (destruct (map f s)))\n (Computation.destruct (Computation.map (Option.map (Prod.map f (map f))) (destruct s)))",
"tactic": "induction' s using WSeq.recOn with a s s <;> simp"
},
{
"state_after": "no goals",
"state_before": "case bisim.intro.h3\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns✝¹ : WSeq α\nc1 c2 : Computation (Option (β × WSeq β))\ns✝ : WSeq α\nh✝ : c1 = destruct (map f s✝) ∧ c2 = Computation.map (Option.map (Prod.map f (map f))) (destruct s✝)\ns : WSeq α\nh : c1 = destruct (map f (think s)) ∧ c2 = Computation.map (Option.map (Prod.map f (map f))) (destruct (think s))\n⊢ ∃ s_1,\n destruct (map f s) = destruct (map f s_1) ∧\n Computation.map (Option.map (Prod.map f (map f))) (destruct s) =\n Computation.map (Option.map (Prod.map f (map f))) (destruct s_1)",
"tactic": "exact ⟨s, rfl, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case r\nα : Type u\nβ : Type v\nγ : Type w\nf : α → β\ns : WSeq α\n⊢ ∃ s_1,\n destruct (map f s) = destruct (map f s_1) ∧\n Computation.map (Option.map (Prod.map f (map f))) (destruct s) =\n Computation.map (Option.map (Prod.map f (map f))) (destruct s_1)",
"tactic": "exact ⟨s, rfl, rfl⟩"
}
] |
[
1492,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1480,
1
] |
Mathlib/Algebra/Group/TypeTags.lean
|
ofMul_div
|
[] |
[
342,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
341,
1
] |
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