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/Order/Filter/Basic.lean
|
Filter.EventuallyLE.mul_le_mul'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.213025\nι : Sort x\ninst✝³ : Mul β\ninst✝² : Preorder β\ninst✝¹ : CovariantClass β β (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass β β (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nl : Filter α\nf₁ f₂ g₁ g₂ : α → β\nhf : f₁ ≤ᶠ[l] f₂\nhg : g₁ ≤ᶠ[l] g₂\n⊢ f₁ * g₁ ≤ᶠ[l] f₂ * g₂",
"tactic": "filter_upwards [hf, hg] with x hfx hgx using _root_.mul_le_mul' hfx hgx"
}
] |
[
1762,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1759,
1
] |
Mathlib/Analysis/InnerProductSpace/Adjoint.lean
|
ContinuousLinearMap.adjoint_inner_right
|
[] |
[
130,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/GroupTheory/FreeAbelianGroup.lean
|
FreeAbelianGroup.seq_zero
|
[] |
[
303,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
302,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
HasFDerivWithinAt.add
|
[] |
[
138,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
8
] |
Mathlib/Algebra/Order/LatticeGroup.lean
|
LatticeOrderedCommGroup.pos_le_one_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝¹ : Lattice α\ninst✝ : CommGroup α\na : α\n⊢ a⁺ ≤ 1 ↔ a ≤ 1",
"tactic": "rw [m_pos_part_def, sup_le_iff, and_iff_left le_rfl]"
}
] |
[
214,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.coe_singletonMonoidHom
|
[] |
[
917,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
916,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.sub_lt_self
|
[
{
"state_after": "a b : Nat\nh₀ : 0 < a\nh₁ : a ≤ b\n⊢ 0 < b",
"state_before": "a b : Nat\nh₀ : 0 < a\nh₁ : a ≤ b\n⊢ b - a < b",
"tactic": "apply sub_lt _ h₀"
},
{
"state_after": "no goals",
"state_before": "a b : Nat\nh₀ : 0 < a\nh₁ : a ≤ b\n⊢ 0 < b",
"tactic": "apply Nat.lt_of_lt_of_le h₀ h₁"
}
] |
[
452,
33
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
450,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.eq_right_or_mem_Ioo_of_mem_Ioc
|
[] |
[
944,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
943,
1
] |
Mathlib/Data/Finset/Interval.lean
|
Finset.Ico_eq_filter_ssubsets
|
[] |
[
57,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/Data/Real/Sqrt.lean
|
Real.sqrt_le_sqrt_iff
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\nhy : 0 ≤ y\n⊢ sqrt x ≤ sqrt y ↔ x ≤ y",
"tactic": "rw [sqrt, sqrt, NNReal.coe_le_coe, NNReal.sqrt_le_sqrt_iff, Real.toNNReal_le_toNNReal_iff hy]"
}
] |
[
263,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/Geometry/Manifold/ChartedSpace.lean
|
StructureGroupoid.le_iff
|
[] |
[
216,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/CategoryTheory/Extensive.lean
|
CategoryTheory.IsInitial.isVanKampenColimit
|
[
{
"state_after": "J : Type v'\ninst✝² : Category J\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasStrictInitialObjects C\nX : C\nh : IsInitial X\nF' : Discrete PEmpty ⥤ C\nc' : Cocone F'\nα : F' ⟶ Functor.empty C\nf : c'.pt ⟶ (asEmptyCocone X).pt\nhf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty)).map f\nhα : NatTrans.Equifibered α\n⊢ Nonempty (IsColimit c') ↔ ∀ (j : Discrete PEmpty), IsPullback (c'.ι.app j) (α.app j) f ((asEmptyCocone X).ι.app j)",
"state_before": "J : Type v'\ninst✝² : Category J\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasStrictInitialObjects C\nX : C\nh : IsInitial X\n⊢ IsVanKampenColimit (asEmptyCocone X)",
"tactic": "intro F' c' α f hf hα"
},
{
"state_after": "J : Type v'\ninst✝² : Category J\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasStrictInitialObjects C\nX : C\nh : IsInitial X\nF' : Discrete PEmpty ⥤ C\nc' : Cocone F'\nα : F' ⟶ Functor.empty C\nf : c'.pt ⟶ (asEmptyCocone X).pt\nhf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty)).map f\nhα : NatTrans.Equifibered α\nthis : F' = Functor.empty C\n⊢ Nonempty (IsColimit c') ↔ ∀ (j : Discrete PEmpty), IsPullback (c'.ι.app j) (α.app j) f ((asEmptyCocone X).ι.app j)",
"state_before": "J : Type v'\ninst✝² : Category J\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasStrictInitialObjects C\nX : C\nh : IsInitial X\nF' : Discrete PEmpty ⥤ C\nc' : Cocone F'\nα : F' ⟶ Functor.empty C\nf : c'.pt ⟶ (asEmptyCocone X).pt\nhf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty)).map f\nhα : NatTrans.Equifibered α\n⊢ Nonempty (IsColimit c') ↔ ∀ (j : Discrete PEmpty), IsPullback (c'.ι.app j) (α.app j) f ((asEmptyCocone X).ι.app j)",
"tactic": "have : F' = Functor.empty C := by apply Functor.hext <;> rintro ⟨⟨⟩⟩"
},
{
"state_after": "J : Type v'\ninst✝² : Category J\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasStrictInitialObjects C\nX : C\nh : IsInitial X\nc' : Cocone (Functor.empty C)\nα : Functor.empty C ⟶ Functor.empty C\nf : c'.pt ⟶ (asEmptyCocone X).pt\nhf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty)).map f\nhα : NatTrans.Equifibered α\n⊢ Nonempty (IsColimit c') ↔ ∀ (j : Discrete PEmpty), IsPullback (c'.ι.app j) (α.app j) f ((asEmptyCocone X).ι.app j)",
"state_before": "J : Type v'\ninst✝² : Category J\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasStrictInitialObjects C\nX : C\nh : IsInitial X\nF' : Discrete PEmpty ⥤ C\nc' : Cocone F'\nα : F' ⟶ Functor.empty C\nf : c'.pt ⟶ (asEmptyCocone X).pt\nhf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty)).map f\nhα : NatTrans.Equifibered α\nthis : F' = Functor.empty C\n⊢ Nonempty (IsColimit c') ↔ ∀ (j : Discrete PEmpty), IsPullback (c'.ι.app j) (α.app j) f ((asEmptyCocone X).ι.app j)",
"tactic": "subst this"
},
{
"state_after": "J : Type v'\ninst✝² : Category J\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasStrictInitialObjects C\nX : C\nh : IsInitial X\nc' : Cocone (Functor.empty C)\nα : Functor.empty C ⟶ Functor.empty C\nf : c'.pt ⟶ (asEmptyCocone X).pt\nhf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty)).map f\nhα : NatTrans.Equifibered α\nthis : IsIso f\n⊢ Nonempty (IsColimit c') ↔ ∀ (j : Discrete PEmpty), IsPullback (c'.ι.app j) (α.app j) f ((asEmptyCocone X).ι.app j)",
"state_before": "J : Type v'\ninst✝² : Category J\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasStrictInitialObjects C\nX : C\nh : IsInitial X\nc' : Cocone (Functor.empty C)\nα : Functor.empty C ⟶ Functor.empty C\nf : c'.pt ⟶ (asEmptyCocone X).pt\nhf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty)).map f\nhα : NatTrans.Equifibered α\n⊢ Nonempty (IsColimit c') ↔ ∀ (j : Discrete PEmpty), IsPullback (c'.ι.app j) (α.app j) f ((asEmptyCocone X).ι.app j)",
"tactic": "haveI := h.isIso_to f"
},
{
"state_after": "no goals",
"state_before": "J : Type v'\ninst✝² : Category J\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasStrictInitialObjects C\nX : C\nh : IsInitial X\nc' : Cocone (Functor.empty C)\nα : Functor.empty C ⟶ Functor.empty C\nf : c'.pt ⟶ (asEmptyCocone X).pt\nhf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty)).map f\nhα : NatTrans.Equifibered α\nthis : IsIso f\n⊢ Nonempty (IsColimit c') ↔ ∀ (j : Discrete PEmpty), IsPullback (c'.ι.app j) (α.app j) f ((asEmptyCocone X).ι.app j)",
"tactic": "refine' ⟨by rintro _ ⟨⟨⟩⟩,\n fun _ => ⟨IsColimit.ofIsoColimit h (Cocones.ext (asIso f).symm <| by rintro ⟨⟨⟩⟩)⟩⟩"
},
{
"state_after": "no goals",
"state_before": "J : Type v'\ninst✝² : Category J\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasStrictInitialObjects C\nX : C\nh : IsInitial X\nF' : Discrete PEmpty ⥤ C\nc' : Cocone F'\nα : F' ⟶ Functor.empty C\nf : c'.pt ⟶ (asEmptyCocone X).pt\nhf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty)).map f\nhα : NatTrans.Equifibered α\n⊢ F' = Functor.empty C",
"tactic": "apply Functor.hext <;> rintro ⟨⟨⟩⟩"
},
{
"state_after": "no goals",
"state_before": "J : Type v'\ninst✝² : Category J\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasStrictInitialObjects C\nX : C\nh : IsInitial X\nc' : Cocone (Functor.empty C)\nα : Functor.empty C ⟶ Functor.empty C\nf : c'.pt ⟶ (asEmptyCocone X).pt\nhf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty)).map f\nhα : NatTrans.Equifibered α\nthis : IsIso f\n⊢ Nonempty (IsColimit c') → ∀ (j : Discrete PEmpty), IsPullback (c'.ι.app j) (α.app j) f ((asEmptyCocone X).ι.app j)",
"tactic": "rintro _ ⟨⟨⟩⟩"
},
{
"state_after": "no goals",
"state_before": "J : Type v'\ninst✝² : Category J\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasStrictInitialObjects C\nX : C\nh : IsInitial X\nc' : Cocone (Functor.empty C)\nα : Functor.empty C ⟶ Functor.empty C\nf : c'.pt ⟶ (asEmptyCocone X).pt\nhf : α ≫ (asEmptyCocone X).ι = c'.ι ≫ (Functor.const (Discrete PEmpty)).map f\nhα : NatTrans.Equifibered α\nthis : IsIso f\nx✝ : ∀ (j : Discrete PEmpty), IsPullback (c'.ι.app j) (α.app j) f ((asEmptyCocone X).ι.app j)\n⊢ ∀ (j : Discrete PEmpty), (asEmptyCocone X).ι.app j ≫ (asIso f).symm.hom = c'.ι.app j",
"tactic": "rintro ⟨⟨⟩⟩"
}
] |
[
120,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean
|
ContinuousAffineMap.norm_image_zero_le
|
[] |
[
181,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Subalgebra.nsmul_mem
|
[] |
[
149,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
11
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.union_distrib_right
|
[] |
[
1753,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1752,
1
] |
Mathlib/Algebra/Module/Opposites.lean
|
MulOpposite.opLinearEquiv_symm_toAddEquiv
|
[] |
[
72,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/RingTheory/SimpleModule.lean
|
LinearMap.isCoatom_ker_of_surjective
|
[
{
"state_after": "R : Type u_1\ninst✝⁵ : Ring R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nm : Submodule R M\nN : Type u_2\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : IsSimpleModule R N\nf : M →ₗ[R] N\nhf : Function.Surjective ↑f\n⊢ IsSimpleModule R (M ⧸ ker f)",
"state_before": "R : Type u_1\ninst✝⁵ : Ring R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nm : Submodule R M\nN : Type u_2\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : IsSimpleModule R N\nf : M →ₗ[R] N\nhf : Function.Surjective ↑f\n⊢ IsCoatom (ker f)",
"tactic": "rw [← isSimpleModule_iff_isCoatom]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁵ : Ring R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nm : Submodule R M\nN : Type u_2\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : IsSimpleModule R N\nf : M →ₗ[R] N\nhf : Function.Surjective ↑f\n⊢ IsSimpleModule R (M ⧸ ker f)",
"tactic": "exact IsSimpleModule.congr (f.quotKerEquivOfSurjective hf)"
}
] |
[
171,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/CategoryTheory/Filtered.lean
|
CategoryTheory.IsFiltered.tulip
|
[
{
"state_after": "case intro.intro.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFilteredOrEmpty C\nj₁ j₂ j₃ k₁ k₂ l : C\nf₁ : j₁ ⟶ k₁\nf₂ : j₂ ⟶ k₁\nf₃ : j₂ ⟶ k₂\nf₄ : j₃ ⟶ k₂\ng₁ : j₁ ⟶ l\ng₂ : j₃ ⟶ l\nl' : C\nk₁l : k₁ ⟶ l'\nk₂l : k₂ ⟶ l'\nhl : f₂ ≫ k₁l = f₃ ≫ k₂l\n⊢ ∃ s α β γ, f₁ ≫ α = g₁ ≫ β ∧ f₂ ≫ α = f₃ ≫ γ ∧ f₄ ≫ γ = g₂ ≫ β",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsFilteredOrEmpty C\nj₁ j₂ j₃ k₁ k₂ l : C\nf₁ : j₁ ⟶ k₁\nf₂ : j₂ ⟶ k₁\nf₃ : j₂ ⟶ k₂\nf₄ : j₃ ⟶ k₂\ng₁ : j₁ ⟶ l\ng₂ : j₃ ⟶ l\n⊢ ∃ s α β γ, f₁ ≫ α = g₁ ≫ β ∧ f₂ ≫ α = f₃ ≫ γ ∧ f₄ ≫ γ = g₂ ≫ β",
"tactic": "obtain ⟨l', k₁l, k₂l, hl⟩ := span f₂ f₃"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFilteredOrEmpty C\nj₁ j₂ j₃ k₁ k₂ l : C\nf₁ : j₁ ⟶ k₁\nf₂ : j₂ ⟶ k₁\nf₃ : j₂ ⟶ k₂\nf₄ : j₃ ⟶ k₂\ng₁ : j₁ ⟶ l\ng₂ : j₃ ⟶ l\nl' : C\nk₁l : k₁ ⟶ l'\nk₂l : k₂ ⟶ l'\nhl : f₂ ≫ k₁l = f₃ ≫ k₂l\ns : C\nls : l ⟶ s\nl's : l' ⟶ s\nhs₁ : g₁ ≫ ls = (f₁ ≫ k₁l) ≫ l's\nhs₂ : g₂ ≫ ls = (f₄ ≫ k₂l) ≫ l's\n⊢ ∃ s α β γ, f₁ ≫ α = g₁ ≫ β ∧ f₂ ≫ α = f₃ ≫ γ ∧ f₄ ≫ γ = g₂ ≫ β",
"state_before": "case intro.intro.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFilteredOrEmpty C\nj₁ j₂ j₃ k₁ k₂ l : C\nf₁ : j₁ ⟶ k₁\nf₂ : j₂ ⟶ k₁\nf₃ : j₂ ⟶ k₂\nf₄ : j₃ ⟶ k₂\ng₁ : j₁ ⟶ l\ng₂ : j₃ ⟶ l\nl' : C\nk₁l : k₁ ⟶ l'\nk₂l : k₂ ⟶ l'\nhl : f₂ ≫ k₁l = f₃ ≫ k₂l\n⊢ ∃ s α β γ, f₁ ≫ α = g₁ ≫ β ∧ f₂ ≫ α = f₃ ≫ γ ∧ f₄ ≫ γ = g₂ ≫ β",
"tactic": "obtain ⟨s, ls, l's, hs₁, hs₂⟩ := bowtie g₁ (f₁ ≫ k₁l) g₂ (f₄ ≫ k₂l)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\nC : Type u\ninst✝¹ : Category C\ninst✝ : IsFilteredOrEmpty C\nj₁ j₂ j₃ k₁ k₂ l : C\nf₁ : j₁ ⟶ k₁\nf₂ : j₂ ⟶ k₁\nf₃ : j₂ ⟶ k₂\nf₄ : j₃ ⟶ k₂\ng₁ : j₁ ⟶ l\ng₂ : j₃ ⟶ l\nl' : C\nk₁l : k₁ ⟶ l'\nk₂l : k₂ ⟶ l'\nhl : f₂ ≫ k₁l = f₃ ≫ k₂l\ns : C\nls : l ⟶ s\nl's : l' ⟶ s\nhs₁ : g₁ ≫ ls = (f₁ ≫ k₁l) ≫ l's\nhs₂ : g₂ ≫ ls = (f₄ ≫ k₂l) ≫ l's\n⊢ ∃ s α β γ, f₁ ≫ α = g₁ ≫ β ∧ f₂ ≫ α = f₃ ≫ γ ∧ f₄ ≫ γ = g₂ ≫ β",
"tactic": "refine' ⟨s, k₁l ≫ l's, ls, k₂l ≫ l's, _, by simp only [←Category.assoc, hl], _⟩ <;>\n simp only [hs₁, hs₂, Category.assoc]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : IsFilteredOrEmpty C\nj₁ j₂ j₃ k₁ k₂ l : C\nf₁ : j₁ ⟶ k₁\nf₂ : j₂ ⟶ k₁\nf₃ : j₂ ⟶ k₂\nf₄ : j₃ ⟶ k₂\ng₁ : j₁ ⟶ l\ng₂ : j₃ ⟶ l\nl' : C\nk₁l : k₁ ⟶ l'\nk₂l : k₂ ⟶ l'\nhl : f₂ ≫ k₁l = f₃ ≫ k₂l\ns : C\nls : l ⟶ s\nl's : l' ⟶ s\nhs₁ : g₁ ≫ ls = (f₁ ≫ k₁l) ≫ l's\nhs₂ : g₂ ≫ ls = (f₄ ≫ k₂l) ≫ l's\n⊢ f₂ ≫ k₁l ≫ l's = f₃ ≫ k₂l ≫ l's",
"tactic": "simp only [←Category.assoc, hl]"
}
] |
[
459,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
452,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.get_eq_of_promises
|
[] |
[
499,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
498,
1
] |
Mathlib/GroupTheory/Perm/Sign.lean
|
Equiv.Perm.sign_prodCongrRight
|
[
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\ninst✝³ : DecidableEq α\ninst✝² : Fintype α\ninst✝¹ : DecidableEq β\ninst✝ : Fintype β\nσ : α → Perm β\nl : List α\nhl : List.Nodup l\nmem_l : ∀ (x : α), x ∈ l\n⊢ ↑sign (prodCongrRight σ) = ∏ k : α, ↑sign (σ k)",
"state_before": "α : Type u\nβ : Type v\ninst✝³ : DecidableEq α\ninst✝² : Fintype α\ninst✝¹ : DecidableEq β\ninst✝ : Fintype β\nσ : α → Perm β\n⊢ ↑sign (prodCongrRight σ) = ∏ k : α, ↑sign (σ k)",
"tactic": "obtain ⟨l, hl, mem_l⟩ := Finite.exists_univ_list α"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\ninst✝³ : DecidableEq α\ninst✝² : Fintype α\ninst✝¹ : DecidableEq β\ninst✝ : Fintype β\nσ : α → Perm β\nl : List α\nhl : List.Nodup l\nmem_l : ∀ (x : α), x ∈ l\nl_to_finset : List.toFinset l = univ\n⊢ ↑sign (prodCongrRight σ) = ∏ k : α, ↑sign (σ k)",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\ninst✝³ : DecidableEq α\ninst✝² : Fintype α\ninst✝¹ : DecidableEq β\ninst✝ : Fintype β\nσ : α → Perm β\nl : List α\nhl : List.Nodup l\nmem_l : ∀ (x : α), x ∈ l\n⊢ ↑sign (prodCongrRight σ) = ∏ k : α, ↑sign (σ k)",
"tactic": "have l_to_finset : l.toFinset = Finset.univ := by\n apply eq_top_iff.mpr\n intro b _\n exact List.mem_toFinset.mpr (mem_l b)"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\ninst✝³ : DecidableEq α\ninst✝² : Fintype α\ninst✝¹ : DecidableEq β\ninst✝ : Fintype β\nσ : α → Perm β\nl : List α\nhl : List.Nodup l\nmem_l : ∀ (x : α), x ∈ l\nl_to_finset : List.toFinset l = univ\n⊢ List.prod (List.map (↑sign ∘ fun a => prodExtendRight a (σ a)) l) = List.prod (List.map (fun k => ↑sign (σ k)) l)",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\ninst✝³ : DecidableEq α\ninst✝² : Fintype α\ninst✝¹ : DecidableEq β\ninst✝ : Fintype β\nσ : α → Perm β\nl : List α\nhl : List.Nodup l\nmem_l : ∀ (x : α), x ∈ l\nl_to_finset : List.toFinset l = univ\n⊢ ↑sign (prodCongrRight σ) = ∏ k : α, ↑sign (σ k)",
"tactic": "rw [← prod_prodExtendRight σ hl mem_l, sign.map_list_prod, List.map_map, ← l_to_finset,\n List.prod_toFinset _ hl]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\ninst✝³ : DecidableEq α\ninst✝² : Fintype α\ninst✝¹ : DecidableEq β\ninst✝ : Fintype β\nσ : α → Perm β\nl : List α\nhl : List.Nodup l\nmem_l : ∀ (x : α), x ∈ l\nl_to_finset : List.toFinset l = univ\n⊢ List.prod (List.map (↑sign ∘ fun a => prodExtendRight a (σ a)) l) = List.prod (List.map (fun k => ↑sign (σ k)) l)",
"tactic": "simp_rw [← fun a => sign_prodExtendRight a (σ a), Function.comp]"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝³ : DecidableEq α\ninst✝² : Fintype α\ninst✝¹ : DecidableEq β\ninst✝ : Fintype β\nσ : α → Perm β\nl : List α\nhl : List.Nodup l\nmem_l : ∀ (x : α), x ∈ l\n⊢ ⊤ ≤ List.toFinset l",
"state_before": "α : Type u\nβ : Type v\ninst✝³ : DecidableEq α\ninst✝² : Fintype α\ninst✝¹ : DecidableEq β\ninst✝ : Fintype β\nσ : α → Perm β\nl : List α\nhl : List.Nodup l\nmem_l : ∀ (x : α), x ∈ l\n⊢ List.toFinset l = univ",
"tactic": "apply eq_top_iff.mpr"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝³ : DecidableEq α\ninst✝² : Fintype α\ninst✝¹ : DecidableEq β\ninst✝ : Fintype β\nσ : α → Perm β\nl : List α\nhl : List.Nodup l\nmem_l : ∀ (x : α), x ∈ l\nb : α\na✝ : b ∈ ⊤\n⊢ b ∈ List.toFinset l",
"state_before": "α : Type u\nβ : Type v\ninst✝³ : DecidableEq α\ninst✝² : Fintype α\ninst✝¹ : DecidableEq β\ninst✝ : Fintype β\nσ : α → Perm β\nl : List α\nhl : List.Nodup l\nmem_l : ∀ (x : α), x ∈ l\n⊢ ⊤ ≤ List.toFinset l",
"tactic": "intro b _"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝³ : DecidableEq α\ninst✝² : Fintype α\ninst✝¹ : DecidableEq β\ninst✝ : Fintype β\nσ : α → Perm β\nl : List α\nhl : List.Nodup l\nmem_l : ∀ (x : α), x ∈ l\nb : α\na✝ : b ∈ ⊤\n⊢ b ∈ List.toFinset l",
"tactic": "exact List.mem_toFinset.mpr (mem_l b)"
}
] |
[
756,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
748,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.diag_apply
|
[] |
[
613,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
612,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.equivFunOnFintype_symm_coe
|
[] |
[
626,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
625,
1
] |
Mathlib/Order/SymmDiff.lean
|
bihimp_right_injective
|
[] |
[
669,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
668,
1
] |
Mathlib/CategoryTheory/Subobject/MonoOver.lean
|
CategoryTheory.MonoOver.lift_obj_arrow
|
[] |
[
189,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergenceTopology.lean
|
UniformOnFun.tendsto_iff_tendstoUniformlyOn
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.103872\nι : Type u_3\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nF : ι → α →ᵤ[𝔖] β\nf : α →ᵤ[𝔖] β\n⊢ (∀ (i : Set α), Tendsto F p (𝓝 f)) ↔ ∀ (s : Set α), s ∈ 𝔖 → TendstoUniformlyOn F f p s",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.103872\nι : Type u_3\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nF : ι → α →ᵤ[𝔖] β\nf : α →ᵤ[𝔖] β\n⊢ Tendsto F p (𝓝 f) ↔ ∀ (s : Set α), s ∈ 𝔖 → TendstoUniformlyOn F f p s",
"tactic": "rw [UniformOnFun.topologicalSpace_eq, nhds_iInf, tendsto_iInf]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.103872\nι : Type u_3\ns✝ s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nF : ι → α →ᵤ[𝔖] β\nf : α →ᵤ[𝔖] β\ns : Set α\n⊢ Tendsto F p (𝓝 f) ↔ s ∈ 𝔖 → TendstoUniformlyOn F f p s",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.103872\nι : Type u_3\ns s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nF : ι → α →ᵤ[𝔖] β\nf : α →ᵤ[𝔖] β\n⊢ (∀ (i : Set α), Tendsto F p (𝓝 f)) ↔ ∀ (s : Set α), s ∈ 𝔖 → TendstoUniformlyOn F f p s",
"tactic": "refine' forall_congr' fun s => _"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.103872\nι : Type u_3\ns✝ s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nF : ι → α →ᵤ[𝔖] β\nf : α →ᵤ[𝔖] β\ns : Set α\n⊢ s ∈ 𝔖 → Tendsto F p (𝓝 f) ↔ s ∈ 𝔖 → TendstoUniformlyOn F f p s",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.103872\nι : Type u_3\ns✝ s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nF : ι → α →ᵤ[𝔖] β\nf : α →ᵤ[𝔖] β\ns : Set α\n⊢ Tendsto F p (𝓝 f) ↔ s ∈ 𝔖 → TendstoUniformlyOn F f p s",
"tactic": "rw [nhds_iInf, tendsto_iInf]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.103872\nι : Type u_3\ns✝ s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nF : ι → α →ᵤ[𝔖] β\nf : α →ᵤ[𝔖] β\ns : Set α\nhs : s ∈ 𝔖\n⊢ Tendsto F p (𝓝 f) ↔ TendstoUniformlyOn F f p s",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.103872\nι : Type u_3\ns✝ s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nF : ι → α →ᵤ[𝔖] β\nf : α →ᵤ[𝔖] β\ns : Set α\n⊢ s ∈ 𝔖 → Tendsto F p (𝓝 f) ↔ s ∈ 𝔖 → TendstoUniformlyOn F f p s",
"tactic": "refine' forall_congr' fun hs => _"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.103872\nι : Type u_3\ns✝ s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nF : ι → α →ᵤ[𝔖] β\nf : α →ᵤ[𝔖] β\ns : Set α\nhs : s ∈ 𝔖\n⊢ TendstoUniformly ((restrict s ∘ ↑UniformFun.toFun) ∘ F) ((restrict s ∘ ↑UniformFun.toFun) f) p ↔\n TendstoUniformly (fun i x => F i ↑x) (f ∘ Subtype.val) p",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.103872\nι : Type u_3\ns✝ s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nF : ι → α →ᵤ[𝔖] β\nf : α →ᵤ[𝔖] β\ns : Set α\nhs : s ∈ 𝔖\n⊢ Tendsto F p (𝓝 f) ↔ TendstoUniformlyOn F f p s",
"tactic": "rw [nhds_induced (T := _), tendsto_comap_iff, tendstoUniformlyOn_iff_tendstoUniformly_comp_coe,\n UniformFun.tendsto_iff_tendstoUniformly]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.103872\nι : Type u_3\ns✝ s' : Set α\nx : α\np : Filter ι\ng : ι → α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nF : ι → α →ᵤ[𝔖] β\nf : α →ᵤ[𝔖] β\ns : Set α\nhs : s ∈ 𝔖\n⊢ TendstoUniformly ((restrict s ∘ ↑UniformFun.toFun) ∘ F) ((restrict s ∘ ↑UniformFun.toFun) f) p ↔\n TendstoUniformly (fun i x => F i ↑x) (f ∘ Subtype.val) p",
"tactic": "rfl"
}
] |
[
879,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
871,
11
] |
Mathlib/LinearAlgebra/BilinearMap.lean
|
LinearMap.mk₂'_apply
|
[] |
[
115,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Topology/Algebra/Ring/Ideal.lean
|
Ideal.closure_eq_of_isClosed
|
[] |
[
43,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
42,
1
] |
Mathlib/Topology/LocalExtr.lean
|
IsLocalMaxOn.sub
|
[] |
[
437,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
435,
8
] |
Mathlib/Analysis/Normed/Group/Seminorm.lean
|
NonarchAddGroupSeminorm.toZeroHom_eq_coe
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.127014\nR : Type ?u.127017\nR' : Type ?u.127020\nE : Type u_1\nF : Type ?u.127026\nG : Type ?u.127029\ninst✝² : AddGroup E\ninst✝¹ : AddGroup F\ninst✝ : AddGroup G\np q : NonarchAddGroupSeminorm E\n⊢ ↑p.toZeroHom = ↑p",
"tactic": "rfl"
}
] |
[
526,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
525,
1
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.succ_lt_succ
|
[] |
[
210,
15
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
209,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
inv_eq_iff_mul_eq_one
|
[] |
[
687,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
686,
1
] |
Mathlib/Order/SuccPred/IntervalSucc.lean
|
Monotone.pairwise_disjoint_on_Ioo_succ
|
[] |
[
75,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean
|
Real.differentiableAt_tan_of_mem_Ioo
|
[] |
[
84,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/Data/List/Basic.lean
|
List.nthLe_cons
|
[
{
"state_after": "case inl\nι : Type ?u.48326\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\na : α\nn : ℕ\nhl : n < length (a :: l)\nh : n = 0\n⊢ nthLe (a :: l) n hl = a\n\ncase inr\nι : Type ?u.48326\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\na : α\nn : ℕ\nhl : n < length (a :: l)\nh : ¬n = 0\n⊢ nthLe (a :: l) n hl = nthLe l (n - 1) (_ : n - 1 < length l)",
"state_before": "ι : Type ?u.48326\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\na : α\nn : ℕ\nhl : n < length (a :: l)\n⊢ nthLe (a :: l) n hl = if hn : n = 0 then a else nthLe l (n - 1) (_ : n - 1 < length l)",
"tactic": "split_ifs with h"
},
{
"state_after": "case inr.nil\nι : Type ?u.48326\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nn : ℕ\nh : ¬n = 0\nhl : n < length [a]\n⊢ nthLe [a] n hl = nthLe [] (n - 1) (_ : n - 1 < length [])\n\ncase inr.cons\nι : Type ?u.48326\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nn : ℕ\nh : ¬n = 0\nhead✝ : α\ntail✝ : List α\nhl : n < length (a :: head✝ :: tail✝)\n⊢ nthLe (a :: head✝ :: tail✝) n hl = nthLe (head✝ :: tail✝) (n - 1) (_ : n - 1 < length (head✝ :: tail✝))",
"state_before": "case inr\nι : Type ?u.48326\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\na : α\nn : ℕ\nhl : n < length (a :: l)\nh : ¬n = 0\n⊢ nthLe (a :: l) n hl = nthLe l (n - 1) (_ : n - 1 < length l)",
"tactic": "cases l"
},
{
"state_after": "case inr.cons.zero\nι : Type ?u.48326\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na head✝ : α\ntail✝ : List α\nh : ¬zero = 0\nhl : zero < length (a :: head✝ :: tail✝)\n⊢ nthLe (a :: head✝ :: tail✝) zero hl = nthLe (head✝ :: tail✝) (zero - 1) (_ : zero - 1 < length (head✝ :: tail✝))\n\ncase inr.cons.succ\nι : Type ?u.48326\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na head✝ : α\ntail✝ : List α\nn✝ : ℕ\nh : ¬succ n✝ = 0\nhl : succ n✝ < length (a :: head✝ :: tail✝)\n⊢ nthLe (a :: head✝ :: tail✝) (succ n✝) hl =\n nthLe (head✝ :: tail✝) (succ n✝ - 1) (_ : succ n✝ - 1 < length (head✝ :: tail✝))",
"state_before": "case inr.cons\nι : Type ?u.48326\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nn : ℕ\nh : ¬n = 0\nhead✝ : α\ntail✝ : List α\nhl : n < length (a :: head✝ :: tail✝)\n⊢ nthLe (a :: head✝ :: tail✝) n hl = nthLe (head✝ :: tail✝) (n - 1) (_ : n - 1 < length (head✝ :: tail✝))",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case inr.cons.succ\nι : Type ?u.48326\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na head✝ : α\ntail✝ : List α\nn✝ : ℕ\nh : ¬succ n✝ = 0\nhl : succ n✝ < length (a :: head✝ :: tail✝)\n⊢ nthLe (a :: head✝ :: tail✝) (succ n✝) hl =\n nthLe (head✝ :: tail✝) (succ n✝ - 1) (_ : succ n✝ - 1 < length (head✝ :: tail✝))",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Type ?u.48326\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\na : α\nn : ℕ\nhl : n < length (a :: l)\nh : n = 0\n⊢ nthLe (a :: l) n hl = a",
"tactic": "simp [nthLe, h]"
},
{
"state_after": "case inr.nil\nι : Type ?u.48326\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nn : ℕ\nh : ¬n = 0\nhl✝ : n < length [a]\nhl : n = 0\n⊢ nthLe [a] n hl✝ = nthLe [] (n - 1) (_ : n - 1 < length [])",
"state_before": "case inr.nil\nι : Type ?u.48326\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nn : ℕ\nh : ¬n = 0\nhl : n < length [a]\n⊢ nthLe [a] n hl = nthLe [] (n - 1) (_ : n - 1 < length [])",
"tactic": "rw [length_singleton, lt_succ_iff, nonpos_iff_eq_zero] at hl"
},
{
"state_after": "no goals",
"state_before": "case inr.nil\nι : Type ?u.48326\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na : α\nn : ℕ\nh : ¬n = 0\nhl✝ : n < length [a]\nhl : n = 0\n⊢ nthLe [a] n hl✝ = nthLe [] (n - 1) (_ : n - 1 < length [])",
"tactic": "contradiction"
},
{
"state_after": "no goals",
"state_before": "case inr.cons.zero\nι : Type ?u.48326\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\na head✝ : α\ntail✝ : List α\nh : ¬zero = 0\nhl : zero < length (a :: head✝ :: tail✝)\n⊢ nthLe (a :: head✝ :: tail✝) zero hl = nthLe (head✝ :: tail✝) (zero - 1) (_ : zero - 1 < length (head✝ :: tail✝))",
"tactic": "contradiction"
}
] |
[
971,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
962,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.le_log2
|
[
{
"state_after": "no goals",
"state_before": "n k : Nat\nh : n ≠ 0\n⊢ 0 ≤ log2 n ↔ 2 ^ 0 ≤ n",
"tactic": "simp [show 1 ≤ n from Nat.pos_of_ne_zero h]"
},
{
"state_after": "n k✝ : Nat\nh : n ≠ 0\nk : Nat\n⊢ (k + 1 ≤ if n ≥ 2 then log2 (n / 2) + 1 else 0) ↔ 2 ^ (k + 1) ≤ n",
"state_before": "n k✝ : Nat\nh : n ≠ 0\nk : Nat\n⊢ k + 1 ≤ log2 n ↔ 2 ^ (k + 1) ≤ n",
"tactic": "rw [log2]"
},
{
"state_after": "case inl\nn k✝ : Nat\nh : n ≠ 0\nk : Nat\nh✝ : n ≥ 2\n⊢ k + 1 ≤ log2 (n / 2) + 1 ↔ 2 ^ (k + 1) ≤ n\n\ncase inr\nn k✝ : Nat\nh : n ≠ 0\nk : Nat\nh✝ : ¬n ≥ 2\n⊢ k + 1 ≤ 0 ↔ 2 ^ (k + 1) ≤ n",
"state_before": "n k✝ : Nat\nh : n ≠ 0\nk : Nat\n⊢ (k + 1 ≤ if n ≥ 2 then log2 (n / 2) + 1 else 0) ↔ 2 ^ (k + 1) ≤ n",
"tactic": "split"
},
{
"state_after": "case inl\nn k✝ : Nat\nh : n ≠ 0\nk : Nat\nh✝ : n ≥ 2\nn0 : 0 < n / 2\n⊢ k + 1 ≤ log2 (n / 2) + 1 ↔ 2 ^ (k + 1) ≤ n",
"state_before": "case inl\nn k✝ : Nat\nh : n ≠ 0\nk : Nat\nh✝ : n ≥ 2\n⊢ k + 1 ≤ log2 (n / 2) + 1 ↔ 2 ^ (k + 1) ≤ n",
"tactic": "have n0 : 0 < n / 2 := (Nat.le_div_iff_mul_le (by decide)).2 ‹_›"
},
{
"state_after": "no goals",
"state_before": "case inl\nn k✝ : Nat\nh : n ≠ 0\nk : Nat\nh✝ : n ≥ 2\nn0 : 0 < n / 2\n⊢ k + 1 ≤ log2 (n / 2) + 1 ↔ 2 ^ (k + 1) ≤ n",
"tactic": "simp [Nat.add_le_add_iff_le_right, le_log2 (Nat.ne_of_gt n0), le_div_iff_mul_le, Nat.pow_succ]"
},
{
"state_after": "no goals",
"state_before": "n k✝ : Nat\nh : n ≠ 0\nk : Nat\nh✝ : n ≥ 2\n⊢ 0 < 2",
"tactic": "decide"
},
{
"state_after": "case inr\nn k✝ : Nat\nh : n ≠ 0\nk : Nat\nh✝ : ¬n ≥ 2\n⊢ ¬2 ^ (k + 1) ≤ n",
"state_before": "case inr\nn k✝ : Nat\nh : n ≠ 0\nk : Nat\nh✝ : ¬n ≥ 2\n⊢ k + 1 ≤ 0 ↔ 2 ^ (k + 1) ≤ n",
"tactic": "simp only [le_zero_eq, succ_ne_zero, false_iff]"
},
{
"state_after": "case inr\nn k✝ : Nat\nh : n ≠ 0\nk : Nat\nh✝ : ¬n ≥ 2\n⊢ 2 ≤ 2 ^ (k + 1)",
"state_before": "case inr\nn k✝ : Nat\nh : n ≠ 0\nk : Nat\nh✝ : ¬n ≥ 2\n⊢ ¬2 ^ (k + 1) ≤ n",
"tactic": "refine mt (Nat.le_trans ?_) ‹_›"
},
{
"state_after": "no goals",
"state_before": "case inr\nn k✝ : Nat\nh : n ≠ 0\nk : Nat\nh✝ : ¬n ≥ 2\n⊢ 2 ≤ 2 ^ (k + 1)",
"tactic": "exact Nat.pow_le_pow_of_le_right (Nat.succ_pos 1) (Nat.le_add_left 1 k)"
}
] |
[
820,
78
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
811,
1
] |
Mathlib/FieldTheory/IsAlgClosed/Spectrum.lean
|
spectrum.exists_mem_of_not_isUnit_aeval_prod
|
[
{
"state_after": "R : Type u\nA : Type v\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : IsDomain R\np : R[X]\na : A\nh : ¬IsUnit (List.prod (List.map (↑(aeval a)) (Multiset.toList (Multiset.map (fun x => X - ↑C x) (roots p)))))\n⊢ ∃ k, k ∈ σ a ∧ eval k p = 0",
"state_before": "R : Type u\nA : Type v\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : IsDomain R\np : R[X]\na : A\nh : ¬IsUnit (↑(aeval a) (Multiset.prod (Multiset.map (fun x => X - ↑C x) (roots p))))\n⊢ ∃ k, k ∈ σ a ∧ eval k p = 0",
"tactic": "rw [← Multiset.prod_toList, AlgHom.map_list_prod] at h"
},
{
"state_after": "R : Type u\nA : Type v\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : IsDomain R\np : R[X]\na : A\nh :\n ¬∀ (m : (fun x => A) (List.prod (Multiset.toList (Multiset.map (fun x => X - ↑C x) (roots p))))),\n m ∈ List.map (↑(aeval a)) (Multiset.toList (Multiset.map (fun x => X - ↑C x) (roots p))) → IsUnit m\n⊢ ∃ k, k ∈ σ a ∧ eval k p = 0",
"state_before": "R : Type u\nA : Type v\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : IsDomain R\np : R[X]\na : A\nh : ¬IsUnit (List.prod (List.map (↑(aeval a)) (Multiset.toList (Multiset.map (fun x => X - ↑C x) (roots p)))))\n⊢ ∃ k, k ∈ σ a ∧ eval k p = 0",
"tactic": "replace h := mt List.prod_isUnit h"
},
{
"state_after": "R : Type u\nA : Type v\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : IsDomain R\np : R[X]\na : A\nh : ∃ a_1, a_1 ∈ roots p ∧ ¬IsUnit (a - ↑↑ₐ a_1)\n⊢ ∃ k, k ∈ σ a ∧ eval k p = 0",
"state_before": "R : Type u\nA : Type v\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : IsDomain R\np : R[X]\na : A\nh :\n ¬∀ (m : (fun x => A) (List.prod (Multiset.toList (Multiset.map (fun x => X - ↑C x) (roots p))))),\n m ∈ List.map (↑(aeval a)) (Multiset.toList (Multiset.map (fun x => X - ↑C x) (roots p))) → IsUnit m\n⊢ ∃ k, k ∈ σ a ∧ eval k p = 0",
"tactic": "simp only [not_forall, exists_prop, aeval_C, Multiset.mem_toList, List.mem_map, aeval_X,\n exists_exists_and_eq_and, Multiset.mem_map, AlgHom.map_sub] at h"
},
{
"state_after": "case intro.intro\nR : Type u\nA : Type v\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : IsDomain R\np : R[X]\na : A\nr : R\nr_mem : r ∈ roots p\nr_nu : ¬IsUnit (a - ↑↑ₐ r)\n⊢ ∃ k, k ∈ σ a ∧ eval k p = 0",
"state_before": "R : Type u\nA : Type v\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : IsDomain R\np : R[X]\na : A\nh : ∃ a_1, a_1 ∈ roots p ∧ ¬IsUnit (a - ↑↑ₐ a_1)\n⊢ ∃ k, k ∈ σ a ∧ eval k p = 0",
"tactic": "rcases h with ⟨r, r_mem, r_nu⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u\nA : Type v\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : IsDomain R\np : R[X]\na : A\nr : R\nr_mem : r ∈ roots p\nr_nu : ¬IsUnit (a - ↑↑ₐ r)\n⊢ ∃ k, k ∈ σ a ∧ eval k p = 0",
"tactic": "exact ⟨r, by rwa [mem_iff, ← IsUnit.sub_iff], (mem_roots'.1 r_mem).2⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nA : Type v\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : IsDomain R\np : R[X]\na : A\nr : R\nr_mem : r ∈ roots p\nr_nu : ¬IsUnit (a - ↑↑ₐ r)\n⊢ r ∈ σ a",
"tactic": "rwa [mem_iff, ← IsUnit.sub_iff]"
}
] |
[
69,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
isIrreducible_singleton
|
[] |
[
1715,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1714,
1
] |
Mathlib/Data/Nat/Hyperoperation.lean
|
hyperoperation_ge_two_eq_self
|
[
{
"state_after": "case zero\nm : ℕ\n⊢ hyperoperation (Nat.zero + 2) m 1 = m\n\ncase succ\nm nn : ℕ\nnih : hyperoperation (nn + 2) m 1 = m\n⊢ hyperoperation (Nat.succ nn + 2) m 1 = m",
"state_before": "n m : ℕ\n⊢ hyperoperation (n + 2) m 1 = m",
"tactic": "induction' n with nn nih"
},
{
"state_after": "case zero\nm : ℕ\n⊢ (fun x x_1 => x * x_1) m 1 = m",
"state_before": "case zero\nm : ℕ\n⊢ hyperoperation (Nat.zero + 2) m 1 = m",
"tactic": "rw [hyperoperation_two]"
},
{
"state_after": "no goals",
"state_before": "case zero\nm : ℕ\n⊢ (fun x x_1 => x * x_1) m 1 = m",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "case succ\nm nn : ℕ\nnih : hyperoperation (nn + 2) m 1 = m\n⊢ hyperoperation (Nat.succ nn + 2) m 1 = m",
"tactic": "rw [hyperoperation_recursion, hyperoperation_ge_three_eq_one, nih]"
}
] |
[
100,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/Data/FinEnum.lean
|
FinEnum.mem_toList
|
[
{
"state_after": "α : Type u\nβ : α → Type v\ninst✝ : FinEnum α\nx : α\n⊢ ∃ a, ↑Equiv.symm a = x",
"state_before": "α : Type u\nβ : α → Type v\ninst✝ : FinEnum α\nx : α\n⊢ x ∈ toList α",
"tactic": "simp [toList]"
},
{
"state_after": "α : Type u\nβ : α → Type v\ninst✝ : FinEnum α\nx : α\n⊢ ↑Equiv.symm (↑Equiv x) = x",
"state_before": "α : Type u\nβ : α → Type v\ninst✝ : FinEnum α\nx : α\n⊢ ∃ a, ↑Equiv.symm a = x",
"tactic": "exists Equiv x"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\ninst✝ : FinEnum α\nx : α\n⊢ ↑Equiv.symm (↑Equiv x) = x",
"tactic": "simp"
}
] |
[
77,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/Data/List/Duplicate.lean
|
List.Duplicate.ne_singleton
|
[
{
"state_after": "case cons_mem\nα : Type u_1\nl : List α\nx y : α\nl' : List α\nh : x ∈ l'\n⊢ x :: l' ≠ [y]\n\ncase cons_duplicate\nα : Type u_1\nl : List α\nx y z : α\nl' : List α\nh : x ∈+ l'\na_ih✝ : l' ≠ [y]\n⊢ z :: l' ≠ [y]",
"state_before": "α : Type u_1\nl : List α\nx : α\nh : x ∈+ l\ny : α\n⊢ l ≠ [y]",
"tactic": "induction' h with l' h z l' h _"
},
{
"state_after": "no goals",
"state_before": "case cons_mem\nα : Type u_1\nl : List α\nx y : α\nl' : List α\nh : x ∈ l'\n⊢ x :: l' ≠ [y]",
"tactic": "simp [ne_nil_of_mem h]"
},
{
"state_after": "no goals",
"state_before": "case cons_duplicate\nα : Type u_1\nl : List α\nx y z : α\nl' : List α\nh : x ∈+ l'\na_ih✝ : l' ≠ [y]\n⊢ z :: l' ≠ [y]",
"tactic": "simp [ne_nil_of_mem h.mem]"
}
] |
[
77,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.L1.SimpleFunc.setToL1S_mono_left
|
[] |
[
836,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
834,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
|
CategoryTheory.Limits.coprodComparison_inv_natural
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁷ : Category C\nX Y : C\nD : Type u₂\ninst✝⁶ : Category D\nF : C ⥤ D\nA A' B B' : C\ninst✝⁵ : HasBinaryCoproduct A B\ninst✝⁴ : HasBinaryCoproduct A' B'\ninst✝³ : HasBinaryCoproduct (F.obj A) (F.obj B)\ninst✝² : HasBinaryCoproduct (F.obj A') (F.obj B')\nf : A ⟶ A'\ng : B ⟶ B'\ninst✝¹ : IsIso (coprodComparison F A B)\ninst✝ : IsIso (coprodComparison F A' B')\n⊢ inv (coprodComparison F A B) ≫ coprod.map (F.map f) (F.map g) =\n F.map (coprod.map f g) ≫ inv (coprodComparison F A' B')",
"tactic": "rw [IsIso.eq_comp_inv, Category.assoc, IsIso.inv_comp_eq, coprodComparison_natural]"
}
] |
[
1388,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1384,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
Padic.coe_sub
|
[] |
[
556,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
555,
1
] |
Mathlib/Data/List/AList.lean
|
AList.insertRec_insert_mk
|
[] |
[
394,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
390,
1
] |
Mathlib/Algebra/Lie/Abelian.lean
|
LieModule.maxTrivEquiv_of_equiv_symm_eq_symm
|
[] |
[
211,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
209,
1
] |
Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean
|
Gamma1_in_Gamma0
|
[
{
"state_after": "N✝ N : ℕ\nx : SL(2, ℤ)\nHA : x ∈ Gamma1 N\n⊢ x ∈ Gamma0 N",
"state_before": "N✝ N : ℕ\n⊢ Gamma1 N ≤ Gamma0 N",
"tactic": "intro x HA"
},
{
"state_after": "N✝ N : ℕ\nx : SL(2, ℤ)\nHA : ↑(↑x 0 0) = 1 ∧ ↑(↑x 1 1) = 1 ∧ ↑(↑x 1 0) = 0\n⊢ ↑(↑x 1 0) = 0",
"state_before": "N✝ N : ℕ\nx : SL(2, ℤ)\nHA : x ∈ Gamma1 N\n⊢ x ∈ Gamma0 N",
"tactic": "simp only [Gamma0_mem, Gamma1_mem, coe_matrix_coe, Int.coe_castRingHom, map_apply] at *"
},
{
"state_after": "no goals",
"state_before": "N✝ N : ℕ\nx : SL(2, ℤ)\nHA : ↑(↑x 0 0) = 1 ∧ ↑(↑x 1 1) = 1 ∧ ↑(↑x 1 0) = 0\n⊢ ↑(↑x 1 0) = 0",
"tactic": "exact HA.2.2"
}
] |
[
203,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
ConvexOn.sub
|
[] |
[
888,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
887,
1
] |
Mathlib/AlgebraicGeometry/StructureSheaf.lean
|
AlgebraicGeometry.StructureSheaf.to_global_factors
|
[
{
"state_after": "R : Type u\ninst✝ : CommRing R\n⊢ toOpen R ⊤ =\n (CommRingCat.ofHom (algebraMap R (Localization.Away 1)) ≫ toBasicOpen R 1) ≫\n (structureSheaf R).val.map (eqToHom (_ : ⊤ = PrimeSpectrum.basicOpen 1)).op",
"state_before": "R : Type u\ninst✝ : CommRing R\n⊢ toOpen R ⊤ =\n CommRingCat.ofHom (algebraMap R (Localization.Away 1)) ≫\n toBasicOpen R 1 ≫ (structureSheaf R).val.map (eqToHom (_ : ⊤ = PrimeSpectrum.basicOpen 1)).op",
"tactic": "rw [← Category.assoc]"
},
{
"state_after": "R : Type u\ninst✝ : CommRing R\n⊢ toOpen R ⊤ =\n CommRingCat.ofHom (RingHom.comp (toBasicOpen R 1) (algebraMap R (Localization.Away 1))) ≫\n (structureSheaf R).val.map (eqToHom (_ : ⊤ = PrimeSpectrum.basicOpen 1)).op",
"state_before": "R : Type u\ninst✝ : CommRing R\n⊢ toOpen R ⊤ =\n (CommRingCat.ofHom (algebraMap R (Localization.Away 1)) ≫ toBasicOpen R 1) ≫\n (structureSheaf R).val.map (eqToHom (_ : ⊤ = PrimeSpectrum.basicOpen 1)).op",
"tactic": "change toOpen R ⊤ =\n (CommRingCat.ofHom <| (toBasicOpen R 1).comp (algebraMap R (Localization.Away 1))) ≫\n (structureSheaf R).1.map (eqToHom _).op"
},
{
"state_after": "R : Type u\ninst✝ : CommRing R\n⊢ toOpen R ⊤ =\n RingHom.comp (toBasicOpen R 1) (algebraMap R (Localization.Away 1)) ≫\n (structureSheaf R).val.map (eqToHom (_ : ⊤ = PrimeSpectrum.basicOpen 1)).op",
"state_before": "R : Type u\ninst✝ : CommRing R\n⊢ toOpen R ⊤ =\n CommRingCat.ofHom (RingHom.comp (toBasicOpen R 1) (algebraMap R (Localization.Away 1))) ≫\n (structureSheaf R).val.map (eqToHom (_ : ⊤ = PrimeSpectrum.basicOpen 1)).op",
"tactic": "unfold CommRingCat.ofHom"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\n⊢ toOpen R ⊤ =\n RingHom.comp (toBasicOpen R 1) (algebraMap R (Localization.Away 1)) ≫\n (structureSheaf R).val.map (eqToHom (_ : ⊤ = PrimeSpectrum.basicOpen 1)).op",
"tactic": "rw [localization_toBasicOpen R, toOpen_res]"
}
] |
[
996,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
986,
1
] |
Mathlib/Order/Lattice.lean
|
ofDual_inf
|
[] |
[
953,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
952,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
StarSubalgebra.map_le_iff_le_comap
|
[] |
[
245,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/Data/Seq/Seq.lean
|
Stream'.Seq.get?_cons_zero
|
[] |
[
94,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Finset.lean
|
Finset.Nonempty.cSup_mem
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.8944\nγ : Type ?u.8947\ninst✝ : ConditionallyCompleteLinearOrder α\ns✝ t : Set α\na b : α\ns : Finset α\nh : Finset.Nonempty s\n⊢ max' s h ∈ s",
"state_before": "α : Type u_1\nβ : Type ?u.8944\nγ : Type ?u.8947\ninst✝ : ConditionallyCompleteLinearOrder α\ns✝ t : Set α\na b : α\ns : Finset α\nh : Finset.Nonempty s\n⊢ sSup ↑s ∈ s",
"tactic": "rw [h.cSup_eq_max']"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.8944\nγ : Type ?u.8947\ninst✝ : ConditionallyCompleteLinearOrder α\ns✝ t : Set α\na b : α\ns : Finset α\nh : Finset.Nonempty s\n⊢ max' s h ∈ s",
"tactic": "exact s.max'_mem _"
}
] |
[
54,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Order/CompleteLattice.lean
|
biSup_prod
|
[
{
"state_after": "α : Type u_3\nβ : Type u_1\nβ₂ : Type ?u.166426\nγ : Type u_2\nι : Sort ?u.166432\nι' : Sort ?u.166435\nκ : ι → Sort ?u.166440\nκ' : ι' → Sort ?u.166445\ninst✝ : CompleteLattice α\nf✝ g s✝ t✝ : ι → α\na b : α\nf : β × γ → α\ns : Set β\nt : Set γ\n⊢ (⨆ (i : β) (j : γ) (_ : i ∈ s) (_ : j ∈ t), f (i, j)) = ⨆ (a : β) (_ : a ∈ s) (b : γ) (_ : b ∈ t), f (a, b)",
"state_before": "α : Type u_3\nβ : Type u_1\nβ₂ : Type ?u.166426\nγ : Type u_2\nι : Sort ?u.166432\nι' : Sort ?u.166435\nκ : ι → Sort ?u.166440\nκ' : ι' → Sort ?u.166445\ninst✝ : CompleteLattice α\nf✝ g s✝ t✝ : ι → α\na b : α\nf : β × γ → α\ns : Set β\nt : Set γ\n⊢ (⨆ (x : β × γ) (_ : x ∈ s ×ˢ t), f x) = ⨆ (a : β) (_ : a ∈ s) (b : γ) (_ : b ∈ t), f (a, b)",
"tactic": "simp_rw [iSup_prod, mem_prod, iSup_and]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_1\nβ₂ : Type ?u.166426\nγ : Type u_2\nι : Sort ?u.166432\nι' : Sort ?u.166435\nκ : ι → Sort ?u.166440\nκ' : ι' → Sort ?u.166445\ninst✝ : CompleteLattice α\nf✝ g s✝ t✝ : ι → α\na b : α\nf : β × γ → α\ns : Set β\nt : Set γ\n⊢ (⨆ (i : β) (j : γ) (_ : i ∈ s) (_ : j ∈ t), f (i, j)) = ⨆ (a : β) (_ : a ∈ s) (b : γ) (_ : b ∈ t), f (a, b)",
"tactic": "exact iSup_congr fun _ => iSup_comm"
}
] |
[
1559,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1556,
1
] |
Mathlib/Algebra/Order/Group/Abs.lean
|
abs_pos
|
[
{
"state_after": "case inl\nα : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na b c : α\nha : a < 0\n⊢ 0 < abs a ↔ a ≠ 0\n\ncase inr.inl\nα : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nb c : α\n⊢ 0 < abs 0 ↔ 0 ≠ 0\n\ncase inr.inr\nα : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na b c : α\nha : 0 < a\n⊢ 0 < abs a ↔ a ≠ 0",
"state_before": "α : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na b c : α\n⊢ 0 < abs a ↔ a ≠ 0",
"tactic": "rcases lt_trichotomy a 0 with (ha | rfl | ha)"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na b c : α\nha : a < 0\n⊢ 0 < abs a ↔ a ≠ 0",
"tactic": "simp [abs_of_neg ha, neg_pos, ha.ne, ha]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nα : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nb c : α\n⊢ 0 < abs 0 ↔ 0 ≠ 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nα : Type u_1\ninst✝² : AddGroup α\ninst✝¹ : LinearOrder α\ninst✝ : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na b c : α\nha : 0 < a\n⊢ 0 < abs a ↔ a ≠ 0",
"tactic": "simp [abs_of_pos ha, ha, ha.ne.symm]"
}
] |
[
140,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
expSeries_hasSum_exp_of_mem_ball
|
[] |
[
231,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
1
] |
Mathlib/Combinatorics/HalesJewett.lean
|
Combinatorics.Line.map_apply
|
[
{
"state_after": "α : Type u_1\nα' : Type u_2\nι : Type u_3\nf : α → α'\nl : Line α ι\nx : α\n⊢ (fun i => f (Option.getD (idxFun l i) x)) = f ∘ fun i => Option.getD (idxFun l i) x",
"state_before": "α : Type u_1\nα' : Type u_2\nι : Type u_3\nf : α → α'\nl : Line α ι\nx : α\n⊢ (fun x i => Option.getD (idxFun (map f l) i) x) (f x) = f ∘ (fun x i => Option.getD (idxFun l i) x) x",
"tactic": "simp only [Line.apply, Line.map, Option.getD_map]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nα' : Type u_2\nι : Type u_3\nf : α → α'\nl : Line α ι\nx : α\n⊢ (fun i => f (Option.getD (idxFun l i) x)) = f ∘ fun i => Option.getD (idxFun l i) x",
"tactic": "rfl"
}
] |
[
196,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Algebra/Ring/BooleanRing.lean
|
BooleanRing.le_sup_inf
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.18933\nγ : Type ?u.18936\ninst✝² : BooleanRing α\ninst✝¹ : BooleanRing β\ninst✝ : BooleanRing γ\na b c : α\n⊢ (a + b + a * b) * (a + c + a * c) + (a + b * c + a * (b * c)) +\n (a + b + a * b) * (a + c + a * c) * (a + b * c + a * (b * c)) =\n a + b * c + a * (b * c)",
"tactic": "rw [le_sup_inf_aux, add_self, mul_self, zero_add]"
}
] |
[
240,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Mathlib/Data/Option/NAry.lean
|
Option.map₂_some_some
|
[] |
[
54,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.map_algebraMap
|
[
{
"state_after": "l : Type ?u.584026\nm : Type ?u.584029\nn : Type u_2\no : Type ?u.584035\nm' : o → Type ?u.584040\nn' : o → Type ?u.584045\nR : Type u_1\nS : Type ?u.584051\nα : Type v\nβ : Type w\nγ : Type ?u.584058\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq n\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring α\ninst✝² : Semiring β\ninst✝¹ : Algebra R α\ninst✝ : Algebra R β\nr : R\nf : α → β\nhf : f 0 = 0\nhf₂ : f (↑(algebraMap R α) r) = ↑(algebraMap R β) r\n⊢ (diagonal fun m => f (↑(algebraMap R (n → α)) r m)) = diagonal (↑(algebraMap R (n → β)) r)",
"state_before": "l : Type ?u.584026\nm : Type ?u.584029\nn : Type u_2\no : Type ?u.584035\nm' : o → Type ?u.584040\nn' : o → Type ?u.584045\nR : Type u_1\nS : Type ?u.584051\nα : Type v\nβ : Type w\nγ : Type ?u.584058\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq n\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring α\ninst✝² : Semiring β\ninst✝¹ : Algebra R α\ninst✝ : Algebra R β\nr : R\nf : α → β\nhf : f 0 = 0\nhf₂ : f (↑(algebraMap R α) r) = ↑(algebraMap R β) r\n⊢ map (↑(algebraMap R (Matrix n n α)) r) f = ↑(algebraMap R (Matrix n n β)) r",
"tactic": "rw [algebraMap_eq_diagonal, algebraMap_eq_diagonal, diagonal_map hf]"
},
{
"state_after": "l : Type ?u.584026\nm : Type ?u.584029\nn : Type u_2\no : Type ?u.584035\nm' : o → Type ?u.584040\nn' : o → Type ?u.584045\nR : Type u_1\nS : Type ?u.584051\nα : Type v\nβ : Type w\nγ : Type ?u.584058\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq n\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring α\ninst✝² : Semiring β\ninst✝¹ : Algebra R α\ninst✝ : Algebra R β\nr : R\nf : α → β\nhf : f 0 = 0\nhf₂ : f (↑(algebraMap R α) r) = ↑(algebraMap R β) r\n⊢ n → f (↑(algebraMap R α) r) = ↑(algebraMap R β) r",
"state_before": "l : Type ?u.584026\nm : Type ?u.584029\nn : Type u_2\no : Type ?u.584035\nm' : o → Type ?u.584040\nn' : o → Type ?u.584045\nR : Type u_1\nS : Type ?u.584051\nα : Type v\nβ : Type w\nγ : Type ?u.584058\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq n\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring α\ninst✝² : Semiring β\ninst✝¹ : Algebra R α\ninst✝ : Algebra R β\nr : R\nf : α → β\nhf : f 0 = 0\nhf₂ : f (↑(algebraMap R α) r) = ↑(algebraMap R β) r\n⊢ (diagonal fun m => f (↑(algebraMap R (n → α)) r m)) = diagonal (↑(algebraMap R (n → β)) r)",
"tactic": "simp only [Pi.algebraMap_apply, diagonal_eq_diagonal_iff]"
},
{
"state_after": "l : Type ?u.584026\nm : Type ?u.584029\nn : Type u_2\no : Type ?u.584035\nm' : o → Type ?u.584040\nn' : o → Type ?u.584045\nR : Type u_1\nS : Type ?u.584051\nα : Type v\nβ : Type w\nγ : Type ?u.584058\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq n\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring α\ninst✝² : Semiring β\ninst✝¹ : Algebra R α\ninst✝ : Algebra R β\nr : R\nf : α → β\nhf : f 0 = 0\nhf₂ : f (↑(algebraMap R α) r) = ↑(algebraMap R β) r\ni✝ : n\n⊢ f (↑(algebraMap R α) r) = ↑(algebraMap R β) r",
"state_before": "l : Type ?u.584026\nm : Type ?u.584029\nn : Type u_2\no : Type ?u.584035\nm' : o → Type ?u.584040\nn' : o → Type ?u.584045\nR : Type u_1\nS : Type ?u.584051\nα : Type v\nβ : Type w\nγ : Type ?u.584058\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq n\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring α\ninst✝² : Semiring β\ninst✝¹ : Algebra R α\ninst✝ : Algebra R β\nr : R\nf : α → β\nhf : f 0 = 0\nhf₂ : f (↑(algebraMap R α) r) = ↑(algebraMap R β) r\n⊢ n → f (↑(algebraMap R α) r) = ↑(algebraMap R β) r",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.584026\nm : Type ?u.584029\nn : Type u_2\no : Type ?u.584035\nm' : o → Type ?u.584040\nn' : o → Type ?u.584045\nR : Type u_1\nS : Type ?u.584051\nα : Type v\nβ : Type w\nγ : Type ?u.584058\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq n\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring α\ninst✝² : Semiring β\ninst✝¹ : Algebra R α\ninst✝ : Algebra R β\nr : R\nf : α → β\nhf : f 0 = 0\nhf₂ : f (↑(algebraMap R α) r) = ↑(algebraMap R β) r\ni✝ : n\n⊢ f (↑(algebraMap R α) r) = ↑(algebraMap R β) r",
"tactic": "rw [hf₂]"
}
] |
[
1346,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1334,
1
] |
Mathlib/Analysis/Calculus/Deriv/Comp.lean
|
derivWithin.scomp
|
[] |
[
115,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.preimage_inter_range
|
[] |
[
842,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
841,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.eval₂Hom_map_hom
|
[] |
[
1334,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1332,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.coe_clamp
|
[] |
[
2496,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2495,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.lt_of_lt_of_equiv
|
[] |
[
841,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
840,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.uIcc_injective_left
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.193752\nα : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : LocallyFiniteOrder α\na✝ a₁ a₂ b b₁ b₂ c x a : α\n⊢ Injective (uIcc a)",
"tactic": "simpa only [uIcc_comm] using uIcc_injective_right a"
}
] |
[
982,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
981,
1
] |
Mathlib/Data/Polynomial/Coeff.lean
|
Polynomial.coeff_mul_X_pow'
|
[
{
"state_after": "case inl\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d : ℕ\nh : n ≤ d\n⊢ coeff (p * X ^ n) d = coeff p (d - n)\n\ncase inr\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d : ℕ\nh : ¬n ≤ d\n⊢ coeff (p * X ^ n) d = 0",
"state_before": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d : ℕ\n⊢ coeff (p * X ^ n) d = if n ≤ d then coeff p (d - n) else 0",
"tactic": "split_ifs with h"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d : ℕ\nh : n ≤ d\n⊢ coeff (p * X ^ n) d = coeff p (d - n)",
"tactic": "rw [← tsub_add_cancel_of_le h, coeff_mul_X_pow, add_tsub_cancel_right]"
},
{
"state_after": "case inr\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d : ℕ\nh : ¬n ≤ d\nx : ℕ × ℕ\nhx : x ∈ Nat.antidiagonal d\n⊢ coeff p x.fst * coeff (X ^ n) x.snd = 0",
"state_before": "case inr\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d : ℕ\nh : ¬n ≤ d\n⊢ coeff (p * X ^ n) d = 0",
"tactic": "refine' (coeff_mul _ _ _).trans (Finset.sum_eq_zero fun x hx => _)"
},
{
"state_after": "case inr.hnc\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d : ℕ\nh : ¬n ≤ d\nx : ℕ × ℕ\nhx : x ∈ Nat.antidiagonal d\n⊢ ¬x.snd = n",
"state_before": "case inr\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d : ℕ\nh : ¬n ≤ d\nx : ℕ × ℕ\nhx : x ∈ Nat.antidiagonal d\n⊢ coeff p x.fst * coeff (X ^ n) x.snd = 0",
"tactic": "rw [coeff_X_pow, if_neg, mul_zero]"
},
{
"state_after": "no goals",
"state_before": "case inr.hnc\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d : ℕ\nh : ¬n ≤ d\nx : ℕ × ℕ\nhx : x ∈ Nat.antidiagonal d\n⊢ ¬x.snd = n",
"tactic": "exact ((le_of_add_le_right (Finset.Nat.mem_antidiagonal.mp hx).le).trans_lt <| not_le.mp h).ne"
}
] |
[
255,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
1
] |
Mathlib/GroupTheory/Submonoid/Operations.lean
|
Submonoid.bot_or_nontrivial
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_1\nN : Type ?u.166126\nP : Type ?u.166129\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\ninst✝¹ : MulOneClass P\nS✝ : Submonoid M\nA : Type ?u.166150\ninst✝ : SetLike A M\nhA : SubmonoidClass A M\nS' : A\nS : Submonoid M\n⊢ S = ⊥ ∨ Nontrivial { x // x ∈ S }",
"tactic": "simp only [eq_bot_iff_forall, nontrivial_iff_exists_ne_one, ← not_forall, ← not_imp, Classical.em]"
}
] |
[
1380,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1379,
1
] |
Mathlib/Algebra/Hom/GroupInstances.lean
|
AddMonoidHom.coe_mul
|
[] |
[
317,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
316,
1
] |
Mathlib/Algebra/Associated.lean
|
associated_one_iff_isUnit
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.150247\nγ : Type ?u.150250\nδ : Type ?u.150253\ninst✝ : Monoid α\na : α\nx✝ : IsUnit a\nc : αˣ\nh : ↑c = a\n⊢ 1 * ↑c = a",
"tactic": "simp [h]"
}
] |
[
433,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
428,
1
] |
Mathlib/Algebra/Order/Group/MinMax.lean
|
min_div_div_left'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedCommGroup α\na✝ b✝ c✝ a b c : α\n⊢ min (a / b) (a / c) = a / max b c",
"tactic": "simp only [div_eq_mul_inv, min_mul_mul_left, min_inv_inv']"
}
] |
[
68,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.const_injective
|
[] |
[
55,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Algebra/Module/LocalizedModule.lean
|
IsLocalizedModule.mk'_add
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁷ : CommRing R\nS : Submonoid R\nM : Type u_3\nM' : Type u_2\nM'' : Type ?u.847575\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid M'\ninst✝⁴ : AddCommMonoid M''\ninst✝³ : Module R M\ninst✝² : Module R M'\ninst✝¹ : Module R M''\nf : M →ₗ[R] M'\ng : M →ₗ[R] M''\ninst✝ : IsLocalizedModule S f\nm₁ m₂ : M\ns : { x // x ∈ S }\n⊢ mk' f (m₁ + m₂) s = mk' f m₁ s + mk' f m₂ s",
"tactic": "rw [mk'_add_mk', ← smul_add, mk'_cancel_left]"
}
] |
[
973,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
972,
1
] |
Mathlib/Algebra/Order/Monoid/WithTop.lean
|
WithTop.untop_one
|
[] |
[
53,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Order/SuccPred/Basic.lean
|
Order.succ_ne_succ_iff
|
[] |
[
487,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
486,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
IsClosed.rightCoset
|
[] |
[
146,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Std/Data/RBMap/Alter.lean
|
Std.RBNode.Path.insertNew_eq_insert
|
[] |
[
70,
41
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
68,
1
] |
Mathlib/RingTheory/Finiteness.lean
|
Module.Finite.equiv
|
[] |
[
612,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
611,
1
] |
Mathlib/Topology/Sets/Compacts.lean
|
TopologicalSpace.CompactOpens.coe_compl
|
[] |
[
569,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
568,
1
] |
Mathlib/LinearAlgebra/Dimension.lean
|
Submodule.rank_sup_add_rank_inf_eq
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1002876\ninst✝¹⁰ : DivisionRing K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\ninst✝⁷ : AddCommGroup V'\ninst✝⁶ : Module K V'\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module K V₁\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\ns t : Submodule K V\n⊢ ⊤ ≤ LinearMap.range (ofLe (_ : s ≤ s ⊔ t)) ⊔ LinearMap.range (ofLe (_ : t ≤ s ⊔ t))",
"tactic": "rw [← map_le_map_iff' (ker_subtype <| s ⊔ t), Submodule.map_sup, Submodule.map_top, ←\n LinearMap.range_comp, ← LinearMap.range_comp, subtype_comp_ofLe, subtype_comp_ofLe,\n range_subtype, range_subtype, range_subtype]"
},
{
"state_after": "case h.mk.a\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1002876\ninst✝¹⁰ : DivisionRing K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\ninst✝⁷ : AddCommGroup V'\ninst✝⁶ : Module K V'\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module K V₁\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\ns t : Submodule K V\nx : V\nhx : x ∈ s ⊓ t\n⊢ ↑(↑(LinearMap.comp (ofLe (_ : s ≤ s ⊔ t)) (ofLe (_ : s ⊓ t ≤ s))) { val := x, property := hx }) =\n ↑(↑(LinearMap.comp (ofLe (_ : t ≤ s ⊔ t)) (ofLe (_ : s ⊓ t ≤ t))) { val := x, property := hx })",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1002876\ninst✝¹⁰ : DivisionRing K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\ninst✝⁷ : AddCommGroup V'\ninst✝⁶ : Module K V'\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module K V₁\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\ns t : Submodule K V\n⊢ LinearMap.comp (ofLe (_ : s ≤ s ⊔ t)) (ofLe (_ : s ⊓ t ≤ s)) =\n LinearMap.comp (ofLe (_ : t ≤ s ⊔ t)) (ofLe (_ : s ⊓ t ≤ t))",
"tactic": "ext ⟨x, hx⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mk.a\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1002876\ninst✝¹⁰ : DivisionRing K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\ninst✝⁷ : AddCommGroup V'\ninst✝⁶ : Module K V'\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module K V₁\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\ns t : Submodule K V\nx : V\nhx : x ∈ s ⊓ t\n⊢ ↑(↑(LinearMap.comp (ofLe (_ : s ≤ s ⊔ t)) (ofLe (_ : s ⊓ t ≤ s))) { val := x, property := hx }) =\n ↑(↑(LinearMap.comp (ofLe (_ : t ≤ s ⊔ t)) (ofLe (_ : s ⊓ t ≤ t))) { val := x, property := hx })",
"tactic": "rfl"
},
{
"state_after": "case mk.mk\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1002876\ninst✝¹⁰ : DivisionRing K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\ninst✝⁷ : AddCommGroup V'\ninst✝⁶ : Module K V'\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module K V₁\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\ns t : Submodule K V\nb₁ : V\nhb₁ : b₁ ∈ s\nb₂ : V\nhb₂ : b₂ ∈ t\neq : ↑(ofLe (_ : s ≤ s ⊔ t)) { val := b₁, property := hb₁ } = ↑(ofLe (_ : t ≤ s ⊔ t)) { val := b₂, property := hb₂ }\n⊢ ∃ c,\n ↑(ofLe (_ : s ⊓ t ≤ s)) c = { val := b₁, property := hb₁ } ∧\n ↑(ofLe (_ : s ⊓ t ≤ t)) c = { val := b₂, property := hb₂ }",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1002876\ninst✝¹⁰ : DivisionRing K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\ninst✝⁷ : AddCommGroup V'\ninst✝⁶ : Module K V'\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module K V₁\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\ns t : Submodule K V\n⊢ ∀ (d : { x // x ∈ s }) (e : { x // x ∈ t }),\n ↑(ofLe (_ : s ≤ s ⊔ t)) d = ↑(ofLe (_ : t ≤ s ⊔ t)) e →\n ∃ c, ↑(ofLe (_ : s ⊓ t ≤ s)) c = d ∧ ↑(ofLe (_ : s ⊓ t ≤ t)) c = e",
"tactic": "rintro ⟨b₁, hb₁⟩ ⟨b₂, hb₂⟩ eq"
},
{
"state_after": "case mk.mk\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1002876\ninst✝¹⁰ : DivisionRing K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\ninst✝⁷ : AddCommGroup V'\ninst✝⁶ : Module K V'\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module K V₁\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\ns t : Submodule K V\nb₁ : V\nhb₁ : b₁ ∈ s\nhb₂ : b₁ ∈ t\neq : ↑(ofLe (_ : s ≤ s ⊔ t)) { val := b₁, property := hb₁ } = ↑(ofLe (_ : t ≤ s ⊔ t)) { val := b₁, property := hb₂ }\n⊢ ∃ c,\n ↑(ofLe (_ : s ⊓ t ≤ s)) c = { val := b₁, property := hb₁ } ∧\n ↑(ofLe (_ : s ⊓ t ≤ t)) c = { val := b₁, property := hb₂ }",
"state_before": "case mk.mk\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1002876\ninst✝¹⁰ : DivisionRing K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\ninst✝⁷ : AddCommGroup V'\ninst✝⁶ : Module K V'\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module K V₁\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\ns t : Submodule K V\nb₁ : V\nhb₁ : b₁ ∈ s\nb₂ : V\nhb₂ : b₂ ∈ t\neq : ↑(ofLe (_ : s ≤ s ⊔ t)) { val := b₁, property := hb₁ } = ↑(ofLe (_ : t ≤ s ⊔ t)) { val := b₂, property := hb₂ }\n⊢ ∃ c,\n ↑(ofLe (_ : s ⊓ t ≤ s)) c = { val := b₁, property := hb₁ } ∧\n ↑(ofLe (_ : s ⊓ t ≤ t)) c = { val := b₂, property := hb₂ }",
"tactic": "obtain rfl : b₁ = b₂ := congr_arg Subtype.val eq"
},
{
"state_after": "no goals",
"state_before": "case mk.mk\nK : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1002876\ninst✝¹⁰ : DivisionRing K\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module K V\ninst✝⁷ : AddCommGroup V'\ninst✝⁶ : Module K V'\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module K V₁\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : AddCommGroup V₃\ninst✝ : Module K V₃\ns t : Submodule K V\nb₁ : V\nhb₁ : b₁ ∈ s\nhb₂ : b₁ ∈ t\neq : ↑(ofLe (_ : s ≤ s ⊔ t)) { val := b₁, property := hb₁ } = ↑(ofLe (_ : t ≤ s ⊔ t)) { val := b₁, property := hb₂ }\n⊢ ∃ c,\n ↑(ofLe (_ : s ⊓ t ≤ s)) c = { val := b₁, property := hb₁ } ∧\n ↑(ofLe (_ : s ⊓ t ≤ t)) c = { val := b₁, property := hb₂ }",
"tactic": "exact ⟨⟨b₁, hb₁, hb₂⟩, rfl, rfl⟩"
}
] |
[
1152,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1140,
1
] |
Mathlib/Analysis/Normed/Field/UnitBall.lean
|
coe_one_unitSphere
|
[] |
[
161,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/Algebra/Hom/GroupInstances.lean
|
MonoidHom.compr₂_apply
|
[] |
[
279,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/Algebra/Order/Positive/Ring.lean
|
Positive.val_one
|
[] |
[
130,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean
|
Euclidean.isCompact_closedBall
|
[
{
"state_after": "E : Type u_1\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : T2Space E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : FiniteDimensional ℝ E\nx : E\nr : ℝ\n⊢ IsCompact (↑(ContinuousLinearEquiv.symm toEuclidean) '' Metric.closedBall (↑toEuclidean x) r)",
"state_before": "E : Type u_1\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : T2Space E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : FiniteDimensional ℝ E\nx : E\nr : ℝ\n⊢ IsCompact (closedBall x r)",
"tactic": "rw [closedBall_eq_image]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalAddGroup E\ninst✝³ : T2Space E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : FiniteDimensional ℝ E\nx : E\nr : ℝ\n⊢ IsCompact (↑(ContinuousLinearEquiv.symm toEuclidean) '' Metric.closedBall (↑toEuclidean x) r)",
"tactic": "exact (isCompact_closedBall _ _).image toEuclidean.symm.continuous"
}
] |
[
92,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
8
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
inner_map_complex
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.4079769\nE : Type ?u.4079772\nF : Type ?u.4079775\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nG : Type u_1\ninst✝¹ : NormedAddCommGroup G\ninst✝ : InnerProductSpace ℝ G\nf : G ≃ₗᵢ[ℝ] ℂ\nx y : G\n⊢ inner x y = (↑(starRingEnd ℂ) (↑f x) * ↑f y).re",
"tactic": "rw [← Complex.inner, f.inner_map_map]"
}
] |
[
2241,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2240,
1
] |
Mathlib/Algebra/IsPrimePow.lean
|
IsPrimePow.one_lt
|
[] |
[
128,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.coe_mk
|
[] |
[
187,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
LinearIsometryEquiv.contDiff
|
[] |
[
168,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean
|
lt_of_mul_lt_of_one_le_of_nonneg_left
|
[] |
[
915,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
913,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.coe_nonpos
|
[] |
[
393,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
392,
11
] |
Mathlib/Data/Nat/Choose/Basic.lean
|
Nat.choose_le_succ_of_lt_half_left
|
[
{
"state_after": "r n : ℕ\nh : r < n / 2\n⊢ choose n r * (n - r) ≤ choose n (r + 1) * (n - r)",
"state_before": "r n : ℕ\nh : r < n / 2\n⊢ choose n r ≤ choose n (r + 1)",
"tactic": "refine' le_of_mul_le_mul_right _ (lt_tsub_iff_left.mpr (lt_of_lt_of_le h (n.div_le_self 2)))"
},
{
"state_after": "r n : ℕ\nh : r < n / 2\n⊢ choose n (r + 1) * (r + 1) ≤ choose n (r + 1) * (n - r)",
"state_before": "r n : ℕ\nh : r < n / 2\n⊢ choose n r * (n - r) ≤ choose n (r + 1) * (n - r)",
"tactic": "rw [← choose_succ_right_eq]"
},
{
"state_after": "case h\nr n : ℕ\nh : r < n / 2\n⊢ r + 1 ≤ n - r",
"state_before": "r n : ℕ\nh : r < n / 2\n⊢ choose n (r + 1) * (r + 1) ≤ choose n (r + 1) * (n - r)",
"tactic": "apply Nat.mul_le_mul_left"
},
{
"state_after": "case h\nr n : ℕ\nh : r < n / 2\n⊢ r * 2 < n",
"state_before": "case h\nr n : ℕ\nh : r < n / 2\n⊢ r + 1 ≤ n - r",
"tactic": "rw [← Nat.lt_iff_add_one_le, lt_tsub_iff_left, ← mul_two]"
},
{
"state_after": "no goals",
"state_before": "case h\nr n : ℕ\nh : r < n / 2\n⊢ r * 2 < n",
"tactic": "exact lt_of_lt_of_le (mul_lt_mul_of_pos_right h zero_lt_two) (n.div_mul_le_self 2)"
}
] |
[
286,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
1
] |
Mathlib/GroupTheory/Sylow.lean
|
card_sylow_modEq_one
|
[
{
"state_after": "p : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fintype (Sylow p G)\nP : Sylow p G\n⊢ card (Sylow p G) ≡ 1 [MOD p]",
"state_before": "p : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fintype (Sylow p G)\n⊢ card (Sylow p G) ≡ 1 [MOD p]",
"tactic": "refine' Sylow.nonempty.elim fun P : Sylow p G => _"
},
{
"state_after": "p : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fintype (Sylow p G)\nP : Sylow p G\nthis : fixedPoints { x // x ∈ ↑P } (Sylow p G) = {P}\n⊢ card (Sylow p G) ≡ 1 [MOD p]",
"state_before": "p : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fintype (Sylow p G)\nP : Sylow p G\n⊢ card (Sylow p G) ≡ 1 [MOD p]",
"tactic": "have : fixedPoints P.1 (Sylow p G) = {P} :=\n Set.ext fun Q : Sylow p G =>\n calc\n Q ∈ fixedPoints P (Sylow p G) ↔ P.1 ≤ Q := P.2.sylow_mem_fixedPoints_iff\n _ ↔ Q.1 = P.1 := ⟨P.3 Q.2, ge_of_eq⟩\n _ ↔ Q ∈ {P} := Sylow.ext_iff.symm.trans Set.mem_singleton_iff.symm"
},
{
"state_after": "p : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fintype (Sylow p G)\nP : Sylow p G\nthis : fixedPoints { x // x ∈ ↑P } (Sylow p G) = {P}\nfin : Fintype ↑(fixedPoints { x // x ∈ ↑P } (Sylow p G))\n⊢ card (Sylow p G) ≡ 1 [MOD p]",
"state_before": "p : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fintype (Sylow p G)\nP : Sylow p G\nthis : fixedPoints { x // x ∈ ↑P } (Sylow p G) = {P}\n⊢ card (Sylow p G) ≡ 1 [MOD p]",
"tactic": "have fin : Fintype (fixedPoints P.1 (Sylow p G)) := by\n rw [this]\n infer_instance"
},
{
"state_after": "p : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fintype (Sylow p G)\nP : Sylow p G\nthis✝ : fixedPoints { x // x ∈ ↑P } (Sylow p G) = {P}\nfin : Fintype ↑(fixedPoints { x // x ∈ ↑P } (Sylow p G))\nthis : card ↑(fixedPoints { x // x ∈ ↑P } (Sylow p G)) = 1\n⊢ card (Sylow p G) ≡ 1 [MOD p]",
"state_before": "p : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fintype (Sylow p G)\nP : Sylow p G\nthis : fixedPoints { x // x ∈ ↑P } (Sylow p G) = {P}\nfin : Fintype ↑(fixedPoints { x // x ∈ ↑P } (Sylow p G))\n⊢ card (Sylow p G) ≡ 1 [MOD p]",
"tactic": "have : card (fixedPoints P.1 (Sylow p G)) = 1 := by simp [this]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fintype (Sylow p G)\nP : Sylow p G\nthis✝ : fixedPoints { x // x ∈ ↑P } (Sylow p G) = {P}\nfin : Fintype ↑(fixedPoints { x // x ∈ ↑P } (Sylow p G))\nthis : card ↑(fixedPoints { x // x ∈ ↑P } (Sylow p G)) = 1\n⊢ card (Sylow p G) ≡ 1 [MOD p]",
"tactic": "exact (P.2.card_modEq_card_fixedPoints (Sylow p G)).trans (by rw [this])"
},
{
"state_after": "p : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fintype (Sylow p G)\nP : Sylow p G\nthis : fixedPoints { x // x ∈ ↑P } (Sylow p G) = {P}\n⊢ Fintype ↑{P}",
"state_before": "p : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fintype (Sylow p G)\nP : Sylow p G\nthis : fixedPoints { x // x ∈ ↑P } (Sylow p G) = {P}\n⊢ Fintype ↑(fixedPoints { x // x ∈ ↑P } (Sylow p G))",
"tactic": "rw [this]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fintype (Sylow p G)\nP : Sylow p G\nthis : fixedPoints { x // x ∈ ↑P } (Sylow p G) = {P}\n⊢ Fintype ↑{P}",
"tactic": "infer_instance"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fintype (Sylow p G)\nP : Sylow p G\nthis : fixedPoints { x // x ∈ ↑P } (Sylow p G) = {P}\nfin : Fintype ↑(fixedPoints { x // x ∈ ↑P } (Sylow p G))\n⊢ card ↑(fixedPoints { x // x ∈ ↑P } (Sylow p G)) = 1",
"tactic": "simp [this]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Fintype (Sylow p G)\nP : Sylow p G\nthis✝ : fixedPoints { x // x ∈ ↑P } (Sylow p G) = {P}\nfin : Fintype ↑(fixedPoints { x // x ∈ ↑P } (Sylow p G))\nthis : card ↑(fixedPoints { x // x ∈ ↑P } (Sylow p G)) = 1\n⊢ card ↑(fixedPoints { x // x ∈ ↑P } (Sylow p G)) ≡ 1 [MOD p]",
"tactic": "rw [this]"
}
] |
[
341,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
327,
1
] |
Mathlib/Data/Int/Cast/Lemmas.lean
|
Pi.coe_int
|
[] |
[
361,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
360,
1
] |
Mathlib/CategoryTheory/ConcreteCategory/Basic.lean
|
CategoryTheory.coe_comp
|
[] |
[
136,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.ofNat_mul_negSucc
|
[] |
[
373,
91
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
373,
9
] |
Mathlib/Analysis/Calculus/Series.lean
|
iteratedFDeriv_tsum_apply
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_4\nβ : Type ?u.117040\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x✝ : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nk : ℕ\nhk : ↑k ≤ N\nx : E\n⊢ iteratedFDeriv 𝕜 k (fun y => ∑' (n : α), f n y) x = ∑' (n : α), iteratedFDeriv 𝕜 k (f n) x",
"tactic": "rw [iteratedFDeriv_tsum hf hv h'f hk]"
}
] |
[
225,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
1
] |
Mathlib/Analysis/Convex/Cone/Dual.lean
|
pointed_innerDualCone
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.9011\nE : Type ?u.9014\nF : Type ?u.9017\nG : Type ?u.9020\nH : Type u_1\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\ns t : Set H\nx : H\nx✝ : x ∈ s\n⊢ 0 ≤ inner x 0",
"tactic": "rw [inner_zero_right]"
}
] |
[
85,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Data/Dfinsupp/NeLocus.lean
|
Dfinsupp.neLocus_self_add_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nN : α → Type u_2\ninst✝² : DecidableEq α\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → AddGroup (N a)\nf f₁ f₂ g g₁ g₂ : Π₀ (a : α), N a\n⊢ neLocus f (f + g) = support g",
"tactic": "rw [← neLocus_zero_left, ← @neLocus_add_left α N _ _ _ f 0 g, add_zero]"
}
] |
[
164,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/Order/BoundedOrder.lean
|
exists_le_and_iff_exists
|
[] |
[
615,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
613,
1
] |
Mathlib/NumberTheory/ArithmeticFunction.lean
|
Nat.ArithmeticFunction.pmul_apply
|
[] |
[
513,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
512,
1
] |
Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean
|
AlgebraicTopology.AlternatingFaceMapComplex.obj_d_eq
|
[
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX✝ Y X : SimplicialObject C\nn : ℕ\n⊢ HomologicalComplex.d (obj X) (n + 1) n = ∑ i : Fin (n + 2), (-1) ^ ↑i • SimplicialObject.δ X i",
"tactic": "apply ChainComplex.of_d"
}
] |
[
141,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
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