file_path
stringlengths 11
79
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stringlengths 2
100
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stringclasses 4
values | url
stringclasses 4
values | start
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|---|---|---|---|---|---|---|
Mathlib/CategoryTheory/Types.lean
|
CategoryTheory.epi_iff_surjective
|
[
{
"state_after": "case mp\nX Y : Type u\nf : X ⟶ Y\n⊢ Epi f → Function.Surjective f\n\ncase mpr\nX Y : Type u\nf : X ⟶ Y\n⊢ Function.Surjective f → Epi f",
"state_before": "X Y : Type u\nf : X ⟶ Y\n⊢ Epi f ↔ Function.Surjective f",
"tactic": "constructor"
},
{
"state_after": "case mp.mk\nX Y : Type u\nf : X ⟶ Y\nH : ∀ {Z : Type u} (g h : Y ⟶ Z), f ≫ g = f ≫ h → g = h\n⊢ Function.Surjective f",
"state_before": "case mp\nX Y : Type u\nf : X ⟶ Y\n⊢ Epi f → Function.Surjective f",
"tactic": "rintro ⟨H⟩"
},
{
"state_after": "case mp.mk\nX Y : Type u\nf : X ⟶ Y\nH : ∀ {Z : Type u} (g h : Y ⟶ Z), f ≫ g = f ≫ h → g = h\ng₁ g₂ : Y → Prop\nhg : g₁ ∘ f = g₂ ∘ f\n⊢ g₁ = g₂",
"state_before": "case mp.mk\nX Y : Type u\nf : X ⟶ Y\nH : ∀ {Z : Type u} (g h : Y ⟶ Z), f ≫ g = f ≫ h → g = h\n⊢ Function.Surjective f",
"tactic": "refine' Function.surjective_of_right_cancellable_Prop fun g₁ g₂ hg => _"
},
{
"state_after": "case mp.mk\nX Y : Type u\nf : X ⟶ Y\nH : ∀ {Z : Type u} (g h : Y ⟶ Z), f ≫ g = f ≫ h → g = h\ng₁ g₂ : Y → Prop\nhg : g₁ ∘ f = g₂ ∘ f\n⊢ (fun x x_1 => x ∘ x_1) (↑Equiv.ulift.symm) g₁ = (fun x x_1 => x ∘ x_1) (↑Equiv.ulift.symm) g₂",
"state_before": "case mp.mk\nX Y : Type u\nf : X ⟶ Y\nH : ∀ {Z : Type u} (g h : Y ⟶ Z), f ≫ g = f ≫ h → g = h\ng₁ g₂ : Y → Prop\nhg : g₁ ∘ f = g₂ ∘ f\n⊢ g₁ = g₂",
"tactic": "rw [← Equiv.ulift.symm.injective.comp_left.eq_iff]"
},
{
"state_after": "case mp.mk.a\nX Y : Type u\nf : X ⟶ Y\nH : ∀ {Z : Type u} (g h : Y ⟶ Z), f ≫ g = f ≫ h → g = h\ng₁ g₂ : Y → Prop\nhg : g₁ ∘ f = g₂ ∘ f\n⊢ f ≫ (fun x x_1 => x ∘ x_1) (↑Equiv.ulift.symm) g₁ = f ≫ (fun x x_1 => x ∘ x_1) (↑Equiv.ulift.symm) g₂",
"state_before": "case mp.mk\nX Y : Type u\nf : X ⟶ Y\nH : ∀ {Z : Type u} (g h : Y ⟶ Z), f ≫ g = f ≫ h → g = h\ng₁ g₂ : Y → Prop\nhg : g₁ ∘ f = g₂ ∘ f\n⊢ (fun x x_1 => x ∘ x_1) (↑Equiv.ulift.symm) g₁ = (fun x x_1 => x ∘ x_1) (↑Equiv.ulift.symm) g₂",
"tactic": "apply H"
},
{
"state_after": "case mp.mk.a\nX Y : Type u\nf : X ⟶ Y\nH : ∀ {Z : Type u} (g h : Y ⟶ Z), f ≫ g = f ≫ h → g = h\ng₁ g₂ : Y → Prop\nhg : g₁ ∘ f = g₂ ∘ f\n⊢ ULift.up ∘ g₁ ∘ f = ULift.up ∘ g₂ ∘ f",
"state_before": "case mp.mk.a\nX Y : Type u\nf : X ⟶ Y\nH : ∀ {Z : Type u} (g h : Y ⟶ Z), f ≫ g = f ≫ h → g = h\ng₁ g₂ : Y → Prop\nhg : g₁ ∘ f = g₂ ∘ f\n⊢ f ≫ (fun x x_1 => x ∘ x_1) (↑Equiv.ulift.symm) g₁ = f ≫ (fun x x_1 => x ∘ x_1) (↑Equiv.ulift.symm) g₂",
"tactic": "change ULift.up ∘ g₁ ∘ f = ULift.up ∘ g₂ ∘ f"
},
{
"state_after": "no goals",
"state_before": "case mp.mk.a\nX Y : Type u\nf : X ⟶ Y\nH : ∀ {Z : Type u} (g h : Y ⟶ Z), f ≫ g = f ≫ h → g = h\ng₁ g₂ : Y → Prop\nhg : g₁ ∘ f = g₂ ∘ f\n⊢ ULift.up ∘ g₁ ∘ f = ULift.up ∘ g₂ ∘ f",
"tactic": "rw [hg]"
},
{
"state_after": "no goals",
"state_before": "case mpr\nX Y : Type u\nf : X ⟶ Y\n⊢ Function.Surjective f → Epi f",
"tactic": "exact fun H => ⟨fun g g' h => H.injective_comp_right h⟩"
}
] |
[
268,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
Associates.map_subtype_coe_factors'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na : α\n⊢ map Subtype.val (factors' a) = map Associates.mk (factors a)",
"tactic": "simp [factors', Multiset.map_pmap, Multiset.pmap_eq_map]"
}
] |
[
1405,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1403,
1
] |
Mathlib/Topology/Order/Basic.lean
|
Monotone.map_iSup_of_continuousAt'
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁷ : CompleteLinearOrder α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : CompleteLinearOrder β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderClosedTopology β\ninst✝¹ : Nonempty γ\nι : Sort u_1\ninst✝ : Nonempty ι\nf : α → β\ng : ι → α\nCf : ContinuousAt f (iSup g)\nMf : Monotone f\n⊢ sSup (range (f ∘ g)) = sSup (range fun i => f (g i))",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁷ : CompleteLinearOrder α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : CompleteLinearOrder β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderClosedTopology β\ninst✝¹ : Nonempty γ\nι : Sort u_1\ninst✝ : Nonempty ι\nf : α → β\ng : ι → α\nCf : ContinuousAt f (iSup g)\nMf : Monotone f\n⊢ f (⨆ (i : ι), g i) = ⨆ (i : ι), f (g i)",
"tactic": "rw [iSup, Mf.map_sSup_of_continuousAt' Cf (range_nonempty g), ← range_comp, iSup]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁷ : CompleteLinearOrder α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : CompleteLinearOrder β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderClosedTopology β\ninst✝¹ : Nonempty γ\nι : Sort u_1\ninst✝ : Nonempty ι\nf : α → β\ng : ι → α\nCf : ContinuousAt f (iSup g)\nMf : Monotone f\n⊢ sSup (range (f ∘ g)) = sSup (range fun i => f (g i))",
"tactic": "rfl"
}
] |
[
2666,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2664,
1
] |
Mathlib/Order/Cover.lean
|
Prod.mk_wcovby_mk_iff_right
|
[] |
[
525,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
524,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean
|
LinearPMap.map_zero
|
[] |
[
81,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
uniformity_subtype
|
[] |
[
1476,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1474,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.preimage_mul_left_one
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.676324\nα : Type u_1\nβ : Type ?u.676330\nγ : Type ?u.676333\ninst✝ : Group α\ns t : Finset α\na b : α\n⊢ preimage 1 ((fun x x_1 => x * x_1) a) (_ : Set.InjOn ((fun x x_1 => x * x_1) a) ((fun x x_1 => x * x_1) a ⁻¹' ↑1)) =\n {a⁻¹}",
"tactic": "classical rw [← image_mul_left', image_one, mul_one]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.676324\nα : Type u_1\nβ : Type ?u.676330\nγ : Type ?u.676333\ninst✝ : Group α\ns t : Finset α\na b : α\n⊢ preimage 1 ((fun x x_1 => x * x_1) a) (_ : Set.InjOn ((fun x x_1 => x * x_1) a) ((fun x x_1 => x * x_1) a ⁻¹' ↑1)) =\n {a⁻¹}",
"tactic": "rw [← image_mul_left', image_one, mul_one]"
}
] |
[
1227,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1226,
1
] |
Mathlib/FieldTheory/Subfield.lean
|
SubfieldClass.coe_rat_cast
|
[] |
[
103,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/CategoryTheory/Limits/Constructions/Over/Connected.lean
|
CategoryTheory.Over.CreatesConnected.raised_cone_lowers_to_original
|
[
{
"state_after": "no goals",
"state_before": "J : Type v\ninst✝² : SmallCategory J\nC : Type u\ninst✝¹ : Category C\nX : C\ninst✝ : IsConnected J\nB : C\nF : J ⥤ Over B\nc : Cone (F ⋙ forget B)\n⊢ (forget B).mapCone (raiseCone c) = c",
"tactic": "aesop_cat"
}
] |
[
67,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/LinearAlgebra/Multilinear/Basic.lean
|
MultilinearMap.coe_currySumEquiv
|
[] |
[
1514,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1513,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
mulLECancellable_one
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.83868\ninst✝¹ : Monoid α\ninst✝ : LE α\na b : α\n⊢ 1 * a ≤ 1 * b → a ≤ b",
"tactic": "simpa only [one_mul] using id"
}
] |
[
1595,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1594,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
|
QuadraticForm.associated_isSymm
|
[
{
"state_after": "no goals",
"state_before": "S : Type u_3\nR : Type u_1\nR₁ : Type ?u.403144\nM : Type u_2\ninst✝⁷ : Ring R\ninst✝⁶ : CommRing R₁\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module R₁ M\ninst✝² : CommSemiring S\ninst✝¹ : Algebra S R\ninst✝ : Invertible 2\nB₁ : BilinForm R M\nQ : QuadraticForm R M\nx y : M\n⊢ bilin (↑(associatedHom S) Q) x y = bilin (↑(associatedHom S) Q) y x",
"tactic": "simp only [associated_apply, add_comm, add_left_comm, sub_eq_add_neg, add_assoc]"
}
] |
[
786,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
785,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
Real.differentiable_cosh
|
[] |
[
667,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
667,
1
] |
Mathlib/GroupTheory/Coset.lean
|
QuotientGroup.leftRel_apply
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\n⊢ (∃ a, y * ↑a = x) ↔ ∃ a, x⁻¹ * y = ↑a⁻¹",
"tactic": "simp only [inv_mul_eq_iff_eq_mul, Subgroup.coe_inv, eq_mul_inv_iff_mul_eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\nx y : α\n⊢ (∃ a, x⁻¹ * y = ↑a⁻¹) ↔ x⁻¹ * y ∈ s",
"tactic": "simp [exists_inv_mem_iff_exists_mem]"
}
] |
[
330,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
324,
1
] |
Mathlib/Algebra/FreeAlgebra.lean
|
FreeAlgebra.algebraMap_inj
|
[] |
[
423,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
421,
1
] |
Mathlib/Data/MvPolynomial/Variables.lean
|
MvPolynomial.eval₂Hom_congr'
|
[
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\n⊢ ↑(eval₂Hom f₁ g₁) p₁ = ↑(eval₂Hom f₁ g₂) p₁",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ f₂ : R →+* S\ng₁ g₂ : σ → S\np₁ p₂ : MvPolynomial σ R\n⊢ f₁ = f₂ → (∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₂ → g₁ i = g₂ i) → p₁ = p₂ → ↑(eval₂Hom f₁ g₁) p₁ = ↑(eval₂Hom f₂ g₂) p₂",
"tactic": "rintro rfl h rfl"
},
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\n⊢ ↑(eval₂Hom f₁ g₁) (∑ v in support p₁, ↑(monomial v) (coeff v p₁)) =\n ↑(eval₂Hom f₁ g₂) (∑ v in support p₁, ↑(monomial v) (coeff v p₁))",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\n⊢ ↑(eval₂Hom f₁ g₁) p₁ = ↑(eval₂Hom f₁ g₂) p₁",
"tactic": "rw [p₁.as_sum]"
},
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\n⊢ (∑ x in support p₁, ↑f₁ (coeff x p₁) * Finsupp.prod x fun i k => g₁ i ^ k) =\n ∑ x in support p₁, ↑f₁ (coeff x p₁) * Finsupp.prod x fun i k => g₂ i ^ k",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\n⊢ ↑(eval₂Hom f₁ g₁) (∑ v in support p₁, ↑(monomial v) (coeff v p₁)) =\n ↑(eval₂Hom f₁ g₂) (∑ v in support p₁, ↑(monomial v) (coeff v p₁))",
"tactic": "simp only [map_sum, eval₂Hom_monomial]"
},
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\n⊢ ∀ (x : σ →₀ ℕ),\n x ∈ support p₁ →\n (↑f₁ (coeff x p₁) * Finsupp.prod x fun i k => g₁ i ^ k) = ↑f₁ (coeff x p₁) * Finsupp.prod x fun i k => g₂ i ^ k",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\n⊢ (∑ x in support p₁, ↑f₁ (coeff x p₁) * Finsupp.prod x fun i k => g₁ i ^ k) =\n ∑ x in support p₁, ↑f₁ (coeff x p₁) * Finsupp.prod x fun i k => g₂ i ^ k",
"tactic": "apply Finset.sum_congr rfl"
},
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\nd : σ →₀ ℕ\nhd : d ∈ support p₁\n⊢ (↑f₁ (coeff d p₁) * Finsupp.prod d fun i k => g₁ i ^ k) = ↑f₁ (coeff d p₁) * Finsupp.prod d fun i k => g₂ i ^ k",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\n⊢ ∀ (x : σ →₀ ℕ),\n x ∈ support p₁ →\n (↑f₁ (coeff x p₁) * Finsupp.prod x fun i k => g₁ i ^ k) = ↑f₁ (coeff x p₁) * Finsupp.prod x fun i k => g₂ i ^ k",
"tactic": "intro d hd"
},
{
"state_after": "case e_a\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\nd : σ →₀ ℕ\nhd : d ∈ support p₁\n⊢ (Finsupp.prod d fun i k => g₁ i ^ k) = Finsupp.prod d fun i k => g₂ i ^ k",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\nd : σ →₀ ℕ\nhd : d ∈ support p₁\n⊢ (↑f₁ (coeff d p₁) * Finsupp.prod d fun i k => g₁ i ^ k) = ↑f₁ (coeff d p₁) * Finsupp.prod d fun i k => g₂ i ^ k",
"tactic": "congr 1"
},
{
"state_after": "case e_a\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\nd : σ →₀ ℕ\nhd : d ∈ support p₁\n⊢ ∏ x in d.support, g₁ x ^ ↑d x = ∏ x in d.support, g₂ x ^ ↑d x",
"state_before": "case e_a\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\nd : σ →₀ ℕ\nhd : d ∈ support p₁\n⊢ (Finsupp.prod d fun i k => g₁ i ^ k) = Finsupp.prod d fun i k => g₂ i ^ k",
"tactic": "simp only [Finsupp.prod]"
},
{
"state_after": "case e_a\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\nd : σ →₀ ℕ\nhd : d ∈ support p₁\n⊢ ∀ (x : σ), x ∈ d.support → g₁ x ^ ↑d x = g₂ x ^ ↑d x",
"state_before": "case e_a\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\nd : σ →₀ ℕ\nhd : d ∈ support p₁\n⊢ ∏ x in d.support, g₁ x ^ ↑d x = ∏ x in d.support, g₂ x ^ ↑d x",
"tactic": "apply Finset.prod_congr rfl"
},
{
"state_after": "case e_a\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\nd : σ →₀ ℕ\nhd : d ∈ support p₁\ni : σ\nhi : i ∈ d.support\n⊢ g₁ i ^ ↑d i = g₂ i ^ ↑d i",
"state_before": "case e_a\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\nd : σ →₀ ℕ\nhd : d ∈ support p₁\n⊢ ∀ (x : σ), x ∈ d.support → g₁ x ^ ↑d x = g₂ x ^ ↑d x",
"tactic": "intro i hi"
},
{
"state_after": "case e_a\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\nd : σ →₀ ℕ\nhd : d ∈ support p₁\ni : σ\nhi : i ∈ d.support\nthis : i ∈ vars p₁\n⊢ g₁ i ^ ↑d i = g₂ i ^ ↑d i",
"state_before": "case e_a\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\nd : σ →₀ ℕ\nhd : d ∈ support p₁\ni : σ\nhi : i ∈ d.support\n⊢ g₁ i ^ ↑d i = g₂ i ^ ↑d i",
"tactic": "have : i ∈ p₁.vars := by\n rw [mem_vars]\n exact ⟨d, hd, hi⟩"
},
{
"state_after": "no goals",
"state_before": "case e_a\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\nd : σ →₀ ℕ\nhd : d ∈ support p₁\ni : σ\nhi : i ∈ d.support\nthis : i ∈ vars p₁\n⊢ g₁ i ^ ↑d i = g₂ i ^ ↑d i",
"tactic": "rw [h i this this]"
},
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\nd : σ →₀ ℕ\nhd : d ∈ support p₁\ni : σ\nhi : i ∈ d.support\n⊢ ∃ d x, i ∈ d.support",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\nd : σ →₀ ℕ\nhd : d ∈ support p₁\ni : σ\nhi : i ∈ d.support\n⊢ i ∈ vars p₁",
"tactic": "rw [mem_vars]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.629877\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf₁ : R →+* S\ng₁ g₂ : σ → S\np₁ : MvPolynomial σ R\nh : ∀ (i : σ), i ∈ vars p₁ → i ∈ vars p₁ → g₁ i = g₂ i\nd : σ →₀ ℕ\nhd : d ∈ support p₁\ni : σ\nhi : i ∈ d.support\n⊢ ∃ d x, i ∈ d.support",
"tactic": "exact ⟨d, hd, hi⟩"
}
] |
[
857,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
841,
1
] |
Mathlib/Topology/VectorBundle/Basic.lean
|
Trivialization.mk_coordChangeL
|
[
{
"state_after": "case h₁\nR : Type u_4\nB : Type u_1\nF : Type u_2\nE : B → Type u_3\ninst✝⁹ : Semiring R\ninst✝⁸ : TopologicalSpace F\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : TopologicalSpace (TotalSpace E)\ne✝ : Trivialization F TotalSpace.proj\nx : TotalSpace E\nb✝ : B\ny✝ : E b✝\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module R F\ninst✝³ : (x : B) → AddCommMonoid (E x)\ninst✝² : (x : B) → Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear R e\ninst✝ : Trivialization.IsLinear R e'\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\ny : F\n⊢ (b, ↑(coordChangeL R e e' b) y).fst = (↑e' (totalSpaceMk b (Trivialization.symm e b y))).fst\n\ncase h₂\nR : Type u_4\nB : Type u_1\nF : Type u_2\nE : B → Type u_3\ninst✝⁹ : Semiring R\ninst✝⁸ : TopologicalSpace F\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : TopologicalSpace (TotalSpace E)\ne✝ : Trivialization F TotalSpace.proj\nx : TotalSpace E\nb✝ : B\ny✝ : E b✝\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module R F\ninst✝³ : (x : B) → AddCommMonoid (E x)\ninst✝² : (x : B) → Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear R e\ninst✝ : Trivialization.IsLinear R e'\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\ny : F\n⊢ (b, ↑(coordChangeL R e e' b) y).snd = (↑e' (totalSpaceMk b (Trivialization.symm e b y))).snd",
"state_before": "R : Type u_4\nB : Type u_1\nF : Type u_2\nE : B → Type u_3\ninst✝⁹ : Semiring R\ninst✝⁸ : TopologicalSpace F\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : TopologicalSpace (TotalSpace E)\ne✝ : Trivialization F TotalSpace.proj\nx : TotalSpace E\nb✝ : B\ny✝ : E b✝\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module R F\ninst✝³ : (x : B) → AddCommMonoid (E x)\ninst✝² : (x : B) → Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear R e\ninst✝ : Trivialization.IsLinear R e'\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\ny : F\n⊢ (b, ↑(coordChangeL R e e' b) y) = ↑e' (totalSpaceMk b (Trivialization.symm e b y))",
"tactic": "ext"
},
{
"state_after": "case h₁\nR : Type u_4\nB : Type u_1\nF : Type u_2\nE : B → Type u_3\ninst✝⁹ : Semiring R\ninst✝⁸ : TopologicalSpace F\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : TopologicalSpace (TotalSpace E)\ne✝ : Trivialization F TotalSpace.proj\nx : TotalSpace E\nb✝ : B\ny✝ : E b✝\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module R F\ninst✝³ : (x : B) → AddCommMonoid (E x)\ninst✝² : (x : B) → Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear R e\ninst✝ : Trivialization.IsLinear R e'\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\ny : F\n⊢ TotalSpace.proj (↑(LocalHomeomorph.symm e.toLocalHomeomorph) (b, y)) ∈ e'.baseSet",
"state_before": "case h₁\nR : Type u_4\nB : Type u_1\nF : Type u_2\nE : B → Type u_3\ninst✝⁹ : Semiring R\ninst✝⁸ : TopologicalSpace F\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : TopologicalSpace (TotalSpace E)\ne✝ : Trivialization F TotalSpace.proj\nx : TotalSpace E\nb✝ : B\ny✝ : E b✝\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module R F\ninst✝³ : (x : B) → AddCommMonoid (E x)\ninst✝² : (x : B) → Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear R e\ninst✝ : Trivialization.IsLinear R e'\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\ny : F\n⊢ (b, ↑(coordChangeL R e e' b) y).fst = (↑e' (totalSpaceMk b (Trivialization.symm e b y))).fst",
"tactic": "rw [e.mk_symm hb.1 y, e'.coe_fst', e.proj_symm_apply' hb.1]"
},
{
"state_after": "case h₁\nR : Type u_4\nB : Type u_1\nF : Type u_2\nE : B → Type u_3\ninst✝⁹ : Semiring R\ninst✝⁸ : TopologicalSpace F\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : TopologicalSpace (TotalSpace E)\ne✝ : Trivialization F TotalSpace.proj\nx : TotalSpace E\nb✝ : B\ny✝ : E b✝\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module R F\ninst✝³ : (x : B) → AddCommMonoid (E x)\ninst✝² : (x : B) → Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear R e\ninst✝ : Trivialization.IsLinear R e'\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\ny : F\n⊢ b ∈ e'.baseSet",
"state_before": "case h₁\nR : Type u_4\nB : Type u_1\nF : Type u_2\nE : B → Type u_3\ninst✝⁹ : Semiring R\ninst✝⁸ : TopologicalSpace F\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : TopologicalSpace (TotalSpace E)\ne✝ : Trivialization F TotalSpace.proj\nx : TotalSpace E\nb✝ : B\ny✝ : E b✝\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module R F\ninst✝³ : (x : B) → AddCommMonoid (E x)\ninst✝² : (x : B) → Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear R e\ninst✝ : Trivialization.IsLinear R e'\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\ny : F\n⊢ TotalSpace.proj (↑(LocalHomeomorph.symm e.toLocalHomeomorph) (b, y)) ∈ e'.baseSet",
"tactic": "rw [e.proj_symm_apply' hb.1]"
},
{
"state_after": "no goals",
"state_before": "case h₁\nR : Type u_4\nB : Type u_1\nF : Type u_2\nE : B → Type u_3\ninst✝⁹ : Semiring R\ninst✝⁸ : TopologicalSpace F\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : TopologicalSpace (TotalSpace E)\ne✝ : Trivialization F TotalSpace.proj\nx : TotalSpace E\nb✝ : B\ny✝ : E b✝\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module R F\ninst✝³ : (x : B) → AddCommMonoid (E x)\ninst✝² : (x : B) → Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear R e\ninst✝ : Trivialization.IsLinear R e'\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\ny : F\n⊢ b ∈ e'.baseSet",
"tactic": "exact hb.2"
},
{
"state_after": "no goals",
"state_before": "case h₂\nR : Type u_4\nB : Type u_1\nF : Type u_2\nE : B → Type u_3\ninst✝⁹ : Semiring R\ninst✝⁸ : TopologicalSpace F\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : TopologicalSpace (TotalSpace E)\ne✝ : Trivialization F TotalSpace.proj\nx : TotalSpace E\nb✝ : B\ny✝ : E b✝\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module R F\ninst✝³ : (x : B) → AddCommMonoid (E x)\ninst✝² : (x : B) → Module R (E x)\ne e' : Trivialization F TotalSpace.proj\ninst✝¹ : Trivialization.IsLinear R e\ninst✝ : Trivialization.IsLinear R e'\nb : B\nhb : b ∈ e.baseSet ∩ e'.baseSet\ny : F\n⊢ (b, ↑(coordChangeL R e e' b) y).snd = (↑e' (totalSpaceMk b (Trivialization.symm e b y))).snd",
"tactic": "exact e.coordChangeL_apply e' hb y"
}
] |
[
342,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
335,
1
] |
Mathlib/Topology/Order/LowerTopology.lean
|
WithLowerTopology.of_withLowerTopology_symm_eq
|
[] |
[
82,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/RingTheory/Localization/Integral.lean
|
IsLocalization.integerNormalization_spec
|
[
{
"state_after": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.15519\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\np : S[X]\n⊢ ∀ (i : ℕ),\n ↑(algebraMap R S) (coeff (integerNormalization M p) i) =\n ↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) • coeff p i",
"state_before": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.15519\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\np : S[X]\n⊢ ∃ b, ∀ (i : ℕ), ↑(algebraMap R S) (coeff (integerNormalization M p) i) = ↑b • coeff p i",
"tactic": "use Classical.choose (exist_integer_multiples_of_finset M (p.support.image p.coeff))"
},
{
"state_after": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.15519\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\np : S[X]\ni : ℕ\n⊢ ↑(algebraMap R S) (coeff (integerNormalization M p) i) =\n ↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) • coeff p i",
"state_before": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.15519\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\np : S[X]\n⊢ ∀ (i : ℕ),\n ↑(algebraMap R S) (coeff (integerNormalization M p) i) =\n ↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) • coeff p i",
"tactic": "intro i"
},
{
"state_after": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.15519\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\np : S[X]\ni : ℕ\n⊢ ↑(algebraMap R S)\n (if hi : i ∈ support p then\n choose\n (_ :\n IsInteger R\n (↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) •\n coeff p i))\n else 0) =\n ↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) • coeff p i",
"state_before": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.15519\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\np : S[X]\ni : ℕ\n⊢ ↑(algebraMap R S) (coeff (integerNormalization M p) i) =\n ↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) • coeff p i",
"tactic": "rw [integerNormalization_coeff, coeffIntegerNormalization]"
},
{
"state_after": "case inl\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.15519\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\np : S[X]\ni : ℕ\nhi : i ∈ support p\n⊢ ↑(algebraMap R S)\n (if hi : i ∈ support p then\n choose\n (_ :\n IsInteger R\n (↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) •\n coeff p i))\n else 0) =\n ↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) • coeff p i\n\ncase inr\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.15519\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\np : S[X]\ni : ℕ\nhi : ¬i ∈ support p\n⊢ ↑(algebraMap R S)\n (if hi : i ∈ support p then\n choose\n (_ :\n IsInteger R\n (↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) •\n coeff p i))\n else 0) =\n ↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) • coeff p i",
"state_before": "R : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.15519\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\np : S[X]\ni : ℕ\n⊢ ↑(algebraMap R S)\n (if hi : i ∈ support p then\n choose\n (_ :\n IsInteger R\n (↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) •\n coeff p i))\n else 0) =\n ↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) • coeff p i",
"tactic": "split_ifs with hi"
},
{
"state_after": "case inl\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.15519\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\np : S[X]\ni : ℕ\nhi : i ∈ support p\n⊢ ↑(algebraMap R S)\n (choose\n (_ :\n IsInteger R\n (↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) •\n coeff p i))) =\n ↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) • coeff p i",
"state_before": "case inl\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.15519\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\np : S[X]\ni : ℕ\nhi : i ∈ support p\n⊢ ↑(algebraMap R S)\n (if hi : i ∈ support p then\n choose\n (_ :\n IsInteger R\n (↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) •\n coeff p i))\n else 0) =\n ↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) • coeff p i",
"tactic": "rw [dif_pos hi]"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.15519\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\np : S[X]\ni : ℕ\nhi : i ∈ support p\n⊢ ↑(algebraMap R S)\n (choose\n (_ :\n IsInteger R\n (↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) •\n coeff p i))) =\n ↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) • coeff p i",
"tactic": "exact\n Classical.choose_spec\n (Classical.choose_spec (exist_integer_multiples_of_finset M (p.support.image p.coeff))\n (p.coeff i) (Finset.mem_image.mpr ⟨i, hi, rfl⟩))"
},
{
"state_after": "case inr\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.15519\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\np : S[X]\ni : ℕ\nhi : ¬i ∈ support p\n⊢ ↑(algebraMap R S) 0 =\n ↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) • coeff p i",
"state_before": "case inr\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.15519\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\np : S[X]\ni : ℕ\nhi : ¬i ∈ support p\n⊢ ↑(algebraMap R S)\n (if hi : i ∈ support p then\n choose\n (_ :\n IsInteger R\n (↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) •\n coeff p i))\n else 0) =\n ↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) • coeff p i",
"tactic": "rw [dif_neg hi]"
},
{
"state_after": "no goals",
"state_before": "case inr\nR : Type u_2\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.15519\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\np : S[X]\ni : ℕ\nhi : ¬i ∈ support p\n⊢ ↑(algebraMap R S) 0 =\n ↑(choose (_ : ∃ b, ∀ (a : S), a ∈ Finset.image (coeff p) (support p) → IsInteger R (↑b • a))) • coeff p i",
"tactic": "rw [RingHom.map_zero, not_mem_support_iff.mp hi, smul_zero]"
}
] |
[
96,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean
|
Real.hasStrictDerivAt_tan
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\nh : cos x ≠ 0\n⊢ HasStrictDerivAt tan (↑1 / cos x ^ 2) x",
"tactic": "exact_mod_cast (Complex.hasStrictDerivAt_tan (by exact_mod_cast h)).real_of_complex"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nh : cos x ≠ 0\n⊢ Complex.cos ↑x ≠ 0",
"tactic": "exact_mod_cast h"
}
] |
[
32,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
31,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.mem_inter_iff
|
[] |
[
895,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
894,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.top_mul_of_neg
|
[
{
"state_after": "x : EReal\nh : x < 0\n⊢ x * ⊤ = ⊥",
"state_before": "x : EReal\nh : x < 0\n⊢ ⊤ * x = ⊥",
"tactic": "rw [EReal.mul_comm]"
},
{
"state_after": "no goals",
"state_before": "x : EReal\nh : x < 0\n⊢ x * ⊤ = ⊥",
"tactic": "exact mul_top_of_neg h"
}
] |
[
941,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
939,
1
] |
Mathlib/Analysis/Convex/Strict.lean
|
StrictConvex.add_left
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\n𝕝 : Type ?u.85155\nE : Type u_2\nF : Type ?u.85161\nβ : Type ?u.85164\ninst✝⁷ : OrderedSemiring 𝕜\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : AddCommGroup E\ninst✝³ : AddCommGroup F\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 F\ninst✝ : ContinuousAdd E\ns t : Set E\nhs : StrictConvex 𝕜 s\nz : E\n⊢ StrictConvex 𝕜 ((fun x => z + x) '' s)",
"tactic": "simpa only [singleton_add] using (strictConvex_singleton z).add hs"
}
] |
[
264,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.nonempty_Ico_subtype
|
[] |
[
314,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
313,
1
] |
Mathlib/Data/PNat/Defs.lean
|
PNat.succPNat_natPred
|
[] |
[
97,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.bot_eq_zero
|
[] |
[
396,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
395,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.HasBasis.sup
|
[] |
[
597,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
594,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.exp_sub_sinh
|
[] |
[
1417,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1416,
1
] |
Mathlib/Algebra/Module/Submodule/Basic.lean
|
Submodule.injective_subtype
|
[] |
[
381,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
380,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.integral_add_measure
|
[
{
"state_after": "α : Type u_2\nE : Type u_1\nF : Type ?u.1290399\n𝕜 : Type ?u.1290402\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1293093\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nν : Measure α\nf : α → E\nhμ : Integrable f\nhν : Integrable f\nhfi : Integrable f\n⊢ (∫ (x : α), f x ∂μ + ν) = (∫ (x : α), f x ∂μ) + ∫ (x : α), f x ∂ν",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.1290399\n𝕜 : Type ?u.1290402\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1293093\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nν : Measure α\nf : α → E\nhμ : Integrable f\nhν : Integrable f\n⊢ (∫ (x : α), f x ∂μ + ν) = (∫ (x : α), f x ∂μ) + ∫ (x : α), f x ∂ν",
"tactic": "have hfi := hμ.add_measure hν"
},
{
"state_after": "α : Type u_2\nE : Type u_1\nF : Type ?u.1290399\n𝕜 : Type ?u.1290402\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1293093\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nν : Measure α\nf : α → E\nhμ : Integrable f\nhν : Integrable f\nhfi : Integrable f\n⊢ (setToFun (μ + ν) (weightedSMul (μ + ν)) (_ : DominatedFinMeasAdditive (μ + ν) (weightedSMul (μ + ν)) 1) fun a =>\n f a) =\n (setToFun μ (weightedSMul μ) (_ : DominatedFinMeasAdditive μ (weightedSMul μ) 1) fun a => f a) +\n setToFun ν (weightedSMul ν) (_ : DominatedFinMeasAdditive ν (weightedSMul ν) 1) fun a => f a",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.1290399\n𝕜 : Type ?u.1290402\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1293093\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nν : Measure α\nf : α → E\nhμ : Integrable f\nhν : Integrable f\nhfi : Integrable f\n⊢ (∫ (x : α), f x ∂μ + ν) = (∫ (x : α), f x ∂μ) + ∫ (x : α), f x ∂ν",
"tactic": "simp_rw [integral_eq_setToFun]"
},
{
"state_after": "α : Type u_2\nE : Type u_1\nF : Type ?u.1290399\n𝕜 : Type ?u.1290402\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1293093\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nν : Measure α\nf : α → E\nhμ : Integrable f\nhν : Integrable f\nhfi : Integrable f\nhμ_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul μ) 1\n⊢ (setToFun (μ + ν) (weightedSMul (μ + ν)) (_ : DominatedFinMeasAdditive (μ + ν) (weightedSMul (μ + ν)) 1) fun a =>\n f a) =\n (setToFun μ (weightedSMul μ) (_ : DominatedFinMeasAdditive μ (weightedSMul μ) 1) fun a => f a) +\n setToFun ν (weightedSMul ν) (_ : DominatedFinMeasAdditive ν (weightedSMul ν) 1) fun a => f a",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.1290399\n𝕜 : Type ?u.1290402\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1293093\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nν : Measure α\nf : α → E\nhμ : Integrable f\nhν : Integrable f\nhfi : Integrable f\n⊢ (setToFun (μ + ν) (weightedSMul (μ + ν)) (_ : DominatedFinMeasAdditive (μ + ν) (weightedSMul (μ + ν)) 1) fun a =>\n f a) =\n (setToFun μ (weightedSMul μ) (_ : DominatedFinMeasAdditive μ (weightedSMul μ) 1) fun a => f a) +\n setToFun ν (weightedSMul ν) (_ : DominatedFinMeasAdditive ν (weightedSMul ν) 1) fun a => f a",
"tactic": "have hμ_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul μ : Set α → E →L[ℝ] E) 1 :=\n DominatedFinMeasAdditive.add_measure_right μ ν (dominatedFinMeasAdditive_weightedSMul μ)\n zero_le_one"
},
{
"state_after": "α : Type u_2\nE : Type u_1\nF : Type ?u.1290399\n𝕜 : Type ?u.1290402\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1293093\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nν : Measure α\nf : α → E\nhμ : Integrable f\nhν : Integrable f\nhfi : Integrable f\nhμ_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul μ) 1\nhν_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul ν) 1\n⊢ (setToFun (μ + ν) (weightedSMul (μ + ν)) (_ : DominatedFinMeasAdditive (μ + ν) (weightedSMul (μ + ν)) 1) fun a =>\n f a) =\n (setToFun μ (weightedSMul μ) (_ : DominatedFinMeasAdditive μ (weightedSMul μ) 1) fun a => f a) +\n setToFun ν (weightedSMul ν) (_ : DominatedFinMeasAdditive ν (weightedSMul ν) 1) fun a => f a",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.1290399\n𝕜 : Type ?u.1290402\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1293093\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nν : Measure α\nf : α → E\nhμ : Integrable f\nhν : Integrable f\nhfi : Integrable f\nhμ_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul μ) 1\n⊢ (setToFun (μ + ν) (weightedSMul (μ + ν)) (_ : DominatedFinMeasAdditive (μ + ν) (weightedSMul (μ + ν)) 1) fun a =>\n f a) =\n (setToFun μ (weightedSMul μ) (_ : DominatedFinMeasAdditive μ (weightedSMul μ) 1) fun a => f a) +\n setToFun ν (weightedSMul ν) (_ : DominatedFinMeasAdditive ν (weightedSMul ν) 1) fun a => f a",
"tactic": "have hν_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul ν : Set α → E →L[ℝ] E) 1 :=\n DominatedFinMeasAdditive.add_measure_left μ ν (dominatedFinMeasAdditive_weightedSMul ν)\n zero_le_one"
},
{
"state_after": "α : Type u_2\nE : Type u_1\nF : Type ?u.1290399\n𝕜 : Type ?u.1290402\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1293093\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nν : Measure α\nf : α → E\nhμ : Integrable f\nhν : Integrable f\nhfi : Integrable f\nhμ_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul μ) 1\nhν_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul ν) 1\n⊢ (setToFun (μ + ν) (weightedSMul (μ + ν)) (_ : DominatedFinMeasAdditive (μ + ν) (weightedSMul (μ + ν)) 1) fun a =>\n f a) =\n setToFun (μ + ν) (weightedSMul μ) hμ_dfma f + setToFun (μ + ν) (weightedSMul ν) hν_dfma f",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.1290399\n𝕜 : Type ?u.1290402\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1293093\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nν : Measure α\nf : α → E\nhμ : Integrable f\nhν : Integrable f\nhfi : Integrable f\nhμ_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul μ) 1\nhν_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul ν) 1\n⊢ (setToFun (μ + ν) (weightedSMul (μ + ν)) (_ : DominatedFinMeasAdditive (μ + ν) (weightedSMul (μ + ν)) 1) fun a =>\n f a) =\n (setToFun μ (weightedSMul μ) (_ : DominatedFinMeasAdditive μ (weightedSMul μ) 1) fun a => f a) +\n setToFun ν (weightedSMul ν) (_ : DominatedFinMeasAdditive ν (weightedSMul ν) 1) fun a => f a",
"tactic": "rw [← setToFun_congr_measure_of_add_right hμ_dfma (dominatedFinMeasAdditive_weightedSMul μ) f hfi,\n ← setToFun_congr_measure_of_add_left hν_dfma (dominatedFinMeasAdditive_weightedSMul ν) f hfi]"
},
{
"state_after": "α : Type u_2\nE : Type u_1\nF : Type ?u.1290399\n𝕜 : Type ?u.1290402\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1293093\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nν : Measure α\nf : α → E\nhμ : Integrable f\nhν : Integrable f\nhfi : Integrable f\nhμ_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul μ) 1\nhν_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul ν) 1\ns : Set α\nx✝ : MeasurableSet s\nhμνs : ↑↑(μ + ν) s < ⊤\n⊢ weightedSMul (μ + ν) s = weightedSMul μ s + weightedSMul ν s",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.1290399\n𝕜 : Type ?u.1290402\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1293093\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nν : Measure α\nf : α → E\nhμ : Integrable f\nhν : Integrable f\nhfi : Integrable f\nhμ_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul μ) 1\nhν_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul ν) 1\n⊢ (setToFun (μ + ν) (weightedSMul (μ + ν)) (_ : DominatedFinMeasAdditive (μ + ν) (weightedSMul (μ + ν)) 1) fun a =>\n f a) =\n setToFun (μ + ν) (weightedSMul μ) hμ_dfma f + setToFun (μ + ν) (weightedSMul ν) hν_dfma f",
"tactic": "refine' setToFun_add_left' _ _ _ (fun s _ hμνs => _) f"
},
{
"state_after": "α : Type u_2\nE : Type u_1\nF : Type ?u.1290399\n𝕜 : Type ?u.1290402\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1293093\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nν : Measure α\nf : α → E\nhμ : Integrable f\nhν : Integrable f\nhfi : Integrable f\nhμ_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul μ) 1\nhν_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul ν) 1\ns : Set α\nx✝ : MeasurableSet s\nhμνs : ↑↑μ s < ⊤ ∧ ↑↑ν s < ⊤\n⊢ weightedSMul (μ + ν) s = weightedSMul μ s + weightedSMul ν s",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.1290399\n𝕜 : Type ?u.1290402\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1293093\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nν : Measure α\nf : α → E\nhμ : Integrable f\nhν : Integrable f\nhfi : Integrable f\nhμ_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul μ) 1\nhν_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul ν) 1\ns : Set α\nx✝ : MeasurableSet s\nhμνs : ↑↑(μ + ν) s < ⊤\n⊢ weightedSMul (μ + ν) s = weightedSMul μ s + weightedSMul ν s",
"tactic": "rw [Measure.coe_add, Pi.add_apply, add_lt_top] at hμνs"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.1290399\n𝕜 : Type ?u.1290402\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1293093\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nν : Measure α\nf : α → E\nhμ : Integrable f\nhν : Integrable f\nhfi : Integrable f\nhμ_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul μ) 1\nhν_dfma : DominatedFinMeasAdditive (μ + ν) (weightedSMul ν) 1\ns : Set α\nx✝ : MeasurableSet s\nhμνs : ↑↑μ s < ⊤ ∧ ↑↑ν s < ⊤\n⊢ weightedSMul (μ + ν) s = weightedSMul μ s + weightedSMul ν s",
"tactic": "rw [weightedSMul, weightedSMul, weightedSMul, ← add_smul, Measure.coe_add, Pi.add_apply,\n toReal_add hμνs.1.ne hμνs.2.ne]"
}
] |
[
1418,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1403,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean
|
NormedAddGroupHom.NormNoninc.neg_iff
|
[
{
"state_after": "no goals",
"state_before": "V : Type ?u.497709\nW : Type ?u.497712\nV₁ : Type u_1\nV₂ : Type u_2\nV₃ : Type ?u.497721\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : SeminormedAddCommGroup W\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf✝ : NormedAddGroupHom V W\nf : NormedAddGroupHom V₁ V₂\nh : NormNoninc (-f)\nx : V₁\n⊢ ‖↑f x‖ ≤ ‖x‖",
"tactic": "simpa using h x"
}
] |
[
847,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
846,
1
] |
Mathlib/Data/List/Basic.lean
|
List.reduceOption_length_lt_iff
|
[
{
"state_after": "ι : Type ?u.354270\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nl : List (Option α)\n⊢ (¬∀ (x : Option α), x ∈ l → Option.isSome x = true) ↔ none ∈ l",
"state_before": "ι : Type ?u.354270\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nl : List (Option α)\n⊢ length (reduceOption l) < length l ↔ none ∈ l",
"tactic": "rw [(reduceOption_length_le l).lt_iff_ne, Ne, reduceOption_length_eq_iff]"
},
{
"state_after": "case cons\nι : Type ?u.354270\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nhead✝ : Option α\ntail✝ : List (Option α)\ntail_ih✝ : (¬∀ (x : Option α), x ∈ tail✝ → Option.isSome x = true) ↔ none ∈ tail✝\n⊢ (Option.isSome head✝ = true → ∃ x, x ∈ tail✝ ∧ Option.isNone x = true) ↔ none = head✝ ∨ none ∈ tail✝",
"state_before": "ι : Type ?u.354270\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nl : List (Option α)\n⊢ (¬∀ (x : Option α), x ∈ l → Option.isSome x = true) ↔ none ∈ l",
"tactic": "induction l <;> simp [*]"
},
{
"state_after": "case cons\nι : Type ?u.354270\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nhead✝ : Option α\ntail✝ : List (Option α)\ntail_ih✝ : (¬∀ (x : Option α), x ∈ tail✝ → Option.isSome x = true) ↔ none ∈ tail✝\n⊢ (¬Option.isSome head✝ = true ∨ ∃ x, x ∈ tail✝ ∧ Option.isNone x = true) ↔ ¬Option.isSome head✝ = true ∨ none ∈ tail✝",
"state_before": "case cons\nι : Type ?u.354270\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nhead✝ : Option α\ntail✝ : List (Option α)\ntail_ih✝ : (¬∀ (x : Option α), x ∈ tail✝ → Option.isSome x = true) ↔ none ∈ tail✝\n⊢ (Option.isSome head✝ = true → ∃ x, x ∈ tail✝ ∧ Option.isNone x = true) ↔ none = head✝ ∨ none ∈ tail✝",
"tactic": "rw [@eq_comm _ none, ← Option.not_isSome_iff_eq_none, Decidable.imp_iff_not_or]"
},
{
"state_after": "no goals",
"state_before": "case cons\nι : Type ?u.354270\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nhead✝ : Option α\ntail✝ : List (Option α)\ntail_ih✝ : (¬∀ (x : Option α), x ∈ tail✝ → Option.isSome x = true) ↔ none ∈ tail✝\n⊢ (¬Option.isSome head✝ = true ∨ ∃ x, x ∈ tail✝ ∧ Option.isNone x = true) ↔ ¬Option.isSome head✝ = true ∨ none ∈ tail✝",
"tactic": "simp [Option.isNone_iff_eq_none]"
}
] |
[
3477,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3472,
1
] |
Mathlib/RingTheory/PowerBasis.lean
|
PowerBasis.equivOfMinpoly_map
|
[] |
[
496,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
494,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.smul_coeff
|
[] |
[
503,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
502,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.imLm_coe
|
[] |
[
910,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
909,
1
] |
Mathlib/CategoryTheory/Subobject/FactorThru.lean
|
CategoryTheory.Subobject.factors_of_le
|
[
{
"state_after": "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\n⊢ MonoOver.Factors (representative.obj P) f → MonoOver.Factors (representative.obj Q) f",
"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\n⊢ Factors P f → Factors Q f",
"tactic": "simp only [factors_iff]"
},
{
"state_after": "no goals",
"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\n⊢ MonoOver.Factors (representative.obj P) f → MonoOver.Factors (representative.obj Q) f",
"tactic": "exact fun ⟨u, hu⟩ => ⟨u ≫ ofLE _ _ h, by simp [← hu]⟩"
},
{
"state_after": "no goals",
"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\nx✝ : MonoOver.Factors (representative.obj P) f\nu : Z ⟶ (representative.obj P).obj.left\nhu : u ≫ MonoOver.arrow (representative.obj P) = f\n⊢ (u ≫ ofLE P Q h) ≫ MonoOver.arrow (representative.obj Q) = f",
"tactic": "simp [← hu]"
}
] |
[
114,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
Module.finrank_le_one_iff_top_isPrincipal
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\n⊢ finrank K V ≤ 1 ↔ IsPrincipal ⊤",
"tactic": "rw [← Module.rank_le_one_iff_top_isPrincipal, ← finrank_eq_rank, ← Cardinal.natCast_le,\n Nat.cast_one]"
}
] |
[
1316,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1313,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.withDensity_tsum
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.1767013\nγ : Type ?u.1767016\nδ : Type ?u.1767019\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nh : ∀ (i : ℕ), Measurable (f i)\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑(withDensity μ (∑' (n : ℕ), f n)) s = ↑↑(sum fun n => withDensity μ (f n)) s",
"state_before": "α : Type u_1\nβ : Type ?u.1767013\nγ : Type ?u.1767016\nδ : Type ?u.1767019\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nh : ∀ (i : ℕ), Measurable (f i)\n⊢ withDensity μ (∑' (n : ℕ), f n) = sum fun n => withDensity μ (f n)",
"tactic": "ext1 s hs"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.1767013\nγ : Type ?u.1767016\nδ : Type ?u.1767019\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nh : ∀ (i : ℕ), Measurable (f i)\ns : Set α\nhs : MeasurableSet s\n⊢ (∫⁻ (a : α) in s, tsum (fun n => f n) a ∂μ) = ∑' (i : ℕ), ∫⁻ (a : α) in s, f i a ∂μ",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.1767013\nγ : Type ?u.1767016\nδ : Type ?u.1767019\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nh : ∀ (i : ℕ), Measurable (f i)\ns : Set α\nhs : MeasurableSet s\n⊢ ↑↑(withDensity μ (∑' (n : ℕ), f n)) s = ↑↑(sum fun n => withDensity μ (f n)) s",
"tactic": "simp_rw [sum_apply _ hs, withDensity_apply _ hs]"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.1767013\nγ : Type ?u.1767016\nδ : Type ?u.1767019\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nh : ∀ (i : ℕ), Measurable (f i)\ns : Set α\nhs : MeasurableSet s\n⊢ (∫⁻ (x : α) in s, tsum (fun n => f n) x ∂μ) = ∑' (i : ℕ), ∫⁻ (x : α) in s, f i x ∂μ",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.1767013\nγ : Type ?u.1767016\nδ : Type ?u.1767019\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nh : ∀ (i : ℕ), Measurable (f i)\ns : Set α\nhs : MeasurableSet s\n⊢ (∫⁻ (a : α) in s, tsum (fun n => f n) a ∂μ) = ∑' (i : ℕ), ∫⁻ (a : α) in s, f i a ∂μ",
"tactic": "change (∫⁻ x in s, (∑' n, f n) x ∂μ) = ∑' i : ℕ, ∫⁻ x, f i x ∂μ.restrict s"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.1767013\nγ : Type ?u.1767016\nδ : Type ?u.1767019\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nh : ∀ (i : ℕ), Measurable (f i)\ns : Set α\nhs : MeasurableSet s\n⊢ (∫⁻ (x : α) in s, tsum (fun n => f n) x ∂μ) = ∫⁻ (a : α) in s, ∑' (i : ℕ), f i a ∂μ",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.1767013\nγ : Type ?u.1767016\nδ : Type ?u.1767019\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nh : ∀ (i : ℕ), Measurable (f i)\ns : Set α\nhs : MeasurableSet s\n⊢ (∫⁻ (x : α) in s, tsum (fun n => f n) x ∂μ) = ∑' (i : ℕ), ∫⁻ (x : α) in s, f i x ∂μ",
"tactic": "rw [← lintegral_tsum fun i => (h i).aemeasurable]"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.1767013\nγ : Type ?u.1767016\nδ : Type ?u.1767019\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\nh : ∀ (i : ℕ), Measurable (f i)\ns : Set α\nhs : MeasurableSet s\n⊢ (∫⁻ (x : α) in s, tsum (fun n => f n) x ∂μ) = ∫⁻ (a : α) in s, ∑' (i : ℕ), f i a ∂μ",
"tactic": "refine' lintegral_congr fun x => tsum_apply (Pi.summable.2 fun _ => ENNReal.summable)"
}
] |
[
1633,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1627,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.toNat_congr
|
[
{
"state_after": "no goals",
"state_before": "α β✝ : Type u\nβ : Type v\ne : α ≃ β\n⊢ ↑toNat (#α) = ↑toNat (#β)",
"tactic": "rw [← toNat_lift, (lift_mk_eq.{_,_,v}).mpr ⟨e⟩, toNat_lift]"
}
] |
[
1788,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1786,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
Real.abs_rpow_le_exp_log_mul
|
[
{
"state_after": "x y : ℝ\n⊢ abs x ^ y ≤ exp (log x * y)",
"state_before": "x y : ℝ\n⊢ abs (x ^ y) ≤ exp (log x * y)",
"tactic": "refine' (abs_rpow_le_abs_rpow x y).trans _"
},
{
"state_after": "case pos\nx y : ℝ\nhx : x = 0\n⊢ abs x ^ y ≤ exp (log x * y)\n\ncase neg\nx y : ℝ\nhx : ¬x = 0\n⊢ abs x ^ y ≤ exp (log x * y)",
"state_before": "x y : ℝ\n⊢ abs x ^ y ≤ exp (log x * y)",
"tactic": "by_cases hx : x = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\nx y : ℝ\nhx : x = 0\n⊢ abs x ^ y ≤ exp (log x * y)",
"tactic": "by_cases hy : y = 0 <;> simp [hx, hy, zero_le_one]"
},
{
"state_after": "no goals",
"state_before": "case neg\nx y : ℝ\nhx : ¬x = 0\n⊢ abs x ^ y ≤ exp (log x * y)",
"tactic": "rw [rpow_def_of_pos (abs_pos.2 hx), log_abs]"
}
] |
[
166,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/GroupTheory/Index.lean
|
Subgroup.index_mul_card
|
[
{
"state_after": "G : Type u_1\ninst✝¹ : Group G\nH K L : Subgroup G\ninst✝ : Fintype G\nhH : Fintype { x // x ∈ H }\n⊢ relindex ⊥ H * index H = index ⊥",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH K L : Subgroup G\ninst✝ : Fintype G\nhH : Fintype { x // x ∈ H }\n⊢ index H * Fintype.card { x // x ∈ H } = Fintype.card G",
"tactic": "rw [← relindex_bot_left_eq_card, ← index_bot_eq_card, mul_comm]"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH K L : Subgroup G\ninst✝ : Fintype G\nhH : Fintype { x // x ∈ H }\n⊢ relindex ⊥ H * index H = index ⊥",
"tactic": "exact relindex_mul_index bot_le"
}
] |
[
367,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
364,
1
] |
Mathlib/Data/Set/Sups.lean
|
Set.sups_singleton
|
[] |
[
153,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.equivOfEq_trans
|
[] |
[
208,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/Algebra/Hom/Units.lean
|
IsUnit.inv_mul_cancel_right
|
[] |
[
309,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
308,
11
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
Subsemigroup.le_comap_map
|
[] |
[
294,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
293,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
zpow_le_zpow_iff
|
[] |
[
361,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
360,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean
|
NormedAddGroupHom.SurjectiveOnWith.exists_pos
|
[
{
"state_after": "case refine'_1\nV : Type ?u.219726\nV₁ : Type u_1\nV₂ : Type u_2\nV₃ : Type ?u.219735\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf✝ g f : NormedAddGroupHom V₁ V₂\nK : AddSubgroup V₂\nC : ℝ\nh : SurjectiveOnWith f K C\n⊢ abs C + 1 > 0\n\ncase refine'_2\nV : Type ?u.219726\nV₁ : Type u_1\nV₂ : Type u_2\nV₃ : Type ?u.219735\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf✝ g f : NormedAddGroupHom V₁ V₂\nK : AddSubgroup V₂\nC : ℝ\nh : SurjectiveOnWith f K C\n⊢ SurjectiveOnWith f K (abs C + 1)",
"state_before": "V : Type ?u.219726\nV₁ : Type u_1\nV₂ : Type u_2\nV₃ : Type ?u.219735\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf✝ g f : NormedAddGroupHom V₁ V₂\nK : AddSubgroup V₂\nC : ℝ\nh : SurjectiveOnWith f K C\n⊢ ∃ C', C' > 0 ∧ SurjectiveOnWith f K C'",
"tactic": "refine' ⟨|C| + 1, _, _⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nV : Type ?u.219726\nV₁ : Type u_1\nV₂ : Type u_2\nV₃ : Type ?u.219735\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf✝ g f : NormedAddGroupHom V₁ V₂\nK : AddSubgroup V₂\nC : ℝ\nh : SurjectiveOnWith f K C\n⊢ abs C + 1 > 0",
"tactic": "linarith [abs_nonneg C]"
},
{
"state_after": "case refine'_2\nV : Type ?u.219726\nV₁ : Type u_1\nV₂ : Type u_2\nV₃ : Type ?u.219735\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf✝ g f : NormedAddGroupHom V₁ V₂\nK : AddSubgroup V₂\nC : ℝ\nh : SurjectiveOnWith f K C\n⊢ C ≤ abs C + 1",
"state_before": "case refine'_2\nV : Type ?u.219726\nV₁ : Type u_1\nV₂ : Type u_2\nV₃ : Type ?u.219735\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf✝ g f : NormedAddGroupHom V₁ V₂\nK : AddSubgroup V₂\nC : ℝ\nh : SurjectiveOnWith f K C\n⊢ SurjectiveOnWith f K (abs C + 1)",
"tactic": "apply h.mono"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nV : Type ?u.219726\nV₁ : Type u_1\nV₂ : Type u_2\nV₃ : Type ?u.219735\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf✝ g f : NormedAddGroupHom V₁ V₂\nK : AddSubgroup V₂\nC : ℝ\nh : SurjectiveOnWith f K C\n⊢ C ≤ abs C + 1",
"tactic": "linarith [le_abs_self C]"
}
] |
[
200,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Data/Rat/Cast.lean
|
Rat.cast_comm
|
[] |
[
74,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/Combinatorics/Additive/Behrend.lean
|
Behrend.mem_box
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.474\nβ : Type ?u.477\nn d k N : ℕ\nx : Fin n → ℕ\n⊢ x ∈ box n d ↔ ∀ (i : Fin n), x i < d",
"tactic": "simp only [box, Fintype.mem_piFinset, mem_range]"
}
] |
[
77,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.algebraMap_apply'
|
[] |
[
1319,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1317,
1
] |
Mathlib/Data/Prod/Basic.lean
|
Prod.forall
|
[] |
[
34,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
33,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.coe_compRingHom
|
[] |
[
1092,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1091,
1
] |
Mathlib/Order/GaloisConnection.lean
|
GaloisCoinsertion.u_l_leftInverse
|
[] |
[
777,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
775,
1
] |
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
|
WithSeminorms.toLocallyConvexSpace
|
[
{
"state_after": "case hbasis\n𝕜 : Type u_1\n𝕜₂ : Type ?u.678312\n𝕝 : Type ?u.678315\n𝕝₂ : Type ?u.678318\nE : Type u_2\nF : Type ?u.678324\nG : Type ?u.678327\nι : Type u_3\nι' : Type ?u.678333\ninst✝⁸ : Nonempty ι\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : NormedSpace ℝ 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module ℝ E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\n⊢ Filter.HasBasis (𝓝 0) (fun s => s ∈ SeminormFamily.basisSets p) id\n\ncase hconvex\n𝕜 : Type u_1\n𝕜₂ : Type ?u.678312\n𝕝 : Type ?u.678315\n𝕝₂ : Type ?u.678318\nE : Type u_2\nF : Type ?u.678324\nG : Type ?u.678327\nι : Type u_3\nι' : Type ?u.678333\ninst✝⁸ : Nonempty ι\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : NormedSpace ℝ 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module ℝ E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\n⊢ ∀ (i : Set E), i ∈ SeminormFamily.basisSets p → Convex ℝ (id i)",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type ?u.678312\n𝕝 : Type ?u.678315\n𝕝₂ : Type ?u.678318\nE : Type u_2\nF : Type ?u.678324\nG : Type ?u.678327\nι : Type u_3\nι' : Type ?u.678333\ninst✝⁸ : Nonempty ι\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : NormedSpace ℝ 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module ℝ E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\n⊢ LocallyConvexSpace ℝ E",
"tactic": "apply ofBasisZero ℝ E id fun s => s ∈ p.basisSets"
},
{
"state_after": "case hbasis\n𝕜 : Type u_1\n𝕜₂ : Type ?u.678312\n𝕝 : Type ?u.678315\n𝕝₂ : Type ?u.678318\nE : Type u_2\nF : Type ?u.678324\nG : Type ?u.678327\nι : Type u_3\nι' : Type ?u.678333\ninst✝⁸ : Nonempty ι\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : NormedSpace ℝ 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module ℝ E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\n⊢ Filter.HasBasis (FilterBasis.filter AddGroupFilterBasis.toFilterBasis) (fun s => s ∈ SeminormFamily.basisSets p) id",
"state_before": "case hbasis\n𝕜 : Type u_1\n𝕜₂ : Type ?u.678312\n𝕝 : Type ?u.678315\n𝕝₂ : Type ?u.678318\nE : Type u_2\nF : Type ?u.678324\nG : Type ?u.678327\nι : Type u_3\nι' : Type ?u.678333\ninst✝⁸ : Nonempty ι\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : NormedSpace ℝ 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module ℝ E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\n⊢ Filter.HasBasis (𝓝 0) (fun s => s ∈ SeminormFamily.basisSets p) id",
"tactic": "rw [hp.1, AddGroupFilterBasis.nhds_eq _, AddGroupFilterBasis.N_zero]"
},
{
"state_after": "no goals",
"state_before": "case hbasis\n𝕜 : Type u_1\n𝕜₂ : Type ?u.678312\n𝕝 : Type ?u.678315\n𝕝₂ : Type ?u.678318\nE : Type u_2\nF : Type ?u.678324\nG : Type ?u.678327\nι : Type u_3\nι' : Type ?u.678333\ninst✝⁸ : Nonempty ι\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : NormedSpace ℝ 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module ℝ E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\n⊢ Filter.HasBasis (FilterBasis.filter AddGroupFilterBasis.toFilterBasis) (fun s => s ∈ SeminormFamily.basisSets p) id",
"tactic": "exact FilterBasis.hasBasis _"
},
{
"state_after": "case hconvex\n𝕜 : Type u_1\n𝕜₂ : Type ?u.678312\n𝕝 : Type ?u.678315\n𝕝₂ : Type ?u.678318\nE : Type u_2\nF : Type ?u.678324\nG : Type ?u.678327\nι : Type u_3\nι' : Type ?u.678333\ninst✝⁸ : Nonempty ι\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : NormedSpace ℝ 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module ℝ E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\ns : Set E\nhs : s ∈ SeminormFamily.basisSets p\n⊢ Convex ℝ (id s)",
"state_before": "case hconvex\n𝕜 : Type u_1\n𝕜₂ : Type ?u.678312\n𝕝 : Type ?u.678315\n𝕝₂ : Type ?u.678318\nE : Type u_2\nF : Type ?u.678324\nG : Type ?u.678327\nι : Type u_3\nι' : Type ?u.678333\ninst✝⁸ : Nonempty ι\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : NormedSpace ℝ 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module ℝ E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\n⊢ ∀ (i : Set E), i ∈ SeminormFamily.basisSets p → Convex ℝ (id i)",
"tactic": "intro s hs"
},
{
"state_after": "case hconvex\n𝕜 : Type u_1\n𝕜₂ : Type ?u.678312\n𝕝 : Type ?u.678315\n𝕝₂ : Type ?u.678318\nE : Type u_2\nF : Type ?u.678324\nG : Type ?u.678327\nι : Type u_3\nι' : Type ?u.678333\ninst✝⁸ : Nonempty ι\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : NormedSpace ℝ 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module ℝ E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\ns : Set E\nhs : s ∈ ⋃ (s : Finset ι) (r : ℝ) (_ : 0 < r), {ball (Finset.sup s p) 0 r}\n⊢ Convex ℝ (id s)",
"state_before": "case hconvex\n𝕜 : Type u_1\n𝕜₂ : Type ?u.678312\n𝕝 : Type ?u.678315\n𝕝₂ : Type ?u.678318\nE : Type u_2\nF : Type ?u.678324\nG : Type ?u.678327\nι : Type u_3\nι' : Type ?u.678333\ninst✝⁸ : Nonempty ι\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : NormedSpace ℝ 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module ℝ E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\ns : Set E\nhs : s ∈ SeminormFamily.basisSets p\n⊢ Convex ℝ (id s)",
"tactic": "change s ∈ Set.iUnion _ at hs"
},
{
"state_after": "case hconvex\n𝕜 : Type u_1\n𝕜₂ : Type ?u.678312\n𝕝 : Type ?u.678315\n𝕝₂ : Type ?u.678318\nE : Type u_2\nF : Type ?u.678324\nG : Type ?u.678327\nι : Type u_3\nι' : Type ?u.678333\ninst✝⁸ : Nonempty ι\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : NormedSpace ℝ 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module ℝ E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\ns : Set E\nhs : ∃ i i_1 h, s = ball (Finset.sup i p) 0 i_1\n⊢ Convex ℝ (id s)",
"state_before": "case hconvex\n𝕜 : Type u_1\n𝕜₂ : Type ?u.678312\n𝕝 : Type ?u.678315\n𝕝₂ : Type ?u.678318\nE : Type u_2\nF : Type ?u.678324\nG : Type ?u.678327\nι : Type u_3\nι' : Type ?u.678333\ninst✝⁸ : Nonempty ι\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : NormedSpace ℝ 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module ℝ E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\ns : Set E\nhs : s ∈ ⋃ (s : Finset ι) (r : ℝ) (_ : 0 < r), {ball (Finset.sup s p) 0 r}\n⊢ Convex ℝ (id s)",
"tactic": "simp_rw [Set.mem_iUnion, Set.mem_singleton_iff] at hs"
},
{
"state_after": "case hconvex.intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type ?u.678312\n𝕝 : Type ?u.678315\n𝕝₂ : Type ?u.678318\nE : Type u_2\nF : Type ?u.678324\nG : Type ?u.678327\nι : Type u_3\nι' : Type ?u.678333\ninst✝⁸ : Nonempty ι\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : NormedSpace ℝ 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module ℝ E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\nI : Finset ι\nr : ℝ\nw✝ : 0 < r\n⊢ Convex ℝ (id (ball (Finset.sup I p) 0 r))",
"state_before": "case hconvex\n𝕜 : Type u_1\n𝕜₂ : Type ?u.678312\n𝕝 : Type ?u.678315\n𝕝₂ : Type ?u.678318\nE : Type u_2\nF : Type ?u.678324\nG : Type ?u.678327\nι : Type u_3\nι' : Type ?u.678333\ninst✝⁸ : Nonempty ι\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : NormedSpace ℝ 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module ℝ E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\ns : Set E\nhs : ∃ i i_1 h, s = ball (Finset.sup i p) 0 i_1\n⊢ Convex ℝ (id s)",
"tactic": "rcases hs with ⟨I, r, _, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case hconvex.intro.intro.intro\n𝕜 : Type u_1\n𝕜₂ : Type ?u.678312\n𝕝 : Type ?u.678315\n𝕝₂ : Type ?u.678318\nE : Type u_2\nF : Type ?u.678324\nG : Type ?u.678327\nι : Type u_3\nι' : Type ?u.678333\ninst✝⁸ : Nonempty ι\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : NormedSpace ℝ 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module ℝ E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalAddGroup E\np : SeminormFamily 𝕜 E ι\nhp : WithSeminorms p\nI : Finset ι\nr : ℝ\nw✝ : 0 < r\n⊢ Convex ℝ (id (ball (Finset.sup I p) 0 r))",
"tactic": "exact convex_ball _ _ _"
}
] |
[
684,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
675,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
Subsemigroup.comap_equiv_eq_map_symm
|
[] |
[
718,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
716,
1
] |
Mathlib/Algebra/Order/Pointwise.lean
|
LinearOrderedField.smul_Ico
|
[
{
"state_after": "case h\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\n⊢ x ∈ r • Ico a b ↔ x ∈ Ico (r • a) (r • b)",
"state_before": "α : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\n⊢ r • Ico a b = Ico (r • a) (r • b)",
"tactic": "ext x"
},
{
"state_after": "case h\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\n⊢ (∃ y, (a ≤ y ∧ y < b) ∧ r * y = x) ↔ r * a ≤ x ∧ x < r * b",
"state_before": "case h\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\n⊢ x ∈ r • Ico a b ↔ x ∈ Ico (r • a) (r • b)",
"tactic": "simp only [mem_smul_set, smul_eq_mul, mem_Ico]"
},
{
"state_after": "case h.mp\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\n⊢ (∃ y, (a ≤ y ∧ y < b) ∧ r * y = x) → r * a ≤ x ∧ x < r * b\n\ncase h.mpr\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\n⊢ r * a ≤ x ∧ x < r * b → ∃ y, (a ≤ y ∧ y < b) ∧ r * y = x",
"state_before": "case h\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\n⊢ (∃ y, (a ≤ y ∧ y < b) ∧ r * y = x) ↔ r * a ≤ x ∧ x < r * b",
"tactic": "constructor"
},
{
"state_after": "case h.mp.intro.intro.intro\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na✝ b r : K\nhr : 0 < r\na : K\na_h_left_left : a✝ ≤ a\na_h_left_right : a < b\n⊢ r * a✝ ≤ r * a ∧ r * a < r * b",
"state_before": "case h.mp\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\n⊢ (∃ y, (a ≤ y ∧ y < b) ∧ r * y = x) → r * a ≤ x ∧ x < r * b",
"tactic": "rintro ⟨a, ⟨a_h_left_left, a_h_left_right⟩, rfl⟩"
},
{
"state_after": "case h.mp.intro.intro.intro.left\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na✝ b r : K\nhr : 0 < r\na : K\na_h_left_left : a✝ ≤ a\na_h_left_right : a < b\n⊢ r * a✝ ≤ r * a\n\ncase h.mp.intro.intro.intro.right\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na✝ b r : K\nhr : 0 < r\na : K\na_h_left_left : a✝ ≤ a\na_h_left_right : a < b\n⊢ r * a < r * b",
"state_before": "case h.mp.intro.intro.intro\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na✝ b r : K\nhr : 0 < r\na : K\na_h_left_left : a✝ ≤ a\na_h_left_right : a < b\n⊢ r * a✝ ≤ r * a ∧ r * a < r * b",
"tactic": "constructor"
},
{
"state_after": "case h.mp.intro.intro.intro.right\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na✝ b r : K\nhr : 0 < r\na : K\na_h_left_left : a✝ ≤ a\na_h_left_right : a < b\n⊢ r * a < r * b",
"state_before": "case h.mp.intro.intro.intro.left\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na✝ b r : K\nhr : 0 < r\na : K\na_h_left_left : a✝ ≤ a\na_h_left_right : a < b\n⊢ r * a✝ ≤ r * a\n\ncase h.mp.intro.intro.intro.right\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na✝ b r : K\nhr : 0 < r\na : K\na_h_left_left : a✝ ≤ a\na_h_left_right : a < b\n⊢ r * a < r * b",
"tactic": "exact (mul_le_mul_left hr).mpr a_h_left_left"
},
{
"state_after": "no goals",
"state_before": "case h.mp.intro.intro.intro.right\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na✝ b r : K\nhr : 0 < r\na : K\na_h_left_left : a✝ ≤ a\na_h_left_right : a < b\n⊢ r * a < r * b",
"tactic": "exact (mul_lt_mul_left hr).mpr a_h_left_right"
},
{
"state_after": "case h.mpr.intro\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\na_left : r * a ≤ x\na_right : x < r * b\n⊢ ∃ y, (a ≤ y ∧ y < b) ∧ r * y = x",
"state_before": "case h.mpr\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\n⊢ r * a ≤ x ∧ x < r * b → ∃ y, (a ≤ y ∧ y < b) ∧ r * y = x",
"tactic": "rintro ⟨a_left, a_right⟩"
},
{
"state_after": "case h.mpr.intro\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\na_left : r * a ≤ x\na_right : x < r * b\n⊢ (a ≤ x / r ∧ x / r < b) ∧ r * (x / r) = x",
"state_before": "case h.mpr.intro\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\na_left : r * a ≤ x\na_right : x < r * b\n⊢ ∃ y, (a ≤ y ∧ y < b) ∧ r * y = x",
"tactic": "use x / r"
},
{
"state_after": "case h.mpr.intro\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\na_left : r * a ≤ x\na_right : x < r * b\n⊢ r * (x / r) = x",
"state_before": "case h.mpr.intro\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\na_left : r * a ≤ x\na_right : x < r * b\n⊢ (a ≤ x / r ∧ x / r < b) ∧ r * (x / r) = x",
"tactic": "refine' ⟨⟨(le_div_iff' hr).mpr a_left, (div_lt_iff' hr).mpr a_right⟩, _⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.intro\nα : Type ?u.56727\nK : Type u_1\ninst✝ : LinearOrderedField K\na b r : K\nhr : 0 < r\nx : K\na_left : r * a ≤ x\na_right : x < r * b\n⊢ r * (x / r) = x",
"tactic": "rw [mul_div_cancel' _ (ne_of_gt hr)]"
}
] |
[
227,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
HasFDerivWithinAt.const_sub
|
[] |
[
610,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
608,
8
] |
Mathlib/NumberTheory/PellMatiyasevic.lean
|
Pell.pellZd_succ
|
[
{
"state_after": "no goals",
"state_before": "a : ℕ\na1 : 1 < a\nn : ℕ\n⊢ pellZd a1 (n + 1) = pellZd a1 n * { re := ↑a, im := 1 }",
"tactic": "simp [Zsqrtd.ext]"
}
] |
[
223,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.parallelPair_obj_one
|
[] |
[
231,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/Order/FixedPoints.lean
|
OrderHom.map_gfp
|
[] |
[
128,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/NumberTheory/LucasLehmer.lean
|
LucasLehmer.Int.coe_nat_two_pow_pred
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\n⊢ 0 < 2",
"tactic": "decide"
}
] |
[
137,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/Algebra/Group/Commute.lean
|
Commute.units_val_iff
|
[] |
[
259,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/GroupTheory/GroupAction/SubMulAction.lean
|
SubMulAction.SMulMemClass.coeSubtype
|
[] |
[
227,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
226,
11
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.degree_lt_degree_mul_X
|
[
{
"state_after": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.707375\nhp : p ≠ 0\nthis : Nontrivial R\n⊢ degree p < degree (p * X)",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.707375\nhp : p ≠ 0\n⊢ degree p < degree (p * X)",
"tactic": "haveI := Nontrivial.of_polynomial_ne hp"
},
{
"state_after": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.707375\nhp : p ≠ 0\nthis✝ : Nontrivial R\nthis : leadingCoeff p * leadingCoeff X ≠ 0\n⊢ degree p < degree (p * X)",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.707375\nhp : p ≠ 0\nthis : Nontrivial R\n⊢ degree p < degree (p * X)",
"tactic": "have : leadingCoeff p * leadingCoeff X ≠ 0 := by simpa"
},
{
"state_after": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.707375\nhp : p ≠ 0\nthis✝ : Nontrivial R\nthis : leadingCoeff p * leadingCoeff X ≠ 0\n⊢ ↑(natDegree p) < ↑(natDegree p) + 1",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.707375\nhp : p ≠ 0\nthis✝ : Nontrivial R\nthis : leadingCoeff p * leadingCoeff X ≠ 0\n⊢ degree p < degree (p * X)",
"tactic": "erw [degree_mul' this, degree_eq_natDegree hp, degree_X, ← WithBot.coe_one,\n ← WithBot.coe_add, WithBot.coe_lt_coe]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.707375\nhp : p ≠ 0\nthis✝ : Nontrivial R\nthis : leadingCoeff p * leadingCoeff X ≠ 0\n⊢ ↑(natDegree p) < ↑(natDegree p) + 1",
"tactic": "exact Nat.lt_succ_self _"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.707375\nhp : p ≠ 0\nthis : Nontrivial R\n⊢ leadingCoeff p * leadingCoeff X ≠ 0",
"tactic": "simpa"
}
] |
[
1109,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1105,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.nnnorm_const_le
|
[] |
[
1020,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1019,
1
] |
Mathlib/Algebra/Group/Commute.lean
|
Commute.mul_div_mul_comm
|
[] |
[
336,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
11
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.simpleFunc_bot'
|
[
{
"state_after": "α✝ : Type ?u.21990\nβ : Type u_2\nγ : Type ?u.21996\nδ : Type ?u.21999\ninst✝¹ : MeasurableSpace α✝\nα : Type u_1\ninst✝ : Nonempty β\nf : α →ₛ β\nthis : MeasurableSpace α := ⊥\n⊢ ∃ c, f = const α c",
"state_before": "α✝ : Type ?u.21990\nβ : Type u_2\nγ : Type ?u.21996\nδ : Type ?u.21999\ninst✝¹ : MeasurableSpace α✝\nα : Type u_1\ninst✝ : Nonempty β\nf : α →ₛ β\n⊢ ∃ c, f = const α c",
"tactic": "letI : MeasurableSpace α := ⊥"
},
{
"state_after": "case intro\nα✝ : Type ?u.21990\nβ : Type u_2\nγ : Type ?u.21996\nδ : Type ?u.21999\ninst✝¹ : MeasurableSpace α✝\nα : Type u_1\ninst✝ : Nonempty β\nf : α →ₛ β\nthis : MeasurableSpace α := ⊥\nc : β\nh_eq : ∀ (x : α), ↑f x = c\n⊢ ∃ c, f = const α c",
"state_before": "α✝ : Type ?u.21990\nβ : Type u_2\nγ : Type ?u.21996\nδ : Type ?u.21999\ninst✝¹ : MeasurableSpace α✝\nα : Type u_1\ninst✝ : Nonempty β\nf : α →ₛ β\nthis : MeasurableSpace α := ⊥\n⊢ ∃ c, f = const α c",
"tactic": "obtain ⟨c, h_eq⟩ := simpleFunc_bot f"
},
{
"state_after": "case intro\nα✝ : Type ?u.21990\nβ : Type u_2\nγ : Type ?u.21996\nδ : Type ?u.21999\ninst✝¹ : MeasurableSpace α✝\nα : Type u_1\ninst✝ : Nonempty β\nf : α →ₛ β\nthis : MeasurableSpace α := ⊥\nc : β\nh_eq : ∀ (x : α), ↑f x = c\n⊢ f = const α c",
"state_before": "case intro\nα✝ : Type ?u.21990\nβ : Type u_2\nγ : Type ?u.21996\nδ : Type ?u.21999\ninst✝¹ : MeasurableSpace α✝\nα : Type u_1\ninst✝ : Nonempty β\nf : α →ₛ β\nthis : MeasurableSpace α := ⊥\nc : β\nh_eq : ∀ (x : α), ↑f x = c\n⊢ ∃ c, f = const α c",
"tactic": "refine' ⟨c, _⟩"
},
{
"state_after": "case intro.H\nα✝ : Type ?u.21990\nβ : Type u_2\nγ : Type ?u.21996\nδ : Type ?u.21999\ninst✝¹ : MeasurableSpace α✝\nα : Type u_1\ninst✝ : Nonempty β\nf : α →ₛ β\nthis : MeasurableSpace α := ⊥\nc : β\nh_eq : ∀ (x : α), ↑f x = c\nx : α\n⊢ ↑f x = ↑(const α c) x",
"state_before": "case intro\nα✝ : Type ?u.21990\nβ : Type u_2\nγ : Type ?u.21996\nδ : Type ?u.21999\ninst✝¹ : MeasurableSpace α✝\nα : Type u_1\ninst✝ : Nonempty β\nf : α →ₛ β\nthis : MeasurableSpace α := ⊥\nc : β\nh_eq : ∀ (x : α), ↑f x = c\n⊢ f = const α c",
"tactic": "ext1 x"
},
{
"state_after": "no goals",
"state_before": "case intro.H\nα✝ : Type ?u.21990\nβ : Type u_2\nγ : Type ?u.21996\nδ : Type ?u.21999\ninst✝¹ : MeasurableSpace α✝\nα : Type u_1\ninst✝ : Nonempty β\nf : α →ₛ β\nthis : MeasurableSpace α := ⊥\nc : β\nh_eq : ∀ (x : α), ↑f x = c\nx : α\n⊢ ↑f x = ↑(const α c) x",
"tactic": "rw [h_eq x, SimpleFunc.coe_const, Function.const]"
}
] |
[
193,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/ModelTheory/LanguageMap.lean
|
FirstOrder.Language.LHom.sumMap_comp_inl
|
[] |
[
216,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.ofNat_natAbs_of_nonpos
|
[
{
"state_after": "no goals",
"state_before": "a : Int\nH : a ≤ 0\n⊢ ↑(natAbs a) = -a",
"tactic": "rw [← natAbs_neg, natAbs_of_nonneg (Int.neg_nonneg_of_nonpos H)]"
}
] |
[
1248,
67
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1247,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.nnnorm_const_eq
|
[] |
[
1025,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1024,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.sub_le_sub
|
[] |
[
865,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
864,
1
] |
Mathlib/Combinatorics/Additive/Energy.lean
|
Finset.multiplicativeEnergy_mono_right
|
[] |
[
70,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/InformationTheory/Hamming.lean
|
hammingNorm_nonneg
|
[] |
[
180,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/Analysis/Calculus/Deriv/Mul.lean
|
deriv_mul_const
|
[] |
[
231,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.coe_pi
|
[] |
[
1219,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1218,
1
] |
Mathlib/CategoryTheory/Sites/SheafOfTypes.lean
|
CategoryTheory.Presieve.isSheaf_bot
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX✝ Y : C\nS : Sieve X✝\nR : Presieve X✝\nJ J₂ : GrothendieckTopology C\nX : C\n⊢ ∀ (S : Sieve X), S ∈ GrothendieckTopology.sieves ⊥ X → IsSheafFor P S.arrows",
"tactic": "simp [isSheafFor_top_sieve]"
}
] |
[
783,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
782,
1
] |
Mathlib/Combinatorics/Quiver/Cast.lean
|
Quiver.Hom.cast_heq
|
[
{
"state_after": "U : Type u_1\ninst✝ : Quiver U\nu' v' : U\ne : u' ⟶ v'\n⊢ HEq (cast (_ : u' = u') (_ : v' = v') e) e",
"state_before": "U : Type u_1\ninst✝ : Quiver U\nu v u' v' : U\nhu : u = u'\nhv : v = v'\ne : u ⟶ v\n⊢ HEq (cast hu hv e) e",
"tactic": "subst_vars"
},
{
"state_after": "no goals",
"state_before": "U : Type u_1\ninst✝ : Quiver U\nu' v' : U\ne : u' ⟶ v'\n⊢ HEq (cast (_ : u' = u') (_ : v' = v') e) e",
"tactic": "rfl"
}
] |
[
63,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Order/WellFoundedSet.lean
|
Set.isWf_singleton
|
[] |
[
471,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
471,
9
] |
Std/Data/Int/DivMod.lean
|
Int.add_mul_emod_self_left
|
[
{
"state_after": "no goals",
"state_before": "a b c : Int\n⊢ (a + b * c) % b = a % b",
"tactic": "rw [Int.mul_comm, Int.add_mul_emod_self]"
}
] |
[
412,
43
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
411,
9
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.comap_neBot_iff_frequently
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.263265\nι : Sort x\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm✝ : α → β\nm' : β → γ\ns : Set α\nt : Set β\nf : Filter β\nm : α → β\n⊢ NeBot (comap m f) ↔ ∃ᶠ (y : β) in f, y ∈ range m",
"tactic": "simp only [comap_neBot_iff, frequently_iff, mem_range, @and_comm (_ ∈ _), exists_exists_eq_and]"
}
] |
[
2348,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2346,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean
|
CategoryTheory.Limits.WidePullback.hom_ext
|
[
{
"state_after": "J : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : D\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng1 g2 : X ⟶ widePullback B (fun j => objs j) arrows\nh1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j\nh2 : g1 ≫ base arrows = g2 ≫ base arrows\n⊢ g1 = g2",
"state_before": "J : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : D\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng1 g2 : X ⟶ widePullback B (fun j => objs j) arrows\n⊢ (∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j) → g1 ≫ base arrows = g2 ≫ base arrows → g1 = g2",
"tactic": "intro h1 h2"
},
{
"state_after": "case w\nJ : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : D\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng1 g2 : X ⟶ widePullback B (fun j => objs j) arrows\nh1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j\nh2 : g1 ≫ base arrows = g2 ≫ base arrows\n⊢ ∀ (j : WidePullbackShape J),\n g1 ≫ limit.π (WidePullbackShape.wideCospan B (fun j => objs j) arrows) j =\n g2 ≫ limit.π (WidePullbackShape.wideCospan B (fun j => objs j) arrows) j",
"state_before": "J : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : D\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng1 g2 : X ⟶ widePullback B (fun j => objs j) arrows\nh1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j\nh2 : g1 ≫ base arrows = g2 ≫ base arrows\n⊢ g1 = g2",
"tactic": "apply limit.hom_ext"
},
{
"state_after": "case w.none\nJ : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : D\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng1 g2 : X ⟶ widePullback B (fun j => objs j) arrows\nh1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j\nh2 : g1 ≫ base arrows = g2 ≫ base arrows\n⊢ g1 ≫ limit.π (WidePullbackShape.wideCospan B (fun j => objs j) arrows) none =\n g2 ≫ limit.π (WidePullbackShape.wideCospan B (fun j => objs j) arrows) none\n\ncase w.some\nJ : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : D\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng1 g2 : X ⟶ widePullback B (fun j => objs j) arrows\nh1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j\nh2 : g1 ≫ base arrows = g2 ≫ base arrows\nval✝ : J\n⊢ g1 ≫ limit.π (WidePullbackShape.wideCospan B (fun j => objs j) arrows) (some val✝) =\n g2 ≫ limit.π (WidePullbackShape.wideCospan B (fun j => objs j) arrows) (some val✝)",
"state_before": "case w\nJ : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : D\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng1 g2 : X ⟶ widePullback B (fun j => objs j) arrows\nh1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j\nh2 : g1 ≫ base arrows = g2 ≫ base arrows\n⊢ ∀ (j : WidePullbackShape J),\n g1 ≫ limit.π (WidePullbackShape.wideCospan B (fun j => objs j) arrows) j =\n g2 ≫ limit.π (WidePullbackShape.wideCospan B (fun j => objs j) arrows) j",
"tactic": "rintro (_ | _)"
},
{
"state_after": "no goals",
"state_before": "case w.none\nJ : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : D\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng1 g2 : X ⟶ widePullback B (fun j => objs j) arrows\nh1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j\nh2 : g1 ≫ base arrows = g2 ≫ base arrows\n⊢ g1 ≫ limit.π (WidePullbackShape.wideCospan B (fun j => objs j) arrows) none =\n g2 ≫ limit.π (WidePullbackShape.wideCospan B (fun j => objs j) arrows) none",
"tactic": "apply h2"
},
{
"state_after": "no goals",
"state_before": "case w.some\nJ : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : D\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng1 g2 : X ⟶ widePullback B (fun j => objs j) arrows\nh1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j\nh2 : g1 ≫ base arrows = g2 ≫ base arrows\nval✝ : J\n⊢ g1 ≫ limit.π (WidePullbackShape.wideCospan B (fun j => objs j) arrows) (some val✝) =\n g2 ≫ limit.π (WidePullbackShape.wideCospan B (fun j => objs j) arrows) (some val✝)",
"tactic": "apply h1"
}
] |
[
378,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
372,
1
] |
Mathlib/Analysis/Normed/Group/BallSphere.lean
|
coe_neg_closedBall
|
[] |
[
61,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
9
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.canonicalEquiv_trans_canonicalEquiv
|
[] |
[
941,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
938,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.conj_re
|
[] |
[
340,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
339,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
TensorProduct.map_comp
|
[] |
[
757,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
755,
1
] |
Std/Data/RBMap/Lemmas.lean
|
Std.RBNode.Path.fill_toList
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nt : RBNode α\np : Path α\n⊢ toList (fill p t) = withList p (toList t)",
"tactic": "induction p generalizing t <;> simp [*]"
}
] |
[
485,
42
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
484,
9
] |
Mathlib/Order/Bounds/Basic.lean
|
IsGLB.of_image
|
[] |
[
1558,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1555,
1
] |
Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean
|
Asymptotics.SuperpolynomialDecay.trans_abs_le
|
[] |
[
194,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.seq_assoc
|
[
{
"state_after": "case refine'_1\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.294746\nι : Sort x\nx : Filter α\ng : Filter (α → β)\nh : Filter (β → γ)\ns : Set (α → γ)\nhs : s ∈ seq (map (fun x x_1 => x ∘ x_1) h) g\nt : Set α\nht : t ∈ x\n⊢ Set.seq s t ∈ seq h (seq g x)\n\ncase refine'_2\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.294746\nι : Sort x\nx : Filter α\ng : Filter (α → β)\nh : Filter (β → γ)\ns : Set (β → γ)\nhs : s ∈ h\nt : Set β\nht : t ∈ seq g x\n⊢ Set.seq s t ∈ seq (seq (map (fun x x_1 => x ∘ x_1) h) g) x",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.294746\nι : Sort x\nx : Filter α\ng : Filter (α → β)\nh : Filter (β → γ)\n⊢ seq h (seq g x) = seq (seq (map (fun x x_1 => x ∘ x_1) h) g) x",
"tactic": "refine' le_antisymm (le_seq fun s hs t ht => _) (le_seq fun s hs t ht => _)"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.294746\nι : Sort x\nx : Filter α\ng : Filter (α → β)\nh : Filter (β → γ)\ns : Set (α → γ)\nhs✝ : s ∈ seq (map (fun x x_1 => x ∘ x_1) h) g\nt : Set α\nht : t ∈ x\nu : Set ((α → β) → α → γ)\nhu : u ∈ map (fun x x_1 => x ∘ x_1) h\nv : Set (α → β)\nhv : v ∈ g\nhs : Set.seq u v ⊆ s\n⊢ Set.seq s t ∈ seq h (seq g x)",
"state_before": "case refine'_1\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.294746\nι : Sort x\nx : Filter α\ng : Filter (α → β)\nh : Filter (β → γ)\ns : Set (α → γ)\nhs : s ∈ seq (map (fun x x_1 => x ∘ x_1) h) g\nt : Set α\nht : t ∈ x\n⊢ Set.seq s t ∈ seq h (seq g x)",
"tactic": "rcases mem_seq_iff.1 hs with ⟨u, hu, v, hv, hs⟩"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.294746\nι : Sort x\nx : Filter α\ng : Filter (α → β)\nh : Filter (β → γ)\ns : Set (α → γ)\nhs✝ : s ∈ seq (map (fun x x_1 => x ∘ x_1) h) g\nt : Set α\nht : t ∈ x\nu : Set ((α → β) → α → γ)\nhu✝ : u ∈ map (fun x x_1 => x ∘ x_1) h\nv : Set (α → β)\nhv : v ∈ g\nhs : Set.seq u v ⊆ s\nw : Set (β → γ)\nhw : w ∈ h\nhu : (fun x x_1 => x ∘ x_1) '' w ⊆ u\n⊢ Set.seq s t ∈ seq h (seq g x)",
"state_before": "case refine'_1.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.294746\nι : Sort x\nx : Filter α\ng : Filter (α → β)\nh : Filter (β → γ)\ns : Set (α → γ)\nhs✝ : s ∈ seq (map (fun x x_1 => x ∘ x_1) h) g\nt : Set α\nht : t ∈ x\nu : Set ((α → β) → α → γ)\nhu : u ∈ map (fun x x_1 => x ∘ x_1) h\nv : Set (α → β)\nhv : v ∈ g\nhs : Set.seq u v ⊆ s\n⊢ Set.seq s t ∈ seq h (seq g x)",
"tactic": "rcases mem_map_iff_exists_image.1 hu with ⟨w, hw, hu⟩"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.294746\nι : Sort x\nx : Filter α\ng : Filter (α → β)\nh : Filter (β → γ)\ns : Set (α → γ)\nhs✝ : s ∈ seq (map (fun x x_1 => x ∘ x_1) h) g\nt : Set α\nht : t ∈ x\nu : Set ((α → β) → α → γ)\nhu✝ : u ∈ map (fun x x_1 => x ∘ x_1) h\nv : Set (α → β)\nhv : v ∈ g\nhs : Set.seq u v ⊆ s\nw : Set (β → γ)\nhw : w ∈ h\nhu : (fun x x_1 => x ∘ x_1) '' w ⊆ u\n⊢ Set.seq (Set.seq ((fun x x_1 => x ∘ x_1) '' w) v) t ∈ seq h (seq g x)",
"state_before": "case refine'_1.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.294746\nι : Sort x\nx : Filter α\ng : Filter (α → β)\nh : Filter (β → γ)\ns : Set (α → γ)\nhs✝ : s ∈ seq (map (fun x x_1 => x ∘ x_1) h) g\nt : Set α\nht : t ∈ x\nu : Set ((α → β) → α → γ)\nhu✝ : u ∈ map (fun x x_1 => x ∘ x_1) h\nv : Set (α → β)\nhv : v ∈ g\nhs : Set.seq u v ⊆ s\nw : Set (β → γ)\nhw : w ∈ h\nhu : (fun x x_1 => x ∘ x_1) '' w ⊆ u\n⊢ Set.seq s t ∈ seq h (seq g x)",
"tactic": "refine' mem_of_superset _ (Set.seq_mono ((Set.seq_mono hu Subset.rfl).trans hs) Subset.rfl)"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.294746\nι : Sort x\nx : Filter α\ng : Filter (α → β)\nh : Filter (β → γ)\ns : Set (α → γ)\nhs✝ : s ∈ seq (map (fun x x_1 => x ∘ x_1) h) g\nt : Set α\nht : t ∈ x\nu : Set ((α → β) → α → γ)\nhu✝ : u ∈ map (fun x x_1 => x ∘ x_1) h\nv : Set (α → β)\nhv : v ∈ g\nhs : Set.seq u v ⊆ s\nw : Set (β → γ)\nhw : w ∈ h\nhu : (fun x x_1 => x ∘ x_1) '' w ⊆ u\n⊢ Set.seq w (Set.seq v t) ∈ seq h (seq g x)",
"state_before": "case refine'_1.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.294746\nι : Sort x\nx : Filter α\ng : Filter (α → β)\nh : Filter (β → γ)\ns : Set (α → γ)\nhs✝ : s ∈ seq (map (fun x x_1 => x ∘ x_1) h) g\nt : Set α\nht : t ∈ x\nu : Set ((α → β) → α → γ)\nhu✝ : u ∈ map (fun x x_1 => x ∘ x_1) h\nv : Set (α → β)\nhv : v ∈ g\nhs : Set.seq u v ⊆ s\nw : Set (β → γ)\nhw : w ∈ h\nhu : (fun x x_1 => x ∘ x_1) '' w ⊆ u\n⊢ Set.seq (Set.seq ((fun x x_1 => x ∘ x_1) '' w) v) t ∈ seq h (seq g x)",
"tactic": "rw [← Set.seq_seq]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.294746\nι : Sort x\nx : Filter α\ng : Filter (α → β)\nh : Filter (β → γ)\ns : Set (α → γ)\nhs✝ : s ∈ seq (map (fun x x_1 => x ∘ x_1) h) g\nt : Set α\nht : t ∈ x\nu : Set ((α → β) → α → γ)\nhu✝ : u ∈ map (fun x x_1 => x ∘ x_1) h\nv : Set (α → β)\nhv : v ∈ g\nhs : Set.seq u v ⊆ s\nw : Set (β → γ)\nhw : w ∈ h\nhu : (fun x x_1 => x ∘ x_1) '' w ⊆ u\n⊢ Set.seq w (Set.seq v t) ∈ seq h (seq g x)",
"tactic": "exact seq_mem_seq hw (seq_mem_seq hv ht)"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.294746\nι : Sort x\nx : Filter α\ng : Filter (α → β)\nh : Filter (β → γ)\ns : Set (β → γ)\nhs : s ∈ h\nt : Set β\nht✝ : t ∈ seq g x\nu : Set (α → β)\nhu : u ∈ g\nv : Set α\nhv : v ∈ x\nht : Set.seq u v ⊆ t\n⊢ Set.seq s t ∈ seq (seq (map (fun x x_1 => x ∘ x_1) h) g) x",
"state_before": "case refine'_2\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.294746\nι : Sort x\nx : Filter α\ng : Filter (α → β)\nh : Filter (β → γ)\ns : Set (β → γ)\nhs : s ∈ h\nt : Set β\nht : t ∈ seq g x\n⊢ Set.seq s t ∈ seq (seq (map (fun x x_1 => x ∘ x_1) h) g) x",
"tactic": "rcases mem_seq_iff.1 ht with ⟨u, hu, v, hv, ht⟩"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.294746\nι : Sort x\nx : Filter α\ng : Filter (α → β)\nh : Filter (β → γ)\ns : Set (β → γ)\nhs : s ∈ h\nt : Set β\nht✝ : t ∈ seq g x\nu : Set (α → β)\nhu : u ∈ g\nv : Set α\nhv : v ∈ x\nht : Set.seq u v ⊆ t\n⊢ Set.seq s (Set.seq u v) ∈ seq (seq (map (fun x x_1 => x ∘ x_1) h) g) x",
"state_before": "case refine'_2.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.294746\nι : Sort x\nx : Filter α\ng : Filter (α → β)\nh : Filter (β → γ)\ns : Set (β → γ)\nhs : s ∈ h\nt : Set β\nht✝ : t ∈ seq g x\nu : Set (α → β)\nhu : u ∈ g\nv : Set α\nhv : v ∈ x\nht : Set.seq u v ⊆ t\n⊢ Set.seq s t ∈ seq (seq (map (fun x x_1 => x ∘ x_1) h) g) x",
"tactic": "refine' mem_of_superset _ (Set.seq_mono Subset.rfl ht)"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.294746\nι : Sort x\nx : Filter α\ng : Filter (α → β)\nh : Filter (β → γ)\ns : Set (β → γ)\nhs : s ∈ h\nt : Set β\nht✝ : t ∈ seq g x\nu : Set (α → β)\nhu : u ∈ g\nv : Set α\nhv : v ∈ x\nht : Set.seq u v ⊆ t\n⊢ Set.seq (Set.seq ((fun x x_1 => x ∘ x_1) '' s) u) v ∈ seq (seq (map (fun x x_1 => x ∘ x_1) h) g) x",
"state_before": "case refine'_2.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.294746\nι : Sort x\nx : Filter α\ng : Filter (α → β)\nh : Filter (β → γ)\ns : Set (β → γ)\nhs : s ∈ h\nt : Set β\nht✝ : t ∈ seq g x\nu : Set (α → β)\nhu : u ∈ g\nv : Set α\nhv : v ∈ x\nht : Set.seq u v ⊆ t\n⊢ Set.seq s (Set.seq u v) ∈ seq (seq (map (fun x x_1 => x ∘ x_1) h) g) x",
"tactic": "rw [Set.seq_seq]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.294746\nι : Sort x\nx : Filter α\ng : Filter (α → β)\nh : Filter (β → γ)\ns : Set (β → γ)\nhs : s ∈ h\nt : Set β\nht✝ : t ∈ seq g x\nu : Set (α → β)\nhu : u ∈ g\nv : Set α\nhv : v ∈ x\nht : Set.seq u v ⊆ t\n⊢ Set.seq (Set.seq ((fun x x_1 => x ∘ x_1) '' s) u) v ∈ seq (seq (map (fun x x_1 => x ∘ x_1) h) g) x",
"tactic": "exact seq_mem_seq (seq_mem_seq (image_mem_map hs) hu) hv"
}
] |
[
2677,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2666,
1
] |
Mathlib/Topology/Algebra/WithZeroTopology.lean
|
WithZeroTopology.isClosed_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.91489\nΓ₀ : Type u_1\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nγ γ₁ γ₂ : Γ₀\nl : Filter α\nf : α → Γ₀\ns : Set Γ₀\n⊢ IsClosed s ↔ 0 ∈ s ∨ ∃ γ, γ ≠ 0 ∧ s ⊆ Ici γ",
"tactic": "simp only [← isOpen_compl_iff, isOpen_iff, mem_compl_iff, not_not, ← compl_Ici,\n compl_subset_compl]"
}
] |
[
147,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
le_csInf_iff'
|
[] |
[
979,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
978,
1
] |
Mathlib/Data/Fin/Tuple/Basic.lean
|
Fin.update_cons_zero
|
[
{
"state_after": "case h\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\n⊢ update (cons x p) 0 z j = cons z p j",
"state_before": "m n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\n⊢ update (cons x p) 0 z = cons z p",
"tactic": "ext j"
},
{
"state_after": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : j = 0\n⊢ update (cons x p) 0 z j = cons z p j\n\ncase neg\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : ¬j = 0\n⊢ update (cons x p) 0 z j = cons z p j",
"state_before": "case h\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\n⊢ update (cons x p) 0 z j = cons z p j",
"tactic": "by_cases h : j = 0"
},
{
"state_after": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : j = 0\n⊢ update (cons x p) 0 z 0 = cons z p 0",
"state_before": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : j = 0\n⊢ update (cons x p) 0 z j = cons z p j",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : j = 0\n⊢ update (cons x p) 0 z 0 = cons z p 0",
"tactic": "simp"
},
{
"state_after": "case neg\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : ¬j = 0\n⊢ cons x p j = cons z p j",
"state_before": "case neg\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : ¬j = 0\n⊢ update (cons x p) 0 z j = cons z p j",
"tactic": "simp only [h, update_noteq, Ne.def, not_false_iff]"
},
{
"state_after": "case neg\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : ¬j = 0\nj' : Fin n := pred j h\n⊢ cons x p j = cons z p j",
"state_before": "case neg\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : ¬j = 0\n⊢ cons x p j = cons z p j",
"tactic": "let j' := pred j h"
},
{
"state_after": "case neg\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : ¬j = 0\nj' : Fin n := pred j h\nthis : succ j' = j\n⊢ cons x p j = cons z p j",
"state_before": "case neg\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : ¬j = 0\nj' : Fin n := pred j h\n⊢ cons x p j = cons z p j",
"tactic": "have : j'.succ = j := succ_pred j h"
},
{
"state_after": "no goals",
"state_before": "case neg\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α 0\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (succ i)\ni : Fin n\ny : α (succ i)\nz : α 0\nj : Fin (n + 1)\nh : ¬j = 0\nj' : Fin n := pred j h\nthis : succ j' = j\n⊢ cons x p j = cons z p j",
"tactic": "rw [← this, cons_succ, cons_succ]"
}
] |
[
131,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Topology/UniformSpace/Separation.lean
|
separated_equiv
|
[] |
[
108,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.coe_deriv₂
|
[] |
[
1378,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1376,
1
] |
Mathlib/Order/Hom/Bounded.lean
|
BotHom.copy_eq
|
[] |
[
418,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
417,
1
] |
Std/Data/List/Lemmas.lean
|
List.next?_nil
|
[] |
[
456,
54
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
456,
9
] |
Mathlib/Algebra/Category/ModuleCat/Images.lean
|
ModuleCat.image.fac
|
[] |
[
59,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.mapDomain_equiv_apply
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.172677\nι : Type ?u.172680\nM : Type u_3\nM' : Type ?u.172686\nN : Type ?u.172689\nP : Type ?u.172692\nG : Type ?u.172695\nH : Type ?u.172698\nR : Type ?u.172701\nS : Type ?u.172704\ninst✝ : AddCommMonoid M\nv v₁ v₂ : α →₀ M\nf : α ≃ β\nx : α →₀ M\na : β\n⊢ ↑(mapDomain (↑f) x) (↑f (↑f.symm a)) = ↑x (↑f.symm a)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.172677\nι : Type ?u.172680\nM : Type u_3\nM' : Type ?u.172686\nN : Type ?u.172689\nP : Type ?u.172692\nG : Type ?u.172695\nH : Type ?u.172698\nR : Type ?u.172701\nS : Type ?u.172704\ninst✝ : AddCommMonoid M\nv v₁ v₂ : α →₀ M\nf : α ≃ β\nx : α →₀ M\na : β\n⊢ ↑(mapDomain (↑f) x) a = ↑x (↑f.symm a)",
"tactic": "conv_lhs => rw [← f.apply_symm_apply a]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.172677\nι : Type ?u.172680\nM : Type u_3\nM' : Type ?u.172686\nN : Type ?u.172689\nP : Type ?u.172692\nG : Type ?u.172695\nH : Type ?u.172698\nR : Type ?u.172701\nS : Type ?u.172704\ninst✝ : AddCommMonoid M\nv v₁ v₂ : α →₀ M\nf : α ≃ β\nx : α →₀ M\na : β\n⊢ ↑(mapDomain (↑f) x) (↑f (↑f.symm a)) = ↑x (↑f.symm a)",
"tactic": "exact mapDomain_apply f.injective _ _"
}
] |
[
506,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
503,
1
] |
Mathlib/LinearAlgebra/Dual.lean
|
Basis.toDual_inj
|
[
{
"state_after": "R : Type u\nM : Type v\nK : Type u₁\nV : Type u₂\nι : Type w\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : DecidableEq ι\nb : Basis ι R M\nm : M\na : ↑(toDual b) m = 0\n⊢ ↑b.repr m = 0",
"state_before": "R : Type u\nM : Type v\nK : Type u₁\nV : Type u₂\nι : Type w\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : DecidableEq ι\nb : Basis ι R M\nm : M\na : ↑(toDual b) m = 0\n⊢ m = 0",
"tactic": "rw [← mem_bot R, ← b.repr.ker, mem_ker, LinearEquiv.coe_coe]"
},
{
"state_after": "case h\nR : Type u\nM : Type v\nK : Type u₁\nV : Type u₂\nι : Type w\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : DecidableEq ι\nb : Basis ι R M\nm : M\na : ↑(toDual b) m = 0\n⊢ ∀ (a : ι), ↑(↑b.repr m) a = ↑0 a",
"state_before": "R : Type u\nM : Type v\nK : Type u₁\nV : Type u₂\nι : Type w\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : DecidableEq ι\nb : Basis ι R M\nm : M\na : ↑(toDual b) m = 0\n⊢ ↑b.repr m = 0",
"tactic": "apply Finsupp.ext"
},
{
"state_after": "case h\nR : Type u\nM : Type v\nK : Type u₁\nV : Type u₂\nι : Type w\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : DecidableEq ι\nb✝ : Basis ι R M\nm : M\na : ↑(toDual b✝) m = 0\nb : ι\n⊢ ↑(↑b✝.repr m) b = ↑0 b",
"state_before": "case h\nR : Type u\nM : Type v\nK : Type u₁\nV : Type u₂\nι : Type w\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : DecidableEq ι\nb : Basis ι R M\nm : M\na : ↑(toDual b) m = 0\n⊢ ∀ (a : ι), ↑(↑b.repr m) a = ↑0 a",
"tactic": "intro b"
},
{
"state_after": "case h\nR : Type u\nM : Type v\nK : Type u₁\nV : Type u₂\nι : Type w\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : DecidableEq ι\nb✝ : Basis ι R M\nm : M\na : ↑(toDual b✝) m = 0\nb : ι\n⊢ ↑0 (↑b✝ b) = ↑0 b",
"state_before": "case h\nR : Type u\nM : Type v\nK : Type u₁\nV : Type u₂\nι : Type w\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : DecidableEq ι\nb✝ : Basis ι R M\nm : M\na : ↑(toDual b✝) m = 0\nb : ι\n⊢ ↑(↑b✝.repr m) b = ↑0 b",
"tactic": "rw [← toDual_eq_repr, a]"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\nM : Type v\nK : Type u₁\nV : Type u₂\nι : Type w\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : DecidableEq ι\nb✝ : Basis ι R M\nm : M\na : ↑(toDual b✝) m = 0\nb : ι\n⊢ ↑0 (↑b✝ b) = ↑0 b",
"tactic": "rfl"
}
] |
[
358,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.insert_inj
|
[] |
[
1172,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1171,
1
] |
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