file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Analysis/Normed/Group/Hom.lean
|
NormedAddGroupHom.Equalizer.map_normNoninc
|
[] |
[
998,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
996,
1
] |
Mathlib/Analysis/Convex/Exposed.lean
|
exposed_point_def
|
[] |
[
210,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
isLUB_empty
|
[] |
[
896,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
895,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.not_mem_union
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.141391\nγ : Type ?u.141394\ninst✝ : DecidableEq α\ns s₁ s₂ t t₁ t₂ u v : Finset α\na b : α\n⊢ ¬a ∈ s ∪ t ↔ ¬a ∈ s ∧ ¬a ∈ t",
"tactic": "rw [mem_union, not_or]"
}
] |
[
1362,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1362,
1
] |
Mathlib/GroupTheory/QuotientGroup.lean
|
QuotientGroup.map_id
|
[] |
[
260,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Real.nndist_eq'
|
[] |
[
1360,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1359,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
dist_dist_dist_le_left
|
[] |
[
1777,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1776,
1
] |
Mathlib/LinearAlgebra/Matrix/Determinant.lean
|
Matrix.det_updateColumn_smul
|
[
{
"state_after": "m : Type ?u.1649973\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nM : Matrix n n R\nj : n\ns : R\nu : n → R\n⊢ s * det (updateRow Mᵀ j u) = s * det (updateColumn M j u)",
"state_before": "m : Type ?u.1649973\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nM : Matrix n n R\nj : n\ns : R\nu : n → R\n⊢ det (updateColumn M j (s • u)) = s * det (updateColumn M j u)",
"tactic": "rw [← det_transpose, ← updateRow_transpose, det_updateRow_smul]"
},
{
"state_after": "no goals",
"state_before": "m : Type ?u.1649973\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nM : Matrix n n R\nj : n\ns : R\nu : n → R\n⊢ s * det (updateRow Mᵀ j u) = s * det (updateColumn M j u)",
"tactic": "simp [updateRow_transpose, det_transpose]"
}
] |
[
410,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.mulVec_smul
|
[
{
"state_after": "case h\nl : Type ?u.867277\nm : Type u_4\nn : Type u_1\no : Type ?u.867286\nm' : o → Type ?u.867291\nn' : o → Type ?u.867296\nR : Type u_2\nS : Type u_3\nα : Type v\nβ : Type w\nγ : Type ?u.867309\ninst✝⁵ : NonUnitalNonAssocSemiring α\ninst✝⁴ : Fintype n\ninst✝³ : Monoid R\ninst✝² : NonUnitalNonAssocSemiring S\ninst✝¹ : DistribMulAction R S\ninst✝ : SMulCommClass R S S\nM : Matrix m n S\nb : R\nv : n → S\ni : m\n⊢ mulVec M (b • v) i = (b • mulVec M v) i",
"state_before": "l : Type ?u.867277\nm : Type u_4\nn : Type u_1\no : Type ?u.867286\nm' : o → Type ?u.867291\nn' : o → Type ?u.867296\nR : Type u_2\nS : Type u_3\nα : Type v\nβ : Type w\nγ : Type ?u.867309\ninst✝⁵ : NonUnitalNonAssocSemiring α\ninst✝⁴ : Fintype n\ninst✝³ : Monoid R\ninst✝² : NonUnitalNonAssocSemiring S\ninst✝¹ : DistribMulAction R S\ninst✝ : SMulCommClass R S S\nM : Matrix m n S\nb : R\nv : n → S\n⊢ mulVec M (b • v) = b • mulVec M v",
"tactic": "ext i"
},
{
"state_after": "no goals",
"state_before": "case h\nl : Type ?u.867277\nm : Type u_4\nn : Type u_1\no : Type ?u.867286\nm' : o → Type ?u.867291\nn' : o → Type ?u.867296\nR : Type u_2\nS : Type u_3\nα : Type v\nβ : Type w\nγ : Type ?u.867309\ninst✝⁵ : NonUnitalNonAssocSemiring α\ninst✝⁴ : Fintype n\ninst✝³ : Monoid R\ninst✝² : NonUnitalNonAssocSemiring S\ninst✝¹ : DistribMulAction R S\ninst✝ : SMulCommClass R S S\nM : Matrix m n S\nb : R\nv : n → S\ni : m\n⊢ mulVec M (b • v) i = (b • mulVec M v) i",
"tactic": "simp only [mulVec, dotProduct, Finset.smul_sum, Pi.smul_apply, mul_smul_comm]"
}
] |
[
1783,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1779,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.sum_add
|
[] |
[
999,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
997,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
|
derivWithin_rpow_const
|
[] |
[
595,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
592,
1
] |
Mathlib/Data/Multiset/LocallyFinite.lean
|
Multiset.Ioc_self
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ Ioc a a = 0",
"tactic": "rw [Ioc, Finset.Ioc_self, Finset.empty_val]"
}
] |
[
108,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/RingTheory/OreLocalization/Basic.lean
|
OreLocalization.add_left_neg
|
[
{
"state_after": "case c\nR : Type u_1\ninst✝¹ : Ring R\nS : Submonoid R\ninst✝ : OreSet S\nr : R\ns : { x // x ∈ S }\n⊢ -(r /ₒ s) + r /ₒ s = 0",
"state_before": "R : Type u_1\ninst✝¹ : Ring R\nS : Submonoid R\ninst✝ : OreSet S\nx : OreLocalization R S\n⊢ -x + x = 0",
"tactic": "induction' x using OreLocalization.ind with r s"
},
{
"state_after": "no goals",
"state_before": "case c\nR : Type u_1\ninst✝¹ : Ring R\nS : Submonoid R\ninst✝ : OreSet S\nr : R\ns : { x // x ∈ S }\n⊢ -(r /ₒ s) + r /ₒ s = 0",
"tactic": "simp"
}
] |
[
852,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
851,
11
] |
Mathlib/Data/Set/Function.lean
|
Function.Injective.injOn_range
|
[
{
"state_after": "case intro.intro\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\nι : Sort ?u.34009\nπ : α → Type ?u.34014\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nh : Injective (g ∘ f)\nx y : α\nH : g (f x) = g (f y)\n⊢ f x = f y",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nι : Sort ?u.34009\nπ : α → Type ?u.34014\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nh : Injective (g ∘ f)\n⊢ InjOn g (range f)",
"tactic": "rintro _ ⟨x, rfl⟩ _ ⟨y, rfl⟩ H"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nβ : Type u_3\nγ : Type u_2\nι : Sort ?u.34009\nπ : α → Type ?u.34014\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\nh : Injective (g ∘ f)\nx y : α\nH : g (f x) = g (f y)\n⊢ f x = f y",
"tactic": "exact congr_arg f (h H)"
}
] |
[
679,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
677,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.uniformity_basis_dist_pow
|
[] |
[
767,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
763,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.submatrix_diagonal
|
[
{
"state_after": "l : Type u_2\nm : Type u_1\nn : Type ?u.1089606\no : Type ?u.1089609\nm' : o → Type ?u.1089614\nn' : o → Type ?u.1089619\nR : Type ?u.1089622\nS : Type ?u.1089625\nα : Type v\nβ : Type w\nγ : Type ?u.1089632\ninst✝² : Zero α\ninst✝¹ : DecidableEq m\ninst✝ : DecidableEq l\nd : m → α\ne : l → m\nhe : Function.Injective e\ni j : l\n⊢ diagonal d (e i) (e j) = diagonal (d ∘ e) i j",
"state_before": "l : Type u_2\nm : Type u_1\nn : Type ?u.1089606\no : Type ?u.1089609\nm' : o → Type ?u.1089614\nn' : o → Type ?u.1089619\nR : Type ?u.1089622\nS : Type ?u.1089625\nα : Type v\nβ : Type w\nγ : Type ?u.1089632\ninst✝² : Zero α\ninst✝¹ : DecidableEq m\ninst✝ : DecidableEq l\nd : m → α\ne : l → m\nhe : Function.Injective e\ni j : l\n⊢ submatrix (diagonal d) e e i j = diagonal (d ∘ e) i j",
"tactic": "rw [submatrix_apply]"
},
{
"state_after": "case pos\nl : Type u_2\nm : Type u_1\nn : Type ?u.1089606\no : Type ?u.1089609\nm' : o → Type ?u.1089614\nn' : o → Type ?u.1089619\nR : Type ?u.1089622\nS : Type ?u.1089625\nα : Type v\nβ : Type w\nγ : Type ?u.1089632\ninst✝² : Zero α\ninst✝¹ : DecidableEq m\ninst✝ : DecidableEq l\nd : m → α\ne : l → m\nhe : Function.Injective e\ni j : l\nh : i = j\n⊢ diagonal d (e i) (e j) = diagonal (d ∘ e) i j\n\ncase neg\nl : Type u_2\nm : Type u_1\nn : Type ?u.1089606\no : Type ?u.1089609\nm' : o → Type ?u.1089614\nn' : o → Type ?u.1089619\nR : Type ?u.1089622\nS : Type ?u.1089625\nα : Type v\nβ : Type w\nγ : Type ?u.1089632\ninst✝² : Zero α\ninst✝¹ : DecidableEq m\ninst✝ : DecidableEq l\nd : m → α\ne : l → m\nhe : Function.Injective e\ni j : l\nh : ¬i = j\n⊢ diagonal d (e i) (e j) = diagonal (d ∘ e) i j",
"state_before": "l : Type u_2\nm : Type u_1\nn : Type ?u.1089606\no : Type ?u.1089609\nm' : o → Type ?u.1089614\nn' : o → Type ?u.1089619\nR : Type ?u.1089622\nS : Type ?u.1089625\nα : Type v\nβ : Type w\nγ : Type ?u.1089632\ninst✝² : Zero α\ninst✝¹ : DecidableEq m\ninst✝ : DecidableEq l\nd : m → α\ne : l → m\nhe : Function.Injective e\ni j : l\n⊢ diagonal d (e i) (e j) = diagonal (d ∘ e) i j",
"tactic": "by_cases h : i = j"
},
{
"state_after": "case pos\nl : Type u_2\nm : Type u_1\nn : Type ?u.1089606\no : Type ?u.1089609\nm' : o → Type ?u.1089614\nn' : o → Type ?u.1089619\nR : Type ?u.1089622\nS : Type ?u.1089625\nα : Type v\nβ : Type w\nγ : Type ?u.1089632\ninst✝² : Zero α\ninst✝¹ : DecidableEq m\ninst✝ : DecidableEq l\nd : m → α\ne : l → m\nhe : Function.Injective e\ni j : l\nh : i = j\n⊢ d (e j) = (d ∘ e) j",
"state_before": "case pos\nl : Type u_2\nm : Type u_1\nn : Type ?u.1089606\no : Type ?u.1089609\nm' : o → Type ?u.1089614\nn' : o → Type ?u.1089619\nR : Type ?u.1089622\nS : Type ?u.1089625\nα : Type v\nβ : Type w\nγ : Type ?u.1089632\ninst✝² : Zero α\ninst✝¹ : DecidableEq m\ninst✝ : DecidableEq l\nd : m → α\ne : l → m\nhe : Function.Injective e\ni j : l\nh : i = j\n⊢ diagonal d (e i) (e j) = diagonal (d ∘ e) i j",
"tactic": "rw [h, diagonal_apply_eq, diagonal_apply_eq]"
},
{
"state_after": "no goals",
"state_before": "case pos\nl : Type u_2\nm : Type u_1\nn : Type ?u.1089606\no : Type ?u.1089609\nm' : o → Type ?u.1089614\nn' : o → Type ?u.1089619\nR : Type ?u.1089622\nS : Type ?u.1089625\nα : Type v\nβ : Type w\nγ : Type ?u.1089632\ninst✝² : Zero α\ninst✝¹ : DecidableEq m\ninst✝ : DecidableEq l\nd : m → α\ne : l → m\nhe : Function.Injective e\ni j : l\nh : i = j\n⊢ d (e j) = (d ∘ e) j",
"tactic": "simp only [Function.comp_apply]"
},
{
"state_after": "no goals",
"state_before": "case neg\nl : Type u_2\nm : Type u_1\nn : Type ?u.1089606\no : Type ?u.1089609\nm' : o → Type ?u.1089614\nn' : o → Type ?u.1089619\nR : Type ?u.1089622\nS : Type ?u.1089625\nα : Type v\nβ : Type w\nγ : Type ?u.1089632\ninst✝² : Zero α\ninst✝¹ : DecidableEq m\ninst✝ : DecidableEq l\nd : m → α\ne : l → m\nhe : Function.Injective e\ni j : l\nh : ¬i = j\n⊢ diagonal d (e i) (e j) = diagonal (d ∘ e) i j",
"tactic": "rw [diagonal_apply_ne _ h, diagonal_apply_ne _ (he.ne h)]"
}
] |
[
2452,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2445,
1
] |
Mathlib/Data/Finset/NAry.lean
|
Finset.image₂_singleton_left'
|
[] |
[
164,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/Topology/ContinuousFunction/Basic.lean
|
ContinuousMap.const_comp
|
[] |
[
267,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/Dynamics/PeriodicPts.lean
|
Function.IsPeriodicPt.const_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.2402\nf fa : α → α\nfb : β → β\nx y : α\nm n✝ : ℕ\nhm : IsPeriodicPt f m x\nn : ℕ\n⊢ IsPeriodicPt f (n * m) x",
"tactic": "simp only [mul_comm n, hm.mul_const n]"
}
] |
[
130,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
11
] |
Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean
|
ProjectiveSpectrum.mem_compl_zeroLocus_iff_not_mem
|
[
{
"state_after": "R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nI : ProjectiveSpectrum 𝒜\n⊢ ¬f ∈ ↑I.asHomogeneousIdeal ↔ ¬f ∈ I.asHomogeneousIdeal",
"state_before": "R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nI : ProjectiveSpectrum 𝒜\n⊢ I ∈ zeroLocus 𝒜 {f}ᶜ ↔ ¬f ∈ I.asHomogeneousIdeal",
"tactic": "rw [Set.mem_compl_iff, mem_zeroLocus, Set.singleton_subset_iff]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nf : A\nI : ProjectiveSpectrum 𝒜\n⊢ ¬f ∈ ↑I.asHomogeneousIdeal ↔ ¬f ∈ I.asHomogeneousIdeal",
"tactic": "rfl"
}
] |
[
323,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
321,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Submodule.smul_induction_on'
|
[
{
"state_after": "R : Type u\nM : Type v\nF : Type ?u.42169\nG : Type ?u.42172\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nI J : Ideal R\nN P : Submodule R M\nx : M\nhx : x ∈ I • N\np : (x : M) → x ∈ I • N → Prop\nHb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (_ : r • n ∈ I • N)\nH1 : ∀ (x : M) (hx : x ∈ I • N) (y : M) (hy : y ∈ I • N), p x hx → p y hy → p (x + y) (_ : x + y ∈ I • N)\n⊢ ∃ x_1, p x x_1",
"state_before": "R : Type u\nM : Type v\nF : Type ?u.42169\nG : Type ?u.42172\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nI J : Ideal R\nN P : Submodule R M\nx : M\nhx : x ∈ I • N\np : (x : M) → x ∈ I • N → Prop\nHb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (_ : r • n ∈ I • N)\nH1 : ∀ (x : M) (hx : x ∈ I • N) (y : M) (hy : y ∈ I • N), p x hx → p y hy → p (x + y) (_ : x + y ∈ I • N)\n⊢ p x hx",
"tactic": "refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nM : Type v\nF : Type ?u.42169\nG : Type ?u.42172\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nI J : Ideal R\nN P : Submodule R M\nx : M\nhx : x ∈ I • N\np : (x : M) → x ∈ I • N → Prop\nHb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (_ : r • n ∈ I • N)\nH1 : ∀ (x : M) (hx : x ∈ I • N) (y : M) (hy : y ∈ I • N), p x hx → p y hy → p (x + y) (_ : x + y ∈ I • N)\n⊢ ∃ x_1, p x x_1",
"tactic": "exact\n smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ =>\n ⟨_, H1 _ _ _ _ hx hy⟩"
}
] |
[
134,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
Summable.indicator
|
[] |
[
1174,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1173,
11
] |
Mathlib/Data/Real/Pointwise.lean
|
Real.mul_iInf_of_nonpos
|
[] |
[
127,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/RingTheory/DedekindDomain/Factorization.lean
|
Ideal.finprod_not_dvd
|
[
{
"state_after": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.235854\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nhf : Set.Finite (mulSupport fun v => maxPowDividing v I)\n⊢ ¬v.asIdeal ^ (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) + 1) ∣\n ∏ᶠ (v : HeightOneSpectrum R), maxPowDividing v I",
"state_before": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.235854\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\n⊢ ¬v.asIdeal ^ (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) + 1) ∣\n ∏ᶠ (v : HeightOneSpectrum R), maxPowDividing v I",
"tactic": "have hf := finite_mulSupport hI"
},
{
"state_after": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.235854\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nhf : Set.Finite (mulSupport fun v => maxPowDividing v I)\nh_ne_zero : maxPowDividing v I ≠ 0\n⊢ ¬v.asIdeal ^ (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) + 1) ∣\n ∏ᶠ (v : HeightOneSpectrum R), maxPowDividing v I",
"state_before": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.235854\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nhf : Set.Finite (mulSupport fun v => maxPowDividing v I)\n⊢ ¬v.asIdeal ^ (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) + 1) ∣\n ∏ᶠ (v : HeightOneSpectrum R), maxPowDividing v I",
"tactic": "have h_ne_zero : v.maxPowDividing I ≠ 0 := pow_ne_zero _ v.ne_bot"
},
{
"state_after": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.235854\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nhf : Set.Finite (mulSupport fun v => maxPowDividing v I)\nh_ne_zero : maxPowDividing v I ≠ 0\n⊢ ¬v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) * v.asIdeal ∣\n maxPowDividing v I * ∏ i in Finset.erase (Finite.toFinset hf) v, maxPowDividing i I",
"state_before": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.235854\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nhf : Set.Finite (mulSupport fun v => maxPowDividing v I)\nh_ne_zero : maxPowDividing v I ≠ 0\n⊢ ¬v.asIdeal ^ (Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) + 1) ∣\n ∏ᶠ (v : HeightOneSpectrum R), maxPowDividing v I",
"tactic": "rw [← mul_finprod_cond_ne v hf, pow_add, pow_one, finprod_cond_ne _ _ hf]"
},
{
"state_after": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.235854\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nhf : Set.Finite (mulSupport fun v => maxPowDividing v I)\nh_ne_zero : maxPowDividing v I ≠ 0\nh_contr :\n v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) * v.asIdeal ∣\n maxPowDividing v I * ∏ i in Finset.erase (Finite.toFinset hf) v, maxPowDividing i I\n⊢ False",
"state_before": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.235854\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nhf : Set.Finite (mulSupport fun v => maxPowDividing v I)\nh_ne_zero : maxPowDividing v I ≠ 0\n⊢ ¬v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) * v.asIdeal ∣\n maxPowDividing v I * ∏ i in Finset.erase (Finite.toFinset hf) v, maxPowDividing i I",
"tactic": "intro h_contr"
},
{
"state_after": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.235854\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nhf : Set.Finite (mulSupport fun v => maxPowDividing v I)\nh_ne_zero : maxPowDividing v I ≠ 0\nh_contr :\n v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) * v.asIdeal ∣\n maxPowDividing v I * ∏ i in Finset.erase (Finite.toFinset hf) v, maxPowDividing i I\nhv_prime : Prime v.asIdeal\n⊢ False",
"state_before": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.235854\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nhf : Set.Finite (mulSupport fun v => maxPowDividing v I)\nh_ne_zero : maxPowDividing v I ≠ 0\nh_contr :\n v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) * v.asIdeal ∣\n maxPowDividing v I * ∏ i in Finset.erase (Finite.toFinset hf) v, maxPowDividing i I\n⊢ False",
"tactic": "have hv_prime : Prime v.asIdeal := Ideal.prime_of_isPrime v.ne_bot v.IsPrime"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.235854\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nhf : Set.Finite (mulSupport fun v => maxPowDividing v I)\nh_ne_zero : maxPowDividing v I ≠ 0\nh_contr :\n v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) * v.asIdeal ∣\n maxPowDividing v I * ∏ i in Finset.erase (Finite.toFinset hf) v, maxPowDividing i I\nhv_prime : Prime v.asIdeal\nw : HeightOneSpectrum R\nhw : w ∈ Finset.erase (Finite.toFinset hf) v\nhvw' : v.asIdeal ∣ maxPowDividing w I\n⊢ False",
"state_before": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.235854\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nhf : Set.Finite (mulSupport fun v => maxPowDividing v I)\nh_ne_zero : maxPowDividing v I ≠ 0\nh_contr :\n v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) * v.asIdeal ∣\n maxPowDividing v I * ∏ i in Finset.erase (Finite.toFinset hf) v, maxPowDividing i I\nhv_prime : Prime v.asIdeal\n⊢ False",
"tactic": "obtain ⟨w, hw, hvw'⟩ :=\n Prime.exists_mem_finset_dvd hv_prime ((mul_dvd_mul_iff_left h_ne_zero).mp h_contr)"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.235854\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nhf : Set.Finite (mulSupport fun v => maxPowDividing v I)\nh_ne_zero : maxPowDividing v I ≠ 0\nh_contr :\n v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) * v.asIdeal ∣\n maxPowDividing v I * ∏ i in Finset.erase (Finite.toFinset hf) v, maxPowDividing i I\nhv_prime : Prime v.asIdeal\nw : HeightOneSpectrum R\nhw : w ∈ Finset.erase (Finite.toFinset hf) v\nhvw' : v.asIdeal ∣ maxPowDividing w I\nhw_prime : Prime w.asIdeal\n⊢ False",
"state_before": "case intro.intro\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.235854\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nhf : Set.Finite (mulSupport fun v => maxPowDividing v I)\nh_ne_zero : maxPowDividing v I ≠ 0\nh_contr :\n v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) * v.asIdeal ∣\n maxPowDividing v I * ∏ i in Finset.erase (Finite.toFinset hf) v, maxPowDividing i I\nhv_prime : Prime v.asIdeal\nw : HeightOneSpectrum R\nhw : w ∈ Finset.erase (Finite.toFinset hf) v\nhvw' : v.asIdeal ∣ maxPowDividing w I\n⊢ False",
"tactic": "have hw_prime : Prime w.asIdeal := Ideal.prime_of_isPrime w.ne_bot w.IsPrime"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.235854\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nhf : Set.Finite (mulSupport fun v => maxPowDividing v I)\nh_ne_zero : maxPowDividing v I ≠ 0\nh_contr :\n v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) * v.asIdeal ∣\n maxPowDividing v I * ∏ i in Finset.erase (Finite.toFinset hf) v, maxPowDividing i I\nhv_prime : Prime v.asIdeal\nw : HeightOneSpectrum R\nhw : w ∈ Finset.erase (Finite.toFinset hf) v\nhvw' : v.asIdeal ∣ maxPowDividing w I\nhw_prime : Prime w.asIdeal\nhvw : v.asIdeal ∣ w.asIdeal\n⊢ False",
"state_before": "case intro.intro\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.235854\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nhf : Set.Finite (mulSupport fun v => maxPowDividing v I)\nh_ne_zero : maxPowDividing v I ≠ 0\nh_contr :\n v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) * v.asIdeal ∣\n maxPowDividing v I * ∏ i in Finset.erase (Finite.toFinset hf) v, maxPowDividing i I\nhv_prime : Prime v.asIdeal\nw : HeightOneSpectrum R\nhw : w ∈ Finset.erase (Finite.toFinset hf) v\nhvw' : v.asIdeal ∣ maxPowDividing w I\nhw_prime : Prime w.asIdeal\n⊢ False",
"tactic": "have hvw := Prime.dvd_of_dvd_pow hv_prime hvw'"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.235854\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nhf : Set.Finite (mulSupport fun v => maxPowDividing v I)\nh_ne_zero : maxPowDividing v I ≠ 0\nh_contr :\n v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) * v.asIdeal ∣\n maxPowDividing v I * ∏ i in Finset.erase (Finite.toFinset hf) v, maxPowDividing i I\nhv_prime : Prime v.asIdeal\nw : HeightOneSpectrum R\nhw : w ∈ Finset.erase (Finite.toFinset hf) v\nhvw' : v.asIdeal ∣ maxPowDividing w I\nhw_prime : Prime w.asIdeal\nhvw : v.asIdeal = w.asIdeal\n⊢ False",
"state_before": "case intro.intro\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.235854\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nhf : Set.Finite (mulSupport fun v => maxPowDividing v I)\nh_ne_zero : maxPowDividing v I ≠ 0\nh_contr :\n v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) * v.asIdeal ∣\n maxPowDividing v I * ∏ i in Finset.erase (Finite.toFinset hf) v, maxPowDividing i I\nhv_prime : Prime v.asIdeal\nw : HeightOneSpectrum R\nhw : w ∈ Finset.erase (Finite.toFinset hf) v\nhvw' : v.asIdeal ∣ maxPowDividing w I\nhw_prime : Prime w.asIdeal\nhvw : v.asIdeal ∣ w.asIdeal\n⊢ False",
"tactic": "rw [Prime.dvd_prime_iff_associated hv_prime hw_prime, associated_iff_eq] at hvw"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDedekindDomain R\nK : Type ?u.235854\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nI : Ideal R\nhI : I ≠ 0\nhf : Set.Finite (mulSupport fun v => maxPowDividing v I)\nh_ne_zero : maxPowDividing v I ≠ 0\nh_contr :\n v.asIdeal ^ Associates.count (Associates.mk v.asIdeal) (Associates.factors (Associates.mk I)) * v.asIdeal ∣\n maxPowDividing v I * ∏ i in Finset.erase (Finite.toFinset hf) v, maxPowDividing i I\nhv_prime : Prime v.asIdeal\nw : HeightOneSpectrum R\nhw : w ∈ Finset.erase (Finite.toFinset hf) v\nhvw' : v.asIdeal ∣ maxPowDividing w I\nhw_prime : Prime w.asIdeal\nhvw : v.asIdeal = w.asIdeal\n⊢ False",
"tactic": "exact (Finset.mem_erase.mp hw).1 (HeightOneSpectrum.ext w v (Eq.symm hvw))"
}
] |
[
123,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/MeasureTheory/MeasurableSpaceDef.lean
|
measurable_const
|
[] |
[
567,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
566,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ico_union_Ici
|
[
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.83670\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhab : a ≤ b\nh : c ≤ b\n⊢ Ico a b ∪ Ici c = Ici (min a c)\n\ncase inr\nα : Type u_1\nβ : Type ?u.83670\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhab : b ≤ a\nh : c ≤ a\n⊢ Ico a b ∪ Ici c = Ici (min a c)",
"state_before": "α : Type u_1\nβ : Type ?u.83670\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh : c ≤ max a b\n⊢ Ico a b ∪ Ici c = Ici (min a c)",
"tactic": "cases' le_total a b with hab hab <;> simp [hab] at h"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.83670\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhab : a ≤ b\nh : c ≤ b\n⊢ Ico a b ∪ Ici c = Ici (min a c)",
"tactic": "exact Ico_union_Ici' h"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.83670\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhab : b ≤ a\nh : c ≤ a\n⊢ Ico a b ∪ Ici c = Ici (min a c)",
"tactic": "simp [*]"
}
] |
[
1287,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1284,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.map₂_vsub
|
[] |
[
1085,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1084,
1
] |
Mathlib/Order/SuccPred/Basic.lean
|
Order.bot_lt_succ
|
[] |
[
553,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
552,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.infiniteNeg_mul_infinitePos
|
[] |
[
879,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
877,
1
] |
Std/Data/AssocList.lean
|
Std.AssocList.foldlM_eq
|
[
{
"state_after": "no goals",
"state_before": "m : Type u_1 → Type u_2\nδ : Type u_1\nα : Type u_3\nβ : Type u_4\ninst✝ : Monad m\nf : δ → α → β → m δ\ninit : δ\nl : AssocList α β\n⊢ foldlM f init l =\n List.foldlM\n (fun d x =>\n match x with\n | (a, b) => f d a b)\n init (toList l)",
"tactic": "induction l generalizing init <;> simp [*, foldlM]"
}
] |
[
53,
53
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
51,
9
] |
Mathlib/Topology/Algebra/Order/LeftRightLim.lean
|
Monotone.rightLim_le_leftLim
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\nh : x < y\nthis : TopologicalSpace α := Preorder.topology α\n⊢ rightLim f x ≤ leftLim f y",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\nh : x < y\n⊢ rightLim f x ≤ leftLim f y",
"tactic": "letI : TopologicalSpace α := Preorder.topology α"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\nh : x < y\nthis✝ : TopologicalSpace α := Preorder.topology α\nthis : OrderTopology α\n⊢ rightLim f x ≤ leftLim f y",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\nh : x < y\nthis : TopologicalSpace α := Preorder.topology α\n⊢ rightLim f x ≤ leftLim f y",
"tactic": "haveI : OrderTopology α := ⟨rfl⟩"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\nh : x < y\nthis✝ : TopologicalSpace α := Preorder.topology α\nthis : OrderTopology α\nh' : 𝓝[Iio y] y = ⊥\n⊢ rightLim f x ≤ leftLim f y\n\ncase inr\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\nh : x < y\nthis✝ : TopologicalSpace α := Preorder.topology α\nthis : OrderTopology α\nh' : 𝓝[Iio y] y ≠ ⊥\n⊢ rightLim f x ≤ leftLim f y",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\nh : x < y\nthis✝ : TopologicalSpace α := Preorder.topology α\nthis : OrderTopology α\n⊢ rightLim f x ≤ leftLim f y",
"tactic": "rcases eq_or_ne (𝓝[<] y) ⊥ with (h' | h')"
},
{
"state_after": "case inr.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\nh : x < y\nthis✝ : TopologicalSpace α := Preorder.topology α\nthis : OrderTopology α\nh' : 𝓝[Iio y] y ≠ ⊥\na : α\nxa : x < a\nay : a < y\n⊢ rightLim f x ≤ leftLim f y",
"state_before": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\nh : x < y\nthis✝ : TopologicalSpace α := Preorder.topology α\nthis : OrderTopology α\nh' : 𝓝[Iio y] y ≠ ⊥\n⊢ rightLim f x ≤ leftLim f y",
"tactic": "obtain ⟨a, ⟨xa, ay⟩⟩ : (Ioo x y).Nonempty :=\n forall_mem_nonempty_iff_neBot.2 (neBot_iff.2 h') (Ioo x y)\n (Ioo_mem_nhdsWithin_Iio ⟨h, le_refl _⟩)"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\nh : x < y\nthis✝ : TopologicalSpace α := Preorder.topology α\nthis : OrderTopology α\nh' : 𝓝[Iio y] y ≠ ⊥\na : α\nxa : x < a\nay : a < y\n⊢ rightLim f x ≤ leftLim f y",
"tactic": "calc\n rightLim f x ≤ f a := hf.rightLim_le xa\n _ ≤ leftLim f y := hf.le_leftLim ay"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\nh : x < y\nthis✝ : TopologicalSpace α := Preorder.topology α\nthis : OrderTopology α\nh' : 𝓝[Iio y] y = ⊥\n⊢ rightLim f x ≤ f y",
"state_before": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\nh : x < y\nthis✝ : TopologicalSpace α := Preorder.topology α\nthis : OrderTopology α\nh' : 𝓝[Iio y] y = ⊥\n⊢ rightLim f x ≤ leftLim f y",
"tactic": "simp [leftLim, h']"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\nh : x < y\nthis✝ : TopologicalSpace α := Preorder.topology α\nthis : OrderTopology α\nh' : 𝓝[Iio y] y = ⊥\n⊢ rightLim f x ≤ f y",
"tactic": "exact rightLim_le hf h"
}
] |
[
162,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/GroupTheory/Complement.lean
|
Subgroup.range_mem_rightTransversals
|
[
{
"state_after": "case mk.intro.mk.intro\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nf : Quotient (QuotientGroup.rightRel H) → G\nhf : ∀ (q : Quotient (QuotientGroup.rightRel H)), Quotient.mk'' (f q) = q\nq₁ q₂ : Quotient (QuotientGroup.rightRel H)\nh :\n Set.restrict (Set.range f) Quotient.mk'' { val := f q₁, property := (_ : ∃ y, f y = f q₁) } =\n Set.restrict (Set.range f) Quotient.mk'' { val := f q₂, property := (_ : ∃ y, f y = f q₂) }\n⊢ { val := f q₁, property := (_ : ∃ y, f y = f q₁) } = { val := f q₂, property := (_ : ∃ y, f y = f q₂) }",
"state_before": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nf : Quotient (QuotientGroup.rightRel H) → G\nhf : ∀ (q : Quotient (QuotientGroup.rightRel H)), Quotient.mk'' (f q) = q\n⊢ Function.Injective (Set.restrict (Set.range f) Quotient.mk'')",
"tactic": "rintro ⟨-, q₁, rfl⟩ ⟨-, q₂, rfl⟩ h"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.mk.intro\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nS T : Set G\nf : Quotient (QuotientGroup.rightRel H) → G\nhf : ∀ (q : Quotient (QuotientGroup.rightRel H)), Quotient.mk'' (f q) = q\nq₁ q₂ : Quotient (QuotientGroup.rightRel H)\nh :\n Set.restrict (Set.range f) Quotient.mk'' { val := f q₁, property := (_ : ∃ y, f y = f q₁) } =\n Set.restrict (Set.range f) Quotient.mk'' { val := f q₂, property := (_ : ∃ y, f y = f q₂) }\n⊢ { val := f q₁, property := (_ : ∃ y, f y = f q₁) } = { val := f q₂, property := (_ : ∃ y, f y = f q₂) }",
"tactic": "exact Subtype.ext $ congr_arg f $ ((hf q₁).symm.trans h).trans (hf q₂)"
}
] |
[
310,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
305,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
exp_eq_exp
|
[
{
"state_after": "case h\n𝕂 : Type u_2\n𝕂' : Type u_3\n𝔸 : Type u_1\ninst✝⁶ : Field 𝕂\ninst✝⁵ : Field 𝕂'\ninst✝⁴ : Ring 𝔸\ninst✝³ : Algebra 𝕂 𝔸\ninst✝² : Algebra 𝕂' 𝔸\ninst✝¹ : TopologicalSpace 𝔸\ninst✝ : TopologicalRing 𝔸\nx : 𝔸\n⊢ exp 𝕂 x = exp 𝕂' x",
"state_before": "𝕂 : Type u_2\n𝕂' : Type u_3\n𝔸 : Type u_1\ninst✝⁶ : Field 𝕂\ninst✝⁵ : Field 𝕂'\ninst✝⁴ : Ring 𝔸\ninst✝³ : Algebra 𝕂 𝔸\ninst✝² : Algebra 𝕂' 𝔸\ninst✝¹ : TopologicalSpace 𝔸\ninst✝ : TopologicalRing 𝔸\n⊢ exp 𝕂 = exp 𝕂'",
"tactic": "ext x"
},
{
"state_after": "case h\n𝕂 : Type u_2\n𝕂' : Type u_3\n𝔸 : Type u_1\ninst✝⁶ : Field 𝕂\ninst✝⁵ : Field 𝕂'\ninst✝⁴ : Ring 𝔸\ninst✝³ : Algebra 𝕂 𝔸\ninst✝² : Algebra 𝕂' 𝔸\ninst✝¹ : TopologicalSpace 𝔸\ninst✝ : TopologicalRing 𝔸\nx : 𝔸\n⊢ FormalMultilinearSeries.sum (expSeries 𝕂 𝔸) x = FormalMultilinearSeries.sum (expSeries 𝕂' 𝔸) x",
"state_before": "case h\n𝕂 : Type u_2\n𝕂' : Type u_3\n𝔸 : Type u_1\ninst✝⁶ : Field 𝕂\ninst✝⁵ : Field 𝕂'\ninst✝⁴ : Ring 𝔸\ninst✝³ : Algebra 𝕂 𝔸\ninst✝² : Algebra 𝕂' 𝔸\ninst✝¹ : TopologicalSpace 𝔸\ninst✝ : TopologicalRing 𝔸\nx : 𝔸\n⊢ exp 𝕂 x = exp 𝕂' x",
"tactic": "rw [exp, exp]"
},
{
"state_after": "case h\n𝕂 : Type u_2\n𝕂' : Type u_3\n𝔸 : Type u_1\ninst✝⁶ : Field 𝕂\ninst✝⁵ : Field 𝕂'\ninst✝⁴ : Ring 𝔸\ninst✝³ : Algebra 𝕂 𝔸\ninst✝² : Algebra 𝕂' 𝔸\ninst✝¹ : TopologicalSpace 𝔸\ninst✝ : TopologicalRing 𝔸\nx : 𝔸\nn : ℕ\n⊢ (↑(expSeries 𝕂 𝔸 n) fun x_1 => x) = ↑(expSeries 𝕂' 𝔸 n) fun x_1 => x",
"state_before": "case h\n𝕂 : Type u_2\n𝕂' : Type u_3\n𝔸 : Type u_1\ninst✝⁶ : Field 𝕂\ninst✝⁵ : Field 𝕂'\ninst✝⁴ : Ring 𝔸\ninst✝³ : Algebra 𝕂 𝔸\ninst✝² : Algebra 𝕂' 𝔸\ninst✝¹ : TopologicalSpace 𝔸\ninst✝ : TopologicalRing 𝔸\nx : 𝔸\n⊢ FormalMultilinearSeries.sum (expSeries 𝕂 𝔸) x = FormalMultilinearSeries.sum (expSeries 𝕂' 𝔸) x",
"tactic": "refine' tsum_congr fun n => _"
},
{
"state_after": "no goals",
"state_before": "case h\n𝕂 : Type u_2\n𝕂' : Type u_3\n𝔸 : Type u_1\ninst✝⁶ : Field 𝕂\ninst✝⁵ : Field 𝕂'\ninst✝⁴ : Ring 𝔸\ninst✝³ : Algebra 𝕂 𝔸\ninst✝² : Algebra 𝕂' 𝔸\ninst✝¹ : TopologicalSpace 𝔸\ninst✝ : TopologicalRing 𝔸\nx : 𝔸\nn : ℕ\n⊢ (↑(expSeries 𝕂 𝔸 n) fun x_1 => x) = ↑(expSeries 𝕂' 𝔸 n) fun x_1 => x",
"tactic": "rw [expSeries_eq_expSeries 𝕂 𝕂' 𝔸 n x]"
}
] |
[
664,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
660,
1
] |
Mathlib/Order/Filter/Cofinite.lean
|
Filter.disjoint_cofinite_right
|
[] |
[
139,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.addOrderOf_coe
|
[
{
"state_after": "case zero\nn : ℕ\nn0 : n ≠ 0\n⊢ addOrderOf ↑Nat.zero = n / Nat.gcd n Nat.zero\n\ncase succ\nn : ℕ\nn0 : n ≠ 0\na : ℕ\n⊢ addOrderOf ↑(Nat.succ a) = n / Nat.gcd n (Nat.succ a)",
"state_before": "a n : ℕ\nn0 : n ≠ 0\n⊢ addOrderOf ↑a = n / Nat.gcd n a",
"tactic": "cases' a with a"
},
{
"state_after": "case succ\nn : ℕ\nn0 : n ≠ 0\na : ℕ\n⊢ addOrderOf ↑(Nat.succ a) = n / Nat.gcd n (Nat.succ a)",
"state_before": "case zero\nn : ℕ\nn0 : n ≠ 0\n⊢ addOrderOf ↑Nat.zero = n / Nat.gcd n Nat.zero\n\ncase succ\nn : ℕ\nn0 : n ≠ 0\na : ℕ\n⊢ addOrderOf ↑(Nat.succ a) = n / Nat.gcd n (Nat.succ a)",
"tactic": "simp [Nat.pos_of_ne_zero n0]"
},
{
"state_after": "no goals",
"state_before": "case succ\nn : ℕ\nn0 : n ≠ 0\na : ℕ\n⊢ addOrderOf ↑(Nat.succ a) = n / Nat.gcd n (Nat.succ a)",
"tactic": "rw [← Nat.smul_one_eq_coe, addOrderOf_nsmul' _ a.succ_ne_zero, ZMod.addOrderOf_one]"
}
] |
[
114,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Std/Data/BinomialHeap.lean
|
Std.BinomialHeapImp.Heap.WellFormed.deleteMin
|
[
{
"state_after": "case cons.refl\nα : Type u_1\nle : α → α → Bool\nn r : Nat\na : α\nc : HeapNode α\ns : Heap α\nh : WellFormed le n (cons r a c s)\n⊢ WellFormed le 0\n (Heap.merge le\n (HeapNode.toHeap (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)\n (FindMin.before (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next))",
"state_before": "α : Type u_1\nle : α → α → Bool\nn : Nat\na : α\ns' s : Heap α\nh : WellFormed le n s\neq : Heap.deleteMin le s = some (a, s')\n⊢ WellFormed le 0 s'",
"tactic": "cases s with cases eq | cons r a c s => ?_"
},
{
"state_after": "case cons.refl\nα : Type u_1\nle : α → α → Bool\nn r : Nat\na : α\nc : HeapNode α\ns : Heap α\nh : WellFormed le n (cons r a c s)\nthis :\n Nonempty (FindMin.WellFormed le (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }))\n⊢ WellFormed le 0\n (Heap.merge le\n (HeapNode.toHeap (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)\n (FindMin.before (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next))",
"state_before": "case cons.refl\nα : Type u_1\nle : α → α → Bool\nn r : Nat\na : α\nc : HeapNode α\ns : Heap α\nh : WellFormed le n (cons r a c s)\n⊢ WellFormed le 0\n (Heap.merge le\n (HeapNode.toHeap (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)\n (FindMin.before (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next))",
"tactic": "have : Nonempty ((s.findMin le (cons r a c) ⟨id, a, c, s⟩).WellFormed le) :=\n let ⟨_, h₂, h₃⟩ := h\n h₃.findMin ⟨_, fun h => h.of_le (Nat.zero_le _), h₂, h₃⟩\n fun h => ⟨Nat.zero_le _, h₂, h⟩"
},
{
"state_after": "case cons.refl\nα : Type u_1\nle : α → α → Bool\nn r : Nat\na : α\nc : HeapNode α\ns : Heap α\nh : WellFormed le n (cons r a c s)\n⊢ Nonempty (FindMin.WellFormed le (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s })) →\n WellFormed le 0\n (Heap.merge le\n (HeapNode.toHeap (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)\n (FindMin.before (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next))",
"state_before": "case cons.refl\nα : Type u_1\nle : α → α → Bool\nn r : Nat\na : α\nc : HeapNode α\ns : Heap α\nh : WellFormed le n (cons r a c s)\nthis :\n Nonempty (FindMin.WellFormed le (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }))\n⊢ WellFormed le 0\n (Heap.merge le\n (HeapNode.toHeap (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)\n (FindMin.before (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next))",
"tactic": "revert this"
},
{
"state_after": "no goals",
"state_before": "case cons.refl\nα : Type u_1\nle : α → α → Bool\nn r : Nat\na : α\nc : HeapNode α\ns : Heap α\nh : WellFormed le n (cons r a c s)\n⊢ Nonempty (FindMin.WellFormed le (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s })) →\n WellFormed le 0\n (Heap.merge le\n (HeapNode.toHeap (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).node)\n (FindMin.before (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s })\n (Heap.findMin le (cons r a c) s { before := id, val := a, node := c, next := s }).next))",
"tactic": "match s.findMin le (cons r a c) ⟨id, a, c, s⟩ with\n| { before, val, node, next } =>\n intro ⟨⟨_, hk, ih₁, ih₂⟩⟩\n exact ih₁.toHeap.merge <| hk (ih₂.of_le (Nat.le_succ _))"
},
{
"state_after": "α : Type u_1\nle : α → α → Bool\nn r : Nat\na : α\nc : HeapNode α\ns : Heap α\nh : WellFormed le n (cons r a c s)\nbefore : Heap α → Heap α\nval : α\nnode : HeapNode α\nnext : Heap α\nrank✝ : Nat\nhk :\n ∀ {s : Heap α},\n WellFormed le rank✝ s →\n WellFormed le 0 (FindMin.before { before := before, val := val, node := node, next := next } s)\nih₁ :\n HeapNode.WellFormed le { before := before, val := val, node := node, next := next }.val\n { before := before, val := val, node := node, next := next }.node rank✝\nih₂ : WellFormed le (rank✝ + 1) { before := before, val := val, node := node, next := next }.next\n⊢ WellFormed le 0\n (Heap.merge le (HeapNode.toHeap { before := before, val := val, node := node, next := next }.node)\n (FindMin.before { before := before, val := val, node := node, next := next }\n { before := before, val := val, node := node, next := next }.next))",
"state_before": "α : Type u_1\nle : α → α → Bool\nn r : Nat\na : α\nc : HeapNode α\ns : Heap α\nh : WellFormed le n (cons r a c s)\nbefore : Heap α → Heap α\nval : α\nnode : HeapNode α\nnext : Heap α\n⊢ Nonempty (FindMin.WellFormed le { before := before, val := val, node := node, next := next }) →\n WellFormed le 0\n (Heap.merge le (HeapNode.toHeap { before := before, val := val, node := node, next := next }.node)\n (FindMin.before { before := before, val := val, node := node, next := next }\n { before := before, val := val, node := node, next := next }.next))",
"tactic": "intro ⟨⟨_, hk, ih₁, ih₂⟩⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nle : α → α → Bool\nn r : Nat\na : α\nc : HeapNode α\ns : Heap α\nh : WellFormed le n (cons r a c s)\nbefore : Heap α → Heap α\nval : α\nnode : HeapNode α\nnext : Heap α\nrank✝ : Nat\nhk :\n ∀ {s : Heap α},\n WellFormed le rank✝ s →\n WellFormed le 0 (FindMin.before { before := before, val := val, node := node, next := next } s)\nih₁ :\n HeapNode.WellFormed le { before := before, val := val, node := node, next := next }.val\n { before := before, val := val, node := node, next := next }.node rank✝\nih₂ : WellFormed le (rank✝ + 1) { before := before, val := val, node := node, next := next }.next\n⊢ WellFormed le 0\n (Heap.merge le (HeapNode.toHeap { before := before, val := val, node := node, next := next }.node)\n (FindMin.before { before := before, val := val, node := node, next := next }\n { before := before, val := val, node := node, next := next }.next))",
"tactic": "exact ih₁.toHeap.merge <| hk (ih₂.of_le (Nat.le_succ _))"
}
] |
[
497,
61
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
486,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.mapDomain_mapRange
|
[] |
[
651,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
644,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
TensorProduct.sum_tmul
|
[
{
"state_after": "case empty\nR : Type u_4\ninst✝¹⁶ : CommSemiring R\nR' : Type ?u.457403\ninst✝¹⁵ : Monoid R'\nR'' : Type ?u.457409\ninst✝¹⁴ : Semiring R''\nM : Type u_2\nN : Type u_3\nP : Type ?u.457421\nQ : Type ?u.457424\nS : Type ?u.457427\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : AddCommMonoid Q\ninst✝⁹ : AddCommMonoid S\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\ninst✝⁶ : Module R P\ninst✝⁵ : Module R Q\ninst✝⁴ : Module R S\ninst✝³ : DistribMulAction R' M\ninst✝² : Module R'' M\ninst✝¹ : SMulCommClass R R' M\ninst✝ : SMulCommClass R R'' M\nα : Type u_1\nm : α → M\nn : N\n⊢ (∑ a in ∅, m a) ⊗ₜ[R] n = ∑ a in ∅, m a ⊗ₜ[R] n\n\ncase insert\nR : Type u_4\ninst✝¹⁶ : CommSemiring R\nR' : Type ?u.457403\ninst✝¹⁵ : Monoid R'\nR'' : Type ?u.457409\ninst✝¹⁴ : Semiring R''\nM : Type u_2\nN : Type u_3\nP : Type ?u.457421\nQ : Type ?u.457424\nS : Type ?u.457427\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : AddCommMonoid Q\ninst✝⁹ : AddCommMonoid S\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\ninst✝⁶ : Module R P\ninst✝⁵ : Module R Q\ninst✝⁴ : Module R S\ninst✝³ : DistribMulAction R' M\ninst✝² : Module R'' M\ninst✝¹ : SMulCommClass R R' M\ninst✝ : SMulCommClass R R'' M\nα : Type u_1\nm : α → M\nn : N\na : α\ns : Finset α\nhas : ¬a ∈ s\nih : (∑ a in s, m a) ⊗ₜ[R] n = ∑ a in s, m a ⊗ₜ[R] n\n⊢ (∑ a in insert a s, m a) ⊗ₜ[R] n = ∑ a in insert a s, m a ⊗ₜ[R] n",
"state_before": "R : Type u_4\ninst✝¹⁶ : CommSemiring R\nR' : Type ?u.457403\ninst✝¹⁵ : Monoid R'\nR'' : Type ?u.457409\ninst✝¹⁴ : Semiring R''\nM : Type u_2\nN : Type u_3\nP : Type ?u.457421\nQ : Type ?u.457424\nS : Type ?u.457427\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : AddCommMonoid Q\ninst✝⁹ : AddCommMonoid S\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\ninst✝⁶ : Module R P\ninst✝⁵ : Module R Q\ninst✝⁴ : Module R S\ninst✝³ : DistribMulAction R' M\ninst✝² : Module R'' M\ninst✝¹ : SMulCommClass R R' M\ninst✝ : SMulCommClass R R'' M\nα : Type u_1\ns : Finset α\nm : α → M\nn : N\n⊢ (∑ a in s, m a) ⊗ₜ[R] n = ∑ a in s, m a ⊗ₜ[R] n",
"tactic": "induction' s using Finset.induction with a s has ih h"
},
{
"state_after": "no goals",
"state_before": "case empty\nR : Type u_4\ninst✝¹⁶ : CommSemiring R\nR' : Type ?u.457403\ninst✝¹⁵ : Monoid R'\nR'' : Type ?u.457409\ninst✝¹⁴ : Semiring R''\nM : Type u_2\nN : Type u_3\nP : Type ?u.457421\nQ : Type ?u.457424\nS : Type ?u.457427\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : AddCommMonoid Q\ninst✝⁹ : AddCommMonoid S\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\ninst✝⁶ : Module R P\ninst✝⁵ : Module R Q\ninst✝⁴ : Module R S\ninst✝³ : DistribMulAction R' M\ninst✝² : Module R'' M\ninst✝¹ : SMulCommClass R R' M\ninst✝ : SMulCommClass R R'' M\nα : Type u_1\nm : α → M\nn : N\n⊢ (∑ a in ∅, m a) ⊗ₜ[R] n = ∑ a in ∅, m a ⊗ₜ[R] n",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case insert\nR : Type u_4\ninst✝¹⁶ : CommSemiring R\nR' : Type ?u.457403\ninst✝¹⁵ : Monoid R'\nR'' : Type ?u.457409\ninst✝¹⁴ : Semiring R''\nM : Type u_2\nN : Type u_3\nP : Type ?u.457421\nQ : Type ?u.457424\nS : Type ?u.457427\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : AddCommMonoid Q\ninst✝⁹ : AddCommMonoid S\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\ninst✝⁶ : Module R P\ninst✝⁵ : Module R Q\ninst✝⁴ : Module R S\ninst✝³ : DistribMulAction R' M\ninst✝² : Module R'' M\ninst✝¹ : SMulCommClass R R' M\ninst✝ : SMulCommClass R R'' M\nα : Type u_1\nm : α → M\nn : N\na : α\ns : Finset α\nhas : ¬a ∈ s\nih : (∑ a in s, m a) ⊗ₜ[R] n = ∑ a in s, m a ⊗ₜ[R] n\n⊢ (∑ a in insert a s, m a) ⊗ₜ[R] n = ∑ a in insert a s, m a ⊗ₜ[R] n",
"tactic": "simp [Finset.sum_insert has, add_tmul, ih]"
}
] |
[
394,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
389,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
Submodule.coe_iSup_of_chain
|
[] |
[
326,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
325,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.blockDiagonal'_zero
|
[
{
"state_after": "case a.h\nl : Type ?u.203967\nm : Type ?u.203970\nn : Type ?u.203973\no : Type u_1\np : Type ?u.203979\nq : Type ?u.203982\nm' : o → Type u_2\nn' : o → Type u_3\np' : o → Type ?u.203997\nR : Type ?u.204000\nS : Type ?u.204003\nα : Type u_4\nβ : Type ?u.204009\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\ni✝ : (i : o) × m' i\nx✝ : (i : o) × n' i\n⊢ blockDiagonal' 0 i✝ x✝ = OfNat.ofNat 0 i✝ x✝",
"state_before": "l : Type ?u.203967\nm : Type ?u.203970\nn : Type ?u.203973\no : Type u_1\np : Type ?u.203979\nq : Type ?u.203982\nm' : o → Type u_2\nn' : o → Type u_3\np' : o → Type ?u.203997\nR : Type ?u.204000\nS : Type ?u.204003\nα : Type u_4\nβ : Type ?u.204009\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\n⊢ blockDiagonal' 0 = 0",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a.h\nl : Type ?u.203967\nm : Type ?u.203970\nn : Type ?u.203973\no : Type u_1\np : Type ?u.203979\nq : Type ?u.203982\nm' : o → Type u_2\nn' : o → Type u_3\np' : o → Type ?u.203997\nR : Type ?u.204000\nS : Type ?u.204003\nα : Type u_4\nβ : Type ?u.204009\ninst✝² : DecidableEq o\ninst✝¹ : Zero α\ninst✝ : Zero β\ni✝ : (i : o) × m' i\nx✝ : (i : o) × n' i\n⊢ blockDiagonal' 0 i✝ x✝ = OfNat.ofNat 0 i✝ x✝",
"tactic": "simp [blockDiagonal'_apply]"
}
] |
[
697,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
695,
1
] |
Mathlib/LinearAlgebra/CrossProduct.lean
|
cross_self
|
[
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\nv : Fin 3 → R\n⊢ v 1 * v 2 - v 1 * v 2 = 0 ∧ v 0 * v 2 - v 0 * v 2 = 0 ∧ v 0 * v 1 - v 0 * v 1 = 0 ∧ ![] = 0",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nv : Fin 3 → R\n⊢ ↑(↑crossProduct v) v = 0",
"tactic": "simp_rw [cross_apply, mul_comm, cons_eq_zero_iff]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nv : Fin 3 → R\n⊢ v 1 * v 2 - v 1 * v 2 = 0 ∧ v 0 * v 2 - v 0 * v 2 = 0 ∧ v 0 * v 1 - v 0 * v 1 = 0 ∧ ![] = 0",
"tactic": "exact ⟨sub_self _, sub_self _, sub_self _, zero_empty.symm⟩"
}
] |
[
91,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/GroupTheory/Congruence.lean
|
Con.mul
|
[] |
[
162,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
11
] |
Mathlib/Deprecated/Group.lean
|
Additive.isAddHom
|
[] |
[
446,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
444,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
|
Real.log_le_iff_le_exp
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\nhx : 0 < x\n⊢ log x ≤ y ↔ x ≤ exp y",
"tactic": "rw [← exp_le_exp, exp_log hx]"
}
] |
[
158,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Topology/Basic.lean
|
nhds_bind_nhds
|
[] |
[
982,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
981,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.tsum_eq_add_tsum_ite
|
[] |
[
1029,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1027,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
csSup_singleton
|
[] |
[
652,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
651,
1
] |
Mathlib/RingTheory/Derivation/Lie.lean
|
Derivation.commutator_apply
|
[] |
[
49,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
48,
1
] |
Std/Logic.lean
|
Bool.eq_false_or_eq_true
|
[] |
[
765,
22
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
763,
1
] |
Mathlib/Topology/Constructions.lean
|
Filter.HasBasis.prod_nhds'
|
[] |
[
552,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
548,
1
] |
Mathlib/Topology/Category/TopCat/OpenNhds.lean
|
TopologicalSpace.OpenNhds.map_id_obj_unop
|
[
{
"state_after": "no goals",
"state_before": "X Y : TopCat\nf : X ⟶ Y\nx : ↑X\nU : (OpenNhds x)ᵒᵖ\n⊢ (map (𝟙 X) x).obj U.unop = U.unop",
"tactic": "simp"
}
] |
[
127,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.fg_of_noetherian
|
[] |
[
942,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
941,
1
] |
Mathlib/Analysis/InnerProductSpace/l2Space.lean
|
IsHilbertSum.linearIsometryEquiv_symm_apply_single
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\nhV : IsHilbertSum 𝕜 G V\ni : ι\nx : G i\n⊢ ↑(LinearIsometryEquiv.symm (linearIsometryEquiv hV)) (lp.single 2 i x) = ↑(V i) x",
"tactic": "simp [IsHilbertSum.linearIsometryEquiv, OrthogonalFamily.linearIsometry_apply_single]"
}
] |
[
354,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
352,
11
] |
Mathlib/Algebra/GroupPower/Ring.lean
|
sub_sq'
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.130363\nM : Type ?u.130366\ninst✝ : CommRing R\na b : R\n⊢ (a - b) ^ 2 = a ^ 2 + b ^ 2 - 2 * a * b",
"tactic": "rw [sub_eq_add_neg, add_sq', neg_sq, mul_neg, ← sub_eq_add_neg]"
}
] |
[
291,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
290,
1
] |
Mathlib/Analysis/BoxIntegral/Box/Basic.lean
|
BoxIntegral.Box.mem_coe
|
[] |
[
113,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/Analysis/Convex/StrictConvexBetween.lean
|
Sbtw.dist_lt_max_dist
|
[
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nh : Sbtw ℝ p₁ p₂ p₃\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\n⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nh : Sbtw ℝ p₁ p₂ p₃\n⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)",
"tactic": "have hp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p := by simpa using h.left_ne_right"
},
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nh : (p₂ -ᵥ p = p₁ -ᵥ p ∨ p₂ -ᵥ p = p₃ -ᵥ p ∨ p₂ -ᵥ p ∈ openSegment ℝ (p₁ -ᵥ p) (p₃ -ᵥ p)) ∧ p₂ ≠ p₁ ∧ p₂ ≠ p₃\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\n⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nh : Sbtw ℝ p₁ p₂ p₃\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\n⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)",
"tactic": "rw [Sbtw, ← wbtw_vsub_const_iff p, Wbtw, affineSegment_eq_segment, ← insert_endpoints_openSegment,\n Set.mem_insert_iff, Set.mem_insert_iff] at h"
},
{
"state_after": "case intro.inl.intro\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\nh : p₂ -ᵥ p = p₁ -ᵥ p\nhp₂p₁ : p₂ ≠ p₁\nhp₂p₃ : p₂ ≠ p₃\n⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)\n\ncase intro.inr.inl.intro\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\nh : p₂ -ᵥ p = p₃ -ᵥ p\nhp₂p₁ : p₂ ≠ p₁\nhp₂p₃ : p₂ ≠ p₃\n⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)\n\ncase intro.inr.inr.intro\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\nh : p₂ -ᵥ p ∈ openSegment ℝ (p₁ -ᵥ p) (p₃ -ᵥ p)\nhp₂p₁ : p₂ ≠ p₁\nhp₂p₃ : p₂ ≠ p₃\n⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nh : (p₂ -ᵥ p = p₁ -ᵥ p ∨ p₂ -ᵥ p = p₃ -ᵥ p ∨ p₂ -ᵥ p ∈ openSegment ℝ (p₁ -ᵥ p) (p₃ -ᵥ p)) ∧ p₂ ≠ p₁ ∧ p₂ ≠ p₃\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\n⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)",
"tactic": "rcases h with ⟨h | h | h, hp₂p₁, hp₂p₃⟩"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nh : Sbtw ℝ p₁ p₂ p₃\n⊢ p₁ -ᵥ p ≠ p₃ -ᵥ p",
"tactic": "simpa using h.left_ne_right"
},
{
"state_after": "case intro.inl.intro\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\nh : p₂ = p₁\nhp₂p₁ : p₂ ≠ p₁\nhp₂p₃ : p₂ ≠ p₃\n⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)",
"state_before": "case intro.inl.intro\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\nh : p₂ -ᵥ p = p₁ -ᵥ p\nhp₂p₁ : p₂ ≠ p₁\nhp₂p₃ : p₂ ≠ p₃\n⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)",
"tactic": "rw [vsub_left_cancel_iff] at h"
},
{
"state_after": "no goals",
"state_before": "case intro.inl.intro\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\nh : p₂ = p₁\nhp₂p₁ : p₂ ≠ p₁\nhp₂p₃ : p₂ ≠ p₃\n⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)",
"tactic": "exact False.elim (hp₂p₁ h)"
},
{
"state_after": "case intro.inr.inl.intro\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\nh : p₂ = p₃\nhp₂p₁ : p₂ ≠ p₁\nhp₂p₃ : p₂ ≠ p₃\n⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)",
"state_before": "case intro.inr.inl.intro\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\nh : p₂ -ᵥ p = p₃ -ᵥ p\nhp₂p₁ : p₂ ≠ p₁\nhp₂p₃ : p₂ ≠ p₃\n⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)",
"tactic": "rw [vsub_left_cancel_iff] at h"
},
{
"state_after": "no goals",
"state_before": "case intro.inr.inl.intro\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\nh : p₂ = p₃\nhp₂p₁ : p₂ ≠ p₁\nhp₂p₃ : p₂ ≠ p₃\n⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)",
"tactic": "exact False.elim (hp₂p₃ h)"
},
{
"state_after": "case intro.inr.inr.intro\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\nh : ∃ x, x ∈ Set.Ioo 0 1 ∧ (1 - x) • (p₁ -ᵥ p) + x • (p₃ -ᵥ p) = p₂ -ᵥ p\nhp₂p₁ : p₂ ≠ p₁\nhp₂p₃ : p₂ ≠ p₃\n⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)",
"state_before": "case intro.inr.inr.intro\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\nh : p₂ -ᵥ p ∈ openSegment ℝ (p₁ -ᵥ p) (p₃ -ᵥ p)\nhp₂p₁ : p₂ ≠ p₁\nhp₂p₃ : p₂ ≠ p₃\n⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)",
"tactic": "rw [openSegment_eq_image, Set.mem_image] at h"
},
{
"state_after": "case intro.inr.inr.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\nhp₂p₁ : p₂ ≠ p₁\nhp₂p₃ : p₂ ≠ p₃\nr : ℝ\nhr : (1 - r) • (p₁ -ᵥ p) + r • (p₃ -ᵥ p) = p₂ -ᵥ p\nhr0 : 0 < r\nhr1 : r < 1\n⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)",
"state_before": "case intro.inr.inr.intro\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\nh : ∃ x, x ∈ Set.Ioo 0 1 ∧ (1 - x) • (p₁ -ᵥ p) + x • (p₃ -ᵥ p) = p₂ -ᵥ p\nhp₂p₁ : p₂ ≠ p₁\nhp₂p₃ : p₂ ≠ p₃\n⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)",
"tactic": "rcases h with ⟨r, ⟨hr0, hr1⟩, hr⟩"
},
{
"state_after": "case intro.inr.inr.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\nhp₂p₁ : p₂ ≠ p₁\nhp₂p₃ : p₂ ≠ p₃\nr : ℝ\nhr : (1 - r) • (p₁ -ᵥ p) + r • (p₃ -ᵥ p) = p₂ -ᵥ p\nhr0 : 0 < r\nhr1 : r < 1\n⊢ ‖(1 - r) • (p₁ -ᵥ p) + r • (p₃ -ᵥ p)‖ < max ‖p₁ -ᵥ p‖ ‖p₃ -ᵥ p‖",
"state_before": "case intro.inr.inr.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\nhp₂p₁ : p₂ ≠ p₁\nhp₂p₃ : p₂ ≠ p₃\nr : ℝ\nhr : (1 - r) • (p₁ -ᵥ p) + r • (p₃ -ᵥ p) = p₂ -ᵥ p\nhr0 : 0 < r\nhr1 : r < 1\n⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)",
"tactic": "simp_rw [@dist_eq_norm_vsub V, ← hr]"
},
{
"state_after": "no goals",
"state_before": "case intro.inr.inr.intro.intro.intro.intro\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\nhp₂p₁ : p₂ ≠ p₁\nhp₂p₃ : p₂ ≠ p₃\nr : ℝ\nhr : (1 - r) • (p₁ -ᵥ p) + r • (p₃ -ᵥ p) = p₂ -ᵥ p\nhr0 : 0 < r\nhr1 : r < 1\n⊢ ‖(1 - r) • (p₁ -ᵥ p) + r • (p₃ -ᵥ p)‖ < max ‖p₁ -ᵥ p‖ ‖p₃ -ᵥ p‖",
"tactic": "exact\n norm_combo_lt_of_ne (le_max_left _ _) (le_max_right _ _) hp₁p₃ (sub_pos.2 hr1) hr0 (by abel)"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : StrictConvexSpace ℝ V\np p₁ p₂ p₃ : P\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\nhp₂p₁ : p₂ ≠ p₁\nhp₂p₃ : p₂ ≠ p₃\nr : ℝ\nhr : (1 - r) • (p₁ -ᵥ p) + r • (p₃ -ᵥ p) = p₂ -ᵥ p\nhr0 : 0 < r\nhr1 : r < 1\n⊢ 1 - r + r = 1",
"tactic": "abel"
}
] |
[
41,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
27,
1
] |
Mathlib/LinearAlgebra/Ray.lean
|
units_inv_smul
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : LinearOrderedCommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nu : Rˣ\nv : Module.Ray R M\nthis : 0 < ↑u * ↑u\n⊢ 0 < ↑(u * u)",
"tactic": "exact this"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : LinearOrderedCommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nu : Rˣ\nv : Module.Ray R M\nthis : 0 < ↑u * ↑u\n⊢ (u * u) • u⁻¹ • v = u • v",
"tactic": "rw [mul_smul, smul_inv_smul]"
}
] |
[
525,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
521,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearMap.eqLocus_toAddSubmonoid
|
[] |
[
1274,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1272,
1
] |
Std/Data/Rat/Lemmas.lean
|
Rat.one_mul
|
[
{
"state_after": "no goals",
"state_before": "a : Rat\n⊢ 1 * a = a",
"tactic": "simp [mul_def, normalize_self]"
}
] |
[
262,
93
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
262,
19
] |
Mathlib/Logic/Basic.lean
|
forall_true_iff
|
[] |
[
713,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
713,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
IsDedekindDomainInv.mul_inv_eq_one
|
[] |
[
280,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
279,
1
] |
Mathlib/CategoryTheory/Yoneda.lean
|
CategoryTheory.Functor.reprW_app_hom
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nF : Cᵒᵖ ⥤ Type v₁\ninst✝ : Representable F\nX : Cᵒᵖ\nf : X.unop ⟶ reprX F\n⊢ (reprF F).app X f = ((reprF F).app (reprX F).op ≫ F.map f.op) (𝟙 (reprX F))",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nF : Cᵒᵖ ⥤ Type v₁\ninst✝ : Representable F\nX : Cᵒᵖ\nf : X.unop ⟶ reprX F\n⊢ ((reprW F).app X).hom f = F.map f.op (reprx F)",
"tactic": "change F.reprF.app X f = (F.reprF.app (op F.reprX) ≫ F.map f.op) (𝟙 F.reprX)"
},
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nF : Cᵒᵖ ⥤ Type v₁\ninst✝ : Representable F\nX : Cᵒᵖ\nf : X.unop ⟶ reprX F\n⊢ (reprF F).app X f = ((yoneda.obj (reprX F)).map f.op ≫ (reprF F).app X.unop.op) (𝟙 (reprX F))",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nF : Cᵒᵖ ⥤ Type v₁\ninst✝ : Representable F\nX : Cᵒᵖ\nf : X.unop ⟶ reprX F\n⊢ (reprF F).app X f = ((reprF F).app (reprX F).op ≫ F.map f.op) (𝟙 (reprX F))",
"tactic": "rw [← F.reprF.naturality]"
},
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nF : Cᵒᵖ ⥤ Type v₁\ninst✝ : Representable F\nX : Cᵒᵖ\nf : X.unop ⟶ reprX F\n⊢ (reprF F).app X f = (reprF F).app X (f ≫ 𝟙 (reprX F))",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nF : Cᵒᵖ ⥤ Type v₁\ninst✝ : Representable F\nX : Cᵒᵖ\nf : X.unop ⟶ reprX F\n⊢ (reprF F).app X f = ((yoneda.obj (reprX F)).map f.op ≫ (reprF F).app X.unop.op) (𝟙 (reprX F))",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nF : Cᵒᵖ ⥤ Type v₁\ninst✝ : Representable F\nX : Cᵒᵖ\nf : X.unop ⟶ reprX F\n⊢ (reprF F).app X f = (reprF F).app X (f ≫ 𝟙 (reprX F))",
"tactic": "simp"
}
] |
[
228,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
Homeomorph.transLocalHomeomorph_eq_trans
|
[] |
[
924,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
922,
1
] |
Mathlib/Data/MvPolynomial/Equiv.lean
|
MvPolynomial.finSuccEquiv_coeff_coeff
|
[
{
"state_after": "case h1\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\ni : ℕ\n⊢ coeff m (Polynomial.coeff (↑(finSuccEquiv R n) (↑(monomial j) r)) i) = coeff (cons i m) (↑(monomial j) r)\n\ncase h2\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\np q : MvPolynomial (Fin (n + 1)) R\nhp : ∀ (m : Fin n →₀ ℕ) (i : ℕ), coeff m (Polynomial.coeff (↑(finSuccEquiv R n) p) i) = coeff (cons i m) p\nhq : ∀ (m : Fin n →₀ ℕ) (i : ℕ), coeff m (Polynomial.coeff (↑(finSuccEquiv R n) q) i) = coeff (cons i m) q\nm : Fin n →₀ ℕ\ni : ℕ\n⊢ coeff m (Polynomial.coeff (↑(finSuccEquiv R n) (p + q)) i) = coeff (cons i m) (p + q)",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm : Fin n →₀ ℕ\nf : MvPolynomial (Fin (n + 1)) R\ni : ℕ\n⊢ coeff m (Polynomial.coeff (↑(finSuccEquiv R n) f) i) = coeff (cons i m) f",
"tactic": "induction' f using MvPolynomial.induction_on' with j r p q hp hq generalizing i m"
},
{
"state_after": "case h2\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\np q : MvPolynomial (Fin (n + 1)) R\nhp : ∀ (m : Fin n →₀ ℕ) (i : ℕ), coeff m (Polynomial.coeff (↑(finSuccEquiv R n) p) i) = coeff (cons i m) p\nhq : ∀ (m : Fin n →₀ ℕ) (i : ℕ), coeff m (Polynomial.coeff (↑(finSuccEquiv R n) q) i) = coeff (cons i m) q\nm : Fin n →₀ ℕ\ni : ℕ\n⊢ coeff m (Polynomial.coeff (↑(finSuccEquiv R n) (p + q)) i) = coeff (cons i m) (p + q)\n\ncase h1\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\ni : ℕ\n⊢ coeff m (Polynomial.coeff (↑(finSuccEquiv R n) (↑(monomial j) r)) i) = coeff (cons i m) (↑(monomial j) r)",
"state_before": "case h1\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\ni : ℕ\n⊢ coeff m (Polynomial.coeff (↑(finSuccEquiv R n) (↑(monomial j) r)) i) = coeff (cons i m) (↑(monomial j) r)\n\ncase h2\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\np q : MvPolynomial (Fin (n + 1)) R\nhp : ∀ (m : Fin n →₀ ℕ) (i : ℕ), coeff m (Polynomial.coeff (↑(finSuccEquiv R n) p) i) = coeff (cons i m) p\nhq : ∀ (m : Fin n →₀ ℕ) (i : ℕ), coeff m (Polynomial.coeff (↑(finSuccEquiv R n) q) i) = coeff (cons i m) q\nm : Fin n →₀ ℕ\ni : ℕ\n⊢ coeff m (Polynomial.coeff (↑(finSuccEquiv R n) (p + q)) i) = coeff (cons i m) (p + q)",
"tactic": "swap"
},
{
"state_after": "case h1\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\ni : ℕ\n⊢ r * coeff m (Polynomial.coeff (Polynomial.X ^ ↑j 0 * ↑Polynomial.C (∏ x : Fin n, X x ^ ↑j (Fin.succ x))) i) =\n if j = cons i m then r else 0",
"state_before": "case h1\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\ni : ℕ\n⊢ coeff m (Polynomial.coeff (↑(finSuccEquiv R n) (↑(monomial j) r)) i) = coeff (cons i m) (↑(monomial j) r)",
"tactic": "simp only [finSuccEquiv_apply, coe_eval₂Hom, eval₂_monomial, RingHom.coe_comp, prod_pow,\n Polynomial.coeff_C_mul, coeff_C_mul, coeff_monomial, Fin.prod_univ_succ, Fin.cases_zero,\n Fin.cases_succ, ← map_prod, ← RingHom.map_pow, Function.comp_apply]"
},
{
"state_after": "case h1\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\ni : ℕ\n⊢ r * coeff m (if i = ↑j 0 then ∏ x : Fin n, X x ^ ↑j (Fin.succ x) else 0) = r * if j = cons i m then 1 else 0",
"state_before": "case h1\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\ni : ℕ\n⊢ r * coeff m (Polynomial.coeff (Polynomial.X ^ ↑j 0 * ↑Polynomial.C (∏ x : Fin n, X x ^ ↑j (Fin.succ x))) i) =\n if j = cons i m then r else 0",
"tactic": "rw [← mul_boole, mul_comm (Polynomial.X ^ j 0), Polynomial.coeff_C_mul_X_pow]"
},
{
"state_after": "case h1.e_a\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\ni : ℕ\n⊢ coeff m (if i = ↑j 0 then ∏ x : Fin n, X x ^ ↑j (Fin.succ x) else 0) = if j = cons i m then 1 else 0",
"state_before": "case h1\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\ni : ℕ\n⊢ r * coeff m (if i = ↑j 0 then ∏ x : Fin n, X x ^ ↑j (Fin.succ x) else 0) = r * if j = cons i m then 1 else 0",
"tactic": "congr 1"
},
{
"state_after": "case h1.e_a.inl\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\nr : R\nm : Fin n →₀ ℕ\ni : ℕ\n⊢ coeff m (if i = ↑(cons i m) 0 then ∏ x : Fin n, X x ^ ↑(cons i m) (Fin.succ x) else 0) =\n if cons i m = cons i m then 1 else 0\n\ncase h1.e_a.inr\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\ni : ℕ\nhjmi : j ≠ cons i m\n⊢ coeff m (if i = ↑j 0 then ∏ x : Fin n, X x ^ ↑j (Fin.succ x) else 0) = if j = cons i m then 1 else 0",
"state_before": "case h1.e_a\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\ni : ℕ\n⊢ coeff m (if i = ↑j 0 then ∏ x : Fin n, X x ^ ↑j (Fin.succ x) else 0) = if j = cons i m then 1 else 0",
"tactic": "obtain rfl | hjmi := eq_or_ne j (m.cons i)"
},
{
"state_after": "no goals",
"state_before": "case h2\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\np q : MvPolynomial (Fin (n + 1)) R\nhp : ∀ (m : Fin n →₀ ℕ) (i : ℕ), coeff m (Polynomial.coeff (↑(finSuccEquiv R n) p) i) = coeff (cons i m) p\nhq : ∀ (m : Fin n →₀ ℕ) (i : ℕ), coeff m (Polynomial.coeff (↑(finSuccEquiv R n) q) i) = coeff (cons i m) q\nm : Fin n →₀ ℕ\ni : ℕ\n⊢ coeff m (Polynomial.coeff (↑(finSuccEquiv R n) (p + q)) i) = coeff (cons i m) (p + q)",
"tactic": "simp only [(finSuccEquiv R n).map_add, Polynomial.coeff_add, coeff_add, hp, hq]"
},
{
"state_after": "no goals",
"state_before": "case h1.e_a.inl\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\nr : R\nm : Fin n →₀ ℕ\ni : ℕ\n⊢ coeff m (if i = ↑(cons i m) 0 then ∏ x : Fin n, X x ^ ↑(cons i m) (Fin.succ x) else 0) =\n if cons i m = cons i m then 1 else 0",
"tactic": "simpa only [cons_zero, cons_succ, if_pos rfl, monomial_eq, C_1, one_mul, prod_pow] using\n coeff_monomial m m (1 : R)"
},
{
"state_after": "case h1.e_a.inr\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\ni : ℕ\nhjmi : j ≠ cons i m\n⊢ coeff m (if i = ↑j 0 then ∏ x : Fin n, X x ^ ↑j (Fin.succ x) else 0) = 0",
"state_before": "case h1.e_a.inr\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\ni : ℕ\nhjmi : j ≠ cons i m\n⊢ coeff m (if i = ↑j 0 then ∏ x : Fin n, X x ^ ↑j (Fin.succ x) else 0) = if j = cons i m then 1 else 0",
"tactic": "simp only [hjmi, if_false]"
},
{
"state_after": "case h1.e_a.inr.inl\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\ni : ℕ\nhjmi : j ≠ cons i m\nhij : i ≠ ↑j 0\n⊢ coeff m (if i = ↑j 0 then ∏ x : Fin n, X x ^ ↑j (Fin.succ x) else 0) = 0\n\ncase h1.e_a.inr.inr\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\nhjmi : j ≠ cons (↑j 0) m\n⊢ coeff m (if ↑j 0 = ↑j 0 then ∏ x : Fin n, X x ^ ↑j (Fin.succ x) else 0) = 0",
"state_before": "case h1.e_a.inr\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\ni : ℕ\nhjmi : j ≠ cons i m\n⊢ coeff m (if i = ↑j 0 then ∏ x : Fin n, X x ^ ↑j (Fin.succ x) else 0) = 0",
"tactic": "obtain hij | rfl := ne_or_eq i (j 0)"
},
{
"state_after": "case h1.e_a.inr.inr\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\nhjmi : j ≠ cons (↑j 0) m\n⊢ coeff m (∏ x : Fin n, X x ^ ↑j (Fin.succ x)) = 0",
"state_before": "case h1.e_a.inr.inr\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\nhjmi : j ≠ cons (↑j 0) m\n⊢ coeff m (if ↑j 0 = ↑j 0 then ∏ x : Fin n, X x ^ ↑j (Fin.succ x) else 0) = 0",
"tactic": "simp only [eq_self_iff_true, if_true]"
},
{
"state_after": "case h1.e_a.inr.inr\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\nhjmi : j ≠ cons (↑j 0) m\nhmj : m ≠ tail j\n⊢ coeff m (∏ x : Fin n, X x ^ ↑j (Fin.succ x)) = 0",
"state_before": "case h1.e_a.inr.inr\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\nhjmi : j ≠ cons (↑j 0) m\n⊢ coeff m (∏ x : Fin n, X x ^ ↑j (Fin.succ x)) = 0",
"tactic": "have hmj : m ≠ j.tail := by\n rintro rfl\n rw [cons_tail] at hjmi\n contradiction"
},
{
"state_after": "no goals",
"state_before": "case h1.e_a.inr.inr\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\nhjmi : j ≠ cons (↑j 0) m\nhmj : m ≠ tail j\n⊢ coeff m (∏ x : Fin n, X x ^ ↑j (Fin.succ x)) = 0",
"tactic": "simpa only [monomial_eq, C_1, one_mul, prod_pow, Finsupp.tail_apply, if_neg hmj.symm] using\n coeff_monomial m j.tail (1 : R)"
},
{
"state_after": "no goals",
"state_before": "case h1.e_a.inr.inl\nR : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni✝ : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\ni : ℕ\nhjmi : j ≠ cons i m\nhij : i ≠ ↑j 0\n⊢ coeff m (if i = ↑j 0 then ∏ x : Fin n, X x ^ ↑j (Fin.succ x) else 0) = 0",
"tactic": "simp only [hij, if_false, coeff_zero]"
},
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm : Fin n →₀ ℕ\ni : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nhjmi : j ≠ cons (↑j 0) (tail j)\n⊢ False",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm✝ : Fin n →₀ ℕ\ni : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nm : Fin n →₀ ℕ\nhjmi : j ≠ cons (↑j 0) m\n⊢ m ≠ tail j",
"tactic": "rintro rfl"
},
{
"state_after": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm : Fin n →₀ ℕ\ni : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nhjmi : j ≠ j\n⊢ False",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm : Fin n →₀ ℕ\ni : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nhjmi : j ≠ cons (↑j 0) (tail j)\n⊢ False",
"tactic": "rw [cons_tail] at hjmi"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type ?u.1032018\na a' a₁ a₂ : R\ne : ℕ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nn : ℕ\nm : Fin n →₀ ℕ\ni : ℕ\nj : Fin (n + 1) →₀ ℕ\nr : R\nhjmi : j ≠ j\n⊢ False",
"tactic": "contradiction"
}
] |
[
389,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
368,
1
] |
Mathlib/Algebra/GroupRingAction/Basic.lean
|
toRingHom_injective
|
[] |
[
69,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean
|
MeasureTheory.ae_eq_zero_of_forall_dual
|
[] |
[
117,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/Logic/Function/Conjugate.lean
|
Function.Semiconj₂.isIdempotent_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.3984\nf : α → β\nga : α → α → α\ngb : β → β → β\ninst✝ : IsIdempotent β gb\nh : Semiconj₂ f ga gb\nh_inj : Injective f\nx : α\n⊢ f (ga x x) = f x",
"tactic": "rw [h.eq, @IsIdempotent.idempotent _ gb]"
}
] |
[
181,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/Algebra/Ring/Aut.lean
|
RingAut.smul_def
|
[] |
[
130,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
11
] |
Mathlib/Analysis/Convex/Hull.lean
|
convexHull_mono
|
[] |
[
85,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Topology/Basic.lean
|
mem_closure_iff_clusterPt
|
[] |
[
1291,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1290,
1
] |
Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean
|
GeneralizedContinuedFraction.of_s_tail
|
[] |
[
322,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
321,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Nat.floor_add_nat
|
[
{
"state_after": "F : Type ?u.78159\nα : Type u_1\nβ : Type ?u.78165\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na : α\nn✝ : ℕ\nha : 0 ≤ a\nn b : ℕ\n⊢ ↑b ≤ a + ↑n ↔ b ≤ ⌊a⌋₊ + n",
"state_before": "F : Type ?u.78159\nα : Type u_1\nβ : Type ?u.78165\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na : α\nn✝ : ℕ\nha : 0 ≤ a\nn b : ℕ\n⊢ b ≤ ⌊a + ↑n⌋₊ ↔ b ≤ ⌊a⌋₊ + n",
"tactic": "rw [le_floor_iff (add_nonneg ha n.cast_nonneg)]"
},
{
"state_after": "case inl\nF : Type ?u.78159\nα : Type u_1\nβ : Type ?u.78165\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na : α\nn✝ : ℕ\nha : 0 ≤ a\nn b : ℕ\nhb : n ≤ b\n⊢ ↑b ≤ a + ↑n ↔ b ≤ ⌊a⌋₊ + n\n\ncase inr\nF : Type ?u.78159\nα : Type u_1\nβ : Type ?u.78165\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na : α\nn✝ : ℕ\nha : 0 ≤ a\nn b : ℕ\nhb : b ≤ n\n⊢ ↑b ≤ a + ↑n ↔ b ≤ ⌊a⌋₊ + n",
"state_before": "F : Type ?u.78159\nα : Type u_1\nβ : Type ?u.78165\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na : α\nn✝ : ℕ\nha : 0 ≤ a\nn b : ℕ\n⊢ ↑b ≤ a + ↑n ↔ b ≤ ⌊a⌋₊ + n",
"tactic": "obtain hb | hb := le_total n b"
},
{
"state_after": "case inl.intro\nF : Type ?u.78159\nα : Type u_1\nβ : Type ?u.78165\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na : α\nn✝ : ℕ\nha : 0 ≤ a\nn d : ℕ\nhb : n ≤ n + d\n⊢ ↑(n + d) ≤ a + ↑n ↔ n + d ≤ ⌊a⌋₊ + n",
"state_before": "case inl\nF : Type ?u.78159\nα : Type u_1\nβ : Type ?u.78165\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na : α\nn✝ : ℕ\nha : 0 ≤ a\nn b : ℕ\nhb : n ≤ b\n⊢ ↑b ≤ a + ↑n ↔ b ≤ ⌊a⌋₊ + n",
"tactic": "obtain ⟨d, rfl⟩ := exists_add_of_le hb"
},
{
"state_after": "no goals",
"state_before": "case inl.intro\nF : Type ?u.78159\nα : Type u_1\nβ : Type ?u.78165\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na : α\nn✝ : ℕ\nha : 0 ≤ a\nn d : ℕ\nhb : n ≤ n + d\n⊢ ↑(n + d) ≤ a + ↑n ↔ n + d ≤ ⌊a⌋₊ + n",
"tactic": "rw [Nat.cast_add, add_comm n, add_comm (n : α), add_le_add_iff_right, add_le_add_iff_right,\n le_floor_iff ha]"
},
{
"state_after": "case inr.intro\nF : Type ?u.78159\nα : Type u_1\nβ : Type ?u.78165\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\nb d : ℕ\nhb : b ≤ b + d\n⊢ ↑b ≤ a + ↑(b + d) ↔ b ≤ ⌊a⌋₊ + (b + d)",
"state_before": "case inr\nF : Type ?u.78159\nα : Type u_1\nβ : Type ?u.78165\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na : α\nn✝ : ℕ\nha : 0 ≤ a\nn b : ℕ\nhb : b ≤ n\n⊢ ↑b ≤ a + ↑n ↔ b ≤ ⌊a⌋₊ + n",
"tactic": "obtain ⟨d, rfl⟩ := exists_add_of_le hb"
},
{
"state_after": "case inr.intro\nF : Type ?u.78159\nα : Type u_1\nβ : Type ?u.78165\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\nb d : ℕ\nhb : b ≤ b + d\n⊢ ↑b ≤ ↑b + (a + ↑d) ↔ b ≤ b + (⌊a⌋₊ + d)",
"state_before": "case inr.intro\nF : Type ?u.78159\nα : Type u_1\nβ : Type ?u.78165\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\nb d : ℕ\nhb : b ≤ b + d\n⊢ ↑b ≤ a + ↑(b + d) ↔ b ≤ ⌊a⌋₊ + (b + d)",
"tactic": "rw [Nat.cast_add, add_left_comm _ b, add_left_comm _ (b : α)]"
},
{
"state_after": "case inr.intro\nF : Type ?u.78159\nα : Type u_1\nβ : Type ?u.78165\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\nb d : ℕ\nhb : b ≤ b + d\n⊢ ↑b ≤ ↑b + (a + ↑d)",
"state_before": "case inr.intro\nF : Type ?u.78159\nα : Type u_1\nβ : Type ?u.78165\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\nb d : ℕ\nhb : b ≤ b + d\n⊢ ↑b ≤ ↑b + (a + ↑d) ↔ b ≤ b + (⌊a⌋₊ + d)",
"tactic": "refine' iff_of_true _ le_self_add"
},
{
"state_after": "no goals",
"state_before": "case inr.intro\nF : Type ?u.78159\nα : Type u_1\nβ : Type ?u.78165\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\nha : 0 ≤ a\nb d : ℕ\nhb : b ≤ b + d\n⊢ ↑b ≤ ↑b + (a + ↑d)",
"tactic": "exact le_add_of_nonneg_right <| ha.trans <| le_add_of_nonneg_right d.cast_nonneg"
}
] |
[
448,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
438,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
neg_convexOn_iff
|
[
{
"state_after": "case mp\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.627009\nα : Type ?u.627012\nβ : Type u_3\nι : Type ?u.627018\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf g : E → β\n⊢ ConvexOn 𝕜 s (-f) → ConcaveOn 𝕜 s f\n\ncase mpr\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.627009\nα : Type ?u.627012\nβ : Type u_3\nι : Type ?u.627018\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf g : E → β\n⊢ ConcaveOn 𝕜 s f → ConvexOn 𝕜 s (-f)",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.627009\nα : Type ?u.627012\nβ : Type u_3\nι : Type ?u.627018\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf g : E → β\n⊢ ConvexOn 𝕜 s (-f) ↔ ConcaveOn 𝕜 s f",
"tactic": "constructor"
},
{
"state_after": "case mp.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.627009\nα : Type ?u.627012\nβ : Type u_3\nι : Type ?u.627018\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf g : E → β\nhconv : Convex 𝕜 s\nh :\n ∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (-f) (a • x + b • y) ≤ a • (-f) x + b • (-f) y\n⊢ ConcaveOn 𝕜 s f",
"state_before": "case mp\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.627009\nα : Type ?u.627012\nβ : Type u_3\nι : Type ?u.627018\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf g : E → β\n⊢ ConvexOn 𝕜 s (-f) → ConcaveOn 𝕜 s f",
"tactic": "rintro ⟨hconv, h⟩"
},
{
"state_after": "case mp.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.627009\nα : Type ?u.627012\nβ : Type u_3\nι : Type ?u.627018\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf g : E → β\nhconv : Convex 𝕜 s\nh :\n ∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (-f) (a • x + b • y) ≤ a • (-f) x + b • (-f) y\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a • f x + b • f y ≤ f (a • x + b • y)",
"state_before": "case mp.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.627009\nα : Type ?u.627012\nβ : Type u_3\nι : Type ?u.627018\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf g : E → β\nhconv : Convex 𝕜 s\nh :\n ∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (-f) (a • x + b • y) ≤ a • (-f) x + b • (-f) y\n⊢ ConcaveOn 𝕜 s f",
"tactic": "refine' ⟨hconv, fun x hx y hy a b ha hb hab => _⟩"
},
{
"state_after": "case mp.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.627009\nα : Type ?u.627012\nβ : Type u_3\nι : Type ?u.627018\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf g : E → β\nhconv : Convex 𝕜 s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nh : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • f x + b • f y ≤ f (a • x + b • y)\n⊢ a • f x + b • f y ≤ f (a • x + b • y)",
"state_before": "case mp.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.627009\nα : Type ?u.627012\nβ : Type u_3\nι : Type ?u.627018\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf g : E → β\nhconv : Convex 𝕜 s\nh :\n ∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → (-f) (a • x + b • y) ≤ a • (-f) x + b • (-f) y\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a • f x + b • f y ≤ f (a • x + b • y)",
"tactic": "simp [neg_apply, neg_le, add_comm] at h"
},
{
"state_after": "no goals",
"state_before": "case mp.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.627009\nα : Type ?u.627012\nβ : Type u_3\nι : Type ?u.627018\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf g : E → β\nhconv : Convex 𝕜 s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nh : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • f x + b • f y ≤ f (a • x + b • y)\n⊢ a • f x + b • f y ≤ f (a • x + b • y)",
"tactic": "exact h hx hy ha hb hab"
},
{
"state_after": "case mpr.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.627009\nα : Type ?u.627012\nβ : Type u_3\nι : Type ?u.627018\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf g : E → β\nhconv : Convex 𝕜 s\nh : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • f x + b • f y ≤ f (a • x + b • y)\n⊢ ConvexOn 𝕜 s (-f)",
"state_before": "case mpr\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.627009\nα : Type ?u.627012\nβ : Type u_3\nι : Type ?u.627018\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf g : E → β\n⊢ ConcaveOn 𝕜 s f → ConvexOn 𝕜 s (-f)",
"tactic": "rintro ⟨hconv, h⟩"
},
{
"state_after": "case mpr.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.627009\nα : Type ?u.627012\nβ : Type u_3\nι : Type ?u.627018\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf g : E → β\nhconv : Convex 𝕜 s\nh : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • f x + b • f y ≤ f (a • x + b • y)\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ (-f) (a • x + b • y) ≤ a • (-f) x + b • (-f) y",
"state_before": "case mpr.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.627009\nα : Type ?u.627012\nβ : Type u_3\nι : Type ?u.627018\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf g : E → β\nhconv : Convex 𝕜 s\nh : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • f x + b • f y ≤ f (a • x + b • y)\n⊢ ConvexOn 𝕜 s (-f)",
"tactic": "refine' ⟨hconv, fun x hx y hy a b ha hb hab => _⟩"
},
{
"state_after": "case mpr.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.627009\nα : Type ?u.627012\nβ : Type u_3\nι : Type ?u.627018\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf g : E → β\nhconv : Convex 𝕜 s\nh : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • f x + b • f y ≤ f (a • x + b • y)\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ -(a • (-f) x + b • (-f) y) ≤ -(-f) (a • x + b • y)",
"state_before": "case mpr.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.627009\nα : Type ?u.627012\nβ : Type u_3\nι : Type ?u.627018\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf g : E → β\nhconv : Convex 𝕜 s\nh : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • f x + b • f y ≤ f (a • x + b • y)\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ (-f) (a • x + b • y) ≤ a • (-f) x + b • (-f) y",
"tactic": "rw [← neg_le_neg_iff]"
},
{
"state_after": "case mpr.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.627009\nα : Type ?u.627012\nβ : Type u_3\nι : Type ?u.627018\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf g : E → β\nhconv : Convex 𝕜 s\nh : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • f x + b • f y ≤ f (a • x + b • y)\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a • f x + b • f y ≤ f (a • x + b • y)",
"state_before": "case mpr.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.627009\nα : Type ?u.627012\nβ : Type u_3\nι : Type ?u.627018\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf g : E → β\nhconv : Convex 𝕜 s\nh : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • f x + b • f y ≤ f (a • x + b • y)\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ -(a • (-f) x + b • (-f) y) ≤ -(-f) (a • x + b • y)",
"tactic": "simp_rw [neg_add, Pi.neg_apply, smul_neg, neg_neg]"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.627009\nα : Type ?u.627012\nβ : Type u_3\nι : Type ?u.627018\ninst✝⁵ : OrderedSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid F\ninst✝² : OrderedAddCommGroup β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf g : E → β\nhconv : Convex 𝕜 s\nh : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • f x + b • f y ≤ f (a • x + b • y)\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a • f x + b • f y ≤ f (a • x + b • y)",
"tactic": "exact h hx hy ha hb hab"
}
] |
[
845,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
835,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
FormalMultilinearSeries.le_radius_of_bound
|
[] |
[
136,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/GroupTheory/FreeAbelianGroup.lean
|
FreeAbelianGroup.map_id
|
[] |
[
376,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.succ_succAbove_succ
|
[
{
"state_after": "n✝ m n : ℕ\ni : Fin (n + 1)\nj : Fin n\nh : ↑castSucc j < i\nh' : ↑castSucc (succ j) < succ i\n⊢ ↑(succAbove (succ i)) (succ j) = succ (↑(succAbove i) j)",
"state_before": "n✝ m n : ℕ\ni : Fin (n + 1)\nj : Fin n\nh : ↑castSucc j < i\n⊢ ↑(succAbove (succ i)) (succ j) = succ (↑(succAbove i) j)",
"tactic": "have h' : castSucc j.succ < i.succ := by simpa [lt_iff_val_lt_val] using h"
},
{
"state_after": "case h\nn✝ m n : ℕ\ni : Fin (n + 1)\nj : Fin n\nh : ↑castSucc j < i\nh' : ↑castSucc (succ j) < succ i\n⊢ ↑(↑(succAbove (succ i)) (succ j)) = ↑(succ (↑(succAbove i) j))",
"state_before": "n✝ m n : ℕ\ni : Fin (n + 1)\nj : Fin n\nh : ↑castSucc j < i\nh' : ↑castSucc (succ j) < succ i\n⊢ ↑(succAbove (succ i)) (succ j) = succ (↑(succAbove i) j)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nn✝ m n : ℕ\ni : Fin (n + 1)\nj : Fin n\nh : ↑castSucc j < i\nh' : ↑castSucc (succ j) < succ i\n⊢ ↑(↑(succAbove (succ i)) (succ j)) = ↑(succ (↑(succAbove i) j))",
"tactic": "simp [succAbove_below _ _ h, succAbove_below _ _ h']"
},
{
"state_after": "no goals",
"state_before": "n✝ m n : ℕ\ni : Fin (n + 1)\nj : Fin n\nh : ↑castSucc j < i\n⊢ ↑castSucc (succ j) < succ i",
"tactic": "simpa [lt_iff_val_lt_val] using h"
},
{
"state_after": "n✝ m n : ℕ\ni : Fin (n + 1)\nj : Fin n\nh : ↑castSucc j ≥ i\nh' : succ i ≤ ↑castSucc (succ j)\n⊢ ↑(succAbove (succ i)) (succ j) = succ (↑(succAbove i) j)",
"state_before": "n✝ m n : ℕ\ni : Fin (n + 1)\nj : Fin n\nh : ↑castSucc j ≥ i\n⊢ ↑(succAbove (succ i)) (succ j) = succ (↑(succAbove i) j)",
"tactic": "have h' : i.succ ≤ castSucc j.succ := by simpa [le_iff_val_le_val] using h"
},
{
"state_after": "case h\nn✝ m n : ℕ\ni : Fin (n + 1)\nj : Fin n\nh : ↑castSucc j ≥ i\nh' : succ i ≤ ↑castSucc (succ j)\n⊢ ↑(↑(succAbove (succ i)) (succ j)) = ↑(succ (↑(succAbove i) j))",
"state_before": "n✝ m n : ℕ\ni : Fin (n + 1)\nj : Fin n\nh : ↑castSucc j ≥ i\nh' : succ i ≤ ↑castSucc (succ j)\n⊢ ↑(succAbove (succ i)) (succ j) = succ (↑(succAbove i) j)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nn✝ m n : ℕ\ni : Fin (n + 1)\nj : Fin n\nh : ↑castSucc j ≥ i\nh' : succ i ≤ ↑castSucc (succ j)\n⊢ ↑(↑(succAbove (succ i)) (succ j)) = ↑(succ (↑(succAbove i) j))",
"tactic": "simp [succAbove_above _ _ h, succAbove_above _ _ h']"
},
{
"state_after": "no goals",
"state_before": "n✝ m n : ℕ\ni : Fin (n + 1)\nj : Fin n\nh : ↑castSucc j ≥ i\n⊢ succ i ≤ ↑castSucc (succ j)",
"tactic": "simpa [le_iff_val_le_val] using h"
}
] |
[
2226,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2216,
1
] |
Mathlib/Analysis/Complex/Basic.lean
|
Complex.ofReal_tsum
|
[] |
[
603,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
602,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieSubmodule.incl_coe
|
[] |
[
566,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
565,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.piecewise_le_piecewise'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.310085\nγ : Type ?u.310088\nδ✝ : α → Sort ?u.310093\ns : Finset α\nf✝ g✝ : (i : α) → δ✝ i\ninst✝¹ : (j : α) → Decidable (j ∈ s)\nδ : α → Type u_1\ninst✝ : (i : α) → Preorder (δ i)\nf g f' g' : (i : α) → δ i\nHf : ∀ (x : α), x ∈ s → f x ≤ f' x\nHg : ∀ (x : α), ¬x ∈ s → g x ≤ g' x\nx : α\n⊢ piecewise s f g x ≤ piecewise s f' g' x",
"tactic": "by_cases hx : x ∈ s <;> simp [hx, *]"
}
] |
[
2569,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2567,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
tsum_iUnion_decode₂
|
[] |
[
726,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
724,
1
] |
Mathlib/Algebra/DirectSum/Module.lean
|
DirectSum.apply_eq_component
|
[] |
[
206,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.le_iff_exists_add
|
[] |
[
662,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
657,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.ListBlank.induction_on
|
[] |
[
208,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
11
] |
Mathlib/GroupTheory/Congruence.lean
|
Con.refl
|
[] |
[
144,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
11
] |
Mathlib/LinearAlgebra/Quotient.lean
|
Submodule.mkQ_surjective
|
[
{
"state_after": "case mk\nR : Type u_1\nM : Type u_2\nr : R\nx✝ y : M\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\np p' : Submodule R M\nM₂ : Type ?u.182338\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R M₂\nA : Submodule R M\nb✝ : M ⧸ A\nx : M\n⊢ ∃ a, ↑(mkQ A) a = Quot.mk Setoid.r x",
"state_before": "R : Type u_1\nM : Type u_2\nr : R\nx y : M\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\np p' : Submodule R M\nM₂ : Type ?u.182338\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R M₂\nA : Submodule R M\n⊢ Function.Surjective ↑(mkQ A)",
"tactic": "rintro ⟨x⟩"
},
{
"state_after": "no goals",
"state_before": "case mk\nR : Type u_1\nM : Type u_2\nr : R\nx✝ y : M\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\np p' : Submodule R M\nM₂ : Type ?u.182338\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R M₂\nA : Submodule R M\nb✝ : M ⧸ A\nx : M\n⊢ ∃ a, ↑(mkQ A) a = Quot.mk Setoid.r x",
"tactic": "exact ⟨x, rfl⟩"
}
] |
[
331,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
330,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Cardinal.mul_le_max
|
[
{
"state_after": "case inl\nb : Cardinal\n⊢ 0 * b ≤ max (max 0 b) ℵ₀\n\ncase inr\na b : Cardinal\nha0 : a ≠ 0\n⊢ a * b ≤ max (max a b) ℵ₀",
"state_before": "a b : Cardinal\n⊢ a * b ≤ max (max a b) ℵ₀",
"tactic": "rcases eq_or_ne a 0 with (rfl | ha0)"
},
{
"state_after": "case inr.inl\na : Cardinal\nha0 : a ≠ 0\n⊢ a * 0 ≤ max (max a 0) ℵ₀\n\ncase inr.inr\na b : Cardinal\nha0 : a ≠ 0\nhb0 : b ≠ 0\n⊢ a * b ≤ max (max a b) ℵ₀",
"state_before": "case inr\na b : Cardinal\nha0 : a ≠ 0\n⊢ a * b ≤ max (max a b) ℵ₀",
"tactic": "rcases eq_or_ne b 0 with (rfl | hb0)"
},
{
"state_after": "case inr.inr.inl\na b : Cardinal\nha0 : a ≠ 0\nhb0 : b ≠ 0\nha : ℵ₀ ≤ a\n⊢ a * b ≤ max (max a b) ℵ₀\n\ncase inr.inr.inr\na b : Cardinal\nha0 : a ≠ 0\nhb0 : b ≠ 0\nha : a < ℵ₀\n⊢ a * b ≤ max (max a b) ℵ₀",
"state_before": "case inr.inr\na b : Cardinal\nha0 : a ≠ 0\nhb0 : b ≠ 0\n⊢ a * b ≤ max (max a b) ℵ₀",
"tactic": "cases' le_or_lt ℵ₀ a with ha ha"
},
{
"state_after": "no goals",
"state_before": "case inl\nb : Cardinal\n⊢ 0 * b ≤ max (max 0 b) ℵ₀",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\na : Cardinal\nha0 : a ≠ 0\n⊢ a * 0 ≤ max (max a 0) ℵ₀",
"tactic": "simp"
},
{
"state_after": "case inr.inr.inl\na b : Cardinal\nha0 : a ≠ 0\nhb0 : b ≠ 0\nha : ℵ₀ ≤ a\n⊢ max a b ≤ max (max a b) ℵ₀",
"state_before": "case inr.inr.inl\na b : Cardinal\nha0 : a ≠ 0\nhb0 : b ≠ 0\nha : ℵ₀ ≤ a\n⊢ a * b ≤ max (max a b) ℵ₀",
"tactic": "rw [mul_eq_max_of_aleph0_le_left ha hb0]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inl\na b : Cardinal\nha0 : a ≠ 0\nhb0 : b ≠ 0\nha : ℵ₀ ≤ a\n⊢ max a b ≤ max (max a b) ℵ₀",
"tactic": "exact le_max_left _ _"
},
{
"state_after": "case inr.inr.inr.inl\na b : Cardinal\nha0 : a ≠ 0\nhb0 : b ≠ 0\nha : a < ℵ₀\nhb : ℵ₀ ≤ b\n⊢ a * b ≤ max (max a b) ℵ₀\n\ncase inr.inr.inr.inr\na b : Cardinal\nha0 : a ≠ 0\nhb0 : b ≠ 0\nha : a < ℵ₀\nhb : b < ℵ₀\n⊢ a * b ≤ max (max a b) ℵ₀",
"state_before": "case inr.inr.inr\na b : Cardinal\nha0 : a ≠ 0\nhb0 : b ≠ 0\nha : a < ℵ₀\n⊢ a * b ≤ max (max a b) ℵ₀",
"tactic": "cases' le_or_lt ℵ₀ b with hb hb"
},
{
"state_after": "case inr.inr.inr.inl\na b : Cardinal\nha0 : a ≠ 0\nhb0 : b ≠ 0\nha : a < ℵ₀\nhb : ℵ₀ ≤ b\n⊢ max a b ≤ max (max a b) ℵ₀",
"state_before": "case inr.inr.inr.inl\na b : Cardinal\nha0 : a ≠ 0\nhb0 : b ≠ 0\nha : a < ℵ₀\nhb : ℵ₀ ≤ b\n⊢ a * b ≤ max (max a b) ℵ₀",
"tactic": "rw [mul_comm, mul_eq_max_of_aleph0_le_left hb ha0, max_comm]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inr.inl\na b : Cardinal\nha0 : a ≠ 0\nhb0 : b ≠ 0\nha : a < ℵ₀\nhb : ℵ₀ ≤ b\n⊢ max a b ≤ max (max a b) ℵ₀",
"tactic": "exact le_max_left _ _"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inr.inr\na b : Cardinal\nha0 : a ≠ 0\nhb0 : b ≠ 0\nha : a < ℵ₀\nhb : b < ℵ₀\n⊢ a * b ≤ max (max a b) ℵ₀",
"tactic": "exact le_max_of_le_right (mul_lt_aleph0 ha hb).le"
}
] |
[
648,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
639,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean
|
CategoryTheory.IsPushout.unop
|
[] |
[
864,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
859,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiff.hasStrictDerivAt
|
[] |
[
1987,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1985,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.sum_lt_top_iff
|
[] |
[
1227,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1226,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean
|
CategoryTheory.Limits.coequalizer_as_cokernel
|
[] |
[
692,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
691,
1
] |
Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean
|
MeasureTheory.ProbabilityMeasure.tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds
|
[] |
[
270,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/Data/Nat/Parity.lean
|
Nat.Odd.of_mul_right
|
[] |
[
169,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/Data/Nat/Fib.lean
|
Nat.fib_mono
|
[] |
[
102,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Data/MvPolynomial/Variables.lean
|
MvPolynomial.degreeOf_rename_of_injective
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q p : MvPolynomial σ R\nf : σ → τ\nh : Injective f\ni : σ\n⊢ degreeOf (f i) (↑(rename f) p) = degreeOf i p",
"tactic": "classical\nsimp only [degreeOf, degrees_rename_of_injective h, Multiset.count_map_eq_count' f p.degrees h]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np✝ q p : MvPolynomial σ R\nf : σ → τ\nh : Injective f\ni : σ\n⊢ degreeOf (f i) (↑(rename f) p) = degreeOf i p",
"tactic": "simp only [degreeOf, degrees_rename_of_injective h, Multiset.count_map_eq_count' f p.degrees h]"
}
] |
[
582,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
579,
1
] |
Mathlib/Order/CompleteLattice.lean
|
sSup_univ
|
[] |
[
495,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
494,
1
] |
Mathlib/Logic/Equiv/Set.lean
|
Equiv.symm_preimage_preimage
|
[] |
[
108,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Std/Data/Int/DivMod.lean
|
Int.mod_zero
|
[] |
[
256,
18
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
254,
9
] |
Mathlib/Order/Basic.lean
|
Preorder.ext
|
[
{
"state_after": "case a.le.h.h.a\nι : Type ?u.19676\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.19687\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : Preorder α\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nx y : α\n⊢ x ≤ y ↔ x ≤ y",
"state_before": "ι : Type ?u.19676\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.19687\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : Preorder α\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\n⊢ A = B",
"tactic": "ext x y"
},
{
"state_after": "no goals",
"state_before": "case a.le.h.h.a\nι : Type ?u.19676\nα✝ : Type u\nβ : Type v\nγ : Type w\nπ : ι → Type ?u.19687\nr : α✝ → α✝ → Prop\nα : Type u_1\nA B : Preorder α\nH : ∀ (x y : α), x ≤ y ↔ x ≤ y\nx y : α\n⊢ x ≤ y ↔ x ≤ y",
"tactic": "exact H x y"
}
] |
[
653,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
650,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.inter_smul_union_subset_union
|
[] |
[
220,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
src/lean/Init/SizeOfLemmas.lean
|
Subtype.sizeOf
|
[
{
"state_after": "case mk\nα : Sort u_1\np : α → Prop\nval✝ : α\nproperty✝ : p val✝\n⊢ SizeOf.sizeOf { val := val✝, property := property✝ } = SizeOf.sizeOf { val := val✝, property := property✝ }.val + 1",
"state_before": "α : Sort u_1\np : α → Prop\ns : Subtype p\n⊢ SizeOf.sizeOf s = SizeOf.sizeOf s.val + 1",
"tactic": "cases s"
},
{
"state_after": "no goals",
"state_before": "case mk\nα : Sort u_1\np : α → Prop\nval✝ : α\nproperty✝ : p val✝\n⊢ SizeOf.sizeOf { val := val✝, property := property✝ } = SizeOf.sizeOf { val := val✝, property := property✝ }.val + 1",
"tactic": "simp_arith"
}
] |
[
33,
22
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
32,
9
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.