file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Order/Hom/CompleteLattice.lean
|
FrameHom.id_comp
|
[] |
[
637,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
636,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.biUnion_subset_iff_forall_subset
|
[] |
[
3629,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3625,
1
] |
Mathlib/Combinatorics/Composition.lean
|
Composition.single_blocks
|
[] |
[
583,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
582,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
ContinuousMultilinearMap.continuous_restrictScalars
|
[] |
[
612,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
609,
1
] |
Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean
|
SimpleGraph.bot_strongly_regular
|
[
{
"state_after": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nv w : V\n_h : v ≠ w\n⊢ filter (fun x => x ∈ commonNeighbors ⊥ v w) univ = ∅",
"state_before": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nv w : V\n_h : v ≠ w\n⊢ ¬Adj ⊥ v w → Fintype.card ↑(commonNeighbors ⊥ v w) = 0",
"tactic": "simp only [card_eq_zero, Fintype.card_ofFinset, forall_true_left, not_false_iff, bot_adj]"
},
{
"state_after": "case a\nV : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nv w : V\n_h : v ≠ w\na✝ : V\n⊢ a✝ ∈ filter (fun x => x ∈ commonNeighbors ⊥ v w) univ ↔ a✝ ∈ ∅",
"state_before": "V : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nv w : V\n_h : v ≠ w\n⊢ filter (fun x => x ∈ commonNeighbors ⊥ v w) univ = ∅",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nV : Type u\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nn k ℓ μ : ℕ\nv w : V\n_h : v ≠ w\na✝ : V\n⊢ a✝ ∈ filter (fun x => x ∈ commonNeighbors ⊥ v w) univ ↔ a✝ ∈ ∅",
"tactic": "simp [mem_commonNeighbors]"
}
] |
[
68,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
1
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.eq_add_of_sub_eq
|
[
{
"state_after": "no goals",
"state_before": "a b c : Nat\nhle : b ≤ a\nh : a - b = c\n⊢ a = c + b",
"tactic": "rw [h.symm, Nat.sub_add_cancel hle]"
}
] |
[
608,
38
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
607,
11
] |
Mathlib/Algebra/Group/Basic.lean
|
inv_involutive
|
[] |
[
241,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.mem_prod
|
[] |
[
1052,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1050,
1
] |
Mathlib/Algebra/Homology/Homotopy.lean
|
prevD_comp_right
|
[
{
"state_after": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g✝ : C ⟶ D\nh k : D ⟶ E\ni : ι\nf : (i j : ι) → X C i ⟶ X D j\ng : D ⟶ E\nj : ι\n⊢ (f j (ComplexShape.prev c j) ≫ Hom.f g (ComplexShape.prev c j)) ≫ d E (ComplexShape.prev c j) j =\n (f j (ComplexShape.prev c j) ≫ d D (ComplexShape.prev c j) j) ≫ Hom.f g j",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g✝ : C ⟶ D\nh k : D ⟶ E\ni : ι\nf : (i j : ι) → X C i ⟶ X D j\ng : D ⟶ E\nj : ι\n⊢ (↑(prevD j) fun i j => f i j ≫ Hom.f g j) = ↑(prevD j) f ≫ Hom.f g j",
"tactic": "dsimp [prevD]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g✝ : C ⟶ D\nh k : D ⟶ E\ni : ι\nf : (i j : ι) → X C i ⟶ X D j\ng : D ⟶ E\nj : ι\n⊢ (f j (ComplexShape.prev c j) ≫ Hom.f g (ComplexShape.prev c j)) ≫ d E (ComplexShape.prev c j) j =\n (f j (ComplexShape.prev c j) ≫ d D (ComplexShape.prev c j) j) ≫ Hom.f g j",
"tactic": "simp only [Category.assoc, g.comm]"
}
] |
[
107,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
FractionalIdeal.isPrincipal_inv
|
[
{
"state_after": "R : Type ?u.84677\nA : Type ?u.84680\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.85691\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nh : I ≠ 0\n⊢ ∃ x, I⁻¹ = spanSingleton R₁⁰ x",
"state_before": "R : Type ?u.84677\nA : Type ?u.84680\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.85691\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nh : I ≠ 0\n⊢ IsPrincipal ↑I⁻¹",
"tactic": "rw [val_eq_coe, isPrincipal_iff]"
},
{
"state_after": "R : Type ?u.84677\nA : Type ?u.84680\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.85691\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nh : I ≠ 0\n⊢ I⁻¹ = spanSingleton R₁⁰ (generator ↑I)⁻¹",
"state_before": "R : Type ?u.84677\nA : Type ?u.84680\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.85691\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nh : I ≠ 0\n⊢ ∃ x, I⁻¹ = spanSingleton R₁⁰ x",
"tactic": "use (generator (I : Submodule R₁ K))⁻¹"
},
{
"state_after": "case hI\nR : Type ?u.84677\nA : Type ?u.84680\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.85691\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nh : I ≠ 0\n⊢ I * spanSingleton R₁⁰ (generator ↑I)⁻¹ = 1\n\nR : Type ?u.84677\nA : Type ?u.84680\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.85691\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nh : I ≠ 0\nhI : I * spanSingleton R₁⁰ (generator ↑I)⁻¹ = 1\n⊢ I⁻¹ = spanSingleton R₁⁰ (generator ↑I)⁻¹",
"state_before": "R : Type ?u.84677\nA : Type ?u.84680\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.85691\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nh : I ≠ 0\n⊢ I⁻¹ = spanSingleton R₁⁰ (generator ↑I)⁻¹",
"tactic": "have hI : I * spanSingleton _ (generator (I : Submodule R₁ K))⁻¹ = 1"
},
{
"state_after": "R : Type ?u.84677\nA : Type ?u.84680\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.85691\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nh : I ≠ 0\nhI : I * spanSingleton R₁⁰ (generator ↑I)⁻¹ = 1\n⊢ I⁻¹ = spanSingleton R₁⁰ (generator ↑I)⁻¹",
"state_before": "case hI\nR : Type ?u.84677\nA : Type ?u.84680\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.85691\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nh : I ≠ 0\n⊢ I * spanSingleton R₁⁰ (generator ↑I)⁻¹ = 1\n\nR : Type ?u.84677\nA : Type ?u.84680\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.85691\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nh : I ≠ 0\nhI : I * spanSingleton R₁⁰ (generator ↑I)⁻¹ = 1\n⊢ I⁻¹ = spanSingleton R₁⁰ (generator ↑I)⁻¹",
"tactic": "apply mul_generator_self_inv _ I h"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.84677\nA : Type ?u.84680\nK : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : IsDomain A\nR₁ : Type u_1\ninst✝⁷ : CommRing R₁\ninst✝⁶ : IsDomain R₁\ninst✝⁵ : Algebra R₁ K\ninst✝⁴ : IsFractionRing R₁ K\nI✝ J : FractionalIdeal R₁⁰ K\nK' : Type ?u.85691\ninst✝³ : Field K'\ninst✝² : Algebra R₁ K'\ninst✝¹ : IsFractionRing R₁ K'\nI : FractionalIdeal R₁⁰ K\ninst✝ : IsPrincipal ↑I\nh : I ≠ 0\nhI : I * spanSingleton R₁⁰ (generator ↑I)⁻¹ = 1\n⊢ I⁻¹ = spanSingleton R₁⁰ (generator ↑I)⁻¹",
"tactic": "exact (right_inverse_eq _ I (spanSingleton _ (generator (I : Submodule R₁ K))⁻¹) hI).symm"
}
] |
[
236,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
230,
1
] |
Mathlib/Algebra/Hom/Equiv/Units/Basic.lean
|
Group.mulLeft_bijective
|
[] |
[
150,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.Fork.app_one_eq_ι_comp_right
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : C\nf g : X ⟶ Y\ns : Fork f g\n⊢ s.π.app one = ι s ≫ g",
"tactic": "rw [← s.app_zero_eq_ι, ← s.w right, parallelPair_map_right]"
}
] |
[
346,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
345,
1
] |
Mathlib/ModelTheory/LanguageMap.lean
|
FirstOrder.Language.LHom.comp_assoc
|
[] |
[
170,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/RingTheory/Ideal/LocalRing.lean
|
LocalRing.ResidueField.map_residue
|
[] |
[
436,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
434,
1
] |
Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean
|
continuous_of_linear_of_bound
|
[] |
[
97,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Data/Nat/Order/Lemmas.lean
|
Nat.not_two_dvd_bit1
|
[
{
"state_after": "a b m n✝ k n : ℕ\n⊢ ¬2 = 1",
"state_before": "a b m n✝ k n : ℕ\n⊢ ¬2 ∣ bit1 n",
"tactic": "rw [bit1, Nat.dvd_add_right two_dvd_bit0, Nat.dvd_one]"
},
{
"state_after": "no goals",
"state_before": "a b m n✝ k n : ℕ\n⊢ ¬2 = 1",
"tactic": "decide"
}
] |
[
106,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
11
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_piecewise
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.770262\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g✝ : α → β\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq α\ns t : Finset α\nf g : α → β\n⊢ ∏ x in s, piecewise t f g x = (∏ x in s ∩ t, f x) * ∏ x in s \\ t, g x",
"tactic": "erw [prod_ite, filter_mem_eq_inter, ← sdiff_eq_filter]"
}
] |
[
1539,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1537,
1
] |
Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean
|
SimpleGraph.nonuniformWitnesses_spec
|
[
{
"state_after": "α : Type u_1\n𝕜 : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns t : Finset α\na b : α\nh : ¬IsUniform G ε s t\n⊢ ε ≤\n ↑(abs\n (edgeDensity G\n (Exists.choose\n (_ :\n ∃ s',\n s' ⊆ s ∧\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤ ↑(card s') ∧\n ↑(card t) * ε ≤ ↑(card t') ∧ ε ≤ ↑(abs (edgeDensity G s' t' - edgeDensity G s t))),\n Exists.choose\n (_ :\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤\n ↑(card\n (Exists.choose\n (_ :\n ∃ s',\n s' ⊆ s ∧\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤ ↑(card s') ∧\n ↑(card t) * ε ≤ ↑(card t') ∧\n ε ≤ ↑(abs (edgeDensity G s' t' - edgeDensity G s t))))) ∧\n ↑(card t) * ε ≤ ↑(card t') ∧\n ε ≤\n ↑(abs\n (edgeDensity G\n (Exists.choose\n (_ :\n ∃ s',\n s' ⊆ s ∧\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤ ↑(card s') ∧\n ↑(card t) * ε ≤ ↑(card t') ∧\n ε ≤ ↑(abs (edgeDensity G s' t' - edgeDensity G s t))))\n t' -\n edgeDensity G s t)))).fst\n (Exists.choose\n (_ :\n ∃ s',\n s' ⊆ s ∧\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤ ↑(card s') ∧\n ↑(card t) * ε ≤ ↑(card t') ∧ ε ≤ ↑(abs (edgeDensity G s' t' - edgeDensity G s t))),\n Exists.choose\n (_ :\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤\n ↑(card\n (Exists.choose\n (_ :\n ∃ s',\n s' ⊆ s ∧\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤ ↑(card s') ∧\n ↑(card t) * ε ≤ ↑(card t') ∧\n ε ≤ ↑(abs (edgeDensity G s' t' - edgeDensity G s t))))) ∧\n ↑(card t) * ε ≤ ↑(card t') ∧\n ε ≤\n ↑(abs\n (edgeDensity G\n (Exists.choose\n (_ :\n ∃ s',\n s' ⊆ s ∧\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤ ↑(card s') ∧\n ↑(card t) * ε ≤ ↑(card t') ∧\n ε ≤ ↑(abs (edgeDensity G s' t' - edgeDensity G s t))))\n t' -\n edgeDensity G s t)))).snd -\n edgeDensity G s t))",
"state_before": "α : Type u_1\n𝕜 : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns t : Finset α\na b : α\nh : ¬IsUniform G ε s t\n⊢ ε ≤ ↑(abs (edgeDensity G (nonuniformWitnesses G ε s t).fst (nonuniformWitnesses G ε s t).snd - edgeDensity G s t))",
"tactic": "rw [nonuniformWitnesses, dif_pos h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\n𝕜 : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : 𝕜\ns t : Finset α\na b : α\nh : ¬IsUniform G ε s t\n⊢ ε ≤\n ↑(abs\n (edgeDensity G\n (Exists.choose\n (_ :\n ∃ s',\n s' ⊆ s ∧\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤ ↑(card s') ∧\n ↑(card t) * ε ≤ ↑(card t') ∧ ε ≤ ↑(abs (edgeDensity G s' t' - edgeDensity G s t))),\n Exists.choose\n (_ :\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤\n ↑(card\n (Exists.choose\n (_ :\n ∃ s',\n s' ⊆ s ∧\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤ ↑(card s') ∧\n ↑(card t) * ε ≤ ↑(card t') ∧\n ε ≤ ↑(abs (edgeDensity G s' t' - edgeDensity G s t))))) ∧\n ↑(card t) * ε ≤ ↑(card t') ∧\n ε ≤\n ↑(abs\n (edgeDensity G\n (Exists.choose\n (_ :\n ∃ s',\n s' ⊆ s ∧\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤ ↑(card s') ∧\n ↑(card t) * ε ≤ ↑(card t') ∧\n ε ≤ ↑(abs (edgeDensity G s' t' - edgeDensity G s t))))\n t' -\n edgeDensity G s t)))).fst\n (Exists.choose\n (_ :\n ∃ s',\n s' ⊆ s ∧\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤ ↑(card s') ∧\n ↑(card t) * ε ≤ ↑(card t') ∧ ε ≤ ↑(abs (edgeDensity G s' t' - edgeDensity G s t))),\n Exists.choose\n (_ :\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤\n ↑(card\n (Exists.choose\n (_ :\n ∃ s',\n s' ⊆ s ∧\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤ ↑(card s') ∧\n ↑(card t) * ε ≤ ↑(card t') ∧\n ε ≤ ↑(abs (edgeDensity G s' t' - edgeDensity G s t))))) ∧\n ↑(card t) * ε ≤ ↑(card t') ∧\n ε ≤\n ↑(abs\n (edgeDensity G\n (Exists.choose\n (_ :\n ∃ s',\n s' ⊆ s ∧\n ∃ t',\n t' ⊆ t ∧\n ↑(card s) * ε ≤ ↑(card s') ∧\n ↑(card t) * ε ≤ ↑(card t') ∧\n ε ≤ ↑(abs (edgeDensity G s' t' - edgeDensity G s t))))\n t' -\n edgeDensity G s t)))).snd -\n edgeDensity G s t))",
"tactic": "exact (not_isUniform_iff.1 h).choose_spec.2.choose_spec.2.2.2"
}
] |
[
157,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
Polynomial.coe_def
|
[] |
[
2540,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2539,
1
] |
Std/Data/List/Init/Lemmas.lean
|
List.all_cons
|
[] |
[
37,
66
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
37,
9
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
|
CategoryTheory.Limits.map_inl_inv_coprodComparison
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁶ : Category C\nX Y : C\nD : Type u₂\ninst✝⁵ : Category D\nF : C ⥤ D\nA A' B B' : C\ninst✝⁴ : HasBinaryCoproduct A B\ninst✝³ : HasBinaryCoproduct A' B'\ninst✝² : HasBinaryCoproduct (F.obj A) (F.obj B)\ninst✝¹ : HasBinaryCoproduct (F.obj A') (F.obj B')\ninst✝ : IsIso (coprodComparison F A B)\n⊢ F.map coprod.inl ≫ inv (coprodComparison F A B) = coprod.inl",
"tactic": "simp [IsIso.inv_comp_eq]"
}
] |
[
1374,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1373,
1
] |
Mathlib/Data/Sigma/Interval.lean
|
Sigma.Ioo_mk_mk
|
[] |
[
100,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/Order/Partition/Equipartition.lean
|
Finpartition.IsEquipartition.average_le_card_part
|
[
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP : Finpartition s\nhP : IsEquipartition P\nht : t ∈ P.parts\n⊢ (Finset.sum P.parts fun i => Finset.card i) / Finset.card P.parts ≤ Finset.card t",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP : Finpartition s\nhP : IsEquipartition P\nht : t ∈ P.parts\n⊢ Finset.card s / Finset.card P.parts ≤ Finset.card t",
"tactic": "rw [← P.sum_card_parts]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nP : Finpartition s\nhP : IsEquipartition P\nht : t ∈ P.parts\n⊢ (Finset.sum P.parts fun i => Finset.card i) / Finset.card P.parts ≤ Finset.card t",
"tactic": "exact Finset.EquitableOn.le hP ht"
}
] |
[
59,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
UniqueFactorizationMonoid.exists_associated_prime_pow_of_unique_normalized_factor
|
[
{
"state_after": "α : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\np r : α\nh : ∀ {m : α}, m ∈ normalizedFactors r → m = p\nhr : r ≠ 0\n⊢ p ^ ZeroHom.toFun (↑Multiset.card) (normalizedFactors r) ~ᵤ r",
"state_before": "α : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\np r : α\nh : ∀ {m : α}, m ∈ normalizedFactors r → m = p\nhr : r ≠ 0\n⊢ ∃ i, p ^ i ~ᵤ r",
"tactic": "use Multiset.card.toFun (normalizedFactors r)"
},
{
"state_after": "α : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\np r : α\nh : ∀ {m : α}, m ∈ normalizedFactors r → m = p\nhr : r ≠ 0\nthis : Multiset.prod (normalizedFactors r) ~ᵤ r\n⊢ p ^ ZeroHom.toFun (↑Multiset.card) (normalizedFactors r) ~ᵤ r",
"state_before": "α : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\np r : α\nh : ∀ {m : α}, m ∈ normalizedFactors r → m = p\nhr : r ≠ 0\n⊢ p ^ ZeroHom.toFun (↑Multiset.card) (normalizedFactors r) ~ᵤ r",
"tactic": "have := UniqueFactorizationMonoid.normalizedFactors_prod hr"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : CancelCommMonoidWithZero α\ninst✝² : DecidableEq α\ninst✝¹ : NormalizationMonoid α\ninst✝ : UniqueFactorizationMonoid α\np r : α\nh : ∀ {m : α}, m ∈ normalizedFactors r → m = p\nhr : r ≠ 0\nthis : Multiset.prod (normalizedFactors r) ~ᵤ r\n⊢ p ^ ZeroHom.toFun (↑Multiset.card) (normalizedFactors r) ~ᵤ r",
"tactic": "rwa [Multiset.eq_replicate_of_mem fun b => h, Multiset.prod_replicate] at this"
}
] |
[
760,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
756,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.bsup_eq_blsub_iff_succ
|
[
{
"state_after": "α : Type ?u.356089\nβ : Type ?u.356092\nγ : Type ?u.356095\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\n⊢ sup (familyOfBFamily o f) = lsub (familyOfBFamily o f) ↔\n ∀ (a : Ordinal), a < lsub (familyOfBFamily o f) → succ a < lsub (familyOfBFamily o f)",
"state_before": "α : Type ?u.356089\nβ : Type ?u.356092\nγ : Type ?u.356095\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\n⊢ bsup o f = blsub o f ↔ ∀ (a : Ordinal), a < blsub o f → succ a < blsub o f",
"tactic": "rw [← sup_eq_bsup, ← lsub_eq_blsub]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.356089\nβ : Type ?u.356092\nγ : Type ?u.356095\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\n⊢ sup (familyOfBFamily o f) = lsub (familyOfBFamily o f) ↔\n ∀ (a : Ordinal), a < lsub (familyOfBFamily o f) → succ a < lsub (familyOfBFamily o f)",
"tactic": "apply sup_eq_lsub_iff_succ"
}
] |
[
1860,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1857,
1
] |
Mathlib/Computability/TMToPartrec.lean
|
Turing.PartrecToTM2.codeSupp_comp
|
[
{
"state_after": "f g : Code\nk : Cont'\n⊢ trStmts₁ (trNormal g (Cont'.comp f k)) ∪\n (codeSupp' g (Cont'.comp f k) ∪ (trStmts₁ (trNormal f k) ∪ (codeSupp' f k ∪ contSupp k))) =\n trStmts₁ (trNormal g (Cont'.comp f k)) ∪ (codeSupp' g (Cont'.comp f k) ∪ (codeSupp' f k ∪ contSupp k))",
"state_before": "f g : Code\nk : Cont'\n⊢ codeSupp (Code.comp f g) k = trStmts₁ (trNormal (Code.comp f g) k) ∪ codeSupp g (Cont'.comp f k)",
"tactic": "simp [codeSupp, codeSupp', contSupp, Finset.union_assoc]"
},
{
"state_after": "no goals",
"state_before": "f g : Code\nk : Cont'\n⊢ trStmts₁ (trNormal g (Cont'.comp f k)) ∪\n (codeSupp' g (Cont'.comp f k) ∪ (trStmts₁ (trNormal f k) ∪ (codeSupp' f k ∪ contSupp k))) =\n trStmts₁ (trNormal g (Cont'.comp f k)) ∪ (codeSupp' g (Cont'.comp f k) ∪ (codeSupp' f k ∪ contSupp k))",
"tactic": "rw [← Finset.union_assoc _ _ (contSupp k),\n Finset.union_eq_right_iff_subset.2 (codeSupp'_self _ _)]"
}
] |
[
1845,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1840,
1
] |
Mathlib/Analysis/Convex/Star.lean
|
StarConvex.prod
|
[] |
[
146,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/NumberTheory/PellMatiyasevic.lean
|
Pell.xn_modEq_x2n_sub
|
[
{
"state_after": "a : ℕ\na1 : 1 < a\nn j : ℕ\nh : j ≤ 2 * n\njn : n ≤ j\nthis : 2 * n - j + j ≤ n + j\n⊢ xn a1 (2 * n - j) + xn a1 j ≡ 0 [MOD xn a1 n]",
"state_before": "a : ℕ\na1 : 1 < a\nn j : ℕ\nh : j ≤ 2 * n\njn : n ≤ j\n⊢ xn a1 (2 * n - j) + xn a1 j ≡ 0 [MOD xn a1 n]",
"tactic": "have : 2 * n - j + j ≤ n + j := by\n rw [tsub_add_cancel_of_le h, two_mul]; exact Nat.add_le_add_left jn _"
},
{
"state_after": "a : ℕ\na1 : 1 < a\nn j : ℕ\nh : j ≤ 2 * n\njn : n ≤ j\nthis : 2 * n - j + j ≤ n + j\nt : xn a1 (2 * n - (2 * n - j)) + xn a1 (2 * n - j) ≡ 0 [MOD xn a1 n] :=\n xn_modEq_x2n_sub_lem a1 (Nat.le_of_add_le_add_right this)\n⊢ xn a1 (2 * n - j) + xn a1 j ≡ 0 [MOD xn a1 n]",
"state_before": "a : ℕ\na1 : 1 < a\nn j : ℕ\nh : j ≤ 2 * n\njn : n ≤ j\nthis : 2 * n - j + j ≤ n + j\n⊢ xn a1 (2 * n - j) + xn a1 j ≡ 0 [MOD xn a1 n]",
"tactic": "let t := xn_modEq_x2n_sub_lem a1 (Nat.le_of_add_le_add_right this)"
},
{
"state_after": "no goals",
"state_before": "a : ℕ\na1 : 1 < a\nn j : ℕ\nh : j ≤ 2 * n\njn : n ≤ j\nthis : 2 * n - j + j ≤ n + j\nt : xn a1 (2 * n - (2 * n - j)) + xn a1 (2 * n - j) ≡ 0 [MOD xn a1 n] :=\n xn_modEq_x2n_sub_lem a1 (Nat.le_of_add_le_add_right this)\n⊢ xn a1 (2 * n - j) + xn a1 j ≡ 0 [MOD xn a1 n]",
"tactic": "rwa [tsub_tsub_cancel_of_le h, add_comm] at t"
},
{
"state_after": "a : ℕ\na1 : 1 < a\nn j : ℕ\nh : j ≤ 2 * n\njn : n ≤ j\n⊢ n + n ≤ n + j",
"state_before": "a : ℕ\na1 : 1 < a\nn j : ℕ\nh : j ≤ 2 * n\njn : n ≤ j\n⊢ 2 * n - j + j ≤ n + j",
"tactic": "rw [tsub_add_cancel_of_le h, two_mul]"
},
{
"state_after": "no goals",
"state_before": "a : ℕ\na1 : 1 < a\nn j : ℕ\nh : j ≤ 2 * n\njn : n ≤ j\n⊢ n + n ≤ n + j",
"tactic": "exact Nat.add_le_add_left jn _"
}
] |
[
629,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
624,
1
] |
Mathlib/CategoryTheory/Sites/Grothendieck.lean
|
CategoryTheory.GrothendieckTopology.top_covering
|
[] |
[
335,
5
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Complex.tan_add_int_mul_pi
|
[] |
[
1331,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1330,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegral.integral_const
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.13713922\n𝕜 : Type ?u.13713925\nE : Type u_1\nF : Type ?u.13713931\nA : Type ?u.13713934\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\nc : E\n⊢ (∫ (x : ℝ) in a..b, c) = (b - a) • c",
"tactic": "simp only [integral_const', Real.volume_Ioc, ENNReal.toReal_ofReal', ← neg_sub b,\n max_zero_sub_eq_self]"
}
] |
[
642,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
640,
1
] |
Mathlib/RingTheory/AlgebraicIndependent.lean
|
AlgebraicIndependent.restrict_of_comp_subtype
|
[] |
[
262,
5
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/Data/SetLike/Basic.lean
|
SetLike.not_le_iff_exists
|
[] |
[
211,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/Data/Multiset/Range.lean
|
Multiset.range_succ
|
[
{
"state_after": "n : ℕ\n⊢ ↑[n] + ↑(List.range n) = n ::ₘ range n",
"state_before": "n : ℕ\n⊢ range (succ n) = n ::ₘ range n",
"tactic": "rw [range, List.range_succ, ← coe_add, add_comm]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ ↑[n] + ↑(List.range n) = n ::ₘ range n",
"tactic": "rfl"
}
] |
[
39,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
38,
1
] |
Mathlib/Order/Hom/CompleteLattice.lean
|
map_iSup
|
[
{
"state_after": "no goals",
"state_before": "F : Type u_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.3707\nδ : Type ?u.3710\nι : Sort u_4\nκ : ι → Sort ?u.3718\ninst✝² : SupSet α\ninst✝¹ : SupSet β\ninst✝ : sSupHomClass F α β\nf : F\ng : ι → α\n⊢ ↑f (⨆ (i : ι), g i) = ⨆ (i : ι), ↑f (g i)",
"tactic": "simp [iSup, ← Set.range_comp, Function.comp]"
}
] |
[
132,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Combinatorics/Composition.lean
|
Composition.boundary_zero
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\nc : Composition n\n⊢ ↑(boundary c) 0 = 0",
"tactic": "simp [boundary, Fin.ext_iff]"
}
] |
[
266,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.inter_insert_of_mem
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.184956\nγ : Type ?u.184959\ninst✝ : DecidableEq α\ns s₁✝ s₂✝ t t₁ t₂ u v : Finset α\na✝ b : α\ns₁ s₂ : Finset α\na : α\nh : a ∈ s₁\n⊢ s₁ ∩ insert a s₂ = insert a (s₁ ∩ s₂)",
"tactic": "rw [inter_comm, insert_inter_of_mem h, inter_comm]"
}
] |
[
1658,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1657,
1
] |
Mathlib/Order/Lattice.lean
|
sup_left_right_swap
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : SemilatticeSup α\na✝ b✝ c✝ d a b c : α\n⊢ a ⊔ b ⊔ c = c ⊔ b ⊔ a",
"tactic": "rw [sup_comm, @sup_comm _ _ a, sup_assoc]"
}
] |
[
262,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/GroupTheory/Subsemigroup/Center.lean
|
Subsemigroup.mem_center_iff
|
[] |
[
163,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/CategoryTheory/Adjunction/Mates.lean
|
CategoryTheory.transferNatTransSelf_id
|
[
{
"state_after": "case w.h\nC : Type u₁\nD : Type u₂\ninst✝¹ : Category C\ninst✝ : Category D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\nx✝ : D\n⊢ (↑(transferNatTransSelf adj₁ adj₁) (𝟙 L₁)).app x✝ = (𝟙 R₁).app x✝",
"state_before": "C : Type u₁\nD : Type u₂\ninst✝¹ : Category C\ninst✝ : Category D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\n⊢ ↑(transferNatTransSelf adj₁ adj₁) (𝟙 L₁) = 𝟙 R₁",
"tactic": "ext"
},
{
"state_after": "case w.h\nC : Type u₁\nD : Type u₂\ninst✝¹ : Category C\ninst✝ : Category D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\nx✝ : D\n⊢ 𝟙 (R₁.obj x✝) ≫\n (adj₁.unit.app (R₁.obj x✝) ≫\n R₁.map ((𝟙 (L₁.obj (R₁.obj x✝)) ≫ 𝟙 (L₁.obj (R₁.obj x✝)) ≫ 𝟙 (L₁.obj (R₁.obj x✝))) ≫ adj₁.counit.app x✝)) ≫\n 𝟙 (R₁.obj x✝) =\n 𝟙 (R₁.obj x✝)",
"state_before": "case w.h\nC : Type u₁\nD : Type u₂\ninst✝¹ : Category C\ninst✝ : Category D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\nx✝ : D\n⊢ (↑(transferNatTransSelf adj₁ adj₁) (𝟙 L₁)).app x✝ = (𝟙 R₁).app x✝",
"tactic": "dsimp [transferNatTransSelf, transferNatTrans]"
},
{
"state_after": "no goals",
"state_before": "case w.h\nC : Type u₁\nD : Type u₂\ninst✝¹ : Category C\ninst✝ : Category D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\nx✝ : D\n⊢ 𝟙 (R₁.obj x✝) ≫\n (adj₁.unit.app (R₁.obj x✝) ≫\n R₁.map ((𝟙 (L₁.obj (R₁.obj x✝)) ≫ 𝟙 (L₁.obj (R₁.obj x✝)) ≫ 𝟙 (L₁.obj (R₁.obj x✝))) ≫ adj₁.counit.app x✝)) ≫\n 𝟙 (R₁.obj x✝) =\n 𝟙 (R₁.obj x✝)",
"tactic": "simp"
}
] |
[
188,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.PullbackCone.π_app_left
|
[] |
[
531,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
531,
1
] |
Mathlib/Order/Bounded.lean
|
Set.unbounded_le_univ
|
[] |
[
152,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Topology/MetricSpace/Isometry.lean
|
IsometryEquiv.nndist_eq
|
[] |
[
360,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
358,
11
] |
Mathlib/Tactic/NormNum/Core.lean
|
Mathlib.Meta.NormNum.IsRat.to_isNat
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Ring α\na✝ : ℕ\ninv : Invertible ↑1\nthis : Invertible 1\n⊢ ↑(Int.ofNat a✝) * ⅟↑1 = ↑a✝",
"tactic": "simp"
}
] |
[
151,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Algebra/Associated.lean
|
Associates.isUnit_iff_eq_one
|
[] |
[
886,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
885,
1
] |
Mathlib/CategoryTheory/Generator.lean
|
CategoryTheory.isSeparator_unop_iff
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nG : Cᵒᵖ\n⊢ IsSeparator G.unop ↔ IsCoseparator G",
"tactic": "rw [IsSeparator, IsCoseparator, ← isSeparating_unop_iff, Set.singleton_unop]"
}
] |
[
419,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
418,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
MeasureTheory.set_integral_nonpos_le
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.297696\nE : Type ?u.297699\nF : Type ?u.297702\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\ns✝ s : Set α\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhfi : Integrable f\n⊢ (∫ (x : α), indicator {a | f a ≤ 0} (fun x => f x) x ∂μ) ≤ ∫ (x : α), indicator s (fun x => f x) x ∂μ",
"state_before": "α : Type u_1\nβ : Type ?u.297696\nE : Type ?u.297699\nF : Type ?u.297702\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\ns✝ s : Set α\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhfi : Integrable f\n⊢ (∫ (x : α) in {y | f y ≤ 0}, f x ∂μ) ≤ ∫ (x : α) in s, f x ∂μ",
"tactic": "rw [← integral_indicator hs, ←\n integral_indicator (hf.measurableSet_le stronglyMeasurable_const)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.297696\nE : Type ?u.297699\nF : Type ?u.297702\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\ns✝ s : Set α\nhs : MeasurableSet s\nhf : StronglyMeasurable f\nhfi : Integrable f\n⊢ (∫ (x : α), indicator {a | f a ≤ 0} (fun x => f x) x ∂μ) ≤ ∫ (x : α), indicator s (fun x => f x) x ∂μ",
"tactic": "exact\n integral_mono (hfi.indicator (hf.measurableSet_le stronglyMeasurable_const))\n (hfi.indicator hs) (indicator_nonpos_le_indicator s f)"
}
] |
[
778,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
772,
1
] |
Mathlib/Algebra/Algebra/Unitization.lean
|
Unitization.fst_one
|
[] |
[
360,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
359,
1
] |
Mathlib/Topology/VectorBundle/Basic.lean
|
VectorBundleCore.localTriv_symm_fst
|
[] |
[
719,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
717,
1
] |
Mathlib/Algebra/Algebra/Hom.lean
|
AlgHom.comp_algebraMap
|
[] |
[
242,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
241,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.update_zero_eq_erase
|
[
{
"state_after": "case a\nR : Type u\na b : R\nm n✝¹ : ℕ\ninst✝ : Semiring R\np✝ q p : R[X]\nn n✝ : ℕ\n⊢ coeff (update p n 0) n✝ = coeff (erase n p) n✝",
"state_before": "R : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np✝ q p : R[X]\nn : ℕ\n⊢ update p n 0 = erase n p",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u\na b : R\nm n✝¹ : ℕ\ninst✝ : Semiring R\np✝ q p : R[X]\nn n✝ : ℕ\n⊢ coeff (update p n 0) n✝ = coeff (erase n p) n✝",
"tactic": "rw [coeff_update_apply, coeff_erase]"
}
] |
[
1108,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1106,
1
] |
Std/Data/List/Lemmas.lean
|
List.isPrefix.length_le
|
[] |
[
1635,
22
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1634,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Concrete.lean
|
Equiv.Perm.length_toList
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx : α\n⊢ length (toList p x) = Finset.card (support (cycleOf p x))",
"tactic": "simp [toList]"
}
] |
[
233,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Std/Data/RBMap/WF.lean
|
Std.RBNode.Balanced.append
|
[
{
"state_after": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nl r : RBNode α\nhl : Balanced l c₁ n\nhr : Balanced r c₂ n\n⊢ RedRed (c₁ = black → c₂ ≠ black)\n (match l, r with\n | nil, x => x\n | x, nil => x\n | node red a x b, node red c y d =>\n match append b c with\n | node red b' z c' => node red (node red a x b') z (node red c' y d)\n | bc => node red a x (node red bc y d)\n | node black a x b, node black c y d =>\n match append b c with\n | node red b' z c' => node red (node black a x b') z (node black c' y d)\n | bc => balLeft a x (node black bc y d)\n | a@h:(node black l v r), node red b x c => node red (append a b) x c\n | node red a x b, c@h:(node black l v r) => node red a x (append b c))\n n",
"state_before": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nl r : RBNode α\nhl : Balanced l c₁ n\nhr : Balanced r c₂ n\n⊢ RedRed (c₁ = black → c₂ ≠ black) (append l r) n",
"tactic": "unfold append"
},
{
"state_after": "case h_1\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nr : RBNode α\nhr : Balanced r c₂ n\nx✝¹ x✝ : RBNode α\nhl : Balanced nil c₁ n\n⊢ RedRed (c₁ = black → c₂ ≠ black) r n\n\ncase h_2\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nl : RBNode α\nhl : Balanced l c₁ n\nx✝² x✝¹ : RBNode α\nx✝ : l = nil → False\nhr : Balanced nil c₂ n\n⊢ RedRed (c₁ = black → c₂ ≠ black) l n\n\ncase h_3\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb✝ c✝ : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node red a✝ x✝ b✝) c₁ n\nhr : Balanced (node red c✝ y✝ d✝) c₂ n\n⊢ RedRed (c₁ = black → c₂ ≠ black)\n (match append b✝ c✝ with\n | node red b' z c' => node red (node red a✝ x✝ b') z (node red c' y✝ d✝)\n | bc => node red a✝ x✝ (node red bc y✝ d✝))\n n\n\ncase h_4\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb✝ c✝ : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node black a✝ x✝ b✝) c₁ n\nhr : Balanced (node black c✝ y✝ d✝) c₂ n\n⊢ RedRed (c₁ = black → c₂ ≠ black)\n (match append b✝ c✝ with\n | node red b' z c' => node red (node black a✝ x✝ b') z (node black c' y✝ d✝)\n | bc => balLeft a✝ x✝ (node black bc y✝ d✝))\n n\n\ncase h_5\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ l✝ : RBNode α\nv✝ : α\nr✝ b✝ : RBNode α\nx✝ : α\nc✝ : RBNode α\nhl : Balanced (node black l✝ v✝ r✝) c₁ n\nhr : Balanced (node red b✝ x✝ c✝) c₂ n\n⊢ RedRed (c₁ = black → c₂ ≠ black) (node red (append (node black l✝ v✝ r✝) b✝) x✝ c✝) n\n\ncase h_6\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb✝ l✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nhl : Balanced (node red a✝ x✝ b✝) c₁ n\nhr : Balanced (node black l✝ v✝ r✝) c₂ n\n⊢ RedRed (c₁ = black → c₂ ≠ black) (node red a✝ x✝ (append b✝ (node black l✝ v✝ r✝))) n",
"state_before": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nl r : RBNode α\nhl : Balanced l c₁ n\nhr : Balanced r c₂ n\n⊢ RedRed (c₁ = black → c₂ ≠ black)\n (match l, r with\n | nil, x => x\n | x, nil => x\n | node red a x b, node red c y d =>\n match append b c with\n | node red b' z c' => node red (node red a x b') z (node red c' y d)\n | bc => node red a x (node red bc y d)\n | node black a x b, node black c y d =>\n match append b c with\n | node red b' z c' => node red (node black a x b') z (node black c' y d)\n | bc => balLeft a x (node black bc y d)\n | a@h:(node black l v r), node red b x c => node red (append a b) x c\n | node red a x b, c@h:(node black l v r) => node red a x (append b c))\n n",
"tactic": "split"
},
{
"state_after": "no goals",
"state_before": "case h_1\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nr : RBNode α\nhr : Balanced r c₂ n\nx✝¹ x✝ : RBNode α\nhl : Balanced nil c₁ n\n⊢ RedRed (c₁ = black → c₂ ≠ black) r n",
"tactic": "exact .balanced hr"
},
{
"state_after": "no goals",
"state_before": "case h_2\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nl : RBNode α\nhl : Balanced l c₁ n\nx✝² x✝¹ : RBNode α\nx✝ : l = nil → False\nhr : Balanced nil c₂ n\n⊢ RedRed (c₁ = black → c₂ ≠ black) l n",
"tactic": "exact .balanced hl"
},
{
"state_after": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node red a✝ x✝ b) c₁ n\nhr : Balanced (node red c y✝ d✝) c₂ n\nha : Balanced a✝ black n\nhb : Balanced b black n\n⊢ RedRed (red = black → c₂ ≠ black)\n (match append b c with\n | node red b' z c' => node red (node red a✝ x✝ b') z (node red c' y✝ d✝)\n | bc => node red a✝ x✝ (node red bc y✝ d✝))\n n",
"state_before": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node red a✝ x✝ b) c₁ n\nhr : Balanced (node red c y✝ d✝) c₂ n\n⊢ RedRed (c₁ = black → c₂ ≠ black)\n (match append b c with\n | node red b' z c' => node red (node red a✝ x✝ b') z (node red c' y✝ d✝)\n | bc => node red a✝ x✝ (node red bc y✝ d✝))\n n",
"tactic": "have .red ha hb := hl"
},
{
"state_after": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node red a✝ x✝ b) c₁ n\nhr : Balanced (node red c y✝ d✝) c₂ n\nha : Balanced a✝ black n\nhb : Balanced b black n\nhc : Balanced c black n\nhd : Balanced d✝ black n\n⊢ RedRed (red = black → red ≠ black)\n (match append b c with\n | node red b' z c' => node red (node red a✝ x✝ b') z (node red c' y✝ d✝)\n | bc => node red a✝ x✝ (node red bc y✝ d✝))\n n",
"state_before": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node red a✝ x✝ b) c₁ n\nhr : Balanced (node red c y✝ d✝) c₂ n\nha : Balanced a✝ black n\nhb : Balanced b black n\n⊢ RedRed (red = black → c₂ ≠ black)\n (match append b c with\n | node red b' z c' => node red (node red a✝ x✝ b') z (node red c' y✝ d✝)\n | bc => node red a✝ x✝ (node red bc y✝ d✝))\n n",
"tactic": "have .red hc hd := hr"
},
{
"state_after": "case h_1\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝³ x✝² a✝¹ : RBNode α\nx✝¹ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node red a✝¹ x✝¹ b) c₁ n\nhr : Balanced (node red c y✝ d✝) c₂ n\nha : Balanced a✝¹ black n\nhb : Balanced b black n\nhc : Balanced c black n\nhd : Balanced d✝ black n\nw✝ : RBColor\nIH : Balanced (append b c) w✝ n\nl✝ a✝ : RBNode α\nx✝ : α\nb✝ : RBNode α\nheq✝ : append b c = node red a✝ x✝ b✝\n⊢ RedRed (red = black → red ≠ black) (node red (node red a✝¹ x✝¹ a✝) x✝ (node red b✝ y✝ d✝)) n\n\ncase h_2\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝³ x✝² a✝ : RBNode α\nx✝¹ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node red a✝ x✝¹ b) c₁ n\nhr : Balanced (node red c y✝ d✝) c₂ n\nha : Balanced a✝ black n\nhb : Balanced b black n\nhc : Balanced c black n\nhd : Balanced d✝ black n\nw✝ : RBColor\nIH : Balanced (append b c) w✝ n\nl✝ : RBNode α\nx✝ : ∀ (a : RBNode α) (x : α) (b_1 : RBNode α), append b c = node red a x b_1 → False\n⊢ RedRed (red = black → red ≠ black) (node red a✝ x✝¹ (node red (append b c) y✝ d✝)) n",
"state_before": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node red a✝ x✝ b) c₁ n\nhr : Balanced (node red c y✝ d✝) c₂ n\nha : Balanced a✝ black n\nhb : Balanced b black n\nhc : Balanced c black n\nhd : Balanced d✝ black n\nw✝ : RBColor\nIH : Balanced (append b c) w✝ n\n⊢ RedRed (red = black → red ≠ black)\n (match append b c with\n | node red b' z c' => node red (node red a✝ x✝ b') z (node red c' y✝ d✝)\n | bc => node red a✝ x✝ (node red bc y✝ d✝))\n n",
"tactic": "split"
},
{
"state_after": "no goals",
"state_before": "case h_1\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝³ x✝² a✝¹ : RBNode α\nx✝¹ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node red a✝¹ x✝¹ b) c₁ n\nhr : Balanced (node red c y✝ d✝) c₂ n\nha : Balanced a✝¹ black n\nhb : Balanced b black n\nhc : Balanced c black n\nhd : Balanced d✝ black n\nw✝ : RBColor\nIH : Balanced (append b c) w✝ n\nl✝ a✝ : RBNode α\nx✝ : α\nb✝ : RBNode α\nheq✝ : append b c = node red a✝ x✝ b✝\n⊢ RedRed (red = black → red ≠ black) (node red (node red a✝¹ x✝¹ a✝) x✝ (node red b✝ y✝ d✝)) n",
"tactic": "next e =>\nhave .red hb' hc' := e ▸ IH\nexact .redred (fun.) (.red ha hb') (.red hc' hd)"
},
{
"state_after": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝³ x✝² a✝¹ : RBNode α\nx✝¹ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node red a✝¹ x✝¹ b) c₁ n\nhr : Balanced (node red c y✝ d✝) c₂ n\nha : Balanced a✝¹ black n\nhb : Balanced b black n\nhc : Balanced c black n\nhd : Balanced d✝ black n\nl✝ a✝ : RBNode α\nx✝ : α\nb✝ : RBNode α\ne : append b c = node red a✝ x✝ b✝\nhb' : Balanced a✝ black n\nhc' : Balanced b✝ black n\nIH : Balanced (append b c) red n\n⊢ RedRed (red = black → red ≠ black) (node red (node red a✝¹ x✝¹ a✝) x✝ (node red b✝ y✝ d✝)) n",
"state_before": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝³ x✝² a✝¹ : RBNode α\nx✝¹ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node red a✝¹ x✝¹ b) c₁ n\nhr : Balanced (node red c y✝ d✝) c₂ n\nha : Balanced a✝¹ black n\nhb : Balanced b black n\nhc : Balanced c black n\nhd : Balanced d✝ black n\nw✝ : RBColor\nIH : Balanced (append b c) w✝ n\nl✝ a✝ : RBNode α\nx✝ : α\nb✝ : RBNode α\ne : append b c = node red a✝ x✝ b✝\n⊢ RedRed (red = black → red ≠ black) (node red (node red a✝¹ x✝¹ a✝) x✝ (node red b✝ y✝ d✝)) n",
"tactic": "have .red hb' hc' := e ▸ IH"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝³ x✝² a✝¹ : RBNode α\nx✝¹ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node red a✝¹ x✝¹ b) c₁ n\nhr : Balanced (node red c y✝ d✝) c₂ n\nha : Balanced a✝¹ black n\nhb : Balanced b black n\nhc : Balanced c black n\nhd : Balanced d✝ black n\nl✝ a✝ : RBNode α\nx✝ : α\nb✝ : RBNode α\ne : append b c = node red a✝ x✝ b✝\nhb' : Balanced a✝ black n\nhc' : Balanced b✝ black n\nIH : Balanced (append b c) red n\n⊢ RedRed (red = black → red ≠ black) (node red (node red a✝¹ x✝¹ a✝) x✝ (node red b✝ y✝ d✝)) n",
"tactic": "exact .redred (fun.) (.red ha hb') (.red hc' hd)"
},
{
"state_after": "no goals",
"state_before": "case h_2\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝³ x✝² a✝ : RBNode α\nx✝¹ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node red a✝ x✝¹ b) c₁ n\nhr : Balanced (node red c y✝ d✝) c₂ n\nha : Balanced a✝ black n\nhb : Balanced b black n\nhc : Balanced c black n\nhd : Balanced d✝ black n\nw✝ : RBColor\nIH : Balanced (append b c) w✝ n\nl✝ : RBNode α\nx✝ : ∀ (a : RBNode α) (x : α) (b_1 : RBNode α), append b c = node red a x b_1 → False\n⊢ RedRed (red = black → red ≠ black) (node red a✝ x✝¹ (node red (append b c) y✝ d✝)) n",
"tactic": "next bcc _ H =>\nmatch bcc, append b c, IH, H with\n| black, _, IH, _ => exact .redred (fun.) ha (.red IH hd)\n| red, _, .red .., H => cases H _ _ _ rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node red a✝ x✝ b) c₁ n\nhr : Balanced (node red c y✝ d✝) c₂ n\nha : Balanced a✝ black n\nhb : Balanced b black n\nhc : Balanced c black n\nhd : Balanced d✝ black n\nbcc : RBColor\nIH : Balanced (append b c) bcc n\nl✝ : RBNode α\nH : ∀ (a : RBNode α) (x : α) (b_1 : RBNode α), append b c = node red a x b_1 → False\n⊢ RedRed (red = black → red ≠ black) (node red a✝ x✝ (node red (append b c) y✝ d✝)) n",
"tactic": "match bcc, append b c, IH, H with\n| black, _, IH, _ => exact .redred (fun.) ha (.red IH hd)\n| red, _, .red .., H => cases H _ _ _ rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝⁴ x✝³ a✝ : RBNode α\nx✝² : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node red a✝ x✝² b) c₁ n\nhr : Balanced (node red c y✝ d✝) c₂ n\nha : Balanced a✝ black n\nhb : Balanced b black n\nhc : Balanced c black n\nhd : Balanced d✝ black n\nbcc : RBColor\nIH✝ : Balanced (append b c) bcc n\nl✝ : RBNode α\nH : ∀ (a : RBNode α) (x : α) (b_1 : RBNode α), append b c = node red a x b_1 → False\nx✝¹ : RBNode α\nIH : Balanced x✝¹ black n\nx✝ : ∀ (a : RBNode α) (x : α) (b : RBNode α), x✝¹ = node red a x b → False\n⊢ RedRed (red = black → red ≠ black) (node red a✝ x✝² (node red x✝¹ y✝ d✝)) n",
"tactic": "exact .redred (fun.) ha (.red IH hd)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝³ x✝² a✝² : RBNode α\nx✝¹ : α\nb c : RBNode α\ny✝¹ : α\nd✝ : RBNode α\nhl : Balanced (node red a✝² x✝¹ b) c₁ n\nhr : Balanced (node red c y✝¹ d✝) c₂ n\nha : Balanced a✝² black n\nhb : Balanced b black n\nhc : Balanced c black n\nhd : Balanced d✝ black n\nbcc : RBColor\nIH : Balanced (append b c) bcc n\nl✝ : RBNode α\nH✝ : ∀ (a : RBNode α) (x : α) (b_1 : RBNode α), append b c = node red a x b_1 → False\nx✝ y✝ : RBNode α\nv✝ : α\na✝¹ : Balanced x✝ black n\na✝ : Balanced y✝ black n\nH : ∀ (a : RBNode α) (x : α) (b : RBNode α), node red x✝ v✝ y✝ = node red a x b → False\n⊢ RedRed (red = black → red ≠ black) (node red a✝² x✝¹ (node red (node red x✝ v✝ y✝) y✝¹ d✝)) n",
"tactic": "cases H _ _ _ rfl"
},
{
"state_after": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node black a✝ x✝ b) c₁ n\nc₁✝ : RBColor\nn✝ : Nat\nc₂✝ : RBColor\nha : Balanced a✝ c₁✝ n✝\nhb : Balanced b c₂✝ n✝\nhr : Balanced (node black c y✝ d✝) c₂ (n✝ + 1)\n⊢ RedRed (black = black → c₂ ≠ black)\n (match append b c with\n | node red b' z c' => node red (node black a✝ x✝ b') z (node black c' y✝ d✝)\n | bc => balLeft a✝ x✝ (node black bc y✝ d✝))\n (n✝ + 1)",
"state_before": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node black a✝ x✝ b) c₁ n\nhr : Balanced (node black c y✝ d✝) c₂ n\n⊢ RedRed (c₁ = black → c₂ ≠ black)\n (match append b c with\n | node red b' z c' => node red (node black a✝ x✝ b') z (node black c' y✝ d✝)\n | bc => balLeft a✝ x✝ (node black bc y✝ d✝))\n n",
"tactic": "have .black ha hb := hl"
},
{
"state_after": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node black a✝ x✝ b) c₁ n\nc₁✝¹ : RBColor\nn✝ : Nat\nc₂✝¹ : RBColor\nha : Balanced a✝ c₁✝¹ n✝\nhb : Balanced b c₂✝¹ n✝\nhr : Balanced (node black c y✝ d✝) c₂ (n✝ + 1)\nc₁✝ c₂✝ : RBColor\nhc : Balanced c c₁✝ n✝\nhd : Balanced d✝ c₂✝ n✝\n⊢ RedRed (black = black → black ≠ black)\n (match append b c with\n | node red b' z c' => node red (node black a✝ x✝ b') z (node black c' y✝ d✝)\n | bc => balLeft a✝ x✝ (node black bc y✝ d✝))\n (n✝ + 1)",
"state_before": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node black a✝ x✝ b) c₁ n\nc₁✝ : RBColor\nn✝ : Nat\nc₂✝ : RBColor\nha : Balanced a✝ c₁✝ n✝\nhb : Balanced b c₂✝ n✝\nhr : Balanced (node black c y✝ d✝) c₂ (n✝ + 1)\n⊢ RedRed (black = black → c₂ ≠ black)\n (match append b c with\n | node red b' z c' => node red (node black a✝ x✝ b') z (node black c' y✝ d✝)\n | bc => balLeft a✝ x✝ (node black bc y✝ d✝))\n (n✝ + 1)",
"tactic": "have .black hc hd := hr"
},
{
"state_after": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node black a✝ x✝ b) c₁ n\nc₁✝¹ : RBColor\nn✝ : Nat\nc₂✝¹ : RBColor\nha : Balanced a✝ c₁✝¹ n✝\nhb : Balanced b c₂✝¹ n✝\nhr : Balanced (node black c y✝ d✝) c₂ (n✝ + 1)\nc₁✝ c₂✝ : RBColor\nhc : Balanced c c₁✝ n✝\nhd : Balanced d✝ c₂✝ n✝\nIH : RedRed (c₂✝¹ = black → c₁✝ ≠ black) (append b c) n✝\n⊢ RedRed (black = black → black ≠ black)\n (match append b c with\n | node red b' z c' => node red (node black a✝ x✝ b') z (node black c' y✝ d✝)\n | bc => balLeft a✝ x✝ (node black bc y✝ d✝))\n (n✝ + 1)",
"state_before": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node black a✝ x✝ b) c₁ n\nc₁✝¹ : RBColor\nn✝ : Nat\nc₂✝¹ : RBColor\nha : Balanced a✝ c₁✝¹ n✝\nhb : Balanced b c₂✝¹ n✝\nhr : Balanced (node black c y✝ d✝) c₂ (n✝ + 1)\nc₁✝ c₂✝ : RBColor\nhc : Balanced c c₁✝ n✝\nhd : Balanced d✝ c₂✝ n✝\n⊢ RedRed (black = black → black ≠ black)\n (match append b c with\n | node red b' z c' => node red (node black a✝ x✝ b') z (node black c' y✝ d✝)\n | bc => balLeft a✝ x✝ (node black bc y✝ d✝))\n (n✝ + 1)",
"tactic": "have IH := hb.append hc"
},
{
"state_after": "case h_1\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝³ x✝² a✝¹ : RBNode α\nx✝¹ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node black a✝¹ x✝¹ b) c₁ n\nc₁✝¹ : RBColor\nn✝ : Nat\nc₂✝¹ : RBColor\nha : Balanced a✝¹ c₁✝¹ n✝\nhb : Balanced b c₂✝¹ n✝\nhr : Balanced (node black c y✝ d✝) c₂ (n✝ + 1)\nc₁✝ c₂✝ : RBColor\nhc : Balanced c c₁✝ n✝\nhd : Balanced d✝ c₂✝ n✝\nIH : RedRed (c₂✝¹ = black → c₁✝ ≠ black) (append b c) n✝\nl✝ a✝ : RBNode α\nx✝ : α\nb✝ : RBNode α\nheq✝ : append b c = node red a✝ x✝ b✝\n⊢ RedRed (black = black → black ≠ black) (node red (node black a✝¹ x✝¹ a✝) x✝ (node black b✝ y✝ d✝)) (n✝ + 1)\n\ncase h_2\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝³ x✝² a✝ : RBNode α\nx✝¹ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node black a✝ x✝¹ b) c₁ n\nc₁✝¹ : RBColor\nn✝ : Nat\nc₂✝¹ : RBColor\nha : Balanced a✝ c₁✝¹ n✝\nhb : Balanced b c₂✝¹ n✝\nhr : Balanced (node black c y✝ d✝) c₂ (n✝ + 1)\nc₁✝ c₂✝ : RBColor\nhc : Balanced c c₁✝ n✝\nhd : Balanced d✝ c₂✝ n✝\nIH : RedRed (c₂✝¹ = black → c₁✝ ≠ black) (append b c) n✝\nl✝ : RBNode α\nx✝ : ∀ (a : RBNode α) (x : α) (b_1 : RBNode α), append b c = node red a x b_1 → False\n⊢ RedRed (black = black → black ≠ black) (balLeft a✝ x✝¹ (node black (append b c) y✝ d✝)) (n✝ + 1)",
"state_before": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node black a✝ x✝ b) c₁ n\nc₁✝¹ : RBColor\nn✝ : Nat\nc₂✝¹ : RBColor\nha : Balanced a✝ c₁✝¹ n✝\nhb : Balanced b c₂✝¹ n✝\nhr : Balanced (node black c y✝ d✝) c₂ (n✝ + 1)\nc₁✝ c₂✝ : RBColor\nhc : Balanced c c₁✝ n✝\nhd : Balanced d✝ c₂✝ n✝\nIH : RedRed (c₂✝¹ = black → c₁✝ ≠ black) (append b c) n✝\n⊢ RedRed (black = black → black ≠ black)\n (match append b c with\n | node red b' z c' => node red (node black a✝ x✝ b') z (node black c' y✝ d✝)\n | bc => balLeft a✝ x✝ (node black bc y✝ d✝))\n (n✝ + 1)",
"tactic": "split"
},
{
"state_after": "no goals",
"state_before": "case h_1\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝³ x✝² a✝¹ : RBNode α\nx✝¹ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node black a✝¹ x✝¹ b) c₁ n\nc₁✝¹ : RBColor\nn✝ : Nat\nc₂✝¹ : RBColor\nha : Balanced a✝¹ c₁✝¹ n✝\nhb : Balanced b c₂✝¹ n✝\nhr : Balanced (node black c y✝ d✝) c₂ (n✝ + 1)\nc₁✝ c₂✝ : RBColor\nhc : Balanced c c₁✝ n✝\nhd : Balanced d✝ c₂✝ n✝\nIH : RedRed (c₂✝¹ = black → c₁✝ ≠ black) (append b c) n✝\nl✝ a✝ : RBNode α\nx✝ : α\nb✝ : RBNode α\nheq✝ : append b c = node red a✝ x✝ b✝\n⊢ RedRed (black = black → black ≠ black) (node red (node black a✝¹ x✝¹ a✝) x✝ (node black b✝ y✝ d✝)) (n✝ + 1)",
"tactic": "next e => match e ▸ IH with\n| .balanced (.red hb' hc') | .redred _ hb' hc' =>\n exact .balanced (.red (.black ha hb') (.black hc' hd))"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝³ x✝² a✝¹ : RBNode α\nx✝¹ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node black a✝¹ x✝¹ b) c₁ n\nc₁✝¹ : RBColor\nn✝ : Nat\nc₂✝¹ : RBColor\nha : Balanced a✝¹ c₁✝¹ n✝\nhb : Balanced b c₂✝¹ n✝\nhr : Balanced (node black c y✝ d✝) c₂ (n✝ + 1)\nc₁✝ c₂✝ : RBColor\nhc : Balanced c c₁✝ n✝\nhd : Balanced d✝ c₂✝ n✝\nIH : RedRed (c₂✝¹ = black → c₁✝ ≠ black) (append b c) n✝\nl✝ a✝ : RBNode α\nx✝ : α\nb✝ : RBNode α\ne : append b c = node red a✝ x✝ b✝\n⊢ RedRed (black = black → black ≠ black) (node red (node black a✝¹ x✝¹ a✝) x✝ (node black b✝ y✝ d✝)) (n✝ + 1)",
"tactic": "match e ▸ IH with\n| .balanced (.red hb' hc') | .redred _ hb' hc' =>\nexact .balanced (.red (.black ha hb') (.black hc' hd))"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝³ x✝² a✝² : RBNode α\nx✝¹ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node black a✝² x✝¹ b) c₁ n\nc₁✝² : RBColor\nn✝ : Nat\nc₂✝² : RBColor\nha : Balanced a✝² c₁✝² n✝\nhb : Balanced b c₂✝² n✝\nhr : Balanced (node black c y✝ d✝) c₂ (n✝ + 1)\nc₁✝¹ c₂✝¹ : RBColor\nhc : Balanced c c₁✝¹ n✝\nhd : Balanced d✝ c₂✝¹ n✝\nIH : RedRed (c₂✝² = black → c₁✝¹ ≠ black) (append b c) n✝\nl✝ a✝¹ : RBNode α\nx✝ : α\nb✝ : RBNode α\ne : append b c = node red a✝¹ x✝ b✝\nc₁✝ c₂✝ : RBColor\na✝ : c₂✝² = black → c₁✝¹ ≠ black\nhb' : Balanced a✝¹ c₁✝ n✝\nhc' : Balanced b✝ c₂✝ n✝\n⊢ RedRed (black = black → black ≠ black) (node red (node black a✝² x✝¹ a✝¹) x✝ (node black b✝ y✝ d✝)) (n✝ + 1)",
"tactic": "exact .balanced (.red (.black ha hb') (.black hc' hd))"
},
{
"state_after": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝³ x✝² a✝ : RBNode α\nx✝¹ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node black a✝ x✝¹ b) c₁ n\nc₁✝¹ : RBColor\nn✝ : Nat\nc₂✝¹ : RBColor\nha : Balanced a✝ c₁✝¹ n✝\nhb : Balanced b c₂✝¹ n✝\nhr : Balanced (node black c y✝ d✝) c₂ (n✝ + 1)\nc₁✝ c₂✝ : RBColor\nhc : Balanced c c₁✝ n✝\nhd : Balanced d✝ c₂✝ n✝\nIH : RedRed (c₂✝¹ = black → c₁✝ ≠ black) (append b c) n✝\nl✝ : RBNode α\nH : ∀ (a : RBNode α) (x : α) (b_1 : RBNode α), append b c = node red a x b_1 → False\nbc : RBNode α\nc✝ : RBColor\nhbc : Balanced bc c✝ n✝\nx✝ : ∀ (a : RBNode α) (x : α) (b : RBNode α), bc = node red a x b → False\n⊢ RedRed (black = black → black ≠ black)\n (match a✝ with\n | node red a x b => node red (node black a x b) x✝¹ (node black bc y✝ d✝)\n | l =>\n match node black bc y✝ d✝ with\n | node black a y b => balance2 l x✝¹ (node red a y b)\n | node red (node black a y b) z c => node red (node black l x✝¹ a) y (balance2 b z (setRed c))\n | r => node red l x✝¹ r)\n (n✝ + 1)",
"state_before": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝³ x✝² a✝ : RBNode α\nx✝¹ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node black a✝ x✝¹ b) c₁ n\nc₁✝¹ : RBColor\nn✝ : Nat\nc₂✝¹ : RBColor\nha : Balanced a✝ c₁✝¹ n✝\nhb : Balanced b c₂✝¹ n✝\nhr : Balanced (node black c y✝ d✝) c₂ (n✝ + 1)\nc₁✝ c₂✝ : RBColor\nhc : Balanced c c₁✝ n✝\nhd : Balanced d✝ c₂✝ n✝\nIH : RedRed (c₂✝¹ = black → c₁✝ ≠ black) (append b c) n✝\nl✝ : RBNode α\nH : ∀ (a : RBNode α) (x : α) (b_1 : RBNode α), append b c = node red a x b_1 → False\nbc : RBNode α\nc✝ : RBColor\nhbc : Balanced bc c✝ n✝\nx✝ : ∀ (a : RBNode α) (x : α) (b : RBNode α), bc = node red a x b → False\n⊢ RedRed (black = black → black ≠ black) (balLeft a✝ x✝¹ (node black bc y✝ d✝)) (n✝ + 1)",
"tactic": "unfold balLeft"
},
{
"state_after": "case h_1\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝⁴ x✝³ : RBNode α\nx✝² : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nc₁✝¹ : RBColor\nn✝ : Nat\nc₂✝¹ : RBColor\nhb : Balanced b c₂✝¹ n✝\nhr : Balanced (node black c y✝ d✝) c₂ (n✝ + 1)\nc₁✝ c₂✝ : RBColor\nhc : Balanced c c₁✝ n✝\nhd : Balanced d✝ c₂✝ n✝\nIH : RedRed (c₂✝¹ = black → c₁✝ ≠ black) (append b c) n✝\nl✝¹ : RBNode α\nH : ∀ (a : RBNode α) (x : α) (b_1 : RBNode α), append b c = node red a x b_1 → False\nbc : RBNode α\nc✝ : RBColor\nhbc : Balanced bc c✝ n✝\nx✝¹ : ∀ (a : RBNode α) (x : α) (b : RBNode α), bc = node red a x b → False\nl✝ a✝ : RBNode α\nx✝ : α\nb✝ : RBNode α\nhl : Balanced (node black (node red a✝ x✝ b✝) x✝² b) c₁ n\nha : Balanced (node red a✝ x✝ b✝) c₁✝¹ n✝\n⊢ RedRed (black = black → black ≠ black) (node red (node black a✝ x✝ b✝) x✝² (node black bc y✝ d✝)) (n✝ + 1)\n\ncase h_2\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝⁴ x✝³ a✝ : RBNode α\nx✝² : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node black a✝ x✝² b) c₁ n\nc₁✝¹ : RBColor\nn✝ : Nat\nc₂✝¹ : RBColor\nha : Balanced a✝ c₁✝¹ n✝\nhb : Balanced b c₂✝¹ n✝\nhr : Balanced (node black c y✝ d✝) c₂ (n✝ + 1)\nc₁✝ c₂✝ : RBColor\nhc : Balanced c c₁✝ n✝\nhd : Balanced d✝ c₂✝ n✝\nIH : RedRed (c₂✝¹ = black → c₁✝ ≠ black) (append b c) n✝\nl✝¹ : RBNode α\nH : ∀ (a : RBNode α) (x : α) (b_1 : RBNode α), append b c = node red a x b_1 → False\nbc : RBNode α\nc✝ : RBColor\nhbc : Balanced bc c✝ n✝\nx✝¹ : ∀ (a : RBNode α) (x : α) (b : RBNode α), bc = node red a x b → False\nl✝ : RBNode α\nx✝ : ∀ (a : RBNode α) (x : α) (b : RBNode α), a✝ = node red a x b → False\n⊢ RedRed (black = black → black ≠ black)\n (match node black bc y✝ d✝ with\n | node black a y b => balance2 a✝ x✝² (node red a y b)\n | node red (node black a y b) z c => node red (node black a✝ x✝² a) y (balance2 b z (setRed c))\n | r => node red a✝ x✝² r)\n (n✝ + 1)",
"state_before": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝³ x✝² a✝ : RBNode α\nx✝¹ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node black a✝ x✝¹ b) c₁ n\nc₁✝¹ : RBColor\nn✝ : Nat\nc₂✝¹ : RBColor\nha : Balanced a✝ c₁✝¹ n✝\nhb : Balanced b c₂✝¹ n✝\nhr : Balanced (node black c y✝ d✝) c₂ (n✝ + 1)\nc₁✝ c₂✝ : RBColor\nhc : Balanced c c₁✝ n✝\nhd : Balanced d✝ c₂✝ n✝\nIH : RedRed (c₂✝¹ = black → c₁✝ ≠ black) (append b c) n✝\nl✝ : RBNode α\nH : ∀ (a : RBNode α) (x : α) (b_1 : RBNode α), append b c = node red a x b_1 → False\nbc : RBNode α\nc✝ : RBColor\nhbc : Balanced bc c✝ n✝\nx✝ : ∀ (a : RBNode α) (x : α) (b : RBNode α), bc = node red a x b → False\n⊢ RedRed (black = black → black ≠ black)\n (match a✝ with\n | node red a x b => node red (node black a x b) x✝¹ (node black bc y✝ d✝)\n | l =>\n match node black bc y✝ d✝ with\n | node black a y b => balance2 l x✝¹ (node red a y b)\n | node red (node black a y b) z c => node red (node black l x✝¹ a) y (balance2 b z (setRed c))\n | r => node red l x✝¹ r)\n (n✝ + 1)",
"tactic": "split"
},
{
"state_after": "case h_1\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝⁴ x✝³ : RBNode α\nx✝² : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nc₁✝¹ : RBColor\nn✝ : Nat\nc₂✝¹ : RBColor\nhb : Balanced b c₂✝¹ n✝\nhr : Balanced (node black c y✝ d✝) c₂ (n✝ + 1)\nc₁✝ c₂✝ : RBColor\nhc : Balanced c c₁✝ n✝\nhd : Balanced d✝ c₂✝ n✝\nIH : RedRed (c₂✝¹ = black → c₁✝ ≠ black) (append b c) n✝\nl✝¹ : RBNode α\nH : ∀ (a : RBNode α) (x : α) (b_1 : RBNode α), append b c = node red a x b_1 → False\nbc : RBNode α\nc✝ : RBColor\nhbc : Balanced bc c✝ n✝\nx✝¹ : ∀ (a : RBNode α) (x : α) (b : RBNode α), bc = node red a x b → False\nl✝ a✝ : RBNode α\nx✝ : α\nb✝ : RBNode α\nhl : Balanced (node black (node red a✝ x✝ b✝) x✝² b) c₁ n\nha : Balanced (node red a✝ x✝ b✝) c₁✝¹ n✝\nha' : Balanced a✝ black n✝\nhb' : Balanced b✝ black n✝\n⊢ RedRed (black = black → black ≠ black) (node red (node black a✝ x✝ b✝) x✝² (node black bc y✝ d✝)) (n✝ + 1)",
"state_before": "case h_1\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝⁴ x✝³ : RBNode α\nx✝² : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nc₁✝¹ : RBColor\nn✝ : Nat\nc₂✝¹ : RBColor\nhb : Balanced b c₂✝¹ n✝\nhr : Balanced (node black c y✝ d✝) c₂ (n✝ + 1)\nc₁✝ c₂✝ : RBColor\nhc : Balanced c c₁✝ n✝\nhd : Balanced d✝ c₂✝ n✝\nIH : RedRed (c₂✝¹ = black → c₁✝ ≠ black) (append b c) n✝\nl✝¹ : RBNode α\nH : ∀ (a : RBNode α) (x : α) (b_1 : RBNode α), append b c = node red a x b_1 → False\nbc : RBNode α\nc✝ : RBColor\nhbc : Balanced bc c✝ n✝\nx✝¹ : ∀ (a : RBNode α) (x : α) (b : RBNode α), bc = node red a x b → False\nl✝ a✝ : RBNode α\nx✝ : α\nb✝ : RBNode α\nhl : Balanced (node black (node red a✝ x✝ b✝) x✝² b) c₁ n\nha : Balanced (node red a✝ x✝ b✝) c₁✝¹ n✝\n⊢ RedRed (black = black → black ≠ black) (node red (node black a✝ x✝ b✝) x✝² (node black bc y✝ d✝)) (n✝ + 1)",
"tactic": "have .red ha' hb' := ha"
},
{
"state_after": "no goals",
"state_before": "case h_1\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝⁴ x✝³ : RBNode α\nx✝² : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nc₁✝¹ : RBColor\nn✝ : Nat\nc₂✝¹ : RBColor\nhb : Balanced b c₂✝¹ n✝\nhr : Balanced (node black c y✝ d✝) c₂ (n✝ + 1)\nc₁✝ c₂✝ : RBColor\nhc : Balanced c c₁✝ n✝\nhd : Balanced d✝ c₂✝ n✝\nIH : RedRed (c₂✝¹ = black → c₁✝ ≠ black) (append b c) n✝\nl✝¹ : RBNode α\nH : ∀ (a : RBNode α) (x : α) (b_1 : RBNode α), append b c = node red a x b_1 → False\nbc : RBNode α\nc✝ : RBColor\nhbc : Balanced bc c✝ n✝\nx✝¹ : ∀ (a : RBNode α) (x : α) (b : RBNode α), bc = node red a x b → False\nl✝ a✝ : RBNode α\nx✝ : α\nb✝ : RBNode α\nhl : Balanced (node black (node red a✝ x✝ b✝) x✝² b) c₁ n\nha : Balanced (node red a✝ x✝ b✝) c₁✝¹ n✝\nha' : Balanced a✝ black n✝\nhb' : Balanced b✝ black n✝\n⊢ RedRed (black = black → black ≠ black) (node red (node black a✝ x✝ b✝) x✝² (node black bc y✝ d✝)) (n✝ + 1)",
"tactic": "exact .balanced (.red (.black ha' hb') (.black hbc hd))"
},
{
"state_after": "no goals",
"state_before": "case h_2\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝⁴ x✝³ a✝ : RBNode α\nx✝² : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node black a✝ x✝² b) c₁ n\nc₁✝¹ : RBColor\nn✝ : Nat\nc₂✝¹ : RBColor\nha : Balanced a✝ c₁✝¹ n✝\nhb : Balanced b c₂✝¹ n✝\nhr : Balanced (node black c y✝ d✝) c₂ (n✝ + 1)\nc₁✝ c₂✝ : RBColor\nhc : Balanced c c₁✝ n✝\nhd : Balanced d✝ c₂✝ n✝\nIH : RedRed (c₂✝¹ = black → c₁✝ ≠ black) (append b c) n✝\nl✝¹ : RBNode α\nH : ∀ (a : RBNode α) (x : α) (b_1 : RBNode α), append b c = node red a x b_1 → False\nbc : RBNode α\nc✝ : RBColor\nhbc : Balanced bc c✝ n✝\nx✝¹ : ∀ (a : RBNode α) (x : α) (b : RBNode α), bc = node red a x b → False\nl✝ : RBNode α\nx✝ : ∀ (a : RBNode α) (x : α) (b : RBNode α), a✝ = node red a x b → False\n⊢ RedRed (black = black → black ≠ black)\n (match node black bc y✝ d✝ with\n | node black a y b => balance2 a✝ x✝² (node red a y b)\n | node red (node black a y b) z c => node red (node black a✝ x✝² a) y (balance2 b z (setRed c))\n | r => node red a✝ x✝² r)\n (n✝ + 1)",
"tactic": "exact have ⟨c, h⟩ := RedRed.balance2 ha (.redred trivial hbc hd); .balanced h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝³ x✝² a✝⁴ : RBNode α\nx✝¹ : α\nb c : RBNode α\ny✝ : α\nd✝ : RBNode α\nhl : Balanced (node black a✝⁴ x✝¹ b) c₁ n\nc₁✝² : RBColor\nn✝ : Nat\nc₂✝² : RBColor\nha : Balanced a✝⁴ c₁✝² n✝\nhb : Balanced b c₂✝² n✝\nhr : Balanced (node black c y✝ d✝) c₂ (n✝ + 1)\nc₁✝¹ c₂✝¹ : RBColor\nhc : Balanced c c₁✝¹ n✝\nhd : Balanced d✝ c₂✝¹ n✝\nIH : RedRed (c₂✝² = black → c₁✝¹ ≠ black) (append b c) n✝\nl✝ : RBNode α\nH✝ : ∀ (a : RBNode α) (x : α) (b_1 : RBNode α), append b c = node red a x b_1 → False\na✝³ : RBNode α\nc₁✝ : RBColor\nb✝ : RBNode α\nc₂✝ : RBColor\nx✝ : α\na✝² : c₂✝² = black → c₁✝¹ ≠ black\na✝¹ : Balanced a✝³ c₁✝ n✝\na✝ : Balanced b✝ c₂✝ n✝\nH : ∀ (a : RBNode α) (x : α) (b : RBNode α), node red a✝³ x✝ b✝ = node red a x b → False\n⊢ RedRed (black = black → black ≠ black) (balLeft a✝⁴ x✝¹ (node black (node red a✝³ x✝ b✝) y✝ d✝)) (n✝ + 1)",
"tactic": "cases H _ _ _ rfl"
},
{
"state_after": "case h_5\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ l✝ : RBNode α\nv✝ : α\nr✝ b✝ : RBNode α\nx✝ : α\nc✝ : RBNode α\nhl : Balanced (node black l✝ v✝ r✝) c₁ n\nhr : Balanced (node red b✝ x✝ c✝) c₂ n\nhc : Balanced b✝ black n\nhd : Balanced c✝ black n\n⊢ RedRed (c₁ = black → red ≠ black) (node red (append (node black l✝ v✝ r✝) b✝) x✝ c✝) n",
"state_before": "case h_5\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ l✝ : RBNode α\nv✝ : α\nr✝ b✝ : RBNode α\nx✝ : α\nc✝ : RBNode α\nhl : Balanced (node black l✝ v✝ r✝) c₁ n\nhr : Balanced (node red b✝ x✝ c✝) c₂ n\n⊢ RedRed (c₁ = black → c₂ ≠ black) (node red (append (node black l✝ v✝ r✝) b✝) x✝ c✝) n",
"tactic": "have .red hc hd := hr"
},
{
"state_after": "case h_5\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ l✝ : RBNode α\nv✝ : α\nr✝ b✝ : RBNode α\nx✝ : α\nc✝ : RBNode α\nhl : Balanced (node black l✝ v✝ r✝) c₁ n\nhr : Balanced (node red b✝ x✝ c✝) c₂ n\nhc : Balanced b✝ black n\nhd : Balanced c✝ black n\nIH : RedRed (c₁ = black → black ≠ black) (append (node black l✝ v✝ r✝) b✝) n\n⊢ RedRed (c₁ = black → red ≠ black) (node red (append (node black l✝ v✝ r✝) b✝) x✝ c✝) n",
"state_before": "case h_5\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ l✝ : RBNode α\nv✝ : α\nr✝ b✝ : RBNode α\nx✝ : α\nc✝ : RBNode α\nhl : Balanced (node black l✝ v✝ r✝) c₁ n\nhr : Balanced (node red b✝ x✝ c✝) c₂ n\nhc : Balanced b✝ black n\nhd : Balanced c✝ black n\n⊢ RedRed (c₁ = black → red ≠ black) (node red (append (node black l✝ v✝ r✝) b✝) x✝ c✝) n",
"tactic": "have IH := hl.append hc"
},
{
"state_after": "case h_5\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ l✝ : RBNode α\nv✝ : α\nr✝ b✝ : RBNode α\nx✝ : α\nc✝ : RBNode α\nhl : Balanced (node black l✝ v✝ r✝) c₁ n\nc₁✝ : RBColor\nn✝ : Nat\nc₂✝ : RBColor\nha : Balanced l✝ c₁✝ n✝\nhb : Balanced r✝ c₂✝ n✝\nhr : Balanced (node red b✝ x✝ c✝) c₂ (n✝ + 1)\nhc : Balanced b✝ black (n✝ + 1)\nhd : Balanced c✝ black (n✝ + 1)\nIH : RedRed (black = black → black ≠ black) (append (node black l✝ v✝ r✝) b✝) (n✝ + 1)\n⊢ RedRed (black = black → red ≠ black) (node red (append (node black l✝ v✝ r✝) b✝) x✝ c✝) (n✝ + 1)",
"state_before": "case h_5\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ l✝ : RBNode α\nv✝ : α\nr✝ b✝ : RBNode α\nx✝ : α\nc✝ : RBNode α\nhl : Balanced (node black l✝ v✝ r✝) c₁ n\nhr : Balanced (node red b✝ x✝ c✝) c₂ n\nhc : Balanced b✝ black n\nhd : Balanced c✝ black n\nIH : RedRed (c₁ = black → black ≠ black) (append (node black l✝ v✝ r✝) b✝) n\n⊢ RedRed (c₁ = black → red ≠ black) (node red (append (node black l✝ v✝ r✝) b✝) x✝ c✝) n",
"tactic": "have .black ha hb := hl"
},
{
"state_after": "no goals",
"state_before": "case h_5\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ l✝ : RBNode α\nv✝ : α\nr✝ b✝ : RBNode α\nx✝ : α\nc✝ : RBNode α\nhl : Balanced (node black l✝ v✝ r✝) c₁ n\nc₁✝ : RBColor\nn✝ : Nat\nc₂✝ : RBColor\nha : Balanced l✝ c₁✝ n✝\nhb : Balanced r✝ c₂✝ n✝\nhr : Balanced (node red b✝ x✝ c✝) c₂ (n✝ + 1)\nhc : Balanced b✝ black (n✝ + 1)\nhd : Balanced c✝ black (n✝ + 1)\nIH✝ : RedRed (black = black → black ≠ black) (append (node black l✝ v✝ r✝) b✝) (n✝ + 1)\nc : RBColor\nIH : Balanced (append (node black l✝ v✝ r✝) b✝) c (n✝ + 1)\n⊢ RedRed (black = black → red ≠ black) (node red (append (node black l✝ v✝ r✝) b✝) x✝ c✝) (n✝ + 1)",
"tactic": "exact .redred (fun.) IH hd"
},
{
"state_after": "case h_6\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb✝ l✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nhl : Balanced (node red a✝ x✝ b✝) c₁ n\nhr : Balanced (node black l✝ v✝ r✝) c₂ n\nha : Balanced a✝ black n\nhb : Balanced b✝ black n\n⊢ RedRed (red = black → c₂ ≠ black) (node red a✝ x✝ (append b✝ (node black l✝ v✝ r✝))) n",
"state_before": "case h_6\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb✝ l✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nhl : Balanced (node red a✝ x✝ b✝) c₁ n\nhr : Balanced (node black l✝ v✝ r✝) c₂ n\n⊢ RedRed (c₁ = black → c₂ ≠ black) (node red a✝ x✝ (append b✝ (node black l✝ v✝ r✝))) n",
"tactic": "have .red ha hb := hl"
},
{
"state_after": "case h_6\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb✝ l✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nhl : Balanced (node red a✝ x✝ b✝) c₁ n\nhr : Balanced (node black l✝ v✝ r✝) c₂ n\nha : Balanced a✝ black n\nhb : Balanced b✝ black n\nIH : RedRed (black = black → c₂ ≠ black) (append b✝ (node black l✝ v✝ r✝)) n\n⊢ RedRed (red = black → c₂ ≠ black) (node red a✝ x✝ (append b✝ (node black l✝ v✝ r✝))) n",
"state_before": "case h_6\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb✝ l✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nhl : Balanced (node red a✝ x✝ b✝) c₁ n\nhr : Balanced (node black l✝ v✝ r✝) c₂ n\nha : Balanced a✝ black n\nhb : Balanced b✝ black n\n⊢ RedRed (red = black → c₂ ≠ black) (node red a✝ x✝ (append b✝ (node black l✝ v✝ r✝))) n",
"tactic": "have IH := hb.append hr"
},
{
"state_after": "case h_6\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb✝ l✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nhr : Balanced (node black l✝ v✝ r✝) c₂ n\nc₁✝ : RBColor\nn✝ : Nat\nc₂✝ : RBColor\nhc : Balanced l✝ c₁✝ n✝\nhd : Balanced r✝ c₂✝ n✝\nhl : Balanced (node red a✝ x✝ b✝) c₁ (n✝ + 1)\nha : Balanced a✝ black (n✝ + 1)\nhb : Balanced b✝ black (n✝ + 1)\nIH : RedRed (black = black → black ≠ black) (append b✝ (node black l✝ v✝ r✝)) (n✝ + 1)\n⊢ RedRed (red = black → black ≠ black) (node red a✝ x✝ (append b✝ (node black l✝ v✝ r✝))) (n✝ + 1)",
"state_before": "case h_6\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb✝ l✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nhl : Balanced (node red a✝ x✝ b✝) c₁ n\nhr : Balanced (node black l✝ v✝ r✝) c₂ n\nha : Balanced a✝ black n\nhb : Balanced b✝ black n\nIH : RedRed (black = black → c₂ ≠ black) (append b✝ (node black l✝ v✝ r✝)) n\n⊢ RedRed (red = black → c₂ ≠ black) (node red a✝ x✝ (append b✝ (node black l✝ v✝ r✝))) n",
"tactic": "have .black hc hd := hr"
},
{
"state_after": "no goals",
"state_before": "case h_6\nα : Type u_1\nc₁ : RBColor\nn : Nat\nc₂ : RBColor\nx✝² x✝¹ a✝ : RBNode α\nx✝ : α\nb✝ l✝ : RBNode α\nv✝ : α\nr✝ : RBNode α\nhr : Balanced (node black l✝ v✝ r✝) c₂ n\nc₁✝ : RBColor\nn✝ : Nat\nc₂✝ : RBColor\nhc : Balanced l✝ c₁✝ n✝\nhd : Balanced r✝ c₂✝ n✝\nhl : Balanced (node red a✝ x✝ b✝) c₁ (n✝ + 1)\nha : Balanced a✝ black (n✝ + 1)\nhb : Balanced b✝ black (n✝ + 1)\nIH✝ : RedRed (black = black → black ≠ black) (append b✝ (node black l✝ v✝ r✝)) (n✝ + 1)\nc : RBColor\nIH : Balanced (append b✝ (node black l✝ v✝ r✝)) c (n✝ + 1)\n⊢ RedRed (red = black → black ≠ black) (node red a✝ x✝ (append b✝ (node black l✝ v✝ r✝))) (n✝ + 1)",
"tactic": "exact .redred (fun.) ha IH"
}
] |
[
383,
36
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
347,
11
] |
Mathlib/GroupTheory/Schreier.lean
|
Subgroup.card_commutator_dvd_index_center_pow
|
[
{
"state_after": "case pos\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : index (center G) = 0\n⊢ Nat.card { x // x ∈ _root_.commutator G } ∣ index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G) + 1)\n\ncase neg\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\n⊢ Nat.card { x // x ∈ _root_.commutator G } ∣ index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G) + 1)",
"state_before": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\n⊢ Nat.card { x // x ∈ _root_.commutator G } ∣ index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G) + 1)",
"tactic": "by_cases hG : (center G).index = 0"
},
{
"state_after": "case neg\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis : FiniteIndex (center G)\n⊢ Nat.card { x // x ∈ _root_.commutator G } ∣ index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G) + 1)",
"state_before": "case neg\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\n⊢ Nat.card { x // x ∈ _root_.commutator G } ∣ index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G) + 1)",
"tactic": "haveI : FiniteIndex (center G) := ⟨hG⟩"
},
{
"state_after": "case neg\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis : FiniteIndex (center G)\n⊢ Nat.card { x // x ∈ subgroupOf (center G) (_root_.commutator G) } *\n index (subgroupOf (center G) (_root_.commutator G)) ∣\n index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G)) * index (center G)",
"state_before": "case neg\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis : FiniteIndex (center G)\n⊢ Nat.card { x // x ∈ _root_.commutator G } ∣ index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G) + 1)",
"tactic": "rw [← ((center G).subgroupOf (_root_.commutator G)).card_mul_index, pow_succ']"
},
{
"state_after": "case neg\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) ∣ index (center G)\n⊢ Nat.card { x // x ∈ subgroupOf (center G) (_root_.commutator G) } *\n index (subgroupOf (center G) (_root_.commutator G)) ∣\n index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G)) * index (center G)",
"state_before": "case neg\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis : FiniteIndex (center G)\n⊢ Nat.card { x // x ∈ subgroupOf (center G) (_root_.commutator G) } *\n index (subgroupOf (center G) (_root_.commutator G)) ∣\n index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G)) * index (center G)",
"tactic": "have h1 := relindex_dvd_index_of_normal (center G) (_root_.commutator G)"
},
{
"state_after": "case neg\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) ∣ index (center G)\n⊢ Nat.card { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ∣\n index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G))",
"state_before": "case neg\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) ∣ index (center G)\n⊢ Nat.card { x // x ∈ subgroupOf (center G) (_root_.commutator G) } *\n index (subgroupOf (center G) (_root_.commutator G)) ∣\n index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G)) * index (center G)",
"tactic": "refine' mul_dvd_mul _ h1"
},
{
"state_after": "case neg\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis✝ : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) ∣ index (center G)\nthis : FiniteIndex (subgroupOf (center G) (_root_.commutator G))\n⊢ Nat.card { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ∣\n index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G))",
"state_before": "case neg\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) ∣ index (center G)\n⊢ Nat.card { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ∣\n index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G))",
"tactic": "haveI : FiniteIndex ((center G).subgroupOf (_root_.commutator G)) :=\n ⟨ne_zero_of_dvd_ne_zero hG h1⟩"
},
{
"state_after": "case neg\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis✝ : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) ∣ index (center G)\nthis : FiniteIndex (subgroupOf (center G) (_root_.commutator G))\nh2 :\n Group.rank { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ≤\n index (subgroupOf (center G) (_root_.commutator G)) * Group.rank { x // x ∈ _root_.commutator G }\n⊢ Nat.card { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ∣\n index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G))",
"state_before": "case neg\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis✝ : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) ∣ index (center G)\nthis : FiniteIndex (subgroupOf (center G) (_root_.commutator G))\n⊢ Nat.card { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ∣\n index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G))",
"tactic": "have h2 := rank_le_index_mul_rank ((center G).subgroupOf (_root_.commutator G))"
},
{
"state_after": "case neg\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis✝ : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) ∣ index (center G)\nthis : FiniteIndex (subgroupOf (center G) (_root_.commutator G))\nh2 :\n Group.rank { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ≤\n index (subgroupOf (center G) (_root_.commutator G)) * Group.rank { x // x ∈ _root_.commutator G }\nh3 :\n relindex (center G) (_root_.commutator G) * Group.rank { x // x ∈ _root_.commutator G } ≤\n index (center G) * Nat.card ↑(commutatorSet G)\n⊢ Nat.card { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ∣\n index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G))",
"state_before": "case neg\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis✝ : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) ∣ index (center G)\nthis : FiniteIndex (subgroupOf (center G) (_root_.commutator G))\nh2 :\n Group.rank { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ≤\n index (subgroupOf (center G) (_root_.commutator G)) * Group.rank { x // x ∈ _root_.commutator G }\n⊢ Nat.card { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ∣\n index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G))",
"tactic": "have h3 := Nat.mul_le_mul (Nat.le_of_dvd (Nat.pos_of_ne_zero hG) h1) (rank_commutator_le_card G)"
},
{
"state_after": "case neg\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis✝ : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) ∣ index (center G)\nthis : FiniteIndex (subgroupOf (center G) (_root_.commutator G))\nh2 :\n Group.rank { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ≤\n index (subgroupOf (center G) (_root_.commutator G)) * Group.rank { x // x ∈ _root_.commutator G }\nh3 :\n relindex (center G) (_root_.commutator G) * Group.rank { x // x ∈ _root_.commutator G } ≤\n index (center G) * Nat.card ↑(commutatorSet G)\n⊢ Nat.card { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ∣\n index (center G) ^ Group.rank { x // x ∈ subgroupOf (center G) (_root_.commutator G) }",
"state_before": "case neg\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis✝ : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) ∣ index (center G)\nthis : FiniteIndex (subgroupOf (center G) (_root_.commutator G))\nh2 :\n Group.rank { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ≤\n index (subgroupOf (center G) (_root_.commutator G)) * Group.rank { x // x ∈ _root_.commutator G }\nh3 :\n relindex (center G) (_root_.commutator G) * Group.rank { x // x ∈ _root_.commutator G } ≤\n index (center G) * Nat.card ↑(commutatorSet G)\n⊢ Nat.card { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ∣\n index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G))",
"tactic": "refine' dvd_trans _ (pow_dvd_pow (center G).index (h2.trans h3))"
},
{
"state_after": "case neg.hG\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis✝ : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) ∣ index (center G)\nthis : FiniteIndex (subgroupOf (center G) (_root_.commutator G))\nh2 :\n Group.rank { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ≤\n index (subgroupOf (center G) (_root_.commutator G)) * Group.rank { x // x ∈ _root_.commutator G }\nh3 :\n relindex (center G) (_root_.commutator G) * Group.rank { x // x ∈ _root_.commutator G } ≤\n index (center G) * Nat.card ↑(commutatorSet G)\n⊢ ∀ (g : { x // x ∈ subgroupOf (center G) (_root_.commutator G) }), g ^ index (center G) = 1",
"state_before": "case neg\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis✝ : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) ∣ index (center G)\nthis : FiniteIndex (subgroupOf (center G) (_root_.commutator G))\nh2 :\n Group.rank { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ≤\n index (subgroupOf (center G) (_root_.commutator G)) * Group.rank { x // x ∈ _root_.commutator G }\nh3 :\n relindex (center G) (_root_.commutator G) * Group.rank { x // x ∈ _root_.commutator G } ≤\n index (center G) * Nat.card ↑(commutatorSet G)\n⊢ Nat.card { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ∣\n index (center G) ^ Group.rank { x // x ∈ subgroupOf (center G) (_root_.commutator G) }",
"tactic": "apply card_dvd_exponent_pow_rank'"
},
{
"state_after": "case neg.hG\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis✝ : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) ∣ index (center G)\nthis : FiniteIndex (subgroupOf (center G) (_root_.commutator G))\nh2 :\n Group.rank { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ≤\n index (subgroupOf (center G) (_root_.commutator G)) * Group.rank { x // x ∈ _root_.commutator G }\nh3 :\n relindex (center G) (_root_.commutator G) * Group.rank { x // x ∈ _root_.commutator G } ≤\n index (center G) * Nat.card ↑(commutatorSet G)\ng : { x // x ∈ subgroupOf (center G) (_root_.commutator G) }\n⊢ g ^ index (center G) = 1",
"state_before": "case neg.hG\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis✝ : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) ∣ index (center G)\nthis : FiniteIndex (subgroupOf (center G) (_root_.commutator G))\nh2 :\n Group.rank { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ≤\n index (subgroupOf (center G) (_root_.commutator G)) * Group.rank { x // x ∈ _root_.commutator G }\nh3 :\n relindex (center G) (_root_.commutator G) * Group.rank { x // x ∈ _root_.commutator G } ≤\n index (center G) * Nat.card ↑(commutatorSet G)\n⊢ ∀ (g : { x // x ∈ subgroupOf (center G) (_root_.commutator G) }), g ^ index (center G) = 1",
"tactic": "intro g"
},
{
"state_after": "case neg.hG\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis✝¹ : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) ∣ index (center G)\nthis✝ : FiniteIndex (subgroupOf (center G) (_root_.commutator G))\nh2 :\n Group.rank { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ≤\n index (subgroupOf (center G) (_root_.commutator G)) * Group.rank { x // x ∈ _root_.commutator G }\nh3 :\n relindex (center G) (_root_.commutator G) * Group.rank { x // x ∈ _root_.commutator G } ≤\n index (center G) * Nat.card ↑(commutatorSet G)\ng : { x // x ∈ subgroupOf (center G) (_root_.commutator G) }\nthis : ↑↑g ∈ MonoidHom.ker (MonoidHom.transferCenterPow G)\n⊢ g ^ index (center G) = 1",
"state_before": "case neg.hG\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis✝ : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) ∣ index (center G)\nthis : FiniteIndex (subgroupOf (center G) (_root_.commutator G))\nh2 :\n Group.rank { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ≤\n index (subgroupOf (center G) (_root_.commutator G)) * Group.rank { x // x ∈ _root_.commutator G }\nh3 :\n relindex (center G) (_root_.commutator G) * Group.rank { x // x ∈ _root_.commutator G } ≤\n index (center G) * Nat.card ↑(commutatorSet G)\ng : { x // x ∈ subgroupOf (center G) (_root_.commutator G) }\n⊢ g ^ index (center G) = 1",
"tactic": "have := Abelianization.commutator_subset_ker (MonoidHom.transferCenterPow G) g.1.2"
},
{
"state_after": "no goals",
"state_before": "case neg.hG\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : ¬index (center G) = 0\nthis✝¹ : FiniteIndex (center G)\nh1 : relindex (center G) (_root_.commutator G) ∣ index (center G)\nthis✝ : FiniteIndex (subgroupOf (center G) (_root_.commutator G))\nh2 :\n Group.rank { x // x ∈ subgroupOf (center G) (_root_.commutator G) } ≤\n index (subgroupOf (center G) (_root_.commutator G)) * Group.rank { x // x ∈ _root_.commutator G }\nh3 :\n relindex (center G) (_root_.commutator G) * Group.rank { x // x ∈ _root_.commutator G } ≤\n index (center G) * Nat.card ↑(commutatorSet G)\ng : { x // x ∈ subgroupOf (center G) (_root_.commutator G) }\nthis : ↑↑g ∈ MonoidHom.ker (MonoidHom.transferCenterPow G)\n⊢ g ^ index (center G) = 1",
"tactic": "simpa only [MonoidHom.mem_ker, Subtype.ext_iff] using this"
},
{
"state_after": "no goals",
"state_before": "case pos\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\nR S : Set G\ninst✝ : Finite ↑(commutatorSet G)\nhG : index (center G) = 0\n⊢ Nat.card { x // x ∈ _root_.commutator G } ∣ index (center G) ^ (index (center G) * Nat.card ↑(commutatorSet G) + 1)",
"tactic": "simp_rw [hG, MulZeroClass.zero_mul, zero_add, pow_one, dvd_zero]"
}
] |
[
181,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/RingTheory/GradedAlgebra/Basic.lean
|
DirectSum.coe_decompose_mul_of_left_mem_of_le
|
[
{
"state_after": "case intro\nι : Type u_3\nR : Type ?u.357840\nA : Type u_1\nσ : Type u_2\ninst✝⁸ : Semiring A\ninst✝⁷ : DecidableEq ι\ninst✝⁶ : CanonicallyOrderedAddMonoid ι\ninst✝⁵ : SetLike σ A\ninst✝⁴ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝³ : GradedRing 𝒜\nb : A\nn i : ι\ninst✝² : Sub ι\ninst✝¹ : OrderedSub ι\ninst✝ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : i ≤ n\na : { x // x ∈ 𝒜 i }\n⊢ ↑(↑(↑(decompose 𝒜) (↑a * b)) n) = ↑a * ↑(↑(↑(decompose 𝒜) b) (n - i))",
"state_before": "ι : Type u_3\nR : Type ?u.357840\nA : Type u_1\nσ : Type u_2\ninst✝⁸ : Semiring A\ninst✝⁷ : DecidableEq ι\ninst✝⁶ : CanonicallyOrderedAddMonoid ι\ninst✝⁵ : SetLike σ A\ninst✝⁴ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝³ : GradedRing 𝒜\na b : A\nn i : ι\ninst✝² : Sub ι\ninst✝¹ : OrderedSub ι\ninst✝ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\na_mem : a ∈ 𝒜 i\nh : i ≤ n\n⊢ ↑(↑(↑(decompose 𝒜) (a * b)) n) = a * ↑(↑(↑(decompose 𝒜) b) (n - i))",
"tactic": "lift a to 𝒜 i using a_mem"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Type u_3\nR : Type ?u.357840\nA : Type u_1\nσ : Type u_2\ninst✝⁸ : Semiring A\ninst✝⁷ : DecidableEq ι\ninst✝⁶ : CanonicallyOrderedAddMonoid ι\ninst✝⁵ : SetLike σ A\ninst✝⁴ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝³ : GradedRing 𝒜\nb : A\nn i : ι\ninst✝² : Sub ι\ninst✝¹ : OrderedSub ι\ninst✝ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : i ≤ n\na : { x // x ∈ 𝒜 i }\n⊢ ↑(↑(↑(decompose 𝒜) (↑a * b)) n) = ↑a * ↑(↑(↑(decompose 𝒜) b) (n - i))",
"tactic": "rwa [decompose_mul, decompose_coe, coe_of_mul_apply_of_le]"
}
] |
[
318,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Std/Data/Nat/Gcd.lean
|
Nat.coprime_one_left
|
[] |
[
357,
60
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
357,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
LinearMap.prodMap_one
|
[] |
[
341,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
340,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.mk_of_not_mem
|
[] |
[
592,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
591,
1
] |
Mathlib/MeasureTheory/Constructions/Pi.lean
|
MeasureTheory.measurePreserving_piEquivPiSubtypeProd
|
[
{
"state_after": "ι✝ : Type ?u.5372445\nι' : Type ?u.5372448\nα✝ : ι✝ → Type ?u.5372453\ninst✝³ : Fintype ι✝\nm✝ : (i : ι✝) → OuterMeasure (α✝ i)\nι : Type u\nα : ι → Type v\ninst✝² : Fintype ι\nm : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝¹ : ∀ (i : ι), SigmaFinite (μ i)\np : ι → Prop\ninst✝ : DecidablePred p\ne : ((i : Subtype p) → α ↑i) × ((i : { i // ¬p i }) → α ↑i) ≃ᵐ ((i : ι) → α i) :=\n MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd α p)\n⊢ MeasurePreserving ↑(MeasurableEquiv.piEquivPiSubtypeProd α p)",
"state_before": "ι✝ : Type ?u.5372445\nι' : Type ?u.5372448\nα✝ : ι✝ → Type ?u.5372453\ninst✝³ : Fintype ι✝\nm✝ : (i : ι✝) → OuterMeasure (α✝ i)\nι : Type u\nα : ι → Type v\ninst✝² : Fintype ι\nm : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝¹ : ∀ (i : ι), SigmaFinite (μ i)\np : ι → Prop\ninst✝ : DecidablePred p\n⊢ MeasurePreserving ↑(MeasurableEquiv.piEquivPiSubtypeProd α p)",
"tactic": "set e := (MeasurableEquiv.piEquivPiSubtypeProd α p).symm"
},
{
"state_after": "ι✝ : Type ?u.5372445\nι' : Type ?u.5372448\nα✝ : ι✝ → Type ?u.5372453\ninst✝³ : Fintype ι✝\nm✝ : (i : ι✝) → OuterMeasure (α✝ i)\nι : Type u\nα : ι → Type v\ninst✝² : Fintype ι\nm : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝¹ : ∀ (i : ι), SigmaFinite (μ i)\np : ι → Prop\ninst✝ : DecidablePred p\ne : ((i : Subtype p) → α ↑i) × ((i : { i // ¬p i }) → α ↑i) ≃ᵐ ((i : ι) → α i) :=\n MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd α p)\n⊢ MeasurePreserving ↑e",
"state_before": "ι✝ : Type ?u.5372445\nι' : Type ?u.5372448\nα✝ : ι✝ → Type ?u.5372453\ninst✝³ : Fintype ι✝\nm✝ : (i : ι✝) → OuterMeasure (α✝ i)\nι : Type u\nα : ι → Type v\ninst✝² : Fintype ι\nm : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝¹ : ∀ (i : ι), SigmaFinite (μ i)\np : ι → Prop\ninst✝ : DecidablePred p\ne : ((i : Subtype p) → α ↑i) × ((i : { i // ¬p i }) → α ↑i) ≃ᵐ ((i : ι) → α i) :=\n MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd α p)\n⊢ MeasurePreserving ↑(MeasurableEquiv.piEquivPiSubtypeProd α p)",
"tactic": "refine' MeasurePreserving.symm e _"
},
{
"state_after": "ι✝ : Type ?u.5372445\nι' : Type ?u.5372448\nα✝ : ι✝ → Type ?u.5372453\ninst✝³ : Fintype ι✝\nm✝ : (i : ι✝) → OuterMeasure (α✝ i)\nι : Type u\nα : ι → Type v\ninst✝² : Fintype ι\nm : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝¹ : ∀ (i : ι), SigmaFinite (μ i)\np : ι → Prop\ninst✝ : DecidablePred p\ne : ((i : Subtype p) → α ↑i) × ((i : { i // ¬p i }) → α ↑i) ≃ᵐ ((i : ι) → α i) :=\n MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd α p)\ns : (i : ι) → Set (α i)\nx✝ : ∀ (i : ι), MeasurableSet (s i)\n⊢ ↑↑(Measure.map (↑e) (Measure.prod (Measure.pi fun i => μ ↑i) (Measure.pi fun i => μ ↑i))) (Set.pi univ s) =\n ∏ i : ι, ↑↑(μ i) (s i)",
"state_before": "ι✝ : Type ?u.5372445\nι' : Type ?u.5372448\nα✝ : ι✝ → Type ?u.5372453\ninst✝³ : Fintype ι✝\nm✝ : (i : ι✝) → OuterMeasure (α✝ i)\nι : Type u\nα : ι → Type v\ninst✝² : Fintype ι\nm : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝¹ : ∀ (i : ι), SigmaFinite (μ i)\np : ι → Prop\ninst✝ : DecidablePred p\ne : ((i : Subtype p) → α ↑i) × ((i : { i // ¬p i }) → α ↑i) ≃ᵐ ((i : ι) → α i) :=\n MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd α p)\n⊢ MeasurePreserving ↑e",
"tactic": "refine' ⟨e.measurable, (pi_eq fun s _ => _).symm⟩"
},
{
"state_after": "ι✝ : Type ?u.5372445\nι' : Type ?u.5372448\nα✝ : ι✝ → Type ?u.5372453\ninst✝³ : Fintype ι✝\nm✝ : (i : ι✝) → OuterMeasure (α✝ i)\nι : Type u\nα : ι → Type v\ninst✝² : Fintype ι\nm : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝¹ : ∀ (i : ι), SigmaFinite (μ i)\np : ι → Prop\ninst✝ : DecidablePred p\ne : ((i : Subtype p) → α ↑i) × ((i : { i // ¬p i }) → α ↑i) ≃ᵐ ((i : ι) → α i) :=\n MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd α p)\ns : (i : ι) → Set (α i)\nx✝ : ∀ (i : ι), MeasurableSet (s i)\nthis : ↑e ⁻¹' Set.pi univ s = (Set.pi univ fun i => s ↑i) ×ˢ Set.pi univ fun i => s ↑i\n⊢ ↑↑(Measure.map (↑e) (Measure.prod (Measure.pi fun i => μ ↑i) (Measure.pi fun i => μ ↑i))) (Set.pi univ s) =\n ∏ i : ι, ↑↑(μ i) (s i)",
"state_before": "ι✝ : Type ?u.5372445\nι' : Type ?u.5372448\nα✝ : ι✝ → Type ?u.5372453\ninst✝³ : Fintype ι✝\nm✝ : (i : ι✝) → OuterMeasure (α✝ i)\nι : Type u\nα : ι → Type v\ninst✝² : Fintype ι\nm : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝¹ : ∀ (i : ι), SigmaFinite (μ i)\np : ι → Prop\ninst✝ : DecidablePred p\ne : ((i : Subtype p) → α ↑i) × ((i : { i // ¬p i }) → α ↑i) ≃ᵐ ((i : ι) → α i) :=\n MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd α p)\ns : (i : ι) → Set (α i)\nx✝ : ∀ (i : ι), MeasurableSet (s i)\n⊢ ↑↑(Measure.map (↑e) (Measure.prod (Measure.pi fun i => μ ↑i) (Measure.pi fun i => μ ↑i))) (Set.pi univ s) =\n ∏ i : ι, ↑↑(μ i) (s i)",
"tactic": "have : e ⁻¹' pi univ s =\n (pi univ fun i : { i // p i } => s i) ×ˢ pi univ fun i : { i // ¬p i } => s i :=\n Equiv.preimage_piEquivPiSubtypeProd_symm_pi p s"
},
{
"state_after": "ι✝ : Type ?u.5372445\nι' : Type ?u.5372448\nα✝ : ι✝ → Type ?u.5372453\ninst✝³ : Fintype ι✝\nm✝ : (i : ι✝) → OuterMeasure (α✝ i)\nι : Type u\nα : ι → Type v\ninst✝² : Fintype ι\nm : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝¹ : ∀ (i : ι), SigmaFinite (μ i)\np : ι → Prop\ninst✝ : DecidablePred p\ne : ((i : Subtype p) → α ↑i) × ((i : { i // ¬p i }) → α ↑i) ≃ᵐ ((i : ι) → α i) :=\n MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd α p)\ns : (i : ι) → Set (α i)\nx✝ : ∀ (i : ι), MeasurableSet (s i)\nthis : ↑e ⁻¹' Set.pi univ s = (Set.pi univ fun i => s ↑i) ×ˢ Set.pi univ fun i => s ↑i\n⊢ (∏ i : Subtype p, ↑↑(μ ↑i) (s ↑i)) * ∏ i : { i // ¬p i }, ↑↑(μ ↑i) (s ↑i) = ∏ i : ι, ↑↑(μ i) (s i)",
"state_before": "ι✝ : Type ?u.5372445\nι' : Type ?u.5372448\nα✝ : ι✝ → Type ?u.5372453\ninst✝³ : Fintype ι✝\nm✝ : (i : ι✝) → OuterMeasure (α✝ i)\nι : Type u\nα : ι → Type v\ninst✝² : Fintype ι\nm : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝¹ : ∀ (i : ι), SigmaFinite (μ i)\np : ι → Prop\ninst✝ : DecidablePred p\ne : ((i : Subtype p) → α ↑i) × ((i : { i // ¬p i }) → α ↑i) ≃ᵐ ((i : ι) → α i) :=\n MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd α p)\ns : (i : ι) → Set (α i)\nx✝ : ∀ (i : ι), MeasurableSet (s i)\nthis : ↑e ⁻¹' Set.pi univ s = (Set.pi univ fun i => s ↑i) ×ˢ Set.pi univ fun i => s ↑i\n⊢ ↑↑(Measure.map (↑e) (Measure.prod (Measure.pi fun i => μ ↑i) (Measure.pi fun i => μ ↑i))) (Set.pi univ s) =\n ∏ i : ι, ↑↑(μ i) (s i)",
"tactic": "rw [e.map_apply, this, prod_prod, pi_pi, pi_pi]"
},
{
"state_after": "no goals",
"state_before": "ι✝ : Type ?u.5372445\nι' : Type ?u.5372448\nα✝ : ι✝ → Type ?u.5372453\ninst✝³ : Fintype ι✝\nm✝ : (i : ι✝) → OuterMeasure (α✝ i)\nι : Type u\nα : ι → Type v\ninst✝² : Fintype ι\nm : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝¹ : ∀ (i : ι), SigmaFinite (μ i)\np : ι → Prop\ninst✝ : DecidablePred p\ne : ((i : Subtype p) → α ↑i) × ((i : { i // ¬p i }) → α ↑i) ≃ᵐ ((i : ι) → α i) :=\n MeasurableEquiv.symm (MeasurableEquiv.piEquivPiSubtypeProd α p)\ns : (i : ι) → Set (α i)\nx✝ : ∀ (i : ι), MeasurableSet (s i)\nthis : ↑e ⁻¹' Set.pi univ s = (Set.pi univ fun i => s ↑i) ×ˢ Set.pi univ fun i => s ↑i\n⊢ (∏ i : Subtype p, ↑↑(μ ↑i) (s ↑i)) * ∏ i : { i // ¬p i }, ↑↑(μ ↑i) (s ↑i) = ∏ i : ι, ↑↑(μ i) (s i)",
"tactic": "exact Fintype.prod_subtype_mul_prod_subtype p fun i => μ i (s i)"
}
] |
[
747,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
735,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.sup_vanishingIdeal_le
|
[
{
"state_after": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nt t' : Set (PrimeSpectrum R)\nr : R\n⊢ r ∈ vanishingIdeal t ⊔ vanishingIdeal t' → r ∈ vanishingIdeal (t ∩ t')",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nt t' : Set (PrimeSpectrum R)\n⊢ vanishingIdeal t ⊔ vanishingIdeal t' ≤ vanishingIdeal (t ∩ t')",
"tactic": "intro r"
},
{
"state_after": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nt t' : Set (PrimeSpectrum R)\nr : R\n⊢ (∃ y, y ∈ vanishingIdeal t ∧ ∃ z, z ∈ vanishingIdeal t' ∧ y + z = r) →\n ∀ (x : PrimeSpectrum R), x ∈ t ∩ t' → r ∈ x.asIdeal",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nt t' : Set (PrimeSpectrum R)\nr : R\n⊢ r ∈ vanishingIdeal t ⊔ vanishingIdeal t' → r ∈ vanishingIdeal (t ∩ t')",
"tactic": "rw [Submodule.mem_sup, mem_vanishingIdeal]"
},
{
"state_after": "case intro.intro.intro.intro.intro\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nt t' : Set (PrimeSpectrum R)\nf : R\nhf : f ∈ vanishingIdeal t\ng : R\nhg : g ∈ vanishingIdeal t'\nx : PrimeSpectrum R\nhxt : x ∈ t\nhxt' : x ∈ t'\n⊢ f + g ∈ x.asIdeal",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nt t' : Set (PrimeSpectrum R)\nr : R\n⊢ (∃ y, y ∈ vanishingIdeal t ∧ ∃ z, z ∈ vanishingIdeal t' ∧ y + z = r) →\n ∀ (x : PrimeSpectrum R), x ∈ t ∩ t' → r ∈ x.asIdeal",
"tactic": "rintro ⟨f, hf, g, hg, rfl⟩ x ⟨hxt, hxt'⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nt t' : Set (PrimeSpectrum R)\nf : R\nhf : ∀ (x : PrimeSpectrum R), x ∈ t → f ∈ x.asIdeal\ng : R\nhg : ∀ (x : PrimeSpectrum R), x ∈ t' → g ∈ x.asIdeal\nx : PrimeSpectrum R\nhxt : x ∈ t\nhxt' : x ∈ t'\n⊢ f + g ∈ x.asIdeal",
"state_before": "case intro.intro.intro.intro.intro\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nt t' : Set (PrimeSpectrum R)\nf : R\nhf : f ∈ vanishingIdeal t\ng : R\nhg : g ∈ vanishingIdeal t'\nx : PrimeSpectrum R\nhxt : x ∈ t\nhxt' : x ∈ t'\n⊢ f + g ∈ x.asIdeal",
"tactic": "rw [mem_vanishingIdeal] at hf hg"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\nR : Type u\nS : Type v\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nt t' : Set (PrimeSpectrum R)\nf : R\nhf : ∀ (x : PrimeSpectrum R), x ∈ t → f ∈ x.asIdeal\ng : R\nhg : ∀ (x : PrimeSpectrum R), x ∈ t' → g ∈ x.asIdeal\nx : PrimeSpectrum R\nhxt : x ∈ t\nhxt' : x ∈ t'\n⊢ f + g ∈ x.asIdeal",
"tactic": "apply Submodule.add_mem <;> solve_by_elim"
}
] |
[
396,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
390,
1
] |
Mathlib/Logic/Basic.lean
|
forall_swap
|
[] |
[
653,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
653,
1
] |
Mathlib/Data/Option/Defs.lean
|
Option.iget_some
|
[] |
[
105,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/Order/GameAdd.lean
|
Prod.gameAdd_mk_iff
|
[] |
[
73,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/MeasureTheory/Measure/NullMeasurable.lean
|
MeasureTheory.NullMeasurableSet.compl_iff
|
[] |
[
136,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
AffineIsometryEquiv.coe_inv
|
[] |
[
615,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
614,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.ofReal_prod
|
[] |
[
1338,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1337,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
IsCompact.nonempty_iInter_of_sequence_nonempty_compact_closed
|
[] |
[
307,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
300,
1
] |
Mathlib/Data/Sum/Basic.lean
|
Sum.update_elim_inr
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_1\nδ : Type ?u.16302\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq (α ⊕ β)\nf : α → γ\ng : β → γ\ni : β\nx : γ\n⊢ x = Sum.elim f (update g i x) (inr i)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nα' : Type w\nβ : Type v\nβ' : Type x\nγ : Type u_1\nδ : Type ?u.16302\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq (α ⊕ β)\nf : α → γ\ng : β → γ\ni : β\nx : γ\n⊢ ∀ (x_1 : α ⊕ β), x_1 ≠ inr i → Sum.elim f g x_1 = Sum.elim f (update g i x) x_1",
"tactic": "simp (config := { contextual := true })"
}
] |
[
285,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.HasBasis.exists_iff
|
[] |
[
381,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
376,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Finsupp.mapRange.linearEquiv_apply
|
[] |
[
858,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
857,
1
] |
Mathlib/Order/SymmDiff.lean
|
symmDiff_symmDiff_symmDiff_comm
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.73226\nα : Type u_1\nβ : Type ?u.73232\nπ : ι → Type ?u.73237\ninst✝ : GeneralizedBooleanAlgebra α\na b c d : α\n⊢ a ∆ b ∆ (c ∆ d) = a ∆ c ∆ (b ∆ d)",
"tactic": "simp_rw [symmDiff_assoc, symmDiff_left_comm]"
}
] |
[
493,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
492,
1
] |
Mathlib/Data/Polynomial/Coeff.lean
|
Polynomial.coeff_mul_X_pow
|
[
{
"state_after": "case h₀\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d : ℕ\n⊢ ∀ (b : ℕ × ℕ), b ∈ Nat.antidiagonal (d + n) → b ≠ (d, n) → coeff p b.fst * coeff (X ^ n) b.snd = 0\n\ncase h₁\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d : ℕ\n⊢ ¬(d, n) ∈ Nat.antidiagonal (d + n) → coeff p (d, n).fst * coeff (X ^ n) (d, n).snd = 0",
"state_before": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d : ℕ\n⊢ coeff (p * X ^ n) (d + n) = coeff p d",
"tactic": "rw [coeff_mul, sum_eq_single (d, n), coeff_X_pow, if_pos rfl, mul_one]"
},
{
"state_after": "case h₀.mk\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d i j : ℕ\nh1 : (i, j) ∈ Nat.antidiagonal (d + n)\nh2 : (i, j) ≠ (d, n)\n⊢ coeff p (i, j).fst * coeff (X ^ n) (i, j).snd = 0",
"state_before": "case h₀\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d : ℕ\n⊢ ∀ (b : ℕ × ℕ), b ∈ Nat.antidiagonal (d + n) → b ≠ (d, n) → coeff p b.fst * coeff (X ^ n) b.snd = 0",
"tactic": "rintro ⟨i, j⟩ h1 h2"
},
{
"state_after": "case h₀.mk.hnc\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d i j : ℕ\nh1 : (i, j) ∈ Nat.antidiagonal (d + n)\nh2 : (i, j) ≠ (d, n)\n⊢ ¬(i, j).snd = n",
"state_before": "case h₀.mk\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d i j : ℕ\nh1 : (i, j) ∈ Nat.antidiagonal (d + n)\nh2 : (i, j) ≠ (d, n)\n⊢ coeff p (i, j).fst * coeff (X ^ n) (i, j).snd = 0",
"tactic": "rw [coeff_X_pow, if_neg, mul_zero]"
},
{
"state_after": "case h₀.mk.hnc\nR : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nd i j : ℕ\nh1 : (i, j) ∈ Nat.antidiagonal (d + (i, j).snd)\nh2 : (i, j) ≠ (d, (i, j).snd)\n⊢ False",
"state_before": "case h₀.mk.hnc\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d i j : ℕ\nh1 : (i, j) ∈ Nat.antidiagonal (d + n)\nh2 : (i, j) ≠ (d, n)\n⊢ ¬(i, j).snd = n",
"tactic": "rintro rfl"
},
{
"state_after": "case h₀.mk.hnc\nR : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nd i j : ℕ\nh1 : (i, j) ∈ Nat.antidiagonal (d + (i, j).snd)\nh2 : (i, j) ≠ (d, (i, j).snd)\n⊢ (i, j) = (d, (i, j).snd)",
"state_before": "case h₀.mk.hnc\nR : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nd i j : ℕ\nh1 : (i, j) ∈ Nat.antidiagonal (d + (i, j).snd)\nh2 : (i, j) ≠ (d, (i, j).snd)\n⊢ False",
"tactic": "apply h2"
},
{
"state_after": "case h₀.mk.hnc\nR : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nd i j : ℕ\nh1 : (i, j).fst = d\nh2 : (i, j) ≠ (d, (i, j).snd)\n⊢ (i, j) = (d, (i, j).snd)",
"state_before": "case h₀.mk.hnc\nR : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nd i j : ℕ\nh1 : (i, j) ∈ Nat.antidiagonal (d + (i, j).snd)\nh2 : (i, j) ≠ (d, (i, j).snd)\n⊢ (i, j) = (d, (i, j).snd)",
"tactic": "rw [Nat.mem_antidiagonal, add_right_cancel_iff] at h1"
},
{
"state_after": "case h₀.mk.hnc\nR : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\ni j : ℕ\nh2 : (i, j) ≠ ((i, j).fst, (i, j).snd)\n⊢ (i, j) = ((i, j).fst, (i, j).snd)",
"state_before": "case h₀.mk.hnc\nR : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nd i j : ℕ\nh1 : (i, j).fst = d\nh2 : (i, j) ≠ (d, (i, j).snd)\n⊢ (i, j) = (d, (i, j).snd)",
"tactic": "subst h1"
},
{
"state_after": "no goals",
"state_before": "case h₀.mk.hnc\nR : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\ni j : ℕ\nh2 : (i, j) ≠ ((i, j).fst, (i, j).snd)\n⊢ (i, j) = ((i, j).fst, (i, j).snd)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case h₁\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nn d : ℕ\n⊢ ¬(d, n) ∈ Nat.antidiagonal (d + n) → coeff p (d, n).fst * coeff (X ^ n) (d, n).snd = 0",
"tactic": "exact fun h1 => (h1 (Nat.mem_antidiagonal.2 rfl)).elim"
}
] |
[
241,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/Topology/MetricSpace/Contracting.lean
|
ContractingWith.one_sub_K_ne_zero
|
[] |
[
59,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.edges_map
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf : G →g G'\nf' : G' →g G''\nu v u' v' : V\np : Walk G u v\n⊢ edges (Walk.map f p) = List.map (Sym2.map ↑f) (edges p)",
"tactic": "induction p with\n| nil => rfl\n| cons _ _ ih =>\n simp only [Walk.map_cons, edges_cons, List.map_cons, Sym2.map_pair_eq, List.cons.injEq,\n true_and, ih]"
},
{
"state_after": "no goals",
"state_before": "case nil\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf : G →g G'\nf' : G' →g G''\nu v u' v' : V\np : Walk G u v\nu✝ : V\n⊢ edges (Walk.map f nil) = List.map (Sym2.map ↑f) (edges nil)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf : G →g G'\nf' : G' →g G''\nu v u' v' : V\np : Walk G u v\nu✝ v✝ w✝ : V\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : edges (Walk.map f p✝) = List.map (Sym2.map ↑f) (edges p✝)\n⊢ edges (Walk.map f (cons h✝ p✝)) = List.map (Sym2.map ↑f) (edges (cons h✝ p✝))",
"tactic": "simp only [Walk.map_cons, edges_cons, List.map_cons, Sym2.map_pair_eq, List.cons.injEq,\n true_and, ih]"
}
] |
[
1524,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1519,
1
] |
Mathlib/Algebra/Order/Sub/Defs.lean
|
tsub_le_tsub_right
|
[] |
[
109,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/LinearAlgebra/Matrix/DotProduct.lean
|
Matrix.dotProduct_star_self_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type v\nn : Type w\ninst✝⁴ : Fintype n\ninst✝³ : PartialOrder R\ninst✝² : NonUnitalRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : NoZeroDivisors R\nv : n → R\n⊢ (∀ (i : n), i ∈ Finset.univ → star (v i) * v i = 0) ↔ v = 0",
"tactic": "simp [Function.funext_iff, mul_eq_zero]"
}
] |
[
93,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Data/Int/Dvd/Pow.lean
|
Int.dvd_of_pow_dvd
|
[] |
[
38,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
37,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
DifferentiableWithinAt.csin
|
[] |
[
389,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
387,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.fin_two_eq_of_eq_zero_iff
|
[
{
"state_after": "n m : ℕ\n⊢ ∀ {a b : Fin 2}, (a = 0 ↔ b = 0) → a = b",
"state_before": "n m : ℕ\na b : Fin 2\nh : a = 0 ↔ b = 0\n⊢ a = b",
"tactic": "revert a b"
},
{
"state_after": "no goals",
"state_before": "n m : ℕ\n⊢ ∀ {a b : Fin 2}, (a = 0 ↔ b = 0) → a = b",
"tactic": "simp [forall_fin_two]"
}
] |
[
1777,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1775,
1
] |
Mathlib/NumberTheory/Multiplicity.lean
|
multiplicity.Nat.pow_add_pow
|
[
{
"state_after": "R : Type ?u.840239\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : p ∣ x + y\nhx : ¬p ∣ x\nn : ℕ\nhn : Odd n\n⊢ multiplicity ↑p ↑(x ^ n + y ^ n) = multiplicity ↑p ↑(x + y) + multiplicity p n",
"state_before": "R : Type ?u.840239\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : p ∣ x + y\nhx : ¬p ∣ x\nn : ℕ\nhn : Odd n\n⊢ multiplicity p (x ^ n + y ^ n) = multiplicity p (x + y) + multiplicity p n",
"tactic": "iterate 2 rw [← Int.coe_nat_multiplicity]"
},
{
"state_after": "R : Type ?u.840239\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : ↑p ∣ ↑(x + y)\nhx : ¬↑p ∣ ↑x\nn : ℕ\nhn : Odd n\n⊢ multiplicity ↑p ↑(x ^ n + y ^ n) = multiplicity ↑p ↑(x + y) + multiplicity p n",
"state_before": "R : Type ?u.840239\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : p ∣ x + y\nhx : ¬p ∣ x\nn : ℕ\nhn : Odd n\n⊢ multiplicity ↑p ↑(x ^ n + y ^ n) = multiplicity ↑p ↑(x + y) + multiplicity p n",
"tactic": "rw [← Int.coe_nat_dvd] at hxy hx"
},
{
"state_after": "R : Type ?u.840239\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhx : ¬↑p ∣ ↑x\nn : ℕ\nhn : Odd n\nhxy : ↑p ∣ ↑x + ↑y\n⊢ multiplicity (↑p) (↑x ^ n + ↑y ^ n) = multiplicity (↑p) (↑x + ↑y) + multiplicity p n",
"state_before": "R : Type ?u.840239\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : ↑p ∣ ↑(x + y)\nhx : ¬↑p ∣ ↑x\nn : ℕ\nhn : Odd n\n⊢ multiplicity ↑p ↑(x ^ n + y ^ n) = multiplicity ↑p ↑(x + y) + multiplicity p n",
"tactic": "push_cast at *"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.840239\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhx : ¬↑p ∣ ↑x\nn : ℕ\nhn : Odd n\nhxy : ↑p ∣ ↑x + ↑y\n⊢ multiplicity (↑p) (↑x ^ n + ↑y ^ n) = multiplicity (↑p) (↑x + ↑y) + multiplicity p n",
"tactic": "exact Int.pow_add_pow hp hp1 hxy hx hn"
},
{
"state_after": "R : Type ?u.840239\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : p ∣ x + y\nhx : ¬p ∣ x\nn : ℕ\nhn : Odd n\n⊢ multiplicity ↑p ↑(x ^ n + y ^ n) = multiplicity ↑p ↑(x + y) + multiplicity p n",
"state_before": "R : Type ?u.840239\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ : R\np : ℕ\nhp : Nat.Prime p\nhp1 : Odd p\nx y : ℕ\nhxy : p ∣ x + y\nhx : ¬p ∣ x\nn : ℕ\nhn : Odd n\n⊢ multiplicity ↑p ↑(x ^ n + y ^ n) = multiplicity p (x + y) + multiplicity p n",
"tactic": "rw [← Int.coe_nat_multiplicity]"
}
] |
[
249,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
244,
1
] |
Mathlib/Data/Set/Intervals/Disjoint.lean
|
IsLUB.biUnion_Iio_eq
|
[] |
[
197,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/Algebra/Order/Monoid/MinMax.lean
|
min_mul_mul_left
|
[] |
[
53,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Algebra/GroupPower/Order.lean
|
MonoidHom.map_sub_swap
|
[
{
"state_after": "no goals",
"state_before": "β : Type ?u.300600\nA : Type ?u.300603\nG : Type ?u.300606\nM : Type u_1\nR : Type u_2\ninst✝³ : Ring R\ninst✝² : Monoid M\ninst✝¹ : LinearOrder M\ninst✝ : CovariantClass M M (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\nf : R →* M\nx y : R\n⊢ ↑f (x - y) = ↑f (y - x)",
"tactic": "rw [← map_neg, neg_sub]"
}
] |
[
808,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
808,
1
] |
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
LinearMap.toMatrix₂_compl₂
|
[
{
"state_after": "R : Type u_1\nR₁ : Type ?u.1647539\nR₂ : Type ?u.1647542\nM : Type ?u.1647545\nM₁ : Type u_2\nM₂ : Type u_3\nM₁' : Type ?u.1647554\nM₂' : Type u_4\nn : Type u_5\nm : Type u_7\nn' : Type ?u.1647566\nm' : Type u_6\nι : Type ?u.1647572\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : AddCommMonoid M₁\ninst✝¹⁴ : Module R M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : Module R M₂\ninst✝¹¹ : DecidableEq n\ninst✝¹⁰ : Fintype n\ninst✝⁹ : DecidableEq m\ninst✝⁸ : Fintype m\nb₁ : Basis n R M₁\nb₂ : Basis m R M₂\ninst✝⁷ : AddCommMonoid M₁'\ninst✝⁶ : Module R M₁'\ninst✝⁵ : AddCommMonoid M₂'\ninst✝⁴ : Module R M₂'\nb₁' : Basis n' R M₁'\nb₂' : Basis m' R M₂'\ninst✝³ : Fintype n'\ninst✝² : Fintype m'\ninst✝¹ : DecidableEq n'\ninst✝ : DecidableEq m'\nB : M₁ →ₗ[R] M₂ →ₗ[R] R\nf : M₂' →ₗ[R] M₂\n⊢ (↑(toMatrix b₁ b₁) id)ᵀ ⬝ ↑(toMatrix₂ b₁ b₂) B ⬝ ↑(toMatrix b₂' b₂) f =\n ↑(toMatrix₂ b₁ b₂) (comp B id) ⬝ ↑(toMatrix b₂' b₂) f",
"state_before": "R : Type u_1\nR₁ : Type ?u.1647539\nR₂ : Type ?u.1647542\nM : Type ?u.1647545\nM₁ : Type u_2\nM₂ : Type u_3\nM₁' : Type ?u.1647554\nM₂' : Type u_4\nn : Type u_5\nm : Type u_7\nn' : Type ?u.1647566\nm' : Type u_6\nι : Type ?u.1647572\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : AddCommMonoid M₁\ninst✝¹⁴ : Module R M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : Module R M₂\ninst✝¹¹ : DecidableEq n\ninst✝¹⁰ : Fintype n\ninst✝⁹ : DecidableEq m\ninst✝⁸ : Fintype m\nb₁ : Basis n R M₁\nb₂ : Basis m R M₂\ninst✝⁷ : AddCommMonoid M₁'\ninst✝⁶ : Module R M₁'\ninst✝⁵ : AddCommMonoid M₂'\ninst✝⁴ : Module R M₂'\nb₁' : Basis n' R M₁'\nb₂' : Basis m' R M₂'\ninst✝³ : Fintype n'\ninst✝² : Fintype m'\ninst✝¹ : DecidableEq n'\ninst✝ : DecidableEq m'\nB : M₁ →ₗ[R] M₂ →ₗ[R] R\nf : M₂' →ₗ[R] M₂\n⊢ ↑(toMatrix₂ b₁ b₂') (compl₂ B f) = ↑(toMatrix₂ b₁ b₂) B ⬝ ↑(toMatrix b₂' b₂) f",
"tactic": "rw [← LinearMap.comp_id B, ← LinearMap.compl₁₂, LinearMap.toMatrix₂_compl₁₂ b₁ b₂]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nR₁ : Type ?u.1647539\nR₂ : Type ?u.1647542\nM : Type ?u.1647545\nM₁ : Type u_2\nM₂ : Type u_3\nM₁' : Type ?u.1647554\nM₂' : Type u_4\nn : Type u_5\nm : Type u_7\nn' : Type ?u.1647566\nm' : Type u_6\nι : Type ?u.1647572\ninst✝¹⁶ : CommRing R\ninst✝¹⁵ : AddCommMonoid M₁\ninst✝¹⁴ : Module R M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : Module R M₂\ninst✝¹¹ : DecidableEq n\ninst✝¹⁰ : Fintype n\ninst✝⁹ : DecidableEq m\ninst✝⁸ : Fintype m\nb₁ : Basis n R M₁\nb₂ : Basis m R M₂\ninst✝⁷ : AddCommMonoid M₁'\ninst✝⁶ : Module R M₁'\ninst✝⁵ : AddCommMonoid M₂'\ninst✝⁴ : Module R M₂'\nb₁' : Basis n' R M₁'\nb₂' : Basis m' R M₂'\ninst✝³ : Fintype n'\ninst✝² : Fintype m'\ninst✝¹ : DecidableEq n'\ninst✝ : DecidableEq m'\nB : M₁ →ₗ[R] M₂ →ₗ[R] R\nf : M₂' →ₗ[R] M₂\n⊢ (↑(toMatrix b₁ b₁) id)ᵀ ⬝ ↑(toMatrix₂ b₁ b₂) B ⬝ ↑(toMatrix b₂' b₂) f =\n ↑(toMatrix₂ b₁ b₂) (comp B id) ⬝ ↑(toMatrix b₂' b₂) f",
"tactic": "simp"
}
] |
[
477,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
474,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.iUnion_mul_right_image
|
[] |
[
484,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
483,
1
] |
Mathlib/Order/ModularLattice.lean
|
IsModularLattice.inf_sup_inf_assoc
|
[] |
[
215,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Submodule.mul_annihilator
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nM : Type v\nF : Type ?u.61436\nG : Type ?u.61439\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nI✝ J : Ideal R\nN P : Submodule R M\nI : Ideal R\n⊢ I * annihilator I = ⊥",
"tactic": "rw [mul_comm, annihilator_mul]"
}
] |
[
183,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/Order/Antisymmetrization.lean
|
antisymmRel_iff_eq
|
[] |
[
74,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.pushoutCoconeOfLeftIso_ι_app_right
|
[] |
[
1804,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1804,
1
] |
Mathlib/CategoryTheory/Abelian/Opposite.lean
|
CategoryTheory.image_ι_op_comp_imageUnopOp_hom
|
[
{
"state_after": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\nA B : Cᵒᵖ\ng : A ⟶ B\n⊢ (((((kernel.lift (cokernel.π g.unop) (cokernel.π (cokernel.π g.unop).op).unop\n (_ : (cokernel.π (cokernel.π g.unop).op).unop ≫ cokernel.π g.unop = 0) ≫\n (Abelian.imageIsoImage g.unop).hom) ≫\n image.ι g.unop).op ≫\n (cokernelIsoOfEq\n (_ : (cokernel.π g.unop).op = (cokernelUnopOp g).hom ≫ kernel.ι g ≫ eqToHom (_ : A = A.unop.op))).hom) ≫\n cokernel.desc ((cokernelUnopOp g).hom ≫ kernel.ι g ≫ eqToHom (_ : A = A.unop.op))\n (cokernel.π (kernel.ι g ≫ eqToHom (_ : A = A.unop.op)))\n (_ :\n ((cokernelUnopOp g).hom ≫ kernel.ι g ≫ eqToHom (_ : A = A.unop.op)) ≫\n cokernel.π (kernel.ι g ≫ eqToHom (_ : A = A.unop.op)) =\n 0)) ≫\n cokernel.desc (kernel.ι g ≫ eqToHom (_ : A = A.unop.op))\n (inv (eqToHom (_ : A = A.unop.op)) ≫ cokernel.π (kernel.ι g))\n (_ :\n (kernel.ι g ≫ eqToHom (_ : A = A.unop.op)) ≫ inv (eqToHom (_ : A = A.unop.op)) ≫ cokernel.π (kernel.ι g) =\n 0)) ≫\n cokernel.desc (kernel.ι g) (factorThruImage g) (_ : kernel.ι g ≫ factorThruImage g = 0) =\n factorThruImage g",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\nA B : Cᵒᵖ\ng : A ⟶ B\n⊢ (image.ι g.unop).op ≫ (imageUnopOp g).hom = factorThruImage g",
"tactic": "simp only [imageUnopOp, Iso.trans, Iso.symm, Iso.op, cokernelOpOp_inv, cokernelEpiComp_hom,\n cokernelCompIsIso_hom, Abelian.coimageIsoImage'_hom, ← Category.assoc, ← op_comp]"
},
{
"state_after": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\nA B : Cᵒᵖ\ng : A ⟶ B\n⊢ inv (eqToHom (_ : A = A.unop.op)) ≫ factorThruImage g = factorThruImage g",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\nA B : Cᵒᵖ\ng : A ⟶ B\n⊢ (((((kernel.lift (cokernel.π g.unop) (cokernel.π (cokernel.π g.unop).op).unop\n (_ : (cokernel.π (cokernel.π g.unop).op).unop ≫ cokernel.π g.unop = 0) ≫\n (Abelian.imageIsoImage g.unop).hom) ≫\n image.ι g.unop).op ≫\n (cokernelIsoOfEq\n (_ : (cokernel.π g.unop).op = (cokernelUnopOp g).hom ≫ kernel.ι g ≫ eqToHom (_ : A = A.unop.op))).hom) ≫\n cokernel.desc ((cokernelUnopOp g).hom ≫ kernel.ι g ≫ eqToHom (_ : A = A.unop.op))\n (cokernel.π (kernel.ι g ≫ eqToHom (_ : A = A.unop.op)))\n (_ :\n ((cokernelUnopOp g).hom ≫ kernel.ι g ≫ eqToHom (_ : A = A.unop.op)) ≫\n cokernel.π (kernel.ι g ≫ eqToHom (_ : A = A.unop.op)) =\n 0)) ≫\n cokernel.desc (kernel.ι g ≫ eqToHom (_ : A = A.unop.op))\n (inv (eqToHom (_ : A = A.unop.op)) ≫ cokernel.π (kernel.ι g))\n (_ :\n (kernel.ι g ≫ eqToHom (_ : A = A.unop.op)) ≫ inv (eqToHom (_ : A = A.unop.op)) ≫ cokernel.π (kernel.ι g) =\n 0)) ≫\n cokernel.desc (kernel.ι g) (factorThruImage g) (_ : kernel.ι g ≫ factorThruImage g = 0) =\n factorThruImage g",
"tactic": "simp only [Category.assoc, Abelian.imageIsoImage_hom_comp_image_ι, kernel.lift_ι,\n Quiver.Hom.op_unop, cokernelIsoOfEq_hom_comp_desc_assoc, cokernel.π_desc_assoc,\n cokernel.π_desc]"
},
{
"state_after": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\nA B : Cᵒᵖ\ng : A ⟶ B\n⊢ inv (𝟙 A) ≫ factorThruImage g = factorThruImage g",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\nA B : Cᵒᵖ\ng : A ⟶ B\n⊢ inv (eqToHom (_ : A = A.unop.op)) ≫ factorThruImage g = factorThruImage g",
"tactic": "simp only [eqToHom_refl]"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\nA B : Cᵒᵖ\ng : A ⟶ B\n⊢ inv (𝟙 A) ≫ factorThruImage g = factorThruImage g",
"tactic": "erw [IsIso.inv_id, Category.id_comp]"
}
] |
[
179,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
ContDiffWithinAt.sin
|
[] |
[
1049,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1047,
1
] |
Mathlib/NumberTheory/Padics/PadicNorm.lean
|
padicNorm.one
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\n⊢ padicNorm p 1 = 1",
"tactic": "simp [padicNorm]"
}
] |
[
82,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
11
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.normSq_inv
|
[] |
[
820,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
819,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
|
Real.log_mul
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ exp (log (x * y)) = exp (log x + log y)",
"tactic": "rw [exp_log_eq_abs (mul_ne_zero hx hy), exp_add, exp_log_eq_abs hx, exp_log_eq_abs hy, abs_mul]"
}
] |
[
131,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/ModelTheory/FinitelyGenerated.lean
|
FirstOrder.Language.Substructure.cg_bot
|
[] |
[
142,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/Data/Set/Intervals/Group.lean
|
Set.pairwise_disjoint_Ioc_add_int_cast
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedRing α\na : α\n⊢ Pairwise (Disjoint on fun n => Ioc (a + ↑n) (a + ↑n + 1))",
"tactic": "simpa only [zsmul_one, Int.cast_add, Int.cast_one, ← add_assoc] using\n pairwise_disjoint_Ioc_add_zsmul a (1 : α)"
}
] |
[
240,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/Order/ModularLattice.lean
|
isModularLattice_iff_inf_sup_inf_assoc
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Lattice α\nh : ∀ (x y z : α), x ⊓ z ⊔ y ⊓ z = (x ⊓ z ⊔ y) ⊓ z\nx✝ y z : α\nxz : x✝ ≤ z\n⊢ (x✝ ⊔ y) ⊓ z ≤ x✝ ⊔ y ⊓ z",
"tactic": "rw [← inf_eq_left.2 xz, h]"
}
] |
[
384,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
381,
1
] |
Mathlib/Order/BoundedOrder.lean
|
top_unique
|
[] |
[
163,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Finsupp.linearEquivFunOnFinite_single
|
[] |
[
113,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
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