mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
M : Type u_5
inst✝⁴ : CommRing R
inst✝³ : LinearOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : LinearMap.BilinForm R M
hs : ∀ (x : M), 0 ≤ (B x) x
hB : IsSymm B
W : Submodule R M
hW : Disjoint W (ker B)
hB' : (domRestrict₁₂ B W W).IsRefl
x : M
hx : x ∈ W
h : ∀ (y : ↥W), ((domRestrict₁₂ B W W) ⟨x, hx⟩) y = 0
⊢ ⟨x, hx⟩ = 0

Tactic:
[

Goal state after:
R : Type u_1
M : Type u_5
inst✝⁴ : CommRing R
inst✝³ : LinearOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : LinearMap.BilinForm R M
hs : ∀ (x : M), 0 ≤ (B x) x
hB : IsSymm B
W : Submodule R M
hW : Disjoint W (ker B)
hB' : (domRestrict₁₂ B W W).IsRefl
x : M
hx : x ∈ W
h : ∀ (y : ↥W), ((domRestrict₁₂ B W W) ⟨x, hx⟩) y = 0
⊢ ⟨x, hx⟩ = 0
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
W : Submodule R (Dual R M)
⊢ W.quotDualCoannihilatorToDual.ker = ⊥ ∧ W.quotDualCoannihilatorToDual.flip.ker = ⊥

Tactic:
[

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
W : Submodule R (Dual R M)
⊢ W.quotDualCoannihilatorToDual.ker = ⊥ ∧ W.quotDualCoannihilatorToDual.flip.ker = ⊥
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: K : Type u_1
V₁ : Type u_2
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V₁
inst✝¹ : Module K V₁
f : Dual K V₁
hf : f ≠ 0
inst✝ : FiniteDimensional K V₁
this : finrank K ↥(LinearMap.range f) = 1
⊢ 1 = finrank K ↥(LinearMap.range f)

Tactic:
this

Goal state after:
K : Type u_1
V₁ : Type u_2
inst✝³ : DivisionRing K
inst✝² : AddCommGroup V₁
inst✝¹ : Module K V₁
f : Dual K V₁
hf : f ≠ 0
inst✝ : FiniteDimensional K V₁
this : finrank K ↥(LinearMap.range f) = 1
⊢ 1 = 1
Module: Mathlib.CategoryTheory.Limits.Constructions.Pullbacks

Goal state before:
pp: C : Type u
𝒞 : Category.{v, u} C
X Y Z : C
f : X ⟶ Y
g : X ⟶ Z
inst✝¹ : HasColimit (pair Y Z)
inst✝ : HasColimit (parallelPair (f ≫ coprod.inl) (g ≫ coprod.inr))
ι₁ : Y ⟶ Y ⨿ Z := coprod.inl
ι₂ : Z ⟶ Y ⨿ Z := coprod.inr
c : Y ⨿ Z ⟶ coequalizer (f ≫ ι₁) (g ≫ ι₂) := coequalizer.π (f ≫ ι₁) (g ≫ ι₂)
s : PushoutCocone f g
⊢ (f ≫ ι₁) ≫ coprod.desc (s.ι.app WalkingSpan.left) (s.ι.app WalkingSpan.right) =
    (g ≫ ι₂) ≫ coprod.desc (s.ι.app WalkingSpan.left) (s.ι.app WalkingSpan.right)

Tactic:
rw [Category.assoc, colimit.ι_desc, Category.assoc, colimit.ι_desc]

Goal state after:
C : Type u
𝒞 : Category.{v, u} C
X Y Z : C
f : X ⟶ Y
g : X ⟶ Z
inst✝¹ : HasColimit (pair Y Z)
inst✝ : HasColimit (parallelPair (f ≫ coprod.inl) (g ≫ coprod.inr))
ι₁ : Y ⟶ Y ⨿ Z := coprod.inl
ι₂ : Z ⟶ Y ⨿ Z := coprod.inr
c : Y ⨿ Z ⟶ coequalizer (f ≫ ι₁) (g ≫ ι₂) := coequalizer.π (f ≫ ι₁) (g ≫ ι₂)
s : PushoutCocone f g
⊢ f ≫ (BinaryCofan.mk (s.ι.app WalkingSpan.left) (s.ι.app WalkingSpan.right)).ι.app { as := WalkingPair.left } =
    g ≫ (BinaryCofan.mk (s.ι.app WalkingSpan.left) (s.ι.app WalkingSpan.right)).ι.app { as := WalkingPair.right }
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: K : Type u_1
V₁ : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V₁
inst✝ : Module K V₁
W W' : Subspace K V₁
h : W ⊓ W' = ⊥ ∧ W ⊔ W' = ⊤
⊢ (W ⊔ W').dualAnnihilator = ⊥ ∧ dualAnnihilator ⊥ = ⊤

Tactic:
h.2,

Goal state after:
K : Type u_1
V₁ : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V₁
inst✝ : Module K V₁
W W' : Subspace K V₁
h : W ⊓ W' = ⊥ ∧ W ⊔ W' = ⊤
⊢ dualAnnihilator ⊤ = ⊥ ∧ dualAnnihilator ⊥ = ⊤
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: K : Type u_1
V₁ : Type u_2
V₂ : Type u_3
inst✝⁴ : Field K
inst✝³ : AddCommGroup V₁
inst✝² : Module K V₁
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
f : V₁ →ₗ[K] V₂
⊢ Function.Bijective ⇑f.dualMap ↔ Function.Bijective ⇑f

Tactic:
[

Goal state after:
K : Type u_1
V₁ : Type u_2
V₂ : Type u_3
inst✝⁴ : Field K
inst✝³ : AddCommGroup V₁
inst✝² : Module K V₁
inst✝¹ : AddCommGroup V₂
inst✝ : Module K V₂
f : V₁ →ₗ[K] V₂
⊢ Function.Bijective ⇑f.dualMap ↔ Function.Bijective ⇑f
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: K : Type u_1
V₁ : Type u_2
V₂ : Type u_3
inst✝⁵ : Field K
inst✝⁴ : AddCommGroup V₁
inst✝³ : Module K V₁
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
B : V₁ →ₗ[K] V₂ →ₗ[K] K
inst✝ : FiniteDimensional K V₁
⊢ Bijective ⇑B.flip ↔ Bijective ⇑B

Tactic:
[

Goal state after:
K : Type u_1
V₁ : Type u_2
V₂ : Type u_3
inst✝⁵ : Field K
inst✝⁴ : AddCommGroup V₁
inst✝³ : Module K V₁
inst✝² : AddCommGroup V₂
inst✝¹ : Module K V₂
B : V₁ →ₗ[K] V₂ →ₗ[K] K
inst✝ : FiniteDimensional K V₁
⊢ Bijective ⇑B.flip ↔ Bijective ⇑B
Module: Mathlib.CategoryTheory.Abelian.NonPreadditive

Goal state before:
pp: case refine_1
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : NonPreadditiveAbelian C
A : C
hlp : prod.lift (𝟙 A) 0 ≫ prod.snd = 0
⊢ ∀ (s : Fork prod.snd 0), (s.ι ≫ prod.fst) ≫ Fork.ι (KernelFork.ofι (prod.lift (𝟙 A) 0) hlp) = s.ι

Tactic:
intro s
      apply Limits.prod.hom_ext <;> simp

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Basic

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasPullbacks C
W X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : Epi f
R : C
e : Y ⟶ R
h : pullback.snd f g ≫ e = 0
u : X ⊞ Y ⟶ R := biprod.desc 0 e
hu : PullbackToBiproductIsKernel.pullbackToBiproduct f g ≫ u = 0
this✝¹ : IsColimit (CokernelCofork.ofπ (biprod.desc f (-g)) ⋯)
d : Z ⟶ R
hd : biprod.desc f (-g) ≫ d = biprod.desc 0 e
this✝ : f ≫ d = 0
this : d = 0
⊢ biprod.inr ≫ biprod.desc f (-g) ≫ d = biprod.inr ≫ biprod.desc f (-g) ≫ 0

Tactic:
this

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasPullbacks C
W X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : Epi f
R : C
e : Y ⟶ R
h : pullback.snd f g ≫ e = 0
u : X ⊞ Y ⟶ R := biprod.desc 0 e
hu : PullbackToBiproductIsKernel.pullbackToBiproduct f g ≫ u = 0
this✝¹ : IsColimit (CokernelCofork.ofπ (biprod.desc f (-g)) ⋯)
d : Z ⟶ R
hd : biprod.desc f (-g) ≫ d = biprod.desc 0 e
this✝ : f ≫ d = 0
this : d = 0
⊢ biprod.inr ≫ biprod.desc f (-g) ≫ 0 = biprod.inr ≫ biprod.desc f (-g) ≫ 0
Module: Mathlib.CategoryTheory.Abelian.Basic

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasPullbacks C
W X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : Epi g
R : C
e : X ⟶ R
h : pullback.fst f g ≫ e = 0
u : X ⊞ Y ⟶ R := biprod.desc e 0
hu : PullbackToBiproductIsKernel.pullbackToBiproduct f g ≫ u = 0
this✝¹ : IsColimit (CokernelCofork.ofπ (biprod.desc f (-g)) ⋯)
d : Z ⟶ R
hd : biprod.desc f (-g) ≫ d = biprod.desc e 0
this✝ : (-g) ≫ d = 0
this : d = 0
⊢ biprod.inl ≫ biprod.desc f (-g) ≫ d = biprod.inl ≫ biprod.desc f (-g) ≫ 0

Tactic:
this

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasPullbacks C
W X Y Z : C
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : Epi g
R : C
e : X ⟶ R
h : pullback.fst f g ≫ e = 0
u : X ⊞ Y ⟶ R := biprod.desc e 0
hu : PullbackToBiproductIsKernel.pullbackToBiproduct f g ≫ u = 0
this✝¹ : IsColimit (CokernelCofork.ofπ (biprod.desc f (-g)) ⋯)
d : Z ⟶ R
hd : biprod.desc f (-g) ≫ d = biprod.desc e 0
this✝ : (-g) ≫ d = 0
this : d = 0
⊢ biprod.inl ≫ biprod.desc f (-g) ≫ 0 = biprod.inl ≫ biprod.desc f (-g) ≫ 0
Module: Mathlib.Algebra.Category.Grp.EpiMono

Goal state before:
pp: case neg
A B : GrpCat
f : A ⟶ B
a : ↑A
y : ↑B
m : y ∉ (Hom.hom f).range
⊢ (h ((ConcreteCategory.hom f) a)) (fromCoset ⟨(fun x ↦ x • ↑(Hom.hom f).range) y, ⋯⟩) =
    fromCoset ⟨(ConcreteCategory.hom f) a • ↑⟨(fun x ↦ x • ↑(Hom.hom f).range) y, ⋯⟩, ⋯⟩

Tactic:
· rw [h_apply_fromCoset_nin_range f (f a) ⟨_, rfl⟩ _ m]
        simp only [leftCoset_assoc]

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Basic

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasPushouts C
W X Y Z : C
f : X ⟶ Y
g : X ⟶ Z
inst✝ : Mono f
R : C
e : R ⟶ Z
h : e ≫ pushout.inr f g = 0
u : R ⟶ Y ⊞ Z := biprod.lift 0 e
hu : u ≫ BiproductToPushoutIsCokernel.biproductToPushout f g = 0
this✝¹ : IsLimit (KernelFork.ofι (biprod.lift f (-g)) ⋯)
d : R ⟶ X
hd : d ≫ biprod.lift f (-g) = biprod.lift 0 e
this✝ : d ≫ f = 0
this : d = 0
⊢ (d ≫ biprod.lift f (-g)) ≫ biprod.snd = (0 ≫ biprod.lift f (-g)) ≫ biprod.snd

Tactic:
this

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasPushouts C
W X Y Z : C
f : X ⟶ Y
g : X ⟶ Z
inst✝ : Mono f
R : C
e : R ⟶ Z
h : e ≫ pushout.inr f g = 0
u : R ⟶ Y ⊞ Z := biprod.lift 0 e
hu : u ≫ BiproductToPushoutIsCokernel.biproductToPushout f g = 0
this✝¹ : IsLimit (KernelFork.ofι (biprod.lift f (-g)) ⋯)
d : R ⟶ X
hd : d ≫ biprod.lift f (-g) = biprod.lift 0 e
this✝ : d ≫ f = 0
this : d = 0
⊢ (0 ≫ biprod.lift f (-g)) ≫ biprod.snd = (0 ≫ biprod.lift f (-g)) ≫ biprod.snd
Module: Mathlib.Algebra.Category.Grp.EpiMono

Goal state before:
pp: A B : GrpCat
f : A ⟶ B
a : ↑A
x✝ : X'
this : (ConcreteCategory.hom f) a ∈ {b | h b = g b}
⊢ (g ((Hom.hom f) a)) x✝ = (h ((Hom.hom f) a)) x✝

Tactic:
this

Goal state after:
A B : GrpCat
f : A ⟶ B
a : ↑A
x✝ : X'
this : (ConcreteCategory.hom f) a ∈ {b | h b = g b}
⊢ (g ((Hom.hom f) a)) x✝ = (g ((ConcreteCategory.hom f) a)) x✝
Module: Mathlib.CategoryTheory.Abelian.Basic

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasPushouts C
W X Y Z : C
f : X ⟶ Y
g : X ⟶ Z
inst✝ : Mono g
R : C
e : R ⟶ Y
h : e ≫ pushout.inl f g = 0
u : R ⟶ Y ⊞ Z := biprod.lift e 0
hu : u ≫ BiproductToPushoutIsCokernel.biproductToPushout f g = 0
this✝¹ : IsLimit (KernelFork.ofι (biprod.lift f (-g)) ⋯)
d : R ⟶ X
hd : d ≫ biprod.lift f (-g) = biprod.lift e 0
this✝ : d ≫ (-g) = 0
this : d = 0
⊢ (d ≫ biprod.lift f (-g)) ≫ biprod.fst = (0 ≫ biprod.lift f (-g)) ≫ biprod.fst

Tactic:
this

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasPushouts C
W X Y Z : C
f : X ⟶ Y
g : X ⟶ Z
inst✝ : Mono g
R : C
e : R ⟶ Y
h : e ≫ pushout.inl f g = 0
u : R ⟶ Y ⊞ Z := biprod.lift e 0
hu : u ≫ BiproductToPushoutIsCokernel.biproductToPushout f g = 0
this✝¹ : IsLimit (KernelFork.ofι (biprod.lift f (-g)) ⋯)
d : R ⟶ X
hd : d ≫ biprod.lift f (-g) = biprod.lift e 0
this✝ : d ≫ (-g) = 0
this : d = 0
⊢ (0 ≫ biprod.lift f (-g)) ≫ biprod.fst = (0 ≫ biprod.lift f (-g)) ≫ biprod.fst
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroMorphisms C
S₁ S₂ S₃ S₄ : ShortComplex C
inst✝³ : S₁.HasHomology
inst✝² : S₂.HasHomology
inst✝¹ : S₃.HasHomology
inst✝ : S₄.HasHomology
φ : S₁ ⟶ S₂
φ' : S₃ ⟶ S₄
e : Arrow.mk φ ≅ Arrow.mk φ'
hφ : QuasiIso φ
α : S₃ ⟶ S₁ := Arrow.Hom.left e.inv
β : S₂ ⟶ S₄ := Arrow.Hom.right e.hom
this : φ' = α ≫ φ ≫ β
⊢ QuasiIso φ'

Tactic:
this

Goal state after:
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : HasZeroMorphisms C
S₁ S₂ S₃ S₄ : ShortComplex C
inst✝³ : S₁.HasHomology
inst✝² : S₂.HasHomology
inst✝¹ : S₃.HasHomology
inst✝ : S₄.HasHomology
φ : S₁ ⟶ S₂
φ' : S₃ ⟶ S₄
e : Arrow.mk φ ≅ Arrow.mk φ'
hφ : QuasiIso φ
α : S₃ ⟶ S₁ := Arrow.Hom.left e.inv
β : S₂ ⟶ S₄ := Arrow.Hom.right e.hom
this : φ' = α ≫ φ ≫ β
⊢ QuasiIso (α ≫ φ ≫ β)
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasLeftHomology
A : C
f₁ f₂ : S.leftHomology ⟶ A
⊢ f₁ = f₂ ↔ S.leftHomologyπ ≫ f₁ = S.leftHomologyπ ≫ f₂

Tactic:
cancel_epi

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasLeftHomology
A : C
f₁ f₂ : S.leftHomology ⟶ A
⊢ f₁ = f₂ ↔ f₁ = f₂
Module: Mathlib.Algebra.Homology.ShortComplex.Homology

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
h : S.HomologyData
inst✝ : S.HasHomology
⊢ h.iso.hom = h.left.homologyIso.inv ≫ h.right.homologyIso.hom

Tactic:
rw [← leftRightHomologyComparison'_fac, leftRightHomologyComparison'_eq]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
inst✝ : S.HasRightHomology
⊢ Epi S.pOpcycles

Tactic:
dsimp only [pOpcycles]
  infer_instance

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasRightHomology
A : C
f₁ f₂ : S.opcycles ⟶ A
⊢ f₁ = f₂ ↔ S.pOpcycles ≫ f₁ = S.pOpcycles ≫ f₂

Tactic:
cancel_epi

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasRightHomology
A : C
f₁ f₂ : S.opcycles ⟶ A
⊢ f₁ = f₂ ↔ f₁ = f₂
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ S₃ : ShortComplex C
φ₁ : S₁ ⟶ S₂
φ₂ : S₂ ⟶ S₃
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
h₃ : S₃.RightHomologyData
γ₁ : RightHomologyMapData φ₁ h₁ h₂ := rightHomologyMapData φ₁ h₁ h₂
γ₂ : RightHomologyMapData φ₂ h₂ h₃ := rightHomologyMapData φ₂ h₂ h₃
⊢ (γ₁.comp γ₂).φH = γ₁.φH ≫ γ₂.φH

Tactic:
RightHomologyMapData.comp_φH

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ S₃ : ShortComplex C
φ₁ : S₁ ⟶ S₂
φ₂ : S₂ ⟶ S₃
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
h₃ : S₃.RightHomologyData
γ₁ : RightHomologyMapData φ₁ h₁ h₂ := rightHomologyMapData φ₁ h₁ h₂
γ₂ : RightHomologyMapData φ₂ h₂ h₃ := rightHomologyMapData φ₂ h₂ h₃
⊢ γ₁.φH ≫ γ₂.φH = γ₁.φH ≫ γ₂.φH
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : S₂.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
wi : i ≫ S₁.g = 0
hi : IsLimit (KernelFork.ofι i wi)
f' : (KernelFork.ofι S₁.f ⋯).pt ⟶ (KernelFork.ofι i wi).pt := hi.lift (KernelFork.ofι S₁.f ⋯)
hf' : f' ≫ i = S₁.f
⊢ f' = φ.τ₁ ≫ h.f'

Tactic:
rw [← cancel_mono h.i, ← cancel_mono (inv φ.τ₂), assoc, assoc, assoc, hf', f'_i_assoc,
      φ.comm₁₂_assoc, IsIso.hom_inv_id, comp_id]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S : ShortComplex C
inst✝¹ : S.HasLeftHomology
inst✝ : HasKernel S.g
h : S.LeftHomologyData := S.leftHomologyData
⊢ HasCokernel (kernel.lift S.g S.f ⋯)

Tactic:
have : HasColimit (parallelPair h.f' 0) := ⟨⟨⟨_, h.hπ'⟩⟩⟩

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S : ShortComplex C
inst✝¹ : S.HasLeftHomology
inst✝ : HasKernel S.g
h : S.LeftHomologyData := S.leftHomologyData
this : HasColimit (parallelPair h.f' 0)
⊢ HasCokernel (kernel.lift S.g S.f ⋯)
Module: Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
inst✝⁴ : HasZeroMorphisms C
inst✝³ : HasZeroMorphisms D
S : ShortComplex C
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : S.HasLeftHomology
inst✝ : F.PreservesLeftHomologyOf S
⊢ (S.mapCyclesIso F).hom ≫ F.map S.iCycles = (S.map F).iCycles

Tactic:
by
  apply LeftHomologyData.cyclesIso_hom_comp_i

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Limits

Goal state before:
pp: case h₂
J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
inst✝ : HasZeroMorphisms C
F : J ⥤ ShortComplex C
c : Cocone F
h₁ : IsColimit (π₁.mapCocone c)
h₂ : IsColimit (π₂.mapCocone c)
h₃ : IsColimit (π₃.mapCocone c)
s : Cocone F
m : c.pt ⟶ s.pt
hm : ∀ (j : J), c.ι.app j ≫ m = s.ι.app j
⊢ m.τ₂ =
    { τ₁ := h₁.desc (π₁.mapCocone s), τ₂ := h₂.desc (π₂.mapCocone s), τ₃ := h₃.desc (π₃.mapCocone s), comm₁₂ := ⋯,
        comm₂₃ := ⋯ }.τ₂

pp: case h₃
J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
inst✝ : HasZeroMorphisms C
F : J ⥤ ShortComplex C
c : Cocone F
h₁ : IsColimit (π₁.mapCocone c)
h₂ : IsColimit (π₂.mapCocone c)
h₃ : IsColimit (π₃.mapCocone c)
s : Cocone F
m : c.pt ⟶ s.pt
hm : ∀ (j : J), c.ι.app j ≫ m = s.ι.app j
⊢ m.τ₃ =
    { τ₁ := h₁.desc (π₁.mapCocone s), τ₂ := h₂.desc (π₂.mapCocone s), τ₃ := h₃.desc (π₃.mapCocone s), comm₁₂ := ⋯,
        comm₂₃ := ⋯ }.τ₃

Tactic:
· exact h₂.uniq (π₂.mapCocone s) _ (fun j => π₂.congr_map (hm j))

Goal state after:
case h₃
J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
inst✝ : HasZeroMorphisms C
F : J ⥤ ShortComplex C
c : Cocone F
h₁ : IsColimit (π₁.mapCocone c)
h₂ : IsColimit (π₂.mapCocone c)
h₃ : IsColimit (π₃.mapCocone c)
s : Cocone F
m : c.pt ⟶ s.pt
hm : ∀ (j : J), c.ι.app j ≫ m = s.ι.app j
⊢ m.τ₃ =
    { τ₁ := h₁.desc (π₁.mapCocone s), τ₂ := h₂.desc (π₂.mapCocone s), τ₃ := h₃.desc (π₃.mapCocone s), comm₁₂ := ⋯,
        comm₂₃ := ⋯ }.τ₃
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ S₃ : ShortComplex C
φ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
h' : Homotopy φ₃ φ₄
⊢ (φ₁ - φ₃).τ₃ = h.h₃ - h'.h₃ + (h.h₂ - h'.h₂) ≫ S₂.g + (φ₂ - φ₄).τ₃

Tactic:
rw [sub_τ₃, sub_τ₃, h.comm₃, h'.comm₃, sub_comp]; abel

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.Opposite

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
A B : Cᵒᵖ
g : A ⟶ B
⊢ (imageUnopOp g).inv ≫ (factorThruImage g.unop).op = image.ι g

Tactic:
by
  rw [Iso.inv_comp_eq, imageUnopOp_hom_comp_image_ι]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.ShortExact

Goal state before:
pp: case mpr
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex C
hS : S.ShortExact
inst✝ : Balanced C
this✝¹ : HasZeroObject C
this✝ : Mono S.f
this : Epi S.g
hX₃ : IsZero S.X₃
⊢ IsIso S.f

Tactic:
have : Epi S.f := (S.exact_iff_epi (hX₃.eq_of_tgt _ _)).1 hS.exact

Goal state after:
case mpr
C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
S : ShortComplex C
hS : S.ShortExact
inst✝ : Balanced C
this✝² : HasZeroObject C
this✝¹ : Mono S.f
this✝ : Epi S.g
hX₃ : IsZero S.X₃
this : Epi S.f
⊢ IsIso S.f
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasHomology
h : S.LeftHomologyData
⊢ S.Exact ↔ IsZero h.H

Tactic:
by
  rw [S.exact_iff_isZero_homology]
  exact Iso.isZero_iff h.homologyIso

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : HasZeroMorphisms C
S : ShortComplex C
inst✝ : S.HasHomology
h : S.RightHomologyData
⊢ S.Exact ↔ IsZero h.H

Tactic:
by
  rw [S.exact_iff_isZero_homology]
  exact Iso.isZero_iff h.homologyIso

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
e : S₁ ≅ S₂
this : Mono S₁.f ↔ Mono S₂.f
⊢ S₂.Exact ∧ Mono S₁.f ↔ S₂.Exact ∧ Mono S₂.f

Tactic:
this

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
e : S₁ ≅ S₂
this : Mono S₁.f ↔ Mono S₂.f
⊢ S₂.Exact ∧ Mono S₂.f ↔ S₂.Exact ∧ Mono S₂.f
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
e : S₁ ≅ S₂
this : Epi S₁.g ↔ Epi S₂.g
⊢ S₂.Exact ∧ Epi S₁.g ↔ S₂.Exact ∧ Epi S₂.g

Tactic:
this

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
e : S₁ ≅ S₂
this : Epi S₁.g ↔ Epi S₂.g
⊢ S₂.Exact ∧ Epi S₂.g ↔ S₂.Exact ∧ Epi S₂.g
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

Goal state before:
pp: case mpr
C : Type u_1
D : Type u_2
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Category.{v_2, u_2} D
inst✝⁶ : HasZeroMorphisms C
inst✝⁵ : HasZeroMorphisms D
S : ShortComplex C
inst✝⁴ : S.HasHomology
F : C ⥤ D
inst✝³ : F.PreservesZeroMorphisms
inst✝² : F.PreservesLeftHomologyOf S
inst✝¹ : F.PreservesRightHomologyOf S
inst✝ : F.Faithful
⊢ S.Exact → (S.map F).Exact

Tactic:
intro h
    exact h.map F

Goal state after:
Module: Mathlib.CategoryTheory.Subobject.MonoOver

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
X Y Z : C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : HasImages C
f : X ⟶ Y
inst✝ : Mono f
⊢ (X_1 : MonoOver X) → («exists» f).obj X_1 ≅ (map f).obj X_1

Tactic:
by
    intro Z
    suffices (forget _).obj ((«exists» f).obj Z) ≅ (forget _).obj ((map f).obj Z) by
      apply (forget _).preimageIso this
    apply Over.isoMk _ _
    · apply imageMonoIsoSource (Z.arrow ≫ f)
    · apply imageMonoIsoSource_hom_self

Goal state after:
Module: Mathlib.CategoryTheory.Subobject.Basic

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X Y Z : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : X ≅ Y
g : Subobject X
⊢ (map e.inv).obj ((map e.hom).obj g) = g

Tactic:
[

Goal state after:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X Y Z : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : X ≅ Y
g : Subobject X
⊢ (map e.inv).obj ((map e.hom).obj g) = g
Module: Mathlib.CategoryTheory.Subobject.Basic

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X Y Z : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : X ≅ Y
g : Subobject Y
⊢ (map e.hom).obj ((map e.inv).obj g) = g

Tactic:
[

Goal state after:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X Y Z : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : X ≅ Y
g : Subobject Y
⊢ (map e.hom).obj ((map e.inv).obj g) = g
Module: Mathlib.CategoryTheory.Subobject.Lattice

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
X Y Z : C
D : Type u₂
inst✝² : Category.{v₂, u₂} D
inst✝¹ : LocallySmall.{w, v₁, u₁} C
inst✝ : WellPowered.{w, v₁, u₁} C
A : C
s : Set (Subobject A)
f : Subobject A
k : ∀ g ∈ s, f ≤ g
g : Subobject A
m : g ∈ s
⊢ (equivShrink (Subobject A)).symm ↑⟨(equivShrink (Subobject A)) g, ⋯⟩ ∈ s

Tactic:
simpa using m

Goal state after:
Module: Mathlib.CategoryTheory.Subobject.Lattice

Goal state before:
pp: α : Type u_1
β : Type u_2
e : α ≃ β
s : Set α
a : α
m : a ∈ s
⊢ e.symm (e a) ∈ s

Tactic:
simpa using m

Goal state after:
Module: Mathlib.CategoryTheory.Subobject.Lattice

Goal state before:
pp: case f.refine_1
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : LocallySmall.{w, v₁, u₁} C
inst✝² : WellPowered.{w, v₁, u₁} C
inst✝¹ : HasCoproducts C
inst✝ : HasImages C
A : C
s : Set (Subobject A)
f : Subobject A
k : ∀ g ∈ s, g ≤ f
g : Shrink.{w, max u₁ v₁} (Subobject A)
m : g ∈ ⇑(equivShrink (Subobject A)) '' s
⊢ (equivShrink (Subobject A)).symm ↑⟨g, m⟩ ∈ s

Tactic:
simpa using m

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
α : Type u_1
Z : α → C
f f' : Fan Z
c : Cofan fun x ↦ op (Z x)
hf : IsLimit f
hf' : IsLimit f'
hc : IsColimit c
j : α
⊢ ((opProductIsoCoproduct' hf hc).hom ≫ (opProductIsoCoproduct' hf' hc).inv).unop ≫ f.π.app { as := j } =
    (hf.conePointUniqueUpToIso hf').op.inv.unop ≫ f.π.app { as := j }

Tactic:
change _ ≫ f.proj _ = _

Goal state after:
C : Type u₁
inst✝ : Category.{v₁, u₁} C
α : Type u_1
Z : α → C
f f' : Fan Z
c : Cofan fun x ↦ op (Z x)
hf : IsLimit f
hf' : IsLimit f'
hc : IsColimit c
j : α
⊢ ((opProductIsoCoproduct' hf hc).hom ≫ (opProductIsoCoproduct' hf' hc).inv).unop ≫ f.proj j =
    (hf.conePointUniqueUpToIso hf').op.inv.unop ≫ f.π.app { as := j }
Module: Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A B : C
inst✝ : HasBinaryProduct A B
⊢ (opProdIsoCoprod A B).inv.unop ≫ coprod.inr.unop = prod.snd

Tactic:
by
  rw [← unop_comp, inr_opProdIsoCoprod_inv, Quiver.Hom.unop_op]

Goal state after: