mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ (0 ≫ biprod.lift f (-g)) ≫ biprod.fst = (0 ≫ biprod.lift f (-g)) ≫ biprod.fst

Tactic:
]

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 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

Tactic:
]

Goal state after:
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' : S₂ ⟶ S₃
hφ : IsIso (homologyMap φ)
hφ' : IsIso (homologyMap φ')
⊢ IsIso (homologyMap (φ ≫ φ'))

Tactic:
]

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' : S₂ ⟶ S₃
hφ : IsIso (homologyMap φ)
hφ' : IsIso (homologyMap φ')
⊢ IsIso (homologyMap (φ ≫ φ'))
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' : S₂ ⟶ S₃
hφ : IsIso (homologyMap φ)
hφ' : IsIso (homologyMap φ')
⊢ IsIso (homologyMap φ ≫ homologyMap φ')

Tactic:
]

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' : S₂ ⟶ S₃
hφ : IsIso (homologyMap φ)
hφ' : IsIso (homologyMap φ')
⊢ IsIso (homologyMap φ ≫ homologyMap φ')
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S₁ S₂ S₃ : ShortComplex C
inst✝² : S₁.HasHomology
inst✝¹ : S₂.HasHomology
inst✝ : S₃.HasHomology
φ : S₁ ⟶ S₂
φ' : S₂ ⟶ S₃
hφ : IsIso (homologyMap φ)
hφφ' : IsIso (homologyMap (φ ≫ φ'))
⊢ IsIso (homologyMap φ')

Tactic:
]

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S₁ S₂ S₃ : ShortComplex C
inst✝² : S₁.HasHomology
inst✝¹ : S₂.HasHomology
inst✝ : S₃.HasHomology
φ : S₁ ⟶ S₂
φ' : S₂ ⟶ S₃
hφ : IsIso (homologyMap φ)
hφφ' : IsIso (homologyMap (φ ≫ φ'))
⊢ IsIso (homologyMap φ')
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S₁ S₂ S₃ : ShortComplex C
inst✝² : S₁.HasHomology
inst✝¹ : S₂.HasHomology
inst✝ : S₃.HasHomology
φ : S₁ ⟶ S₂
φ' : S₂ ⟶ S₃
hφ : IsIso (homologyMap φ)
hφφ' : IsIso (homologyMap φ ≫ homologyMap φ')
⊢ IsIso (homologyMap φ')

Tactic:
]

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S₁ S₂ S₃ : ShortComplex C
inst✝² : S₁.HasHomology
inst✝¹ : S₂.HasHomology
inst✝ : S₃.HasHomology
φ : S₁ ⟶ S₂
φ' : S₂ ⟶ S₃
hφ : IsIso (homologyMap φ)
hφφ' : IsIso (homologyMap φ ≫ homologyMap φ')
⊢ IsIso (homologyMap φ')
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S₁ S₂ S₃ : ShortComplex C
inst✝² : S₁.HasHomology
inst✝¹ : S₂.HasHomology
inst✝ : S₃.HasHomology
φ : S₁ ⟶ S₂
φ' : S₂ ⟶ S₃
hφ' : IsIso (homologyMap φ')
hφφ' : IsIso (homologyMap (φ ≫ φ'))
⊢ IsIso (homologyMap φ)

Tactic:
]

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S₁ S₂ S₃ : ShortComplex C
inst✝² : S₁.HasHomology
inst✝¹ : S₂.HasHomology
inst✝ : S₃.HasHomology
φ : S₁ ⟶ S₂
φ' : S₂ ⟶ S₃
hφ' : IsIso (homologyMap φ')
hφφ' : IsIso (homologyMap (φ ≫ φ'))
⊢ IsIso (homologyMap φ)
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S₁ S₂ S₃ : ShortComplex C
inst✝² : S₁.HasHomology
inst✝¹ : S₂.HasHomology
inst✝ : S₃.HasHomology
φ : S₁ ⟶ S₂
φ' : S₂ ⟶ S₃
hφ' : IsIso (homologyMap φ')
hφφ' : IsIso (homologyMap φ ≫ homologyMap φ')
⊢ IsIso (homologyMap φ)

Tactic:
]

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S₁ S₂ S₃ : ShortComplex C
inst✝² : S₁.HasHomology
inst✝¹ : S₂.HasHomology
inst✝ : S₃.HasHomology
φ : S₁ ⟶ S₂
φ' : S₂ ⟶ S₃
hφ' : IsIso (homologyMap φ')
hφφ' : IsIso (homologyMap φ ≫ homologyMap φ')
⊢ IsIso (homologyMap φ)
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 (α ≫ φ ≫ β)

Tactic:
]

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.QuasiIso

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
γ : LeftHomologyMapData φ h₁ h₂
⊢ IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv) ↔ IsIso γ.φH

Tactic:
]

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
γ : LeftHomologyMapData φ h₁ h₂
⊢ IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv) ↔ IsIso γ.φH
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
⊢ IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv) ↔ IsIso γ.φH

Tactic:
]

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
⊢ IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv) ↔ IsIso γ.φH
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
γ : LeftHomologyMapData φ h₁ h₂
⊢ IsIso γ.φH ↔ IsIso γ.φH

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
⊢ IsIso γ.φH ↔ IsIso γ.φH

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝⁴ : S₁.HasHomology
inst✝³ : S₂.HasHomology
φ : S₁ ⟶ S₂
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
⊢ IsIso (LeftHomologyMapData.ofEpiOfIsIsoOfMono φ S₁.leftHomologyData).φH

Tactic:
]

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝⁴ : S₁.HasHomology
inst✝³ : S₂.HasHomology
φ : S₁ ⟶ S₂
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
⊢ IsIso (LeftHomologyMapData.ofEpiOfIsIsoOfMono φ S₁.leftHomologyData).φH
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
γ : HomologyMapData φ S₁.homologyData S₂.homologyData
⊢ IsIso γ.op.right.φH ↔ IsIso γ.left.φH

Tactic:
]

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
γ : HomologyMapData φ S₁.homologyData S₂.homologyData
⊢ IsIso γ.op.right.φH ↔ IsIso γ.left.φH
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝² : S₁.HasHomology
inst✝¹ : S₂.HasHomology
φ : S₁ ⟶ S₂
inst✝ : QuasiIso φ
⊢ QuasiIso φ

Tactic:
]

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝² : S₁.HasHomology
inst✝¹ : S₂.HasHomology
φ : S₁ ⟶ S₂
inst✝ : QuasiIso φ
⊢ QuasiIso φ
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasZeroMorphisms C
S₁ S₂ : ShortComplex Cᵒᵖ
inst✝⁴ : S₁.HasHomology
inst✝³ : S₂.HasHomology
inst✝² : S₁.unop.HasHomology
inst✝¹ : S₂.unop.HasHomology
φ : S₁ ⟶ S₂
inst✝ : QuasiIso φ
⊢ QuasiIso (opMap (unopMap φ))

Tactic:
]

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : HasZeroMorphisms C
S₁ S₂ : ShortComplex Cᵒᵖ
inst✝⁴ : S₁.HasHomology
inst✝³ : S₂.HasHomology
inst✝² : S₁.unop.HasHomology
inst✝¹ : S₂.unop.HasHomology
φ : S₁ ⟶ S₂
inst✝ : QuasiIso φ
⊢ QuasiIso (opMap (unopMap φ))
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: C : Type ?u.2
inst✝⁵ : Category.{v_1, ?u.2} 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
hf₁ : S₁.f = 0
hg₁ : S₁.g = 0
hf₂ : S₂.f = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
hf₁ : S₁.f = 0
hg₁ : S₁.g = 0
hf₂ : S₂.f = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type ?u.2
inst✝⁵ : Category.{v_1, ?u.2} 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
hg₁ : S₁.g = 0
hf₂ : S₂.f = 0
hg₂ : S₂.g = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
hg₁ : S₁.g = 0
hf₂ : S₂.f = 0
hg₂ : S₂.g = 0
⊢ 0 = 0

Tactic:
]

Goal state after:
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
S S₁ S₂ S₃ : ShortComplex C
h : S.RightHomologyData
A : C
k : S.X₂ ⟶ A
x : S.X₃ ⟶ A
hx : k = S.g ≫ x
⊢ 0 = 0

Tactic:
]

Goal state after:
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
S S₁ S₂ S₃ : ShortComplex C
h : S.RightHomologyData
A : C
k : S.X₂ ⟶ A
x : S.X₃ ⟶ A
hx : k = S.g ≫ x
⊢ S.f ≫ k = 0

Tactic:
rw [hx, S.zero_assoc, zero_comp]

Goal state after:
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
S : ShortComplex C
h : S.RightHomologyData
A : C
k : S.X₂ ⟶ A
x : S.X₃ ⟶ A
hx : k = S.g ≫ x
⊢ h.ι ≫ h.descQ k ⋯ = h.ι ≫ h.g' ≫ x

Tactic:
]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex C
h : S.RightHomologyData
A : C
k : S.X₂ ⟶ A
x : S.X₃ ⟶ A
hx : k = S.g ≫ x
⊢ h.ι ≫ h.descQ k ⋯ = h.ι ≫ h.g' ≫ x
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
S : ShortComplex C
h : S.RightHomologyData
hf : S.f = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
h : S.LeftHomologyData
A : C
k : A ⟶ S.X₂
x : A ⟶ S.X₁
hx : k = x ≫ S.f
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex C
h : S.LeftHomologyData
A : C
k : A ⟶ S.X₂
x : A ⟶ S.X₁
hx : k = x ≫ S.f
⊢ h.liftK k ⋯ ≫ h.π = (x ≫ h.f') ≫ h.π

Tactic:
]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex C
h : S.LeftHomologyData
A : C
k : A ⟶ S.X₂
x : A ⟶ S.X₁
hx : k = x ≫ S.f
⊢ h.liftK k ⋯ ≫ h.π = (x ≫ h.f') ≫ h.π
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
S : ShortComplex C
h : S.RightHomologyData
hg : S.g = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex C
h : S.LeftHomologyData
hg : S.g = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex C
h : S.LeftHomologyData
hf : S.f = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex C
h : S.LeftHomologyData
hf : S.f = 0
φ : (CokernelCofork.ofπ h.π ⋯).pt ⟶ h.K
hφ : h.π ≫ φ = 𝟙 h.K
⊢ h.π = h.π

Tactic:
]

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
γ₁ γ₂ : HomologyMapData φ h₁ h₂
eq : γ₁ = γ₂
⊢ γ₂.left.φH = γ₂.left.φH

Tactic:
]

Goal state after:
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
S : ShortComplex C
h : S.RightHomologyData
hg : S.g = 0
φ : h.Q ⟶ (KernelFork.ofι h.ι ⋯).pt
hφ : φ ≫ h.ι = 𝟙 h.Q
⊢ h.ι = h.ι

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
h : S.LeftHomologyData
A : C
hg : S.g = 0
c : CokernelCofork S.f
hc : IsColimit c
⊢ 0 = 0

Tactic:
]

Goal state after:
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
S S₁ S₂ S₃ : ShortComplex C
h : S.RightHomologyData
A : C
hf : S.f = 0
c : KernelFork S.g
hc : IsLimit c
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex C
hg : S.g = 0
c : CokernelCofork S.f
hc : IsColimit c
T : C
φ : T ⟶ S.X₂
⊢ φ = φ ≫ (ofIsColimitCokernelCofork S hg c hc).i

Tactic:
]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex C
hg : S.g = 0
c : CokernelCofork S.f
hc : IsColimit c
T : C
φ : T ⟶ S.X₂
⊢ φ = φ ≫ (ofIsColimitCokernelCofork S hg c hc).i
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
inst✝ : S.HasHomology
⊢ leftHomologyMap' (𝟙 S) S.homologyData.left S.leftHomologyData =
    leftHomologyMap' (𝟙 S) S.homologyData.left S.leftHomologyData

Tactic:
]

Goal state after:
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
S : ShortComplex C
hf : S.f = 0
c : KernelFork S.g
hc : IsLimit c
⊢ S.g = S.g

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex C
hf : S.f = 0
c : KernelFork S.g
hc : IsLimit c
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
h : S.LeftHomologyData
A : C
hf : S.f = 0
hg : S.g = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex C
hf : S.f = 0
hg : S.g = 0
⊢ 0 = 0

Tactic:
]

Goal state after:
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
S : ShortComplex C
hf : S.f = 0
c : KernelFork S.g
hc : IsLimit c
T : C
φ : S.X₂ ⟶ T
⊢ φ = (ofIsLimitKernelFork S hf c hc).p ≫ φ

Tactic:
]

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex C
hf : S.f = 0
c : KernelFork S.g
hc : IsLimit c
T : C
φ : S.X₂ ⟶ T
⊢ φ = (ofIsLimitKernelFork S hf c hc).p ≫ φ
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
h : S.LeftHomologyData
A K' H' : C
eK : K' ≅ h.K
eH : H' ≅ h.H
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
ψ₁ ψ₂ : LeftHomologyMapData φ h₁ h₂
⊢ h₁.i ≫ φ.τ₂ = h₁.i ≫ φ.τ₂

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
ψ₁ ψ₂ : LeftHomologyMapData φ h₁ h₂
hK : ψ₁.φK = ψ₂.φK
⊢ ψ₂.φK ≫ h₂.π = ψ₂.φK ≫ h₂.π

Tactic:
]

Goal state after:
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
S : ShortComplex C
hg : S.g = 0
c : CokernelCofork S.f
hc : IsColimit c
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ S₄ : ShortComplex C
φ : S₁ ⟶ S₂
h₁✝ : S₁.HomologyData
h₂✝ : S₂.HomologyData
e : S₁ ≅ S₂
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
⊢ 𝟙 h₁.left.H = 𝟙 h₁.left.H

Tactic:
]

Goal state after:
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
S S₁ S₂ S₃ : ShortComplex C
h : S.RightHomologyData
A : C
hf : S.f = 0
hg : S.g = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
φK : h₁.K ⟶ h₂.K := h₂.liftK (h₁.i ≫ φ.τ₂) ⋯
⊢ S₁.f ≫ φ.τ₂ = S₁.f ≫ φ.τ₂

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
φK : h₁.K ⟶ h₂.K := h₂.liftK (h₁.i ≫ φ.τ₂) ⋯
commf' : h₁.f' ≫ φK = φ.τ₁ ≫ h₂.f'
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
S S₁ S₂ S₃ S₄ : ShortComplex C
φ : S₁ ⟶ S₂
h₁✝ : S₁.HomologyData
h₂✝ : S₂.HomologyData
e : S₁ ≅ S₂
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
⊢ 𝟙 h₂.left.H = 𝟙 h₂.left.H

Tactic:
]

Goal state after:
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
S : ShortComplex C
hf : S.f = 0
hg : S.g = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
γ₁ γ₂ : LeftHomologyMapData φ h₁ h₂
eq : γ₁ = γ₂
⊢ γ₂.φH = γ₂.φH

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
γ₁ γ₂ : LeftHomologyMapData φ h₁ h₂
eq : γ₁ = γ₂
⊢ γ₂.φK = γ₂.φK

Tactic:
]

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
S S₁ S₂ S₃ S₄ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
e : S₁ ≅ S₂
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
⊢ 𝟙 S₁.homology = 𝟙 S₁.homology

Tactic:
]

Goal state after:
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
S S₁ S₂ S₃ : ShortComplex C
h : S.RightHomologyData
A Q' H' : C
eQ : Q' ≅ h.Q
eH : H' ≅ h.H
⊢ 0 = 0

Tactic:
]

Goal state after:
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 ↔ f₁ = f₂

Tactic:
]

Goal state after: