mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ (parallelPair (h.hp.desc (CokernelCofork.ofπ S.g ⋯)) 0).map WalkingParallelPairHom.left ≫ (Iso.refl S.X₃).hom =
    eQ.symm.hom ≫
      (parallelPair ((IsCokernel.cokernelIso S.f (h.p ≫ eQ.inv) h.hp eQ.symm ⋯).desc (CokernelCofork.ofπ S.g ⋯)) 0).map
        WalkingParallelPairHom.left

Tactic:
by

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 S₃ : ShortComplex C
φ₁ : S₁ ⟶ S₂
φ₂ : S₂ ⟶ S₃
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
h₃ : S₃.LeftHomologyData
⊢ cyclesMap' (φ₁ ≫ φ₂) h₁ h₃ = cyclesMap' φ₁ h₁ h₂ ≫ cyclesMap' φ₂ h₂ h₃

Tactic:
by

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
e : S₁ ≅ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ leftHomologyMap' e.hom h₁ h₂ ≫ leftHomologyMap' e.inv h₂ h₁ = 𝟙 h₁.H

Tactic:
by

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
e : S₁ ≅ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ leftHomologyMap' e.inv h₂ h₁ ≫ leftHomologyMap' e.hom h₁ h₂ = 𝟙 h₂.H

Tactic:
by

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
e : S₁ ≅ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ cyclesMap' e.hom h₁ h₂ ≫ cyclesMap' e.inv h₂ h₁ = 𝟙 h₁.K

Tactic:
by

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
e : S₁ ≅ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ cyclesMap' e.inv h₂ h₁ ≫ cyclesMap' e.hom h₁ h₂ = 𝟙 h₂.K

Tactic:
by

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
⊢ (parallelPair (h.hp.desc (CokernelCofork.ofπ S.g ⋯)) 0).map WalkingParallelPairHom.right ≫ (Iso.refl S.X₃).hom =
    eQ.symm.hom ≫
      (parallelPair ((IsCokernel.cokernelIso S.f (h.p ≫ eQ.inv) h.hp eQ.symm ⋯).desc (CokernelCofork.ofπ S.g ⋯)) 0).map
        WalkingParallelPairHom.right

Tactic:
by

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 S₁ S₂ S₃ : ShortComplex C
e : S₁ ≅ S₂
inst✝¹ : S₁.HasLeftHomology
inst✝ : S₂.HasLeftHomology
⊢ leftHomologyMap e.hom ≫ leftHomologyMap e.inv = 𝟙 S₁.leftHomology

Tactic:
by

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 S₁ S₂ S₃ : ShortComplex C
e : S₁ ≅ S₂
inst✝¹ : S₁.HasLeftHomology
inst✝ : S₂.HasLeftHomology
⊢ leftHomologyMap e.inv ≫ leftHomologyMap e.hom = 𝟙 S₂.leftHomology

Tactic:
by

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 S₁ S₂ S₃ : ShortComplex C
e : S₁ ≅ S₂
inst✝¹ : S₁.HasLeftHomology
inst✝ : S₂.HasLeftHomology
⊢ cyclesMap e.hom ≫ cyclesMap e.inv = 𝟙 S₁.cycles

Tactic:
by

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
⊢ ((Cone.postcompose (parallelPair.ext eQ.symm (Iso.refl S.X₃) ⋯ ⋯).hom).obj (KernelFork.ofι h.ι ⋯)).pt ≅
    (KernelFork.ofι (eH.hom ≫ h.ι ≫ eQ.inv) ⋯).pt

Tactic:
by

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 S₁ S₂ S₃ : ShortComplex C
e : S₁ ≅ S₂
inst✝¹ : S₁.HasLeftHomology
inst✝ : S₂.HasLeftHomology
⊢ cyclesMap e.inv ≫ cyclesMap e.hom = 𝟙 S₂.cycles

Tactic:
by

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
h : S.LeftHomologyData
inst✝ : S.HasLeftHomology
⊢ h.cyclesIso.hom ≫ h.i = S.iCycles

Tactic:
by

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
h : S.LeftHomologyData
inst✝ : S.HasLeftHomology
⊢ h.cyclesIso.inv ≫ S.iCycles = h.i

Tactic:
by

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 : ShortComplex C
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
⊢ h₁.π ≫ leftRightHomologyComparison' h₁ h₂ ≫ h₂.ι = h₁.i ≫ h₂.p

Tactic:
by

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
h : S.LeftHomologyData
inst✝ : S.HasLeftHomology
⊢ S.leftHomologyπ ≫ h.leftHomologyIso.hom = h.cyclesIso.hom ≫ h.π

Tactic:
by

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
h : S.LeftHomologyData
inst✝ : S.HasLeftHomology
⊢ h.π ≫ h.leftHomologyIso.inv = h.cyclesIso.inv ≫ S.leftHomologyπ

Tactic:
by

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
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
⊢ (h₁.i ≫ h₂.p) ≫ h₂.g' = 0

Tactic:
by

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
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
γ : LeftHomologyMapData φ h₁ h₂
inst✝¹ : S₁.HasLeftHomology
inst✝ : S₂.HasLeftHomology
⊢ leftHomologyMap φ = h₁.leftHomologyIso.hom ≫ γ.φH ≫ h₂.leftHomologyIso.inv

Tactic:
by

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
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
γ : LeftHomologyMapData φ h₁ h₂
inst✝¹ : S₁.HasLeftHomology
inst✝ : S₂.HasLeftHomology
⊢ cyclesMap φ = h₁.cyclesIso.hom ≫ γ.φK ≫ h₂.cyclesIso.inv

Tactic:
by

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
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
γ : LeftHomologyMapData φ h₁ h₂
inst✝¹ : S₁.HasLeftHomology
inst✝ : S₂.HasLeftHomology
⊢ leftHomologyMap φ ≫ h₂.leftHomologyIso.hom = h₁.leftHomologyIso.hom ≫ γ.φH

Tactic:
by

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
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
γ : LeftHomologyMapData φ h₁ h₂
inst✝¹ : S₁.HasLeftHomology
inst✝ : S₂.HasLeftHomology
⊢ cyclesMap φ ≫ h₂.cyclesIso.hom = h₁.cyclesIso.hom ≫ γ.φK

Tactic:
by

Goal state after:
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 : S₁.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂
⊢ i ≫ S₂.g = 0

Tactic:
by

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
⊢ ∀ (j : WalkingParallelPair),
    ((Cone.postcompose (parallelPair.ext eQ.symm (Iso.refl S.X₃) ⋯ ⋯).hom).obj (KernelFork.ofι h.ι ⋯)).π.app j =
      eH.symm.hom ≫ (KernelFork.ofι (eH.hom ≫ h.ι ≫ eQ.inv) ⋯).π.app j

Tactic:
by

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
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
⊢ h₁.f' ≫ h₂.liftH (h₁.i ≫ h₂.p) ⋯ = 0

Tactic:
by

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 : ShortComplex C
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
⊢ leftRightHomologyComparison' h₁ h₂ = h₁.descH (h₂.liftH (h₁.i ≫ h₂.p) ⋯) ⋯

Tactic:
by

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.LeftHomologyData
h₂ : S₁.RightHomologyData
h₁' : S₂.LeftHomologyData
h₂' : S₂.RightHomologyData
⊢ leftHomologyMap' φ h₁ h₁' ≫ leftRightHomologyComparison' h₁' h₂' =
    leftRightHomologyComparison' h₁ h₂ ≫ rightHomologyMap' φ h₂ h₂'

Tactic:
by

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
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
ψ₁ ψ₂ : RightHomologyMapData φ h₁ h₂
⊢ ψ₁.φQ = ψ₂.φQ

Tactic:
by

Goal state after:
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 : S₁.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂
wi : i ≫ S₂.g = 0
W'✝ : C
x : W'✝ ⟶ S₂.X₂
hx : x ≫ S₂.g = 0
⊢ (x ≫ inv φ.τ₂) ≫ S₁.g = 0

Tactic:
by

Goal state after:
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 : S₁.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂
wi : i ≫ S₂.g = 0
W'✝ : C
x : W'✝ ⟶ S₂.X₂
hx : x ≫ S₂.g = 0
⊢ h.liftK (x ≫ inv φ.τ₂) ⋯ ≫ i = x

Tactic:
by

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 : ShortComplex C
h₁ h₁' : S.LeftHomologyData
h₂ h₂' : S.RightHomologyData
⊢ leftRightHomologyComparison' h₁ h₂ =
    leftHomologyMap' (𝟙 S) h₁ h₁' ≫ leftRightHomologyComparison' h₁' h₂' ≫ rightHomologyMap' (𝟙 S) h₂' h₂

Tactic:
by

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 : ShortComplex C
h : S.HomologyData
⊢ leftRightHomologyComparison' h.left h.right = h.iso.hom

Tactic:
by

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
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
ψ₁ ψ₂ : RightHomologyMapData φ h₁ h₂
hQ : ψ₁.φQ = ψ₂.φQ
⊢ ψ₁.φH = ψ₂.φH

Tactic:
by

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
h : S.HomologyData
⊢ IsIso (leftRightHomologyComparison' h.left h.right)

Tactic:
by

Goal state after:
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 : S₁.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂
wi : i ≫ S₂.g = 0
W'✝ : C
x : W'✝ ⟶ S₂.X₂
hx : x ≫ S₂.g = 0
b : W'✝ ⟶ h.K
hb : b ≫ i = x
⊢ b = h.liftK (x ≫ inv φ.τ₂) ⋯

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasZeroMorphisms C
S S₁ S₂ S₃ S₄ : ShortComplex C
φ : S₁ ⟶ S₂
h₁✝ : S₁.HomologyData
h₂✝ : S₂.HomologyData
inst✝ : S.HasHomology
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
⊢ IsIso (leftRightHomologyComparison' h₁ h₂)

Tactic:
by

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

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

Tactic:
dsimp only [leftRightHomologyComparison]
  infer_instance

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

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

Tactic:
by

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
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
ψ₁ ψ₂ : RightHomologyMapData φ h₁ h₂
⊢ ψ₁ = ψ₂

Tactic:
by

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 : ShortComplex C
h : S.HomologyData
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
⊢ leftRightHomologyComparison' h₁ h₂ = leftHomologyMap' (𝟙 S) h₁ h.left ≫ h.iso.hom ≫ rightHomologyMap' (𝟙 S) h.right h₂

Tactic:
by

Goal state after:
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 : S₁.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂
wi : i ≫ S₂.g = 0
hi : IsLimit (KernelFork.ofι i wi)
f' : (KernelFork.ofι S₂.f ⋯).pt ⟶ (KernelFork.ofι i wi).pt := hi.lift (KernelFork.ofι S₂.f ⋯)
⊢ φ.τ₁ ≫ f' = h.f'

Tactic:
by

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
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ S₁.f ≫ φ.τ₂ ≫ h₂.p = 0

Tactic:
by

Goal state after:
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 : S₁.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂
wi : i ≫ S₂.g = 0
hi : IsLimit (KernelFork.ofι i wi)
f' : (KernelFork.ofι S₂.f ⋯).pt ⟶ (KernelFork.ofι i wi).pt := hi.lift (KernelFork.ofι S₂.f ⋯)
hf' : φ.τ₁ ≫ f' = h.f'
⊢ f' ≫ h.π = 0

Tactic:
by

Goal state after:
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
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
inst✝ : S.HasHomology
⊢ leftRightHomologyComparison' h₁ h₂ = h₁.homologyIso.inv ≫ h₂.homologyIso.hom

Tactic:
by

Goal state after:
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
⊢ S.leftRightHomologyComparison = S.leftHomologyIso.hom ≫ S.rightHomologyIso.inv

Tactic:
by

Goal state after:
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
this : h.iso = h.left.homologyIso.symm ≪≫ h.right.homologyIso
⊢ h.right.homologyIso = h.left.homologyIso ≪≫ h.iso

Tactic:
by

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
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
φQ : h₁.Q ⟶ h₂.Q := h₁.descQ (φ.τ₂ ≫ h₂.p) ⋯
⊢ φQ ≫ h₂.g' = h₁.g' ≫ φ.τ₃

Tactic:
by

Goal state after:
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 : S₁.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂
wi : i ≫ S₂.g = 0
hi : IsLimit (KernelFork.ofι i wi)
f' : (KernelFork.ofι S₂.f ⋯).pt ⟶ (KernelFork.ofι i wi).pt := hi.lift (KernelFork.ofι S₂.f ⋯)
hf' : φ.τ₁ ≫ f' = h.f'
wπ : f' ≫ h.π = 0
Z'✝ : C
x : h.K ⟶ Z'✝
hx : f' ≫ x = 0
⊢ h.f' ≫ x = 0

Tactic:
by

Goal state after:
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 : S₁.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂
wi : i ≫ S₂.g = 0
hi : IsLimit (KernelFork.ofι i wi)
f' : (KernelFork.ofι S₂.f ⋯).pt ⟶ (KernelFork.ofι i wi).pt := hi.lift (KernelFork.ofι S₂.f ⋯)
hf' : φ.τ₁ ≫ f' = h.f'
wπ : f' ≫ h.π = 0
Z'✝ : C
x : h.K ⟶ Z'✝
hx : f' ≫ x = 0
⊢ h.π ≫ h.descH x ⋯ = x

Tactic:
by

Goal state after:
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.right.homologyIso = h.left.homologyIso ≪≫ h.iso

Tactic:
by

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
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
φQ : h₁.Q ⟶ h₂.Q := h₁.descQ (φ.τ₂ ≫ h₂.p) ⋯
commg' : φQ ≫ h₂.g' = h₁.g' ≫ φ.τ₃
⊢ (h₁.ι ≫ φQ) ≫ h₂.g' = 0

Tactic:
by

Goal state after:
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.left.homologyIso = h.right.homologyIso ≪≫ h.iso.symm

Tactic:
by

Goal state after:
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 : S₁.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂
wi : i ≫ S₂.g = 0
hi : IsLimit (KernelFork.ofι i wi)
f' : (KernelFork.ofι S₂.f ⋯).pt ⟶ (KernelFork.ofι i wi).pt := hi.lift (KernelFork.ofι S₂.f ⋯)
hf' : φ.τ₁ ≫ f' = h.f'
wπ : f' ≫ h.π = 0
Z'✝ : C
x : h.K ⟶ Z'✝
hx : f' ≫ x = 0
b : h.H ⟶ Z'✝
hb : h.π ≫ b = x
⊢ b = h.descH x ⋯

Tactic:
by

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 : ShortComplex C
inst✝¹ : S.HasLeftHomology
inst✝ : S.HasRightHomology
h : IsIso S.leftRightHomologyComparison
⊢ S.HasHomology

Tactic:
by

Goal state after:
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 : S₁.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
⊢ S₂.LeftHomologyData

Tactic:
by

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
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
φQ : h₁.Q ⟶ h₂.Q := h₁.descQ (φ.τ₂ ≫ h₂.p) ⋯
commg' : φQ ≫ h₂.g' = h₁.g' ≫ φ.τ₃
φH : h₁.H ⟶ h₂.H := h₂.liftH (h₁.ι ≫ φQ) ⋯
⊢ h₁.p ≫ φQ = φ.τ₂ ≫ h₂.p

Tactic:
by

Goal state after:
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
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h : S₁.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
⊢ φ.τ₁ ≫ (ofEpiOfIsIsoOfMono φ h).f' = h.f'

Tactic:
by

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
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ h₁.homologyIso.hom ≫ leftHomologyMap' φ h₁ h₂ = homologyMap φ ≫ h₂.homologyIso.hom

Tactic:
by

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
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
φQ : h₁.Q ⟶ h₂.Q := h₁.descQ (φ.τ₂ ≫ h₂.p) ⋯
commg' : φQ ≫ h₂.g' = h₁.g' ≫ φ.τ₃
φH : h₁.H ⟶ h₂.H := h₂.liftH (h₁.ι ≫ φQ) ⋯
⊢ φH ≫ h₂.ι = h₁.ι ≫ φQ

Tactic:
by

Goal state after:
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
⊢ i ≫ S₁.g = 0

Tactic:
by

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
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ h₁.homologyIso.inv ≫ homologyMap φ = leftHomologyMap' φ h₁ h₂ ≫ h₂.homologyIso.inv

Tactic:
by

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
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ RightHomologyMapData φ h₁ h₂

Tactic:
by

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
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
⊢ S₁.leftHomologyIso.hom ≫ homologyMap φ = leftHomologyMap φ ≫ S₂.leftHomologyIso.hom

Tactic:
by

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
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
⊢ S₁.leftHomologyIso.inv ≫ leftHomologyMap φ = homologyMap φ ≫ S₂.leftHomologyIso.inv

Tactic:
by

Goal state after: