mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
h : S.LeftHomologyData
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : h.IsPreservedBy F
this✝ : PreservesLimit (parallelPair S.g 0) F
this : PreservesColimit (parallelPair h.f' 0) F
wi : F.map h.i ≫ F.map S.g = 0
hi : IsLimit ((KernelFork.ofι h.i ⋯).map F)
f' : F.obj S.X₁ ⟶ F.obj h.K := hi.lift (KernelFork.ofι (S.map F).f ⋯)
hf' : f' = F.map h.f'
⊢ (S.map F).LeftHomologyData

Tactic:
have wπ : f' ≫ F.map h.π = 0 := by rw [hf', ← F.map_comp, f'_π, F.map_zero]

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
h : S.LeftHomologyData
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : h.IsPreservedBy F
this✝ : PreservesLimit (parallelPair S.g 0) F
this : PreservesColimit (parallelPair h.f' 0) F
wi : F.map h.i ≫ F.map S.g = 0
hi : IsLimit ((KernelFork.ofι h.i ⋯).map F)
f' : F.obj S.X₁ ⟶ F.obj h.K := hi.lift (KernelFork.ofι (S.map F).f ⋯)
hf' : f' = F.map h.f'
wπ : f' ≫ F.map h.π = 0
⊢ (S.map F).LeftHomologyData
Module: Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
h : S.LeftHomologyData
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : h.IsPreservedBy F
this✝ : PreservesLimit (parallelPair S.g 0) F
this : PreservesColimit (parallelPair h.f' 0) F
wi : F.map h.i ≫ F.map S.g = 0
hi : IsLimit ((KernelFork.ofι h.i ⋯).map F)
f' : F.obj S.X₁ ⟶ F.obj h.K := hi.lift (KernelFork.ofι (S.map F).f ⋯)
hf' : f' = F.map h.f'
wπ : f' ≫ F.map h.π = 0
⊢ (parallelPair f' 0).map WalkingParallelPairHom.left ≫
      (Iso.refl ((parallelPair f' 0).obj WalkingParallelPair.one)).hom =
    (Iso.refl ((parallelPair f' 0).obj WalkingParallelPair.zero)).hom ≫
      (parallelPair (F.map h.f') 0).map WalkingParallelPairHom.left

Tactic:
simpa using hf'

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

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.RightHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
this✝ : Epi (opMap φ).τ₁
this : IsIso (opMap φ).τ₂
⊢ Mono (opMap φ).τ₃

Tactic:
dsimp

Goal state after:
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.RightHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
this✝ : Epi (opMap φ).τ₁
this : IsIso (opMap φ).τ₂
⊢ Mono φ.τ₁.op
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

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.RightHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
⊢ Epi (opMap φ).τ₁

Tactic:
dsimp

Goal state after:
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.RightHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
⊢ Epi φ.τ₃.op
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

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.RightHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
this : Epi (opMap φ).τ₁
⊢ IsIso (opMap φ).τ₂

Tactic:
dsimp

Goal state after:
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.RightHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
this : Epi (opMap φ).τ₁
⊢ IsIso φ.τ₂.op
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

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.RightHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
this✝ : Epi (opMap φ).τ₁
this : IsIso (opMap φ).τ₂
⊢ Mono (opMap φ).τ₃

Tactic:
dsimp

Goal state after:
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.RightHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
this✝ : Epi (opMap φ).τ₁
this : IsIso (opMap φ).τ₂
⊢ Mono φ.τ₁.op
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

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
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
h₂' : S₂.RightHomologyData := RightHomologyData.ofEpiOfIsIsoOfMono φ h₁
⊢ IsIso (RightHomologyMapData.ofEpiOfIsIsoOfMono φ h₁).φH

Tactic:
dsimp

Goal state after:
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
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
h₂' : S₂.RightHomologyData := RightHomologyData.ofEpiOfIsIsoOfMono φ h₁
⊢ IsIso (𝟙 h₁.H)
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
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : S.HasLeftHomology
inst✝ : F.PreservesLeftHomologyOf S
⊢ (F.mapShortComplex.obj S).HasLeftHomology

Tactic:
dsimp

Goal state after:
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
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : S.HasLeftHomology
inst✝ : F.PreservesLeftHomologyOf S
⊢ (S.map F).HasLeftHomology
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
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : S.HasRightHomology
inst✝ : F.PreservesRightHomologyOf S
⊢ (F.mapShortComplex.obj S).HasRightHomology

Tactic:
dsimp

Goal state after:
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
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : S.HasRightHomology
inst✝ : F.PreservesRightHomologyOf S
⊢ (S.map F).HasRightHomology
Module: Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : HasZeroMorphisms C
inst✝⁴ : HasZeroMorphisms D
S : ShortComplex C
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝³ : F.PreservesZeroMorphisms
inst✝² : S.HasHomology
inst✝¹ : F.PreservesLeftHomologyOf S
inst✝ : F.PreservesRightHomologyOf S
⊢ (F.mapShortComplex.obj S).HasHomology

Tactic:
dsimp

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁷ : Category.{v_1, u_1} C
inst✝⁶ : Category.{v_2, u_2} D
inst✝⁵ : HasZeroMorphisms C
inst✝⁴ : HasZeroMorphisms D
S : ShortComplex C
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝³ : F.PreservesZeroMorphisms
inst✝² : S.HasHomology
inst✝¹ : F.PreservesLeftHomologyOf S
inst✝ : F.PreservesRightHomologyOf S
⊢ (S.map F).HasHomology
Module: Mathlib.Algebra.Homology.ShortComplex.Abelian

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Abelian C
S : ShortComplex C
D : Type u'
inst✝³ : Category.{v', u'} D
inst✝² : HasZeroMorphisms D
kf : KernelFork S.g
cc : CokernelCofork S.f
hkf : IsLimit kf
hcc : IsColimit cc
H : C
π : kf.pt ⟶ H
ι : H ⟶ cc.pt
fac : Fork.ι kf ≫ Cofork.π cc = π ≫ ι
inst✝¹ : Epi π
inst✝ : Mono ι
⊢ (hkf.ofIsoLimit (Fork.ext (Iso.refl kf.pt) ⋯)).lift (KernelFork.ofι S.f ⋯) ≫ π = 0

Tactic:
dsimp

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Abelian C
S : ShortComplex C
D : Type u'
inst✝³ : Category.{v', u'} D
inst✝² : HasZeroMorphisms D
kf : KernelFork S.g
cc : CokernelCofork S.f
hkf : IsLimit kf
hcc : IsColimit cc
H : C
π : kf.pt ⟶ H
ι : H ⟶ cc.pt
fac : Fork.ι kf ≫ Cofork.π cc = π ≫ ι
inst✝¹ : Epi π
inst✝ : Mono ι
⊢ (hkf.lift (KernelFork.ofι S.f ⋯) ≫ 𝟙 kf.pt) ≫ π = 0
Module: Mathlib.Algebra.Homology.ShortComplex.Abelian

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Abelian C
S : ShortComplex C
D : Type u'
inst✝³ : Category.{v', u'} D
inst✝² : HasZeroMorphisms D
kf : KernelFork S.g
cc : CokernelCofork S.f
hkf : IsLimit kf
hcc : IsColimit cc
H : C
π : kf.pt ⟶ H
ι : H ⟶ cc.pt
fac : Fork.ι kf ≫ Cofork.π cc = π ≫ ι
inst✝¹ : Epi π
inst✝ : Mono ι
⊢ ι ≫ (hcc.ofIsoColimit (Cofork.ext (Iso.refl cc.pt) ⋯)).desc (CokernelCofork.ofπ S.g ⋯) = 0

Tactic:
dsimp

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : Abelian C
S : ShortComplex C
D : Type u'
inst✝³ : Category.{v', u'} D
inst✝² : HasZeroMorphisms D
kf : KernelFork S.g
cc : CokernelCofork S.f
hkf : IsLimit kf
hcc : IsColimit cc
H : C
π : kf.pt ⟶ H
ι : H ⟶ cc.pt
fac : Fork.ι kf ≫ Cofork.π cc = π ≫ ι
inst✝¹ : Epi π
inst✝ : Mono ι
⊢ ι ≫ 𝟙 cc.pt ≫ hcc.desc (CokernelCofork.ofπ S.g ⋯) = 0
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
⊢ (opMap φ₁).τ₁.unop = (S₂.op.f ≫ h.h₂.op + h.h₃.op + (opMap φ₂).τ₁).unop

Tactic:
dsimp

Goal state after:
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
⊢ φ₁.τ₃ = h.h₂ ≫ S₂.g + h.h₃ + φ₂.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
⊢ (opMap φ₁).τ₂.unop = (S₂.op.g ≫ h.h₁.op + h.h₂.op ≫ S₁.op.f + (opMap φ₂).τ₂).unop

Tactic:
dsimp

Goal state after:
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
⊢ φ₁.τ₂ = h.h₁ ≫ S₂.f + S₁.g ≫ h.h₂ + φ₂.τ₂
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
⊢ (opMap φ₁).τ₃.unop = (h.h₀.op + h.h₁.op ≫ S₁.op.g + (opMap φ₂).τ₃).unop

Tactic:
dsimp

Goal state after:
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
⊢ φ₁.τ₁ = h.h₀ + S₁.f ≫ h.h₁ + φ₂.τ₁
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✝
S₁ S₂ : ShortComplex Cᵒᵖ
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
⊢ (unopMap φ₁).τ₁.op = (S₂.unop.f ≫ h.h₂.unop + h.h₃.unop + (unopMap φ₂).τ₁).op

Tactic:
dsimp

Goal state after:
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✝
S₁ S₂ : ShortComplex Cᵒᵖ
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
⊢ φ₁.τ₃ = h.h₂ ≫ S₂.g + h.h₃ + φ₂.τ₃
Module: Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝¹¹ : Category.{v_1, u_1} C
inst✝¹⁰ : Category.{v_2, u_2} D
inst✝⁹ : HasZeroMorphisms C
inst✝⁸ : HasZeroMorphisms D
S : ShortComplex C
h₁ : S.LeftHomologyData
h₂ : S.RightHomologyData
F : C ⥤ D
inst✝⁷ : F.PreservesZeroMorphisms
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hr₁ : S₁.RightHomologyData
hl₂ : S₂.LeftHomologyData
hr₂ : S₂.RightHomologyData
ψl : LeftHomologyMapData φ hl₁ hl₂
ψr : RightHomologyMapData φ hr₁ hr₂
inst✝⁶ : S₁.HasHomology
inst✝⁵ : S₂.HasHomology
inst✝⁴ : (F.mapShortComplex.obj S₁).HasHomology
inst✝³ : (F.mapShortComplex.obj S₂).HasHomology
inst✝² : F.PreservesRightHomologyOf S₁
inst✝¹ : F.PreservesRightHomologyOf S₂
inst✝ : QuasiIso φ
⊢ QuasiIso (F.mapShortComplex.map φ)

Tactic:
by
  have γ : RightHomologyMapData φ S₁.rightHomologyData S₂.rightHomologyData := default
  have : IsIso γ.φH := by
    rw [← γ.quasiIso_iff]
    infer_instance
  rw [(γ.map F).quasiIso_iff, RightHomologyMapData.map_φH]
  infer_instance

Goal state after:
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✝
S₁ S₂ : ShortComplex Cᵒᵖ
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
⊢ (unopMap φ₁).τ₂.op = (S₂.unop.g ≫ h.h₁.unop + h.h₂.unop ≫ S₁.unop.f + (unopMap φ₂).τ₂).op

Tactic:
dsimp

Goal state after:
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✝
S₁ S₂ : ShortComplex Cᵒᵖ
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
⊢ φ₁.τ₂ = h.h₁ ≫ S₂.f + S₁.g ≫ h.h₂ + φ₂.τ₂
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✝
S₁ S₂ : ShortComplex Cᵒᵖ
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
⊢ (unopMap φ₁).τ₃.op = (h.h₀.unop + h.h₁.unop ≫ S₁.unop.g + (unopMap φ₂).τ₃).op

Tactic:
dsimp

Goal state after:
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✝
S₁ S₂ : ShortComplex Cᵒᵖ
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
⊢ φ₁.τ₁ = h.h₀ + S₁.f ≫ h.h₁ + φ₂.τ₁
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: case h₁
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
⊢ φ₁.τ₁ = (φ₂ + S₁.nullHomotopic S₂ h.h₀ ⋯ h.h₁ h.h₂ h.h₃ ⋯).τ₁

Tactic:
dsimp

Goal state after:
case h₁
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
⊢ φ₁.τ₁ = φ₂.τ₁ + (h.h₀ + S₁.f ≫ h.h₁)
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: case h₂
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
⊢ φ₁.τ₂ = (φ₂ + S₁.nullHomotopic S₂ h.h₀ ⋯ h.h₁ h.h₂ h.h₃ ⋯).τ₂

Tactic:
dsimp

Goal state after:
case h₂
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
⊢ φ₁.τ₂ = φ₂.τ₂ + (h.h₁ ≫ S₂.f + S₁.g ≫ h.h₂)
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: case h₃
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
⊢ φ₁.τ₃ = (φ₂ + S₁.nullHomotopic S₂ h.h₀ ⋯ h.h₁ h.h₂ h.h₃ ⋯).τ₃

Tactic:
dsimp

Goal state after:
case h₃
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ : ShortComplex C
φ₁ φ₂ : S₁ ⟶ S₂
h : Homotopy φ₁ φ₂
⊢ φ₁.τ₃ = φ₂.τ₃ + (h.h₂ ≫ S₂.g + h.h₃)
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
h₀ : S₁.X₁ ⟶ S₂.X₁
h₀_f : h₀ ≫ S₂.f = 0
h₁ : S₁.X₂ ⟶ S₂.X₁
h₂ : S₁.X₃ ⟶ S₂.X₂
h₃ : S₁.X₃ ⟶ S₂.X₃
g_h₃ : S₁.g ≫ h₃ = 0
⊢ (S₁.nullHomotopic S₂ h₀ h₀_f h₁ h₂ h₃ g_h₃).τ₂ = S₁.g ≫ h₂ + h₁ ≫ S₂.f + Hom.τ₂ 0

Tactic:
rw [nullHomotopic_τ₂, zero_τ₂, add_zero]; abel

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex C
h : S.ShortExact
this : Epi S.g
⊢ Mono S.op.f

Tactic:
dsimp

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex C
h : S.ShortExact
this : Epi S.g
⊢ Mono S.g.op
Module: Mathlib.Algebra.Homology.ShortComplex.ShortExact

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex C
h : S.ShortExact
this : Mono S.f
⊢ Epi S.op.g

Tactic:
dsimp

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex C
h : S.ShortExact
this : Mono S.f
⊢ Epi S.f.op
Module: Mathlib.Algebra.Homology.ShortComplex.ShortExact

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex Cᵒᵖ
h : S.ShortExact
this : Epi S.g
⊢ Mono S.unop.f

Tactic:
dsimp

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex Cᵒᵖ
h : S.ShortExact
this : Epi S.g
⊢ Mono S.g.unop
Module: Mathlib.Algebra.Homology.ShortComplex.ShortExact

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex Cᵒᵖ
h : S.ShortExact
this : Mono S.f
⊢ Epi S.unop.g

Tactic:
dsimp

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S : ShortComplex Cᵒᵖ
h : S.ShortExact
this : Mono S.f
⊢ Epi S.f.unop
Module: Mathlib.Algebra.Homology.ShortComplex.ShortExact

Goal state before:
pp: case mp
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
⊢ IsIso S.g → IsZero S.X₁

Tactic:
intro hf
    simp only [IsZero.iff_id_eq_zero, ← cancel_mono S.f, ← cancel_mono S.g,
      S.zero, zero_comp, assoc, comp_zero]

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

Goal state before:
pp: S : ShortComplex Ab
x : ↑S.X₁
⊢ (AddCommGrpCat.Hom.hom
        ((AddCommGrpCat.kernelIsLimit S.g).lift (Limits.KernelFork.ofι S.f ⋯) ≫
          AddCommGrpCat.ofHom (QuotientAddGroup.mk' S.abToCycles.range)))
      x =
    (AddCommGrpCat.Hom.hom 0) x

Tactic:
dsimp

Goal state after:
S : ShortComplex Ab
x : ↑S.X₁
⊢ ↑((AddCommGrpCat.Hom.hom ((AddCommGrpCat.kernelIsLimit S.g).lift (Limits.KernelFork.ofι S.f ⋯))) x) = 0
Module: Mathlib.CategoryTheory.Adjunction.Reflective

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
j : C ⥤ D
inst✝¹ : Coreflective j
A : D
inst✝ : IsSplitEpi ((coreflectorAdjunction j).counit.app A)
ε : coreflector j ⋙ j ⟶ 𝟭 D := (coreflectorAdjunction j).counit
this : IsIso (ε.app (j.obj ((coreflector j).obj A)))
⊢ Mono (ε.app A ≫ section_ (ε.app A))

Tactic:
show ε.app A ≫ section_ _ = _ from (ε.naturality (section_ (ε.app A))).symm

Goal state after:
C : Type u₁
D : Type u₂
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
j : C ⥤ D
inst✝¹ : Coreflective j
A : D
inst✝ : IsSplitEpi ((coreflectorAdjunction j).counit.app A)
ε : coreflector j ⋙ j ⟶ 𝟭 D := (coreflectorAdjunction j).counit
this : IsIso (ε.app (j.obj ((coreflector j).obj A)))
⊢ Mono ((coreflector j ⋙ j).map (section_ (ε.app A)) ≫ ε.app ((coreflector j ⋙ j).obj A))
Module: Mathlib.CategoryTheory.Adjunction.Reflective

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝³ : Category.{v₁, u₁} C
inst✝² : Category.{v₂, u₂} D
j : C ⥤ D
inst✝¹ : Coreflective j
A : D
inst✝ : IsSplitEpi ((coreflectorAdjunction j).counit.app A)
⊢ j.essImage A

Tactic:
let ε : coreflector j ⋙ j ⟶ 𝟭 D := (coreflectorAdjunction j).counit
  have : IsIso (ε.app (j.obj ((coreflector j).obj A))) :=
    Functor.essImage.counit_isIso ((j.obj_mem_essImage _))
  have : Mono (ε.app A) := by
    refine @mono_of_mono _ _ _ _ _ (ε.app A) (section_ (ε.app A)) ?_
    rw [show ε.app A ≫ section_ _ = _ from (ε.naturality (section_ (ε.app A))).symm]
    apply mono_comp _ (ε.app (j.obj ((coreflector j).obj A)))
  have := isIso_of_mono_of_isSplitEpi (ε.app A)
  exact (coreflectorAdjunction j).mem_essImage_of_counit_isIso A

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✝ : Preadditive C
S : ShortComplex C
hS : S.Exact
h : S.LeftHomologyData
this : S.HasHomology
⊢ IsZero (CokernelCofork.ofπ h.π ⋯).pt

Tactic:
dsimp

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S : ShortComplex C
hS : S.Exact
h : S.LeftHomologyData
this : S.HasHomology
⊢ IsZero h.H
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S : ShortComplex C
hS : S.Exact
h : S.RightHomologyData
this : S.HasHomology
⊢ IsZero (KernelFork.ofι h.ι ⋯).pt

Tactic:
dsimp

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S : ShortComplex C
hS : S.Exact
h : S.RightHomologyData
this : S.HasHomology
⊢ IsZero h.H
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
S : ShortComplex C
inst✝ : HasZeroObject C
s : S.Splitting
⊢ S.RightHomologyData

Tactic:
have hp := CokernelCofork.IsColimit.ofπ S.g S.zero
    (fun x _ => s.s ≫ x)
    (fun x hx => by simp only [s.g_s_assoc, sub_comp, id_comp, sub_eq_self, assoc, hx, comp_zero])
    (fun x _ b hb => by simp only [← hb, s.s_g_assoc])

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
S : ShortComplex C
inst✝ : HasZeroObject C
s : S.Splitting
hp : IsColimit (CokernelCofork.ofπ S.g ⋯)
⊢ S.RightHomologyData
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
S : ShortComplex C
inst✝ : HasZeroObject C
s : S.Splitting
hp : IsColimit (CokernelCofork.ofπ S.g ⋯)
g' : (CokernelCofork.ofπ S.g ⋯).pt ⟶ (CokernelCofork.ofπ S.g ⋯).pt := hp.desc (CokernelCofork.ofπ S.g ⋯)
hg' : g' = 𝟙 (CokernelCofork.ofπ S.g ⋯).pt
wι : 0 ≫ g' = 0
⊢ Mono (𝟙 (CokernelCofork.ofπ S.g ⋯).pt)

Tactic:
dsimp

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_2, u_2} D
inst✝² : Preadditive C
inst✝¹ : Preadditive D
S : ShortComplex C
inst✝ : HasZeroObject C
s : S.Splitting
hp : IsColimit (CokernelCofork.ofπ S.g ⋯)
g' : (CokernelCofork.ofπ S.g ⋯).pt ⟶ (CokernelCofork.ofπ S.g ⋯).pt := hp.desc (CokernelCofork.ofπ S.g ⋯)
hg' : g' = 𝟙 (CokernelCofork.ofπ S.g ⋯).pt
wι : 0 ≫ g' = 0
⊢ Mono (𝟙 S.X₃)
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : Balanced C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₂ : S₂.Exact
inst✝³ : Epi S₁.g
inst✝² : Epi S₂.g
inst✝¹ : Epi φ.τ₁
inst✝ : Epi φ.τ₃
⊢ Mono S₁.op.f

Tactic:
dsimp

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : Balanced C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₂ : S₂.Exact
inst✝³ : Epi S₁.g
inst✝² : Epi S₂.g
inst✝¹ : Epi φ.τ₁
inst✝ : Epi φ.τ₃
⊢ Mono S₁.g.op
Module: Mathlib.Algebra.Homology.ShortComplex.Exact

Goal state before:
pp: C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : Balanced C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₂ : S₂.Exact
inst✝³ : Epi S₁.g
inst✝² : Epi S₂.g
inst✝¹ : Epi φ.τ₁
inst✝ : Epi φ.τ₃
this : Mono S₁.op.f
⊢ Mono S₂.op.f

Tactic:
dsimp

Goal state after:
C : Type u_1
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Preadditive C
inst✝⁴ : Balanced C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₂ : S₂.Exact
inst✝³ : Epi S₁.g
inst✝² : Epi S₂.g
inst✝¹ : Epi φ.τ₁
inst✝ : Epi φ.τ₃
this : Mono S₁.op.f
⊢ Mono S₂.g.op