mathlib-initiative/mathlib-tactics huggingface dataset:
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 : ShortComplex C
φ φ' : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ opcyclesMap' (φ + φ') h₁ h₂ = opcyclesMap' φ h₁ h₂ + opcyclesMap' φ' h₁ h₂

Tactic:
by

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 : ShortComplex C
φ φ' : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ rightHomologyMap' (φ - φ') h₁ h₂ = rightHomologyMap' φ h₁ h₂ - rightHomologyMap' φ' h₁ h₂

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
S : ShortComplex C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroMorphisms D
γ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f
f' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯
hf' : f' = kernel.lift γ f' ⋯ ≫ kernel.ι γ
wπ : f' ≫ cokernel.π (kernel.ι γ) = 0
e : Abelian.image S.f ≅ kernel γ :=
  S.abelianImageToKernelIsKernel.conePointUniqueUpToIso (limit.isLimit (parallelPair (kernel.ι S.g ≫ cokernel.π S.f) 0))
he : e.hom ≫ kernel.ι γ = S.abelianImageToKernel
fac : f' = Abelian.factorThruImage S.f ≫ e.hom ≫ kernel.ι γ
Z'✝ : C
x : kernel S.g ⟶ Z'✝
hx : f' ≫ x = 0
b : cokernel (kernel.ι γ) ⟶ Z'✝
hb : cokernel.π (kernel.ι γ) ≫ b = x
⊢ coequalizer.π (kernel.ι γ) 0 ≫ b = coequalizer.π (kernel.ι γ) 0 ≫ cokernel.desc (kernel.ι γ) x ⋯

Tactic:
by

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 : ShortComplex C
φ φ' : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ opcyclesMap' (φ - φ') h₁ h₂ = opcyclesMap' φ h₁ h₂ - opcyclesMap' φ' h₁ h₂

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
S : ShortComplex C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroMorphisms D
⊢ S.LeftHomologyData

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
S : ShortComplex C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroMorphisms D
⊢ S.f ≫ Abelian.coimage.π S.g = 0

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
S : ShortComplex C
⊢ (kernel.ι S.g ≫ cokernel.π S.f) ≫ S.cokernelToAbelianCoimage = 0

Tactic:
by

Goal state after:
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
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hl₂ : S₂.LeftHomologyData
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : hl₁.IsPreservedBy F
inst✝ : hl₂.IsPreservedBy F
⊢ F.map (cyclesMap' φ hl₁ hl₂) = cyclesMap' (F.mapShortComplex.map φ) (hl₁.map F) (hl₂.map F)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
S : ShortComplex C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroMorphisms D
Z'✝ : C
k : cokernel S.f ⟶ Z'✝
hk : (kernel.ι S.g ≫ cokernel.π S.f) ≫ k = 0
⊢ kernel.ι S.g ≫ cokernel.π S.f ≫ k = 0

Tactic:
by

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
h : Homotopy φ₁ φ₂
⊢ φ₂.τ₁ = S₁.f ≫ (-h.h₁) + -h.h₀ + φ₁.τ₁

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
S : ShortComplex C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroMorphisms D
Z'✝ : C
k : cokernel S.f ⟶ Z'✝
hk : (kernel.ι S.g ≫ cokernel.π S.f) ≫ k = 0
⊢ S.cokernelToAbelianCoimage ≫ cokernel.desc (kernel.ι S.g) (cokernel.π S.f ≫ k) ⋯ = k

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
S : ShortComplex C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroMorphisms D
Z'✝ : C
k : cokernel S.f ⟶ Z'✝
hk : (kernel.ι S.g ≫ cokernel.π S.f) ≫ k = 0
b : Abelian.coimage S.g ⟶ Z'✝
hb : S.cokernelToAbelianCoimage ≫ b = k
⊢ b = cokernel.desc (kernel.ι S.g) (cokernel.π S.f ≫ k) ⋯

Tactic:
by

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
h : Homotopy φ₁ φ₂
⊢ φ₂.τ₂ = S₁.g ≫ (-h.h₂) + (-h.h₁) ≫ S₂.f + (S₁.g ≫ h.h₂ + h.h₁ ≫ S₂.f + φ₂.τ₂)

Tactic:
comp_neg,

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
⊢ φ₂.τ₂ = -S₁.g ≫ h.h₂ + (-h.h₁) ≫ S₂.f + (S₁.g ≫ h.h₂ + h.h₁ ≫ S₂.f + φ₂.τ₂)
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
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hl₁ : S₁.LeftHomologyData
hl₂ : S₂.LeftHomologyData
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : hl₁.IsPreservedBy F
inst✝ : hl₂.IsPreservedBy F
⊢ F.map (leftHomologyMap' φ hl₁ hl₂) = leftHomologyMap' (F.mapShortComplex.map φ) (hl₁.map F) (hl₂.map F)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
S : ShortComplex C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroMorphisms D
γ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f
g' : cokernel S.f ⟶ S.X₃ := cokernel.desc S.f S.g ⋯
⊢ γ ≫ g' = 0

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
S : ShortComplex C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroMorphisms D
γ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f
g' : cokernel S.f ⟶ S.X₃ := cokernel.desc S.f S.g ⋯
⊢ g' = cokernel.π γ ≫ cokernel.desc γ g' ⋯

Tactic:
by

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
h : Homotopy φ₁ φ₂
⊢ φ₂.τ₂ = S₁.g ≫ (-h.h₂) + (-h.h₁) ≫ S₂.f + φ₁.τ₂

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
S : ShortComplex C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroMorphisms D
γ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f
g' : cokernel S.f ⟶ S.X₃ := cokernel.desc S.f S.g ⋯
hg' : g' = cokernel.π γ ≫ cokernel.desc γ g' ⋯
⊢ kernel.ι (cokernel.π γ) ≫ g' = 0

Tactic:
by

Goal state after:
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
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hr₁ : S₁.RightHomologyData
hr₂ : S₂.RightHomologyData
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : hr₁.IsPreservedBy F
inst✝ : hr₂.IsPreservedBy F
⊢ F.map (opcyclesMap' φ hr₁ hr₂) = opcyclesMap' (F.mapShortComplex.map φ) (hr₁.map F) (hr₂.map F)

Tactic:
by

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
h : Homotopy φ₁ φ₂
⊢ φ₂.τ₃ = -h.h₃ + (-h.h₂) ≫ S₂.g + φ₁.τ₃

Tactic:
by

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
h : Homotopy φ₁ φ₂
⊢ (-φ₁).τ₁ = S₁.f ≫ (-h.h₁) + -h.h₀ + (-φ₂).τ₁

Tactic:
by

Goal state after:
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
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
hr₁ : S₁.RightHomologyData
hr₂ : S₂.RightHomologyData
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : hr₁.IsPreservedBy F
inst✝ : hr₂.IsPreservedBy F
⊢ F.map (rightHomologyMap' φ hr₁ hr₂) = rightHomologyMap' (F.mapShortComplex.map φ) (hr₁.map F) (hr₂.map F)

Tactic:
by

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
h : Homotopy φ₁ φ₂
⊢ -φ₂.τ₂ + -(S₁.g ≫ h.h₂ + h.h₁ ≫ S₂.f) = S₁.g ≫ (-h.h₂) + (-h.h₁) ≫ S₂.f + -φ₂.τ₂

Tactic:
comp_neg,

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
⊢ -φ₂.τ₂ + -(S₁.g ≫ h.h₂ + h.h₁ ≫ S₂.f) = -S₁.g ≫ h.h₂ + (-h.h₁) ≫ S₂.f + -φ₂.τ₂
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

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
S : ShortComplex C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroMorphisms D
γ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f
g' : cokernel S.f ⟶ S.X₃ := cokernel.desc S.f S.g ⋯
hg' : g' = cokernel.π γ ≫ cokernel.desc γ g' ⋯
wι : kernel.ι (cokernel.π γ) ≫ g' = 0
e : cokernel γ ≅ Abelian.coimage S.g :=
  (colimit.isColimit (parallelPair (kernel.ι S.g ≫ cokernel.π S.f) 0)).coconePointUniqueUpToIso
    S.cokernelToAbelianCoimageIsCokernel
he : cokernel.π γ ≫ e.hom = S.cokernelToAbelianCoimage
⊢ g' = cokernel.π γ ≫ e.hom ≫ Abelian.factorThruCoimage S.g

Tactic:
by

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

Goal state before:
pp: 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
j : J
⊢ (π₂.mapCocone c).ι.app j ≫ h₂.desc (π₂.mapCocone s) ≫ s.pt.g =
    (π₂.mapCocone c).ι.app j ≫ c.pt.g ≫ h₃.desc (π₃.mapCocone s)

Tactic:
by

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
h : Homotopy φ₁ φ₂
⊢ (-φ₁).τ₂ = S₁.g ≫ (-h.h₂) + (-h.h₁) ≫ S₂.f + (-φ₂).τ₂

Tactic:
by

Goal state after:
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
hl : S.LeftHomologyData
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : S.HasLeftHomology
inst✝ : F.PreservesLeftHomologyOf S
⊢ S.mapCyclesIso F = (hl.map F).cyclesIso ≪≫ F.mapIso hl.cyclesIso.symm

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
S : ShortComplex C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroMorphisms D
γ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f
g' : cokernel S.f ⟶ S.X₃ := cokernel.desc S.f S.g ⋯
hg' : g' = cokernel.π γ ≫ cokernel.desc γ g' ⋯
wι : kernel.ι (cokernel.π γ) ≫ g' = 0
e : cokernel γ ≅ Abelian.coimage S.g :=
  (colimit.isColimit (parallelPair (kernel.ι S.g ≫ cokernel.π S.f) 0)).coconePointUniqueUpToIso
    S.cokernelToAbelianCoimageIsCokernel
he : cokernel.π γ ≫ e.hom = S.cokernelToAbelianCoimage
fac : g' = cokernel.π γ ≫ e.hom ≫ Abelian.factorThruCoimage S.g
W'✝ : C
x : W'✝ ⟶ cokernel S.f
hx : x ≫ g' = 0
⊢ x ≫ cokernel.π γ = 0

Tactic:
by

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
h : Homotopy φ₁ φ₂
⊢ (-φ₁).τ₃ = -h.h₃ + (-h.h₂) ≫ S₂.g + (-φ₂).τ₃

Tactic:
by

Goal state after:
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
hl : S.LeftHomologyData
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : S.HasLeftHomology
inst✝ : F.PreservesLeftHomologyOf S
⊢ S.mapLeftHomologyIso F = (hl.map F).leftHomologyIso ≪≫ F.mapIso hl.leftHomologyIso.symm

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
S : ShortComplex C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroMorphisms D
γ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f
g' : cokernel S.f ⟶ S.X₃ := cokernel.desc S.f S.g ⋯
hg' : g' = cokernel.π γ ≫ cokernel.desc γ g' ⋯
wι : kernel.ι (cokernel.π γ) ≫ g' = 0
e : cokernel γ ≅ Abelian.coimage S.g :=
  (colimit.isColimit (parallelPair (kernel.ι S.g ≫ cokernel.π S.f) 0)).coconePointUniqueUpToIso
    S.cokernelToAbelianCoimageIsCokernel
he : cokernel.π γ ≫ e.hom = S.cokernelToAbelianCoimage
fac : g' = cokernel.π γ ≫ e.hom ≫ Abelian.factorThruCoimage S.g
W'✝ : C
x : W'✝ ⟶ cokernel S.f
hx : x ≫ g' = 0
b : W'✝ ⟶ kernel (cokernel.π γ)
hb : b ≫ kernel.ι (cokernel.π γ) = x
⊢ b ≫ equalizer.ι (cokernel.π γ) 0 = kernel.lift (cokernel.π γ) x ⋯ ≫ equalizer.ι (cokernel.π γ) 0

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
S : ShortComplex C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroMorphisms D
⊢ S.RightHomologyData

Tactic:
by

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
h₁₂ : Homotopy φ₁ φ₂
h₂₃ : Homotopy φ₂ φ₃
⊢ φ₁.τ₁ = S₁.f ≫ (h₁₂.h₁ + h₂₃.h₁) + (h₁₂.h₀ + h₂₃.h₀) + φ₃.τ₁

Tactic:
by

Goal state after:
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
hr : S.RightHomologyData
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : S.HasRightHomology
inst✝ : F.PreservesRightHomologyOf S
⊢ S.mapOpcyclesIso F = (hr.map F).opcyclesIso ≪≫ F.mapIso hr.opcyclesIso.symm

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
S : ShortComplex C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroMorphisms D
X✝ Y✝ : ShortComplex C
i : X✝ ⟶ Y✝
x✝ : Mono i
⊢ NormalMono i

Tactic:
by

Goal state after:
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
hr : S.RightHomologyData
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : S.HasRightHomology
inst✝ : F.PreservesRightHomologyOf S
⊢ S.mapRightHomologyIso F = (hr.map F).rightHomologyIso ≪≫ F.mapIso hr.rightHomologyIso.symm

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
S : ShortComplex C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroMorphisms D
X✝ Y✝ : ShortComplex C
p : X✝ ⟶ Y✝
x✝ : Epi p
⊢ NormalEpi p

Tactic:
by

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

Goal state before:
pp: 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
j : J
⊢ c.ι.app j ≫
      { τ₁ := h₁.desc (π₁.mapCocone s), τ₂ := h₂.desc (π₂.mapCocone s), τ₃ := h₃.desc (π₃.mapCocone s), comm₁₂ := ⋯,
        comm₂₃ := ⋯ } =
    s.ι.app j

Tactic:
by

Goal state after:
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
⊢ ((S.isoCyclesOfIsLimit hkf).inv ≫ π) ≫ ι ≫ (S.isoOpcyclesOfIsColimit hcc).hom = S.iCycles ≫ S.pOpcycles

Tactic:
by

Goal state after:
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
hl : S.LeftHomologyData
F : C ⥤ D
inst✝³ : F.PreservesZeroMorphisms
inst✝² : S.HasHomology
inst✝¹ : (S.map F).HasHomology
inst✝ : F.PreservesLeftHomologyOf S
⊢ S.mapHomologyIso F = (hl.map F).homologyIso ≪≫ F.mapIso hl.homologyIso.symm

Tactic:
by

Goal state after:
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
⊢ H ≅ image (S.iCycles ≫ S.pOpcycles)

Tactic:
by

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
h₁₂ : Homotopy φ₁ φ₂
h₂₃ : Homotopy φ₂ φ₃
⊢ φ₁.τ₂ = S₁.g ≫ (h₁₂.h₂ + h₂₃.h₂) + (h₁₂.h₁ + h₂₃.h₁) ≫ S₂.f + φ₃.τ₂

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
S : ShortComplex C
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
⊢ (isoImage S hkf hcc fac).hom ≫ image.ι (S.iCycles ≫ S.pOpcycles) = ι ≫ (S.isoOpcyclesOfIsColimit hcc).hom

Tactic:
by

Goal state after:
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
hr : S.RightHomologyData
F : C ⥤ D
inst✝³ : F.PreservesZeroMorphisms
inst✝² : S.HasHomology
inst✝¹ : (S.map F).HasHomology
inst✝ : F.PreservesRightHomologyOf S
⊢ S.mapHomologyIso' F = (hr.map F).homologyIso ≪≫ F.mapIso hr.homologyIso.symm

Tactic:
by

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

Goal state before:
pp: 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
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
F : C ⥤ D
inst✝⁴ : F.PreservesZeroMorphisms
inst✝³ : S₁.HasLeftHomology
inst✝² : S₂.HasLeftHomology
inst✝¹ : F.PreservesLeftHomologyOf S₁
inst✝ : F.PreservesLeftHomologyOf S₂
⊢ cyclesMap (F.mapShortComplex.map φ) ≫ (S₂.mapCyclesIso F).hom = (S₁.mapCyclesIso F).hom ≫ F.map (cyclesMap φ)

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
S : ShortComplex C
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
⊢ π ≫ (isoHomology S hkf hcc fac).hom = (S.isoCyclesOfIsLimit hkf).hom ≫ S.homologyπ

Tactic:
by

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

Goal state before:
pp: 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 =
    { τ₁ := h₁.desc (π₁.mapCocone s), τ₂ := h₂.desc (π₂.mapCocone s), τ₃ := h₃.desc (π₃.mapCocone s), comm₁₂ := ⋯,
      comm₂₃ := ⋯ }

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
S : ShortComplex C
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
⊢ (isoHomology S hkf hcc fac).inv ≫ ι = S.homologyι ≫ (S.isoOpcyclesOfIsColimit hcc).inv

Tactic:
by

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
h₁₂ : Homotopy φ₁ φ₂
h₂₃ : Homotopy φ₂ φ₃
⊢ φ₁.τ₃ = h₁₂.h₃ + h₂₃.h₃ + (h₁₂.h₂ + h₂₃.h₂) ≫ S₂.g + φ₃.τ₃

Tactic:
by

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

Goal state before:
pp: J : Type u_1
C : Type u_2
inst✝⁵ : Category.{v_1, u_1} J
inst✝⁴ : Category.{v_2, u_2} C
inst✝³ : HasZeroMorphisms C
F : J ⥤ ShortComplex C
inst✝² : HasColimit (F ⋙ π₁)
inst✝¹ : HasColimit (F ⋙ π₂)
inst✝ : HasColimit (F ⋙ π₃)
⊢ colimMap (F.whiskerLeft π₁Toπ₂) ≫ colimMap (F.whiskerLeft π₂Toπ₃) = 0

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Abelian C
S : ShortComplex C
kf : KernelFork S.g
hkf : IsLimit kf
⊢ hkf.lift (KernelFork.ofι S.f ⋯) = S.toCycles ≫ (S.isoCyclesOfIsLimit hkf).inv

Tactic:
by

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

Goal state before:
pp: J : Type u_1
C : Type u_2
inst✝⁵ : Category.{v_1, u_1} J
inst✝⁴ : Category.{v_2, u_2} C
inst✝³ : HasZeroMorphisms C
F : J ⥤ ShortComplex C
inst✝² : HasColimit (F ⋙ π₁)
inst✝¹ : HasColimit (F ⋙ π₂)
inst✝ : HasColimit (F ⋙ π₃)
j : J
⊢ colimit.ι (F ⋙ π₁) j ≫
      (((const J).obj
              { X₁ := colimit (F ⋙ π₁), X₂ := colimit (F ⋙ π₂), X₃ := colimit (F ⋙ π₃),
                f := colimMap (F.whiskerLeft π₁Toπ₂), g := colimMap (F.whiskerLeft π₂Toπ₃), zero := ⋯ }).obj
          j).f =
    (F.obj j).f ≫ colimit.ι (F ⋙ π₂) j

Tactic:
by

Goal state after: