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

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

Tactic:
·

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

Goal state before:
pp: case refine_1
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₂✝ : IsIso φ.τ₂
h₁✝ : Epi φ.τ₁
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ S₂.f ≫ inv φ.τ₂ ≫ h₁.p = 0

pp: case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₂✝ : IsIso φ.τ₂
h₁✝ : Epi φ.τ₁
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ opcyclesMap' φ h₁ h₂ ≫ h₂.descQ (inv φ.τ₂ ≫ h₁.p) ?refine_1 = 𝟙 h₁.Q

pp: case refine_3
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₂✝ : IsIso φ.τ₂
h₁✝ : Epi φ.τ₁
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ h₂.descQ (inv φ.τ₂ ≫ h₁.p) ?refine_1 ≫ opcyclesMap' φ h₁ h₂ = 𝟙 h₂.Q

Tactic:
·

Goal state after:
case refine_1
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₂✝ : IsIso φ.τ₂
h₁✝ : Epi φ.τ₁
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ S₂.f ≫ inv φ.τ₂ ≫ h₁.p = 0
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

Goal state before:
pp: case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₂✝ : IsIso φ.τ₂
h₁✝ : Epi φ.τ₁
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ opcyclesMap' φ h₁ h₂ ≫ h₂.descQ (inv φ.τ₂ ≫ h₁.p) ⋯ = 𝟙 h₁.Q

pp: case refine_3
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₂✝ : IsIso φ.τ₂
h₁✝ : Epi φ.τ₁
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ h₂.descQ (inv φ.τ₂ ≫ h₁.p) ⋯ ≫ opcyclesMap' φ h₁ h₂ = 𝟙 h₂.Q

Tactic:
·

Goal state after:
case refine_2
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₂✝ : IsIso φ.τ₂
h₁✝ : Epi φ.τ₁
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ opcyclesMap' φ h₁ h₂ ≫ h₂.descQ (inv φ.τ₂ ≫ h₁.p) ⋯ = 𝟙 h₁.Q
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

Goal state before:
pp: case refine_3
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₂✝ : IsIso φ.τ₂
h₁✝ : Epi φ.τ₁
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ h₂.descQ (inv φ.τ₂ ≫ h₁.p) ⋯ ≫ opcyclesMap' φ h₁ h₂ = 𝟙 h₂.Q

Tactic:
·

Goal state after:
case refine_3
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₂✝ : IsIso φ.τ₂
h₁✝ : Epi φ.τ₁
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
⊢ h₂.descQ (inv φ.τ₂ ≫ h₁.p) ⋯ ≫ opcyclesMap' φ h₁ h₂ = 𝟙 h₂.Q
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
hₗ : S.LeftHomologyData
hᵣ : S.RightHomologyData
inst✝¹ : hₗ.IsPreservedBy F
inst✝ : hᵣ.IsPreservedBy F
⊢ (Cofork.π (CokernelCofork.ofπ (hₗ.map F).π ⋯) ≫ F.map (leftRightHomologyComparison' hₗ hᵣ)) ≫
      Fork.ι (KernelFork.ofι (hᵣ.map F).ι ⋯) =
    F.map (hₗ.i ≫ hᵣ.p)

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
hₗ : S.LeftHomologyData
hᵣ : S.RightHomologyData
inst✝¹ : hₗ.IsPreservedBy F
inst✝ : hᵣ.IsPreservedBy F
⊢ F.map (hₗ.i ≫ hᵣ.p) =
    (Cofork.π (CokernelCofork.ofπ (hₗ.map F).π ⋯) ≫ leftRightHomologyComparison' (hₗ.map F) (hᵣ.map F)) ≫
      Fork.ι (KernelFork.ofι (hᵣ.map F).ι ⋯)

Tactic:
·

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
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
hₗ : S.LeftHomologyData
hᵣ : S.RightHomologyData
inst✝¹ : hₗ.IsPreservedBy F
inst✝ : hᵣ.IsPreservedBy F
⊢ (Cofork.π (CokernelCofork.ofπ (hₗ.map F).π ⋯) ≫ F.map (leftRightHomologyComparison' hₗ hᵣ)) ≫
      Fork.ι (KernelFork.ofι (hᵣ.map F).ι ⋯) =
    F.map (hₗ.i ≫ hᵣ.p)
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
hₗ : S.LeftHomologyData
hᵣ : S.RightHomologyData
inst✝¹ : hₗ.IsPreservedBy F
inst✝ : hᵣ.IsPreservedBy F
⊢ F.map (hₗ.i ≫ hᵣ.p) = (hₗ.map F).π ≫ leftRightHomologyComparison' (hₗ.map F) (hᵣ.map F) ≫ (hᵣ.map F).ι

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
hₗ : S.LeftHomologyData
hᵣ : S.RightHomologyData
inst✝¹ : hₗ.IsPreservedBy F
inst✝ : hᵣ.IsPreservedBy F
⊢ (hₗ.map F).π ≫ leftRightHomologyComparison' (hₗ.map F) (hᵣ.map F) ≫ (hᵣ.map F).ι =
    (Cofork.π (CokernelCofork.ofπ (hₗ.map F).π ⋯) ≫ leftRightHomologyComparison' (hₗ.map F) (hᵣ.map F)) ≫
      Fork.ι (KernelFork.ofι (hᵣ.map F).ι ⋯)

Tactic:
·

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
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
hₗ : S.LeftHomologyData
hᵣ : S.RightHomologyData
inst✝¹ : hₗ.IsPreservedBy F
inst✝ : hᵣ.IsPreservedBy F
⊢ F.map (hₗ.i ≫ hᵣ.p) = (hₗ.map F).π ≫ leftRightHomologyComparison' (hₗ.map F) (hᵣ.map F) ≫ (hᵣ.map F).ι
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
hₗ : S.LeftHomologyData
hᵣ : S.RightHomologyData
inst✝¹ : hₗ.IsPreservedBy F
inst✝ : hᵣ.IsPreservedBy F
⊢ (hₗ.map F).π ≫ leftRightHomologyComparison' (hₗ.map F) (hᵣ.map F) ≫ (hᵣ.map F).ι =
    (Cofork.π (CokernelCofork.ofπ (hₗ.map F).π ⋯) ≫ leftRightHomologyComparison' (hₗ.map F) (hᵣ.map F)) ≫
      Fork.ι (KernelFork.ofι (hᵣ.map F).ι ⋯)

Tactic:
·

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
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
hₗ : S.LeftHomologyData
hᵣ : S.RightHomologyData
inst✝¹ : hₗ.IsPreservedBy F
inst✝ : hᵣ.IsPreservedBy F
⊢ (hₗ.map F).π ≫ leftRightHomologyComparison' (hₗ.map F) (hᵣ.map F) ≫ (hᵣ.map F).ι =
    (Cofork.π (CokernelCofork.ofπ (hₗ.map F).π ⋯) ≫ leftRightHomologyComparison' (hₗ.map F) (hᵣ.map F)) ≫
      Fork.ι (KernelFork.ofι (hᵣ.map F).ι ⋯)
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 : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
γ : LeftHomologyMapData φ h₁ h₂
⊢ leftHomologyMap' (-φ) h₁ h₂ = -leftHomologyMap' φ h₁ h₂

Tactic:
simp only [γ.leftHomologyMap'_eq, γ.neg.leftHomologyMap'_eq, LeftHomologyMapData.neg_φH]

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.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ cyclesMap' (φ + φ') h₁ h₂ = cyclesMap' φ h₁ h₂ + cyclesMap' φ' h₁ h₂

Tactic:
by
  have γ : LeftHomologyMapData φ h₁ h₂ := default
  have γ' : LeftHomologyMapData φ' h₁ h₂ := default
  simp only [γ.cyclesMap'_eq, γ'.cyclesMap'_eq,
    (γ.add γ').cyclesMap'_eq, LeftHomologyMapData.add_φK]

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:
simp only [← cancel_epi (cokernel.π S.f),
        cokernel_π_comp_cokernelToAbelianCoimage_assoc, cokernel.π_desc]

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

Goal state before:
pp: case h₁
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 := ⋯ }).τ₁ =
    (s.ι.app j).τ₁

pp: case h₂
J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
inst✝ : HasZeroMorphisms C
F : J ⥤ ShortComplex C
c : Cocone F
h₁ : IsColimit (π₁.mapCocone c)
h₂ : IsColimit (π₂.mapCocone c)
h₃ : IsColimit (π₃.mapCocone c)
s : Cocone F
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 := ⋯ }).τ₂ =
    (s.ι.app j).τ₂

pp: case h₃
J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
inst✝ : HasZeroMorphisms C
F : J ⥤ ShortComplex C
c : Cocone F
h₁ : IsColimit (π₁.mapCocone c)
h₂ : IsColimit (π₂.mapCocone c)
h₃ : IsColimit (π₃.mapCocone c)
s : Cocone F
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 := ⋯ }).τ₃ =
    (s.ι.app j).τ₃

Tactic:
·

Goal state after:
case h₁
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 := ⋯ }).τ₁ =
    (s.ι.app j).τ₁
Module: Mathlib.Algebra.Homology.ShortComplex.Limits

Goal state before:
pp: case h₂
J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
inst✝ : HasZeroMorphisms C
F : J ⥤ ShortComplex C
c : Cocone F
h₁ : IsColimit (π₁.mapCocone c)
h₂ : IsColimit (π₂.mapCocone c)
h₃ : IsColimit (π₃.mapCocone c)
s : Cocone F
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 := ⋯ }).τ₂ =
    (s.ι.app j).τ₂

pp: case h₃
J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
inst✝ : HasZeroMorphisms C
F : J ⥤ ShortComplex C
c : Cocone F
h₁ : IsColimit (π₁.mapCocone c)
h₂ : IsColimit (π₂.mapCocone c)
h₃ : IsColimit (π₃.mapCocone c)
s : Cocone F
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 := ⋯ }).τ₃ =
    (s.ι.app j).τ₃

Tactic:
·

Goal state after:
case h₂
J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
inst✝ : HasZeroMorphisms C
F : J ⥤ ShortComplex C
c : Cocone F
h₁ : IsColimit (π₁.mapCocone c)
h₂ : IsColimit (π₂.mapCocone c)
h₃ : IsColimit (π₃.mapCocone c)
s : Cocone F
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 := ⋯ }).τ₂ =
    (s.ι.app j).τ₂
Module: Mathlib.Algebra.Homology.ShortComplex.Limits

Goal state before:
pp: case h₃
J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
inst✝ : HasZeroMorphisms C
F : J ⥤ ShortComplex C
c : Cocone F
h₁ : IsColimit (π₁.mapCocone c)
h₂ : IsColimit (π₂.mapCocone c)
h₃ : IsColimit (π₃.mapCocone c)
s : Cocone F
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 := ⋯ }).τ₃ =
    (s.ι.app j).τ₃

Tactic:
·

Goal state after:
case h₃
J : Type u_1
C : Type u_2
inst✝² : Category.{v_1, u_1} J
inst✝¹ : Category.{v_2, u_2} C
inst✝ : HasZeroMorphisms C
F : J ⥤ ShortComplex C
c : Cocone F
h₁ : IsColimit (π₁.mapCocone c)
h₂ : IsColimit (π₂.mapCocone c)
h₃ : IsColimit (π₃.mapCocone c)
s : Cocone F
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 := ⋯ }).τ₃ =
    (s.ι.app j).τ₃
Module: Mathlib.Algebra.Homology.ShortComplex.Limits

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

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

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

Tactic:
·

Goal state after:
case h₁
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 =
    { τ₁ := h₁.desc (π₁.mapCocone s), τ₂ := h₂.desc (π₂.mapCocone s), τ₃ := h₃.desc (π₃.mapCocone s), comm₁₂ := ⋯,
        comm₂₃ := ⋯ }.τ₁
Module: Mathlib.Algebra.Homology.ShortComplex.Limits

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

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

Tactic:
·

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

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

Tactic:
·

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

Goal state before:
pp: case h₂
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)
x✝¹ x✝ : J
f : x✝¹ ⟶ x✝
⊢ (F.map f ≫
        { τ₁ := colimit.ι (F ⋙ π₁) x✝, τ₂ := colimit.ι (F ⋙ π₂) x✝, τ₃ := colimit.ι (F ⋙ π₃) x✝, comm₁₂ := ⋯,
          comm₂₃ := ⋯ }).τ₂ =
    ({ τ₁ := colimit.ι (F ⋙ π₁) x✝¹, τ₂ := colimit.ι (F ⋙ π₂) x✝¹, τ₃ := colimit.ι (F ⋙ π₃) x✝¹, comm₁₂ := ⋯,
          comm₂₃ := ⋯ } ≫
        ((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 := ⋯ }).map
          f).τ₂

pp: case h₃
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)
x✝¹ x✝ : J
f : x✝¹ ⟶ x✝
⊢ (F.map f ≫
        { τ₁ := colimit.ι (F ⋙ π₁) x✝, τ₂ := colimit.ι (F ⋙ π₂) x✝, τ₃ := colimit.ι (F ⋙ π₃) x✝, comm₁₂ := ⋯,
          comm₂₃ := ⋯ }).τ₃ =
    ({ τ₁ := colimit.ι (F ⋙ π₁) x✝¹, τ₂ := colimit.ι (F ⋙ π₂) x✝¹, τ₃ := colimit.ι (F ⋙ π₃) x✝¹, comm₁₂ := ⋯,
          comm₂₃ := ⋯ } ≫
        ((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 := ⋯ }).map
          f).τ₃

Tactic:
·

Goal state after:
case h₂
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)
x✝¹ x✝ : J
f : x✝¹ ⟶ x✝
⊢ (F.map f ≫
        { τ₁ := colimit.ι (F ⋙ π₁) x✝, τ₂ := colimit.ι (F ⋙ π₂) x✝, τ₃ := colimit.ι (F ⋙ π₃) x✝, comm₁₂ := ⋯,
          comm₂₃ := ⋯ }).τ₂ =
    ({ τ₁ := colimit.ι (F ⋙ π₁) x✝¹, τ₂ := colimit.ι (F ⋙ π₂) x✝¹, τ₃ := colimit.ι (F ⋙ π₃) x✝¹, comm₁₂ := ⋯,
          comm₂₃ := ⋯ } ≫
        ((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 := ⋯ }).map
          f).τ₂
Module: Mathlib.Algebra.Homology.ShortComplex.Limits

Goal state before:
pp: case h₃
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)
x✝¹ x✝ : J
f : x✝¹ ⟶ x✝
⊢ (F.map f ≫
        { τ₁ := colimit.ι (F ⋙ π₁) x✝, τ₂ := colimit.ι (F ⋙ π₂) x✝, τ₃ := colimit.ι (F ⋙ π₃) x✝, comm₁₂ := ⋯,
          comm₂₃ := ⋯ }).τ₃ =
    ({ τ₁ := colimit.ι (F ⋙ π₁) x✝¹, τ₂ := colimit.ι (F ⋙ π₂) x✝¹, τ₃ := colimit.ι (F ⋙ π₃) x✝¹, comm₁₂ := ⋯,
          comm₂₃ := ⋯ } ≫
        ((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 := ⋯ }).map
          f).τ₃

Tactic:
·

Goal state after:
case h₃
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)
x✝¹ x✝ : J
f : x✝¹ ⟶ x✝
⊢ (F.map f ≫
        { τ₁ := colimit.ι (F ⋙ π₁) x✝, τ₂ := colimit.ι (F ⋙ π₂) x✝, τ₃ := colimit.ι (F ⋙ π₃) x✝, comm₁₂ := ⋯,
          comm₂₃ := ⋯ }).τ₃ =
    ({ τ₁ := colimit.ι (F ⋙ π₁) x✝¹, τ₂ := colimit.ι (F ⋙ π₂) x✝¹, τ₃ := colimit.ι (F ⋙ π₃) x✝¹, comm₁₂ := ⋯,
          comm₂₃ := ⋯ } ≫
        ((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 := ⋯ }).map
          f).τ₃
Module: Mathlib.Algebra.Homology.ShortComplex.Abelian

Goal state before:
pp: case refine_1
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
⊢ parallelPair ((hkf.ofIsoLimit (Fork.ext (Iso.refl kf.pt) ⋯)).lift (KernelFork.ofι S.f ⋯)) 0 ≅
    parallelPair S.toCycles 0

pp: case refine_2
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
⊢ (Cocone.precompose (Iso.inv ?refine_1)).obj (CokernelCofork.ofπ π ⋯) ≅ CokernelCofork.ofπ S.homologyπ ⋯

Tactic:
·

Goal state after:
case refine_1
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
⊢ parallelPair ((hkf.ofIsoLimit (Fork.ext (Iso.refl kf.pt) ⋯)).lift (KernelFork.ofι S.f ⋯)) 0 ≅
    parallelPair S.toCycles 0