mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝¹ : IsIso φ'.hom₁
this✝ : IsIso φ'.hom₂
this : IsIso φ'.hom₃
⊢ IsIso φ'

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

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 : Preadditive C
inst✝² : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
n : ℤ
⊢ n + 1 + 0 = n + 1

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝¹ : IsIso φ'.hom₁
this✝ : IsIso φ'.hom₂
this : IsIso φ'.hom₃
⊢ productTriangle T ∈ distinguishedTriangles

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

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 : Preadditive C
inst✝² : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
⊢ Cochain.ofHom
      (desc (inr φ) 0 (triangle φ).mor₃ ⋯ ≫
        lift (inr φ) (-(Cocycle.ofHom φ).leftShift 1 1 ⋯) (-(inl φ).leftShift 1 0 ⋯) ⋯) =
    δ (-1) 0 (-(snd (inr φ)).comp ((snd φ).comp (inl (inr φ)) ⋯) ⋯) + Cochain.ofHom (𝟙 (mappingCone (inr φ)))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

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 : Preadditive C
inst✝² : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
p : ℤ
⊢ p + 1 + -1 = p

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

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 : Preadditive C
inst✝² : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
p : ℤ
⊢ p + 1 + 0 = p + 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

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 : Preadditive C
inst✝² : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
⊢ Cochain.ofHom ((triangle φ).mor₃ ≫ (rotateHomotopyEquiv φ).hom) =
    δ (-1) 0 (-(snd φ).comp (inl (inr φ)) ⋯) + Cochain.ofHom (inr (inr φ))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
K L : CochainComplex C ℤ
φ : K ⟶ L
⊢ (HomotopyCategory.quotient C (ComplexShape.up ℤ)).map (triangle φ).mor₃ ≫
      (HomotopyCategory.quotient C (ComplexShape.up ℤ)).map (rotateHomotopyEquiv φ).hom =
    (HomotopyCategory.quotient C (ComplexShape.up ℤ)).map (inr (inr φ))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
K L : CochainComplex C ℤ
φ : K ⟶ L
p : ℤ
⊢ p + 1 + 0 = p + 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
K L : CochainComplex C ℤ
φ : K ⟶ L
⊢ (rotateHomotopyEquiv φ).hom ≫ (triangle (inr φ)).mor₃ = -(CategoryTheory.shiftFunctor (CochainComplex C ℤ) 1).map φ

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

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 : Preadditive C
inst✝² : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
⊢ (triangleh φ).rotate.mor₁ ≫ (Iso.refl (triangleh φ).rotate.obj₂).hom =
    (Iso.refl (triangleh φ).rotate.obj₁).hom ≫ (triangleh (inr φ)).mor₁

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

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 : Preadditive C
inst✝² : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
⊢ (triangleh φ).rotate.mor₂ ≫
      ((Functor.commShiftIso (HomotopyCategory.quotient C (ComplexShape.up ℤ)) 1).symm.app K ≪≫
          HomotopyCategory.isoOfHomotopyEquiv (rotateHomotopyEquiv φ)).hom =
    (Iso.refl (triangleh φ).rotate.obj₂).hom ≫ (triangleh (inr φ)).mor₂

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝¹ : IsIso φ'.hom₁
this✝ : IsIso φ'.hom₂
A : C
this : Mono φ'.hom₃
⊢ Function.Injective fun x ↦ x ≫ φ'.hom₃

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

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 : Preadditive C
inst✝² : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
⊢ (triangleh φ).rotate.mor₃ ≫
      (CategoryTheory.shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) 1).map
        (Iso.refl (triangleh φ).rotate.obj₁).hom =
    ((Functor.commShiftIso (HomotopyCategory.quotient C (ComplexShape.up ℤ)) 1).symm.app K ≪≫
          HomotopyCategory.isoOfHomotopyEquiv (rotateHomotopyEquiv φ)).hom ≫
      (triangleh (inr φ)).mor₃

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

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 : Preadditive C
inst✝² : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
n : ℤ
⊢ δ 0 1 ((snd φ).shift n) +
      (↑(n.negOnePow • (fst φ).shift n)).comp
        (Cochain.ofHom ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).map φ)) ⋯ =
    0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

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 : Preadditive C
inst✝² : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
n : ℤ
⊢ δ (-1) 0 (n.negOnePow • (inl φ).shift n) =
    Cochain.ofHom
      ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).map φ ≫
        (CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).map (inr φ))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

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 : Preadditive C
inst✝² : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
n : ℤ
⊢ lift ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).map φ) (n.negOnePow • (fst φ).shift n) ((snd φ).shift n)
        ⋯ ≫
      desc ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).map φ) (n.negOnePow • (inl φ).shift n)
        ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).map (inr φ)) ⋯ =
    𝟙 ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj (mappingCone φ))

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

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 : Preadditive C
inst✝² : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
n : ℤ
⊢ desc ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).map φ) (n.negOnePow • (inl φ).shift n)
        ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).map (inr φ)) ⋯ ≫
      lift ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).map φ) (n.negOnePow • (fst φ).shift n)
        ((snd φ).shift n) ⋯ =
    𝟙 (mappingCone ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).map φ))

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A✝ A : C
f : A ⟶ T'.obj₃
hf : f ≫ φ'.hom₃ = 0
⊢ f ≫ T'.mor₃ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

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 : Preadditive C
inst✝² : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
n p : ℤ
⊢ p + n + 1 = p + 1 + n

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

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 : Preadditive C
inst✝² : HasBinaryBiproducts C
inst✝¹ : Preadditive D
inst✝ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
n : ℤ
⊢ (Triangle.shiftFunctor (CochainComplex C ℤ) n).obj (triangle φ) ≅
    triangle ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).map φ)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A✝ A : C
f : A ⟶ T'.obj₃
hf : f ≫ φ'.hom₃ = 0
hf' : f ≫ T'.mor₃ = 0
g : A ⟶ T'.obj₂
hg : f = g ≫ T'.mor₂
j : J
⊢ g ≫ T'.mor₂ ≫ φ'.hom₃ ≫ Pi.π (fun j ↦ (T j).obj₃) j = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

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 : Preadditive C
inst✝³ : HasBinaryBiproducts C
inst✝² : Preadditive D
inst✝¹ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
G : C ⥤ D
inst✝ : G.Additive
n : ℤ
⊢ 0 + 1 = 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

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 : Preadditive C
inst✝³ : HasBinaryBiproducts C
inst✝² : Preadditive D
inst✝¹ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
G : C ⥤ D
inst✝ : G.Additive
n : ℤ
⊢ n + 0 = n

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

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 : Preadditive C
inst✝³ : HasBinaryBiproducts C
inst✝² : Preadditive D
inst✝¹ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
G : C ⥤ D
inst✝ : G.Additive
n : ℤ
⊢ n + 1 = n + 1

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

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 : Preadditive C
inst✝³ : HasBinaryBiproducts C
inst✝² : Preadditive D
inst✝¹ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
G : C ⥤ D
inst✝ : G.Additive
⊢ (G.mapHomologicalComplex (ComplexShape.up ℤ)).map (triangle φ).mor₃ ≫
      (Functor.commShiftIso (G.mapHomologicalComplex (ComplexShape.up ℤ)) 1).hom.app K =
    (mapHomologicalComplexIso φ G).hom ≫ (triangle ((G.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)).mor₃

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A✝ A : C
f : A ⟶ T'.obj₃
hf : f ≫ φ'.hom₃ = 0
hf' : f ≫ T'.mor₃ = 0
g : A ⟶ T'.obj₂
hg : f = g ≫ T'.mor₂
j : J
⊢ (g ≫ Pi.π (fun j ↦ (T j).obj₂) j) ≫ (T j).mor₂ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

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 : Preadditive C
inst✝³ : HasBinaryBiproducts C
inst✝² : Preadditive D
inst✝¹ : HasBinaryBiproducts D
K L : CochainComplex C ℤ
φ : K ⟶ L
G : C ⥤ D
inst✝ : G.Additive
⊢ (G.mapHomologicalComplex (ComplexShape.up ℤ)).mapTriangle.obj (triangle φ) ≅
    triangle ((G.mapHomologicalComplex (ComplexShape.up ℤ)).map φ)

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
T₁ : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
hT₁ : T₁ ∈ distinguishedTriangles C
T₂ : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
e : T₂ ≅ T₁
⊢ T₂ ∈ distinguishedTriangles C

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasBinaryBiproducts C
inst✝ : HasZeroObject C
X : HomologicalComplex C (ComplexShape.up ℤ)
h : IsZero ((quotient C (ComplexShape.up ℤ)).obj (CochainComplex.mappingCone (𝟙 X)))
⊢ (contractibleTriangle { as := X }).mor₁ ≫ (Iso.refl (contractibleTriangle { as := X }).obj₂).hom =
    (Iso.refl (contractibleTriangle { as := X }).obj₁).hom ≫ (CochainComplex.mappingCone.triangleh (𝟙 X)).mor₁

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasBinaryBiproducts C
inst✝ : HasZeroObject C
X : HomologicalComplex C (ComplexShape.up ℤ)
h : IsZero ((quotient C (ComplexShape.up ℤ)).obj (CochainComplex.mappingCone (𝟙 X)))
⊢ (contractibleTriangle { as := X }).mor₃ ≫
      (shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) 1).map
        (Iso.refl (contractibleTriangle { as := X }).obj₁).hom =
    h.isoZero.symm.hom ≫ (CochainComplex.mappingCone.triangleh (𝟙 X)).mor₃

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasBinaryBiproducts C
inst✝ : HasZeroObject C
X : HomotopyCategory C (ComplexShape.up ℤ)
⊢ contractibleTriangle X ∈ distinguishedTriangles C

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
X Y : HomotopyCategory C (ComplexShape.up ℤ)
f : X ⟶ Y
⊢ ∃ Z g h, Triangle.mk f g h ∈ distinguishedTriangles C

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
T : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
hT : T ∈ distinguishedTriangles C
⊢ T.rotate ∈ distinguishedTriangles C

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
T : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
hT : T ∈ distinguishedTriangles C
n : ℤ
⊢ (Triangle.shiftFunctor (HomotopyCategory C (ComplexShape.up ℤ)) n).obj T ∈ distinguishedTriangles C

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
T : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
⊢ T ∈ distinguishedTriangles C ↔ T.rotate ∈ distinguishedTriangles C

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated

Goal state before:
pp: C : Type u
inst✝⁸ : Category.{v, u} C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
hC : Pretriangulated C
J : Type u_1
T : J → Triangle C
hT : ∀ (j : J), T j ∈ distinguishedTriangles
inst✝³ : HasProduct fun j ↦ (T j).obj₁
inst✝² : HasProduct fun j ↦ (T j).obj₂
inst✝¹ : HasProduct fun j ↦ (T j).obj₃
inst✝ : HasProduct fun j ↦ (shiftFunctor C 1).obj (T j).obj₁
f₁ : (∏ᶜ fun j ↦ (T j).obj₁) ⟶ ∏ᶜ fun j ↦ (T j).obj₂ := Limits.Pi.map fun j ↦ (T j).mor₁
Z : C
f₂ : (∏ᶜ fun j ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ (shiftFunctor C 1).obj (∏ᶜ fun j ↦ (T j).obj₁)
T' : Triangle C := Triangle.mk f₁ f₂ f₃
hT' : T' ∈ distinguishedTriangles
φ : (j : J) → T' ⟶ T j :=
  fun j ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun j ↦ (T j).obj₁) j) (Pi.π (fun j ↦ (T j).obj₂) j) ⋯
φ' : T' ⟶ productTriangle T := productTriangle.lift T φ
h₁ : φ'.hom₁ = 𝟙 T'.obj₁
h₂ : φ'.hom₂ = 𝟙 T'.obj₂
this✝ : IsIso φ'.hom₁
this : IsIso φ'.hom₂
A✝ A : C
f : A ⟶ T'.obj₃
hf : f ≫ φ'.hom₃ = 0
hf' : f ≫ T'.mor₃ = 0
g : A ⟶ T'.obj₂
hg : f = g ≫ T'.mor₂
hg' : ∀ (j : J), (g ≫ Pi.π (fun j ↦ (T j).obj₂) j) ≫ (T j).mor₂ = 0
hg'' : ∀ (j : J), ∃ g_1, g ≫ Pi.π (fun j ↦ (T j).obj₂) j = g_1 ≫ (T j).mor₁
α : (j : J) → A ⟶ (T j).obj₁ := fun j ↦ ⋯.choose
hα : ∀ (j : J), g ≫ Pi.π (fun j ↦ (T j).obj₂) j = α j ≫ (T j).mor₁
⊢ g = Pi.lift α ≫ T'.mor₁

Tactic:
by

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasBinaryBiproducts C
T₁ T₂ : Triangle (HomotopyCategory C (ComplexShape.up ℤ))
a : T₁.obj₁ ⟶ T₂.obj₁
b : T₁.obj₂ ⟶ T₂.obj₂
fac : T₁.mor₁ ≫ b = a ≫ T₂.mor₁
K₁ L₁ : CochainComplex C ℤ
φ₁ : K₁ ⟶ L₁
e₁ : T₁ ≅ triangleh φ₁
K₂ L₂ : CochainComplex C ℤ
φ₂ : K₂ ⟶ L₂
e₂ : T₂ ≅ triangleh φ₂
a'' : K₁ ⟶ K₂
ha' : (quotient C (ComplexShape.up ℤ)).map a'' = e₁.inv.hom₁ ≫ a ≫ e₂.hom.hom₁
b'' : L₁ ⟶ L₂
hb' : (quotient C (ComplexShape.up ℤ)).map b'' = e₁.inv.hom₂ ≫ b ≫ e₂.hom.hom₂
⊢ (quotient C (ComplexShape.up ℤ)).map (φ₁ ≫ b'') = (quotient C (ComplexShape.up ℤ)).map (a'' ≫ φ₂)

Tactic:
by

Goal state after: