awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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
a : A ⟶ (productTriangle T).obj₃
j : J
⊢ (T j).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (T j).mor₁ = 0

Tactic:
exact comp_distTriang_mor_zero₃₁ _ (hT j)

Premises used: CategoryTheory.Pretriangulated.comp_distTriang_mor_zero₃₁
-- Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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
a : A ⟶ (productTriangle T).obj₃
⊢ (productTriangle T).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) T'.mor₁ = 0

Tactic:
have zero : ((productTriangle T).mor₃) ≫ (shiftFunctor C 1).map T'.mor₁ = 0 := by
        rw [← cancel_mono (φ'.hom₂⟦1⟧'), zero_comp, assoc, ← Functor.map_comp, φ'.comm₁, h₁,
          id_comp, productTriangle.zero₃₁]
        intro j
        exact comp_distTriang_mor_zero₃₁ _ (hT j)
-- Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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
a : A ⟶ (productTriangle T).obj₃
zero : (productTriangle T).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) T'.mor₁ = 0
rw : a ≫ (productTriangle T).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) T'.mor₁ = 0
rw₁ : Eq (α := A ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) T'.obj₂) (a ≫ 0) 0
rw₂ rw₃ : Eq.{v + 1} (α := A ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) T'.obj₂) 0 0
⊢ (a ≫ (productTriangle T).mor₃) ≫ Functor.map (shiftFunctor C (A := ℤ) 1) T'.mor₁ = 0

Tactic:
rw [assoc, zero, comp_zero]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Limits.comp_zero
-- Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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
a : A ⟶ (productTriangle T).obj₃
zero : (productTriangle T).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) T'.mor₁ = 0
have :
  ∀ (g : A ⟶ T'.obj₃) (hg : a ≫ (productTriangle T).mor₃ = g ≫ T'.mor₃),
    ∃ (a' : A ⟶ Z), a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃
⊢ ∃ (a' : A ⟶ Z), a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃

Tactic:
have ⟨g, hg⟩ := T'.coyoneda_exact₁ hT' (a ≫ (productTriangle T).mor₃) (by
        rw [assoc, zero, comp_zero])
-- Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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
a : A ⟶ (productTriangle T).obj₃
zero : (productTriangle T).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) T'.mor₁ = 0
g : A ⟶ T'.obj₃
hg : a ≫ (productTriangle T).mor₃ = g ≫ T'.mor₃
⊢ ∃ (a' : A ⟶ Z), a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃

Tactic:
⟨g, hg.symm⟩

Premises used: Exists.intro, Eq.symm
-- Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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
a : A ⟶ (productTriangle T).obj₃
zero : (productTriangle T).mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) T'.mor₁ = 0
g : A ⟶ T'.obj₃
hg : a ≫ (productTriangle T).mor₃ = g ≫ T'.mor₃
⊢ ∃ (a' : A ⟶ Z), a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃

Tactic:
exact ⟨g, hg.symm⟩
-- Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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
a : A ⟶ (productTriangle T).obj₃
refine_2 :
  ∀ (a' : A ⟶ Z) (ha' : a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃),
    ∃ (a_1 : A ⟶ T'.obj₃), (fun (x : A ⟶ T'.obj₃) ↦ x ≫ φ'.hom₃) a_1 = a
⊢ ∃ (a_1 : A ⟶ T'.obj₃), (fun (x : A ⟶ T'.obj₃) ↦ x ≫ φ'.hom₃) a_1 = a

Tactic:
obtain ⟨a', ha'⟩ : ∃ (a' : A ⟶ Z), a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃ := by
      have zero : ((productTriangle T).mor₃) ≫ (shiftFunctor C 1).map T'.mor₁ = 0 := by
        rw [← cancel_mono (φ'.hom₂⟦1⟧'), zero_comp, assoc, ← Functor.map_comp, φ'.comm₁, h₁,
          id_comp, productTriangle.zero₃₁]
        intro j
        exact comp_distTriang_mor_zero₃₁ _ (hT j)
      have ⟨g, hg⟩ := T'.coyoneda_exact₁ hT' (a ≫ (productTriangle T).mor₃) (by
        rw [assoc, zero, comp_zero])
      exact ⟨g, hg.symm⟩

Premises used: CategoryTheory.Pretriangulated.productTriangle, CategoryTheory.Pretriangulated.Triangle.mor₃
-- Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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
a : A ⟶ (productTriangle T).obj₃
a' : A ⟶ Z
ha' : a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃
j : J
simp :
  HSub.hSub (α := A ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁)
      (a ≫ Pi.π (fun (j : J) ↦ (T j).obj₃) j ≫ (T j).mor₃)
      (a' ≫ φ'.hom₃ ≫ Pi.π (fun (j : J) ↦ (T j).obj₃) j ≫ (T j).mor₃) =
    0
⊢ ((a - a' ≫ φ'.hom₃) ≫ Pi.π (fun (j : J) ↦ (T j).obj₃) j) ≫ (T j).mor₃ = 0

Tactic:
simp only [sub_comp, assoc]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Preadditive.sub_comp
-- Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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
a : A ⟶ (productTriangle T).obj₃
a' : A ⟶ Z
ha' : a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃
j : J
erw :
  a ≫
        (productTriangle T).mor₃ ≫
          Functor.map (shiftFunctor C (A := ℤ) 1) (TriangleMorphism.hom₁ (productTriangle.π T j)) -
      a' ≫
        φ'.hom₃ ≫
          (productTriangle T).mor₃ ≫
            Functor.map (shiftFunctor C (A := ℤ) 1) (TriangleMorphism.hom₁ (productTriangle.π T j)) =
    0
⊢ HSub.hSub (α := A ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁)
      (a ≫ Pi.π (fun (j : J) ↦ (T j).obj₃) j ≫ (T j).mor₃)
      (a' ≫ φ'.hom₃ ≫ Pi.π (fun (j : J) ↦ (T j).obj₃) j ≫ (T j).mor₃) =
    0

Tactic:
erw [← (productTriangle.π T j).comm₃]

Premises used: CategoryTheory.Pretriangulated.TriangleMorphism.comm₃, CategoryTheory.Pretriangulated.productTriangle.π
-- Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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
a : A ⟶ (productTriangle T).obj₃
a' : A ⟶ Z
ha' : a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃
j : J
rw :
  a ≫
        (productTriangle T).mor₃ ≫
          Functor.map (shiftFunctor C (A := ℤ) 1) (TriangleMorphism.hom₁ (productTriangle.π T j)) -
      a' ≫
        T'.mor₃ ≫
          Functor.map (shiftFunctor C (A := ℤ) 1) φ'.hom₁ ≫
            Functor.map (shiftFunctor C (A := ℤ) 1) (TriangleMorphism.hom₁ (productTriangle.π T j)) =
    0
⊢ a ≫
        (productTriangle T).mor₃ ≫
          Functor.map (shiftFunctor C (A := ℤ) 1) (TriangleMorphism.hom₁ (productTriangle.π T j)) -
      a' ≫
        φ'.hom₃ ≫
          (productTriangle T).mor₃ ≫
            Functor.map (shiftFunctor C (A := ℤ) 1) (TriangleMorphism.hom₁ (productTriangle.π T j)) =
    0

Tactic:
rw [← φ'.comm₃_assoc]

Premises used: CategoryTheory.Pretriangulated.TriangleMorphism.comm₃_assoc
-- Module: Mathlib.CategoryTheory.Triangulated.Pretriangulated
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasZeroObject.{v, u} C
inst_2 : HasShift.{v, u, 0} C ℤ
inst_3 : Preadditive.{v, u} C
inst_4 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
hC : Pretriangulated.{v, u} C
J : Type u_1
T : J → Triangle.{v, u} C
hT :
  ∀ (j : J), Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) (T j)
inst_5 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₁
inst_6 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₂
inst_7 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ (T j).obj₃
inst_8 : HasProduct.{u_1, v, u} (C := C) fun (j : J) ↦ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (T j).obj₁
f₁ : Quiver.Hom.{v, u} (V := C) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
  (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) :=
  Limits.Pi.map fun (j : J) ↦ (T j).mor₁
Z : C
f₂ : (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₂) ⟶ Z
f₃ : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) (piObj.{u_1, v, u} fun (j : J) ↦ (T j).obj₁)
T' : Triangle.{v, u} C := Triangle.mk f₁ f₂ f₃
hT' : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T'
φ : (j : J) → T' ⟶ T j :=
  fun (j : J) ↦
    completeDistinguishedTriangleMorphism T' (T j) hT' ⋯ (Pi.π (fun (j : J) ↦ (T j).obj₁) j)
      (Pi.π (fun (j : 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
a : A ⟶ (productTriangle T).obj₃
a' : A ⟶ Z
ha' : a' ≫ T'.mor₃ = a ≫ (productTriangle T).mor₃
⊢ ∀ (j : J), ∃ (g : A ⟶ (T j).obj₂), (a - a' ≫ φ'.hom₃) ≫ Pi.π (fun (j : J) ↦ (T j).obj₃) j = g ≫ (T j).mor₂

Tactic:
have ha'' := fun (j : J) => (T j).coyoneda_exact₃ (hT j) ((a - a' ≫ φ'.hom₃) ≫ Pi.π _ j) (by
      simp only [sub_comp, assoc]
      erw [← (productTriangle.π T j).comm₃]
      rw [← φ'.comm₃_assoc]
      rw [reassoc_of% ha', sub_eq_zero, h₁, Functor.map_id, id_comp])