awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory (goal)
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
hf₁ : admissibleMorphism ι f₁
⊢ HasKernel f₁
have obtain : ∀ (X₃ : C) (f₂ : ι.obj X₂ ⟶ X₃) (f₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁))
  (hT :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
      (Triangle.mk (ι.map f₁) f₂ f₃)),
  HasKernel f₁
Tactic: obtain ⟨X₃, f₂, f₃, hT⟩ := distinguished_cocone_triangle (ι.map f₁)
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory (goal)
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
hf₁ : admissibleMorphism ι f₁
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) f₂ f₃)
⊢ HasKernel f₁
have obtain : ∀ (K Q : A) (α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₃) (β : X₃ ⟶ ι.obj Q)
  (γ :
    ι.obj Q ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
        (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K)))
  (hT' :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
      (Triangle.mk α β γ)),
  HasKernel f₁
Tactic: obtain ⟨K, Q, α, β, γ, hT'⟩ := hf₁ f₂ f₃ hT
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory (goal)
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
hf₁ : admissibleMorphism ι f₁
⊢ HasCokernel f₁
have obtain : ∀ (X₃ : C) (f₂ : ι.obj X₂ ⟶ X₃) (f₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁))
  (hT :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
      (Triangle.mk (ι.map f₁) f₂ f₃)),
  HasCokernel f₁
Tactic: obtain ⟨X₃, f₂, f₃, hT⟩ := distinguished_cocone_triangle (ι.map f₁)
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory (goal)
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
hf₁ : admissibleMorphism ι f₁
X₃ : C
f₂ : ι.obj X₂ ⟶ X₃
f₃ : X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) f₂ f₃)
⊢ HasCokernel f₁
have obtain : ∀ (K Q : A) (α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₃) (β : X₃ ⟶ ι.obj Q)
  (γ :
    ι.obj Q ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1)
        (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K)))
  (hT' :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
      (Triangle.mk α β γ)),
  HasCokernel f₁
Tactic: obtain ⟨K, Q, α, β, γ, hT'⟩ := hf₁ f₂ f₃ hT
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory (hypothesis)
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
⊢ ι.map (f₁ ≫ f₂) = ι.map 0
have this : (Triangle.mk (ι.map f₁) (ι.map f₂) f₃).mor₁ ≫ (Triangle.mk (ι.map f₁) (ι.map f₂) f₃).mor₂ = 0
Tactic: have := comp_distTriang_mor_zero₁₂ _ hT
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory (goal)
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
⊢ Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (Z := ι.obj 0) (𝟙 (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁))) 0 0)
have refine : Iso (C := Triangle.{v_1, u_1} C)
  (Triangle.mk (Z := ι.obj 0) (𝟙 (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁))) 0 0)
  (contractibleTriangle ((ι ⋙ shiftFunctor C (A := ℤ) 1).obj X₁))
Tactic: refine isomorphic_distinguished _ (contractible_distinguished
      (((ι ⋙ shiftFunctor C (1 : ℤ)).obj X₁))) _ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory (goal)
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
⊢ IsZero.{v_1, u_1}
    (Triangle.obj₃.{v_1, u_1}
      (Triangle.mk (Z := ι.obj 0) (𝟙 (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁))) 0 0))
have dsimp : IsZero.{v_1, u_1} (ι.obj 0)
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory (goal)
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
⊢ IsZero.{v_1, u_1} (ι.obj 0)
have rw : Eq.{v_1 + 1} (α := ι.obj 0 ⟶ ι.obj 0) (𝟙 (ι.obj 0)) 0
Tactic: rw [IsZero.iff_id_eq_zero, ← ι.map_id, id_zero, ι.map_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory (goal)
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
rw : Eq.{v_1 + 1} (α := ι.obj 0 ⟶ ι.obj 0) (𝟙 (ι.obj 0)) 0
⊢ IsZero.{v_1, u_1} (ι.obj 0)
have rw₁ : Eq (α := ι.obj 0 ⟶ ι.obj 0) (ι.map (𝟙 0)) 0
Tactic: rw [IsZero.iff_id_eq_zero, ← ι.map_id, id_zero, ι.map_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory (goal)
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
rw : Eq.{v_1 + 1} (α := ι.obj 0 ⟶ ι.obj 0) (𝟙 (ι.obj 0)) 0
rw₁ : Eq (α := ι.obj 0 ⟶ ι.obj 0) (ι.map (𝟙 0)) 0
⊢ IsZero.{v_1, u_1} (ι.obj 0)
have rw₂ : Eq (α := ι.obj 0 ⟶ ι.obj 0) (ι.map 0) 0
Tactic: rw [IsZero.iff_id_eq_zero, ← ι.map_id, id_zero, ι.map_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory (goal)
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
rw : Eq.{v_1 + 1} (α := ι.obj 0 ⟶ ι.obj 0) (𝟙 (ι.obj 0)) 0
rw₁ : Eq (α := ι.obj 0 ⟶ ι.obj 0) (ι.map (𝟙 0)) 0
rw₂ : Eq (α := ι.obj 0 ⟶ ι.obj 0) (ι.map 0) 0
⊢ IsZero.{v_1, u_1} (ι.obj 0)
have rw₃ : Eq.{v_1 + 1} (α := ι.obj 0 ⟶ ι.obj 0) 0 0
Tactic: rw [IsZero.iff_id_eq_zero, ← ι.map_id, id_zero, ι.map_zero]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory (hypothesis)
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
⊢ IsLimit (F := parallelPair f₂ 0) (KernelFork.ofι f₁ ⋯)
have hT' : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
  (Triangle.mk (Z := ι.obj 0) (𝟙 (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁))) 0 0)
Tactic: have hT' : Triangle.mk (𝟙 ((ι.obj X₁)⟦(1 : ℤ)⟧)) (0 : _ ⟶ ι.obj 0) 0 ∈ distTriang C := by
    refine isomorphic_distinguished _ (contractible_distinguished
      (((ι ⋙ shiftFunctor C (1 : ℤ)).obj X₁))) _ ?_
    exact Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _) (IsZero.iso (by
      dsimp
      rw [IsZero.iff_id_eq_zero, ← ι.map_id, id_zero, ι.map_zero]) (isZero_zero C))
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory (goal)
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
hT' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (Z := ι.obj 0) (𝟙 (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁))) 0 0)
⊢ IsLimit (F := parallelPair f₂ 0) (KernelFork.ofι f₁ ⋯)
have refine : Iso (C := Cone (parallelPair f₂ 0))
  (KernelFork.ofι
    (ιK (-Functor.map (shiftFunctor C (A := ℤ) 1) (Triangle.mk (ι.map f₁) (ι.map f₂) f₃).mor₁)
      (𝟙 (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁))))
    sorry)
  (KernelFork.ofι f₁ sorry)
Tactic: refine IsLimit.ofIsoLimit (AbelianSubcategory.isLimitKernelFork hι
    (rot_of_distTriang _ hT) hT') ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory (goal)
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
⊢ Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0)
have refine : Iso (C := Triangle.{v_1, u_1} C) (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0)
  (Triangle.invRotate (contractibleTriangle (ι.obj X₃)))
Tactic: refine isomorphic_distinguished _ (inv_rot_of_distTriang _
      (contractible_distinguished (ι.obj X₃))) _ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory (goal)
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
⊢ Iso (C := Triangle.{v_1, u_1} C) (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0)
    (Triangle.invRotate (contractibleTriangle (ι.obj X₃)))
have refine_1 : IsZero.{v_1, u_1}
  (Triangle.obj₁.{v_1, u_1} (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0))
Tactic: refine Triangle.isoMk _ _ (IsZero.iso ?_ ?_) (Iso.refl _) (Iso.refl _) ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory (goal)
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
refine_1 :
  IsZero.{v_1, u_1}
    (Triangle.obj₁.{v_1, u_1} (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0))
⊢ Iso (C := Triangle.{v_1, u_1} C) (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0)
    (Triangle.invRotate (contractibleTriangle (ι.obj X₃)))
have refine_2 : IsZero.{v_1, u_1} (Triangle.obj₁.{v_1, u_1} (Triangle.invRotate (contractibleTriangle (ι.obj X₃))))
Tactic: refine Triangle.isoMk _ _ (IsZero.iso ?_ ?_) (Iso.refl _) (Iso.refl _) ?_ ?_ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.TStructure.AbelianSubcategory (goal)
Goal:
C : Type u_1
A : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : Preadditive.{v_1, u_1} C
inst_3 : HasShift.{v_1, u_1, 0} C ℤ
inst_4 : ∀ (n : ℤ), Functor.Additive.{v_1, u_1, v_1, u_1} (shiftFunctor C n)
inst_5 : Pretriangulated.{v_1, u_1} C
inst_6 : Category.{v_2, u_2} A
ι : Functor.{v_2, v_1, u_2, u_1} A C
hι : ∀ ⦃X Y : A⦄ ⦃n : ℤ⦄ (f : ι.obj X ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C n) (ι.obj Y)), n < 0 → f = 0
inst_7 : Preadditive.{v_2, u_2} A
inst_8 : ι.Full
inst_9 : ι.Faithful
inst_10 : HasFiniteProducts.{v_2, u_2} A
inst_11 : ι.Additive
X₁ X₂ X₃ : A
f₁ : X₁ ⟶ X₂
f₂ : X₂ ⟶ X₃
f₃ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map f₁) (ι.map f₂) f₃)
refine_1 :
  IsZero.{v_1, u_1}
    (Triangle.obj₁.{v_1, u_1} (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0))
refine_2 : IsZero.{v_1, u_1} (Triangle.obj₁.{v_1, u_1} (Triangle.invRotate (contractibleTriangle (ι.obj X₃))))
⊢ Iso (C := Triangle.{v_1, u_1} C) (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0)
    (Triangle.invRotate (contractibleTriangle (ι.obj X₃)))
have refine_3 : Triangle.mor₁ (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0) ≫
    Iso.hom
      (Iso.refl
        (Triangle.obj₂.{v_1, u_1} (Triangle.mk (X := (ι ⋙ shiftFunctor C (A := ℤ) 1).obj 0) 0 (𝟙 (ι.obj X₃)) 0))) =
  Iso.hom (IsZero.iso sorry sorry) ≫ Triangle.mor₁ (Triangle.invRotate (contractibleTriangle (ι.obj X₃)))
Tactic: refine Triangle.isoMk _ _ (IsZero.iso ?_ ?_) (Iso.refl _) (Iso.refl _) ?_ ?_ ?_