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
inst_10 : ι.Additive
hA : admissibleMorphism ι = ⊤
X₂ X₃ : A
π : X₂ ⟶ X₃
inst_11 : Epi π
X₁ : C
f₂ : ι.obj X₃ ⟶ X₁
f₃ : X₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₂)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map π) f₂ f₃)
this : admissibleMorphism ι π
K Q : A
α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₁
β : X₁ ⟶ 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)
hQ : Eq.{v_2 + 1} (α := Q ⟶ Q) (𝟙 Q) 0
⊢ IsZero.{v_1, u_1} (Triangle.mk α β γ).obj₃
have dsimp : IsZero.{v_1, u_1} (ι.obj Q)
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 : ι.Additive
hA : admissibleMorphism ι = ⊤
X₂ X₃ : A
π : X₂ ⟶ X₃
inst_11 : Epi π
X₁ : C
f₂ : ι.obj X₃ ⟶ X₁
f₃ : X₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₂)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map π) f₂ f₃)
this : admissibleMorphism ι π
K Q : A
α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₁
β : X₁ ⟶ 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)
hQ : Eq.{v_2 + 1} (α := Q ⟶ Q) (𝟙 Q) 0
⊢ IsZero.{v_1, u_1} (ι.obj Q)
have rw : Eq.{v_1 + 1} (α := ι.obj Q ⟶ ι.obj Q) (𝟙 (ι.obj Q)) 0
Tactic: rw [IsZero.iff_id_eq_zero, ← ι.map_id, hQ, 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 : ι.Additive
hA : admissibleMorphism ι = ⊤
X₂ X₃ : A
π : X₂ ⟶ X₃
inst_11 : Epi π
X₁ : C
f₂ : ι.obj X₃ ⟶ X₁
f₃ : X₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₂)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map π) f₂ f₃)
this : admissibleMorphism ι π
K Q : A
α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₁
β : X₁ ⟶ 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)
hQ : Eq.{v_2 + 1} (α := Q ⟶ Q) (𝟙 Q) 0
rw : Eq.{v_1 + 1} (α := ι.obj Q ⟶ ι.obj Q) (𝟙 (ι.obj Q)) 0
⊢ IsZero.{v_1, u_1} (ι.obj Q)
have rw₁ : ι.map (𝟙 Q) = 0
Tactic: rw [IsZero.iff_id_eq_zero, ← ι.map_id, hQ, 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 : ι.Additive
hA : admissibleMorphism ι = ⊤
X₂ X₃ : A
π : X₂ ⟶ X₃
inst_11 : Epi π
X₁ : C
f₂ : ι.obj X₃ ⟶ X₁
f₃ : X₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₂)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map π) f₂ f₃)
this : admissibleMorphism ι π
K Q : A
α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₁
β : X₁ ⟶ 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)
hQ : Eq.{v_2 + 1} (α := Q ⟶ Q) (𝟙 Q) 0
rw : Eq.{v_1 + 1} (α := ι.obj Q ⟶ ι.obj Q) (𝟙 (ι.obj Q)) 0
rw₁ : ι.map (𝟙 Q) = 0
⊢ IsZero.{v_1, u_1} (ι.obj Q)
have rw₂ : Eq (α := ι.obj Q ⟶ ι.obj Q) (ι.map 0) 0
Tactic: rw [IsZero.iff_id_eq_zero, ← ι.map_id, hQ, 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 : ι.Additive
hA : admissibleMorphism ι = ⊤
X₂ X₃ : A
π : X₂ ⟶ X₃
inst_11 : Epi π
X₁ : C
f₂ : ι.obj X₃ ⟶ X₁
f₃ : X₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₂)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map π) f₂ f₃)
this : admissibleMorphism ι π
K Q : A
α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₁
β : X₁ ⟶ 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)
hQ : Eq.{v_2 + 1} (α := Q ⟶ Q) (𝟙 Q) 0
rw : Eq.{v_1 + 1} (α := ι.obj Q ⟶ ι.obj Q) (𝟙 (ι.obj Q)) 0
rw₁ : ι.map (𝟙 Q) = 0
rw₂ : Eq (α := ι.obj Q ⟶ ι.obj Q) (ι.map 0) 0
⊢ IsZero.{v_1, u_1} (ι.obj Q)
have rw₃ : Eq.{v_1 + 1} (α := ι.obj Q ⟶ ι.obj Q) 0 0
Tactic: rw [IsZero.iff_id_eq_zero, ← ι.map_id, hQ, 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 : ι.Additive
hA : admissibleMorphism ι = ⊤
X₂ X₃ : A
π : X₂ ⟶ X₃
inst_11 : Epi π
X₁ : C
f₂ : ι.obj X₃ ⟶ X₁
f₃ : X₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₂)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map π) f₂ f₃)
this : admissibleMorphism ι π
K Q : A
α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₁
β : X₁ ⟶ 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)
hQ : Eq.{v_2 + 1} (α := Q ⟶ Q) (𝟙 Q) 0
⊢ ∃ (X₁ : A) (i : X₁ ⟶ X₂) (δ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)),
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
      (Triangle.mk (ι.map i) (ι.map π) δ)
have this₁ : IsIso α
Tactic: have : IsIso α := (Triangle.isZero₃_iff_isIso₁ _ hT').1 (by
    dsimp
    rw [IsZero.iff_id_eq_zero, ← ι.map_id, hQ, 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 : ι.Additive
hA : admissibleMorphism ι = ⊤
X₂ X₃ : A
π : X₂ ⟶ X₃
inst_11 : Epi π
X₁ : C
f₂ : ι.obj X₃ ⟶ X₁
f₃ : X₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₂)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map π) f₂ f₃)
this_1 : admissibleMorphism ι π
K Q : A
α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₁
β : X₁ ⟶ 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)
hQ : Eq.{v_2 + 1} (α := Q ⟶ Q) (𝟙 Q) 0
this : IsIso α
⊢ ∃ (X₁ : A) (i : X₁ ⟶ X₂) (δ : ι.obj X₃ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₁)),
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
      (Triangle.mk (ι.map i) (ι.map π) δ)
have refine : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
  (Triangle.mk (ι.map (-ιK f₃ α)) (ι.map π) (f₂ ≫ inv α))
Tactic: refine ⟨K, -ιK f₃ α, f₂ ≫ inv α, ?_⟩
-- 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 : ι.Additive
hA : admissibleMorphism ι = ⊤
X₂ X₃ : A
π : X₂ ⟶ X₃
inst_11 : Epi π
X₁ : C
f₂ : ι.obj X₃ ⟶ X₁
f₃ : X₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₂)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map π) f₂ f₃)
this_1 : admissibleMorphism ι π
K Q : A
α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₁
β : X₁ ⟶ 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)
hQ : Eq.{v_2 + 1} (α := Q ⟶ Q) (𝟙 Q) 0
this : IsIso α
⊢ Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map (-ιK f₃ α)) (ι.map π) (f₂ ≫ inv α))
have rw : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
  (Triangle.mk (ι.map (-ιK f₃ α)) (ι.map π) (f₂ ≫ inv α)).rotate
Tactic: rw [rotate_distinguished_triangle]
-- 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 : ι.Additive
hA : admissibleMorphism ι = ⊤
X₂ X₃ : A
π : X₂ ⟶ X₃
inst_11 : Epi π
X₁ : C
f₂ : ι.obj X₃ ⟶ X₁
f₃ : X₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj X₂)
hT :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map π) f₂ f₃)
this_1 : admissibleMorphism ι π
K Q : A
α : Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj K) ⟶ X₁
β : X₁ ⟶ 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)
hQ : Eq.{v_2 + 1} (α := Q ⟶ Q) (𝟙 Q) 0
this : IsIso α
⊢ Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
    (Triangle.mk (ι.map (-ιK f₃ α)) (ι.map π) (f₂ ≫ inv α)).rotate
have refine : (Triangle.mk (ι.map (-ιK f₃ α)) (ι.map π) (f₂ ≫ inv α)).rotate ≅ Triangle.mk (ι.map π) f₂ f₃
Tactic: refine isomorphic_distinguished _ 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
hA : admissibleMorphism ι = ⊤
inst_12 : IsTriangulated.{v_1, u_1} C
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
⊢ Nonempty (Abelian.AbelianStruct 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)),
  Nonempty (Abelian.AbelianStruct f₁)
Tactic: obtain ⟨X₃, f₂, f₃, hT⟩ := distinguished_cocone_triangle (ι.map f₁)
-- 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
hA : admissibleMorphism ι = ⊤
inst_12 : IsTriangulated.{v_1, u_1} C
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
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)
⊢ Nonempty (Abelian.AbelianStruct f₁)
have this : admissibleMorphism ι f₁
Tactic: have : admissibleMorphism ι f₁ := by simp [hA]
-- 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
hA : admissibleMorphism ι = ⊤
inst_12 : IsTriangulated.{v_1, u_1} C
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
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)
this : admissibleMorphism ι f₁
⊢ Nonempty (Abelian.AbelianStruct 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)),
  Nonempty (Abelian.AbelianStruct f₁)
Tactic: obtain ⟨K, Q, α, β, γ, hT'⟩ := this 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
hA : admissibleMorphism ι = ⊤
inst_12 : IsTriangulated.{v_1, u_1} C
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
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)
this : admissibleMorphism ι f₁
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)
⊢ Nonempty (Abelian.AbelianStruct f₁)
have this₁ : Epi (πQ f₂ β)
Tactic: have := epi_πQ hι 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
hA : admissibleMorphism ι = ⊤
inst_12 : IsTriangulated.{v_1, u_1} C
X₁ X₂ : A
f₁ : X₁ ⟶ X₂
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)
this_1 : admissibleMorphism ι f₁
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)
this : Epi (πQ f₂ β)
⊢ Nonempty (Abelian.AbelianStruct f₁)
have obtain : ∀ (I : A) (i : I ⟶ X₂) (δ : ι.obj Q ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (ι.obj I))
  (hI :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_2 inst_4 _)
      (Triangle.mk (ι.map i) (ι.map (πQ f₂ β)) δ)),
  Nonempty (Abelian.AbelianStruct f₁)
Tactic: obtain ⟨I, i, δ, hI⟩ := exists_distinguished_triangle_of_epi hι hA (πQ f₂ β)