awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
Y Z_1 Z' : C
f₁ f₂ : Y ⟶ Z_1
s : Z_1 ⟶ Z'
hf₁ : f₁ ≫ s = f₂ ≫ s
Z : C
g : Z ⟶ Z_1
h : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Z
H :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk g s h)
mem : P Z
hf₂ : (f₁ - f₂) ≫ s = 0
q : Y ⟶ (Triangle.mk g s h).obj₁
hq : f₁ - f₂ = q ≫ (Triangle.mk g s h).mor₁
Y' : C
r : Y' ⟶ Y
t : (Triangle.mk g s h).obj₁ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) Y'
mem' :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk r q t)
eq : r ≫ q = 0
refine_2 : r ≫ f₁ - r ≫ f₂ = 0
refine_2₁ : r ≫ (f₁ - f₂) = 0
refine_2₂ : r ≫ q ≫ (Triangle.mk g s h).mor₁ = 0
refine_2₃ : Eq (α := Y' ⟶ Z_1) (0 ≫ (Triangle.mk g s h).mor₁) 0
⊢ r ≫ f₁ = r ≫ f₂
have refine_2₄ : Eq.{v_1 + 1} (α := Y' ⟶ Z_1) 0 0
Tactic: rw [← sub_eq_zero, ← comp_sub, hq, reassoc_of% eq, zero_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
T₁ T₃ : Triangle.{v_1, u_1} C
mem₁ : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T₁
mem₃ : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T₃
a : T₁.obj₁ ⟶ T₃.obj₁
b : T₁.obj₂ ⟶ T₃.obj₂
Z₅ : C
g₅ : T₃.obj₁ ⟶ Z₅
h₅ : Z₅ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₁
mem₅ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk a g₅ h₅)
mem₅' : P Z₅
Z₄ : C
g₄ : T₃.obj₂ ⟶ Z₄
h₄ : Z₄ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₂
mem₄ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk b g₄ h₄)
mem₄' : P Z₄
comm : T₁.mor₁ ≫ b = a ≫ T₃.mor₁
⊢ ∃ (c : T₁.obj₃ ⟶ T₃.obj₃) (_ : P.trW c),
    T₁.mor₂ ≫ c = b ≫ T₃.mor₂ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a = c ≫ T₃.mor₃
have obtain : ∀ (Z₂ : C) (g₂ : T₃.obj₂ ⟶ Z₂) (h₂ : Z₂ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₁)
  (mem₂ :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk (T₁.mor₁ ≫ b) g₂ h₂)),
  ∃ (c : T₁.obj₃ ⟶ T₃.obj₃) (_ : P.trW c),
    T₁.mor₂ ≫ c = b ≫ T₃.mor₂ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a = c ≫ T₃.mor₃
Tactic: obtain ⟨Z₂, g₂, h₂, mem₂⟩ := distinguished_cocone_triangle (T₁.mor₁ ≫ b)
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
T₁ T₃ : Triangle.{v_1, u_1} C
mem₁ : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T₁
mem₃ : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T₃
a : T₁.obj₁ ⟶ T₃.obj₁
b : T₁.obj₂ ⟶ T₃.obj₂
Z₅ : C
g₅ : T₃.obj₁ ⟶ Z₅
h₅ : Z₅ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₁
mem₅ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk a g₅ h₅)
mem₅' : P Z₅
Z₄ : C
g₄ : T₃.obj₂ ⟶ Z₄
h₄ : Z₄ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₂
mem₄ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk b g₄ h₄)
mem₄' : P Z₄
comm : T₁.mor₁ ≫ b = a ≫ T₃.mor₁
Z₂ : C
g₂ : T₃.obj₂ ⟶ Z₂
h₂ : Z₂ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₁
mem₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk (T₁.mor₁ ≫ b) g₂ h₂)
⊢ ∃ (c : T₁.obj₃ ⟶ T₃.obj₃) (_ : P.trW c),
    T₁.mor₂ ≫ c = b ≫ T₃.mor₂ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a = c ≫ T₃.mor₃
have have : ∀ (H : Octahedron sorry mem₁ mem₄ mem₂),
  ∃ (c : T₁.obj₃ ⟶ T₃.obj₃) (_ : P.trW c),
    T₁.mor₂ ≫ c = b ≫ T₃.mor₂ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a = c ≫ T₃.mor₃
Tactic: have H := someOctahedron rfl mem₁ mem₄ mem₂
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : IsTriangulated.{v_1, u_1} C
inst_15 : P.IsTriangulated
T₁ T₃ : Triangle.{v_1, u_1} C
mem₁ : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T₁
mem₃ : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T₃
a : T₁.obj₁ ⟶ T₃.obj₁
b : T₁.obj₂ ⟶ T₃.obj₂
Z₅ : C
g₅ : T₃.obj₁ ⟶ Z₅
h₅ : Z₅ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₁
mem₅ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk a g₅ h₅)
mem₅' : P Z₅
Z₄ : C
g₄ : T₃.obj₂ ⟶ Z₄
h₄ : Z₄ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₂
mem₄ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk b g₄ h₄)
mem₄' : P Z₄
comm : T₁.mor₁ ≫ b = a ≫ T₃.mor₁
Z₂ : C
g₂ : T₃.obj₂ ⟶ Z₂
h₂ : Z₂ ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) T₁.obj₁
mem₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk (T₁.mor₁ ≫ b) g₂ h₂)
H : Octahedron ⋯ mem₁ mem₄ mem₂
⊢ ∃ (c : T₁.obj₃ ⟶ T₃.obj₃) (_ : P.trW c),
    T₁.mor₂ ≫ c = b ≫ T₃.mor₂ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a = c ≫ T₃.mor₃
have have : ∀ (H' : Octahedron sorry mem₅ mem₃ mem₂),
  ∃ (c : T₁.obj₃ ⟶ T₃.obj₃) (_ : P.trW c),
    T₁.mor₂ ≫ c = b ≫ T₃.mor₂ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a = c ≫ T₃.mor₃
Tactic: have H' := someOctahedron comm.symm mem₅ mem₃ mem₂
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P P' : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsTriangulatedClosed₂
inst_15 : P.IsClosedUnderIsomorphisms
inst_16 : P'.IsTriangulatedClosed₂
T : Triangle.{v_1, u_1} C
hT : Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T
h₁ : (P ⊓ P') T.obj₁
h₃ : (P ⊓ P') T.obj₃
⊢ (P ⊓ P').isoClosure T.obj₂
have obtain : ∀ (X₂ : C) (h₂ : P' X₂) (e : T.obj₂ ≅ X₂), (P ⊓ P').isoClosure T.obj₂
Tactic: obtain ⟨X₂, h₂, ⟨e⟩⟩ := P'.ext_of_isTriangulatedClosed₂' T hT h₁.2 h₃.2
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsTriangulated
inst_15 : P.IsClosedUnderIsomorphisms
X : C
p : P.LimitOfShape (Discrete WalkingPair) X
⊢ P X
have refine : Triangle.obj₂.{v_1, u_1}
    (binaryProductTriangle (p.toLimitPresentation (P := P).diag.1 { as := WalkingPair.left })
      (p.toLimitPresentation (P := P).diag.1 { as := WalkingPair.right })) ≅
  X
Tactic: refine P.prop_of_iso ?_ (P.ext_of_isTriangulatedClosed₂ _
      (binaryProductTriangle_distinguished _ _)
      (p.prop_diag_obj (.mk .left)) (p.prop_diag_obj (.mk .right)))
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsTriangulated
⊢ P.trW.IsStableUnderFiniteProducts
have rw : P.isoClosure.trW.IsStableUnderFiniteProducts
Tactic: rw [← trW_isoClosure]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsTriangulated
J : Type
h : Finite J
⊢ P.isoClosure.trW.IsStableUnderProductsOfShape J
have refine : ∀ (X₁ X₂ : J → C) [inst_15 : HasProduct.{0, v_1, u_1} X₁] [inst_16 : HasProduct.{0, v_1, u_1} X₂]
  (f : (j : J) → X₁ j ⟶ X₂ j),
  (∀ (j : J), P.isoClosure.trW (f j)) →
    P.isoClosure.trW (X := piObj.{0, v_1, u_1} X₁) (Y := piObj.{0, v_1, u_1} X₂) (Limits.Pi.map f)
Tactic: refine MorphismProperty.IsStableUnderProductsOfShape.mk _ _ ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsTriangulated
X Y : P.FullSubcategory
f : X ⟶ Y
⊢ ∃ (Z : P.FullSubcategory) (g : Y ⟶ Z) (h :
    Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor P.FullSubcategory (A := ℤ) 1) X),
    Membership.mem (γ := Set (Triangle.{v_1, u_1} P.FullSubcategory))
      (Set.preimage (β := Triangle.{v_1, u_1} C) (Functor.obj P.ι.mapTriangle)
        (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _))
      (Triangle.mk f g h)
have obtain : ∀ (Z' : C) (g' : P.ι.obj Y ⟶ Z') (h' : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (P.ι.obj X))
  (mem :
    Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk (P.ι.map f) g' h')),
  ∃ (Z : P.FullSubcategory) (g : Y ⟶ Z) (h :
    Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor P.FullSubcategory (A := ℤ) 1) X),
    Membership.mem (γ := Set (Triangle.{v_1, u_1} P.FullSubcategory))
      (Set.preimage (β := Triangle.{v_1, u_1} C) (Functor.obj P.ι.mapTriangle)
        (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _))
      (Triangle.mk f g h)
Tactic: obtain ⟨Z', g', h', mem⟩ := distinguished_cocone_triangle (P.ι.map f)
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsTriangulated
X Y : P.FullSubcategory
f : X ⟶ Y
Z' : C
g' : P.ι.obj Y ⟶ Z'
h' : Z' ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (A := ℤ) 1) (P.ι.obj X)
mem :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk (P.ι.map f) g' h')
⊢ ∃ (Z : P.FullSubcategory) (g : Y ⟶ Z) (h :
    Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor P.FullSubcategory (A := ℤ) 1) X),
    Membership.mem (γ := Set (Triangle.{v_1, u_1} P.FullSubcategory))
      (Set.preimage (β := Triangle.{v_1, u_1} C) (Functor.obj P.ι.mapTriangle)
        (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _))
      (Triangle.mk f g h)
have obtain : ∀ (Z'' : C) (hZ'' : P Z'') (e : (Triangle.mk (P.ι.map f) g' h').obj₃ ≅ Z''),
  ∃ (Z : P.FullSubcategory) (g : Y ⟶ Z) (h :
    Z ⟶ Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor P.FullSubcategory (A := ℤ) 1) X),
    Membership.mem (γ := Set (Triangle.{v_1, u_1} P.FullSubcategory))
      (Set.preimage (β := Triangle.{v_1, u_1} C) (Functor.obj P.ι.mapTriangle)
        (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _))
      (Triangle.mk f g h)
Tactic: obtain ⟨Z'', hZ'', ⟨e⟩⟩ := P.ext_of_isTriangulatedClosed₃' _ mem X.2 Y.2
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsTriangulated
T₁ T₂ : Triangle.{v_1, u_1} P.FullSubcategory
hT₁ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} P.FullSubcategory))
    (Set.preimage (β := Triangle.{v_1, u_1} C) (Functor.obj P.ι.mapTriangle)
      (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _))
    T₁
hT₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} P.FullSubcategory))
    (Set.preimage (β := Triangle.{v_1, u_1} C) (Functor.obj P.ι.mapTriangle)
      (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _))
    T₂
a : T₁.obj₁ ⟶ T₂.obj₁
b : T₁.obj₂ ⟶ T₂.obj₂
comm : T₁.mor₁ ≫ b = a ≫ T₂.mor₁
⊢ ∃ (c : T₁.obj₃ ⟶ T₂.obj₃),
    T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor P.FullSubcategory (A := ℤ) 1) a = c ≫ T₂.mor₃
have obtain : ∀ (c : Triangle.obj₃ (Functor.obj P.ι.mapTriangle T₁) ⟶ Triangle.obj₃ (Functor.obj P.ι.mapTriangle T₂))
  (hc₁ :
    Triangle.mor₂ (Functor.obj P.ι.mapTriangle T₁) ≫ c = P.ι.map b ≫ Triangle.mor₂ (Functor.obj P.ι.mapTriangle T₂))
  (hc₂ :
    Triangle.mor₃ (Functor.obj P.ι.mapTriangle T₁) ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (P.ι.map a) =
      c ≫ Triangle.mor₃ (Functor.obj P.ι.mapTriangle T₂)),
  ∃ (c : T₁.obj₃ ⟶ T₂.obj₃),
    T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor P.FullSubcategory (A := ℤ) 1) a = c ≫ T₂.mor₃
Tactic: obtain ⟨c, ⟨hc₁, hc₂⟩⟩ := complete_distinguished_triangle_morphism (P.ι.mapTriangle.obj T₁)
      (P.ι.mapTriangle.obj T₂) hT₁ hT₂ (P.ι.map a) (P.ι.map b)
      (by simpa using P.ι.congr_map comm)
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsTriangulated
T₁ T₂ : Triangle.{v_1, u_1} P.FullSubcategory
hT₁ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} P.FullSubcategory))
    (Set.preimage (β := Triangle.{v_1, u_1} C) (Functor.obj P.ι.mapTriangle)
      (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _))
    T₁
hT₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} P.FullSubcategory))
    (Set.preimage (β := Triangle.{v_1, u_1} C) (Functor.obj P.ι.mapTriangle)
      (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _))
    T₂
a : T₁.obj₁ ⟶ T₂.obj₁
b : T₁.obj₂ ⟶ T₂.obj₂
comm : T₁.mor₁ ≫ b = a ≫ T₂.mor₁
c : Triangle.obj₃ (Functor.obj P.ι.mapTriangle T₁) ⟶ Triangle.obj₃ (Functor.obj P.ι.mapTriangle T₂)
hc₁ : Triangle.mor₂ (Functor.obj P.ι.mapTriangle T₁) ≫ c = P.ι.map b ≫ Triangle.mor₂ (Functor.obj P.ι.mapTriangle T₂)
hc₂ :
  Triangle.mor₃ (Functor.obj P.ι.mapTriangle T₁) ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (P.ι.map a) =
    c ≫ Triangle.mor₃ (Functor.obj P.ι.mapTriangle T₂)
⊢ ∃ (c : T₁.obj₃ ⟶ T₂.obj₃),
    T₁.mor₂ ≫ c = b ≫ T₂.mor₂ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor P.FullSubcategory (A := ℤ) 1) a = c ≫ T₂.mor₃
have refine : T₁.mor₃ ≫ Functor.map (shiftFunctor P.FullSubcategory (A := ℤ) 1) a = P.fullyFaithfulι.preimage c ≫ T₂.mor₃
Tactic: refine ⟨P.fullyFaithfulι.preimage c, ⟨by cat_disch, ?_⟩⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Subcategory (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : HasZeroObject.{v_1, u_1} C
inst_2 : HasShift.{v_1, u_1, 0} C ℤ
inst_3 : Preadditive.{v_1, u_1} C
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
D : Type u_2
inst_6 : Category.{v_2, u_2} D
inst_7 : Preadditive.{v_2, u_2} D
inst_8 : HasZeroObject.{v_2, u_2} D
inst_9 : HasShift.{v_2, u_2, 0} D ℤ
inst_10 : ∀ (n : ℤ), Functor.Additive.{v_2, u_2, v_2, u_2} (shiftFunctor D n)
inst_11 : Pretriangulated.{v_2, u_2} D
E : Type u_3
inst_12 : Category.{v_3, u_3} E
inst_13 : HasShift.{v_3, u_3, 0} E ℤ
P : ObjectProperty.{v_1, u_1} C
inst_14 : P.IsTriangulated
T₁ T₂ : Triangle.{v_1, u_1} P.FullSubcategory
hT₁ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} P.FullSubcategory))
    (Set.preimage (β := Triangle.{v_1, u_1} C) (Functor.obj P.ι.mapTriangle)
      (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _))
    T₁
hT₂ :
  Membership.mem (γ := Set (Triangle.{v_1, u_1} P.FullSubcategory))
    (Set.preimage (β := Triangle.{v_1, u_1} C) (Functor.obj P.ι.mapTriangle)
      (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _))
    T₂
a : T₁.obj₁ ⟶ T₂.obj₁
b : T₁.obj₂ ⟶ T₂.obj₂
comm : T₁.mor₁ ≫ b = a ≫ T₂.mor₁
c : Triangle.obj₃ (Functor.obj P.ι.mapTriangle T₁) ⟶ Triangle.obj₃ (Functor.obj P.ι.mapTriangle T₂)
hc₁ : Triangle.mor₂ (Functor.obj P.ι.mapTriangle T₁) ≫ c = P.ι.map b ≫ Triangle.mor₂ (Functor.obj P.ι.mapTriangle T₂)
hc₂ :
  Triangle.mor₃ (Functor.obj P.ι.mapTriangle T₁) ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (P.ι.map a) =
    c ≫ Triangle.mor₃ (Functor.obj P.ι.mapTriangle T₂)
⊢ T₁.mor₃ ≫ Functor.map (shiftFunctor P.FullSubcategory (A := ℤ) 1) a = P.fullyFaithfulι.preimage c ≫ T₂.mor₃
have h : (T₁.mor₃ ≫ Functor.map (shiftFunctor P.FullSubcategory (A := ℤ) 1) a).hom = (P.fullyFaithfulι.preimage c ≫ T₂.mor₃).hom
Tactic: ext