awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Orthogonal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
P : ObjectProperty.{v, u} C
inst_2 : HasZeroObject.{v, u} C
inst_3 : HasShift.{v, u, 0} C ℤ
inst_4 : Preadditive.{v, u} C
inst_5 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
inst_6 : Pretriangulated.{v, u} C
inst_7 : P.IsTriangulated
inst_8 : IsTriangulated.{v, u} C
Y : C
hY : P.trW.isLocal Y
L : Functor.{v, v', u, u'} C D
inst_9 : L.IsLocalization P.trW
X : C
g : L.obj X ⟶ L.obj Y
φ : P.trW.RightFraction X Y
hφ : g = φ.map (W := P.trW) L ⋯
α : X ⟶ Y
hα : (fun (g : X ⟶ Y) ↦ φ.s (W := P.trW) ≫ g) α = φ.f (W := P.trW)
refine_2 : L.map α = φ.map (W := P.trW) L ⋯
⊢ L.map α = g
have refine_2₁ : L.map (φ.s (W := P.trW)) ≫ L.map α = L.map (φ.s (W := P.trW)) ≫ φ.map (W := P.trW) L sorry
Tactic: rw [hφ, ← cancel_epi (L.map φ.s), MorphismProperty.RightFraction.map_s_comp_map,
      ← hα, Functor.map_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Orthogonal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
P : ObjectProperty.{v, u} C
inst_2 : HasZeroObject.{v, u} C
inst_3 : HasShift.{v, u, 0} C ℤ
inst_4 : Preadditive.{v, u} C
inst_5 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
inst_6 : Pretriangulated.{v, u} C
inst_7 : P.IsTriangulated
inst_8 : IsTriangulated.{v, u} C
Y : C
hY : P.trW.isLocal Y
L : Functor.{v, v', u, u'} C D
inst_9 : L.IsLocalization P.trW
X : C
g : L.obj X ⟶ L.obj Y
φ : P.trW.RightFraction X Y
hφ : g = φ.map (W := P.trW) L ⋯
α : X ⟶ Y
hα : (fun (g : X ⟶ Y) ↦ φ.s (W := P.trW) ≫ g) α = φ.f (W := P.trW)
refine_2 : L.map α = φ.map (W := P.trW) L ⋯
refine_2₁ : L.map (φ.s (W := P.trW)) ≫ L.map α = L.map (φ.s (W := P.trW)) ≫ φ.map (W := P.trW) L ⋯
⊢ L.map α = g
have refine_2₂ : L.map (φ.s (W := P.trW)) ≫ L.map α = L.map (φ.f (W := P.trW))
Tactic: rw [hφ, ← cancel_epi (L.map φ.s), MorphismProperty.RightFraction.map_s_comp_map,
      ← hα, Functor.map_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Orthogonal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
P : ObjectProperty.{v, u} C
inst_2 : HasZeroObject.{v, u} C
inst_3 : HasShift.{v, u, 0} C ℤ
inst_4 : Preadditive.{v, u} C
inst_5 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
inst_6 : Pretriangulated.{v, u} C
inst_7 : P.IsTriangulated
inst_8 : IsTriangulated.{v, u} C
Y : C
hY : P.trW.isLocal Y
L : Functor.{v, v', u, u'} C D
inst_9 : L.IsLocalization P.trW
X : C
g : L.obj X ⟶ L.obj Y
φ : P.trW.RightFraction X Y
hφ : g = φ.map (W := P.trW) L ⋯
α : X ⟶ Y
hα : (fun (g : X ⟶ Y) ↦ φ.s (W := P.trW) ≫ g) α = φ.f (W := P.trW)
refine_2 : L.map α = φ.map (W := P.trW) L ⋯
refine_2₁ : L.map (φ.s (W := P.trW)) ≫ L.map α = L.map (φ.s (W := P.trW)) ≫ φ.map (W := P.trW) L ⋯
refine_2₂ : L.map (φ.s (W := P.trW)) ≫ L.map α = L.map (φ.f (W := P.trW))
⊢ L.map α = g
have refine_2₃ : L.map (φ.s (W := P.trW)) ≫ L.map α = L.map ((fun (g : X ⟶ Y) ↦ φ.s (W := P.trW) ≫ g) α)
Tactic: rw [hφ, ← cancel_epi (L.map φ.s), MorphismProperty.RightFraction.map_s_comp_map,
      ← hα, Functor.map_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Orthogonal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
P : ObjectProperty.{v, u} C
inst_2 : HasZeroObject.{v, u} C
inst_3 : HasShift.{v, u, 0} C ℤ
inst_4 : Preadditive.{v, u} C
inst_5 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
inst_6 : Pretriangulated.{v, u} C
inst_7 : P.IsTriangulated
inst_8 : IsTriangulated.{v, u} C
Y : C
hY : P.trW.isLocal Y
L : Functor.{v, v', u, u'} C D
inst_9 : L.IsLocalization P.trW
X : C
g : L.obj X ⟶ L.obj Y
φ : P.trW.RightFraction X Y
hφ : g = φ.map (W := P.trW) L ⋯
α : X ⟶ Y
hα : (fun (g : X ⟶ Y) ↦ φ.s (W := P.trW) ≫ g) α = φ.f (W := P.trW)
refine_2 : L.map α = φ.map (W := P.trW) L ⋯
refine_2₁ : L.map (φ.s (W := P.trW)) ≫ L.map α = L.map (φ.s (W := P.trW)) ≫ φ.map (W := P.trW) L ⋯
refine_2₂ : L.map (φ.s (W := P.trW)) ≫ L.map α = L.map (φ.f (W := P.trW))
refine_2₃ : L.map (φ.s (W := P.trW)) ≫ L.map α = L.map ((fun (g : X ⟶ Y) ↦ φ.s (W := P.trW) ≫ g) α)
⊢ L.map α = g
have refine_2₄ : L.map (φ.s (W := P.trW)) ≫ L.map α = L.map (φ.s (W := P.trW)) ≫ L.map α
Tactic: rw [hφ, ← cancel_epi (L.map φ.s), MorphismProperty.RightFraction.map_s_comp_map,
      ← hα, Functor.map_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Orthogonal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
P : ObjectProperty.{v, u} C
inst_2 : HasZeroObject.{v, u} C
inst_3 : HasShift.{v, u, 0} C ℤ
inst_4 : Preadditive.{v, u} C
inst_5 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
inst_6 : Pretriangulated.{v, u} C
inst_7 : P.IsTriangulated
inst_8 : IsTriangulated.{v, u} C
X : C
hX : P.leftOrthogonal X
L : Functor.{v, v', u, u'} C D
inst_9 : L.IsLocalization P.trW
Y : C
⊢ Function.Bijective (α := X ⟶ Y) L.map
have rw : ∀ (hX : P.trW.isColocal X) (L : Functor.{v, v', u, u'} C D) [L.IsLocalization P.trW] (Y : C),
  Function.Bijective (α := X ⟶ Y) L.map
Tactic: rw [← isColocal_trW] at hX
Cleared hypotheses: hX, L, inst_9, Y
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Orthogonal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
P : ObjectProperty.{v, u} C
inst_2 : HasZeroObject.{v, u} C
inst_3 : HasShift.{v, u, 0} C ℤ
inst_4 : Preadditive.{v, u} C
inst_5 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
inst_6 : Pretriangulated.{v, u} C
inst_7 : P.IsTriangulated
inst_8 : IsTriangulated.{v, u} C
X : C
hX : P.trW.isColocal X
L : Functor.{v, v', u, u'} C D
inst_9 : L.IsLocalization P.trW
Y : C
⊢ Function.Bijective (α := X ⟶ Y) L.map
have refine_1 : ∀ (f₁ f₂ : X ⟶ Y) (hf : L.map f₁ = L.map f₂), f₁ = f₂
Tactic: refine ⟨fun f₁ f₂ hf ↦ ?_, fun g ↦ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Orthogonal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
P : ObjectProperty.{v, u} C
inst_2 : HasZeroObject.{v, u} C
inst_3 : HasShift.{v, u, 0} C ℤ
inst_4 : Preadditive.{v, u} C
inst_5 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
inst_6 : Pretriangulated.{v, u} C
inst_7 : P.IsTriangulated
inst_8 : IsTriangulated.{v, u} C
X : C
hX : P.trW.isColocal X
L : Functor.{v, v', u, u'} C D
inst_9 : L.IsLocalization P.trW
Y : C
refine_1 : ∀ (f₁ f₂ : X ⟶ Y) (hf : L.map f₁ = L.map f₂), f₁ = f₂
⊢ Function.Bijective (α := X ⟶ Y) L.map
have refine_2 : ∀ (g : L.obj X ⟶ L.obj Y), ∃ (a : X ⟶ Y), L.map a = g
Tactic: refine ⟨fun f₁ f₂ hf ↦ ?_, fun g ↦ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Orthogonal (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
P : ObjectProperty.{v, u} C
inst_2 : HasZeroObject.{v, u} C
inst_3 : HasShift.{v, u, 0} C ℤ
inst_4 : Preadditive.{v, u} C
inst_5 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
inst_6 : Pretriangulated.{v, u} C
inst_7 : P.IsTriangulated
inst_8 : IsTriangulated.{v, u} C
X : C
hX : P.trW.isColocal X
L : Functor.{v, v', u, u'} C D
inst_9 : L.IsLocalization P.trW
Y : C
f₁ f₂ : X ⟶ Y
hf : L.map f₁ = L.map f₂
⊢ f₁ = f₂
have hf₁ : ∃ (Z : C) (s : Y ⟶ Z) (_ : P.trW s), f₁ ≫ s = f₂ ≫ s
Tactic: rw [MorphismProperty.map_eq_iff_postcomp L P.trW] at hf
Cleared hypotheses: hf
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Orthogonal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
P : ObjectProperty.{v, u} C
inst_2 : HasZeroObject.{v, u} C
inst_3 : HasShift.{v, u, 0} C ℤ
inst_4 : Preadditive.{v, u} C
inst_5 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
inst_6 : Pretriangulated.{v, u} C
inst_7 : P.IsTriangulated
inst_8 : IsTriangulated.{v, u} C
X : C
hX : P.trW.isColocal X
L : Functor.{v, v', u, u'} C D
inst_9 : L.IsLocalization P.trW
Y : C
f₁ f₂ : X ⟶ Y
hf : ∃ (Z : C) (s : Y ⟶ Z) (_ : P.trW s), f₁ ≫ s = f₂ ≫ s
⊢ f₁ = f₂
have refine_1 : ∀ (Z : C) (s : Y ⟶ Z) (hs : P.trW s) (eq : f₁ ≫ s = f₂ ≫ s), f₁ = f₂
Tactic: obtain ⟨Z, s, hs, eq⟩ := hf
Cleared hypotheses: hf
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Orthogonal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
P : ObjectProperty.{v, u} C
inst_2 : HasZeroObject.{v, u} C
inst_3 : HasShift.{v, u, 0} C ℤ
inst_4 : Preadditive.{v, u} C
inst_5 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
inst_6 : Pretriangulated.{v, u} C
inst_7 : P.IsTriangulated
inst_8 : IsTriangulated.{v, u} C
X : C
hX : P.trW.isColocal X
L : Functor.{v, v', u, u'} C D
inst_9 : L.IsLocalization P.trW
Y : C
g : L.obj X ⟶ L.obj Y
⊢ ∃ (a : X ⟶ Y), L.map a = g
have refine_2 : ∀ (φ : P.trW.LeftFraction X Y) (hφ : g = φ.map (W := P.trW) L sorry), ∃ (a : X ⟶ Y), L.map a = g
Tactic: obtain ⟨φ, hφ⟩ := Localization.exists_leftFraction L P.trW g
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Orthogonal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
P : ObjectProperty.{v, u} C
inst_2 : HasZeroObject.{v, u} C
inst_3 : HasShift.{v, u, 0} C ℤ
inst_4 : Preadditive.{v, u} C
inst_5 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
inst_6 : Pretriangulated.{v, u} C
inst_7 : P.IsTriangulated
inst_8 : IsTriangulated.{v, u} C
X : C
hX : P.trW.isColocal X
L : Functor.{v, v', u, u'} C D
inst_9 : L.IsLocalization P.trW
Y : C
g : L.obj X ⟶ L.obj Y
φ : P.trW.LeftFraction X Y
hφ : g = φ.map (W := P.trW) L ⋯
⊢ ∃ (a : X ⟶ Y), L.map a = g
have refine_2 : ∀ (α : X ⟶ Y) (hα : (fun (f : X ⟶ Y) ↦ f ≫ φ.s (W := P.trW)) α = φ.f (W := P.trW)), ∃ (a : X ⟶ Y), L.map a = g
Tactic: obtain ⟨α, hα⟩ := (hX _ φ.hs).2 φ.f
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Orthogonal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
P : ObjectProperty.{v, u} C
inst_2 : HasZeroObject.{v, u} C
inst_3 : HasShift.{v, u, 0} C ℤ
inst_4 : Preadditive.{v, u} C
inst_5 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
inst_6 : Pretriangulated.{v, u} C
inst_7 : P.IsTriangulated
inst_8 : IsTriangulated.{v, u} C
X : C
hX : P.trW.isColocal X
L : Functor.{v, v', u, u'} C D
inst_9 : L.IsLocalization P.trW
Y : C
g : L.obj X ⟶ L.obj Y
φ : P.trW.LeftFraction X Y
hφ : g = φ.map (W := P.trW) L ⋯
α : X ⟶ Y
hα : (fun (f : X ⟶ Y) ↦ f ≫ φ.s (W := P.trW)) α = φ.f (W := P.trW)
⊢ ∃ (a : X ⟶ Y), L.map a = g
have refine_2 : L.map α = g
Tactic: refine ⟨α, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Orthogonal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
P : ObjectProperty.{v, u} C
inst_2 : HasZeroObject.{v, u} C
inst_3 : HasShift.{v, u, 0} C ℤ
inst_4 : Preadditive.{v, u} C
inst_5 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
inst_6 : Pretriangulated.{v, u} C
inst_7 : P.IsTriangulated
inst_8 : IsTriangulated.{v, u} C
X : C
hX : P.trW.isColocal X
L : Functor.{v, v', u, u'} C D
inst_9 : L.IsLocalization P.trW
Y : C
g : L.obj X ⟶ L.obj Y
φ : P.trW.LeftFraction X Y
hφ : g = φ.map (W := P.trW) L ⋯
α : X ⟶ Y
hα : (fun (f : X ⟶ Y) ↦ f ≫ φ.s (W := P.trW)) α = φ.f (W := P.trW)
⊢ L.map α = g
have refine_2 : L.map α = φ.map (W := P.trW) L sorry
Tactic: rw [hφ, ← cancel_mono (L.map φ.s), MorphismProperty.LeftFraction.map_comp_map_s,
      ← hα, Functor.map_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Orthogonal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
P : ObjectProperty.{v, u} C
inst_2 : HasZeroObject.{v, u} C
inst_3 : HasShift.{v, u, 0} C ℤ
inst_4 : Preadditive.{v, u} C
inst_5 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
inst_6 : Pretriangulated.{v, u} C
inst_7 : P.IsTriangulated
inst_8 : IsTriangulated.{v, u} C
X : C
hX : P.trW.isColocal X
L : Functor.{v, v', u, u'} C D
inst_9 : L.IsLocalization P.trW
Y : C
g : L.obj X ⟶ L.obj Y
φ : P.trW.LeftFraction X Y
hφ : g = φ.map (W := P.trW) L ⋯
α : X ⟶ Y
hα : (fun (f : X ⟶ Y) ↦ f ≫ φ.s (W := P.trW)) α = φ.f (W := P.trW)
refine_2 : L.map α = φ.map (W := P.trW) L ⋯
⊢ L.map α = g
have refine_2₁ : L.map α ≫ L.map (φ.s (W := P.trW)) = φ.map (W := P.trW) L sorry ≫ L.map (φ.s (W := P.trW))
Tactic: rw [hφ, ← cancel_mono (L.map φ.s), MorphismProperty.LeftFraction.map_comp_map_s,
      ← hα, Functor.map_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Orthogonal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
P : ObjectProperty.{v, u} C
inst_2 : HasZeroObject.{v, u} C
inst_3 : HasShift.{v, u, 0} C ℤ
inst_4 : Preadditive.{v, u} C
inst_5 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
inst_6 : Pretriangulated.{v, u} C
inst_7 : P.IsTriangulated
inst_8 : IsTriangulated.{v, u} C
X : C
hX : P.trW.isColocal X
L : Functor.{v, v', u, u'} C D
inst_9 : L.IsLocalization P.trW
Y : C
g : L.obj X ⟶ L.obj Y
φ : P.trW.LeftFraction X Y
hφ : g = φ.map (W := P.trW) L ⋯
α : X ⟶ Y
hα : (fun (f : X ⟶ Y) ↦ f ≫ φ.s (W := P.trW)) α = φ.f (W := P.trW)
refine_2 : L.map α = φ.map (W := P.trW) L ⋯
refine_2₁ : L.map α ≫ L.map (φ.s (W := P.trW)) = φ.map (W := P.trW) L ⋯ ≫ L.map (φ.s (W := P.trW))
⊢ L.map α = g
have refine_2₂ : L.map α ≫ L.map (φ.s (W := P.trW)) = L.map (φ.f (W := P.trW))
Tactic: rw [hφ, ← cancel_mono (L.map φ.s), MorphismProperty.LeftFraction.map_comp_map_s,
      ← hα, Functor.map_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Orthogonal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
P : ObjectProperty.{v, u} C
inst_2 : HasZeroObject.{v, u} C
inst_3 : HasShift.{v, u, 0} C ℤ
inst_4 : Preadditive.{v, u} C
inst_5 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
inst_6 : Pretriangulated.{v, u} C
inst_7 : P.IsTriangulated
inst_8 : IsTriangulated.{v, u} C
X : C
hX : P.trW.isColocal X
L : Functor.{v, v', u, u'} C D
inst_9 : L.IsLocalization P.trW
Y : C
g : L.obj X ⟶ L.obj Y
φ : P.trW.LeftFraction X Y
hφ : g = φ.map (W := P.trW) L ⋯
α : X ⟶ Y
hα : (fun (f : X ⟶ Y) ↦ f ≫ φ.s (W := P.trW)) α = φ.f (W := P.trW)
refine_2 : L.map α = φ.map (W := P.trW) L ⋯
refine_2₁ : L.map α ≫ L.map (φ.s (W := P.trW)) = φ.map (W := P.trW) L ⋯ ≫ L.map (φ.s (W := P.trW))
refine_2₂ : L.map α ≫ L.map (φ.s (W := P.trW)) = L.map (φ.f (W := P.trW))
⊢ L.map α = g
have refine_2₃ : L.map α ≫ L.map (φ.s (W := P.trW)) = L.map ((fun (f : X ⟶ Y) ↦ f ≫ φ.s (W := P.trW)) α)
Tactic: rw [hφ, ← cancel_mono (L.map φ.s), MorphismProperty.LeftFraction.map_comp_map_s,
      ← hα, Functor.map_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Orthogonal (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u'
inst_1 : Category.{v', u'} D
P : ObjectProperty.{v, u} C
inst_2 : HasZeroObject.{v, u} C
inst_3 : HasShift.{v, u, 0} C ℤ
inst_4 : Preadditive.{v, u} C
inst_5 : ∀ (n : ℤ), Functor.Additive.{v, u, v, u} (shiftFunctor C n)
inst_6 : Pretriangulated.{v, u} C
inst_7 : P.IsTriangulated
inst_8 : IsTriangulated.{v, u} C
X : C
hX : P.trW.isColocal X
L : Functor.{v, v', u, u'} C D
inst_9 : L.IsLocalization P.trW
Y : C
g : L.obj X ⟶ L.obj Y
φ : P.trW.LeftFraction X Y
hφ : g = φ.map (W := P.trW) L ⋯
α : X ⟶ Y
hα : (fun (f : X ⟶ Y) ↦ f ≫ φ.s (W := P.trW)) α = φ.f (W := P.trW)
refine_2 : L.map α = φ.map (W := P.trW) L ⋯
refine_2₁ : L.map α ≫ L.map (φ.s (W := P.trW)) = φ.map (W := P.trW) L ⋯ ≫ L.map (φ.s (W := P.trW))
refine_2₂ : L.map α ≫ L.map (φ.s (W := P.trW)) = L.map (φ.f (W := P.trW))
refine_2₃ : L.map α ≫ L.map (φ.s (W := P.trW)) = L.map ((fun (f : X ⟶ Y) ↦ f ≫ φ.s (W := P.trW)) α)
⊢ L.map α = g
have refine_2₄ : L.map α ≫ L.map (φ.s (W := P.trW)) = L.map α ≫ L.map (φ.s (W := P.trW))
Tactic: rw [hφ, ← cancel_mono (L.map φ.s), MorphismProperty.LeftFraction.map_comp_map_s,
      ← hα, Functor.map_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Pretriangulated (goal)
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
T : Triangle.{v, u} C
H : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T
⊢ T.mor₁ ≫ T.mor₂ = 0
have obtain : ∀ (c : Triangle.obj₃.{v, u} (contractibleTriangle T.obj₁) ⟶ T.obj₃)
  (hc :
    Triangle.mor₂ (contractibleTriangle T.obj₁) ≫ c = T.mor₁ ≫ T.mor₂ ∧
      Triangle.mor₃ (contractibleTriangle T.obj₁) ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (𝟙 T.obj₁) = c ≫ T.mor₃),
  T.mor₁ ≫ T.mor₂ = 0
Tactic: obtain ⟨c, hc⟩ :=
    complete_distinguished_triangle_morphism _ _ (contractible_distinguished T.obj₁) H (𝟙 T.obj₁)
      T.mor₁ rfl
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Pretriangulated (hypothesis)
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
T : Triangle.{v, u} C
H : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T
⊢ T.mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) T.mor₁ = 0
have H₂ : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T.rotate.rotate
Tactic: have H₂ := rot_of_distTriang T.rotate (rot_of_distTriang T H)
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Pretriangulated (goal)
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
Y Z : C
g : Y ⟶ Z
⊢ ∃ (X : C) (f : X ⟶ Y) (h : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) X),
    Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk f g h)
have obtain : ∀ (X' : C) (f' : Z ⟶ X') (g' : X' ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) Y)
  (mem :
    Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk g f' g')),
  ∃ (X : C) (f : X ⟶ Y) (h : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) X),
    Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk f g h)
Tactic: obtain ⟨X', f', g', mem⟩ := distinguished_cocone_triangle g
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Pretriangulated (goal)
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
Z X : C
h : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) X
⊢ ∃ (Y : C) (f : X ⟶ Y) (g : Y ⟶ Z),
    Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk f g h)
have obtain : ∀ (Y' : C) (f' : Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) X ⟶ Y')
  (g' : Y' ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) Z)
  (mem :
    Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk h f' g')),
  ∃ (Y : C) (f : X ⟶ Y) (g : Y ⟶ Z),
    Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk f g h)
Tactic: obtain ⟨Y', f', g', mem⟩ := distinguished_cocone_triangle h
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Pretriangulated (goal)
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
Z X : C
h : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) X
Y' : C
f' : Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) X ⟶ Y'
g' : Y' ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) Z
mem :
  Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk h f' g')
T' : Triangle.{v, u} C := (Triangle.mk h f' g').invRotate.invRotate
⊢ ∃ (Y : C) (f : X ⟶ Y) (g : Y ⟶ Z),
    Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
      (Triangle.mk f g h)
have refine : Triangle.mk (Iso.hom (Iso.app (Equivalence.unitIso (shiftEquiv C (A := ℤ) 1)) X) ≫ T'.mor₁) T'.mor₂ h ≅
  (Triangle.mk h f' g').invRotate.invRotate
Tactic: refine ⟨T'.obj₂, ((shiftEquiv C (1 : ℤ)).unitIso.app X).hom ≫ T'.mor₁, T'.mor₂,
    isomorphic_distinguished _ (inv_rot_of_distTriang _ (inv_rot_of_distTriang _ mem)) _ ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Pretriangulated (goal)
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
Z X : C
h : Z ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) X
Y' : C
f' : Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) X ⟶ Y'
g' : Y' ⟶ Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) Z
mem :
  Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _)
    (Triangle.mk h f' g')
T' : Triangle.{v, u} C := (Triangle.mk h f' g').invRotate.invRotate
⊢ (Triangle.mk (Iso.hom (Iso.app (Equivalence.unitIso (shiftEquiv C (A := ℤ) 1)) X) ≫ T'.mor₁) T'.mor₂ h).mor₃ ≫
      Functor.map (shiftFunctor C (A := ℤ) 1) (Iso.hom (Iso.app (Equivalence.unitIso (shiftEquiv C (A := ℤ) 1)) X)) =
    Iso.hom
        (Iso.refl
          (Triangle.mk (Iso.hom (Iso.app (Equivalence.unitIso (shiftEquiv C (A := ℤ) 1)) X) ≫ T'.mor₁) T'.mor₂
              h).obj₃) ≫
      (Triangle.mk h f' g').invRotate.invRotate.mor₃
have dsimp : h ≫ Functor.map (shiftFunctor C (A := ℤ) 1) (NatTrans.app (Iso.inv (shiftFunctorCompIsoId C 1 (-1) sorry)) X) =
  𝟙 Z ≫
    h ≫
      NatTrans.app (Iso.inv (shiftFunctorCompIsoId C (-1) 1 sorry)) (Functor.obj.{v, v, u, u} (shiftFunctor C (A := ℤ) 1) X)
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Triangulated.Pretriangulated (goal)
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
T₁ T₂ : Triangle.{v, u} C
hT₁ : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T₁
hT₂ : Membership.mem (γ := Set (Triangle.{v, u} C)) (@distinguishedTriangles _ _ inst_1 _ inst_3 inst_4 _) T₂
b : T₁.obj₂ ⟶ T₂.obj₂
c : T₁.obj₃ ⟶ T₂.obj₃
comm : T₁.mor₂ ≫ c = b ≫ T₂.mor₂
⊢ ∃ (a : T₁.obj₁ ⟶ T₂.obj₁),
    T₁.mor₁ ≫ b = a ≫ T₂.mor₁ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a = c ≫ T₂.mor₃
have obtain : ∀ (a : T₁.rotate.obj₃ ⟶ T₂.rotate.obj₃) (ha₁ : T₁.rotate.mor₂ ≫ a = c ≫ T₂.rotate.mor₂)
  (ha₂ : T₁.rotate.mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) b = a ≫ T₂.rotate.mor₃),
  ∃ (a : T₁.obj₁ ⟶ T₂.obj₁),
    T₁.mor₁ ≫ b = a ≫ T₂.mor₁ ∧ T₁.mor₃ ≫ Functor.map (shiftFunctor C (A := ℤ) 1) a = c ≫ T₂.mor₃
Tactic: obtain ⟨a, ⟨ha₁, ha₂⟩⟩ := complete_distinguished_triangle_morphism _ _
    (rot_of_distTriang _ hT₁) (rot_of_distTriang _ hT₂) b c comm