awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
W : MorphismProperty.{v_1, u_1} C
inst_2 : W.HasLeftCalculusOfFractions
X Y Z : C
z₁_1 : Hom W X Y
z₂_1 : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' (W := W) ⟶ U
t₂ : a₂.Y' (W := W) ⟶ U
hst : a₁.s (W := W) ≫ t₁ = a₂.s (W := W) ≫ t₂
hft : a₁.f (W := W) ≫ t₁ = a₂.f (W := W) ≫ t₂
ht : W (a₁.s (W := W) ≫ t₁)
z₁ : W.LeftFraction (a₁.Y' (W := W)) (b.Y' (W := W))
fac₁ :
  (RightFraction.mk (W := W) (a₁.s (W := W)) ⋯ (b.f (W := W))).f (W := W) ≫ z₁.s (W := W) =
    (RightFraction.mk (W := W) (a₁.s (W := W)) ⋯ (b.f (W := W))).s (W := W) ≫ z₁.f (W := W)
z₂ : W.LeftFraction (a₂.Y' (W := W)) (b.Y' (W := W))
fac₂ :
  (RightFraction.mk (W := W) (a₂.s (W := W)) ⋯ (b.f (W := W))).f (W := W) ≫ z₂.s (W := W) =
    (RightFraction.mk (W := W) (a₂.s (W := W)) ⋯ (b.f (W := W))).s (W := W) ≫ z₂.f (W := W)
refine_2 :
  ∀ (w₁ : W.LeftFraction U (z₁.Y' (W := W)))
    (fac₁' :
      (RightFraction.mk (W := W) (a₁.s (W := W) ≫ t₁) ht (b.f (W := W) ≫ z₁.s (W := W))).f (W := W) ≫ w₁.s (W := W) =
        (RightFraction.mk (W := W) (a₁.s (W := W) ≫ t₁) ht (b.f (W := W) ≫ z₁.s (W := W))).s (W := W) ≫ w₁.f (W := W)),
    Eq (α := Hom W X Z) ((fun (a : W.LeftFraction X Y) (b : W.LeftFraction Y Z) ↦ a.comp (W := W) b) a₁ b)
      ((fun (a : W.LeftFraction X Y) (b : W.LeftFraction Y Z) ↦ a.comp (W := W) b) a₂ b)
⊢ Eq (α := Hom W X Z) ((fun (a : W.LeftFraction X Y) (b : W.LeftFraction Y Z) ↦ a.comp (W := W) b) a₁ b)
    ((fun (a : W.LeftFraction X Y) (b : W.LeftFraction Y Z) ↦ a.comp (W := W) b) a₂ b)

Tactic:
obtain ⟨w₁, fac₁'⟩ := exists_leftFraction (RightFraction.mk (a₁.s ≫ t₁) ht (b.f ≫ z₁.s))

Premises used: CategoryTheory.MorphismProperty.RightFraction.mk, CategoryTheory.CategoryStruct.comp, CategoryTheory.MorphismProperty.LeftFraction.f, CategoryTheory.MorphismProperty.LeftFraction.s, CategoryTheory.MorphismProperty.HasLeftCalculusOfFractions.exists_leftFraction
-- Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
W : MorphismProperty.{v_1, u_1} C
inst_2 : W.HasLeftCalculusOfFractions
X Y Z : C
z₁_1 : Hom W X Y
z₂_1 : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' (W := W) ⟶ U
t₂ : a₂.Y' (W := W) ⟶ U
hst : a₁.s (W := W) ≫ t₁ = a₂.s (W := W) ≫ t₂
hft : a₁.f (W := W) ≫ t₁ = a₂.f (W := W) ≫ t₂
ht : W (a₁.s (W := W) ≫ t₁)
z₁ : W.LeftFraction (a₁.Y' (W := W)) (b.Y' (W := W))
fac₁ :
  (RightFraction.mk (W := W) (a₁.s (W := W)) ⋯ (b.f (W := W))).f (W := W) ≫ z₁.s (W := W) =
    (RightFraction.mk (W := W) (a₁.s (W := W)) ⋯ (b.f (W := W))).s (W := W) ≫ z₁.f (W := W)
z₂ : W.LeftFraction (a₂.Y' (W := W)) (b.Y' (W := W))
fac₂ :
  (RightFraction.mk (W := W) (a₂.s (W := W)) ⋯ (b.f (W := W))).f (W := W) ≫ z₂.s (W := W) =
    (RightFraction.mk (W := W) (a₂.s (W := W)) ⋯ (b.f (W := W))).s (W := W) ≫ z₂.f (W := W)
w₁ : W.LeftFraction U (z₁.Y' (W := W))
fac₁' :
  (RightFraction.mk (W := W) (a₁.s (W := W) ≫ t₁) ht (b.f (W := W) ≫ z₁.s (W := W))).f (W := W) ≫ w₁.s (W := W) =
    (RightFraction.mk (W := W) (a₁.s (W := W) ≫ t₁) ht (b.f (W := W) ≫ z₁.s (W := W))).s (W := W) ≫ w₁.f (W := W)
rw : W (a₁.s (W := W) ≫ t₁)
⊢ W (a₂.s (W := W) ≫ t₂)

Tactic:
rw [← hst]
-- Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
W : MorphismProperty.{v_1, u_1} C
inst_2 : W.HasLeftCalculusOfFractions
X Y Z : C
z₁_1 : Hom W X Y
z₂_1 : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' (W := W) ⟶ U
t₂ : a₂.Y' (W := W) ⟶ U
hst : a₁.s (W := W) ≫ t₁ = a₂.s (W := W) ≫ t₂
hft : a₁.f (W := W) ≫ t₁ = a₂.f (W := W) ≫ t₂
ht : W (a₁.s (W := W) ≫ t₁)
z₁ : W.LeftFraction (a₁.Y' (W := W)) (b.Y' (W := W))
fac₁ :
  (RightFraction.mk (W := W) (a₁.s (W := W)) ⋯ (b.f (W := W))).f (W := W) ≫ z₁.s (W := W) =
    (RightFraction.mk (W := W) (a₁.s (W := W)) ⋯ (b.f (W := W))).s (W := W) ≫ z₁.f (W := W)
z₂ : W.LeftFraction (a₂.Y' (W := W)) (b.Y' (W := W))
fac₂ :
  (RightFraction.mk (W := W) (a₂.s (W := W)) ⋯ (b.f (W := W))).f (W := W) ≫ z₂.s (W := W) =
    (RightFraction.mk (W := W) (a₂.s (W := W)) ⋯ (b.f (W := W))).s (W := W) ≫ z₂.f (W := W)
w₁ : W.LeftFraction U (z₁.Y' (W := W))
fac₁' :
  (RightFraction.mk (W := W) (a₁.s (W := W) ≫ t₁) ht (b.f (W := W) ≫ z₁.s (W := W))).f (W := W) ≫ w₁.s (W := W) =
    (RightFraction.mk (W := W) (a₁.s (W := W) ≫ t₁) ht (b.f (W := W) ≫ z₁.s (W := W))).s (W := W) ≫ w₁.f (W := W)
⊢ W (a₁.s (W := W) ≫ t₁)

Tactic:
ht
-- Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
W : MorphismProperty.{v_1, u_1} C
inst_2 : W.HasLeftCalculusOfFractions
X Y Z : C
z₁_1 : Hom W X Y
z₂_1 : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' (W := W) ⟶ U
t₂ : a₂.Y' (W := W) ⟶ U
hst : a₁.s (W := W) ≫ t₁ = a₂.s (W := W) ≫ t₂
hft : a₁.f (W := W) ≫ t₁ = a₂.f (W := W) ≫ t₂
ht : W (a₁.s (W := W) ≫ t₁)
z₁ : W.LeftFraction (a₁.Y' (W := W)) (b.Y' (W := W))
fac₁ :
  (RightFraction.mk (W := W) (a₁.s (W := W)) ⋯ (b.f (W := W))).f (W := W) ≫ z₁.s (W := W) =
    (RightFraction.mk (W := W) (a₁.s (W := W)) ⋯ (b.f (W := W))).s (W := W) ≫ z₁.f (W := W)
z₂ : W.LeftFraction (a₂.Y' (W := W)) (b.Y' (W := W))
fac₂ :
  (RightFraction.mk (W := W) (a₂.s (W := W)) ⋯ (b.f (W := W))).f (W := W) ≫ z₂.s (W := W) =
    (RightFraction.mk (W := W) (a₂.s (W := W)) ⋯ (b.f (W := W))).s (W := W) ≫ z₂.f (W := W)
w₁ : W.LeftFraction U (z₁.Y' (W := W))
fac₁' :
  (RightFraction.mk (W := W) (a₁.s (W := W) ≫ t₁) ht (b.f (W := W) ≫ z₁.s (W := W))).f (W := W) ≫ w₁.s (W := W) =
    (RightFraction.mk (W := W) (a₁.s (W := W) ≫ t₁) ht (b.f (W := W) ≫ z₁.s (W := W))).s (W := W) ≫ w₁.f (W := W)
⊢ W (a₁.s (W := W) ≫ t₁)

Tactic:
exact ht
-- Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
W : MorphismProperty.{v_1, u_1} C
inst_2 : W.HasLeftCalculusOfFractions
X Y Z : C
z₁_1 : Hom W X Y
z₂_1 : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' (W := W) ⟶ U
t₂ : a₂.Y' (W := W) ⟶ U
hst : a₁.s (W := W) ≫ t₁ = a₂.s (W := W) ≫ t₂
hft : a₁.f (W := W) ≫ t₁ = a₂.f (W := W) ≫ t₂
ht : W (a₁.s (W := W) ≫ t₁)
z₁ : W.LeftFraction (a₁.Y' (W := W)) (b.Y' (W := W))
fac₁ :
  (RightFraction.mk (W := W) (a₁.s (W := W)) ⋯ (b.f (W := W))).f (W := W) ≫ z₁.s (W := W) =
    (RightFraction.mk (W := W) (a₁.s (W := W)) ⋯ (b.f (W := W))).s (W := W) ≫ z₁.f (W := W)
z₂ : W.LeftFraction (a₂.Y' (W := W)) (b.Y' (W := W))
fac₂ :
  (RightFraction.mk (W := W) (a₂.s (W := W)) ⋯ (b.f (W := W))).f (W := W) ≫ z₂.s (W := W) =
    (RightFraction.mk (W := W) (a₂.s (W := W)) ⋯ (b.f (W := W))).s (W := W) ≫ z₂.f (W := W)
w₁ : W.LeftFraction U (z₁.Y' (W := W))
fac₁' :
  (RightFraction.mk (W := W) (a₁.s (W := W) ≫ t₁) ht (b.f (W := W) ≫ z₁.s (W := W))).f (W := W) ≫ w₁.s (W := W) =
    (RightFraction.mk (W := W) (a₁.s (W := W) ≫ t₁) ht (b.f (W := W) ≫ z₁.s (W := W))).s (W := W) ≫ w₁.f (W := W)
refine_2 :
  ∀ (w₂ : W.LeftFraction U (z₂.Y' (W := W)))
    (fac₂' :
      Eq (α :=
        RightFraction.X' (W := W) (RightFraction.mk (W := W) (a₂.s (W := W) ≫ t₂) ⋯ (b.f (W := W) ≫ z₂.s (W := W))) ⟶
          w₂.Y' (W := W))
        (RightFraction.f (W := W) (RightFraction.mk (W := W) (a₂.s (W := W) ≫ t₂) ⋯ (b.f (W := W) ≫ z₂.s (W := W))) ≫
          w₂.s (W := W))
        (RightFraction.s (W := W) (RightFraction.mk (W := W) (a₂.s (W := W) ≫ t₂) ⋯ (b.f (W := W) ≫ z₂.s (W := W))) ≫
          w₂.f (W := W))),
    Eq (α := Hom W X Z) ((fun (a : W.LeftFraction X Y) (b : W.LeftFraction Y Z) ↦ a.comp (W := W) b) a₁ b)
      ((fun (a : W.LeftFraction X Y) (b : W.LeftFraction Y Z) ↦ a.comp (W := W) b) a₂ b)
⊢ Eq (α := Hom W X Z) ((fun (a : W.LeftFraction X Y) (b : W.LeftFraction Y Z) ↦ a.comp (W := W) b) a₁ b)
    ((fun (a : W.LeftFraction X Y) (b : W.LeftFraction Y Z) ↦ a.comp (W := W) b) a₂ b)

Tactic:
obtain ⟨w₂, fac₂'⟩ := exists_leftFraction (RightFraction.mk (a₂.s ≫ t₂)
      (show W _ by rw [← hst]; exact ht) (b.f ≫ z₂.s))

Premises used: CategoryTheory.MorphismProperty.RightFraction.mk, CategoryTheory.CategoryStruct.comp, CategoryTheory.MorphismProperty.LeftFraction.f, CategoryTheory.MorphismProperty.LeftFraction.s, CategoryTheory.MorphismProperty.HasLeftCalculusOfFractions.exists_leftFraction
-- Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
W : MorphismProperty.{v_1, u_1} C
inst_2 : W.HasLeftCalculusOfFractions
X Y Z : C
z₁_1 : Hom W X Y
z₂_1 : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' (W := W) ⟶ U
t₂ : a₂.Y' (W := W) ⟶ U
hst : a₁.s (W := W) ≫ t₁ = a₂.s (W := W) ≫ t₂
hft : a₁.f (W := W) ≫ t₁ = a₂.f (W := W) ≫ t₂
ht : W (a₁.s (W := W) ≫ t₁)
z₁ : W.LeftFraction (a₁.Y' (W := W)) (b.Y' (W := W))
fac₁ : b.f (W := W) ≫ z₁.s (W := W) = a₁.s (W := W) ≫ z₁.f (W := W)
z₂ : W.LeftFraction (a₂.Y' (W := W)) (b.Y' (W := W))
fac₂ : b.f (W := W) ≫ z₂.s (W := W) = a₂.s (W := W) ≫ z₂.f (W := W)
w₁ : W.LeftFraction U (z₁.Y' (W := W))
fac₁' : (b.f (W := W) ≫ z₁.s (W := W)) ≫ w₁.s (W := W) = (a₁.s (W := W) ≫ t₁) ≫ w₁.f (W := W)
w₂ : W.LeftFraction U (z₂.Y' (W := W))
fac₂' : (b.f (W := W) ≫ z₂.s (W := W)) ≫ w₂.s (W := W) = (a₂.s (W := W) ≫ t₂) ≫ w₂.f (W := W)
p₁ : W.LeftFraction X Z :=
  LeftFraction.mk (W := W) (a₁.f (W := W) ≫ t₁ ≫ w₁.f (W := W)) (b.s (W := W) ≫ z₁.s (W := W) ≫ w₁.s (W := W)) ⋯
p₂ : W.LeftFraction X Z :=
  LeftFraction.mk (W := W) (a₂.f (W := W) ≫ t₂ ≫ w₂.f (W := W)) (b.s (W := W) ≫ z₂.s (W := W) ≫ w₂.s (W := W)) ⋯
⊢ b.f (W := W) ≫ z₁.s (W := W) ≫ w₁.s (W := W) = a₁.s (W := W) ≫ t₁ ≫ w₁.f (W := W)

Tactic:
simp only [assoc] at fac₁' fac₂'

Premises used: CategoryTheory.Category.assoc
-- Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
W : MorphismProperty.{v_1, u_1} C
inst_2 : W.HasLeftCalculusOfFractions
X Y Z : C
z₁_1 : Hom W X Y
z₂_1 : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' (W := W) ⟶ U
t₂ : a₂.Y' (W := W) ⟶ U
hst : a₁.s (W := W) ≫ t₁ = a₂.s (W := W) ≫ t₂
hft : a₁.f (W := W) ≫ t₁ = a₂.f (W := W) ≫ t₂
ht : W (a₁.s (W := W) ≫ t₁)
z₁ : W.LeftFraction (a₁.Y' (W := W)) (b.Y' (W := W))
fac₁ : b.f (W := W) ≫ z₁.s (W := W) = a₁.s (W := W) ≫ z₁.f (W := W)
z₂ : W.LeftFraction (a₂.Y' (W := W)) (b.Y' (W := W))
fac₂ : b.f (W := W) ≫ z₂.s (W := W) = a₂.s (W := W) ≫ z₂.f (W := W)
w₁ : W.LeftFraction U (z₁.Y' (W := W))
fac₁' : (b.f (W := W) ≫ z₁.s (W := W)) ≫ w₁.s (W := W) = (a₁.s (W := W) ≫ t₁) ≫ w₁.f (W := W)
w₂ : W.LeftFraction U (z₂.Y' (W := W))
fac₂' : (b.f (W := W) ≫ z₂.s (W := W)) ≫ w₂.s (W := W) = (a₂.s (W := W) ≫ t₂) ≫ w₂.f (W := W)
p₁ : W.LeftFraction X Z :=
  LeftFraction.mk (W := W) (a₁.f (W := W) ≫ t₁ ≫ w₁.f (W := W)) (b.s (W := W) ≫ z₁.s (W := W) ≫ w₁.s (W := W)) ⋯
p₂ : W.LeftFraction X Z :=
  LeftFraction.mk (W := W) (a₂.f (W := W) ≫ t₂ ≫ w₂.f (W := W)) (b.s (W := W) ≫ z₂.s (W := W) ≫ w₂.s (W := W)) ⋯
fac₁'₁ : b.f (W := W) ≫ z₁.s (W := W) ≫ w₁.s (W := W) = a₁.s (W := W) ≫ t₁ ≫ w₁.f (W := W)
⊢ b.f (W := W) ≫ z₂.s (W := W) ≫ w₂.s (W := W) = a₂.s (W := W) ≫ t₂ ≫ w₂.f (W := W)

Tactic:
simp only [assoc] at fac₁' fac₂'

Premises used: CategoryTheory.Category.assoc
-- Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
W : MorphismProperty.{v_1, u_1} C
inst_2 : W.HasLeftCalculusOfFractions
X Y Z : C
z₁_1 : Hom W X Y
z₂_1 : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' (W := W) ⟶ U
t₂ : a₂.Y' (W := W) ⟶ U
hst : a₁.s (W := W) ≫ t₁ = a₂.s (W := W) ≫ t₂
hft : a₁.f (W := W) ≫ t₁ = a₂.f (W := W) ≫ t₂
ht : W (a₁.s (W := W) ≫ t₁)
z₁ : W.LeftFraction (a₁.Y' (W := W)) (b.Y' (W := W))
fac₁ : b.f (W := W) ≫ z₁.s (W := W) = a₁.s (W := W) ≫ z₁.f (W := W)
z₂ : W.LeftFraction (a₂.Y' (W := W)) (b.Y' (W := W))
fac₂ : b.f (W := W) ≫ z₂.s (W := W) = a₂.s (W := W) ≫ z₂.f (W := W)
w₁ : W.LeftFraction U (z₁.Y' (W := W))
w₂ : W.LeftFraction U (z₂.Y' (W := W))
p₁ : W.LeftFraction X Z :=
  LeftFraction.mk (W := W) (a₁.f (W := W) ≫ t₁ ≫ w₁.f (W := W)) (b.s (W := W) ≫ z₁.s (W := W) ≫ w₁.s (W := W)) ⋯
p₂ : W.LeftFraction X Z :=
  LeftFraction.mk (W := W) (a₂.f (W := W) ≫ t₂ ≫ w₂.f (W := W)) (b.s (W := W) ≫ z₂.s (W := W) ≫ w₂.s (W := W)) ⋯
fac₁' : b.f (W := W) ≫ z₁.s (W := W) ≫ w₁.s (W := W) = a₁.s (W := W) ≫ t₁ ≫ w₁.f (W := W)
fac₂' : b.f (W := W) ≫ z₂.s (W := W) ≫ w₂.s (W := W) = a₂.s (W := W) ≫ t₂ ≫ w₂.f (W := W)
refine_2 : mk (W := W) (a₁.comp₀ (W := W) b z₁) = mk (W := W) (a₂.comp₀ (W := W) b z₂)
⊢ a₁.comp (W := W) b = a₂.comp (W := W) b

Tactic:
rw [comp_eq _ _ z₁ fac₁, comp_eq _ _ z₂ fac₂]

Premises used: CategoryTheory.MorphismProperty.LeftFraction.comp_eq
-- Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
W : MorphismProperty.{v_1, u_1} C
inst_2 : W.HasLeftCalculusOfFractions
X Y Z : C
z₁_1 : Hom W X Y
z₂_1 : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' (W := W) ⟶ U
t₂ : a₂.Y' (W := W) ⟶ U
hst : a₁.s (W := W) ≫ t₁ = a₂.s (W := W) ≫ t₂
hft : a₁.f (W := W) ≫ t₁ = a₂.f (W := W) ≫ t₂
ht : W (a₁.s (W := W) ≫ t₁)
z₁ : W.LeftFraction (a₁.Y' (W := W)) (b.Y' (W := W))
fac₁ : b.f (W := W) ≫ z₁.s (W := W) = a₁.s (W := W) ≫ z₁.f (W := W)
z₂ : W.LeftFraction (a₂.Y' (W := W)) (b.Y' (W := W))
fac₂ : b.f (W := W) ≫ z₂.s (W := W) = a₂.s (W := W) ≫ z₂.f (W := W)
w₁ : W.LeftFraction U (z₁.Y' (W := W))
w₂ : W.LeftFraction U (z₂.Y' (W := W))
p₁ : W.LeftFraction X Z :=
  LeftFraction.mk (W := W) (a₁.f (W := W) ≫ t₁ ≫ w₁.f (W := W)) (b.s (W := W) ≫ z₁.s (W := W) ≫ w₁.s (W := W)) ⋯
p₂ : W.LeftFraction X Z :=
  LeftFraction.mk (W := W) (a₂.f (W := W) ≫ t₂ ≫ w₂.f (W := W)) (b.s (W := W) ≫ z₂.s (W := W) ≫ w₂.s (W := W)) ⋯
fac₁' : b.f (W := W) ≫ z₁.s (W := W) ≫ w₁.s (W := W) = a₁.s (W := W) ≫ t₁ ≫ w₁.f (W := W)
fac₂' : b.f (W := W) ≫ z₂.s (W := W) ≫ w₂.s (W := W) = a₂.s (W := W) ≫ t₂ ≫ w₂.f (W := W)
refine_2 : LeftFractionRel (W := W) (a₁.comp₀ (W := W) b z₁) (a₂.comp₀ (W := W) b z₂)
⊢ mk (W := W) (a₁.comp₀ (W := W) b z₁) = mk (W := W) (a₂.comp₀ (W := W) b z₂)

Tactic:
apply Quot.sound

Premises used: Quot.sound
-- Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
W : MorphismProperty.{v_1, u_1} C
inst_2 : W.HasLeftCalculusOfFractions
X Y Z : C
z₁_1 : Hom W X Y
z₂_1 : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' (W := W) ⟶ U
t₂ : a₂.Y' (W := W) ⟶ U
hst : a₁.s (W := W) ≫ t₁ = a₂.s (W := W) ≫ t₂
hft : a₁.f (W := W) ≫ t₁ = a₂.f (W := W) ≫ t₂
ht : W (a₁.s (W := W) ≫ t₁)
z₁ : W.LeftFraction (a₁.Y' (W := W)) (b.Y' (W := W))
fac₁ : b.f (W := W) ≫ z₁.s (W := W) = a₁.s (W := W) ≫ z₁.f (W := W)
z₂ : W.LeftFraction (a₂.Y' (W := W)) (b.Y' (W := W))
fac₂ : b.f (W := W) ≫ z₂.s (W := W) = a₂.s (W := W) ≫ z₂.f (W := W)
w₁ : W.LeftFraction U (z₁.Y' (W := W))
w₂ : W.LeftFraction U (z₂.Y' (W := W))
p₁ : W.LeftFraction X Z :=
  LeftFraction.mk (W := W) (a₁.f (W := W) ≫ t₁ ≫ w₁.f (W := W)) (b.s (W := W) ≫ z₁.s (W := W) ≫ w₁.s (W := W)) ⋯
p₂ : W.LeftFraction X Z :=
  LeftFraction.mk (W := W) (a₂.f (W := W) ≫ t₂ ≫ w₂.f (W := W)) (b.s (W := W) ≫ z₂.s (W := W) ≫ w₂.s (W := W)) ⋯
fac₁' : b.f (W := W) ≫ z₁.s (W := W) ≫ w₁.s (W := W) = a₁.s (W := W) ≫ t₁ ≫ w₁.f (W := W)
fac₂' : b.f (W := W) ≫ z₂.s (W := W) ≫ w₂.s (W := W) = a₂.s (W := W) ≫ t₂ ≫ w₂.f (W := W)
⊢ LeftFractionRel (W := W) (a₁.comp₀ (W := W) b z₁) (a₂.comp₀ (W := W) b z₂)

Tactic:
LeftFractionRel.trans ?_ ((?_ : LeftFractionRel p₁ p₂).trans ?_)

Premises used: CategoryTheory.MorphismProperty.LeftFractionRel.trans
-- Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
W : MorphismProperty.{v_1, u_1} C
inst_2 : W.HasLeftCalculusOfFractions
X Y Z : C
z₁_1 : Hom W X Y
z₂_1 : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' (W := W) ⟶ U
t₂ : a₂.Y' (W := W) ⟶ U
hst : a₁.s (W := W) ≫ t₁ = a₂.s (W := W) ≫ t₂
hft : a₁.f (W := W) ≫ t₁ = a₂.f (W := W) ≫ t₂
ht : W (a₁.s (W := W) ≫ t₁)
z₁ : W.LeftFraction (a₁.Y' (W := W)) (b.Y' (W := W))
fac₁ : b.f (W := W) ≫ z₁.s (W := W) = a₁.s (W := W) ≫ z₁.f (W := W)
z₂ : W.LeftFraction (a₂.Y' (W := W)) (b.Y' (W := W))
fac₂ : b.f (W := W) ≫ z₂.s (W := W) = a₂.s (W := W) ≫ z₂.f (W := W)
w₁ : W.LeftFraction U (z₁.Y' (W := W))
w₂ : W.LeftFraction U (z₂.Y' (W := W))
p₁ : W.LeftFraction X Z :=
  LeftFraction.mk (W := W) (a₁.f (W := W) ≫ t₁ ≫ w₁.f (W := W)) (b.s (W := W) ≫ z₁.s (W := W) ≫ w₁.s (W := W)) ⋯
p₂ : W.LeftFraction X Z :=
  LeftFraction.mk (W := W) (a₂.f (W := W) ≫ t₂ ≫ w₂.f (W := W)) (b.s (W := W) ≫ z₂.s (W := W) ≫ w₂.s (W := W)) ⋯
fac₁' : b.f (W := W) ≫ z₁.s (W := W) ≫ w₁.s (W := W) = a₁.s (W := W) ≫ t₁ ≫ w₁.f (W := W)
fac₂' : b.f (W := W) ≫ z₂.s (W := W) ≫ w₂.s (W := W) = a₂.s (W := W) ≫ t₂ ≫ w₂.f (W := W)
refine_2 : LeftFractionRel (W := W) (a₁.comp₀ (W := W) b z₁) p₁
refine_2₁ : LeftFractionRel (W := W) p₁ p₂
refine_2₂ : LeftFractionRel (W := W) p₂ (a₂.comp₀ (W := W) b z₂)
⊢ LeftFractionRel (W := W) (a₁.comp₀ (W := W) b z₁) (a₂.comp₀ (W := W) b z₂)

Tactic:
refine LeftFractionRel.trans ?_ ((?_ : LeftFractionRel p₁ p₂).trans ?_)

Premises used: CategoryTheory.MorphismProperty.LeftFractionRel.trans
-- Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
W : MorphismProperty.{v_1, u_1} C
inst_2 : W.HasLeftCalculusOfFractions
X Y Z : C
z₁_1 : Hom W X Y
z₂_1 : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' (W := W) ⟶ U
t₂ : a₂.Y' (W := W) ⟶ U
hst : a₁.s (W := W) ≫ t₁ = a₂.s (W := W) ≫ t₂
hft : a₁.f (W := W) ≫ t₁ = a₂.f (W := W) ≫ t₂
ht : W (a₁.s (W := W) ≫ t₁)
z₁ : W.LeftFraction (a₁.Y' (W := W)) (b.Y' (W := W))
fac₁ : b.f (W := W) ≫ z₁.s (W := W) = a₁.s (W := W) ≫ z₁.f (W := W)
z₂ : W.LeftFraction (a₂.Y' (W := W)) (b.Y' (W := W))
fac₂ : b.f (W := W) ≫ z₂.s (W := W) = a₂.s (W := W) ≫ z₂.f (W := W)
w₁ : W.LeftFraction U (z₁.Y' (W := W))
w₂ : W.LeftFraction U (z₂.Y' (W := W))
p₁ : W.LeftFraction X Z :=
  LeftFraction.mk (W := W) (a₁.f (W := W) ≫ t₁ ≫ w₁.f (W := W)) (b.s (W := W) ≫ z₁.s (W := W) ≫ w₁.s (W := W)) ⋯
p₂ : W.LeftFraction X Z :=
  LeftFraction.mk (W := W) (a₂.f (W := W) ≫ t₂ ≫ w₂.f (W := W)) (b.s (W := W) ≫ z₂.s (W := W) ≫ w₂.s (W := W)) ⋯
fac₁' : b.f (W := W) ≫ z₁.s (W := W) ≫ w₁.s (W := W) = a₁.s (W := W) ≫ t₁ ≫ w₁.f (W := W)
fac₂' : b.f (W := W) ≫ z₂.s (W := W) ≫ w₂.s (W := W) = a₂.s (W := W) ≫ t₂ ≫ w₂.f (W := W)
rw : a₁.s (W := W) ≫ z₁.f (W := W) ≫ w₁.s (W := W) = b.f (W := W) ≫ z₁.s (W := W) ≫ w₁.s (W := W)
rw₁ rw₂ : a₁.s (W := W) ≫ z₁.f (W := W) ≫ w₁.s (W := W) = a₁.s (W := W) ≫ z₁.f (W := W) ≫ w₁.s (W := W)
⊢ a₁.s (W := W) ≫ z₁.f (W := W) ≫ w₁.s (W := W) = a₁.s (W := W) ≫ t₁ ≫ w₁.f (W := W)

Tactic:
rw [← fac₁', reassoc_of% fac₁]
-- Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
W : MorphismProperty.{v_1, u_1} C
inst_2 : W.HasLeftCalculusOfFractions
X Y Z : C
z₁_1 : Hom W X Y
z₂_1 : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' (W := W) ⟶ U
t₂ : a₂.Y' (W := W) ⟶ U
hst : a₁.s (W := W) ≫ t₁ = a₂.s (W := W) ≫ t₂
hft : a₁.f (W := W) ≫ t₁ = a₂.f (W := W) ≫ t₂
ht : W (a₁.s (W := W) ≫ t₁)
z₁ : W.LeftFraction (a₁.Y' (W := W)) (b.Y' (W := W))
fac₁ : b.f (W := W) ≫ z₁.s (W := W) = a₁.s (W := W) ≫ z₁.f (W := W)
z₂ : W.LeftFraction (a₂.Y' (W := W)) (b.Y' (W := W))
fac₂ : b.f (W := W) ≫ z₂.s (W := W) = a₂.s (W := W) ≫ z₂.f (W := W)
w₁ : W.LeftFraction U (z₁.Y' (W := W))
w₂ : W.LeftFraction U (z₂.Y' (W := W))
p₁ : W.LeftFraction X Z :=
  LeftFraction.mk (W := W) (a₁.f (W := W) ≫ t₁ ≫ w₁.f (W := W)) (b.s (W := W) ≫ z₁.s (W := W) ≫ w₁.s (W := W)) ⋯
p₂ : W.LeftFraction X Z :=
  LeftFraction.mk (W := W) (a₂.f (W := W) ≫ t₂ ≫ w₂.f (W := W)) (b.s (W := W) ≫ z₂.s (W := W) ≫ w₂.s (W := W)) ⋯
fac₁' : b.f (W := W) ≫ z₁.s (W := W) ≫ w₁.s (W := W) = a₁.s (W := W) ≫ t₁ ≫ w₁.f (W := W)
fac₂' : b.f (W := W) ≫ z₂.s (W := W) ≫ w₂.s (W := W) = a₂.s (W := W) ≫ t₂ ≫ w₂.f (W := W)
⊢ a₁.s (W := W) ≫ z₁.f (W := W) ≫ w₁.s (W := W) = a₁.s (W := W) ≫ t₁ ≫ w₁.f (W := W)

Tactic:
have eq : a₁.s ≫ z₁.f ≫ w₁.s = a₁.s ≫ t₁ ≫ w₁.f := by rw [← fac₁', reassoc_of% fac₁]
-- Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
W : MorphismProperty.{v_1, u_1} C
inst_2 : W.HasLeftCalculusOfFractions
X Y Z : C
z₁_1 : Hom W X Y
z₂_1 : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' (W := W) ⟶ U
t₂ : a₂.Y' (W := W) ⟶ U
hst : a₁.s (W := W) ≫ t₁ = a₂.s (W := W) ≫ t₂
hft : a₁.f (W := W) ≫ t₁ = a₂.f (W := W) ≫ t₂
ht : W (a₁.s (W := W) ≫ t₁)
z₁ : W.LeftFraction (a₁.Y' (W := W)) (b.Y' (W := W))
fac₁ : b.f (W := W) ≫ z₁.s (W := W) = a₁.s (W := W) ≫ z₁.f (W := W)
z₂ : W.LeftFraction (a₂.Y' (W := W)) (b.Y' (W := W))
fac₂ : b.f (W := W) ≫ z₂.s (W := W) = a₂.s (W := W) ≫ z₂.f (W := W)
w₁ : W.LeftFraction U (z₁.Y' (W := W))
w₂ : W.LeftFraction U (z₂.Y' (W := W))
p₁ : W.LeftFraction X Z :=
  LeftFraction.mk (W := W) (a₁.f (W := W) ≫ t₁ ≫ w₁.f (W := W)) (b.s (W := W) ≫ z₁.s (W := W) ≫ w₁.s (W := W)) ⋯
p₂ : W.LeftFraction X Z :=
  LeftFraction.mk (W := W) (a₂.f (W := W) ≫ t₂ ≫ w₂.f (W := W)) (b.s (W := W) ≫ z₂.s (W := W) ≫ w₂.s (W := W)) ⋯
fac₁' : b.f (W := W) ≫ z₁.s (W := W) ≫ w₁.s (W := W) = a₁.s (W := W) ≫ t₁ ≫ w₁.f (W := W)
fac₂' : b.f (W := W) ≫ z₂.s (W := W) ≫ w₂.s (W := W) = a₂.s (W := W) ≫ t₂ ≫ w₂.f (W := W)
eq : a₁.s (W := W) ≫ z₁.f (W := W) ≫ w₁.s (W := W) = a₁.s (W := W) ≫ t₁ ≫ w₁.f (W := W)
refine_2 :
  ∀ (Z_1 : C) (u : w₁.Y' (W := W) ⟶ Z_1) (hu : W u)
    (fac₃ : (z₁.f (W := W) ≫ w₁.s (W := W)) ≫ u = (t₁ ≫ w₁.f (W := W)) ≫ u),
    LeftFractionRel (W := W) (a₁.comp₀ (W := W) b z₁) p₁
⊢ LeftFractionRel (W := W) (a₁.comp₀ (W := W) b z₁) p₁

Tactic:
obtain ⟨Z, u, hu, fac₃⟩ := HasLeftCalculusOfFractions.ext _ _ _ a₁.hs eq

Premises used: CategoryTheory.MorphismProperty.LeftFraction.hs, CategoryTheory.MorphismProperty.HasLeftCalculusOfFractions.ext
-- Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
W : MorphismProperty.{v_1, u_1} C
inst_2 : W.HasLeftCalculusOfFractions
X Y Z_1 : C
z₁_1 : Hom W X Y
z₂_1 : Hom W Y Z_1
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z_1
U : C
t₁ : a₁.Y' (W := W) ⟶ U
t₂ : a₂.Y' (W := W) ⟶ U
hst : a₁.s (W := W) ≫ t₁ = a₂.s (W := W) ≫ t₂
hft : a₁.f (W := W) ≫ t₁ = a₂.f (W := W) ≫ t₂
ht : W (a₁.s (W := W) ≫ t₁)
z₁ : W.LeftFraction (a₁.Y' (W := W)) (b.Y' (W := W))
fac₁ : b.f (W := W) ≫ z₁.s (W := W) = a₁.s (W := W) ≫ z₁.f (W := W)
z₂ : W.LeftFraction (a₂.Y' (W := W)) (b.Y' (W := W))
fac₂ : b.f (W := W) ≫ z₂.s (W := W) = a₂.s (W := W) ≫ z₂.f (W := W)
w₁ : W.LeftFraction U (z₁.Y' (W := W))
w₂ : W.LeftFraction U (z₂.Y' (W := W))
p₁ : W.LeftFraction X Z_1 :=
  LeftFraction.mk (W := W) (a₁.f (W := W) ≫ t₁ ≫ w₁.f (W := W)) (b.s (W := W) ≫ z₁.s (W := W) ≫ w₁.s (W := W)) ⋯
p₂ : W.LeftFraction X Z_1 :=
  LeftFraction.mk (W := W) (a₂.f (W := W) ≫ t₂ ≫ w₂.f (W := W)) (b.s (W := W) ≫ z₂.s (W := W) ≫ w₂.s (W := W)) ⋯
fac₁' : b.f (W := W) ≫ z₁.s (W := W) ≫ w₁.s (W := W) = a₁.s (W := W) ≫ t₁ ≫ w₁.f (W := W)
fac₂' : b.f (W := W) ≫ z₂.s (W := W) ≫ w₂.s (W := W) = a₂.s (W := W) ≫ t₂ ≫ w₂.f (W := W)
eq : a₁.s (W := W) ≫ z₁.f (W := W) ≫ w₁.s (W := W) = a₁.s (W := W) ≫ t₁ ≫ w₁.f (W := W)
Z : C
u : w₁.Y' (W := W) ⟶ Z
hu : W u
fac₃ : (z₁.f (W := W) ≫ w₁.s (W := W)) ≫ u = (t₁ ≫ w₁.f (W := W)) ≫ u
⊢ z₁.f (W := W) ≫ w₁.s (W := W) ≫ u = t₁ ≫ w₁.f (W := W) ≫ u

Tactic:
simp only [assoc] at fac₃

Premises used: CategoryTheory.Category.assoc