awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
L : Functor.{v_1, v_2, u_1, u_2} C D
W : MorphismProperty.{v_1, u_1} C
inst_2 : L.IsLocalization W
inst_3 : W.HasLeftCalculusOfFractions
X Y : C
φ ψ : W.LeftFraction₂ X Y
Z : C
t₁ : φ.Y' (W := W) ⟶ Z
t₂ : ψ.Y' (W := W) ⟶ Z
hst : φ.s (W := W) ≫ t₁ = ψ.s (W := W) ≫ t₂
hft : φ.f (W := W) ≫ t₁ = ψ.f (W := W) ≫ t₂
ht : W (φ.s (W := W) ≫ t₁)
Z' : C
t₁' : φ.Y' (W := W) ⟶ Z'
t₂' : ψ.Y' (W := W) ⟶ Z'
hst' : φ.s (W := W) ≫ t₁' = ψ.s (W := W) ≫ t₂'
hft' : φ.f' (W := W) ≫ t₁' = ψ.f' (W := W) ≫ t₂'
ht' : W (φ.s (W := W) ≫ t₁')
α : W.LeftFraction Z Z'
hα : φ.s (W := W) ≫ t₁' ≫ α.s (W := W) = φ.s (W := W) ≫ t₁ ≫ α.f (W := W)
Z'' : C
u : α.Y' (W := W) ⟶ Z''
hu : W u
hα' : ψ.s (W := W) ≫ t₂ ≫ α.f (W := W) ≫ u = ψ.s (W := W) ≫ t₂' ≫ α.s (W := W) ≫ u
Z''' : C
u' : Z'' ⟶ Z'''
hu' : W u'
fac : t₁' ≫ α.s (W := W) ≫ u = t₁ ≫ α.f (W := W) ≫ u
fac' : t₂ ≫ α.f (W := W) ≫ u ≫ u' = t₂' ≫ α.s (W := W) ≫ u ≫ u'
⊢ φ.f (W := W) ≫ t₁' ≫ α.s (W := W) ≫ u ≫ u' = ψ.f (W := W) ≫ t₂' ≫ α.s (W := W) ≫ u ≫ u'
have mp : φ.f (W := W) ≫ t₁ ≫ α.f (W := W) ≫ u ≫ u' = ψ.f (W := W) ≫ t₂' ≫ α.s (W := W) ≫ u ≫ u'
Tactic: rw [reassoc_of% fac, reassoc_of% hft, fac']
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
L : Functor.{v_1, v_2, u_1, u_2} C D
W : MorphismProperty.{v_1, u_1} C
inst_2 : L.IsLocalization W
inst_3 : W.HasLeftCalculusOfFractions
X Y : C
φ ψ : W.LeftFraction₂ X Y
Z : C
t₁ : φ.Y' (W := W) ⟶ Z
t₂ : ψ.Y' (W := W) ⟶ Z
hst : φ.s (W := W) ≫ t₁ = ψ.s (W := W) ≫ t₂
hft : φ.f (W := W) ≫ t₁ = ψ.f (W := W) ≫ t₂
ht : W (φ.s (W := W) ≫ t₁)
Z' : C
t₁' : φ.Y' (W := W) ⟶ Z'
t₂' : ψ.Y' (W := W) ⟶ Z'
hst' : φ.s (W := W) ≫ t₁' = ψ.s (W := W) ≫ t₂'
hft' : φ.f' (W := W) ≫ t₁' = ψ.f' (W := W) ≫ t₂'
ht' : W (φ.s (W := W) ≫ t₁')
α : W.LeftFraction Z Z'
hα : φ.s (W := W) ≫ t₁' ≫ α.s (W := W) = φ.s (W := W) ≫ t₁ ≫ α.f (W := W)
Z'' : C
u : α.Y' (W := W) ⟶ Z''
hu : W u
hα' : ψ.s (W := W) ≫ t₂ ≫ α.f (W := W) ≫ u = ψ.s (W := W) ≫ t₂' ≫ α.s (W := W) ≫ u
Z''' : C
u' : Z'' ⟶ Z'''
hu' : W u'
fac : t₁' ≫ α.s (W := W) ≫ u = t₁ ≫ α.f (W := W) ≫ u
fac' : t₂ ≫ α.f (W := W) ≫ u ≫ u' = t₂' ≫ α.s (W := W) ≫ u ≫ u'
mp : φ.f (W := W) ≫ t₁ ≫ α.f (W := W) ≫ u ≫ u' = ψ.f (W := W) ≫ t₂' ≫ α.s (W := W) ≫ u ≫ u'
⊢ φ.f (W := W) ≫ t₁' ≫ α.s (W := W) ≫ u ≫ u' = ψ.f (W := W) ≫ t₂' ≫ α.s (W := W) ≫ u ≫ u'
have mp₁ : ψ.f (W := W) ≫ t₂ ≫ α.f (W := W) ≫ u ≫ u' = ψ.f (W := W) ≫ t₂' ≫ α.s (W := W) ≫ u ≫ u'
Tactic: rw [reassoc_of% fac, reassoc_of% hft, fac']
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
L : Functor.{v_1, v_2, u_1, u_2} C D
W : MorphismProperty.{v_1, u_1} C
inst_2 : L.IsLocalization W
inst_3 : W.HasLeftCalculusOfFractions
X Y : C
φ ψ : W.LeftFraction₂ X Y
Z : C
t₁ : φ.Y' (W := W) ⟶ Z
t₂ : ψ.Y' (W := W) ⟶ Z
hst : φ.s (W := W) ≫ t₁ = ψ.s (W := W) ≫ t₂
hft : φ.f (W := W) ≫ t₁ = ψ.f (W := W) ≫ t₂
ht : W (φ.s (W := W) ≫ t₁)
Z' : C
t₁' : φ.Y' (W := W) ⟶ Z'
t₂' : ψ.Y' (W := W) ⟶ Z'
hst' : φ.s (W := W) ≫ t₁' = ψ.s (W := W) ≫ t₂'
hft' : φ.f' (W := W) ≫ t₁' = ψ.f' (W := W) ≫ t₂'
ht' : W (φ.s (W := W) ≫ t₁')
α : W.LeftFraction Z Z'
hα : φ.s (W := W) ≫ t₁' ≫ α.s (W := W) = φ.s (W := W) ≫ t₁ ≫ α.f (W := W)
Z'' : C
u : α.Y' (W := W) ⟶ Z''
hu : W u
hα' : ψ.s (W := W) ≫ t₂ ≫ α.f (W := W) ≫ u = ψ.s (W := W) ≫ t₂' ≫ α.s (W := W) ≫ u
Z''' : C
u' : Z'' ⟶ Z'''
hu' : W u'
fac : t₁' ≫ α.s (W := W) ≫ u = t₁ ≫ α.f (W := W) ≫ u
fac' : t₂ ≫ α.f (W := W) ≫ u ≫ u' = t₂' ≫ α.s (W := W) ≫ u ≫ u'
mp : φ.f (W := W) ≫ t₁ ≫ α.f (W := W) ≫ u ≫ u' = ψ.f (W := W) ≫ t₂' ≫ α.s (W := W) ≫ u ≫ u'
mp₁ : ψ.f (W := W) ≫ t₂ ≫ α.f (W := W) ≫ u ≫ u' = ψ.f (W := W) ≫ t₂' ≫ α.s (W := W) ≫ u ≫ u'
⊢ φ.f (W := W) ≫ t₁' ≫ α.s (W := W) ≫ u ≫ u' = ψ.f (W := W) ≫ t₂' ≫ α.s (W := W) ≫ u ≫ u'
have mp₂ : ψ.f (W := W) ≫ t₂' ≫ α.s (W := W) ≫ u ≫ u' = ψ.f (W := W) ≫ t₂' ≫ α.s (W := W) ≫ u ≫ u'
Tactic: rw [reassoc_of% fac, reassoc_of% hft, fac']
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
L : Functor.{v_1, v_2, u_1, u_2} C D
W : MorphismProperty.{v_1, u_1} C
inst_2 : L.IsLocalization W
inst_3 : W.HasLeftCalculusOfFractions
X Y : C
φ ψ : W.LeftFraction₂ X Y
Z : C
t₁ : φ.Y' (W := W) ⟶ Z
t₂ : ψ.Y' (W := W) ⟶ Z
hst : φ.s (W := W) ≫ t₁ = ψ.s (W := W) ≫ t₂
hft : φ.f (W := W) ≫ t₁ = ψ.f (W := W) ≫ t₂
ht : W (φ.s (W := W) ≫ t₁)
Z' : C
t₁' : φ.Y' (W := W) ⟶ Z'
t₂' : ψ.Y' (W := W) ⟶ Z'
hst' : φ.s (W := W) ≫ t₁' = ψ.s (W := W) ≫ t₂'
hft' : φ.f' (W := W) ≫ t₁' = ψ.f' (W := W) ≫ t₂'
ht' : W (φ.s (W := W) ≫ t₁')
α : W.LeftFraction Z Z'
hα : φ.s (W := W) ≫ t₁' ≫ α.s (W := W) = φ.s (W := W) ≫ t₁ ≫ α.f (W := W)
Z'' : C
u : α.Y' (W := W) ⟶ Z''
hu : W u
hα' : ψ.s (W := W) ≫ t₂ ≫ α.f (W := W) ≫ u = ψ.s (W := W) ≫ t₂' ≫ α.s (W := W) ≫ u
Z''' : C
u' : Z'' ⟶ Z'''
hu' : W u'
fac : t₁' ≫ α.s (W := W) ≫ u = t₁ ≫ α.f (W := W) ≫ u
fac' : t₂ ≫ α.f (W := W) ≫ u ≫ u' = t₂' ≫ α.s (W := W) ≫ u ≫ u'
⊢ φ.f' (W := W) ≫ t₁' ≫ α.s (W := W) ≫ u ≫ u' = ψ.f' (W := W) ≫ t₂' ≫ α.s (W := W) ≫ u ≫ u'
have mp : ψ.f' (W := W) ≫ t₂' ≫ α.s (W := W) ≫ u ≫ u' = ψ.f' (W := W) ≫ t₂' ≫ α.s (W := W) ≫ u ≫ u'
Tactic: rw [reassoc_of% hft']
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
L : Functor.{v_1, v_2, u_1, u_2} C D
W : MorphismProperty.{v_1, u_1} C
inst_2 : L.IsLocalization W
inst_3 : W.HasLeftCalculusOfFractions
X Y : C
φ ψ : W.LeftFraction₂ X Y
Z : C
t₁ : φ.Y' (W := W) ⟶ Z
t₂ : ψ.Y' (W := W) ⟶ Z
hst : φ.s (W := W) ≫ t₁ = ψ.s (W := W) ≫ t₂
hft : φ.f (W := W) ≫ t₁ = ψ.f (W := W) ≫ t₂
ht : W (φ.s (W := W) ≫ t₁)
Z' : C
t₁' : φ.Y' (W := W) ⟶ Z'
t₂' : ψ.Y' (W := W) ⟶ Z'
hst' : φ.s (W := W) ≫ t₁' = ψ.s (W := W) ≫ t₂'
hft' : φ.f' (W := W) ≫ t₁' = ψ.f' (W := W) ≫ t₂'
ht' : W (φ.s (W := W) ≫ t₁')
α : W.LeftFraction Z Z'
hα : φ.s (W := W) ≫ t₁' ≫ α.s (W := W) = φ.s (W := W) ≫ t₁ ≫ α.f (W := W)
Z'' : C
u : α.Y' (W := W) ⟶ Z''
hu : W u
hα' : ψ.s (W := W) ≫ t₂ ≫ α.f (W := W) ≫ u = ψ.s (W := W) ≫ t₂' ≫ α.s (W := W) ≫ u
Z''' : C
u' : Z'' ⟶ Z'''
hu' : W u'
fac : t₁' ≫ α.s (W := W) ≫ u = t₁ ≫ α.f (W := W) ≫ u
fac' : t₂ ≫ α.f (W := W) ≫ u ≫ u' = t₂' ≫ α.s (W := W) ≫ u ≫ u'
⊢ W (φ.s (W := W) ≫ t₁' ≫ α.s (W := W) ≫ u ≫ u')
have mp : W ((φ.s (W := W) ≫ t₁') ≫ α.s (W := W) ≫ u ≫ u')
Tactic: rw [← Category.assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
W : MorphismProperty.{v_1, u_1} C
X Y : C
φ : W.RightFraction₂ X Y
inst_1 : W.HasLeftCalculusOfFractions
⊢ ∃ (ψ : W.LeftFraction₂ X Y),
    φ.f (W := W) ≫ ψ.s (W := W) = φ.s (W := W) ≫ ψ.f (W := W) ∧
      φ.f' (W := W) ≫ ψ.s (W := W) = φ.s (W := W) ≫ ψ.f' (W := W)
have obtain : ∀ (ψ₁ : W.LeftFraction X Y)
  (hψ₁ : φ.fst (W := W).f (W := W) ≫ ψ₁.s (W := W) = φ.fst (W := W).s (W := W) ≫ ψ₁.f (W := W)),
  ∃ (ψ : W.LeftFraction₂ X Y),
    φ.f (W := W) ≫ ψ.s (W := W) = φ.s (W := W) ≫ ψ.f (W := W) ∧
      φ.f' (W := W) ≫ ψ.s (W := W) = φ.s (W := W) ≫ ψ.f' (W := W)
Tactic: obtain ⟨ψ₁, hψ₁⟩ := φ.fst.exists_leftFraction
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
W : MorphismProperty.{v_1, u_1} C
X Y : C
φ : W.RightFraction₂ X Y
inst_1 : W.HasLeftCalculusOfFractions
ψ₁ : W.LeftFraction X Y
hψ₁ : φ.fst (W := W).f (W := W) ≫ ψ₁.s (W := W) = φ.fst (W := W).s (W := W) ≫ ψ₁.f (W := W)
⊢ ∃ (ψ : W.LeftFraction₂ X Y),
    φ.f (W := W) ≫ ψ.s (W := W) = φ.s (W := W) ≫ ψ.f (W := W) ∧
      φ.f' (W := W) ≫ ψ.s (W := W) = φ.s (W := W) ≫ ψ.f' (W := W)
have obtain : ∀ (ψ₂ : W.LeftFraction X Y)
  (hψ₂ : φ.snd (W := W).f (W := W) ≫ ψ₂.s (W := W) = φ.snd (W := W).s (W := W) ≫ ψ₂.f (W := W)),
  ∃ (ψ : W.LeftFraction₂ X Y),
    φ.f (W := W) ≫ ψ.s (W := W) = φ.s (W := W) ≫ ψ.f (W := W) ∧
      φ.f' (W := W) ≫ ψ.s (W := W) = φ.s (W := W) ≫ ψ.f' (W := W)
Tactic: obtain ⟨ψ₂, hψ₂⟩ := φ.snd.exists_leftFraction
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
W : MorphismProperty.{v_1, u_1} C
X Y : C
φ : W.RightFraction₂ X Y
inst_1 : W.HasLeftCalculusOfFractions
ψ₁ : W.LeftFraction X Y
hψ₁ : φ.fst (W := W).f (W := W) ≫ ψ₁.s (W := W) = φ.fst (W := W).s (W := W) ≫ ψ₁.f (W := W)
ψ₂ : W.LeftFraction X Y
hψ₂ : φ.snd (W := W).f (W := W) ≫ ψ₂.s (W := W) = φ.snd (W := W).s (W := W) ≫ ψ₂.f (W := W)
⊢ ∃ (ψ : W.LeftFraction₂ X Y),
    φ.f (W := W) ≫ ψ.s (W := W) = φ.s (W := W) ≫ ψ.f (W := W) ∧
      φ.f' (W := W) ≫ ψ.s (W := W) = φ.s (W := W) ≫ ψ.f' (W := W)
have obtain : ∀ (α : W.LeftFraction (ψ₁.Y' (W := W)) (ψ₂.Y' (W := W)))
  (hα :
    (RightFraction.mk (W := W) (ψ₁.s (W := W)) sorry (ψ₂.s (W := W))).f (W := W) ≫ α.s (W := W) =
      (RightFraction.mk (W := W) (ψ₁.s (W := W)) sorry (ψ₂.s (W := W))).s (W := W) ≫ α.f (W := W)),
  ∃ (ψ : W.LeftFraction₂ X Y),
    φ.f (W := W) ≫ ψ.s (W := W) = φ.s (W := W) ≫ ψ.f (W := W) ∧
      φ.f' (W := W) ≫ ψ.s (W := W) = φ.s (W := W) ≫ ψ.f' (W := W)
Tactic: obtain ⟨α, hα⟩ := (RightFraction.mk _ ψ₁.hs ψ₂.s).exists_leftFraction
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
W : MorphismProperty.{v_1, u_1} C
X Y : C
φ : W.RightFraction₂ X Y
inst_1 : W.HasLeftCalculusOfFractions
ψ₁ : W.LeftFraction X Y
hψ₁ : φ.f (W := W) ≫ ψ₁.s (W := W) = φ.s (W := W) ≫ ψ₁.f (W := W)
ψ₂ : W.LeftFraction X Y
hψ₂ : φ.f' (W := W) ≫ ψ₂.s (W := W) = φ.s (W := W) ≫ ψ₂.f (W := W)
α : W.LeftFraction (ψ₁.Y' (W := W)) (ψ₂.Y' (W := W))
hα : ψ₂.s (W := W) ≫ α.s (W := W) = ψ₁.s (W := W) ≫ α.f (W := W)
⊢ ∃ (ψ : W.LeftFraction₂ X Y),
    φ.f (W := W) ≫ ψ.s (W := W) = φ.s (W := W) ≫ ψ.f (W := W) ∧
      φ.f' (W := W) ≫ ψ.s (W := W) = φ.s (W := W) ≫ ψ.f' (W := W)
have refine_1 : φ.f (W := W) ≫
    (LeftFraction₂.mk (W := W) (ψ₁.f (W := W) ≫ α.f (W := W)) (ψ₂.f (W := W) ≫ α.s (W := W))
          (ψ₂.s (W := W) ≫ α.s (W := W)) sorry).s
      (W := W) =
  φ.s (W := W) ≫
    (LeftFraction₂.mk (W := W) (ψ₁.f (W := W) ≫ α.f (W := W)) (ψ₂.f (W := W) ≫ α.s (W := W))
          (ψ₂.s (W := W) ≫ α.s (W := W)) sorry).f
      (W := W)
Tactic: refine ⟨LeftFraction₂.mk (ψ₁.f ≫ α.f) (ψ₂.f ≫ α.s) (ψ₂.s ≫ α.s)
      (W.comp_mem _ _ ψ₂.hs α.hs), ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
W : MorphismProperty.{v_1, u_1} C
X Y : C
φ : W.RightFraction₂ X Y
inst_1 : W.HasLeftCalculusOfFractions
ψ₁ : W.LeftFraction X Y
hψ₁ : φ.f (W := W) ≫ ψ₁.s (W := W) = φ.s (W := W) ≫ ψ₁.f (W := W)
ψ₂ : W.LeftFraction X Y
hψ₂ : φ.f' (W := W) ≫ ψ₂.s (W := W) = φ.s (W := W) ≫ ψ₂.f (W := W)
α : W.LeftFraction (ψ₁.Y' (W := W)) (ψ₂.Y' (W := W))
hα : ψ₂.s (W := W) ≫ α.s (W := W) = ψ₁.s (W := W) ≫ α.f (W := W)
refine_1 :
  φ.f (W := W) ≫
      (LeftFraction₂.mk (W := W) (ψ₁.f (W := W) ≫ α.f (W := W)) (ψ₂.f (W := W) ≫ α.s (W := W))
            (ψ₂.s (W := W) ≫ α.s (W := W)) ⋯).s
        (W := W) =
    φ.s (W := W) ≫
      (LeftFraction₂.mk (W := W) (ψ₁.f (W := W) ≫ α.f (W := W)) (ψ₂.f (W := W) ≫ α.s (W := W))
            (ψ₂.s (W := W) ≫ α.s (W := W)) ⋯).f
        (W := W)
⊢ ∃ (ψ : W.LeftFraction₂ X Y),
    φ.f (W := W) ≫ ψ.s (W := W) = φ.s (W := W) ≫ ψ.f (W := W) ∧
      φ.f' (W := W) ≫ ψ.s (W := W) = φ.s (W := W) ≫ ψ.f' (W := W)
have refine_2 : φ.f' (W := W) ≫
    (LeftFraction₂.mk (W := W) (ψ₁.f (W := W) ≫ α.f (W := W)) (ψ₂.f (W := W) ≫ α.s (W := W))
          (ψ₂.s (W := W) ≫ α.s (W := W)) sorry).s
      (W := W) =
  φ.s (W := W) ≫
    (LeftFraction₂.mk (W := W) (ψ₁.f (W := W) ≫ α.f (W := W)) (ψ₂.f (W := W) ≫ α.s (W := W))
          (ψ₂.s (W := W) ≫ α.s (W := W)) sorry).f'
      (W := W)
Tactic: refine ⟨LeftFraction₂.mk (ψ₁.f ≫ α.f) (ψ₂.f ≫ α.s) (ψ₂.s ≫ α.s)
      (W.comp_mem _ _ ψ₂.hs α.hs), ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
W : MorphismProperty.{v_1, u_1} C
X Y : C
φ : W.RightFraction₂ X Y
inst_1 : W.HasLeftCalculusOfFractions
ψ₁ : W.LeftFraction X Y
hψ₁ : φ.f (W := W) ≫ ψ₁.s (W := W) = φ.s (W := W) ≫ ψ₁.f (W := W)
ψ₂ : W.LeftFraction X Y
hψ₂ : φ.f' (W := W) ≫ ψ₂.s (W := W) = φ.s (W := W) ≫ ψ₂.f (W := W)
α : W.LeftFraction (ψ₁.Y' (W := W)) (ψ₂.Y' (W := W))
hα : ψ₂.s (W := W) ≫ α.s (W := W) = ψ₁.s (W := W) ≫ α.f (W := W)
⊢ φ.f (W := W) ≫
      (LeftFraction₂.mk (W := W) (ψ₁.f (W := W) ≫ α.f (W := W)) (ψ₂.f (W := W) ≫ α.s (W := W))
            (ψ₂.s (W := W) ≫ α.s (W := W)) ⋯).s
        (W := W) =
    φ.s (W := W) ≫
      (LeftFraction₂.mk (W := W) (ψ₁.f (W := W) ≫ α.f (W := W)) (ψ₂.f (W := W) ≫ α.s (W := W))
            (ψ₂.s (W := W) ≫ α.s (W := W)) ⋯).f
        (W := W)
have refine_1 : φ.f (W := W) ≫ ψ₂.s (W := W) ≫ α.s (W := W) = φ.s (W := W) ≫ ψ₁.f (W := W) ≫ α.f (W := W)
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
W : MorphismProperty.{v_1, u_1} C
X Y : C
φ : W.RightFraction₂ X Y
inst_1 : W.HasLeftCalculusOfFractions
ψ₁ : W.LeftFraction X Y
hψ₁ : φ.f (W := W) ≫ ψ₁.s (W := W) = φ.s (W := W) ≫ ψ₁.f (W := W)
ψ₂ : W.LeftFraction X Y
hψ₂ : φ.f' (W := W) ≫ ψ₂.s (W := W) = φ.s (W := W) ≫ ψ₂.f (W := W)
α : W.LeftFraction (ψ₁.Y' (W := W)) (ψ₂.Y' (W := W))
hα : ψ₂.s (W := W) ≫ α.s (W := W) = ψ₁.s (W := W) ≫ α.f (W := W)
⊢ φ.f (W := W) ≫ ψ₂.s (W := W) ≫ α.s (W := W) = φ.s (W := W) ≫ ψ₁.f (W := W) ≫ α.f (W := W)
have refine_1 : φ.f (W := W) ≫ ψ₁.s (W := W) ≫ α.f (W := W) = φ.s (W := W) ≫ ψ₁.f (W := W) ≫ α.f (W := W)
Tactic: rw [hα, reassoc_of% hψ₁]
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
W : MorphismProperty.{v_1, u_1} C
X Y : C
φ : W.RightFraction₂ X Y
inst_1 : W.HasLeftCalculusOfFractions
ψ₁ : W.LeftFraction X Y
hψ₁ : φ.f (W := W) ≫ ψ₁.s (W := W) = φ.s (W := W) ≫ ψ₁.f (W := W)
ψ₂ : W.LeftFraction X Y
hψ₂ : φ.f' (W := W) ≫ ψ₂.s (W := W) = φ.s (W := W) ≫ ψ₂.f (W := W)
α : W.LeftFraction (ψ₁.Y' (W := W)) (ψ₂.Y' (W := W))
hα : ψ₂.s (W := W) ≫ α.s (W := W) = ψ₁.s (W := W) ≫ α.f (W := W)
refine_1 : φ.f (W := W) ≫ ψ₁.s (W := W) ≫ α.f (W := W) = φ.s (W := W) ≫ ψ₁.f (W := W) ≫ α.f (W := W)
⊢ φ.f (W := W) ≫ ψ₂.s (W := W) ≫ α.s (W := W) = φ.s (W := W) ≫ ψ₁.f (W := W) ≫ α.f (W := W)
have refine_1₁ : φ.s (W := W) ≫ ψ₁.f (W := W) ≫ α.f (W := W) = φ.s (W := W) ≫ ψ₁.f (W := W) ≫ α.f (W := W)
Tactic: rw [hα, reassoc_of% hψ₁]
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
W : MorphismProperty.{v_1, u_1} C
X Y : C
φ : W.RightFraction₂ X Y
inst_1 : W.HasLeftCalculusOfFractions
ψ₁ : W.LeftFraction X Y
hψ₁ : φ.f (W := W) ≫ ψ₁.s (W := W) = φ.s (W := W) ≫ ψ₁.f (W := W)
ψ₂ : W.LeftFraction X Y
hψ₂ : φ.f' (W := W) ≫ ψ₂.s (W := W) = φ.s (W := W) ≫ ψ₂.f (W := W)
α : W.LeftFraction (ψ₁.Y' (W := W)) (ψ₂.Y' (W := W))
hα : ψ₂.s (W := W) ≫ α.s (W := W) = ψ₁.s (W := W) ≫ α.f (W := W)
⊢ φ.f' (W := W) ≫
      (LeftFraction₂.mk (W := W) (ψ₁.f (W := W) ≫ α.f (W := W)) (ψ₂.f (W := W) ≫ α.s (W := W))
            (ψ₂.s (W := W) ≫ α.s (W := W)) ⋯).s
        (W := W) =
    φ.s (W := W) ≫
      (LeftFraction₂.mk (W := W) (ψ₁.f (W := W) ≫ α.f (W := W)) (ψ₂.f (W := W) ≫ α.s (W := W))
            (ψ₂.s (W := W) ≫ α.s (W := W)) ⋯).f'
        (W := W)
have refine_2 : φ.s (W := W) ≫ ψ₂.f (W := W) ≫ α.s (W := W) =
  φ.s (W := W) ≫
    (LeftFraction₂.mk (W := W) (ψ₁.f (W := W) ≫ α.f (W := W)) (ψ₂.f (W := W) ≫ α.s (W := W))
          (ψ₂.s (W := W) ≫ α.s (W := W)) sorry).f'
      (W := W)
Tactic: rw [reassoc_of% hψ₂]
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
L : Functor.{v_1, v_2, u_1, u_2} C D
W : MorphismProperty.{v_1, u_1} C
inst_2 : L.IsLocalization W
inst_3 : W.HasLeftCalculusOfFractions
X Y : C
f f' : L.obj X ⟶ L.obj Y
⊢ ∃ (φ : W.LeftFraction₂ X Y), f = φ.fst (W := W).map (W := W) L ⋯ ∧ f' = φ.snd (W := W).map (W := W) L ⋯
have have : ∀ (φ : W.LeftFraction X Y) (hφ : f = φ.map (W := W) L sorry),
  ∃ (φ : W.LeftFraction₂ X Y), f = φ.fst (W := W).map (W := W) L sorry ∧ f' = φ.snd (W := W).map (W := W) L sorry
Tactic: have ⟨φ, hφ⟩ := exists_leftFraction L W f
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
L : Functor.{v_1, v_2, u_1, u_2} C D
W : MorphismProperty.{v_1, u_1} C
inst_2 : L.IsLocalization W
inst_3 : W.HasLeftCalculusOfFractions
X Y : C
f f' : L.obj X ⟶ L.obj Y
φ : W.LeftFraction X Y
hφ : f = φ.map (W := W) L ⋯
⊢ ∃ (φ : W.LeftFraction₂ X Y), f = φ.fst (W := W).map (W := W) L ⋯ ∧ f' = φ.snd (W := W).map (W := W) L ⋯
have have : ∀ (φ' : W.LeftFraction X Y) (hφ' : f' = φ'.map (W := W) L sorry),
  ∃ (φ : W.LeftFraction₂ X Y), f = φ.fst (W := W).map (W := W) L sorry ∧ f' = φ.snd (W := W).map (W := W) L sorry
Tactic: have ⟨φ', hφ'⟩ := exists_leftFraction L W f'
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
L : Functor.{v_1, v_2, u_1, u_2} C D
W : MorphismProperty.{v_1, u_1} C
inst_2 : L.IsLocalization W
inst_3 : W.HasLeftCalculusOfFractions
X Y : C
f f' : L.obj X ⟶ L.obj Y
φ : W.LeftFraction X Y
hφ : f = φ.map (W := W) L ⋯
φ' : W.LeftFraction X Y
hφ' : f' = φ'.map (W := W) L ⋯
⊢ ∃ (φ : W.LeftFraction₂ X Y), f = φ.fst (W := W).map (W := W) L ⋯ ∧ f' = φ.snd (W := W).map (W := W) L ⋯
have obtain : ∀ (α : W.LeftFraction (φ.Y' (W := W)) (φ'.Y' (W := W)))
  (hα :
    (RightFraction.mk (W := W) (φ.s (W := W)) sorry (φ'.s (W := W))).f (W := W) ≫ α.s (W := W) =
      (RightFraction.mk (W := W) (φ.s (W := W)) sorry (φ'.s (W := W))).s (W := W) ≫ α.f (W := W)),
  ∃ (φ : W.LeftFraction₂ X Y), f = φ.fst (W := W).map (W := W) L sorry ∧ f' = φ.snd (W := W).map (W := W) L sorry
Tactic: obtain ⟨α, hα⟩ := (RightFraction.mk _ φ.hs φ'.s).exists_leftFraction
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
L : Functor.{v_1, v_2, u_1, u_2} C D
W : MorphismProperty.{v_1, u_1} C
inst_2 : L.IsLocalization W
inst_3 : W.HasLeftCalculusOfFractions
X Y : C
f f' : L.obj X ⟶ L.obj Y
φ : W.LeftFraction X Y
hφ : f = φ.map (W := W) L ⋯
φ' : W.LeftFraction X Y
hφ' : f' = φ'.map (W := W) L ⋯
α : W.LeftFraction (φ.Y' (W := W)) (φ'.Y' (W := W))
hα :
  (RightFraction.mk (W := W) (φ.s (W := W)) ⋯ (φ'.s (W := W))).f (W := W) ≫ α.s (W := W) =
    (RightFraction.mk (W := W) (φ.s (W := W)) ⋯ (φ'.s (W := W))).s (W := W) ≫ α.f (W := W)
ψ : W.LeftFraction₂ X Y :=
  LeftFraction₂.mk (W := W) (φ.f (W := W) ≫ α.f (W := W)) (φ'.f (W := W) ≫ α.s (W := W)) (φ'.s (W := W) ≫ α.s (W := W))
    ⋯
⊢ IsIso (L.map (φ'.s (W := W) ≫ α.s (W := W)))
have rw : IsIso (L.map (φ'.s (W := W)) ≫ L.map (α.s (W := W)))
Tactic: rw [L.map_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (hypothesis)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
L : Functor.{v_1, v_2, u_1, u_2} C D
W : MorphismProperty.{v_1, u_1} C
inst_2 : L.IsLocalization W
inst_3 : W.HasLeftCalculusOfFractions
X Y : C
f f' : L.obj X ⟶ L.obj Y
φ : W.LeftFraction X Y
hφ : f = φ.map (W := W) L ⋯
φ' : W.LeftFraction X Y
hφ' : f' = φ'.map (W := W) L ⋯
α : W.LeftFraction (φ.Y' (W := W)) (φ'.Y' (W := W))
hα :
  (RightFraction.mk (W := W) (φ.s (W := W)) ⋯ (φ'.s (W := W))).f (W := W) ≫ α.s (W := W) =
    (RightFraction.mk (W := W) (φ.s (W := W)) ⋯ (φ'.s (W := W))).s (W := W) ≫ α.f (W := W)
ψ : W.LeftFraction₂ X Y :=
  LeftFraction₂.mk (W := W) (φ.f (W := W) ≫ α.f (W := W)) (φ'.f (W := W) ≫ α.s (W := W)) (φ'.s (W := W) ≫ α.s (W := W))
    ⋯
⊢ ∃ (φ : W.LeftFraction₂ X Y), f = φ.fst (W := W).map (W := W) L ⋯ ∧ f' = φ.snd (W := W).map (W := W) L ⋯
have this : IsIso (L.map (φ'.s (W := W) ≫ α.s (W := W)))
Tactic: have : IsIso (L.map (φ'.s ≫ α.s)) := by
    rw [L.map_comp]
    infer_instance
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
L : Functor.{v_1, v_2, u_1, u_2} C D
W : MorphismProperty.{v_1, u_1} C
inst_2 : L.IsLocalization W
inst_3 : W.HasLeftCalculusOfFractions
X Y : C
f f' : L.obj X ⟶ L.obj Y
φ : W.LeftFraction X Y
hφ : f = φ.map (W := W) L ⋯
φ' : W.LeftFraction X Y
hφ' : f' = φ'.map (W := W) L ⋯
α : W.LeftFraction (φ.Y' (W := W)) (φ'.Y' (W := W))
hα :
  (RightFraction.mk (W := W) (φ.s (W := W)) ⋯ (φ'.s (W := W))).f (W := W) ≫ α.s (W := W) =
    (RightFraction.mk (W := W) (φ.s (W := W)) ⋯ (φ'.s (W := W))).s (W := W) ≫ α.f (W := W)
ψ : W.LeftFraction₂ X Y :=
  LeftFraction₂.mk (W := W) (φ.f (W := W) ≫ α.f (W := W)) (φ'.f (W := W) ≫ α.s (W := W)) (φ'.s (W := W) ≫ α.s (W := W))
    ⋯
this : IsIso (L.map (φ'.s (W := W) ≫ α.s (W := W)))
⊢ ∃ (φ : W.LeftFraction₂ X Y), f = φ.fst (W := W).map (W := W) L ⋯ ∧ f' = φ.snd (W := W).map (W := W) L ⋯
have refine_1 : f = ψ.fst (W := W).map (W := W) L sorry
Tactic: refine ⟨ψ, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
L : Functor.{v_1, v_2, u_1, u_2} C D
W : MorphismProperty.{v_1, u_1} C
inst_2 : L.IsLocalization W
inst_3 : W.HasLeftCalculusOfFractions
X Y : C
f f' : L.obj X ⟶ L.obj Y
φ : W.LeftFraction X Y
hφ : f = φ.map (W := W) L ⋯
φ' : W.LeftFraction X Y
hφ' : f' = φ'.map (W := W) L ⋯
α : W.LeftFraction (φ.Y' (W := W)) (φ'.Y' (W := W))
hα :
  (RightFraction.mk (W := W) (φ.s (W := W)) ⋯ (φ'.s (W := W))).f (W := W) ≫ α.s (W := W) =
    (RightFraction.mk (W := W) (φ.s (W := W)) ⋯ (φ'.s (W := W))).s (W := W) ≫ α.f (W := W)
ψ : W.LeftFraction₂ X Y :=
  LeftFraction₂.mk (W := W) (φ.f (W := W) ≫ α.f (W := W)) (φ'.f (W := W) ≫ α.s (W := W)) (φ'.s (W := W) ≫ α.s (W := W))
    ⋯
this : IsIso (L.map (φ'.s (W := W) ≫ α.s (W := W)))
refine_1 : f = ψ.fst (W := W).map (W := W) L ⋯
⊢ ∃ (φ : W.LeftFraction₂ X Y), f = φ.fst (W := W).map (W := W) L ⋯ ∧ f' = φ.snd (W := W).map (W := W) L ⋯
have refine_2 : f' = ψ.snd (W := W).map (W := W) L sorry
Tactic: refine ⟨ψ, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.CalculusOfFractions.Fractions (goal)
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
L : Functor.{v_1, v_2, u_1, u_2} C D
W : MorphismProperty.{v_1, u_1} C
inst_2 : L.IsLocalization W
inst_3 : W.HasLeftCalculusOfFractions
X Y : C
f f' : L.obj X ⟶ L.obj Y
φ : W.LeftFraction X Y
hφ : f = φ.map (W := W) L ⋯
φ' : W.LeftFraction X Y
hφ' : f' = φ'.map (W := W) L ⋯
α : W.LeftFraction (φ.Y' (W := W)) (φ'.Y' (W := W))
hα :
  (RightFraction.mk (W := W) (φ.s (W := W)) ⋯ (φ'.s (W := W))).f (W := W) ≫ α.s (W := W) =
    (RightFraction.mk (W := W) (φ.s (W := W)) ⋯ (φ'.s (W := W))).s (W := W) ≫ α.f (W := W)
ψ : W.LeftFraction₂ X Y :=
  LeftFraction₂.mk (W := W) (φ.f (W := W) ≫ α.f (W := W)) (φ'.f (W := W) ≫ α.s (W := W)) (φ'.s (W := W) ≫ α.s (W := W))
    ⋯
this : IsIso (L.map (φ'.s (W := W) ≫ α.s (W := W)))
⊢ f = ψ.fst (W := W).map (W := W) L ⋯
have refine_1 : f ≫ L.map (φ'.s (W := W) ≫ α.s (W := W)) = ψ.fst (W := W).map (W := W) L sorry ≫ L.map (φ'.s (W := W) ≫ α.s (W := W))
Tactic: rw [← cancel_mono (L.map (φ'.s ≫ α.s)), LeftFraction.map_comp_map_s,
      hα, L.map_comp, hφ, LeftFraction.map_comp_map_s_assoc,
      L.map_comp]