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
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 ≫ L.map (φ'.s (W := W) ≫ α.s (W := W)) = ψ.fst (W := W).map (W := W) L ⋯ ≫ L.map (φ'.s (W := W) ≫ α.s (W := W))
⊢ f = ψ.fst (W := W).map (W := W) L ⋯
have refine_1₁ : f ≫ L.map (φ'.s (W := W) ≫ α.s (W := W)) = L.map (ψ.fst (W := W).f (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]
-- 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 ≫ L.map (φ'.s (W := W) ≫ α.s (W := W)) = ψ.fst (W := W).map (W := W) L ⋯ ≫ L.map (φ'.s (W := W) ≫ α.s (W := W))
refine_1₁ : f ≫ L.map (φ'.s (W := W) ≫ α.s (W := W)) = L.map (ψ.fst (W := W).f (W := W))
⊢ f = ψ.fst (W := W).map (W := W) L ⋯
have refine_1₂ : f ≫ L.map ((RightFraction.mk (W := W) (φ.s (W := W)) sorry (φ'.s (W := W))).s (W := W) ≫ α.f (W := W)) =
  L.map (ψ.fst (W := W).f (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]
-- 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 ≫ L.map (φ'.s (W := W) ≫ α.s (W := W)) = ψ.fst (W := W).map (W := W) L ⋯ ≫ L.map (φ'.s (W := W) ≫ α.s (W := W))
refine_1₁ : f ≫ L.map (φ'.s (W := W) ≫ α.s (W := W)) = L.map (ψ.fst (W := W).f (W := W))
refine_1₂ :
  f ≫ L.map ((RightFraction.mk (W := W) (φ.s (W := W)) ⋯ (φ'.s (W := W))).s (W := W) ≫ α.f (W := W)) =
    L.map (ψ.fst (W := W).f (W := W))
⊢ f = ψ.fst (W := W).map (W := W) L ⋯
have refine_1₃ : f ≫ L.map ((RightFraction.mk (W := W) (φ.s (W := W)) sorry (φ'.s (W := W))).s (W := W)) ≫ L.map (α.f (W := W)) =
  L.map (ψ.fst (W := W).f (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]
-- 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 ≫ L.map (φ'.s (W := W) ≫ α.s (W := W)) = ψ.fst (W := W).map (W := W) L ⋯ ≫ L.map (φ'.s (W := W) ≫ α.s (W := W))
refine_1₁ : f ≫ L.map (φ'.s (W := W) ≫ α.s (W := W)) = L.map (ψ.fst (W := W).f (W := W))
refine_1₂ :
  f ≫ L.map ((RightFraction.mk (W := W) (φ.s (W := W)) ⋯ (φ'.s (W := W))).s (W := W) ≫ α.f (W := W)) =
    L.map (ψ.fst (W := W).f (W := W))
refine_1₃ :
  f ≫ L.map ((RightFraction.mk (W := W) (φ.s (W := W)) ⋯ (φ'.s (W := W))).s (W := W)) ≫ L.map (α.f (W := W)) =
    L.map (ψ.fst (W := W).f (W := W))
⊢ f = ψ.fst (W := W).map (W := W) L ⋯
have refine_1₄ : φ.map (W := W) L sorry ≫
    L.map ((RightFraction.mk (W := W) (φ.s (W := W)) sorry (φ'.s (W := W))).s (W := W)) ≫ L.map (α.f (W := W)) =
  L.map (ψ.fst (W := W).f (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]
-- 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 ≫ L.map (φ'.s (W := W) ≫ α.s (W := W)) = ψ.fst (W := W).map (W := W) L ⋯ ≫ L.map (φ'.s (W := W) ≫ α.s (W := W))
refine_1₁ : f ≫ L.map (φ'.s (W := W) ≫ α.s (W := W)) = L.map (ψ.fst (W := W).f (W := W))
refine_1₂ :
  f ≫ L.map ((RightFraction.mk (W := W) (φ.s (W := W)) ⋯ (φ'.s (W := W))).s (W := W) ≫ α.f (W := W)) =
    L.map (ψ.fst (W := W).f (W := W))
refine_1₃ :
  f ≫ L.map ((RightFraction.mk (W := W) (φ.s (W := W)) ⋯ (φ'.s (W := W))).s (W := W)) ≫ L.map (α.f (W := W)) =
    L.map (ψ.fst (W := W).f (W := W))
refine_1₄ :
  φ.map (W := W) L ⋯ ≫
      L.map ((RightFraction.mk (W := W) (φ.s (W := W)) ⋯ (φ'.s (W := W))).s (W := W)) ≫ L.map (α.f (W := W)) =
    L.map (ψ.fst (W := W).f (W := W))
⊢ f = ψ.fst (W := W).map (W := W) L ⋯
have refine_1₅ : L.map (φ.f (W := W)) ≫ L.map (α.f (W := W)) = L.map (ψ.fst (W := W).f (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]
-- 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 ≫ L.map (φ'.s (W := W) ≫ α.s (W := W)) = ψ.fst (W := W).map (W := W) L ⋯ ≫ L.map (φ'.s (W := W) ≫ α.s (W := W))
refine_1₁ : f ≫ L.map (φ'.s (W := W) ≫ α.s (W := W)) = L.map (ψ.fst (W := W).f (W := W))
refine_1₂ :
  f ≫ L.map ((RightFraction.mk (W := W) (φ.s (W := W)) ⋯ (φ'.s (W := W))).s (W := W) ≫ α.f (W := W)) =
    L.map (ψ.fst (W := W).f (W := W))
refine_1₃ :
  f ≫ L.map ((RightFraction.mk (W := W) (φ.s (W := W)) ⋯ (φ'.s (W := W))).s (W := W)) ≫ L.map (α.f (W := W)) =
    L.map (ψ.fst (W := W).f (W := W))
refine_1₄ :
  φ.map (W := W) L ⋯ ≫
      L.map ((RightFraction.mk (W := W) (φ.s (W := W)) ⋯ (φ'.s (W := W))).s (W := W)) ≫ L.map (α.f (W := W)) =
    L.map (ψ.fst (W := W).f (W := W))
refine_1₅ : L.map (φ.f (W := W)) ≫ L.map (α.f (W := W)) = L.map (ψ.fst (W := W).f (W := W))
⊢ f = ψ.fst (W := W).map (W := W) L ⋯
have refine_1₆ : L.map (φ.f (W := W)) ≫ L.map (α.f (W := W)) = L.map (φ.f (W := W)) ≫ L.map (α.f (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]
-- 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.snd (W := W).map (W := W) L ⋯
have refine_2 : φ'.map (W := W) L sorry ≫ L.map (φ'.s (W := W) ≫ α.s (W := W)) =
  ψ.snd (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)), hφ']
-- 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'.map (W := W) L ⋯ ≫ L.map (φ'.s (W := W) ≫ α.s (W := W)) =
    ψ.snd (W := W).map (W := W) L ⋯ ≫ L.map (φ'.s (W := W) ≫ α.s (W := W))
have refine_2 : φ'.map (W := W) L sorry ≫ L.map (φ'.s (W := W)) ≫ L.map (α.s (W := W)) =
  ψ.snd (W := W).map (W := W) L sorry ≫ L.map (φ'.s (W := W) ≫ α.s (W := W))
Tactic: nth_rw 1 [L.map_comp]
-- 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'.map (W := W) L ⋯ ≫ L.map (φ'.s (W := W)) ≫ L.map (α.s (W := W)) =
    ψ.snd (W := W).map (W := W) L ⋯ ≫ L.map (φ'.s (W := W) ≫ α.s (W := W))
have refine_2 : L.map (φ'.f (W := W)) ≫ L.map (α.s (W := W)) = ψ.snd (W := W).map (W := W) L sorry ≫ L.map (φ'.s (W := W) ≫ α.s (W := W))
Tactic: rw [LeftFraction.map_comp_map_s_assoc, LeftFraction.map_comp_map_s,
      L.map_comp]
-- 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_2 :
  L.map (φ'.f (W := W)) ≫ L.map (α.s (W := W)) = ψ.snd (W := W).map (W := W) L ⋯ ≫ L.map (φ'.s (W := W) ≫ α.s (W := W))
⊢ φ'.map (W := W) L ⋯ ≫ L.map (φ'.s (W := W)) ≫ L.map (α.s (W := W)) =
    ψ.snd (W := W).map (W := W) L ⋯ ≫ L.map (φ'.s (W := W) ≫ α.s (W := W))
have refine_2₁ : L.map (φ'.f (W := W)) ≫ L.map (α.s (W := W)) = L.map (ψ.snd (W := W).f (W := W))
Tactic: rw [LeftFraction.map_comp_map_s_assoc, LeftFraction.map_comp_map_s,
      L.map_comp]
-- 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_2 :
  L.map (φ'.f (W := W)) ≫ L.map (α.s (W := W)) = ψ.snd (W := W).map (W := W) L ⋯ ≫ L.map (φ'.s (W := W) ≫ α.s (W := W))
refine_2₁ : L.map (φ'.f (W := W)) ≫ L.map (α.s (W := W)) = L.map (ψ.snd (W := W).f (W := W))
⊢ φ'.map (W := W) L ⋯ ≫ L.map (φ'.s (W := W)) ≫ L.map (α.s (W := W)) =
    ψ.snd (W := W).map (W := W) L ⋯ ≫ L.map (φ'.s (W := W) ≫ α.s (W := W))
have refine_2₂ : L.map (φ'.f (W := W)) ≫ L.map (α.s (W := W)) = L.map (φ'.f (W := W)) ≫ L.map (α.s (W := W))
Tactic: rw [LeftFraction.map_comp_map_s_assoc, LeftFraction.map_comp_map_s,
      L.map_comp]
-- 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' f'' : L.obj X ⟶ L.obj Y
⊢ ∃ (φ : W.LeftFraction₃ X Y),
    f = φ.fst (W := W).map (W := W) L ⋯ ∧ f' = φ.snd (W := W).map (W := W) L ⋯ ∧ f'' = φ.thd (W := W).map (W := W) L ⋯
have obtain : ∀ (α : W.LeftFraction₂ X Y) (hα : f = α.fst (W := W).map (W := W) L sorry) (hα' : f' = α.snd (W := W).map (W := W) L sorry),
  ∃ (φ : W.LeftFraction₃ X Y),
    f = φ.fst (W := W).map (W := W) L sorry ∧ f' = φ.snd (W := W).map (W := W) L sorry ∧ f'' = φ.thd (W := W).map (W := W) L sorry
Tactic: obtain ⟨α, hα, hα'⟩ := exists_leftFraction₂ L W f 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' f'' : L.obj X ⟶ L.obj Y
α : W.LeftFraction₂ X Y
hα : f = α.fst (W := W).map (W := W) L ⋯
hα' : f' = α.snd (W := W).map (W := W) L ⋯
⊢ ∃ (φ : W.LeftFraction₃ X Y),
    f = φ.fst (W := W).map (W := W) L ⋯ ∧ f' = φ.snd (W := W).map (W := W) L ⋯ ∧ f'' = φ.thd (W := W).map (W := W) L ⋯
have have : ∀ (β : W.LeftFraction X Y) (hβ : f'' = β.map (W := W) L sorry),
  ∃ (φ : W.LeftFraction₃ X Y),
    f = φ.fst (W := W).map (W := W) L sorry ∧ f' = φ.snd (W := W).map (W := W) L sorry ∧ f'' = φ.thd (W := W).map (W := W) L sorry
Tactic: have ⟨β, hβ⟩ := 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' f'' : L.obj X ⟶ L.obj Y
α : W.LeftFraction₂ X Y
hα : f = α.fst (W := W).map (W := W) L ⋯
hα' : f' = α.snd (W := W).map (W := W) L ⋯
β : W.LeftFraction X Y
hβ : f'' = β.map (W := W) L ⋯
⊢ ∃ (φ : W.LeftFraction₃ X Y),
    f = φ.fst (W := W).map (W := W) L ⋯ ∧ f' = φ.snd (W := W).map (W := W) L ⋯ ∧ f'' = φ.thd (W := W).map (W := W) L ⋯
have obtain : ∀ (γ : W.LeftFraction (α.Y' (W := W)) (β.Y' (W := W)))
  (hγ :
    (RightFraction.mk (W := W) (α.s (W := W)) sorry (β.s (W := W))).f (W := W) ≫ γ.s (W := W) =
      (RightFraction.mk (W := W) (α.s (W := W)) sorry (β.s (W := W))).s (W := W) ≫ γ.f (W := W)),
  ∃ (φ : W.LeftFraction₃ X Y),
    f = φ.fst (W := W).map (W := W) L sorry ∧ f' = φ.snd (W := W).map (W := W) L sorry ∧ f'' = φ.thd (W := W).map (W := W) L sorry
Tactic: obtain ⟨γ, hγ⟩ := (RightFraction.mk _ α.hs β.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' f'' : L.obj X ⟶ L.obj Y
α : W.LeftFraction₂ X Y
hα : f = α.fst (W := W).map (W := W) L ⋯
hα' : f' = α.snd (W := W).map (W := W) L ⋯
β : W.LeftFraction X Y
hβ : f'' = β.map (W := W) L ⋯
γ : W.LeftFraction (α.Y' (W := W)) (β.Y' (W := W))
hγ : β.s (W := W) ≫ γ.s (W := W) = α.s (W := W) ≫ γ.f (W := W)
ψ : W.LeftFraction₃ X Y :=
  LeftFraction₃.mk (W := W) (α.f (W := W) ≫ γ.f (W := W)) (α.f' (W := W) ≫ γ.f (W := W)) (β.f (W := W) ≫ γ.s (W := W))
    (β.s (W := W) ≫ γ.s (W := W)) ⋯
⊢ IsIso (L.map (β.s (W := W) ≫ γ.s (W := W)))
have rw : IsIso (L.map (β.s (W := W)) ≫ L.map (γ.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' f'' : L.obj X ⟶ L.obj Y
α : W.LeftFraction₂ X Y
hα : f = α.fst (W := W).map (W := W) L ⋯
hα' : f' = α.snd (W := W).map (W := W) L ⋯
β : W.LeftFraction X Y
hβ : f'' = β.map (W := W) L ⋯
γ : W.LeftFraction (α.Y' (W := W)) (β.Y' (W := W))
hγ : β.s (W := W) ≫ γ.s (W := W) = α.s (W := W) ≫ γ.f (W := W)
ψ : W.LeftFraction₃ X Y :=
  LeftFraction₃.mk (W := W) (α.f (W := W) ≫ γ.f (W := W)) (α.f' (W := W) ≫ γ.f (W := W)) (β.f (W := W) ≫ γ.s (W := W))
    (β.s (W := W) ≫ γ.s (W := W)) ⋯
⊢ ∃ (φ : W.LeftFraction₃ X Y),
    f = φ.fst (W := W).map (W := W) L ⋯ ∧ f' = φ.snd (W := W).map (W := W) L ⋯ ∧ f'' = φ.thd (W := W).map (W := W) L ⋯
have this : IsIso (L.map (β.s (W := W) ≫ γ.s (W := W)))
Tactic: have : IsIso (L.map (β.s ≫ γ.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' f'' : L.obj X ⟶ L.obj Y
α : W.LeftFraction₂ X Y
hα : f = α.fst (W := W).map (W := W) L ⋯
hα' : f' = α.snd (W := W).map (W := W) L ⋯
β : W.LeftFraction X Y
hβ : f'' = β.map (W := W) L ⋯
γ : W.LeftFraction (α.Y' (W := W)) (β.Y' (W := W))
hγ : β.s (W := W) ≫ γ.s (W := W) = α.s (W := W) ≫ γ.f (W := W)
ψ : W.LeftFraction₃ X Y :=
  LeftFraction₃.mk (W := W) (α.f (W := W) ≫ γ.f (W := W)) (α.f' (W := W) ≫ γ.f (W := W)) (β.f (W := W) ≫ γ.s (W := W))
    (β.s (W := W) ≫ γ.s (W := W)) ⋯
this : IsIso (L.map (β.s (W := W) ≫ γ.s (W := W)))
⊢ ∃ (φ : W.LeftFraction₃ X Y),
    f = φ.fst (W := W).map (W := W) L ⋯ ∧ f' = φ.snd (W := W).map (W := W) L ⋯ ∧ f'' = φ.thd (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' f'' : L.obj X ⟶ L.obj Y
α : W.LeftFraction₂ X Y
hα : f = α.fst (W := W).map (W := W) L ⋯
hα' : f' = α.snd (W := W).map (W := W) L ⋯
β : W.LeftFraction X Y
hβ : f'' = β.map (W := W) L ⋯
γ : W.LeftFraction (α.Y' (W := W)) (β.Y' (W := W))
hγ : β.s (W := W) ≫ γ.s (W := W) = α.s (W := W) ≫ γ.f (W := W)
ψ : W.LeftFraction₃ X Y :=
  LeftFraction₃.mk (W := W) (α.f (W := W) ≫ γ.f (W := W)) (α.f' (W := W) ≫ γ.f (W := W)) (β.f (W := W) ≫ γ.s (W := W))
    (β.s (W := W) ≫ γ.s (W := W)) ⋯
this : IsIso (L.map (β.s (W := W) ≫ γ.s (W := W)))
refine_1 : f = ψ.fst (W := W).map (W := W) L ⋯
⊢ ∃ (φ : W.LeftFraction₃ X Y),
    f = φ.fst (W := W).map (W := W) L ⋯ ∧ f' = φ.snd (W := W).map (W := W) L ⋯ ∧ f'' = φ.thd (W := W).map (W := W) L ⋯
have refine_2 : f' = ψ.snd (W := W).map (W := W) L sorry
Tactic: refine ⟨ψ, ?_, ?_, ?_⟩