Module: Mathlib.CategoryTheory.Localization.CalculusOfFractions

Goal state before:
pp: case refine_2.refine_2
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y Z : C
z₁✝ : Hom W X Y
z₂✝ : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' ⟶ U
t₂ : a₂.Y' ⟶ U
hst : a₁.s ≫ t₁ = a₂.s ≫ t₂
hft : a₁.f ≫ t₁ = a₂.f ≫ t₂
ht : W (a₁.s ≫ t₁)
z₁ : W.LeftFraction a₁.Y' b.Y'
fac₁ : b.f ≫ z₁.s = a₁.s ≫ z₁.f
z₂ : W.LeftFraction a₂.Y' b.Y'
fac₂ : b.f ≫ z₂.s = a₂.s ≫ z₂.f
w₁ : W.LeftFraction U z₁.Y'
w₂ : W.LeftFraction U z₂.Y'
p₁ : W.LeftFraction X Z := { Y' := w₁.Y', f := a₁.f ≫ t₁ ≫ w₁.f, s := b.s ≫ z₁.s ≫ w₁.s, hs := ⋯ }
p₂ : W.LeftFraction X Z := { Y' := w₂.Y', f := a₂.f ≫ t₂ ≫ w₂.f, s := b.s ≫ z₂.s ≫ w₂.s, hs := ⋯ }
fac₁' : b.f ≫ z₁.s ≫ w₁.s = a₁.s ≫ t₁ ≫ w₁.f
fac₂' : b.f ≫ z₂.s ≫ w₂.s = a₂.s ≫ t₂ ≫ w₂.f
⊢ LeftFractionRel p₁ p₂

pp: case refine_2.refine_3
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y Z : C
z₁✝ : Hom W X Y
z₂✝ : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' ⟶ U
t₂ : a₂.Y' ⟶ U
hst : a₁.s ≫ t₁ = a₂.s ≫ t₂
hft : a₁.f ≫ t₁ = a₂.f ≫ t₂
ht : W (a₁.s ≫ t₁)
z₁ : W.LeftFraction a₁.Y' b.Y'
fac₁ : b.f ≫ z₁.s = a₁.s ≫ z₁.f
z₂ : W.LeftFraction a₂.Y' b.Y'
fac₂ : b.f ≫ z₂.s = a₂.s ≫ z₂.f
w₁ : W.LeftFraction U z₁.Y'
w₂ : W.LeftFraction U z₂.Y'
p₁ : W.LeftFraction X Z := { Y' := w₁.Y', f := a₁.f ≫ t₁ ≫ w₁.f, s := b.s ≫ z₁.s ≫ w₁.s, hs := ⋯ }
p₂ : W.LeftFraction X Z := { Y' := w₂.Y', f := a₂.f ≫ t₂ ≫ w₂.f, s := b.s ≫ z₂.s ≫ w₂.s, hs := ⋯ }
fac₁' : b.f ≫ z₁.s ≫ w₁.s = a₁.s ≫ t₁ ≫ w₁.f
fac₂' : b.f ≫ z₂.s ≫ w₂.s = a₂.s ≫ t₂ ≫ w₂.f
⊢ LeftFractionRel p₂ (a₂.comp₀ b z₂)

Tactic:
· obtain ⟨q, fac₃⟩ := exists_leftFraction (RightFraction.mk (z₁.s ≫ w₁.s)
        (W.comp_mem _ _ z₁.hs w₁.hs) (z₂.s ≫ w₂.s))
      dsimp at fac₃
      simp only [assoc] at fac₃
      have eq : a₁.s ≫ t₁ ≫ w₁.f ≫ q.f = a₁.s ≫ t₁ ≫ w₂.f ≫ q.s := by
        rw [← reassoc_of% fac₁', ← fac₃, reassoc_of% hst, reassoc_of% fac₂']
      obtain ⟨Z, u, hu, fac₄⟩ := HasLeftCalculusOfFractions.ext _ _ _ a₁.hs eq
      simp only [assoc] at fac₄
      refine ⟨Z, q.f ≫ u, q.s ≫ u, ?_, ?_, ?_⟩
      · simp only [p₁, p₂, assoc, reassoc_of% fac₃]
      · rw [assoc, assoc, assoc, assoc, fac₄, reassoc_of% hft]
      · simp only [p₁, assoc, ← reassoc_of% fac₃]
        exact W.comp_mem _ _ b.hs (W.comp_mem _ _ z₂.hs
          (W.comp_mem _ _ w₂.hs (W.comp_mem _ _ q.hs hu)))

Goal state after:
case refine_2.refine_3
C : Type u_1
D : Type u_2
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
W : MorphismProperty C
inst✝ : W.HasLeftCalculusOfFractions
X Y Z : C
z₁✝ : Hom W X Y
z₂✝ : Hom W Y Z
a₁ a₂ : W.LeftFraction X Y
b : W.LeftFraction Y Z
U : C
t₁ : a₁.Y' ⟶ U
t₂ : a₂.Y' ⟶ U
hst : a₁.s ≫ t₁ = a₂.s ≫ t₂
hft : a₁.f ≫ t₁ = a₂.f ≫ t₂
ht : W (a₁.s ≫ t₁)
z₁ : W.LeftFraction a₁.Y' b.Y'
fac₁ : b.f ≫ z₁.s = a₁.s ≫ z₁.f
z₂ : W.LeftFraction a₂.Y' b.Y'
fac₂ : b.f ≫ z₂.s = a₂.s ≫ z₂.f
w₁ : W.LeftFraction U z₁.Y'
w₂ : W.LeftFraction U z₂.Y'
p₁ : W.LeftFraction X Z := { Y' := w₁.Y', f := a₁.f ≫ t₁ ≫ w₁.f, s := b.s ≫ z₁.s ≫ w₁.s, hs := ⋯ }
p₂ : W.LeftFraction X Z := { Y' := w₂.Y', f := a₂.f ≫ t₂ ≫ w₂.f, s := b.s ≫ z₂.s ≫ w₂.s, hs := ⋯ }
fac₁' : b.f ≫ z₁.s ≫ w₁.s = a₁.s ≫ t₁ ≫ w₁.f
fac₂' : b.f ≫ z₂.s ≫ w₂.s = a₂.s ≫ t₂ ≫ w₂.f
⊢ LeftFractionRel p₂ (a₂.comp₀ b z₂)