awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
W₁ : MorphismProperty.{v_1, u_1} C₁
W₂ : MorphismProperty.{v_2, u_2} C₂
Φ : LocalizerMorphism W₁ W₂
inst_2 : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂)
inst_3 : Φ.arrow (W₁ := W₁) (W₂ := W₂).HasRightResolutions (W₁ := W₁.arrow) (W₂ := W₂.arrow)
inst_4 : W₂.ContainsIdentities
D : Type u_3
inst_5 : Category.{v_3, u_3} D
L : Functor.{v_2, v_3, u_2, u_3} C₂ D
inst_6 : L.IsLocalization W₂
X₂ : C₂
X₃ : D
y : L.obj X₂ ⟶ X₃
w : TwoSquare (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D) :=
  TwoSquare.mk (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D)
    (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L).rightUnitor.inv
this_1 : Nonempty (w.CostructuredArrowDownwards y)
this :
  ∀ (X : w.CostructuredArrowDownwards y),
    ∃ (Y : Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂),
      Zigzag X ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L y).obj Y)
X X' :
  (TwoSquare.mk (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D)
        (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L).rightUnitor.inv).CostructuredArrowDownwards
    y
Y : Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂
hX : Zigzag X ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L y).obj Y)
⊢ Zigzag X X'
have obtain : ∀ (Y' : Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂)
  (hX' : Zigzag X' ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L y).obj Y')), Zigzag X X'
Tactic: obtain ⟨Y', hX'⟩ := this X'
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
W₁ : MorphismProperty.{v_1, u_1} C₁
W₂ : MorphismProperty.{v_2, u_2} C₂
Φ : LocalizerMorphism W₁ W₂
inst_2 : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂)
inst_3 : Φ.arrow (W₁ := W₁) (W₂ := W₂).HasRightResolutions (W₁ := W₁.arrow) (W₂ := W₂.arrow)
inst_4 : W₂.ContainsIdentities
D : Type u_3
inst_5 : Category.{v_3, u_3} D
L : Functor.{v_2, v_3, u_2, u_3} C₂ D
inst_6 : L.IsLocalization W₂
X₂ : C₂
X₃ : D
y : L.obj X₂ ⟶ X₃
w : TwoSquare (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D) :=
  TwoSquare.mk (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D)
    (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L).rightUnitor.inv
this : Nonempty (w.CostructuredArrowDownwards y)
⊢ IsConnected
    ((TwoSquare.mk (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D)
          (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L).rightUnitor.inv).CostructuredArrowDownwards
      y)
have suffices : ∀ (X : w.CostructuredArrowDownwards y),
  ∃ (Y : Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂), Zigzag X ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L y).obj Y)
Tactic: suffices ∀ (X : w.CostructuredArrowDownwards y),
      ∃ Y, Zigzag X ((fromRightResolution Φ L y).obj Y) by
    refine zigzag_isConnected (fun X X' => ?_)
    obtain ⟨Y, hX⟩ := this X
    obtain ⟨Y', hX'⟩ := this X'
    exact hX.trans ((zigzag_obj_of_zigzag _ (isPreconnected_zigzag Y Y')).trans hX'.symm)
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
W₁ : MorphismProperty.{v_1, u_1} C₁
W₂ : MorphismProperty.{v_2, u_2} C₂
Φ : LocalizerMorphism W₁ W₂
inst_2 : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂)
inst_3 : Φ.arrow (W₁ := W₁) (W₂ := W₂).HasRightResolutions (W₁ := W₁.arrow) (W₂ := W₂.arrow)
inst_4 : W₂.ContainsIdentities
D : Type u_3
inst_5 : Category.{v_3, u_3} D
L : Functor.{v_2, v_3, u_2, u_3} C₂ D
inst_6 : L.IsLocalization W₂
X₂ : C₂
X₃ : D
y : L.obj X₂ ⟶ X₃
w : TwoSquare (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D) :=
  TwoSquare.mk (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D)
    (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L).rightUnitor.inv
this : Nonempty (w.CostructuredArrowDownwards y)
X : w.CostructuredArrowDownwards y
⊢ ∃ (Y : Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂), Zigzag X ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L y).obj Y)
have obtain : ∀ (c : C₁) (g : X₂ ⟶ Φ.functor (W₁ := W₁) (W₂ := W₂).obj c) (x : (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L).obj c ⟶ X₃)
  (fac : L.map g ≫ w.app c ≫ Functor.map (𝟭 D) x = y),
  ∃ (Y : Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂),
    Zigzag (TwoSquare.CostructuredArrowDownwards.mk w y c g x fac)
      ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L y).obj Y)
Tactic: obtain ⟨c, g, x, fac, rfl⟩ := TwoSquare.CostructuredArrowDownwards.mk_surjective X
Cleared hypotheses: X
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor (hypothesis)
Goal:
C₁ : Type u_1
C₂ : Type u_2
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
W₁ : MorphismProperty.{v_1, u_1} C₁
W₂ : MorphismProperty.{v_2, u_2} C₂
Φ : LocalizerMorphism W₁ W₂
inst_2 : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂)
inst_3 : Φ.arrow (W₁ := W₁) (W₂ := W₂).HasRightResolutions (W₁ := W₁.arrow) (W₂ := W₂.arrow)
inst_4 : W₂.ContainsIdentities
D : Type u_3
inst_5 : Category.{v_3, u_3} D
L : Functor.{v_2, v_3, u_2, u_3} C₂ D
inst_6 : L.IsLocalization W₂
X₂ : C₂
X₃ : D
y : L.obj X₂ ⟶ X₃
w : TwoSquare (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D) :=
  TwoSquare.mk (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D)
    (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L).rightUnitor.inv
this : Nonempty (w.CostructuredArrowDownwards y)
c : C₁
g : X₂ ⟶ Φ.functor (W₁ := W₁) (W₂ := W₂).obj c
x : L.obj (Φ.functor (W₁ := W₁) (W₂ := W₂).obj c) ⟶ X₃
fac : L.map g ≫ 𝟙 (L.obj (Φ.functor (W₁ := W₁) (W₂ := W₂).obj c)) ≫ x = y
⊢ ∃ (Y : Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂),
    Zigzag (TwoSquare.CostructuredArrowDownwards.mk w y c g x fac)
      ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L y).obj Y)
have fac₁ : L.map g ≫ x = y
Tactic: rw [id_comp] at fac
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
W₁ : MorphismProperty.{v_1, u_1} C₁
W₂ : MorphismProperty.{v_2, u_2} C₂
Φ : LocalizerMorphism W₁ W₂
inst_2 : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂)
inst_3 : Φ.arrow (W₁ := W₁) (W₂ := W₂).HasRightResolutions (W₁ := W₁.arrow) (W₂ := W₂.arrow)
inst_4 : W₂.ContainsIdentities
D : Type u_3
inst_5 : Category.{v_3, u_3} D
L : Functor.{v_2, v_3, u_2, u_3} C₂ D
inst_6 : L.IsLocalization W₂
X₂ : C₂
X₃ : D
y : L.obj X₂ ⟶ X₃
w : TwoSquare (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D) :=
  TwoSquare.mk (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D)
    (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L).rightUnitor.inv
this : Nonempty (w.CostructuredArrowDownwards y)
c : C₁
g : X₂ ⟶ Φ.functor (W₁ := W₁) (W₂ := W₂).obj c
x : L.obj (Φ.functor (W₁ := W₁) (W₂ := W₂).obj c) ⟶ X₃
fac_1 : L.map g ≫ 𝟙 (L.obj (Φ.functor (W₁ := W₁) (W₂ := W₂).obj c)) ≫ x = y
fac : L.map g ≫ x = y
ρ : Φ.arrow (W₁ := W₁) (W₂ := W₂).RightResolution (W₁ := W₁.arrow) (W₂ := W₂.arrow) (Arrow.mk g) :=
  Classical.arbitrary (Φ.arrow (W₁ := W₁) (W₂ := W₂).RightResolution (W₁ := W₁.arrow) (W₂ := W₂.arrow) (Arrow.mk g))
⊢ ∃ (Y : Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂),
    Zigzag (TwoSquare.CostructuredArrowDownwards.mk w y c g x fac_1)
      ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L y).obj Y)
have refine : Zigzag (TwoSquare.CostructuredArrowDownwards.mk w y c g x fac_1)
  ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L y).obj
    (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (Arrow.Hom.left (ρ.w (W₁ := W₁.arrow) (W₂ := W₂.arrow))) sorry))
Tactic: refine ⟨RightResolution.mk ρ.w.left ρ.hw.1, ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor (hypothesis)
Goal:
C₁ : Type u_1
C₂ : Type u_2
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
W₁ : MorphismProperty.{v_1, u_1} C₁
W₂ : MorphismProperty.{v_2, u_2} C₂
Φ : LocalizerMorphism W₁ W₂
inst_2 : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂)
inst_3 : Φ.arrow (W₁ := W₁) (W₂ := W₂).HasRightResolutions (W₁ := W₁.arrow) (W₂ := W₂.arrow)
inst_4 : W₂.ContainsIdentities
D : Type u_3
inst_5 : Category.{v_3, u_3} D
L : Functor.{v_2, v_3, u_2, u_3} C₂ D
inst_6 : L.IsLocalization W₂
X₂ : C₂
X₃ : D
y : L.obj X₂ ⟶ X₃
w : TwoSquare (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D) :=
  TwoSquare.mk (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D)
    (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L).rightUnitor.inv
this : Nonempty (w.CostructuredArrowDownwards y)
c : C₁
g : X₂ ⟶ Φ.functor (W₁ := W₁) (W₂ := W₂).obj c
x : L.obj (Φ.functor (W₁ := W₁) (W₂ := W₂).obj c) ⟶ X₃
fac_1 : L.map g ≫ 𝟙 (L.obj (Φ.functor (W₁ := W₁) (W₂ := W₂).obj c)) ≫ x = y
fac : L.map g ≫ x = y
ρ : Φ.arrow (W₁ := W₁) (W₂ := W₂).RightResolution (W₁ := W₁.arrow) (W₂ := W₂.arrow) (Arrow.mk g) :=
  Classical.arbitrary (Φ.arrow (W₁ := W₁) (W₂ := W₂).RightResolution (W₁ := W₁.arrow) (W₂ := W₂.arrow) (Arrow.mk g))
⊢ Zigzag (TwoSquare.CostructuredArrowDownwards.mk w y c g x fac_1)
    ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L y).obj
      (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (Arrow.Hom.left (ρ.w (W₁ := W₁.arrow) (W₂ := W₂.arrow))) ⋯))
have this₁ : Zigzag
  ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
    (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (𝟙 (Φ.functor (W₁ := W₁) (W₂ := W₂).obj c)) sorry))
  ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
    (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (Arrow.Hom.right (ρ.w (W₁ := W₁.arrow) (W₂ := W₂.arrow))) sorry))
Tactic: have := zigzag_obj_of_zigzag
    (fromRightResolution Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac)
      (isPreconnected_zigzag (RightResolution.mk (𝟙 _) (W₂.id_mem _))
        (RightResolution.mk ρ.w.right ρ.hw.2))
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
W₁ : MorphismProperty.{v_1, u_1} C₁
W₂ : MorphismProperty.{v_2, u_2} C₂
Φ : LocalizerMorphism W₁ W₂
inst_2 : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂)
inst_3 : Φ.arrow (W₁ := W₁) (W₂ := W₂).HasRightResolutions (W₁ := W₁.arrow) (W₂ := W₂.arrow)
inst_4 : W₂.ContainsIdentities
D : Type u_3
inst_5 : Category.{v_3, u_3} D
L : Functor.{v_2, v_3, u_2, u_3} C₂ D
inst_6 : L.IsLocalization W₂
X₂ : C₂
X₃ : D
y : L.obj X₂ ⟶ X₃
w : TwoSquare (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D) :=
  TwoSquare.mk (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D)
    (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L).rightUnitor.inv
this_1 : Nonempty (w.CostructuredArrowDownwards y)
c : C₁
g : X₂ ⟶ Φ.functor (W₁ := W₁) (W₂ := W₂).obj c
x : L.obj (Φ.functor (W₁ := W₁) (W₂ := W₂).obj c) ⟶ X₃
fac_1 : L.map g ≫ 𝟙 (L.obj (Φ.functor (W₁ := W₁) (W₂ := W₂).obj c)) ≫ x = y
fac : L.map g ≫ x = y
ρ : Φ.arrow (W₁ := W₁) (W₂ := W₂).RightResolution (W₁ := W₁.arrow) (W₂ := W₂.arrow) (Arrow.mk g) :=
  Classical.arbitrary (Φ.arrow (W₁ := W₁) (W₂ := W₂).RightResolution (W₁ := W₁.arrow) (W₂ := W₂.arrow) (Arrow.mk g))
this :
  Zigzag
    ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
      (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (𝟙 (Φ.functor (W₁ := W₁) (W₂ := W₂).obj c)) ⋯))
    ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
      (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (Arrow.Hom.right (ρ.w (W₁ := W₁.arrow) (W₂ := W₂.arrow))) ⋯))
⊢ Zigzag (TwoSquare.CostructuredArrowDownwards.mk w y c g x fac_1)
    ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L y).obj
      (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (Arrow.Hom.left (ρ.w (W₁ := W₁.arrow) (W₂ := W₂.arrow))) ⋯))
have refine_1 : Zigzag (TwoSquare.CostructuredArrowDownwards.mk w y c g x fac_1)
  ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
    (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (𝟙 (Φ.functor (W₁ := W₁) (W₂ := W₂).obj c)) sorry))
Tactic: refine Zigzag.trans ?_ (Zigzag.trans this ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
W₁ : MorphismProperty.{v_1, u_1} C₁
W₂ : MorphismProperty.{v_2, u_2} C₂
Φ : LocalizerMorphism W₁ W₂
inst_2 : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂)
inst_3 : Φ.arrow (W₁ := W₁) (W₂ := W₂).HasRightResolutions (W₁ := W₁.arrow) (W₂ := W₂.arrow)
inst_4 : W₂.ContainsIdentities
D : Type u_3
inst_5 : Category.{v_3, u_3} D
L : Functor.{v_2, v_3, u_2, u_3} C₂ D
inst_6 : L.IsLocalization W₂
X₂ : C₂
X₃ : D
y : L.obj X₂ ⟶ X₃
w : TwoSquare (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D) :=
  TwoSquare.mk (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D)
    (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L).rightUnitor.inv
this_1 : Nonempty (w.CostructuredArrowDownwards y)
c : C₁
g : X₂ ⟶ Φ.functor (W₁ := W₁) (W₂ := W₂).obj c
x : L.obj (Φ.functor (W₁ := W₁) (W₂ := W₂).obj c) ⟶ X₃
fac_1 : L.map g ≫ 𝟙 (L.obj (Φ.functor (W₁ := W₁) (W₂ := W₂).obj c)) ≫ x = y
fac : L.map g ≫ x = y
ρ : Φ.arrow (W₁ := W₁) (W₂ := W₂).RightResolution (W₁ := W₁.arrow) (W₂ := W₂.arrow) (Arrow.mk g) :=
  Classical.arbitrary (Φ.arrow (W₁ := W₁) (W₂ := W₂).RightResolution (W₁ := W₁.arrow) (W₂ := W₂.arrow) (Arrow.mk g))
this :
  Zigzag
    ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
      (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (𝟙 (Φ.functor (W₁ := W₁) (W₂ := W₂).obj c)) ⋯))
    ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
      (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (Arrow.Hom.right (ρ.w (W₁ := W₁.arrow) (W₂ := W₂.arrow))) ⋯))
refine_1 :
  Zigzag (TwoSquare.CostructuredArrowDownwards.mk w y c g x fac_1)
    ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
      (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (𝟙 (Φ.functor (W₁ := W₁) (W₂ := W₂).obj c)) ⋯))
⊢ Zigzag (TwoSquare.CostructuredArrowDownwards.mk w y c g x fac_1)
    ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L y).obj
      (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (Arrow.Hom.left (ρ.w (W₁ := W₁.arrow) (W₂ := W₂.arrow))) ⋯))
have refine_2 : Zigzag
  ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
    (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (Arrow.Hom.right (ρ.w (W₁ := W₁.arrow) (W₂ := W₂.arrow))) sorry))
  ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L y).obj
    (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (Arrow.Hom.left (ρ.w (W₁ := W₁.arrow) (W₂ := W₂.arrow))) sorry))
Tactic: refine Zigzag.trans ?_ (Zigzag.trans this ?_)
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
W₁ : MorphismProperty.{v_1, u_1} C₁
W₂ : MorphismProperty.{v_2, u_2} C₂
Φ : LocalizerMorphism W₁ W₂
inst_2 : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂)
inst_3 : Φ.arrow (W₁ := W₁) (W₂ := W₂).HasRightResolutions (W₁ := W₁.arrow) (W₂ := W₂.arrow)
inst_4 : W₂.ContainsIdentities
D : Type u_3
inst_5 : Category.{v_3, u_3} D
L : Functor.{v_2, v_3, u_2, u_3} C₂ D
inst_6 : L.IsLocalization W₂
X₂ : C₂
X₃ : D
y : L.obj X₂ ⟶ X₃
w : TwoSquare (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D) :=
  TwoSquare.mk (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D)
    (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L).rightUnitor.inv
this_1 : Nonempty (w.CostructuredArrowDownwards y)
c : C₁
g : X₂ ⟶ Φ.functor (W₁ := W₁) (W₂ := W₂).obj c
x : L.obj (Φ.functor (W₁ := W₁) (W₂ := W₂).obj c) ⟶ X₃
fac_1 : L.map g ≫ 𝟙 (L.obj (Φ.functor (W₁ := W₁) (W₂ := W₂).obj c)) ≫ x = y
fac : L.map g ≫ x = y
ρ : Φ.arrow (W₁ := W₁) (W₂ := W₂).RightResolution (W₁ := W₁.arrow) (W₂ := W₂.arrow) (Arrow.mk g) :=
  Classical.arbitrary (Φ.arrow (W₁ := W₁) (W₂ := W₂).RightResolution (W₁ := W₁.arrow) (W₂ := W₂.arrow) (Arrow.mk g))
this :
  Zigzag
    ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
      (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (𝟙 (Φ.functor (W₁ := W₁) (W₂ := W₂).obj c)) ⋯))
    ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
      (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (Arrow.Hom.right (ρ.w (W₁ := W₁.arrow) (W₂ := W₂.arrow))) ⋯))
⊢ Zigzag
    ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
      (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (Arrow.Hom.right (ρ.w (W₁ := W₁.arrow) (W₂ := W₂.arrow))) ⋯))
    ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L y).obj
      (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (Arrow.Hom.left (ρ.w (W₁ := W₁.arrow) (W₂ := W₂.arrow))) ⋯))
have refine_2 : (fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L y).obj
    (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (Arrow.Hom.left (ρ.w (W₁ := W₁.arrow) (W₂ := W₂.arrow))) sorry) ⟶
  (fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
    (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (Arrow.Hom.right (ρ.w (W₁ := W₁.arrow) (W₂ := W₂.arrow))) sorry)
Tactic: apply Zigzag.of_inv
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
W₁ : MorphismProperty.{v_1, u_1} C₁
W₂ : MorphismProperty.{v_2, u_2} C₂
Φ : LocalizerMorphism W₁ W₂
inst_2 : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂)
inst_3 : Φ.arrow (W₁ := W₁) (W₂ := W₂).HasRightResolutions (W₁ := W₁.arrow) (W₂ := W₂.arrow)
inst_4 : W₂.ContainsIdentities
D : Type u_3
inst_5 : Category.{v_3, u_3} D
L : Functor.{v_2, v_3, u_2, u_3} C₂ D
inst_6 : L.IsLocalization W₂
X₂ : C₂
X₃ : D
y : L.obj X₂ ⟶ X₃
w : TwoSquare (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D) :=
  TwoSquare.mk (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D)
    (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L).rightUnitor.inv
this_1 : Nonempty (w.CostructuredArrowDownwards y)
c : C₁
g : X₂ ⟶ Φ.functor (W₁ := W₁) (W₂ := W₂).obj c
x : L.obj (Φ.functor (W₁ := W₁) (W₂ := W₂).obj c) ⟶ X₃
fac_1 : L.map g ≫ 𝟙 (L.obj (Φ.functor (W₁ := W₁) (W₂ := W₂).obj c)) ≫ x = y
fac : L.map g ≫ x = y
ρ : Φ.arrow (W₁ := W₁) (W₂ := W₂).RightResolution (W₁ := W₁.arrow) (W₂ := W₂.arrow) (Arrow.mk g) :=
  Classical.arbitrary (Φ.arrow (W₁ := W₁) (W₂ := W₂).RightResolution (W₁ := W₁.arrow) (W₂ := W₂.arrow) (Arrow.mk g))
this :
  Zigzag
    ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
      (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (𝟙 (Φ.functor (W₁ := W₁) (W₂ := W₂).obj c)) ⋯))
    ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
      (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (Arrow.Hom.right (ρ.w (W₁ := W₁.arrow) (W₂ := W₂.arrow))) ⋯))
⊢ (fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L y).obj
      (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (Arrow.Hom.left (ρ.w (W₁ := W₁.arrow) (W₂ := W₂.arrow))) ⋯) ⟶
    (fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
      (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (Arrow.Hom.right (ρ.w (W₁ := W₁.arrow) (W₂ := W₂.arrow))) ⋯)
have refine_2 : ((TwoSquare.mk (Φ.functor (W₁ := W₁) (W₂ := W₂)) (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L) L (𝟭 D)
              (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ L).rightUnitor.inv).structuredArrowDownwards
          X₂).map
      (StructuredArrow.homMk ρ.X₁ (W₁ := W₁.arrow) (W₂ := W₂.arrow).hom sorry) ≫
    CostructuredArrow.hom
      ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L x ⋙ w.costructuredArrowDownwardsPrecomp x y g fac).obj
        (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (Arrow.Hom.right (ρ.w (W₁ := W₁.arrow) (W₂ := W₂.arrow))) sorry)) =
  CostructuredArrow.hom
    ((fromRightResolution (W₁ := W₁) (W₂ := W₂) Φ L y).obj
      (RightResolution.mk (W₁ := W₁) (W₂ := W₂) (Arrow.Hom.left (ρ.w (W₁ := W₁.arrow) (W₂ := W₂.arrow))) sorry))
Tactic: refine CostructuredArrow.homMk (StructuredArrow.homMk ρ.X₁.hom (by simp)) ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
W₁ : MorphismProperty.{v_1, u_1} C₁
W₂ : MorphismProperty.{v_2, u_2} C₂
Φ : LocalizerMorphism W₁ W₂
inst_2 : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂)
inst_3 : Φ.arrow (W₁ := W₁) (W₂ := W₂).HasRightResolutions (W₁ := W₁.arrow) (W₂ := W₂.arrow)
inst_4 : W₂.ContainsIdentities
inst_5 : Φ.IsLocalizedEquivalence (W₁ := W₁) (W₂ := W₂)
⊢ Φ.IsRightDerivabilityStructure (W₁ := W₁) (W₂ := W₂)
have rw : ∀ {X₂ : C₂} {X₃ : W₂.Localization} (g : W₂.Q.obj X₂ ⟶ Functor.obj (𝟭 W₂.Localization) X₃),
  IsConnected (TwoSquare.CostructuredArrowDownwards (Φ.functor (W₁ := W₁) (W₂ := W₂) ⋙ W₂.Q).rightUnitor.symm.hom g)
Tactic: rw [Φ.isRightDerivabilityStructure_iff (Φ.functor ⋙ W₂.Q) W₂.Q (𝟭 _)
    (Functor.rightUnitor _).symm, TwoSquare.guitartExact_iff_isConnected_downwards]
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor (goal)
Goal:
C₁ : Type u_1
C₂ : Type u_2
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
W₁ : MorphismProperty.{v_1, u_1} C₁
W₂ : MorphismProperty.{v_2, u_2} C₂
Φ : LocalizerMorphism W₁ W₂
inst_2 : ∀ (X₂ : C₂), IsConnected (Φ.LeftResolution (W₁ := W₁) (W₂ := W₂) X₂)
inst_3 : Φ.arrow (W₁ := W₁) (W₂ := W₂).HasLeftResolutions (W₁ := W₁.arrow) (W₂ := W₂.arrow)
inst_4 : W₂.ContainsIdentities
inst_5 : Φ.IsLocalizedEquivalence (W₁ := W₁) (W₂ := W₂)
⊢ Φ.IsLeftDerivabilityStructure (W₁ := W₁) (W₂ := W₂)
have rw : Φ.op (W₁ := W₁) (W₂ := W₂).IsRightDerivabilityStructure (W₁ := W₁.op) (W₂ := W₂.op)
Tactic: rw [isLeftDerivabilityStructure_iff_op]
-- Lean 4 proof step from Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor (hypothesis)
Goal:
C₁ : Type u_1
C₂ : Type u_2
inst : Category.{v_1, u_1} C₁
inst_1 : Category.{v_2, u_2} C₂
W₁ : MorphismProperty.{v_1, u_1} C₁
W₂ : MorphismProperty.{v_2, u_2} C₂
Φ : LocalizerMorphism W₁ W₂
inst_2 : ∀ (X₂ : C₂), IsConnected (Φ.LeftResolution (W₁ := W₁) (W₂ := W₂) X₂)
inst_3 : Φ.arrow (W₁ := W₁) (W₂ := W₂).HasLeftResolutions (W₁ := W₁.arrow) (W₂ := W₂.arrow)
inst_4 : W₂.ContainsIdentities
inst_5 : Φ.IsLocalizedEquivalence (W₁ := W₁) (W₂ := W₂)
⊢ Φ.op (W₁ := W₁) (W₂ := W₂).IsRightDerivabilityStructure (W₁ := W₁.op) (W₂ := W₂.op)
have this : Φ.op (W₁ := W₁) (W₂ := W₂).arrow (W₁ := W₁.op) (W₂ := W₂.op).HasRightResolutions (W₁ := W₁.op.arrow) (W₂ := W₂.op.arrow)
Tactic: have : Φ.op.arrow.HasRightResolutions := fun f ↦ by
    let R : Φ.arrow.LeftResolution (Arrow.mk f.hom.unop) := Classical.arbitrary _
    exact ⟨{
      X₁ := Arrow.mk R.X₁.hom.op
      w := Arrow.homMk R.w.right.op R.w.left.op (Quiver.Hom.unop_inj R.w.w.symm)
      hw := ⟨R.hw.right, R.hw.left⟩ }⟩