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
R R' : Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂
φ : R ⟶ R'
⊢ StructuredArrow.Hom.right
      (((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 (φ.f (W₁ := W₁) (W₂ := W₂)) ⋯) ≫
        (CostructuredArrow.mk (StructuredArrow.homMk ((isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y) ⋯)).hom) =
    StructuredArrow.Hom.right
      (CostructuredArrow.mk (StructuredArrow.homMk ((isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y) ⋯)).hom
have h : L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
    (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) sorry).inv ≫ y =
  (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) sorry).inv ≫ y
Tactic: dsimp
-- 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
R R' : Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂
φ : R ⟶ R'
⊢ L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
      (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y =
    (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
have h : (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
      (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) sorry).inv) ≫
    y =
  (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) sorry).inv ≫ y
Tactic: rw [← assoc, ← cancel_epi (isoOfHom L W₂ R.w R.hw).hom,
      isoOfHom_hom, isoOfHom_hom_inv_id_assoc, assoc, ← L.map_comp_assoc,
      φ.comm, isoOfHom_hom_inv_id_assoc]
-- 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
R R' : Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂
φ : R ⟶ R'
h :
  (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
        (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
      y =
    (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
⊢ L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
      (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y =
    (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
have h₁ : (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) sorry).hom ≫
    (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
        (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) sorry).inv) ≫
      y =
  (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) sorry).hom ≫ (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) sorry).inv ≫ y
Tactic: rw [← assoc, ← cancel_epi (isoOfHom L W₂ R.w R.hw).hom,
      isoOfHom_hom, isoOfHom_hom_inv_id_assoc, assoc, ← L.map_comp_assoc,
      φ.comm, isoOfHom_hom_inv_id_assoc]
-- 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
R R' : Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂
φ : R ⟶ R'
h :
  (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
        (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
      y =
    (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
h₁ :
  (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).hom ≫
      (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
          (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
        y =
    (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).hom ≫ (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
⊢ L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
      (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y =
    (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
have h₂ : L.map (R.w (W₁ := W₁) (W₂ := W₂)) ≫
    (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
        (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) sorry).inv) ≫
      y =
  L.map (R.w (W₁ := W₁) (W₂ := W₂)) ≫ (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) sorry).inv ≫ y
Tactic: rw [← assoc, ← cancel_epi (isoOfHom L W₂ R.w R.hw).hom,
      isoOfHom_hom, isoOfHom_hom_inv_id_assoc, assoc, ← L.map_comp_assoc,
      φ.comm, isoOfHom_hom_inv_id_assoc]
-- 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
R R' : Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂
φ : R ⟶ R'
h :
  (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
        (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
      y =
    (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
h₁ :
  (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).hom ≫
      (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
          (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
        y =
    (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).hom ≫ (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
h₂ :
  L.map (R.w (W₁ := W₁) (W₂ := W₂)) ≫
      (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
          (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
        y =
    L.map (R.w (W₁ := W₁) (W₂ := W₂)) ≫ (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
⊢ L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
      (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y =
    (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
have h₃ : L.map (R.w (W₁ := W₁) (W₂ := W₂)) ≫
    (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
        (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) sorry).inv) ≫
      y =
  y
Tactic: rw [← assoc, ← cancel_epi (isoOfHom L W₂ R.w R.hw).hom,
      isoOfHom_hom, isoOfHom_hom_inv_id_assoc, assoc, ← L.map_comp_assoc,
      φ.comm, isoOfHom_hom_inv_id_assoc]
-- 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
R R' : Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂
φ : R ⟶ R'
h :
  (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
        (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
      y =
    (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
h₁ :
  (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).hom ≫
      (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
          (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
        y =
    (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).hom ≫ (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
h₂ :
  L.map (R.w (W₁ := W₁) (W₂ := W₂)) ≫
      (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
          (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
        y =
    L.map (R.w (W₁ := W₁) (W₂ := W₂)) ≫ (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
h₃ :
  L.map (R.w (W₁ := W₁) (W₂ := W₂)) ≫
      (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
          (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
        y =
    y
⊢ L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
      (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y =
    (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
have h₄ : L.map (R.w (W₁ := W₁) (W₂ := W₂)) ≫
    L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
      (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) sorry).inv ≫ y =
  y
Tactic: rw [← assoc, ← cancel_epi (isoOfHom L W₂ R.w R.hw).hom,
      isoOfHom_hom, isoOfHom_hom_inv_id_assoc, assoc, ← L.map_comp_assoc,
      φ.comm, isoOfHom_hom_inv_id_assoc]
-- 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
R R' : Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂
φ : R ⟶ R'
h :
  (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
        (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
      y =
    (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
h₁ :
  (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).hom ≫
      (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
          (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
        y =
    (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).hom ≫ (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
h₂ :
  L.map (R.w (W₁ := W₁) (W₂ := W₂)) ≫
      (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
          (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
        y =
    L.map (R.w (W₁ := W₁) (W₂ := W₂)) ≫ (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
h₃ :
  L.map (R.w (W₁ := W₁) (W₂ := W₂)) ≫
      (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
          (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
        y =
    y
h₄ :
  L.map (R.w (W₁ := W₁) (W₂ := W₂)) ≫
      L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
        (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y =
    y
⊢ L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
      (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y =
    (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
have h₅ : L.map (R.w (W₁ := W₁) (W₂ := W₂) ≫ Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
    (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) sorry).inv ≫ y =
  y
Tactic: rw [← assoc, ← cancel_epi (isoOfHom L W₂ R.w R.hw).hom,
      isoOfHom_hom, isoOfHom_hom_inv_id_assoc, assoc, ← L.map_comp_assoc,
      φ.comm, isoOfHom_hom_inv_id_assoc]
-- 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
R R' : Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂
φ : R ⟶ R'
h :
  (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
        (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
      y =
    (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
h₁ :
  (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).hom ≫
      (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
          (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
        y =
    (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).hom ≫ (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
h₂ :
  L.map (R.w (W₁ := W₁) (W₂ := W₂)) ≫
      (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
          (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
        y =
    L.map (R.w (W₁ := W₁) (W₂ := W₂)) ≫ (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
h₃ :
  L.map (R.w (W₁ := W₁) (W₂ := W₂)) ≫
      (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
          (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
        y =
    y
h₄ :
  L.map (R.w (W₁ := W₁) (W₂ := W₂)) ≫
      L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
        (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y =
    y
h₅ :
  L.map (R.w (W₁ := W₁) (W₂ := W₂) ≫ Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
      (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y =
    y
⊢ L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
      (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y =
    (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
have h₆ : L.map (R'.w (W₁ := W₁) (W₂ := W₂)) ≫ (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) sorry).inv ≫ y = y
Tactic: rw [← assoc, ← cancel_epi (isoOfHom L W₂ R.w R.hw).hom,
      isoOfHom_hom, isoOfHom_hom_inv_id_assoc, assoc, ← L.map_comp_assoc,
      φ.comm, isoOfHom_hom_inv_id_assoc]
-- 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
R R' : Φ.RightResolution (W₁ := W₁) (W₂ := W₂) X₂
φ : R ⟶ R'
h :
  (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
        (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
      y =
    (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
h₁ :
  (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).hom ≫
      (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
          (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
        y =
    (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).hom ≫ (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
h₂ :
  L.map (R.w (W₁ := W₁) (W₂ := W₂)) ≫
      (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
          (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
        y =
    L.map (R.w (W₁ := W₁) (W₂ := W₂)) ≫ (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
h₃ :
  L.map (R.w (W₁ := W₁) (W₂ := W₂)) ≫
      (L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
          (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv) ≫
        y =
    y
h₄ :
  L.map (R.w (W₁ := W₁) (W₂ := W₂)) ≫
      L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
        (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y =
    y
h₅ :
  L.map (R.w (W₁ := W₁) (W₂ := W₂) ≫ Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
      (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y =
    y
h₆ : L.map (R'.w (W₁ := W₁) (W₂ := W₂)) ≫ (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y = y
⊢ L.map (Φ.functor (W₁ := W₁) (W₂ := W₂).map (φ.f (W₁ := W₁) (W₂ := W₂))) ≫
      (isoOfHom L W₂ (R'.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y =
    (isoOfHom L W₂ (R.w (W₁ := W₁) (W₂ := W₂)) ⋯).inv ≫ y
have h₇ : y = y
Tactic: rw [← assoc, ← cancel_epi (isoOfHom L W₂ R.w R.hw).hom,
      isoOfHom_hom, isoOfHom_hom_inv_id_assoc, assoc, ← L.map_comp_assoc,
      φ.comm, isoOfHom_hom_inv_id_assoc]
-- 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
⊢ 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 this : Nonempty (w.CostructuredArrowDownwards y)
Tactic: have : Nonempty (w.CostructuredArrowDownwards y) :=
    ⟨(fromRightResolution Φ L y).obj (Classical.arbitrary _)⟩
-- 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)
⊢ 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 refine : ∀
  (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),
  Zigzag X X'
Tactic: refine zigzag_isConnected (fun X 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_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
⊢ 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