awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Grothendieck (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u₁
inst_1 : Category.{v₁, u₁} D
F : C ⥤ Cat
G : Functor.{v, w, u, w + 1} C (Type w)
E : Type u_1
inst_2 : Category.{v_1, u_1} E
c : C
d : (↑(F.obj c) : Type u₂)
⊢ { base := 𝟙 { base := c, fiber := d : Grothendieck F }.base, fiber := eqToHom ⋯ ≫ 𝟙 d :
      { base := c, fiber := d : Grothendieck F }.Hom { base := c, fiber := d : Grothendieck F } } =
    𝟙 { base := c, fiber := d }
have dsimp : { base := 𝟙 c, fiber := eqToHom sorry ≫ 𝟙 d :
    { base := c, fiber := d : Grothendieck F }.Hom { base := c, fiber := d : Grothendieck F } } =
  𝟙 { base := c, fiber := d }
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Grothendieck (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u₁
inst_1 : Category.{v₁, u₁} D
F : C ⥤ Cat
G : Functor.{v, w, u, w + 1} C (Type w)
E : Type u_1
inst_2 : Category.{v_1, u_1} E
c : C
d : (↑(F.obj c) : Type u₂)
⊢ { base := 𝟙 c, fiber := eqToHom ⋯ ≫ 𝟙 d :
      { base := c, fiber := d : Grothendieck F }.Hom { base := c, fiber := d : Grothendieck F } } =
    𝟙 { base := c, fiber := d }
have e_fiber : eqToHom sorry ≫ 𝟙 d = eqToHom sorry
Tactic: congr
-- Lean 4 proof step from Mathlib.CategoryTheory.Grothendieck (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u₁
inst_1 : Category.{v₁, u₁} D
F : C ⥤ Cat
G : Functor.{v, w, u, w + 1} C (Type w)
E : Type u_1
inst_2 : Category.{v_1, u_1} E
c : C
X Y Z : (↑(F.obj c) : Type u₂)
f : X ⟶ Y
g : Y ⟶ Z
⊢ Eq (α := (F.map (𝟙 c ≫ 𝟙 c)).toFunctor.obj X ⟶ Z) ((eqToHom ⋯ ≫ f) ≫ g : (F.map (𝟙 c ≫ 𝟙 c)).toFunctor.obj X ⟶ Z)
    (CategoryStruct.comp (Y := (F.map (𝟙 c)).toFunctor.obj Y) (eqToHom ⋯ ≫ (F.map (𝟙 c)).toFunctor.map f) (eqToHom ⋯) ≫
        g :
      (F.map (𝟙 c ≫ 𝟙 c)).toFunctor.obj X ⟶ Z)
have e_a : Eq (α := (F.map (𝟙 c ≫ 𝟙 c)).toFunctor.obj X ⟶ Y) (eqToHom sorry ≫ f : (F.map (𝟙 c ≫ 𝟙 c)).toFunctor.obj X ⟶ Y)
  (CategoryStruct.comp (Y := (F.map (𝟙 c)).toFunctor.obj Y) (eqToHom sorry ≫ (F.map (𝟙 c)).toFunctor.map f) (eqToHom sorry) :
    (F.map (𝟙 c ≫ 𝟙 c)).toFunctor.obj X ⟶ Y)
Tactic: congr 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Grothendieck (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u₁
inst_1 : Category.{v₁, u₁} D
F : C ⥤ Cat
G : Functor.{v, w, u, w + 1} C (Type w)
E : Type u_1
inst_2 : Category.{v_1, u_1} E
c : C
X Y Z : (↑(F.obj c) : Type u₂)
f : X ⟶ Y
g : Y ⟶ Z
⊢ Eq (α := (F.map (𝟙 c ≫ 𝟙 c)).toFunctor.obj X ⟶ Y) (eqToHom ⋯ ≫ f : (F.map (𝟙 c ≫ 𝟙 c)).toFunctor.obj X ⟶ Y)
    (CategoryStruct.comp (Y := (F.map (𝟙 c)).toFunctor.obj Y) (eqToHom ⋯ ≫ (F.map (𝟙 c)).toFunctor.map f) (eqToHom ⋯) :
      (F.map (𝟙 c ≫ 𝟙 c)).toFunctor.obj X ⟶ Y)
have e_a : f = eqToHom sorry ≫ (F.map (𝟙 c)).toFunctor.map f ≫ eqToHom sorry
Tactic: simp only [eqToHom_comp_iff, Category.assoc, eqToHom_trans_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Grothendieck (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u₁
inst_1 : Category.{v₁, u₁} D
F : C ⥤ Cat
G : Functor.{v, w, u, w + 1} C (Type w)
E : Type u_1
inst_2 : Category.{v_1, u_1} E
c : C
X Y : (↑(F.obj c) : Type u₂)
a₁ a₂ : X ⟶ Y
f : (ι F c).map a₁ = (ι F c).map a₂
⊢ a₁ = a₂
have base_eq : Eq.{v + 1} (α := { base := c, fiber := X : Grothendieck F }.base ⟶ { base := c, fiber := X : Grothendieck F }.base)
  (𝟙 { base := c, fiber := X : Grothendieck F }.base) (𝟙 { base := c, fiber := X : Grothendieck F }.base)
Tactic: injection f with _ f
Cleared hypotheses: f
-- Lean 4 proof step from Mathlib.CategoryTheory.Grothendieck (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u₁
inst_1 : Category.{v₁, u₁} D
F : C ⥤ Cat
G : Functor.{v, w, u, w + 1} C (Type w)
E : Type u_1
inst_2 : Category.{v_1, u_1} E
c : C
X Y : (↑(F.obj c) : Type u₂)
a₁ a₂ : X ⟶ Y
f : (ι F c).map a₁ = (ι F c).map a₂
base_eq :
  Eq.{v + 1} (α := { base := c, fiber := X : Grothendieck F }.base ⟶ { base := c, fiber := X : Grothendieck F }.base)
    (𝟙 { base := c, fiber := X : Grothendieck F }.base) (𝟙 { base := c, fiber := X : Grothendieck F }.base)
⊢ a₁ = a₂
have f₁ : Eq (α :=
  (F.map (𝟙 { base := c, fiber := X : Grothendieck F }.base)).toFunctor.obj
      { base := c, fiber := X : Grothendieck F }.fiber ⟶
    { base := c, fiber := Y }.fiber)
  (eqToHom sorry ≫ a₁) (eqToHom sorry ≫ a₂)
Tactic: injection f with _ f
Cleared hypotheses: f
-- Lean 4 proof step from Mathlib.CategoryTheory.Grothendieck (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u₁
inst_1 : Category.{v₁, u₁} D
F : C ⥤ Cat
G : Functor.{v, w, u, w + 1} C (Type w)
E : Type u_1
inst_2 : Category.{v_1, u_1} E
c : C
X Y : (↑(F.obj c) : Type u₂)
a₁ a₂ : X ⟶ Y
base_eq :
  Eq.{v + 1} (α := { base := c, fiber := X : Grothendieck F }.base ⟶ { base := c, fiber := X : Grothendieck F }.base)
    (𝟙 { base := c, fiber := X : Grothendieck F }.base) (𝟙 { base := c, fiber := X : Grothendieck F }.base)
f :
  Eq (α :=
    (F.map (𝟙 { base := c, fiber := X : Grothendieck F }.base)).toFunctor.obj
        { base := c, fiber := X : Grothendieck F }.fiber ⟶
      { base := c, fiber := Y }.fiber)
    (eqToHom ⋯ ≫ a₁) (eqToHom ⋯ ≫ a₂)
⊢ a₁ = a₂
have rwa : ∀ (f : a₁ = a₂), a₁ = a₂
Tactic: rwa [cancel_epi] at f
Cleared hypotheses: f
-- Lean 4 proof step from Mathlib.CategoryTheory.Grothendieck (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u₁
inst_1 : Category.{v₁, u₁} D
F : C ⥤ Cat
G : Functor.{v, w, u, w + 1} C (Type w)
E : Type u_1
inst_2 : Category.{v_1, u_1} E
X Y : C
f : X ⟶ Y
x x_1 : (↑(F.obj X) : Type u₂)
x_2 : x ⟶ x_1
⊢ (ι F X).map x_2 ≫ { base := f, fiber := 𝟙 ((F.map f).toFunctor.obj ((ι F X).obj x_1).fiber) } =
    { base := f, fiber := 𝟙 ((F.map f).toFunctor.obj ((ι F X).obj x).fiber) :
        ((ι F X).obj x).Hom (((F.map f).toFunctor ⋙ ι F Y).obj x) } ≫
      ((F.map f).toFunctor ⋙ ι F Y).map x_2
have simp : Eq (α := { base := X, fiber := x : Grothendieck F } ⟶ { base := Y, fiber := (F.map f).toFunctor.obj x_1 })
  (CategoryStruct.comp (Y := { base := X, fiber := x_1 }) { base := 𝟙 X, fiber := eqToHom sorry ≫ x_2 }
    { base := f, fiber := 𝟙 ((F.map f).toFunctor.obj x_1) })
  (CategoryStruct.comp (Y := { base := Y, fiber := (F.map f).toFunctor.obj x })
    { base := f, fiber := 𝟙 ((F.map f).toFunctor.obj x) }
    { base := 𝟙 Y, fiber := eqToHom sorry ≫ (F.map f).toFunctor.map x_2 })
Tactic: simp only [ι, Functor.comp_obj, Functor.comp_map]
-- Lean 4 proof step from Mathlib.CategoryTheory.Grothendieck (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u₁
inst_1 : Category.{v₁, u₁} D
F : C ⥤ Cat
G : Functor.{v, w, u, w + 1} C (Type w)
E : Type u_1
inst_2 : Category.{v_1, u_1} E
fib : (c : C) → Functor.{v₂, v_1, u₂, u_1} (↑(F.obj c) : Type u₂) E
hom : {c c' : C} → (f : c ⟶ c') → fib c ⟶ (F.map f).toFunctor ⋙ fib c'
c : C
⊢ fib c = (F.map (𝟙 c)).toFunctor ⋙ fib c
have simp : fib c = Cat.Hom.toFunctor (𝟙 (F.obj c)) ⋙ fib c
Tactic: simp only [Functor.map_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Grothendieck (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : Type u₁
inst_1 : Category.{v₁, u₁} D
F : C ⥤ Cat
G : Functor.{v, w, u, w + 1} C (Type w)
E : Type u_1
inst_2 : Category.{v_1, u_1} E
fib : (c : C) → Functor.{v₂, v_1, u₂, u_1} (↑(F.obj c) : Type u₂) E
hom : {c c' : C} → (f : c ⟶ c') → fib c ⟶ (F.map f).toFunctor ⋙ fib c'
hom_id : ∀ (c : C), hom (𝟙 c) = eqToHom ⋯
c₁ c₂ c₃ : C
f : c₁ ⟶ c₂
g : c₂ ⟶ c₃
⊢ (F.map f).toFunctor ⋙ (F.map g).toFunctor ⋙ fib c₃ = (F.map (f ≫ g)).toFunctor ⋙ fib c₃
have simp : (F.map f).toFunctor ⋙ (F.map g).toFunctor ⋙ fib c₃ = (F.map f ≫ F.map g).toFunctor ⋙ fib c₃
Tactic: simp only [Functor.map_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.Basic (goal)
Goal:
C : Type u_1
inst : Groupoid.{u_2, u_1} C
⊢ Quiver.IsThin.{u_1, u_2} C ↔ ∀ (c : C), Subsingleton.{u_2 + 1} (c ⟶ c)
have refine : ∀ (h : ∀ (c : C), Subsingleton.{u_2 + 1} (c ⟶ c)) (c d : C) (f g : c ⟶ d), f = g
Tactic: refine ⟨fun h c => h c c, fun h c d => Subsingleton.intro fun f g => ?_⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.Basic (hypothesis)
Goal:
C : Type u_1
inst : Groupoid.{u_2, u_1} C
h : ∀ (c : C), Subsingleton.{u_2 + 1} (c ⟶ c)
c d : C
f g : c ⟶ d
⊢ f = g
have this : Subsingleton.{u_2 + 1} (d ⟶ d)
Tactic: haveI := h d
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.Basic (goal)
Goal:
C : Type u_1
inst : Groupoid.{u_2, u_1} C
h : ∀ (c : C), Subsingleton.{u_2 + 1} (c ⟶ c)
c d : C
f g : c ⟶ d
this : Subsingleton.{u_2 + 1} (d ⟶ d)
⊢ f ≫ inv g ≫ g = f ≫ inv f ≫ g
have e_a : inv g ≫ g = inv f ≫ g
Tactic: congr 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.Basic (goal)
Goal:
C : Type u_1
inst : Groupoid.{u_2, u_1} C
h : ∀ (c : C), Subsingleton.{u_2 + 1} (c ⟶ c)
c d : C
f g : c ⟶ d
this : Subsingleton.{u_2 + 1} (d ⟶ d)
⊢ f = g
have f₁ : f = f ≫ inv g ≫ g
Tactic: calc
    f = f ≫ inv g ≫ g := by simp only [inv_eq_inv, IsIso.inv_hom_id, Category.comp_id]
    _ = f ≫ inv f ≫ g := by congr 1
                            simp only [inv_eq_inv, IsIso.inv_hom_id, eq_iff_true_of_subsingleton]
    _ = g := by simp only [inv_eq_inv, IsIso.hom_inv_id_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.Basic (goal)
Goal:
C : Type u_1
inst : Groupoid.{u_2, u_1} C
h : ∀ (c : C), Subsingleton.{u_2 + 1} (c ⟶ c)
c d : C
f g : c ⟶ d
this : Subsingleton.{u_2 + 1} (d ⟶ d)
f₁ : f = f ≫ inv g ≫ g
⊢ f = g
have f₂ : f ≫ inv g ≫ g = f ≫ inv f ≫ g
Tactic: calc
    f = f ≫ inv g ≫ g := by simp only [inv_eq_inv, IsIso.inv_hom_id, Category.comp_id]
    _ = f ≫ inv f ≫ g := by congr 1
                            simp only [inv_eq_inv, IsIso.inv_hom_id, eq_iff_true_of_subsingleton]
    _ = g := by simp only [inv_eq_inv, IsIso.hom_inv_id_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.Basic (goal)
Goal:
C : Type u_1
inst : Groupoid.{u_2, u_1} C
h : ∀ (c : C), Subsingleton.{u_2 + 1} (c ⟶ c)
c d : C
f g : c ⟶ d
this : Subsingleton.{u_2 + 1} (d ⟶ d)
f₁ : f = f ≫ inv g ≫ g
f₂ : f ≫ inv g ≫ g = f ≫ inv f ≫ g
⊢ f = g
have f₃ : f ≫ inv f ≫ g = g
Tactic: calc
    f = f ≫ inv g ≫ g := by simp only [inv_eq_inv, IsIso.inv_hom_id, Category.comp_id]
    _ = f ≫ inv f ≫ g := by congr 1
                            simp only [inv_eq_inv, IsIso.inv_hom_id, eq_iff_true_of_subsingleton]
    _ = g := by simp only [inv_eq_inv, IsIso.hom_inv_id_assoc]
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.FreeGroupoid (goal)
Goal:
V_1 : Type u
inst : Quiver.{v, u} V_1
V : Type u_1
inst_1 : Quiver.{u_2, u_1} V
inst_2 : Nonempty V
⊢ Nonempty.{u_1 + 1} (Quiver.FreeGroupoid.{u_1, u_2} V)
have inhabit : ∀ (inhabited_h : Inhabited V), Nonempty.{u_1 + 1} (Quiver.FreeGroupoid.{u_1, u_2} V)
Tactic: inhabit V
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.FreeGroupoid (goal)
Goal:
V : Type u
inst : Quiver.{v, u} V
X_1 Y_1 X_2 Y_2 X Y : Paths (Symmetrify V)
X_3 Z : Symmetrify V
f : X_3 ⟶ Z
XW : X_1 ⟶ Prefunctor.obj (Paths.of (Symmetrify V)) X_3
WY : Prefunctor.obj (Paths.of (Symmetrify V)) X_3 ⟶ Y_1
⊢ HomRel.CompClosure redStep (X := Y_1) (Y := X_1)
    (Path.reverse WY ≫ 𝟙 (Prefunctor.obj (Paths.of (Symmetrify V)) X_3) ≫ Path.reverse XW)
    (Path.reverse WY ≫ (f.toPath ≫ (reverse f).toPath) ≫ Path.reverse XW)
have h : redStep.{u, v} (𝟙 (Prefunctor.obj (Paths.of (Symmetrify V)) X_3)) (f.toPath ≫ (reverse f).toPath)
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.FreeGroupoid (hypothesis)
Goal:
V : Type u
inst : Quiver.{v, u} V
X_1 Y_1 X_2 Y_2 X Y : Paths (Symmetrify V)
X_3 Z : Symmetrify V
f : X_3 ⟶ Z
XW : X_1 ⟶ Prefunctor.obj (Paths.of (Symmetrify V)) X_3
WY : Prefunctor.obj (Paths.of (Symmetrify V)) X_3 ⟶ Y_1
⊢ HomRel.CompClosure redStep (Path.reverse (XW ≫ 𝟙 (Prefunctor.obj (Paths.of (Symmetrify V)) X_3) ≫ WY))
    (Path.reverse (XW ≫ (f.toPath ≫ (reverse f).toPath) ≫ WY))
have this : HomRel.CompClosure redStep (X := Y_1) (Y := X_1)
  (Path.reverse WY ≫ 𝟙 (Prefunctor.obj (Paths.of (Symmetrify V)) X_3) ≫ Path.reverse XW)
  (Path.reverse WY ≫ (f.toPath ≫ (reverse f).toPath) ≫ Path.reverse XW)
Tactic: have : HomRel.CompClosure redStep (WY.reverse ≫ 𝟙 _ ≫ XW.reverse)
      (WY.reverse ≫ (f.toPath ≫ (Quiver.reverse f).toPath) ≫ XW.reverse) := by
    constructor
    constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.FreeGroupoid (goal)
Goal:
V : Type u
inst : Quiver.{v, u} V
X Y : Paths (Symmetrify V)
p : X ⟶ Y
⊢ Eq (α := Quot (α := X ⟶ X) (HomRel.CompClosure redStep)) (Quot.mk (HomRel.CompClosure redStep) (p ≫ Path.reverse p))
    (Quot.mk (HomRel.CompClosure redStep) (𝟙 X))
have H : EqvGen (HomRel.CompClosure redStep) (p ≫ Path.reverse p) (𝟙 X)
Tactic: apply Quot.eqvGen_sound
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.FreeGroupoid (goal)
Goal:
V : Type u
inst : Quiver.{v, u} V
X Y b c : Paths (Symmetrify V)
q : Path.{v, u} X b
f : b ⟶ c
ih : EqvGen (α := X ⟶ X) (HomRel.CompClosure redStep) (q ≫ q.reverse) (𝟙 X)
⊢ EqvGen (α := X ⟶ X) (HomRel.CompClosure redStep) (q.cons f ≫ (reverse f).toPath.comp q.reverse) (𝟙 X)
have H : X ⟶ X
Tactic: fapply EqvGen.trans
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.FreeGroupoid (goal)
Goal:
V : Type u
inst : Quiver.{v, u} V
X Y b c : Paths (Symmetrify V)
q : Path.{v, u} X b
f : b ⟶ c
ih : EqvGen (α := X ⟶ X) (HomRel.CompClosure redStep) (q ≫ q.reverse) (𝟙 X)
H : X ⟶ X
⊢ EqvGen (α := X ⟶ X) (HomRel.CompClosure redStep) (q.cons f ≫ (reverse f).toPath.comp q.reverse) (𝟙 X)
have H₁ : EqvGen (α := X ⟶ X) (HomRel.CompClosure redStep) (q.cons f ≫ (reverse f).toPath.comp q.reverse) (q ≫ q.reverse)
Tactic: fapply EqvGen.trans
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.FreeGroupoid (goal)
Goal:
V : Type u
inst : Quiver.{v, u} V
X Y b c : Paths (Symmetrify V)
q : Path.{v, u} X b
f : b ⟶ c
ih : EqvGen (α := X ⟶ X) (HomRel.CompClosure redStep) (q ≫ q.reverse) (𝟙 X)
⊢ EqvGen (α := X ⟶ X) (HomRel.CompClosure redStep) (q.cons f ≫ (reverse f).toPath.comp q.reverse) (q ≫ q.reverse)
have H : EqvGen (α := X ⟶ X) (HomRel.CompClosure redStep) (q ≫ q.reverse) (q.cons f ≫ (reverse f).toPath.comp q.reverse)
Tactic: apply EqvGen.symm
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.FreeGroupoid (goal)
Goal:
V : Type u
inst : Quiver.{v, u} V
X Y b c : Paths (Symmetrify V)
q : Path.{v, u} X b
f : b ⟶ c
ih : EqvGen (α := X ⟶ X) (HomRel.CompClosure redStep) (q ≫ q.reverse) (𝟙 X)
⊢ EqvGen (α := X ⟶ X) (HomRel.CompClosure redStep) (q ≫ q.reverse) (q.cons f ≫ (reverse f).toPath.comp q.reverse)
have H : HomRel.CompClosure (C := Paths (Symmetrify V)) redStep (X := X) (Y := X) (q ≫ q.reverse)
  (q.cons f ≫ (reverse f).toPath.comp q.reverse)
Tactic: apply EqvGen.rel
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.FreeGroupoid (goal)
Goal:
V : Type u
inst : Quiver.{v, u} V
X Y b c : Paths (Symmetrify V)
q : Path.{v, u} X b
f : b ⟶ c
ih : EqvGen (α := X ⟶ X) (HomRel.CompClosure redStep) (q ≫ q.reverse) (𝟙 X)
⊢ HomRel.CompClosure redStep (X := X) (Y := X) (q ≫ 𝟙 b ≫ q.reverse) (q ≫ (f.toPath ≫ (reverse f).toPath) ≫ q.reverse)
have h : redStep.{u, v} (𝟙 b) (f.toPath ≫ (reverse f).toPath)
Tactic: apply HomRel.CompClosure.intro
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.FreeGroupoid (hypothesis)
Goal:
V : Type u
inst : Quiver.{v, u} V
X Y b c : Paths (Symmetrify V)
q : Path.{v, u} X b
f : b ⟶ c
ih : EqvGen (α := X ⟶ X) (HomRel.CompClosure redStep) (q ≫ q.reverse) (𝟙 X)
⊢ HomRel.CompClosure (C := Paths (Symmetrify V)) redStep (X := X) (Y := X) (q ≫ q.reverse)
    (q.cons f ≫ (reverse f).toPath.comp q.reverse)
have this : HomRel.CompClosure redStep (X := X) (Y := X) (q ≫ 𝟙 b ≫ q.reverse) (q ≫ (f.toPath ≫ (reverse f).toPath) ≫ q.reverse)
Tactic: have : HomRel.CompClosure redStep (q ≫ 𝟙 _ ≫ Quiver.Path.reverse q)
          (q ≫ (Quiver.Hom.toPath f ≫ Quiver.Hom.toPath (Quiver.reverse f)) ≫
            Quiver.Path.reverse q) := by
        apply HomRel.CompClosure.intro
        apply redStep.step
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.FreeGroupoid (hypothesis)
Goal:
V : Type u
inst : Quiver.{v, u} V
X Y b c : Paths (Symmetrify V)
q : Path.{v, u} X b
f : b ⟶ c
ih : EqvGen (α := X ⟶ X) (HomRel.CompClosure redStep) (q ≫ q.reverse) (𝟙 X)
this :
  HomRel.CompClosure redStep (X := X) (Y := X) (q ≫ 𝟙 b ≫ q.reverse) (q ≫ (f.toPath ≫ (reverse f).toPath) ≫ q.reverse)
⊢ HomRel.CompClosure (C := Paths (Symmetrify V)) redStep (X := X) (Y := X) (q ≫ q.reverse)
    (q.cons f ≫ (reverse f).toPath.comp q.reverse)
have this₁ : HomRel.CompClosure (C := Paths (Symmetrify V)) redStep (X := X) (Y := X) (q ≫ q.reverse)
  (q ≫ f.toPath ≫ (reverse f).toPath ≫ q.reverse)
Tactic: simp only [Category.assoc, Category.id_comp] at this ⊢
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.FreeGroupoid (goal)
Goal:
V : Type u
inst : Quiver.{v, u} V
X Y b c : Paths (Symmetrify V)
q : Path.{v, u} X b
f : b ⟶ c
ih : EqvGen (α := X ⟶ X) (HomRel.CompClosure redStep) (q ≫ q.reverse) (𝟙 X)
this :
  HomRel.CompClosure (C := Paths (Symmetrify V)) redStep (X := X) (Y := X) (q ≫ q.reverse)
    (q ≫ f.toPath ≫ (reverse f).toPath ≫ q.reverse)
⊢ HomRel.CompClosure (C := Paths (Symmetrify V)) redStep (X := X) (Y := X) (q ≫ q.reverse)
    (q.cons f ≫ (reverse f).toPath ≫ q.reverse)
have H : ∀
  (this :
    HomRel.CompClosure (C := Paths (Symmetrify V)) redStep (X := X) (Y := X) (q ≫ q.reverse)
      (((q ≫ f.toPath) ≫ (reverse f).toPath) ≫ q.reverse)),
  HomRel.CompClosure (C := Paths (Symmetrify V)) redStep (X := X) (Y := X) (q ≫ q.reverse)
    ((q.cons f ≫ (reverse f).toPath) ≫ q.reverse)
Tactic: simp only [← Category.assoc] at this ⊢
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.FreeGroupoid (goal)
Goal:
V : Type u
inst : Quiver.{v, u} V
X Y : Paths (Symmetrify V)
p : X ⟶ Y
⊢ EqvGen (HomRel.CompClosure redStep) (p ≫ Path.reverse p) (𝟙 X)
have H : EqvGen (α := X ⟶ X) (HomRel.CompClosure redStep) (Path.nil ≫ Path.reverse Path.nil) (𝟙 X)
Tactic: induction p with
  | nil => apply EqvGen.refl
  | cons q f ih =>
    simp only [Quiver.Path.reverse]
    fapply EqvGen.trans
    -- Porting note: dot notation for `Quiver.Path.*` and `Quiver.Hom.*` not working
    · exact q ≫ Quiver.Path.reverse q
    · apply EqvGen.symm
      apply EqvGen.rel
      have : HomRel.CompClosure redStep (q ≫ 𝟙 _ ≫ Quiver.Path.reverse q)
          (q ≫ (Quiver.Hom.toPath f ≫ Quiver.Hom.toPath (Quiver.reverse f)) ≫
            Quiver.Path.reverse q) := by
        apply HomRel.CompClosure.intro
        apply redStep.step
      simp only [Category.assoc, Category.id_comp] at this ⊢
      -- Porting note: `simp` cannot see how `Quiver.Path.comp_assoc` is relevant, so change to
      -- category notation
      change HomRel.CompClosure redStep (q ≫ Quiver.Path.reverse q)
        (Quiver.Path.cons q f ≫ (Quiver.Hom.toPath (Quiver.reverse f)) ≫ (Quiver.Path.reverse q))
      simp only [← Category.assoc] at this ⊢
      exact this
    · exact ih
Cleared hypotheses: p
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.FreeGroupoid (goal)
Goal:
V : Type u
inst : Quiver.{v, u} V
X Y : Paths (Symmetrify V)
p : X ⟶ Y
H : EqvGen (α := X ⟶ X) (HomRel.CompClosure redStep) (Path.nil ≫ Path.reverse Path.nil) (𝟙 X)
⊢ EqvGen (HomRel.CompClosure redStep) (p ≫ Path.reverse p) (𝟙 X)
have H₁ : ∀ {b c : Paths (Symmetrify V)} (q : Path.{v, u} X b) (f : b ⟶ c)
  (ih : EqvGen (α := X ⟶ X) (HomRel.CompClosure redStep) (q ≫ q.reverse) (𝟙 X)),
  EqvGen (α := X ⟶ X) (HomRel.CompClosure redStep) (q.cons f ≫ (q.cons f).reverse) (𝟙 X)
Tactic: induction p with
  | nil => apply EqvGen.refl
  | cons q f ih =>
    simp only [Quiver.Path.reverse]
    fapply EqvGen.trans
    -- Porting note: dot notation for `Quiver.Path.*` and `Quiver.Hom.*` not working
    · exact q ≫ Quiver.Path.reverse q
    · apply EqvGen.symm
      apply EqvGen.rel
      have : HomRel.CompClosure redStep (q ≫ 𝟙 _ ≫ Quiver.Path.reverse q)
          (q ≫ (Quiver.Hom.toPath f ≫ Quiver.Hom.toPath (Quiver.reverse f)) ≫
            Quiver.Path.reverse q) := by
        apply HomRel.CompClosure.intro
        apply redStep.step
      simp only [Category.assoc, Category.id_comp] at this ⊢
      -- Porting note: `simp` cannot see how `Quiver.Path.comp_assoc` is relevant, so change to
      -- category notation
      change HomRel.CompClosure redStep (q ≫ Quiver.Path.reverse q)
        (Quiver.Path.cons q f ≫ (Quiver.Hom.toPath (Quiver.reverse f)) ≫ (Quiver.Path.reverse q))
      simp only [← Category.assoc] at this ⊢
      exact this
    · exact ih
Cleared hypotheses: p
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.FreeGroupoid (goal)
Goal:
V : Type u
inst : Quiver.{v, u} V
X Y : Paths (Symmetrify V)
p : X ⟶ Y
⊢ Eq (α := Quot (α := Y ⟶ Y) (HomRel.CompClosure redStep)) (Quot.mk (HomRel.CompClosure redStep) (Path.reverse p ≫ p))
    (Quot.mk (HomRel.CompClosure redStep) (𝟙 Y))
have nth_rw : Eq (α := Quot (α := Y ⟶ Y) (HomRel.CompClosure redStep))
  (Quot.mk (HomRel.CompClosure redStep) (Path.reverse p ≫ (Path.reverse p).reverse))
  (Quot.mk (HomRel.CompClosure redStep) (𝟙 Y))
Tactic: nth_rw 2 [← Quiver.Path.reverse_reverse p]
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.FreeGroupoid (goal)
Goal:
V : Type u
inst : Quiver.{v, u} V
V' : Type u'
inst_1 : Groupoid.{u_1, u'} V'
φ : Prefunctor.{v, u_1, u, u'} V V'
X_1 Y_1 : Paths (Symmetrify V)
X Y : Symmetrify V
f : X ⟶ Y
⊢ (Paths.lift (Symmetrify.lift φ)).map (𝟙 (Prefunctor.obj (Paths.of (Symmetrify V)) X)) =
    (Paths.lift (Symmetrify.lift φ)).map (f.toPath ≫ (reverse f).toPath)
have erw : 𝟙 ((Symmetrify.lift φ).obj (Prefunctor.obj (Paths.of (Symmetrify V)) X)) =
  (Symmetrify.lift φ).map f ≫ reverse ((Symmetrify.lift φ).map f)
Tactic: erw [Paths.lift_nil, Paths.lift_cons, Quiver.Path.comp_nil, Paths.lift_toPath,
      Quiver.Symmetrify.lift_reverse]
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.FreeGroupoid (goal)
Goal:
V : Type u
inst : Quiver.{v, u} V
V' : Type u'
inst_1 : Groupoid.{u_1, u'} V'
φ : Prefunctor.{v, u_1, u, u'} V V'
X_1 Y_1 : Paths (Symmetrify V)
X Y : Symmetrify V
f : X ⟶ Y
⊢ 𝟙 ((Symmetrify.lift φ).obj (Prefunctor.obj (Paths.of (Symmetrify V)) X)) =
    (Symmetrify.lift φ).map f ≫ reverse ((Symmetrify.lift φ).map f)
have symm : (Symmetrify.lift φ).map f ≫ reverse ((Symmetrify.lift φ).map f) =
  𝟙 ((Symmetrify.lift φ).obj (Prefunctor.obj (Paths.of (Symmetrify V)) X))
Tactic: symm
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.FreeGroupoid (goal)
Goal:
V : Type u
inst : Quiver.{v, u} V
V' : Type u'
inst_1 : Groupoid.{u_1, u'} V'
φ : Prefunctor.{v, u_1, u, u'} V V'
⊢ of V ⋙q (lift φ).toPrefunctor = φ
have rw : Symmetrify.of ⋙q (Paths.of (Symmetrify V) ⋙q (Quotient.functor redStep ⋙ lift φ).toPrefunctor) = φ
Tactic: rw [of_eq, Prefunctor.comp_assoc, Prefunctor.comp_assoc, Functor.toPrefunctor_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.FreeGroupoid (goal)
Goal:
V : Type u
inst : Quiver.{v, u} V
V' : Type u'
inst_1 : Groupoid.{u_1, u'} V'
φ : Prefunctor.{v, u_1, u, u'} V V'
⊢ Symmetrify.of ⋙q
      (Paths.of (Symmetrify V) ⋙q
        (Quotient.functor redStep ⋙
            CategoryTheory.Quotient.lift redStep (Paths.lift (Symmetrify.lift φ)) ⋯).toPrefunctor) =
    φ
have rw : Symmetrify.of ⋙q (Paths.of (Symmetrify V) ⋙q (Paths.lift (Symmetrify.lift φ)).toPrefunctor) = φ
Tactic: rw [Quotient.lift_spec, Paths.lift_spec, Quiver.Symmetrify.lift_spec]
-- Lean 4 proof step from Mathlib.CategoryTheory.Groupoid.FreeGroupoid (goal)
Goal:
V : Type u
inst : Quiver.{v, u} V
V' : Type u'
inst_1 : Groupoid.{u_1, u'} V'
φ : Prefunctor.{v, u_1, u, u'} V V'
rw : Symmetrify.of ⋙q (Paths.of (Symmetrify V) ⋙q (Paths.lift (Symmetrify.lift φ)).toPrefunctor) = φ
⊢ Symmetrify.of ⋙q
      (Paths.of (Symmetrify V) ⋙q
        (Quotient.functor redStep ⋙
            CategoryTheory.Quotient.lift redStep (Paths.lift (Symmetrify.lift φ)) ⋯).toPrefunctor) =
    φ
have rw₁ : Symmetrify.of ⋙q Symmetrify.lift φ = φ
Tactic: rw [Quotient.lift_spec, Paths.lift_spec, Quiver.Symmetrify.lift_spec]