awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Pointed (goal)
Goal:
X : Type u_1
Y : Pointed
f : typeToPointed.obj X ⟶ Y
x : (typeToPointed.obj X).X
⊢ Pointed.Hom.toFun
      ((fun
            (f :
              X ⟶
                Functor.obj
                  (forget Pointed (FC := fun (X Y : Pointed.{u_1}) ↦ { f : X.X → Y.X // f X.point = Y.point }) (CC :=
                    Pointed.X))
                  Y) ↦
          ({
              toFun := fun (o : (typeToPointed.obj X).X) ↦
                Option.elim o Y.point
                  (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) f) :
                    (a : (fun (X : Type u_1) ↦ X) X) → (fun (X : Type u_1) ↦ X) ((forget Pointed).obj Y)),
              map_point := ⋯ } :
            (typeToPointed.obj X).Hom Y))
        ((fun (f : typeToPointed.obj X ⟶ Y) ↦ ↾(f.toFun ∘ some)) f))
      x =
    f.toFun x
have toFun : Pointed.Hom.toFun (X := typeToPointed.obj X)
    ((fun
          (f :
            X ⟶
              Functor.obj
                (forget Pointed (FC := fun (X Y : Pointed.{u_1}) ↦ { f : X.X → Y.X // f X.point = Y.point }) (CC :=
                  Pointed.X))
                Y) ↦
        ({
            toFun := fun (o : (typeToPointed.obj X).X) ↦
              Option.elim o Y.point
                (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) f) :
                  (a : (fun (X : Type u_1) ↦ X) X) → (fun (X : Type u_1) ↦ X) ((forget Pointed).obj Y)),
            map_point := sorry } :
          (typeToPointed.obj X).Hom Y))
      ((fun (f : typeToPointed.obj X ⟶ Y) ↦ ↾(f.toFun ∘ some)) f))
    none =
  f.toFun none
Tactic: cases x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Pointed (goal)
Goal:
X : Type u_1
Y : Pointed
f : typeToPointed.obj X ⟶ Y
x : (typeToPointed.obj X).X
toFun :
  Pointed.Hom.toFun (X := typeToPointed.obj X)
      ((fun
            (f :
              X ⟶
                Functor.obj
                  (forget Pointed (FC := fun (X Y : Pointed.{u_1}) ↦ { f : X.X → Y.X // f X.point = Y.point }) (CC :=
                    Pointed.X))
                  Y) ↦
          ({
              toFun := fun (o : (typeToPointed.obj X).X) ↦
                Option.elim o Y.point
                  (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) f) :
                    (a : (fun (X : Type u_1) ↦ X) X) → (fun (X : Type u_1) ↦ X) ((forget Pointed).obj Y)),
              map_point := ⋯ } :
            (typeToPointed.obj X).Hom Y))
        ((fun (f : typeToPointed.obj X ⟶ Y) ↦ ↾(f.toFun ∘ some)) f))
      none =
    f.toFun none
⊢ Pointed.Hom.toFun
      ((fun
            (f :
              X ⟶
                Functor.obj
                  (forget Pointed (FC := fun (X Y : Pointed.{u_1}) ↦ { f : X.X → Y.X // f X.point = Y.point }) (CC :=
                    Pointed.X))
                  Y) ↦
          ({
              toFun := fun (o : (typeToPointed.obj X).X) ↦
                Option.elim o Y.point
                  (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) f) :
                    (a : (fun (X : Type u_1) ↦ X) X) → (fun (X : Type u_1) ↦ X) ((forget Pointed).obj Y)),
              map_point := ⋯ } :
            (typeToPointed.obj X).Hom Y))
        ((fun (f : typeToPointed.obj X ⟶ Y) ↦ ↾(f.toFun ∘ some)) f))
      x =
    f.toFun x
have toFun₁ : ∀ (val : X),
  Pointed.Hom.toFun (X := typeToPointed.obj X)
      ((fun
            (f :
              X ⟶
                Functor.obj
                  (forget Pointed (FC := fun (X Y : Pointed.{u_1}) ↦ { f : X.X → Y.X // f X.point = Y.point }) (CC :=
                    Pointed.X))
                  Y) ↦
          ({
              toFun := fun (o : (typeToPointed.obj X).X) ↦
                Option.elim o Y.point
                  (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) f) :
                    (a : (fun (X : Type u_1) ↦ X) X) → (fun (X : Type u_1) ↦ X) ((forget Pointed).obj Y)),
              map_point := sorry } :
            (typeToPointed.obj X).Hom Y))
        ((fun (f : typeToPointed.obj X ⟶ Y) ↦ ↾(f.toFun ∘ some)) f))
      (some val) =
    f.toFun (some val)
Tactic: cases x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Preorder (goal)
Goal:
X : Type u
inst : Preorder X
x y : X
h : x = y
⊢ x ≤ y
have rw : y ≤ y
Tactic: rw [h]
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Preorder (goal)
Goal:
X : Type u
inst : Preorder X
x y : X
h : x = y
⊢ IsIso (homOfLE ⋯)
have subst : IsIso (homOfLE sorry)
Tactic: subst h
Cleared hypotheses: y, h
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Preorder (goal)
Goal:
X : Type u
inst : PartialOrder X
a b : X
f : a ⟶ b
⊢ IsIso f ↔ a = b
have mp : IsIso f → a = b
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Preorder (goal)
Goal:
X : Type u
inst : PartialOrder X
a b : X
f : a ⟶ b
mp : IsIso f → a = b
⊢ IsIso f ↔ a = b
have mpr : a = b → IsIso f
Tactic: constructor
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Preorder (goal)
Goal:
X : Type u
inst : PartialOrder X
a : X
f : a ⟶ a
⊢ IsIso f
have mpr : IsIso (𝟙 a)
Tactic: rw [Subsingleton.elim f (𝟙 _)]
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
V₁ : Type u₁
V₂ : Type u₂
V₃ : Type u₃
inst : Quiver.{v₁, u₁} V₁
inst_1 : Quiver.{v₂, u₂} V₂
inst_2 : Quiver.{v₃, u₃} V₃
F : Prefunctor.{v₁, v₂, u₁, u₂} V₁ V₂
G : Prefunctor.{v₂, v₃, u₂, u₃} V₂ V₃
X Y : Paths V₁
f : X ⟶ Y
⊢ (freeMap (F ⋙q G)).map f ≫
      Iso.hom
        ((fun (x : Paths V₁) ↦
            (Iso.refl ((freeMap (F ⋙q G)).obj x) : (freeMap (F ⋙q G)).obj x ≅ (freeMap (F ⋙q G)).obj x))
          Y) =
    Iso.hom
        ((fun (x : Paths V₁) ↦
            (Iso.refl ((freeMap (F ⋙q G)).obj x) : (freeMap (F ⋙q G)).obj x ≅ (freeMap (F ⋙q G)).obj x))
          X) ≫
      (freeMap F ⋙ freeMap G).map f
have dsimp : Eq (α := G.obj (F.obj X) ⟶ G.obj (F.obj Y)) ((F ⋙q G).mapPath f ≫ 𝟙 (G.obj (F.obj Y)))
  (𝟙 (G.obj (F.obj X)) ≫ G.mapPath (F.mapPath f))
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
V W : Quiv
e : V ≅ W
X Y : (↑W : Type u_2)
f : X ⟶ Y
⊢ e.hom.map (e.inv.map f) = Quiver.homOfEq f ⋯ ⋯
have rw : (e.inv ⋙q e.hom).map f = Quiver.homOfEq f sorry sorry
Tactic: rw [← Prefunctor.comp_map]
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
V W : Type u
inst : Quiver.{u_1, u} V
inst_1 : Quiver.{u_2, u} W
e : V ≃ W
he :
  (X Y : V) → Equiv.{u_1 + 1, u_2 + 1} (X ⟶ Y) (Quiver.Hom (V := W) ((e : (a : V) → W) X : W) ((e : (a : V) → W) Y : W))
X Y : V
f : X ⟶ Y
X' Y' : V
hX : X = X'
hY : Y = Y'
X'' Y'' : W
hX' : (e : (a : V) → W) X' = X''
hY' : (e : (a : V) → W) Y' = Y''
rw : (e : (a : V) → W) X' = X''
⊢ (e : (a : V) → W) X = X''
have rw₁ : X'' = X''
Tactic: rw [hX, hX']
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
V W : Type u
inst : Quiver.{u_1, u} V
inst_1 : Quiver.{u_2, u} W
e : V ≃ W
he :
  (X Y : V) → Equiv.{u_1 + 1, u_2 + 1} (X ⟶ Y) (Quiver.Hom (V := W) ((e : (a : V) → W) X : W) ((e : (a : V) → W) Y : W))
X Y : V
f : X ⟶ Y
X' Y' : V
hX : X = X'
hY : Y = Y'
X'' Y'' : W
hX' : (e : (a : V) → W) X' = X''
hY' rw : (e : (a : V) → W) Y' = Y''
⊢ (e : (a : V) → W) Y = Y''
have rw₁ : Y'' = Y''
Tactic: rw [hY, hY']
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
V W : Type u
inst : Quiver.{u_1, u} V
inst_1 : Quiver.{u_2, u} W
e : V ≃ W
he :
  (X Y : V) → Equiv.{u_1 + 1, u_2 + 1} (X ⟶ Y) (Quiver.Hom (V := W) ((e : (a : V) → W) X : W) ((e : (a : V) → W) Y : W))
X Y : V
f : X ⟶ Y
X' Y' : V
hX : X = X'
hY : Y = Y'
X'' Y'' : W
hX' : (e : (a : V) → W) X' = X''
hY' : (e : (a : V) → W) Y' = Y''
⊢ Eq.{u_2 + 1} (α := X'' ⟶ Y'')
    (Quiver.homOfEq ((he X' Y' : (a : X' ⟶ Y') → (e : (a : V) → W) X' ⟶ (e : (a : V) → W) Y') (Quiver.homOfEq f hX hY))
        hX' hY' :
      X'' ⟶ Y'')
    (Quiver.homOfEq ((he X Y : (a : X ⟶ Y) → (e : (a : V) → W) X ⟶ (e : (a : V) → W) Y) f) ⋯ ⋯ : X'' ⟶ Y'')
have subst : Eq.{u_2 + 1} (α := Quiver.Hom (V := W) ((e : (a : V) → W) X : W) ((e : (a : V) → W) Y : W))
  (Quiver.homOfEq (X := (e : (a : V) → W) X) (Y := (e : (a : V) → W) Y)
      ((he X Y : (a : X ⟶ Y) → (e : (a : V) → W) X ⟶ (e : (a : V) → W) Y) (Quiver.homOfEq f sorry sorry)) sorry sorry :
    Quiver.Hom (V := W) ((e : (a : V) → W) X : W) ((e : (a : V) → W) Y : W))
  (Quiver.homOfEq ((he X Y : (a : X ⟶ Y) → (e : (a : V) → W) X ⟶ (e : (a : V) → W) Y) f) sorry sorry :
    Quiver.Hom (V := W) ((e : (a : V) → W) X : W) ((e : (a : V) → W) Y : W))
Tactic: subst hX hY hX' hY'
Cleared hypotheses: X', Y', hX, hY, X'', Y'', hX', hY'
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
V W : Type u
inst : Quiver.{u_1, u} V
inst_1 : Quiver.{u_1, u} W
e : V ≃ W
he : (X Y : V) → (X ⟶ Y) ≃ Quiver.Hom (V := W) ((e : (a : V) → W) X : W) ((e : (a : V) → W) Y : W)
X Y : (↑(of V) : Type u)
f : X ⟶ Y
⊢ Prefunctor.map
      (CategoryStruct.comp (Y := of W)
        { obj := (⇑e : (a : V) → W),
          map := fun {X Y : (↑(of V) : Type u)} ↦
            (⇑(he X Y) : (a : X ⟶ Y) → (e : (a : V) → W) X ⟶ (e : (a : V) → W) Y) }
        { obj := (⇑e.symm : (a : W) → V),
          map := fun {X Y : (↑(of W) : Type u)} (f : X ⟶ Y) ↦
            (Equiv.symm (he ((e.symm : (a : W) → V) X) ((e.symm : (a : W) → V) Y)) :
                (a : (e : (a : V) → W) ((e.symm : (a : W) → V) X) ⟶ (e : (a : V) → W) ((e.symm : (a : W) → V) Y)) →
                  (e.symm : (a : W) → V) X ⟶ (e.symm : (a : W) → V) Y)
              (Quiver.homOfEq f ⋯ ⋯) })
      f =
    Quiver.homOfEq (Prefunctor.map (𝟙 (of V)) f) ⋯ ⋯
have dsimp : (Equiv.symm (he ((e.symm : (a : W) → V) ((e : (a : V) → W) X : W)) ((e.symm : (a : W) → V) ((e : (a : V) → W) Y : W))) :
      (a :
          (e : (a : V) → W) ((e.symm : (a : W) → V) ((e : (a : V) → W) X)) ⟶
            (e : (a : V) → W) ((e.symm : (a : W) → V) ((e : (a : V) → W) Y))) →
        (e.symm : (a : W) → V) ((e : (a : V) → W) X) ⟶ (e.symm : (a : W) → V) ((e : (a : V) → W) Y))
    (Quiver.homOfEq ((he X Y : (a : X ⟶ Y) → (e : (a : V) → W) X ⟶ (e : (a : V) → W) Y) f) sorry sorry :
      Quiver.Hom (V := W) ((e : (a : V) → W) ((e.symm : (a : W) → V) ((e : (a : V) → W) X : W) : V) : W)
        ((e : (a : V) → W) ((e.symm : (a : W) → V) ((e : (a : V) → W) Y : W) : V) : W)) =
  Quiver.homOfEq f sorry sorry
Tactic: dsimp [Quiv.id_eq_id, Quiv.comp_eq_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
C : Type u
inst : Category.{v, u} C
⊢ Eq (α := Prefunctor.{v, v, u, u} C C)
    { obj := fun (X : C) ↦ Prefunctor.obj (Functor.toPrefunctor (pathComposition C)) (Prefunctor.obj (Paths.of C) X),
      map := fun {X Y : C} (f : X ⟶ Y) ↦
        Prefunctor.map (Functor.toPrefunctor (pathComposition C)) (Prefunctor.map (Paths.of C) f) }
    (𝟭q C)
have e_map : Eq (α :=
  {X Y : C} →
    (X ⟶ Y) →
      (Prefunctor.obj (Functor.toPrefunctor (pathComposition C)) (Prefunctor.obj (Paths.of C) X) ⟶
        Prefunctor.obj (Functor.toPrefunctor (pathComposition C)) (Prefunctor.obj (Paths.of C) Y)))
  (fun {X Y : C} (f : X ⟶ Y) ↦
    Prefunctor.map (Functor.toPrefunctor (pathComposition C)) (Prefunctor.map (Paths.of C) f))
  fun {X Y : C} (f : X ⟶ Y) ↦ f
Tactic: congr
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
C : Type u
inst : Category.{v, u} C
⊢ Eq (α :=
    {X Y : C} →
      (X ⟶ Y) →
        (Prefunctor.obj (Functor.toPrefunctor (pathComposition C)) (Prefunctor.obj (Paths.of C) X) ⟶
          Prefunctor.obj (Functor.toPrefunctor (pathComposition C)) (Prefunctor.obj (Paths.of C) Y)))
    (fun {X Y : C} (f : X ⟶ Y) ↦
      Prefunctor.map (Functor.toPrefunctor (pathComposition C)) (Prefunctor.map (Paths.of C) f))
    fun {X Y : C} (f : X ⟶ Y) ↦ f
have e_map : ∀ (X Y : C) (f : X ⟶ Y), Prefunctor.map (Functor.toPrefunctor (pathComposition C)) (Prefunctor.map (Paths.of C) f) = f
Tactic: funext X Y f
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
⊢ Functor.whiskerRight
        { app := fun (x : Quiv) ↦ Paths.of (↑(Functor.obj (𝟭 Quiv) x) : Type u_2), naturality := adj._proof_1 }
        Cat.free ≫
      (Cat.free.associator forget Cat.free).hom ≫
        Cat.free.whiskerLeft
          { app := fun (C : Cat) ↦ Functor.toCatHom (pathComposition (↑C : Type u_2)), naturality := ⋯ :
            NatTrans (forget ⋙ Cat.free) (𝟭 Cat) } =
    NatTrans.id (𝟭 Quiv ⋙ Cat.free)
have w : ∀ (V : Quiv),
  Cat.Hom.toFunctor
      (NatTrans.app
        (Functor.whiskerRight
            { app := fun (x : Quiv) ↦ Paths.of (↑(Functor.obj (𝟭 Quiv) x) : Type u_2), naturality := adj._proof_1 }
            Cat.free ≫
          (Cat.free.associator forget Cat.free).hom ≫
            Cat.free.whiskerLeft
              { app := fun (C : Cat) ↦ Functor.toCatHom (pathComposition (↑C : Type u_2)), naturality := sorry :
                NatTrans (forget ⋙ Cat.free) (𝟭 Cat) })
        V) =
    ((NatTrans.id (𝟭 Quiv ⋙ Cat.free)).app V).toFunctor
Tactic: ext V
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
⊢ forget.whiskerLeft
        { app := fun (x : Quiv) ↦ Paths.of (↑(Functor.obj (𝟭 Quiv) x) : Type u_2), naturality := adj._proof_1 } ≫
      (forget.associator Cat.free forget).inv ≫
        Functor.whiskerRight
          { app := fun (C : Cat) ↦ Functor.toCatHom (pathComposition (↑C : Type u_2)), naturality := ⋯ :
            NatTrans (forget ⋙ Cat.free) (𝟭 Cat) }
          forget =
    NatTrans.id (forget ⋙ 𝟭 Quiv)
have w : ∀ (C : Cat),
  NatTrans.app
      (forget.whiskerLeft
          { app := fun (x : Quiv) ↦ Paths.of (↑(Functor.obj (𝟭 Quiv) x) : Type u_2), naturality := adj._proof_1 } ≫
        (forget.associator Cat.free forget).inv ≫
          Functor.whiskerRight
            { app := fun (C : Cat) ↦ Functor.toCatHom (pathComposition (↑C : Type u_2)), naturality := sorry :
              NatTrans (forget ⋙ Cat.free) (𝟭 Cat) }
            forget)
      C =
    (NatTrans.id (forget ⋙ 𝟭 Quiv)).app C
Tactic: ext C
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
V : Type u
C : Type u₁
inst : Quiver.{v, u} V
inst_1 : Category.{v₁, u₁} C
F : Paths V ⥤ C
⊢ Cat.freeMap (Paths.of V ⋙q F.toPrefunctor) ⋙ pathComposition C = F
have rw : (Cat.freeMap (Paths.of V) ⋙ Cat.freeMap F.toPrefunctor) ⋙ pathComposition C = F
Tactic: rw [Cat.freeMap_comp, Functor.assoc, pathComposition_naturality, ← Functor.assoc,
      freeMap_pathsOf_pathComposition, Functor.id_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
V : Type u
C : Type u₁
inst : Quiver.{v, u} V
inst_1 : Category.{v₁, u₁} C
F : Paths V ⥤ C
rw : (Cat.freeMap (Paths.of V) ⋙ Cat.freeMap F.toPrefunctor) ⋙ pathComposition C = F
⊢ Cat.freeMap (Paths.of V ⋙q F.toPrefunctor) ⋙ pathComposition C = F
have rw₁ : Cat.freeMap (Paths.of V) ⋙ Cat.freeMap F.toPrefunctor ⋙ pathComposition C = F
Tactic: rw [Cat.freeMap_comp, Functor.assoc, pathComposition_naturality, ← Functor.assoc,
      freeMap_pathsOf_pathComposition, Functor.id_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
V : Type u
C : Type u₁
inst : Quiver.{v, u} V
inst_1 : Category.{v₁, u₁} C
F : Paths V ⥤ C
rw : (Cat.freeMap (Paths.of V) ⋙ Cat.freeMap F.toPrefunctor) ⋙ pathComposition C = F
rw₁ : Cat.freeMap (Paths.of V) ⋙ Cat.freeMap F.toPrefunctor ⋙ pathComposition C = F
⊢ Cat.freeMap (Paths.of V ⋙q F.toPrefunctor) ⋙ pathComposition C = F
have rw₂ : Cat.freeMap (Paths.of V) ⋙ pathComposition (Paths V) ⋙ F = F
Tactic: rw [Cat.freeMap_comp, Functor.assoc, pathComposition_naturality, ← Functor.assoc,
      freeMap_pathsOf_pathComposition, Functor.id_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
V : Type u
C : Type u₁
inst : Quiver.{v, u} V
inst_1 : Category.{v₁, u₁} C
F : Paths V ⥤ C
rw : (Cat.freeMap (Paths.of V) ⋙ Cat.freeMap F.toPrefunctor) ⋙ pathComposition C = F
rw₁ : Cat.freeMap (Paths.of V) ⋙ Cat.freeMap F.toPrefunctor ⋙ pathComposition C = F
rw₂ : Cat.freeMap (Paths.of V) ⋙ pathComposition (Paths V) ⋙ F = F
⊢ Cat.freeMap (Paths.of V ⋙q F.toPrefunctor) ⋙ pathComposition C = F
have rw₃ : (Cat.freeMap (Paths.of V) ⋙ pathComposition (Paths V)) ⋙ F = F
Tactic: rw [Cat.freeMap_comp, Functor.assoc, pathComposition_naturality, ← Functor.assoc,
      freeMap_pathsOf_pathComposition, Functor.id_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
V : Type u
C : Type u₁
inst : Quiver.{v, u} V
inst_1 : Category.{v₁, u₁} C
F : Paths V ⥤ C
rw : (Cat.freeMap (Paths.of V) ⋙ Cat.freeMap F.toPrefunctor) ⋙ pathComposition C = F
rw₁ : Cat.freeMap (Paths.of V) ⋙ Cat.freeMap F.toPrefunctor ⋙ pathComposition C = F
rw₂ : Cat.freeMap (Paths.of V) ⋙ pathComposition (Paths V) ⋙ F = F
rw₃ : (Cat.freeMap (Paths.of V) ⋙ pathComposition (Paths V)) ⋙ F = F
⊢ Cat.freeMap (Paths.of V ⋙q F.toPrefunctor) ⋙ pathComposition C = F
have rw₄ : 𝟭 (Paths V) ⋙ F = F
Tactic: rw [Cat.freeMap_comp, Functor.assoc, pathComposition_naturality, ← Functor.assoc,
      freeMap_pathsOf_pathComposition, Functor.id_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
V : Type u
C : Type u₁
inst : Quiver.{v, u} V
inst_1 : Category.{v₁, u₁} C
F : Paths V ⥤ C
rw : (Cat.freeMap (Paths.of V) ⋙ Cat.freeMap F.toPrefunctor) ⋙ pathComposition C = F
rw₁ : Cat.freeMap (Paths.of V) ⋙ Cat.freeMap F.toPrefunctor ⋙ pathComposition C = F
rw₂ : Cat.freeMap (Paths.of V) ⋙ pathComposition (Paths V) ⋙ F = F
rw₃ : (Cat.freeMap (Paths.of V) ⋙ pathComposition (Paths V)) ⋙ F = F
rw₄ : 𝟭 (Paths V) ⋙ F = F
⊢ Cat.freeMap (Paths.of V ⋙q F.toPrefunctor) ⋙ pathComposition C = F
have rw₅ : F = F
Tactic: rw [Cat.freeMap_comp, Functor.assoc, pathComposition_naturality, ← Functor.assoc,
      freeMap_pathsOf_pathComposition, Functor.id_comp]
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
V : Type u
C : Type u₁
inst : Quiver.{v, u} V
inst_1 : Category.{v₁, u₁} C
G : Prefunctor.{v, v₁, u, u₁} V C
⊢ Paths.of V ⋙q (Cat.freeMap G ⋙ pathComposition C).toPrefunctor = G
have rw : Paths.of V ⋙q ((Cat.freeMap G).toPrefunctor ⋙q Functor.toPrefunctor (pathComposition C)) = G
Tactic: rw [← Functor.toPrefunctor_comp, ← Prefunctor.comp_assoc,
      pathsOf_freeMap_toPrefunctor, Prefunctor.comp_assoc,
      pathsOf_pathComposition_toPrefunctor, Prefunctor.comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
V : Type u
C : Type u₁
inst : Quiver.{v, u} V
inst_1 : Category.{v₁, u₁} C
G : Prefunctor.{v, v₁, u, u₁} V C
rw : Paths.of V ⋙q ((Cat.freeMap G).toPrefunctor ⋙q Functor.toPrefunctor (pathComposition C)) = G
⊢ Paths.of V ⋙q (Cat.freeMap G ⋙ pathComposition C).toPrefunctor = G
have rw₁ : Paths.of V ⋙q (Cat.freeMap G).toPrefunctor ⋙q Functor.toPrefunctor (pathComposition C) = G
Tactic: rw [← Functor.toPrefunctor_comp, ← Prefunctor.comp_assoc,
      pathsOf_freeMap_toPrefunctor, Prefunctor.comp_assoc,
      pathsOf_pathComposition_toPrefunctor, Prefunctor.comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
V : Type u
C : Type u₁
inst : Quiver.{v, u} V
inst_1 : Category.{v₁, u₁} C
G : Prefunctor.{v, v₁, u, u₁} V C
rw : Paths.of V ⋙q ((Cat.freeMap G).toPrefunctor ⋙q Functor.toPrefunctor (pathComposition C)) = G
rw₁ : Paths.of V ⋙q (Cat.freeMap G).toPrefunctor ⋙q Functor.toPrefunctor (pathComposition C) = G
⊢ Paths.of V ⋙q (Cat.freeMap G ⋙ pathComposition C).toPrefunctor = G
have rw₂ : G ⋙q Paths.of C ⋙q Functor.toPrefunctor (pathComposition C) = G
Tactic: rw [← Functor.toPrefunctor_comp, ← Prefunctor.comp_assoc,
      pathsOf_freeMap_toPrefunctor, Prefunctor.comp_assoc,
      pathsOf_pathComposition_toPrefunctor, Prefunctor.comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
V : Type u
C : Type u₁
inst : Quiver.{v, u} V
inst_1 : Category.{v₁, u₁} C
G : Prefunctor.{v, v₁, u, u₁} V C
rw : Paths.of V ⋙q ((Cat.freeMap G).toPrefunctor ⋙q Functor.toPrefunctor (pathComposition C)) = G
rw₁ : Paths.of V ⋙q (Cat.freeMap G).toPrefunctor ⋙q Functor.toPrefunctor (pathComposition C) = G
rw₂ : G ⋙q Paths.of C ⋙q Functor.toPrefunctor (pathComposition C) = G
⊢ Paths.of V ⋙q (Cat.freeMap G ⋙ pathComposition C).toPrefunctor = G
have rw₃ : G ⋙q (Paths.of C ⋙q Functor.toPrefunctor (pathComposition C)) = G
Tactic: rw [← Functor.toPrefunctor_comp, ← Prefunctor.comp_assoc,
      pathsOf_freeMap_toPrefunctor, Prefunctor.comp_assoc,
      pathsOf_pathComposition_toPrefunctor, Prefunctor.comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
V : Type u
C : Type u₁
inst : Quiver.{v, u} V
inst_1 : Category.{v₁, u₁} C
G : Prefunctor.{v, v₁, u, u₁} V C
rw : Paths.of V ⋙q ((Cat.freeMap G).toPrefunctor ⋙q Functor.toPrefunctor (pathComposition C)) = G
rw₁ : Paths.of V ⋙q (Cat.freeMap G).toPrefunctor ⋙q Functor.toPrefunctor (pathComposition C) = G
rw₂ : G ⋙q Paths.of C ⋙q Functor.toPrefunctor (pathComposition C) = G
rw₃ : G ⋙q (Paths.of C ⋙q Functor.toPrefunctor (pathComposition C)) = G
⊢ Paths.of V ⋙q (Cat.freeMap G ⋙ pathComposition C).toPrefunctor = G
have rw₄ : G ⋙q 𝟭q C = G
Tactic: rw [← Functor.toPrefunctor_comp, ← Prefunctor.comp_assoc,
      pathsOf_freeMap_toPrefunctor, Prefunctor.comp_assoc,
      pathsOf_pathComposition_toPrefunctor, Prefunctor.comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.Quiv (goal)
Goal:
V : Type u
C : Type u₁
inst : Quiver.{v, u} V
inst_1 : Category.{v₁, u₁} C
G : Prefunctor.{v, v₁, u, u₁} V C
rw : Paths.of V ⋙q ((Cat.freeMap G).toPrefunctor ⋙q Functor.toPrefunctor (pathComposition C)) = G
rw₁ : Paths.of V ⋙q (Cat.freeMap G).toPrefunctor ⋙q Functor.toPrefunctor (pathComposition C) = G
rw₂ : G ⋙q Paths.of C ⋙q Functor.toPrefunctor (pathComposition C) = G
rw₃ : G ⋙q (Paths.of C ⋙q Functor.toPrefunctor (pathComposition C)) = G
rw₄ : G ⋙q 𝟭q C = G
⊢ Paths.of V ⋙q (Cat.freeMap G ⋙ pathComposition C).toPrefunctor = G
have rw₅ : G = G
Tactic: rw [← Functor.toPrefunctor_comp, ← Prefunctor.comp_assoc,
      pathsOf_freeMap_toPrefunctor, Prefunctor.comp_assoc,
      pathsOf_pathComposition_toPrefunctor, Prefunctor.comp_id]
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.ReflQuiv (goal)
Goal:
C D : Cat
F G : Functor.{v, v, u, u} (↑C : Type u) (↑D : Type u)
hyp : forget.map F.toCatHom = forget.map G.toCatHom
⊢ F = G
have mk : ∀ (obj : (↑C : Type u) → (↑D : Type u)) (map : {X Y : (↑C : Type u)} → (X ⟶ Y) → (obj X ⟶ obj Y))
  (map_id : ∀ (X : (↑C : Type u)), map (𝟙 X) = 𝟙 (obj X))
  (map_comp : ∀ {X Y Z : (↑C : Type u)} (f : X ⟶ Y) (g : Y ⟶ Z), map (f ≫ g) = map f ≫ map g)
  (hyp :
    forget.map
        { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp :
            Functor.{v, v, u, u} (↑C : Type u) (↑D : Type u) }.toCatHom =
      forget.map G.toCatHom),
  { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp :
      Functor.{v, v, u, u} (↑C : Type u) (↑D : Type u) } =
    G
Tactic: cases F
Cleared hypotheses: F, hyp
-- Lean 4 proof step from Mathlib.CategoryTheory.Category.ReflQuiv (goal)
Goal:
C D : Cat
G : Functor.{v, v, u, u} (↑C : Type u) (↑D : Type u)
obj : (↑C : Type u) → (↑D : Type u)
map : {X Y : (↑C : Type u)} → (X ⟶ Y) → (obj X ⟶ obj Y)
map_id : ∀ (X : (↑C : Type u)), map (𝟙 X) = 𝟙 (obj X)
map_comp : ∀ {X Y Z : (↑C : Type u)} (f : X ⟶ Y) (g : Y ⟶ Z), map (f ≫ g) = map f ≫ map g
hyp :
  forget.map
      { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp :
          Functor.{v, v, u, u} (↑C : Type u) (↑D : Type u) }.toCatHom =
    forget.map G.toCatHom
⊢ { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp :
      Functor.{v, v, u, u} (↑C : Type u) (↑D : Type u) } =
    G
have mk : ∀ (obj_1 : (↑C : Type u) → (↑D : Type u)) (map_1 : {X Y : (↑C : Type u)} → (X ⟶ Y) → (obj_1 X ⟶ obj_1 Y))
  (map_id_1 : ∀ (X : (↑C : Type u)), map_1 (𝟙 X) = 𝟙 (obj_1 X))
  (map_comp_1 : ∀ {X Y Z : (↑C : Type u)} (f : X ⟶ Y) (g : Y ⟶ Z), map_1 (f ≫ g) = map_1 f ≫ map_1 g)
  (hyp :
    forget.map
        { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp :
            Functor.{v, v, u, u} (↑C : Type u) (↑D : Type u) }.toCatHom =
      forget.map { obj := obj_1, map := @map_1, map_id := map_id_1, map_comp := @map_comp_1 }.toCatHom),
  { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp :
      Functor.{v, v, u, u} (↑C : Type u) (↑D : Type u) } =
    { obj := obj_1, map := @map_1, map_id := map_id_1, map_comp := @map_comp_1 }
Tactic: cases G
Cleared hypotheses: G, hyp