awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps (goal)
Goal:
X Y : TopCat
f g : C((↑X : Type u), (↑Y : Type u))
H : f.Homotopy g
x₀ x₁ : (↑X : Type u)
p : fromTop x₀ ⟶ fromTop x₁
| ((π.map (TopCat.ofHom H.uliftMap)).map
        (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
          (𝟙 (fromTop { down := 0 }), p)) ≫
      hcast ⋯ ≫ hcast ⋯ :
    Quiver.Hom (V := (↑(π.obj (TopCat.of (↑Y : Type u))) : Type u))
      (fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (0, x₀)))
      (fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (0, x₁))))
have a : ((π.map (TopCat.ofHom H.uliftMap)).map
      (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
        (𝟙 (fromTop { down := 0 }), p)) ≫
    𝟙
      ((π.map (TopCat.ofHom H.uliftMap)).obj
        (Functor.obj (C := (↑(π.obj (TopCat.of (ULift.{u, 0} (↑I : Type)))) : Type u) × (↑(π.obj X) : Type u))
          (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
          ({ as := { down := 0 : ULift.{u, 0} (↑I : Type) } : FundamentalGroupoid (ULift.{u, 0} (↑I : Type)) },
            { as := x₁ : FundamentalGroupoid (↑X : Type u) }))) :
  Quiver.Hom (V := (↑(π.obj (TopCat.of (↑Y : Type u))) : Type u))
    (fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (0, x₀)))
    (fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (0, x₁))))
Tactic: rw [eqToHom_trans, eqToHom_refl]
-- Lean 4 proof step from Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps (goal)
Goal:
X Y : TopCat
f g : C((↑X : Type u), (↑Y : Type u))
H : f.Homotopy g
x₀ x₁ : (↑X : Type u)
p : fromTop x₀ ⟶ fromTop x₁
⊢ CategoryStruct.comp (Y := (π.map (TopCat.ofHom f)).obj (fromTop x₁))
      (hcast ⋯ ≫
        (π.map (TopCat.ofHom H.uliftMap)).map
            (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
              (𝟙 (fromTop { down := 0 }), p)) ≫
          hcast ⋯)
      (hcast ⋯ ≫
        (π.map (TopCat.ofHom H.uliftMap)).map
            (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
              (uhpath01, 𝟙 (fromTop x₁))) ≫
          hcast ⋯) =
    H.diagonalPath' p
have left : hcast sorry ≫
    CategoryStruct.comp (Y := fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (0, x₁)))
        ((π.map (TopCat.ofHom H.uliftMap)).map
              (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
                (𝟙 (fromTop { down := 0 }), p)) ≫
            𝟙
              ((π.map (TopCat.ofHom H.uliftMap)).obj
                (Functor.obj (C := (↑(π.obj (TopCat.of (ULift.{u, 0} (↑I : Type)))) : Type u) × (↑(π.obj X) : Type u))
                  (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
                  ({ as := { down := 0 : ULift.{u, 0} (↑I : Type) } : FundamentalGroupoid (ULift.{u, 0} (↑I : Type)) },
                    { as := x₁ : FundamentalGroupoid (↑X : Type u) }))) :
          Quiver.Hom (V := (↑(π.obj (TopCat.of (↑Y : Type u))) : Type u))
            (fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (0, x₀)))
            (fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (0, x₁))))
        ((π.map (TopCat.ofHom H.uliftMap)).map
            (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
              (uhpath01, 𝟙 (fromTop x₁))) :
          (π.map (TopCat.ofHom H.uliftMap)).obj
              (Functor.obj (C := (↑(π.obj (TopCat.of (ULift.{u, 0} (↑I : Type)))) : Type u) × (↑(π.obj X) : Type u))
                (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
                ({ as := { down := 0 : ULift.{u, 0} (↑I : Type) } : FundamentalGroupoid (ULift.{u, 0} (↑I : Type)) },
                  { as := x₁ : FundamentalGroupoid (↑X : Type u) })) ⟶
            (π.map (TopCat.ofHom H.uliftMap)).obj
              (Functor.obj (C := (↑(π.obj (TopCat.of (ULift.{u, 0} (↑I : Type)))) : Type u) × (↑(π.obj X) : Type u))
                (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X) ({ as := { down := 1 } }, { as := x₁ }))) ≫
      hcast sorry =
  H.diagonalPath' p
Tactic: slice_lhs 2 4 => rw [eqToHom_trans, eqToHom_refl]
-- Lean 4 proof step from Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps (goal)
Goal:
X Y : TopCat
f g : C((↑X : Type u), (↑Y : Type u))
H : f.Homotopy g
x₀ x₁ : (↑X : Type u)
p : fromTop x₀ ⟶ fromTop x₁
| ((π.map (TopCat.ofHom H.uliftMap)).map
        (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
          (𝟙 (fromTop { down := 0 }), p)) ≫
      𝟙
          ((π.map (TopCat.ofHom H.uliftMap)).obj
            (Functor.obj (C := (↑(π.obj (TopCat.of (ULift.{u, 0} (↑I : Type)))) : Type u) × (↑(π.obj X) : Type u))
              (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
              ({ as := { down := 0 : ULift.{u, 0} (↑I : Type) } : FundamentalGroupoid (ULift.{u, 0} (↑I : Type)) },
                { as := x₁ : FundamentalGroupoid (↑X : Type u) }))) ≫
        (π.map (TopCat.ofHom H.uliftMap)).map
          (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
            (uhpath01, 𝟙 (fromTop x₁))) :
    fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (0, x₀)) ⟶
      (π.map (TopCat.ofHom H.uliftMap)).obj
        (Functor.obj (C := (↑(π.obj (TopCat.of (ULift.{u, 0} (↑I : Type)))) : Type u) × (↑(π.obj X) : Type u))
          (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
          ({ as := { down := 1 : ULift.{u, 0} (↑I : Type) } : FundamentalGroupoid (ULift.{u, 0} (↑I : Type)) },
            { as := x₁ : FundamentalGroupoid (↑X : Type u) })))
have a : ((π.map (TopCat.ofHom H.uliftMap)).map
    (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X) (Prod.mkHom uhpath01 p)) :
  (π.map (TopCat.ofHom H.uliftMap)).obj
      { as := ({ down := 0 : ULift.{u, 0} (↑I : Type) }, x₀) :
        FundamentalGroupoid (ULift.{u, 0} (↑I : Type) × (↑X : Type u)) } ⟶
    (π.map (TopCat.ofHom H.uliftMap)).obj { as := ({ down := 1 }, x₁) })
Tactic: simp [← CategoryTheory.Functor.map_comp]
-- Lean 4 proof step from Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps (goal)
Goal:
X Y : TopCat
f g : C((↑X : Type u), (↑Y : Type u))
H : f.Homotopy g
x₀ x₁ : (↑X : Type u)
p : fromTop x₀ ⟶ fromTop x₁
⊢ hcast ⋯ ≫
      CategoryStruct.comp (Y := fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (0, x₁)))
          ((π.map (TopCat.ofHom H.uliftMap)).map
                (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
                  (𝟙 (fromTop { down := 0 }), p)) ≫
              𝟙
                ((π.map (TopCat.ofHom H.uliftMap)).obj
                  (Functor.obj (C := (↑(π.obj (TopCat.of (ULift.{u, 0} (↑I : Type)))) : Type u) × (↑(π.obj X) : Type u))
                    (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
                    ({ as := { down := 0 : ULift.{u, 0} (↑I : Type) } :
                        FundamentalGroupoid (ULift.{u, 0} (↑I : Type)) },
                      { as := x₁ : FundamentalGroupoid (↑X : Type u) }))) :
            Quiver.Hom (V := (↑(π.obj (TopCat.of (↑Y : Type u))) : Type u))
              (fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (0, x₀)))
              (fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (0, x₁))))
          ((π.map (TopCat.ofHom H.uliftMap)).map
              (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
                (uhpath01, 𝟙 (fromTop x₁))) :
            (π.map (TopCat.ofHom H.uliftMap)).obj
                (Functor.obj (C := (↑(π.obj (TopCat.of (ULift.{u, 0} (↑I : Type)))) : Type u) × (↑(π.obj X) : Type u))
                  (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
                  ({ as := { down := 0 : ULift.{u, 0} (↑I : Type) } : FundamentalGroupoid (ULift.{u, 0} (↑I : Type)) },
                    { as := x₁ : FundamentalGroupoid (↑X : Type u) })) ⟶
              (π.map (TopCat.ofHom H.uliftMap)).obj
                (Functor.obj (C := (↑(π.obj (TopCat.of (ULift.{u, 0} (↑I : Type)))) : Type u) × (↑(π.obj X) : Type u))
                  (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X) ({ as := { down := 1 } }, { as := x₁ }))) ≫
        hcast ⋯ =
    H.diagonalPath' p
have left : hcast sorry ≫
    (π.map (TopCat.ofHom H.uliftMap)).map
        (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X) (Prod.mkHom uhpath01 p)) ≫
      hcast sorry =
  H.diagonalPath' p
Tactic: slice_lhs 2 4 => simp [← CategoryTheory.Functor.map_comp]
-- Lean 4 proof step from Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps (goal)
Goal:
X Y : TopCat
f g : C((↑X : Type u), (↑Y : Type u))
H : f.Homotopy g
x₀ x₁ : (↑X : Type u)
p : fromTop x₀ ⟶ fromTop x₁
| ((π.map (TopCat.ofHom H.uliftMap)).map
        (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
          (uhpath01, 𝟙 (fromTop x₀))) ≫
      hcast ⋯ ≫ hcast ⋯ :
    Quiver.Hom (V := (↑(π.obj Y) : Type u)) (fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (0, x₀)))
      (fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (1, x₀))))
have a : ((π.map (TopCat.ofHom H.uliftMap)).map
      (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X) (uhpath01, 𝟙 (fromTop x₀))) ≫
    𝟙
      ((π.map (TopCat.ofHom H.uliftMap)).obj
        (Functor.obj (C := (↑(π.obj (TopCat.of (ULift.{u, 0} (↑I : Type)))) : Type u) × (↑(π.obj X) : Type u))
          (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
          ({ as := { down := 1 : ULift.{u, 0} (↑I : Type) } : FundamentalGroupoid (ULift.{u, 0} (↑I : Type)) },
            { as := x₀ : FundamentalGroupoid (↑X : Type u) }))) :
  Quiver.Hom (V := (↑(π.obj Y) : Type u)) (fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (0, x₀)))
    (fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (1, x₀))))
Tactic: rw [eqToHom_trans, eqToHom_refl]
-- Lean 4 proof step from Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps (goal)
Goal:
X Y : TopCat
f g : C((↑X : Type u), (↑Y : Type u))
H : f.Homotopy g
x₀ x₁ : (↑X : Type u)
p : fromTop x₀ ⟶ fromTop x₁
⊢ (hcast ⋯ ≫
        (π.map (TopCat.ofHom H.uliftMap)).map
            (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
              (uhpath01, 𝟙 (fromTop x₀))) ≫
          hcast ⋯) ≫
      hcast ⋯ ≫
        (π.map (TopCat.ofHom H.uliftMap)).map
            (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
              (𝟙 (fromTop { down := 1 : ULift.{u, 0} (↑I : Type) }), p)) ≫
          hcast ⋯ =
    H.diagonalPath' p
have right : hcast sorry ≫
    CategoryStruct.comp (Y := fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (1, x₀)))
        ((π.map (TopCat.ofHom H.uliftMap)).map
              (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
                (uhpath01, 𝟙 (fromTop x₀))) ≫
            𝟙
              ((π.map (TopCat.ofHom H.uliftMap)).obj
                (Functor.obj (C := (↑(π.obj (TopCat.of (ULift.{u, 0} (↑I : Type)))) : Type u) × (↑(π.obj X) : Type u))
                  (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
                  ({ as := { down := 1 : ULift.{u, 0} (↑I : Type) } : FundamentalGroupoid (ULift.{u, 0} (↑I : Type)) },
                    { as := x₀ : FundamentalGroupoid (↑X : Type u) }))) :
          Quiver.Hom (V := (↑(π.obj Y) : Type u))
            (fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (0, x₀)))
            (fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (1, x₀))))
        ((π.map (TopCat.ofHom H.uliftMap)).map
            (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
              (𝟙 (fromTop { down := 1 }), p)) :
          (π.map (TopCat.ofHom H.uliftMap)).obj
              (Functor.obj (C := (↑(π.obj (TopCat.of (ULift.{u, 0} (↑I : Type)))) : Type u) × (↑(π.obj X) : Type u))
                (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
                ({ as := { down := 1 : ULift.{u, 0} (↑I : Type) } : FundamentalGroupoid (ULift.{u, 0} (↑I : Type)) },
                  { as := x₀ : FundamentalGroupoid (↑X : Type u) })) ⟶
            (π.map (TopCat.ofHom H.uliftMap)).obj
              (Functor.obj (C := (↑(π.obj (TopCat.of (ULift.{u, 0} (↑I : Type)))) : Type u) × (↑(π.obj X) : Type u))
                (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X) ({ as := { down := 1 } }, { as := x₁ }))) ≫
      hcast sorry =
  H.diagonalPath' p
Tactic: slice_lhs 2 4 => rw [eqToHom_trans, eqToHom_refl]
-- Lean 4 proof step from Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps (goal)
Goal:
X Y : TopCat
f g : C((↑X : Type u), (↑Y : Type u))
H : f.Homotopy g
x₀ x₁ : (↑X : Type u)
p : fromTop x₀ ⟶ fromTop x₁
| ((π.map (TopCat.ofHom H.uliftMap)).map
        (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
          (uhpath01, 𝟙 (fromTop x₀))) ≫
      𝟙
          ((π.map (TopCat.ofHom H.uliftMap)).obj
            (Functor.obj (C := (↑(π.obj (TopCat.of (ULift.{u, 0} (↑I : Type)))) : Type u) × (↑(π.obj X) : Type u))
              (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
              ({ as := { down := 1 : ULift.{u, 0} (↑I : Type) } : FundamentalGroupoid (ULift.{u, 0} (↑I : Type)) },
                { as := x₀ : FundamentalGroupoid (↑X : Type u) }))) ≫
        (π.map (TopCat.ofHom H.uliftMap)).map
          (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
            (𝟙 (fromTop { down := 1 : ULift.{u, 0} (↑I : Type) }), p)) :
    fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (0, x₀)) ⟶
      (π.map (TopCat.ofHom H.uliftMap)).obj
        (Functor.obj (C := (↑(π.obj (TopCat.of (ULift.{u, 0} (↑I : Type)))) : Type u) × (↑(π.obj X) : Type u))
          (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
          ({ as := { down := 1 : ULift.{u, 0} (↑I : Type) } : FundamentalGroupoid (ULift.{u, 0} (↑I : Type)) },
            { as := x₁ : FundamentalGroupoid (↑X : Type u) })))
have a : ((π.map (TopCat.ofHom H.uliftMap)).map
    (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X) (Prod.mkHom uhpath01 p)) :
  (π.map (TopCat.ofHom H.uliftMap)).obj
      { as := ({ down := 0 : ULift.{u, 0} (↑I : Type) }, x₀) :
        FundamentalGroupoid (ULift.{u, 0} (↑I : Type) × (↑X : Type u)) } ⟶
    (π.map (TopCat.ofHom H.uliftMap)).obj { as := ({ down := 1 }, x₁) })
Tactic: simp [← CategoryTheory.Functor.map_comp]
-- Lean 4 proof step from Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps (goal)
Goal:
X Y : TopCat
f g : C((↑X : Type u), (↑Y : Type u))
H : f.Homotopy g
x₀ x₁ : (↑X : Type u)
p : fromTop x₀ ⟶ fromTop x₁
⊢ hcast ⋯ ≫
      CategoryStruct.comp (Y := fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (1, x₀)))
          ((π.map (TopCat.ofHom H.uliftMap)).map
                (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
                  (uhpath01, 𝟙 (fromTop x₀))) ≫
              𝟙
                ((π.map (TopCat.ofHom H.uliftMap)).obj
                  (Functor.obj (C := (↑(π.obj (TopCat.of (ULift.{u, 0} (↑I : Type)))) : Type u) × (↑(π.obj X) : Type u))
                    (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
                    ({ as := { down := 1 : ULift.{u, 0} (↑I : Type) } :
                        FundamentalGroupoid (ULift.{u, 0} (↑I : Type)) },
                      { as := x₀ : FundamentalGroupoid (↑X : Type u) }))) :
            Quiver.Hom (V := (↑(π.obj Y) : Type u))
              (fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (0, x₀)))
              (fromTop ((H : (a : (↑I : Type) × (↑X : Type u)) → (↑Y : Type u)) (1, x₀))))
          ((π.map (TopCat.ofHom H.uliftMap)).map
              (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
                (𝟙 (fromTop { down := 1 }), p)) :
            (π.map (TopCat.ofHom H.uliftMap)).obj
                (Functor.obj (C := (↑(π.obj (TopCat.of (ULift.{u, 0} (↑I : Type)))) : Type u) × (↑(π.obj X) : Type u))
                  (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X)
                  ({ as := { down := 1 : ULift.{u, 0} (↑I : Type) } : FundamentalGroupoid (ULift.{u, 0} (↑I : Type)) },
                    { as := x₀ : FundamentalGroupoid (↑X : Type u) })) ⟶
              (π.map (TopCat.ofHom H.uliftMap)).obj
                (Functor.obj (C := (↑(π.obj (TopCat.of (ULift.{u, 0} (↑I : Type)))) : Type u) × (↑(π.obj X) : Type u))
                  (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X) ({ as := { down := 1 } }, { as := x₁ }))) ≫
        hcast ⋯ =
    H.diagonalPath' p
have right : hcast sorry ≫
    (π.map (TopCat.ofHom H.uliftMap)).map
        (CategoryTheory.Functor.map (prodToProdTop (TopCat.of (ULift.{u, 0} (↑I : Type))) X) (Prod.mkHom uhpath01 p)) ≫
      hcast sorry =
  H.diagonalPath' p
Tactic: slice_lhs 2 4 => simp [← CategoryTheory.Functor.map_comp]
-- Lean 4 proof step from Mathlib.AlgebraicTopology.FundamentalGroupoid.PUnit (goal)
Goal:
x y : FundamentalGroupoid PUnit.{u_1 + 1}
⊢ Subsingleton.{u_1 + 1} (x ⟶ y)
have convert_to : Subsingleton.{u_1 + 1} (Path.Homotopic.Quotient PUnit.unit PUnit.unit)
Tactic: convert_to Subsingleton (Path.Homotopic.Quotient PUnit.unit PUnit.unit)
-- Lean 4 proof step from Mathlib.AlgebraicTopology.FundamentalGroupoid.Product (goal)
Goal:
I : Type u
X : I → TopCat
x : (i : I) → (↑(π.obj (X i)) : Type u)
⊢ Path.Homotopic.pi (X := fun (i : I) ↦ (↑(X i) : Type u)) (𝟙 x) = 𝟙 { as := fun (i : I) ↦ (x i).as }
have change : Eq.{u + 1} (α :=
  Path.Homotopic.Quotient (X := (i : I) → (↑(X i) : Type u)) (FundamentalGroupoid.as { as := fun (i : I) ↦ (x i).as })
    (FundamentalGroupoid.as { as := fun (i : I) ↦ (x i).as }))
  (Path.Homotopic.pi (X := fun (i : I) ↦ (↑(X i) : Type u)) fun (i : I) ↦
    Path.Homotopic.Quotient.mk (Path.refl (x i).as))
  (𝟙 { as := fun (i : I) ↦ (x i).as })
Tactic: change (Path.Homotopic.pi fun i => Path.Homotopic.Quotient.mk _) = _
-- Lean 4 proof step from Mathlib.AlgebraicTopology.FundamentalGroupoid.Product (goal)
Goal:
I : Type u
X : I → TopCat
x : (i : I) → (↑(π.obj (X i)) : Type u)
⊢ Eq.{u + 1} (α :=
    Path.Homotopic.Quotient (X := (i : I) → (↑(X i) : Type u)) (FundamentalGroupoid.as { as := fun (i : I) ↦ (x i).as })
      (FundamentalGroupoid.as { as := fun (i : I) ↦ (x i).as }))
    (Path.Homotopic.pi (X := fun (i : I) ↦ (↑(X i) : Type u)) fun (i : I) ↦
      Path.Homotopic.Quotient.mk (Path.refl (x i).as))
    (𝟙 { as := fun (i : I) ↦ (x i).as })
have simp : Eq.{u + 1} (α :=
  Path.Homotopic.Quotient (X := (i : I) → (↑(X i) : Type u)) (fun (i : I) ↦ (x i).as) fun (i : I) ↦ (x i).as)
  (Path.Homotopic.Quotient.mk (Path.pi (χ := fun (i : I) ↦ (↑(X i) : Type u)) fun (i : I) ↦ Path.refl (x i).as))
  (𝟙 { as := fun (i : I) ↦ (x i).as })
Tactic: simp only [Path.Homotopic.pi_lift]
-- Lean 4 proof step from Mathlib.AlgebraicTopology.FundamentalGroupoid.Product (goal)
Goal:
I : Type u
X : I → TopCat
⊢ piToPiTop X ⋙ CategoryTheory.Functor.pi' (C := fun (i : I) ↦ (↑(π.obj (X i)) : Type u)) (proj X) =
    𝟭 ((i : I) → (↑(π.obj (X i)) : Type u))
have apply : ∀ (X_1 : (i : I) → (↑(π.obj (X i)) : Type u)),
  (piToPiTop X ⋙ CategoryTheory.Functor.pi' (C := fun (i : I) ↦ (↑(π.obj (X i)) : Type u)) (proj X)).obj X_1 =
    CategoryTheory.Functor.obj (𝟭 ((i : I) → (↑(π.obj (X i)) : Type u))) X_1
Tactic: apply CategoryTheory.Functor.ext ?_ ?_
-- Lean 4 proof step from Mathlib.AlgebraicTopology.FundamentalGroupoid.Product (goal)
Goal:
I : Type u
X : I → TopCat
apply :
  ∀ (X_1 : (i : I) → (↑(π.obj (X i)) : Type u)),
    (piToPiTop X ⋙ CategoryTheory.Functor.pi' (C := fun (i : I) ↦ (↑(π.obj (X i)) : Type u)) (proj X)).obj X_1 =
      CategoryTheory.Functor.obj (𝟭 ((i : I) → (↑(π.obj (X i)) : Type u))) X_1
⊢ piToPiTop X ⋙ CategoryTheory.Functor.pi' (C := fun (i : I) ↦ (↑(π.obj (X i)) : Type u)) (proj X) =
    𝟭 ((i : I) → (↑(π.obj (X i)) : Type u))
have apply₁ : ∀ (X_1 Y : (i : I) → (↑(π.obj (X i)) : Type u)) (f : X_1 ⟶ Y),
  (piToPiTop X ⋙ CategoryTheory.Functor.pi' (C := fun (i : I) ↦ (↑(π.obj (X i)) : Type u)) (proj X)).map f =
    CategoryTheory.eqToHom sorry ≫
      CategoryTheory.Functor.map (𝟭 ((i : I) → (↑(π.obj (X i)) : Type u))) f ≫ CategoryTheory.eqToHom sorry
Tactic: apply CategoryTheory.Functor.ext ?_ ?_
-- Lean 4 proof step from Mathlib.AlgebraicTopology.FundamentalGroupoid.Product (goal)
Goal:
I : Type u
X : I → TopCat
X_1 Y : (i : I) → (↑(π.obj (X i)) : Type u)
f : X_1 ⟶ Y
⊢ (piToPiTop X ⋙ CategoryTheory.Functor.pi' (C := fun (i : I) ↦ (↑(π.obj (X i)) : Type u)) (proj X)).map f =
    CategoryTheory.eqToHom ⋯ ≫
      CategoryTheory.Functor.map (𝟭 ((i : I) → (↑(π.obj (X i)) : Type u))) f ≫ CategoryTheory.eqToHom ⋯
have w : ∀ (i : I),
  (piToPiTop X ⋙ CategoryTheory.Functor.pi' (C := fun (i : I) ↦ (↑(π.obj (X i)) : Type u)) (proj X)).map f i =
    (CategoryTheory.eqToHom sorry ≫
        CategoryTheory.Functor.map (𝟭 ((i : I) → (↑(π.obj (X i)) : Type u))) f ≫ CategoryTheory.eqToHom sorry)
      i
Tactic: ext
-- Lean 4 proof step from Mathlib.AlgebraicTopology.FundamentalGroupoid.Product (goal)
Goal:
I : Type u
X : I → TopCat
⊢ CategoryTheory.Functor.pi' (C := fun (i : I) ↦ (↑(π.obj (X i)) : Type u)) (proj X) ⋙ piToPiTop X =
    𝟭 (↑(π.obj (TopCat.of ((i : I) → (↑(X i) : Type u)))) : Type u)
have h_map : autoParam
  (∀ (X_1 Y : (↑(π.obj (TopCat.of ((i : I) → (↑(X i) : Type u)))) : Type u)) (f : X_1 ⟶ Y),
    (CategoryTheory.Functor.pi' (C := fun (i : I) ↦ (↑(π.obj (X i)) : Type u)) (proj X) ⋙ piToPiTop X).map f =
      CategoryTheory.eqToHom sorry ≫
        CategoryTheory.Functor.map (𝟭 (↑(π.obj (TopCat.of ((i : I) → (↑(X i) : Type u)))) : Type u)) f ≫
          CategoryTheory.eqToHom sorry)
  CategoryTheory.Functor.ext._auto_1
Tactic: apply CategoryTheory.Functor.ext
-- Lean 4 proof step from Mathlib.AlgebraicTopology.FundamentalGroupoid.Product (goal)
Goal:
I : Type u
X : I → TopCat
h_map :
  autoParam
    (∀ (X_1 Y : (↑(π.obj (TopCat.of ((i : I) → (↑(X i) : Type u)))) : Type u)) (f : X_1 ⟶ Y),
      (CategoryTheory.Functor.pi' (C := fun (i : I) ↦ (↑(π.obj (X i)) : Type u)) (proj X) ⋙ piToPiTop X).map f =
        CategoryTheory.eqToHom ⋯ ≫
          CategoryTheory.Functor.map (𝟭 (↑(π.obj (TopCat.of ((i : I) → (↑(X i) : Type u)))) : Type u)) f ≫
            CategoryTheory.eqToHom ⋯)
    CategoryTheory.Functor.ext._auto_1
⊢ CategoryTheory.Functor.pi' (C := fun (i : I) ↦ (↑(π.obj (X i)) : Type u)) (proj X) ⋙ piToPiTop X =
    𝟭 (↑(π.obj (TopCat.of ((i : I) → (↑(X i) : Type u)))) : Type u)
have h_obj : ∀ (X_1 : (↑(π.obj (TopCat.of ((i : I) → (↑(X i) : Type u)))) : Type u)),
  (CategoryTheory.Functor.pi' (C := fun (i : I) ↦ (↑(π.obj (X i)) : Type u)) (proj X) ⋙ piToPiTop X).obj X_1 =
    CategoryTheory.Functor.obj (𝟭 (↑(π.obj (TopCat.of ((i : I) → (↑(X i) : Type u)))) : Type u)) X_1
Tactic: apply CategoryTheory.Functor.ext
-- Lean 4 proof step from Mathlib.AlgebraicTopology.FundamentalGroupoid.Product (goal)
Goal:
I : Type u
X : I → TopCat
X_1 Y : (↑(π.obj (TopCat.of ((i : I) → (↑(X i) : Type u)))) : Type u)
f : X_1 ⟶ Y
⊢ (CategoryTheory.Functor.pi' (C := fun (i : I) ↦ (↑(π.obj (X i)) : Type u)) (proj X) ⋙ piToPiTop X).map f =
    CategoryTheory.eqToHom ⋯ ≫
      CategoryTheory.Functor.map (𝟭 (↑(π.obj (TopCat.of ((i : I) → (↑(X i) : Type u)))) : Type u)) f ≫
        CategoryTheory.eqToHom ⋯
have h_map : Path.Homotopic.pi (X := fun (i : I) ↦ (↑(X i) : Type u))
    ((CategoryTheory.Functor.pi' (C := fun (i : I) ↦ (↑(π.obj (X i)) : Type u)) (proj X)).map f) =
  f
Tactic: suffices Path.Homotopic.pi ((CategoryTheory.Functor.pi' (proj X)).map f) = f by simpa
-- Lean 4 proof step from Mathlib.AlgebraicTopology.FundamentalGroupoid.Product (goal)
Goal:
I : Type u
X : I → TopCat
⊢ Limits.PreservesLimit (Discrete.functor X) π
have apply : Limits.IsLimit (π.mapCone (TopCat.piFan X))
Tactic: apply Limits.preservesLimit_of_preserves_limit_cone (TopCat.piFanIsLimit X)