awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.AlgebraicTopology.SimplicialSet.Homotopy (goal)
Goal:
X Y : SSet
f g : X ⟶ Y
H : Homotopy f g
n : ℕ
⊢ Eq (α := X _⦋n⦌ ⟶ Y _⦋n⦌)
    ((↾fun (x : X _⦋n⦌) ↦
        (ConcreteCategory.hom (FC := TypeCat.Fun)
              (NatTrans.app ((Equiv.symm yonedaEquiv : (a : X _⦋n⦌) → Δ[n] ⟶ X) x ▷ Δ[1] ≫ H.h) (op ⦋n + 1⦌)) :
            (a : (fun (X : Type u) ↦ X) ((Δ[n] ⊗ Δ[1]) _⦋n + 1⦌)) → (fun (X : Type u) ↦ X) (Y _⦋n + 1⦌))
          (↑((prodStdSimplex.nonDegenerateEquiv₁ :
                  (a : Fin (n + 1)) → (↑((Δ[n] ⊗ Δ[1]).nonDegenerate (n + 1)) : Type u))
                (Fin.last n)) :
            (Δ[n] ⊗ Δ[1]) _⦋n + 1⦌)) ≫
      δ Y (Fin.last (n + 1)))
    (f.app (op ⦋n⦌))
have h : ∀ (x : X _⦋n⦌),
  TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        ((↾fun (x : X _⦋n⦌) ↦
            (ConcreteCategory.hom (FC := TypeCat.Fun)
                  (NatTrans.app ((Equiv.symm yonedaEquiv : (a : X _⦋n⦌) → Δ[n] ⟶ X) x ▷ Δ[1] ≫ H.h) (op ⦋n + 1⦌)) :
                (a : (fun (X : Type u) ↦ X) ((Δ[n] ⊗ Δ[1]) _⦋n + 1⦌)) → (fun (X : Type u) ↦ X) (Y _⦋n + 1⦌))
              (↑((prodStdSimplex.nonDegenerateEquiv₁ :
                      (a : Fin (n + 1)) → (↑((Δ[n] ⊗ Δ[1]).nonDegenerate (n + 1)) : Type u))
                    (Fin.last n)) :
                (Δ[n] ⊗ Δ[1]) _⦋n + 1⦌)) ≫
          δ Y (Fin.last (n + 1))))
      x =
    TypeCat.Fun.toFun (ConcreteCategory.hom (FC := TypeCat.Fun) (f.app (op ⦋n⦌))) x
Tactic: ext x
-- Lean 4 proof step from Mathlib.AlgebraicTopology.SimplicialSet.Homotopy (goal)
Goal:
X Y : SSet
f g : X ⟶ Y
H : Homotopy f g
n : ℕ
x : X _⦋n⦌
⊢ TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        ((↾fun (x : X _⦋n⦌) ↦
            (ConcreteCategory.hom (FC := TypeCat.Fun)
                  (NatTrans.app ((Equiv.symm yonedaEquiv : (a : X _⦋n⦌) → Δ[n] ⟶ X) x ▷ Δ[1] ≫ H.h) (op ⦋n + 1⦌)) :
                (a : (fun (X : Type u) ↦ X) ((Δ[n] ⊗ Δ[1]) _⦋n + 1⦌)) → (fun (X : Type u) ↦ X) (Y _⦋n + 1⦌))
              (↑((prodStdSimplex.nonDegenerateEquiv₁ :
                      (a : Fin (n + 1)) → (↑((Δ[n] ⊗ Δ[1]).nonDegenerate (n + 1)) : Type u))
                    (Fin.last n)) :
                (Δ[n] ⊗ Δ[1]) _⦋n + 1⦌)) ≫
          δ Y (Fin.last (n + 1))))
      x =
    TypeCat.Fun.toFun (ConcreteCategory.hom (FC := TypeCat.Fun) (f.app (op ⦋n⦌))) x
have h : Eq (α := Y _⦋n⦌)
  ((ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app ((Equiv.symm yonedaEquiv : (a : X _⦋n⦌) → Δ[n] ⟶ X) x ▷ Δ[1] ≫ H.h) (op ⦋n⦌)) :
        (a : (Δ[n] ⊗ Δ[1]) _⦋n⦌) → Y _⦋n⦌)
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (δ (tensorObj (𝒞 := Functor.category) Δ[n] Δ[1]) (Fin.last (n + 1))) :
            (a : (Δ[n] ⊗ Δ[1]) _⦋n + 1⦌) → (Δ[n] ⊗ Δ[1]) _⦋n⦌)
          (↑((prodStdSimplex.nonDegenerateEquiv₁ :
                  (a : Fin (n + 1)) → (↑((Δ[n] ⊗ Δ[1]).nonDegenerate (n + 1)) : Type u))
                (Fin.last n)) :
            (Δ[n] ⊗ Δ[1]) _⦋n + 1⦌) :
        tensorObj (𝒞 := Functor.category) Δ[n] Δ[1] _⦋n⦌) :
    Y _⦋n⦌)
  ((ConcreteCategory.hom (FC := TypeCat.Fun) ((ι₀ ≫ H.h).app (op ⦋n⦌)) : (a : X _⦋n⦌) → Y _⦋n⦌) x : Y _⦋n⦌)
Tactic: simp only [TypeCat.Fun.toFun_apply, types_comp_apply, TypeCat.hom_ofHom, TypeCat.Fun.coe_mk,
      ← SSet.δ_naturality_apply, ← H.h₀]
-- Lean 4 proof step from Mathlib.AlgebraicTopology.SimplicialSet.Homotopy (goal)
Goal:
X Y : SSet
f g : X ⟶ Y
H : Homotopy f g
n : ℕ
x : X _⦋n⦌
⊢ Eq (α := Y _⦋n⦌)
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (NatTrans.app ((Equiv.symm yonedaEquiv : (a : X _⦋n⦌) → Δ[n] ⟶ X) x ▷ Δ[1] ≫ H.h) (op ⦋n⦌)) :
          (a : (Δ[n] ⊗ Δ[1]) _⦋n⦌) → Y _⦋n⦌)
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (δ (tensorObj (𝒞 := Functor.category) Δ[n] Δ[1]) (Fin.last (n + 1))) :
              (a : (Δ[n] ⊗ Δ[1]) _⦋n + 1⦌) → (Δ[n] ⊗ Δ[1]) _⦋n⦌)
            (↑((prodStdSimplex.nonDegenerateEquiv₁ :
                    (a : Fin (n + 1)) → (↑((Δ[n] ⊗ Δ[1]).nonDegenerate (n + 1)) : Type u))
                  (Fin.last n)) :
              (Δ[n] ⊗ Δ[1]) _⦋n + 1⦌) :
          tensorObj (𝒞 := Functor.category) Δ[n] Δ[1] _⦋n⦌) :
      Y _⦋n⦌)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((ι₀ ≫ H.h).app (op ⦋n⦌)) : (a : X _⦋n⦌) → Y _⦋n⦌) x : Y _⦋n⦌)
have h : Eq (α := Y _⦋n⦌)
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (H.h.app (op ⦋n⦌)) : (a : X _⦋n⦌ ⊗ Δ[1] _⦋n⦌) → Y _⦋n⦌)
      (((ConcreteCategory.hom (FC := TypeCat.Fun)
                (X.map
                  (Quiver.Hom.op
                    (((stdSimplex.objEquiv : Δ[n] _⦋n⦌ ≃ (unop (op ⦋n⦌) ⟶ ⦋n⦌)) : (a : Δ[n] _⦋n⦌) → ⦋n⦌ ⟶ ⦋n⦌)
                      ((ConcreteCategory.hom (FC := TypeCat.Fun) (δ Δ[n] (Fin.last (n + 1))) :
                          (a : Δ[n] _⦋n + 1⦌) → Δ[n] _⦋n⦌)
                        ((Equiv.symm stdSimplex.objEquiv : (a : ⦋n + 1⦌ ⟶ ⦋n⦌) → Δ[n] _⦋n + 1⦌)
                            (SimplexCategory.σ (Fin.last n)) :
                          Δ[n] _⦋n + 1⦌))))) :
              (a : X _⦋n⦌) → X _⦋n⦌)
            x,
          (ConcreteCategory.hom (FC := TypeCat.Fun) (δ Δ[1] (Fin.last (n + 1))) : (a : Δ[1] _⦋n + 1⦌) → Δ[1] _⦋n⦌)
            (stdSimplex.objMk₁ (Fin.last n).castSucc.succ)) :
        X _⦋n⦌ × Δ[1] _⦋n⦌) :
    Y _⦋n⦌)
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (H.h.app (op ⦋n⦌)) : (a : X _⦋n⦌ ⊗ Δ[1] _⦋n⦌) → Y _⦋n⦌)
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (NatTrans.app ι₀ (op ⦋n⦌)) : (a : X _⦋n⦌) → X _⦋n⦌ ⊗ Δ[1] _⦋n⦌) x :
        tensorObj (𝒞 := types) (X _⦋n⦌) (Δ[1] _⦋n⦌)) :
    Y _⦋n⦌)
Tactic: dsimp
-- Lean 4 proof step from Mathlib.AlgebraicTopology.SimplicialSet.Homotopy (goal)
Goal:
X Y : SSet
f g : X ⟶ Y
H : Homotopy f g
n : ℕ
x : X _⦋n⦌
⊢ Eq (α := Y _⦋n⦌)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (H.h.app (op ⦋n⦌)) : (a : X _⦋n⦌ ⊗ Δ[1] _⦋n⦌) → Y _⦋n⦌)
        (((ConcreteCategory.hom (FC := TypeCat.Fun)
                  (X.map
                    (Quiver.Hom.op
                      (((stdSimplex.objEquiv : Δ[n] _⦋n⦌ ≃ (unop (op ⦋n⦌) ⟶ ⦋n⦌)) : (a : Δ[n] _⦋n⦌) → ⦋n⦌ ⟶ ⦋n⦌)
                        ((ConcreteCategory.hom (FC := TypeCat.Fun) (δ Δ[n] (Fin.last (n + 1))) :
                            (a : Δ[n] _⦋n + 1⦌) → Δ[n] _⦋n⦌)
                          ((Equiv.symm stdSimplex.objEquiv : (a : ⦋n + 1⦌ ⟶ ⦋n⦌) → Δ[n] _⦋n + 1⦌)
                              (SimplexCategory.σ (Fin.last n)) :
                            Δ[n] _⦋n + 1⦌))))) :
                (a : X _⦋n⦌) → X _⦋n⦌)
              x,
            (ConcreteCategory.hom (FC := TypeCat.Fun) (δ Δ[1] (Fin.last (n + 1))) : (a : Δ[1] _⦋n + 1⦌) → Δ[1] _⦋n⦌)
              (stdSimplex.objMk₁ (Fin.last n).castSucc.succ)) :
          X _⦋n⦌ × Δ[1] _⦋n⦌) :
      Y _⦋n⦌)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (H.h.app (op ⦋n⦌)) : (a : X _⦋n⦌ ⊗ Δ[1] _⦋n⦌) → Y _⦋n⦌)
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (NatTrans.app ι₀ (op ⦋n⦌)) : (a : X _⦋n⦌) → X _⦋n⦌ ⊗ Δ[1] _⦋n⦌) x :
          tensorObj (𝒞 := types) (X _⦋n⦌) (Δ[1] _⦋n⦌)) :
      Y _⦋n⦌)
have h : Eq (α := X _⦋n⦌ × Δ[1] _⦋n⦌)
  (((ConcreteCategory.hom (FC := TypeCat.Fun)
            (X.map
              (Quiver.Hom.op
                (((stdSimplex.objEquiv : Δ[n] _⦋n⦌ ≃ (unop (op ⦋n⦌) ⟶ ⦋n⦌)) : (a : Δ[n] _⦋n⦌) → ⦋n⦌ ⟶ ⦋n⦌)
                  ((ConcreteCategory.hom (FC := TypeCat.Fun) (δ Δ[n] (Fin.last (n + 1))) :
                      (a : Δ[n] _⦋n + 1⦌) → Δ[n] _⦋n⦌)
                    ((Equiv.symm stdSimplex.objEquiv : (a : ⦋n + 1⦌ ⟶ ⦋n⦌) → Δ[n] _⦋n + 1⦌)
                        (SimplexCategory.σ (Fin.last n)) :
                      Δ[n] _⦋n + 1⦌))))) :
          (a : X _⦋n⦌) → X _⦋n⦌)
        x,
      (ConcreteCategory.hom (FC := TypeCat.Fun) (δ Δ[1] (Fin.last (n + 1))) : (a : Δ[1] _⦋n + 1⦌) → Δ[1] _⦋n⦌)
        (stdSimplex.objMk₁ (Fin.last n).castSucc.succ)) :
    X _⦋n⦌ × Δ[1] _⦋n⦌)
  ((ConcreteCategory.hom (FC := TypeCat.Fun) (NatTrans.app ι₀ (op ⦋n⦌)) : (a : X _⦋n⦌) → X _⦋n⦌ ⊗ Δ[1] _⦋n⦌) x :
    tensorObj (𝒞 := types) (X _⦋n⦌) (Δ[1] _⦋n⦌))
Tactic: apply congr_arg
-- Lean 4 proof step from Mathlib.AlgebraicTopology.SimplicialSet.Homotopy (goal)
Goal:
X Y : SSet
f g : X ⟶ Y
H : Homotopy f g
n : ℕ
x : X _⦋n⦌
⊢ Eq (α := X _⦋n⦌ × Δ[1] _⦋n⦌)
    (((ConcreteCategory.hom (FC := TypeCat.Fun)
              (X.map
                (Quiver.Hom.op
                  (((stdSimplex.objEquiv : Δ[n] _⦋n⦌ ≃ (unop (op ⦋n⦌) ⟶ ⦋n⦌)) : (a : Δ[n] _⦋n⦌) → ⦋n⦌ ⟶ ⦋n⦌)
                    ((ConcreteCategory.hom (FC := TypeCat.Fun) (δ Δ[n] (Fin.last (n + 1))) :
                        (a : Δ[n] _⦋n + 1⦌) → Δ[n] _⦋n⦌)
                      ((Equiv.symm stdSimplex.objEquiv : (a : ⦋n + 1⦌ ⟶ ⦋n⦌) → Δ[n] _⦋n + 1⦌)
                          (SimplexCategory.σ (Fin.last n)) :
                        Δ[n] _⦋n + 1⦌))))) :
            (a : X _⦋n⦌) → X _⦋n⦌)
          x,
        (ConcreteCategory.hom (FC := TypeCat.Fun) (δ Δ[1] (Fin.last (n + 1))) : (a : Δ[1] _⦋n + 1⦌) → Δ[1] _⦋n⦌)
          (stdSimplex.objMk₁ (Fin.last n).castSucc.succ)) :
      X _⦋n⦌ × Δ[1] _⦋n⦌)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (NatTrans.app ι₀ (op ⦋n⦌)) : (a : X _⦋n⦌) → X _⦋n⦌ ⊗ Δ[1] _⦋n⦌) x :
      tensorObj (𝒞 := types) (X _⦋n⦌) (Δ[1] _⦋n⦌))
have h : Eq (α := X _⦋n⦌)
  (((ConcreteCategory.hom (FC := TypeCat.Fun)
              (X.map
                (Quiver.Hom.op
                  (((stdSimplex.objEquiv : Δ[n] _⦋n⦌ ≃ (unop (op ⦋n⦌) ⟶ ⦋n⦌)) : (a : Δ[n] _⦋n⦌) → ⦋n⦌ ⟶ ⦋n⦌)
                    ((ConcreteCategory.hom (FC := TypeCat.Fun) (δ Δ[n] (Fin.last (n + 1))) :
                        (a : Δ[n] _⦋n + 1⦌) → Δ[n] _⦋n⦌)
                      ((Equiv.symm stdSimplex.objEquiv : (a : ⦋n + 1⦌ ⟶ ⦋n⦌) → Δ[n] _⦋n + 1⦌)
                          (SimplexCategory.σ (Fin.last n)) :
                        Δ[n] _⦋n + 1⦌))))) :
            (a : X _⦋n⦌) → X _⦋n⦌)
          x,
        (ConcreteCategory.hom (FC := TypeCat.Fun) (δ Δ[1] (Fin.last (n + 1))) : (a : Δ[1] _⦋n + 1⦌) → Δ[1] _⦋n⦌)
          (stdSimplex.objMk₁ (Fin.last n).castSucc.succ)).1 :
    X _⦋n⦌)
  (((ConcreteCategory.hom (FC := TypeCat.Fun) (NatTrans.app ι₀ (op ⦋n⦌)) : (a : X _⦋n⦌) → X _⦋n⦌ ⊗ Δ[1] _⦋n⦌) x).1 :
    X _⦋n⦌)
Tactic: ext k
-- Lean 4 proof step from Mathlib.AlgebraicTopology.SimplicialSet.Homotopy (goal)
Goal:
X Y : SSet
f g : X ⟶ Y
H : Homotopy f g
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : i ≤ j.castSucc
⊢ (↾fun (x : X _⦋n + 1⦌) ↦
        (ConcreteCategory.hom (FC := TypeCat.Fun)
              (NatTrans.app ((Equiv.symm yonedaEquiv : (a : X _⦋n + 1⦌) → Δ[n + 1] ⟶ X) x ▷ Δ[1] ≫ H.h)
                (op ⦋n + 1 + 1⦌)) :
            (a : (fun (X : Type u) ↦ X) ((Δ[n + 1] ⊗ Δ[1]) _⦋n + 1 + 1⦌)) → (fun (X : Type u) ↦ X) (Y _⦋n + 1 + 1⦌))
          (↑((prodStdSimplex.nonDegenerateEquiv₁ :
                  (a : Fin (n + 1 + 1)) → (↑((Δ[n + 1] ⊗ Δ[1]).nonDegenerate (n + 1 + 1)) : Type u))
                j.succ) :
            (Δ[n + 1] ⊗ Δ[1]) _⦋n + 1 + 1⦌)) ≫
      δ Y i.castSucc =
    δ X i ≫
      ↾fun (x : X _⦋n⦌) ↦
        (ConcreteCategory.hom (FC := TypeCat.Fun)
              (NatTrans.app ((Equiv.symm yonedaEquiv : (a : X _⦋n⦌) → Δ[n] ⟶ X) x ▷ Δ[1] ≫ H.h) (op ⦋n + 1⦌)) :
            (a : (fun (X : Type u) ↦ X) ((Δ[n] ⊗ Δ[1]) _⦋n + 1⦌)) → (fun (X : Type u) ↦ X) (Y _⦋n + 1⦌))
          (↑((prodStdSimplex.nonDegenerateEquiv₁ :
                  (a : Fin (n + 1)) → (↑((Δ[n] ⊗ Δ[1]).nonDegenerate (n + 1)) : Type u))
                j) :
            (Δ[n] ⊗ Δ[1]) _⦋n + 1⦌)
have h : ∀ (x : X _⦋n + 1⦌),
  TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        ((↾fun (x : X _⦋n + 1⦌) ↦
            (ConcreteCategory.hom (FC := TypeCat.Fun)
                  (NatTrans.app ((Equiv.symm yonedaEquiv : (a : X _⦋n + 1⦌) → Δ[n + 1] ⟶ X) x ▷ Δ[1] ≫ H.h)
                    (op ⦋n + 1 + 1⦌)) :
                (a : (fun (X : Type u) ↦ X) ((Δ[n + 1] ⊗ Δ[1]) _⦋n + 1 + 1⦌)) → (fun (X : Type u) ↦ X) (Y _⦋n + 1 + 1⦌))
              (↑((prodStdSimplex.nonDegenerateEquiv₁ :
                      (a : Fin (n + 1 + 1)) → (↑((Δ[n + 1] ⊗ Δ[1]).nonDegenerate (n + 1 + 1)) : Type u))
                    j.succ) :
                (Δ[n + 1] ⊗ Δ[1]) _⦋n + 1 + 1⦌)) ≫
          δ Y i.castSucc))
      x =
    TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (δ X i ≫
          ↾fun (x : X _⦋n⦌) ↦
            (ConcreteCategory.hom (FC := TypeCat.Fun)
                  (NatTrans.app ((Equiv.symm yonedaEquiv : (a : X _⦋n⦌) → Δ[n] ⟶ X) x ▷ Δ[1] ≫ H.h) (op ⦋n + 1⦌)) :
                (a : (fun (X : Type u) ↦ X) ((Δ[n] ⊗ Δ[1]) _⦋n + 1⦌)) → (fun (X : Type u) ↦ X) (Y _⦋n + 1⦌))
              (↑((prodStdSimplex.nonDegenerateEquiv₁ :
                      (a : Fin (n + 1)) → (↑((Δ[n] ⊗ Δ[1]).nonDegenerate (n + 1)) : Type u))
                    j) :
                (Δ[n] ⊗ Δ[1]) _⦋n + 1⦌)))
      x
Tactic: ext x
-- Lean 4 proof step from Mathlib.AlgebraicTopology.SimplicialSet.Homotopy (goal)
Goal:
X Y : SSet
f g : X ⟶ Y
H : Homotopy f g
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : i ≤ j.castSucc
x : X _⦋n + 1⦌
⊢ TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        ((↾fun (x : X _⦋n + 1⦌) ↦
            (ConcreteCategory.hom (FC := TypeCat.Fun)
                  (NatTrans.app ((Equiv.symm yonedaEquiv : (a : X _⦋n + 1⦌) → Δ[n + 1] ⟶ X) x ▷ Δ[1] ≫ H.h)
                    (op ⦋n + 1 + 1⦌)) :
                (a : (fun (X : Type u) ↦ X) ((Δ[n + 1] ⊗ Δ[1]) _⦋n + 1 + 1⦌)) → (fun (X : Type u) ↦ X) (Y _⦋n + 1 + 1⦌))
              (↑((prodStdSimplex.nonDegenerateEquiv₁ :
                      (a : Fin (n + 1 + 1)) → (↑((Δ[n + 1] ⊗ Δ[1]).nonDegenerate (n + 1 + 1)) : Type u))
                    j.succ) :
                (Δ[n + 1] ⊗ Δ[1]) _⦋n + 1 + 1⦌)) ≫
          δ Y i.castSucc))
      x =
    TypeCat.Fun.toFun
      (ConcreteCategory.hom (FC := TypeCat.Fun)
        (δ X i ≫
          ↾fun (x : X _⦋n⦌) ↦
            (ConcreteCategory.hom (FC := TypeCat.Fun)
                  (NatTrans.app ((Equiv.symm yonedaEquiv : (a : X _⦋n⦌) → Δ[n] ⟶ X) x ▷ Δ[1] ≫ H.h) (op ⦋n + 1⦌)) :
                (a : (fun (X : Type u) ↦ X) ((Δ[n] ⊗ Δ[1]) _⦋n + 1⦌)) → (fun (X : Type u) ↦ X) (Y _⦋n + 1⦌))
              (↑((prodStdSimplex.nonDegenerateEquiv₁ :
                      (a : Fin (n + 1)) → (↑((Δ[n] ⊗ Δ[1]).nonDegenerate (n + 1)) : Type u))
                    j) :
                (Δ[n] ⊗ Δ[1]) _⦋n + 1⦌)))
      x
have h : Eq (α := Y _⦋n + 1⦌)
  ((ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app ((Equiv.symm yonedaEquiv : (a : X _⦋n + 1⦌) → Δ[n + 1] ⟶ X) x ▷ Δ[1] ≫ H.h) (op ⦋n + 1⦌)) :
        (a : (Δ[n + 1] ⊗ Δ[1]) _⦋n + 1⦌) → Y _⦋n + 1⦌)
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (δ (tensorObj (𝒞 := Functor.category) Δ[n + 1] Δ[1]) i.castSucc) :
            (a : (Δ[n + 1] ⊗ Δ[1]) _⦋n + 1 + 1⦌) → (Δ[n + 1] ⊗ Δ[1]) _⦋n + 1⦌)
          (↑((prodStdSimplex.nonDegenerateEquiv₁ :
                  (a : Fin (n + 1 + 1)) → (↑((Δ[n + 1] ⊗ Δ[1]).nonDegenerate (n + 1 + 1)) : Type u))
                j.succ) :
            (Δ[n + 1] ⊗ Δ[1]) _⦋n + 1 + 1⦌) :
        tensorObj (𝒞 := Functor.category) Δ[n + 1] Δ[1] _⦋n + 1⦌) :
    Y _⦋n + 1⦌)
  ((ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app
            ((Equiv.symm yonedaEquiv : (a : X _⦋n⦌) → Δ[n] ⟶ X)
                  ((ConcreteCategory.hom (FC := TypeCat.Fun) (δ X i) : (a : X _⦋n + 1⦌) → X _⦋n⦌) x : X _⦋n⦌) ▷
                Δ[1] ≫
              H.h)
            (op ⦋n + 1⦌)) :
        (a : (Δ[n] ⊗ Δ[1]) _⦋n + 1⦌) → Y _⦋n + 1⦌)
      (↑((prodStdSimplex.nonDegenerateEquiv₁ : (a : Fin (n + 1)) → (↑((Δ[n] ⊗ Δ[1]).nonDegenerate (n + 1)) : Type u))
            j) :
        (Δ[n] ⊗ Δ[1]) _⦋n + 1⦌) :
    Y _⦋n + 1⦌)
Tactic: simp only [TypeCat.Fun.toFun_apply, types_comp_apply, TypeCat.hom_ofHom, TypeCat.Fun.coe_mk,
      ← SSet.δ_naturality_apply]
-- Lean 4 proof step from Mathlib.AlgebraicTopology.SimplicialSet.Homotopy (goal)
Goal:
X Y : SSet
f g : X ⟶ Y
H : Homotopy f g
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : i ≤ j.castSucc
x : X _⦋n + 1⦌
⊢ Eq (α := X _⦋n + 1⦌ × Δ[1] _⦋n + 1⦌)
    (((ConcreteCategory.hom (FC := TypeCat.Fun)
              (X.map
                (Quiver.Hom.op
                  (((stdSimplex.objEquiv : Δ[n + 1] _⦋n + 1⦌ ≃ (unop (op ⦋n + 1⦌) ⟶ ⦋n + 1⦌)) :
                      (a : Δ[n + 1] _⦋n + 1⦌) → ⦋n + 1⦌ ⟶ ⦋n + 1⦌)
                    ((ConcreteCategory.hom (FC := TypeCat.Fun) (δ Δ[n + 1] i.castSucc) :
                        (a : Δ[n + 1] _⦋n + 1 + 1⦌) → Δ[n + 1] _⦋n + 1⦌)
                      ((Equiv.symm stdSimplex.objEquiv : (a : ⦋n + 1 + 1⦌ ⟶ ⦋n + 1⦌) → Δ[n + 1] _⦋n + 1 + 1⦌)
                          (SimplexCategory.σ j.succ) :
                        Δ[n + 1] _⦋n + 1 + 1⦌))))) :
            (a : X _⦋n + 1⦌) → X _⦋n + 1⦌)
          x,
        (ConcreteCategory.hom (FC := TypeCat.Fun) (δ Δ[1] i.castSucc) : (a : Δ[1] _⦋n + 1 + 1⦌) → Δ[1] _⦋n + 1⦌)
          (stdSimplex.objMk₁ j.castSucc.succ.succ)) :
      X _⦋n + 1⦌ × Δ[1] _⦋n + 1⦌)
    (((ConcreteCategory.hom (FC := TypeCat.Fun)
              (NatTrans.app
                ((Equiv.symm yonedaEquiv : (a : X _⦋n⦌) → Δ[n] ⟶ X)
                  ((ConcreteCategory.hom (FC := TypeCat.Fun) (δ X i) : (a : X _⦋n + 1⦌) → X _⦋n⦌) x : X _⦋n⦌))
                (op ⦋n + 1⦌)) :
            (a : Δ[n] _⦋n + 1⦌) → X _⦋n + 1⦌)
          ((Equiv.symm stdSimplex.objEquiv : (a : ⦋n + 1⦌ ⟶ ⦋n⦌) → Δ[n] _⦋n + 1⦌) (SimplexCategory.σ j) :
            Δ[n] _⦋n + 1⦌),
        stdSimplex.objMk₁ j.castSucc.succ) :
      X _⦋n + 1⦌ × Δ[1] _⦋n + 1⦌)
have h : Eq (α := X _⦋n + 1⦌)
  (((ConcreteCategory.hom (FC := TypeCat.Fun)
              (X.map
                (Quiver.Hom.op
                  (((stdSimplex.objEquiv : Δ[n + 1] _⦋n + 1⦌ ≃ (unop (op ⦋n + 1⦌) ⟶ ⦋n + 1⦌)) :
                      (a : Δ[n + 1] _⦋n + 1⦌) → ⦋n + 1⦌ ⟶ ⦋n + 1⦌)
                    ((ConcreteCategory.hom (FC := TypeCat.Fun) (δ Δ[n + 1] i.castSucc) :
                        (a : Δ[n + 1] _⦋n + 1 + 1⦌) → Δ[n + 1] _⦋n + 1⦌)
                      ((Equiv.symm stdSimplex.objEquiv : (a : ⦋n + 1 + 1⦌ ⟶ ⦋n + 1⦌) → Δ[n + 1] _⦋n + 1 + 1⦌)
                          (SimplexCategory.σ j.succ) :
                        Δ[n + 1] _⦋n + 1 + 1⦌))))) :
            (a : X _⦋n + 1⦌) → X _⦋n + 1⦌)
          x,
        (ConcreteCategory.hom (FC := TypeCat.Fun) (δ Δ[1] i.castSucc) : (a : Δ[1] _⦋n + 1 + 1⦌) → Δ[1] _⦋n + 1⦌)
          (stdSimplex.objMk₁ j.castSucc.succ.succ)).1 :
    X _⦋n + 1⦌)
  (((ConcreteCategory.hom (FC := TypeCat.Fun)
              (NatTrans.app
                ((Equiv.symm yonedaEquiv : (a : X _⦋n⦌) → Δ[n] ⟶ X)
                  ((ConcreteCategory.hom (FC := TypeCat.Fun) (δ X i) : (a : X _⦋n + 1⦌) → X _⦋n⦌) x : X _⦋n⦌))
                (op ⦋n + 1⦌)) :
            (a : Δ[n] _⦋n + 1⦌) → X _⦋n + 1⦌)
          ((Equiv.symm stdSimplex.objEquiv : (a : ⦋n + 1⦌ ⟶ ⦋n⦌) → Δ[n] _⦋n + 1⦌) (SimplexCategory.σ j) :
            Δ[n] _⦋n + 1⦌),
        stdSimplex.objMk₁ j.castSucc.succ).1 :
    X _⦋n + 1⦌)
Tactic: ext k : 2
-- Lean 4 proof step from Mathlib.AlgebraicTopology.SimplicialSet.Homotopy (goal)
Goal:
X Y : SSet
f g : X ⟶ Y
H : Homotopy f g
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : i ≤ j.castSucc
x : X _⦋n + 1⦌
k : Fin (n + 1 + 1)
⊢ Eq (α := Fin (HAdd.hAdd (α := ℕ) 1 1))
    (((((ConcreteCategory.hom (FC := TypeCat.Fun)
                      (X.map
                        (Quiver.Hom.op
                          (((stdSimplex.objEquiv : Δ[n + 1] _⦋n + 1⦌ ≃ (unop (op ⦋n + 1⦌) ⟶ ⦋n + 1⦌)) :
                              (a : Δ[n + 1] _⦋n + 1⦌) → ⦋n + 1⦌ ⟶ ⦋n + 1⦌)
                            ((ConcreteCategory.hom (FC := TypeCat.Fun) (δ Δ[n + 1] i.castSucc) :
                                (a : Δ[n + 1] _⦋n + 1 + 1⦌) → Δ[n + 1] _⦋n + 1⦌)
                              ((Equiv.symm stdSimplex.objEquiv : (a : ⦋n + 1 + 1⦌ ⟶ ⦋n + 1⦌) → Δ[n + 1] _⦋n + 1 + 1⦌)
                                  (SimplexCategory.σ j.succ) :
                                Δ[n + 1] _⦋n + 1 + 1⦌))))) :
                    (a : X _⦋n + 1⦌) → X _⦋n + 1⦌)
                  x,
                (ConcreteCategory.hom (FC := TypeCat.Fun) (δ Δ[1] i.castSucc) : (a : Δ[1] _⦋n + 1 + 1⦌) → Δ[1] _⦋n + 1⦌)
                  (stdSimplex.objMk₁ j.castSucc.succ.succ)).2 :
            Δ[1] _⦋n + 1⦌) :
          (a : Fin (n + 1 + 1)) → Fin (1 + 1))
        k :
      Fin (HAdd.hAdd (α := ℕ) 1 1))
    (((((ConcreteCategory.hom (FC := TypeCat.Fun)
                      (NatTrans.app
                        ((Equiv.symm yonedaEquiv : (a : X _⦋n⦌) → Δ[n] ⟶ X)
                          ((ConcreteCategory.hom (FC := TypeCat.Fun) (δ X i) : (a : X _⦋n + 1⦌) → X _⦋n⦌) x : X _⦋n⦌))
                        (op ⦋n + 1⦌)) :
                    (a : Δ[n] _⦋n + 1⦌) → X _⦋n + 1⦌)
                  ((Equiv.symm stdSimplex.objEquiv : (a : ⦋n + 1⦌ ⟶ ⦋n⦌) → Δ[n] _⦋n + 1⦌) (SimplexCategory.σ j) :
                    Δ[n] _⦋n + 1⦌),
                stdSimplex.objMk₁ j.castSucc.succ) :
              X _⦋n + 1⦌ × Δ[1] _⦋n + 1⦌).2 :
          (a : Fin (n + 1 + 1)) → Fin (1 + 1))
        k :
      Fin (HAdd.hAdd (α := ℕ) 1 1))
have h : i.castSucc.castSucc < j.castSucc.succ.succ
Tactic: rw [stdSimplex.δ_objMk₁_of_lt, Fin.pred_succ]
-- Lean 4 proof step from Mathlib.AlgebraicTopology.SimplicialSet.Homotopy (goal)
Goal:
X Y : SSet
f g : X ⟶ Y
H : Homotopy f g
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : i ≤ j.castSucc
x : X _⦋n + 1⦌
k : Fin (n + 1 + 1)
⊢ i.castSucc.castSucc < j.castSucc.succ.succ
have h : i.castSucc ≤ j.succ.castSucc
Tactic: rw [Fin.castSucc_lt_succ_iff, ← Fin.castSucc_succ]
-- Lean 4 proof step from Mathlib.AlgebraicTopology.SimplicialSet.Homotopy (goal)
Goal:
X Y : SSet
f g : X ⟶ Y
H : Homotopy f g
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : i ≤ j.castSucc
x : X _⦋n + 1⦌
k : Fin (n + 1 + 1)
⊢ i.castSucc ≤ j.succ.castSucc
have h : i ≤ j.succ
Tactic: simp only [Fin.castSucc_le_castSucc_iff]