Let x0 of type ι → ι → ι → ι be given.
Let x1 of type ι → ι → ι → ι be given.
Let x2 of type ι → ι → ι → ι be given.
Let x3 of type ι → ι → ι → ο be given.
Let x4 of type ι → ι → ι → ο be given.
Let x5 of type ι → ι → ι → ο be given.
Assume H0:
∀ x6 x7 x8 . x3 x6 x7 x8 ⟶ x0 x6 x7 x8 = add_SNo x6 (mul_SNo 2 x8).
Assume H1: ∀ x6 x7 x8 . x3 x6 x7 x8 ⟶ x1 x6 x7 x8 = x8.
Assume H4: ∀ x6 x7 x8 . x4 x6 x7 x8 ⟶ x1 x6 x7 x8 = x7.
Assume H8: ∀ x6 x7 x8 . x5 x6 x7 x8 ⟶ x2 x6 x7 x8 = x7.
Assume H9:
∀ x6 x7 x8 . add_SNo x6 x8 ∈ x7 ⟶ x3 x6 x7 x8.
Assume H10:
∀ x6 x7 x8 . x3 x6 x7 x8 ⟶ add_SNo x6 x8 ∈ x7.
Assume H11:
∀ x6 x7 x8 . x7 ∈ add_SNo x6 x8 ⟶ x6 ∈ mul_SNo 2 x7 ⟶ x4 x6 x7 x8.
Assume H12:
∀ x6 x7 x8 . x4 x6 x7 x8 ⟶ x7 ∈ add_SNo x6 x8.
Assume H13:
∀ x6 x7 x8 . x4 x6 x7 x8 ⟶ x6 ∈ mul_SNo 2 x7.
Assume H14:
∀ x6 x7 x8 . x7 ∈ add_SNo x6 x8 ⟶ mul_SNo 2 x7 ∈ x6 ⟶ x5 x6 x7 x8.
Assume H15:
∀ x6 x7 x8 . x5 x6 x7 x8 ⟶ x7 ∈ add_SNo x6 x8.
Assume H16:
∀ x6 x7 x8 . x5 x6 x7 x8 ⟶ mul_SNo 2 x7 ∈ x6.
Let x6 of type ι be given.
Let x7 of type ι be given.
Assume H25:
∀ x8 . x8 ∈ ... ⟶ lam 3 (λ x9 . If_i (x9 = 0) (ap x8 0) (If_i (x9 = 1) (ap x8 1) (ap x8 2))) = x8.