Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι be given.

Let x3 of type ι be given.

Assume H0: SNo x0.

Assume H1: SNo x1.

Assume H2: SNo x2.

Assume H3: SNo x3.

set y4 to be add_SNo x0 (add_SNo x1 (add_SNo x2 x3))

set y5 to be add_SNo (add_SNo x1 (add_SNo x2 x3)) y4

Claim L4: ∀ x6 : ι → ο . x6 y5 ⟶ x6 y4

Let x6 of type ι → ο be given.

Assume H4: x6 (add_SNo (add_SNo x2 (add_SNo x3 y4)) y5).

Apply add_SNo_assoc with x2, x3, add_SNo y4 y5, λ x7 . x6 leaving 4 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

Apply SNo_add_SNo with y4, y5 leaving 2 subgoals.

The subproof is completed by applying H2.

The subproof is completed by applying H3.

Apply add_SNo_assoc with add_SNo x2 x3, y4, y5, λ x7 . x6 leaving 4 subgoals.

Apply SNo_add_SNo with x2, x3 leaving 2 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

The subproof is completed by applying H2.

The subproof is completed by applying H3.

set y7 to be add_SNo (add_SNo (add_SNo x2 x3) y4) y5

set y8 to be add_SNo (add_SNo x3 (add_SNo y4 y5)) x6

Claim L5: ∀ x9 : ι → ο . x9 y8 ⟶ x9 y7

Let x9 of type ι → ο be given.

Assume H5: x9 (add_SNo (add_SNo y4 (add_SNo y5 x6)) y7).

set y10 to be λ x10 . x9

set y11 to be λ x11 x12 . y10 (add_SNo x11 y7) (add_SNo x12 y7)

Apply add_SNo_assoc with y4, y5, x6, λ x12 x13 . y11 x13 x12 leaving 4 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

The subproof is completed by applying H2.

The subproof is completed by applying H5.

set y9 to be λ x9 . y8

Apply L5 with λ x10 . y9 x10 y8 ⟶ y9 y8 x10 leaving 2 subgoals.

Assume H6: y9 y8 y8.

The subproof is completed by applying H6.

The subproof is completed by applying L5.

Let x6 of type ι → ι → ο be given.

Apply L4 with λ x7 . x6 x7 y5 ⟶ x6 y5 x7.

Assume H5: x6 y5 y5.

The subproof is completed by applying H5.

■

Proofgold Proof