Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι be given.

Assume H0: SNo x0.

Assume H1: SNo x1.

Assume H2: SNo x2.

set y3 to be mul_SNo x0 (add_SNo x1 x2)

set y4 to be add_SNo (mul_SNo x1 x2) (mul_SNo x1 y3)

Claim L3: ∀ x5 : ι → ο . x5 y4 ⟶ x5 y3

Let x5 of type ι → ο be given.

Assume H3: x5 (add_SNo (mul_SNo x2 y3) (mul_SNo x2 y4)).

Apply mul_SNo_com with x2, add_SNo y3 y4, λ x6 . x5 leaving 3 subgoals.

The subproof is completed by applying H0.

Apply SNo_add_SNo with y3, y4 leaving 2 subgoals.

The subproof is completed by applying H1.

The subproof is completed by applying H2.

Apply mul_SNo_distrR with y3, y4, x2, λ x6 . x5 leaving 4 subgoals.

The subproof is completed by applying H1.

The subproof is completed by applying H2.

The subproof is completed by applying H0.

set y6 to be add_SNo (mul_SNo y3 x2) (mul_SNo y4 x2)

set y7 to be add_SNo (mul_SNo y3 y4) (mul_SNo y3 x5)

Claim L4: ∀ x8 : ι → ο . x8 y7 ⟶ x8 y6

Let x8 of type ι → ο be given.

Assume H4: x8 (add_SNo (mul_SNo y4 x5) (mul_SNo y4 y6)).

set y9 to be λ x9 . x8

Apply mul_SNo_com with x5, y4, λ x10 x11 . y9 (add_SNo x10 (mul_SNo y6 y4)) (add_SNo x11 (mul_SNo y6 y4)) leaving 3 subgoals.

The subproof is completed by applying H1.

The subproof is completed by applying H0.

set y10 to be λ x10 . y9

Apply mul_SNo_com with y7, x5, λ x11 x12 . y10 (add_SNo (mul_SNo x5 y6) x11) (add_SNo (mul_SNo x5 y6) x12) leaving 3 subgoals.

The subproof is completed by applying H2.

The subproof is completed by applying H0.

The subproof is completed by applying H4.

set y8 to be λ x8 . y7

Apply L4 with λ x9 . y8 x9 y7 ⟶ y8 y7 x9 leaving 2 subgoals.

Assume H5: y8 y7 y7.

The subproof is completed by applying H5.

The subproof is completed by applying L4.

Let x5 of type ι → ι → ο be given.

Apply L3 with λ x6 . x5 x6 y4 ⟶ x5 y4 x6.

Assume H4: x5 y4 y4.

The subproof is completed by applying H4.

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Proofgold Proof