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 x1.

Assume H1: SNo x2.

Assume H2: SNo x3.

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

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

Claim L3: ∀ x6 : ι → ο . x6 y5 ⟶ x6 y4

Let x6 of type ι → ο be given.

Assume H3: x6 (add_SNo x2 (add_SNo y4 (add_SNo x3 y5))).

set y7 to be λ x7 . x6

Apply add_SNo_com_3_0_1 with x3, y4, y5, λ x8 x9 . y7 (add_SNo x2 x8) (add_SNo x2 x9) 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 H3.

Let x6 of type ι → ι → ο be given.

Apply L3 with λ x7 . x6 x7 y5 ⟶ x6 y5 x7.

Assume H4: x6 y5 y5.

The subproof is completed by applying H4.

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Proofgold Proof