Let x0 of type ι → ι → ι be given.

Assume H0: ∀ x1 x2 . SNo x1 ⟶ SNo x2 ⟶ SNo (x0 x1 x2).

Assume H1: ∀ x1 x2 x3 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ x0 x1 (add_SNo x2 x3) = add_SNo (x0 x1 x2) (x0 x1 x3).

Assume H2: ∀ x1 x2 x3 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ x0 (add_SNo x1 x2) x3 = add_SNo (x0 x1 x3) (x0 x2 x3).

Assume H3: ∀ x1 x2 . SNo x1 ⟶ SNo x2 ⟶ ∀ x3 . x3 ∈ SNoL (x0 x1 x2) ⟶ ∀ x4 : ο . (∀ x5 . x5 ∈ SNoL x1 ⟶ ∀ x6 . x6 ∈ SNoL x2 ⟶ SNoLe (add_SNo x3 (x0 x5 x6)) (add_SNo (x0 x5 x2) (x0 x1 x6)) ⟶ x4) ⟶ (∀ x5 . x5 ∈ SNoR x1 ⟶ ∀ x6 . x6 ∈ SNoR x2 ⟶ SNoLe (add_SNo x3 (x0 x5 x6)) (add_SNo (x0 x5 x2) (x0 x1 x6)) ⟶ x4) ⟶ x4.

Assume H4: ∀ x1 x2 . SNo x1 ⟶ SNo x2 ⟶ ∀ x3 . x3 ∈ SNoR (x0 x1 x2) ⟶ ∀ x4 : ο . (∀ x5 . x5 ∈ SNoL x1 ⟶ ∀ x6 . x6 ∈ SNoR x2 ⟶ SNoLe (add_SNo (x0 x5 x2) (x0 x1 x6)) (add_SNo x3 (x0 x5 x6)) ⟶ x4) ⟶ (∀ x5 . x5 ∈ SNoR x1 ⟶ ∀ x6 . x6 ∈ SNoL x2 ⟶ SNoLe (add_SNo (x0 x5 x2) (x0 x1 x6)) (add_SNo x3 (x0 x5 x6)) ⟶ x4) ⟶ x4.

Assume H5: ∀ x1 x2 x3 x4 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ SNoLt x3 x1 ⟶ SNoLt x4 x2 ⟶ SNoLt (add_SNo (x0 x3 x2) (x0 x1 x4)) (add_SNo (x0 x1 x2) (x0 x3 x4)).

Assume H6: ∀ x1 x2 x3 x4 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ SNoLe x3 x1 ⟶ SNoLe x4 x2 ⟶ SNoLe (add_SNo (x0 x3 x2) (x0 x1 x4)) (add_SNo (x0 x1 x2) (x0 x3 x4)).

Let x1 of type ι be given.

Let x2 of type ι be given.

Let x3 of type ι be given.

Assume H7: SNo x1.

Assume H8: SNo x2.

Assume H9: SNo x3.

Assume H10: ∀ x4 . ... ⟶ x0 x4 (x0 x2 x3) = x0 (x0 x4 x2) ....

...

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Proofgold Proof