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Proofgold Proof

pf
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.
Let x4 of type ι be given.
Assume H0: SNo x0.
Assume H1: SNo x1.
Assume H2: SNo x2.
Assume H3: SNo x3.
Assume H4: SNo x4.
set y5 to be add_SNo x0 (add_SNo x1 (add_SNo x2 (add_SNo x3 x4)))
set y6 to be add_SNo x4 (add_SNo y5 (add_SNo x1 (add_SNo x2 x3)))
Claim L5: ∀ x7 : ι → ο . x7 y6 ⟶ x7 y5
Let x7 of type ι → ο be given.
Assume H5: x7 (add_SNo y5 (add_SNo y6 (add_SNo x2 (add_SNo x3 x4)))).
Apply add_SNo_rotate_5_1 with x2, x3, x4, y5, y6, λ x8 . x7 leaving 6 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.
The subproof is completed by applying H4.
Apply add_SNo_rotate_5_1 with y6, x2, x3, x4, y5, λ x8 . x7 leaving 6 subgoals.
The subproof is completed by applying H4.
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.
The subproof is completed by applying H5.
Let x7 of type ι → ι → ο be given.
Apply L5 with λ x8 . x7 x8 y6 ⟶ x7 y6 x8.
Assume H6: x7 y6 y6.
The subproof is completed by applying H6.
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