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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.
Assume H0: SNo x0.
Assume H1: SNo x1.
Assume H2: SNo x2.
Assume H3: SNo x3.
Apply minus_add_SNo_distr_3 with x0, x1, add_SNo x2 x3, λ x4 x5 . x5 = add_SNo (minus_SNo x0) (add_SNo (minus_SNo x1) (add_SNo (minus_SNo x2) (minus_SNo x3))) leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply SNo_add_SNo with x2, x3 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
set y4 to be add_SNo (minus_SNo x0) (add_SNo (minus_SNo x1) (add_SNo (minus_SNo x2) (minus_SNo x3)))
Claim L4: ∀ x5 : ι → ο . x5 y4 ⟶ x5 (add_SNo (minus_SNo x0) (add_SNo (minus_SNo x1) (minus_SNo (add_SNo x2 x3))))
Let x5 of type ι → ο be given.
set y6 to be add_SNo (minus_SNo x2) (add_SNo (minus_SNo x3) (minus_SNo y4))
Claim L4: ∀ x7 : ι → ο . x7 y6 ⟶ x7 (add_SNo (minus_SNo x2) (minus_SNo (add_SNo x3 y4)))
Let x7 of type ι → ο be given.
Apply minus_add_SNo_distr with y4, x5, λ x8 x9 . (λ x10 . x7) (add_SNo (minus_SNo x3) x8) (add_SNo (minus_SNo x3) x9) leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
set y7 to be λ x7 x8 . (λ x9 . y6) (add_SNo (minus_SNo x2) x7) (add_SNo (minus_SNo x2) x8)
Apply L4 with λ x8 . y7 x8 y6 ⟶ y7 y6 x8.
Assume H5: y7 y6 y6.
The subproof is completed by applying H5.
Let x5 of type ι → ι → ο be given.
Apply L4 with λ x6 . x5 x6 y4 ⟶ x5 y4 x6.
Assume H5: x5 y4 y4.
The subproof is completed by applying H5.
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