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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: SNo x0.
Assume H1: SNo x1.
set y2 to be add_SNo (add_SNo x0 x1) (minus_SNo x1)
set y3 to be x1
Claim L2: ∀ x4 : ι → ο . x4 y3x4 y2
Let x4 of type ιο be given.
Assume H2: x4 y2.
set y5 to be λ x5 . x4
Apply add_SNo_assoc with y2, y3, minus_SNo y3, λ x6 x7 . y5 x7 x6 leaving 4 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply SNo_minus_SNo with y3.
The subproof is completed by applying H1.
set y6 to be add_SNo y3 (add_SNo x4 (minus_SNo x4))
set y7 to be add_SNo x4 0
Claim L3: ∀ x8 : ι → ο . x8 y7x8 y6
Let x8 of type ιο be given.
Assume H3: x8 (add_SNo y5 0).
set y9 to be λ x9 . x8
Apply add_SNo_minus_SNo_rinv with y6, λ x10 x11 . y9 (add_SNo y5 x10) (add_SNo y5 x11) leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
set y8 to be λ x8 . y7
Apply L3 with λ x9 . y8 x9 y7y8 y7 x9 leaving 2 subgoals.
Assume H4: y8 y7 y7.
The subproof is completed by applying H4.
Apply add_SNo_0R with y6, λ x9 . y8 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying L3.
Let x4 of type ιιο be given.
Apply L2 with λ x5 . x4 x5 y3x4 y3 x5.
Assume H3: x4 y3 y3.
The subproof is completed by applying H3.