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Proofgold Proof

pf
Apply int_SNo_cases with λ x0 . SNoLe 0 x0 ⟶ nat_p x0 leaving 2 subgoals.
Let x0 of type ι be given.
Assume H0: x0 ∈ omega.
Assume H1: SNoLe 0 x0.
Apply omega_nat_p with x0.
The subproof is completed by applying H0.
Let x0 of type ι be given.
Assume H0: x0 ∈ omega.
Assume H1: SNoLe 0 (minus_SNo x0).
Claim L2: x0 = 0
Apply SNoLe_antisym with x0, 0 leaving 4 subgoals.
Apply nat_p_SNo with x0.
Apply omega_nat_p with x0.
The subproof is completed by applying H0.
The subproof is completed by applying SNo_0.
Apply minus_SNo_invol with x0, λ x1 x2 . SNoLe x1 0 leaving 2 subgoals.
Apply nat_p_SNo with x0.
Apply omega_nat_p with x0.
The subproof is completed by applying H0.
Apply minus_SNo_0 with λ x1 x2 . SNoLe (minus_SNo (minus_SNo x0)) x1.
Apply minus_SNo_Le_contra with 0, minus_SNo x0 leaving 3 subgoals.
The subproof is completed by applying SNo_0.
Apply SNo_minus_SNo with x0.
Apply nat_p_SNo with x0.
Apply omega_nat_p with x0.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply unknownprop_72fc13f59561a486e7f04b4e6ad6c40ec1d48eeac6e68c47cb50fa618c19e933 with x0.
Apply omega_nat_p with x0.
The subproof is completed by applying H0.
Apply L2 with λ x1 x2 . nat_p (minus_SNo x2).
Apply minus_SNo_0 with λ x1 x2 . nat_p x2.
The subproof is completed by applying nat_0.
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