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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: nat_p x0.
Apply nat_ind with λ x1 . add_nat x0 x1 = add_nat x1 x0 leaving 2 subgoals.
Apply add_nat_0L with x0, λ x1 x2 . add_nat x0 0 = x2 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying add_nat_0R with x0.
Let x1 of type ι be given.
Assume H1: nat_p x1.
Assume H2: add_nat x0 x1 = add_nat x1 x0.
Apply add_nat_SL with x1, x0, λ x2 x3 . add_nat x0 (ordsucc x1) = x3 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H0.
Apply H2 with λ x2 x3 . add_nat x0 (ordsucc x1) = ordsucc x2.
Apply add_nat_SR with x0, x1.
The subproof is completed by applying H1.