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Proofgold Proof

pf
Let x0 of type ι be given.
Assume H0: x0omega.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply unknownprop_ee6ce5840847ce6b33fd2ab8635c6f4d1cc75cd1ba9d2f744ef438b81a42e563 with λ x3 x4 . 0, λ x3 . λ x4 : ι → ι → ι . λ x5 x6 . b38a5.. (ap x5 0) (ap x6 0) (x4 (stream_rest x5) (stream_rest x6)), x0, λ x3 x4 : ι → ι → ι . x4 x1 x2 = b38a5.. (ap x1 0) (ap x2 0) (df9be.. x0 (stream_rest x1) (stream_rest x2)) leaving 2 subgoals.
Apply omega_nat_p with x0.
The subproof is completed by applying H0.
Let x3 of type ιιο be given.
Assume H1: x3 ((λ x4 . λ x5 : ι → ι → ι . λ x6 x7 . b38a5.. (ap x6 0) (ap x7 0) (x5 (stream_rest x6) (stream_rest x7))) x0 (nat_primrec_iii (λ x4 x5 . 0) (λ x4 . λ x5 : ι → ι → ι . λ x6 x7 . b38a5.. (ap x6 0) (ap x7 0) (x5 (stream_rest x6) (stream_rest x7))) x0) x1 x2) (b38a5.. (ap x1 0) (ap x2 0) (df9be.. x0 (stream_rest x1) (stream_rest x2))).
The subproof is completed by applying H1.