Definition nxtprime n := proj1_sig (prime_above n).
Fact nxtprime_spec1 n : n < nxtprime n.

(* Specifications: *)
Lemma nthprime_inj n m : nthprime n = nthprime m -> n = m.
Record primestream :=
{
str :> nat -> nat;
str_inj : forall n m, str n = str m -> n = m;
str_prime : forall n, prime (str n);
}.