proposition: Theorem update_primes_all_prime_in_first : forall k l l' b, all_prime_in_first k l -> update_primes k l = (l', b) -> all_prime_in_first k l'.
proof: Proof. intros k l l' b Hap Hup p Hplek Hpr. elim (Hap p Hplek Hpr); intros l'1 (l'2, (n', Heq)). apply (update_primes_same_first k l l' b Hup l'1 l'2 p n' Heq). Qed.
imports: List Arith