Corollary alpha_nat_odd n : (rem (alpha_nat (S n)) 2 = 1 \/ rem (alpha_nat n) 2 = 1)%nat.

(* Specifications: *)
Theorem find_odd_alpha u : exists n, (u <= alpha_nat (S n) /\ rem (alpha_nat (S n)) 2 = 1)%nat.
Theorem find_odd_alpha' u : exists n, (u <= alpha_nat n /\ rem (alpha_nat n) 2 = 1)%nat.