Notation pow2 := (Nat.pow 2).Fixpoint Npow2 (n : nat) : N :=
  match n with
    | O => 1
    | S n' => 2 * Npow2 n'
  end%N.

Lemma mod2_cases: forall (n: nat), n mod 2 = 0 \/ n mod 2 = 1.