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

Theorem div2_plus_2 : forall n k, div2 (n + 2 * k) = div2 n + k.