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

Theorem div2_minus_2 : forall n k, 2 * k <= n -> div2 (n - 2 * k) = div2 n - k.