induction n as [n] using strong; intros.
do 2 (destruct n; simpl in *; repeat rewrite untimes2 in *; intuition).
destruct k; simpl in *; intuition.
destruct k; simpl; intuition.
rewrite <- plus_n_Sm.
repeat rewrite untimes2 in *.
simpl; auto.
apply H; omega.
Qed.

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