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

Theorem div2_bound : forall k n,
2 * k <= n
-> k <= div2 n.