intros ? n H; case_eq (mod2 n); intro Heq.
rewrite (div2_odd _ Heq) in H.
omega.
rewrite (div2_even _ Heq) in H.
omega.
Qed.

Lemma div2_compat_lt_l: forall a b, b < 2 * a -> Nat.div2 b < a.
Arguments div2_compat_lt_l {_} {_} _.