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

(* Specifications: *)
Lemma two_times_div2_bound: forall n, 2 * Nat.div2 n <= n.
Lemma div2_compat_lt_l: forall a b, b < 2 * a -> Nat.div2 b < a.
Arguments div2_compat_lt_l {_} {_} _.