induction n as [n] using strong; simpl; intuition.
destruct n as [|n]; simpl in *; intuition.
destruct n as [|n]; simpl in *; intuition.
discriminate.
f_equal.
replace (div2 n + S (div2 n + 0)) with (S (div2 n + (div2 n + 0))); auto.
Qed.

Theorem drop_mod2 : forall n k,
2 * k <= n
-> mod2 (n - 2 * k) = mod2 n.