intros.
unfold sqrd.
replace (S n * S n) with (n * n + 2 * n + 1) by ring.
replace (div2 (n * n) + n) with (div2 (n * n + 2 * n)).
apply div2_preserves_le...
rewrite div2_x_plus_2y...
Qed.

Lemma div2_le_div2_inv x y: div2 x <= div2 y -> x <= S y.
Proof with auto with arith.