intro.
functional induction (ceil_log2_S n)...
apply le_n_S.
apply le_trans with (div2 (S _x))...
apply lt_n_Sm_le.
apply lt_div2...
Qed.

(* Specifications: *)
Lemma log2ceil_lt: forall n, 0 < n -> log2ceil n < n.
Proof with auto.