pattern n.
apply (well_founded_ind lt_wf).
intros.
rewrite Ceq.
destruct x...
simpl pred.
apply Rplus_le_le_0_compat...
apply Rmult_le_pos...
apply Rmult_le_pos...
apply Rsum_nonneg.
intros.
apply in_map_iff in H0. destruct H0. destruct H0. subst.
apply H.
destruct (In_nats_inv _ _ _ H1)...
rewrite map_length. rewrite nats_length...
Qed.

(* Specifications: *)
Lemma C_S n: C n = 2 * pred n / n + S n / n * C (pred n).
Proof with auto with real.