Proof with auto.
unfold C at 1.
destruct CRimpl.
simpl proj1_sig.
induction c...
simpl pred.
f_equal.
f_equal.
f_equal.
generalize H. clear H.
generalize (nats 0 (S n)). intro.
generalize l0. clear n l0.
induction l; intros; inversion_clear H...
simpl.
rewrite <- IHl...
unfold C.
destruct CRimpl.
f_equal.
apply (CRuniq H0 c).
Qed.
(* .. as well as some other properties: *)

Lemma C_nonneg n: 0 <= C n.
Proof with auto with real.