proposition: Lemma finite_cP x: finite_c x <-> (cardinalp x /\ x <> csucc x).
proof: Proof. split. by move => fsx;split; [fprops | case/csucc_inf]; case fsx. move => [cx /csucc_inf xcx]; split; [ exact: (proj1 cx) | by dneg w]. Qed.
imports: mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat)