(** * Study of initial point w.r.t. [succ] (if any). *)
Definition initial n := forall m, n ~= S m.

(* Specifications: *)
Lemma initial_alt : forall n, initial n <-> S (P n) ~= n.
Lemma initial_alt2 : forall n, initial n <-> ~exists m, n == S m.
(** ... then all other points are descendant of it. *)
Lemma initial_ancestor : forall m, exists p, m == (S^p) init.
Section SuccPred.