Section InitialDontExists.
Hypothesis succ_onto : forall n, exists m, n == S m.
End InitialDontExists.
(** To summarize:  S is always injective, P is always surjective  (thanks to [pred_succ]).  I) If S is not surjective, we have an initial point, which is unique. This bottom is below zero: we have N shifted (or not) to the left. P cannot be injective: P init = P (S (P init)). (P init) can be arbitrary.  II) If S is surjective, we have [forall n, S (P n) = n], S and P are bijective and reciprocal.  IIa) if [exists k<>O, 0 == S^k 0], then we have a cyclic structure Z/nZ IIb) otherwise, we have Z *)
(** * An alternative induction principle using [S] and [P]. *)
(** It is weaker than [bi_induction]. For instance it cannot prove that we can go from one point by many [S] _or_ many [P], but only by many [S] mixed with many [P]. Think of a model with two copies of N:  0,  1=S 0,   2=S 1, ... 0', 1'=S 0', 2'=S 1', ...  and P 0 = 0' and P 0' = 0. *)

Lemma bi_induction_pred :
forall A : t -> Prop, Proper (eq==>iff) A ->
A 0 -> (forall n, A n -> A (S n)) -> (forall n, A n -> A (P n)) ->
forall n, A n.