(*=========================================================== PATHS in f-ORBITS: ==========================================================*)
(* REFLEXIVE EXISTENCE OF A PATH: *)
Definition expo(m:fmap)(z t:dart):Prop:=
exd m z /\ exists i:nat, Iter (f m) i z = t.
(* FOR THE DECIDABILITY: ... *)
Definition expo1
(m:fmap)(z t:dart):Prop:=
exd m z /\
let p:= Iter_upb m z in
exists j:nat, (j < p)%nat /\ Iter (f m) j z = t.

(* Specifications: *)
Lemma expo_expo1:
forall(m:fmap)(z t:dart), inv_hmap m ->
(expo m z t <-> expo1 m z t).