(*==========================================================
     SOME LEMMAS, THEN RESULTS ABOUT PERIODICITY
===========================================================*)

(* L2 : *)

Definition PL2(m:fmap)(z t:dart)(x:set):Prop:=
  inv_hmap m -> exd m z -> exds x t ->
    let sr:= Iter_rem_aux m z x in
    let zn0 := (Iter (f m) (ndN m - card x)%nat z) in
    ~exds sr t -> 
       exds x zn0 -> 
        ~ exds (Iter_rem_aux m z (Ds x zn0)) t.

(* Specifications: *)
Lemma L2:forall(m:fmap)(z t:dart)(x:set), inv_hmap m -> exd m z -> exds x t -> let sr:= Iter_rem_aux m z x in let zn0 := (Iter (f m) (ndN m - card x)%nat z) in ~exds sr t -> exds x zn0 -> ~ exds (Iter_rem_aux m z (Ds x zn0)) t.