(*========================================================== 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.