Module Mf(M:Sigf)<:Sigf
with Definition f:=M.f
with Definition f_1:=M.f_1.
Definition f:=M.f.
Definition f_1:=M.f_1.
Definition exd_f:=M.exd_f.
Definition exd_f_1:=M.exd_f_1.
Definition bij_f:=M.bij_f.
Definition bij_f_1:=M.bij_f_1.
Definition f_1_f:=M.f_1_f.
Definition f_f_1:=M.f_f_1.
(*======================================================== ITERATIONS IN ORBITS: WITH NOETHERIAN INDUCTION =======================================================*)
(* REMAINING DARTS DURING f ITERATION ON z STARTING FROM A set sx: *)
(*======================================================== EXISTENCY PROPERTIES OF ITERATED DARTS, upb IN ORBITS...: WITH NOETHERIAN IND. =======================================================*)
(* FOR THE FOLLOWING LEMMA: *)
Definition P1
(m:fmap)(z:dart)(s:set):Prop:=
let sr := Iter_rem_aux m z s in
let n := Iter_upb_aux m z s in
~ exds sr (Iter (f m) n z).
(* THE upb_ITERATED DART BY f DOES NOT REMAIN: *)
(* THE upb_ITERATED DART BY f DOES NOT REMAIN: *)
(* THE upb_ITERATED DART BY f IS IN m: *)

Lemma  exd_Iter_upb:
forall(m:fmap)(z:dart), inv_hmap m ->
exd m z -> exd m (Iter (f m) (Iter_upb m z) z).