(* OK: FOR ALL n0 <= i < j <= nr-1, z^i <> z^j: LENGHT: SYMPLIFY: TOO MUCH omegas... *)
Definition P8(m:fmap)(z:dart)(s:set):Prop:=
inv_hmap m ->
(card s <= ndN m ->
let n0:= ndN m - card s in
let nr:= Iter_upb_aux m z s in
exds s (Iter (f m) n0 z) ->
diff_int m z n0 (nr - 1))%nat.

Lemma LP8:forall(m:fmap)(z:dart)(s:set),
inv_hmap m ->
(card s <= ndN m ->
let n0:= ndN m - card s in
let nr:= Iter_upb_aux m z s in
exds s (Iter (f m) n0 z) ->
diff_int m z n0 (nr - 1))%nat.