Definition P_degree_diff(m:fmap)(z:dart)(n1 n2:nat):
  Prop:= inv_hmap m -> exd m z -> 0 < n1 ->  
    forall i:nat, n1 <= i < n2 -> Iter (f m) i z <> z.

(* Specifications: *)
Lemma degree_diff_aux:forall(m:fmap)(z:dart), P_degree_diff m z 1 (degree m z).
Theorem degree_diff: forall (m:fmap)(z:dart), inv_hmap m -> exd m z -> forall i:nat, 0 < i < (degree m z) -> Iter (f m) i z <> z.
Lemma degree_bound: forall (m:fmap)(z:dart), inv_hmap m -> exd m z -> degree m z <= ndN m.