Fixpoint iteres_0 (A : Set) (f : Map A -> Map A) (x : Map A) 
 (n : nat) {struct n} : prechain A :=
  match n with
  | O => single A x
  | S p =>
      match iteres_0 A f x p with
      | single y => concat A (single A y) (power (Map A) f x (S p))
      | concat z y => concat A (concat A z y) (power (Map A) f x (S p))
      end
  end.

(* Specifications: *)
Lemma iteres_eq_0 : forall (A : Set) (f : Map A -> Map A) (x : Map A) (n : nat), prechain_last A (iteres A f x n) = power (Map A) f x n.
Lemma iteres_eq : forall (A : Set) (f : Map A -> Map A) (x : Map A) (n : nat), iteres A f x n = iteres_0 A f x n.
Lemma iteres_def_ok : forall (A : Set) (T : mEnsemble A) (f : Map A -> Map A) (x : Map A) (n k : nat), def_ok_app A T f -> T (power (Map A) f x n) -> T (power (Map A) f x (n + k)).
Lemma power_def_ok : forall (A : Set) (T : mEnsemble A) (f : Map A -> Map A) (x : Map A) (n : nat), def_ok_app A T f -> T x -> T (power (Map A) f x n).