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.

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.