(***************************************)
(***************************************)
(* Constructions of last_key and fresh *)
(***************************************)
(***************************************)
Variable A : Type.
Implicit Type h : natmap A.
Definition last_key h := last 0 (dom h).
Definition fresh h := (last_key h).+1.

Lemma dom_lastkeyE h a : a < last_key h -> last_key h \in dom h.