(***************************************)
(***************************************)
(* 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 lastkeyPt (x : nat) v : x != 0 -> last_key (x \-> v) = x.