Section FreshLastKey.
Variable A : Type.
Implicit Type h : natmap A.
Definition last_key h := last 0 (dom h).
Definition fresh h := (last_key h).+1.
(* freshness *)
(* pcm induced ordering *)

Lemma umpleq_lastkey h1 h2 :
valid h2 -> [pcm h1 <= h2] -> last_key h1 <= last_key h2.