(* freshness *)

(* Specifications: *)
Lemma dom_ordfresh h x : x \in dom h -> x < fresh h.
Lemma dom_fresh h : fresh h \notin dom h.
Lemma valid_fresh h v : valid (fresh h \-> v \+ h) = valid h.
Lemma lastkey_fresh h v : valid h -> last_key (fresh h \-> v \+ h) = fresh h.
(* pcm induced ordering *)
Lemma umpleq_lastkey h1 h2 :
valid h2 -> [pcm h1 <= h2] -> last_key h1 <= last_key h2.