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

Lemma lastkey_mono h1 h2 :
{subset dom h1 <= dom h2} -> last_key h1 <= last_key h2.