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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 lastkeyUn0 h1 h2 :
valid (h1 \+ h2) ->
last_key h1 = last_key h2 -> (h1 = Unit) * (h2 = Unit).