End FreshLastKey.
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(* Gapless natmaps *)
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Section Gapless.
Variable A : Type.
Implicit Type h : natmap A.
Definition gapless h := forall k, 0 < k <= last_key h -> k \in dom h.

(* Specifications: *)
Lemma gp_undef : gapless um_undef.
Lemma gpPtUn h k u :
valid (k \-> u \+ h) ->
gapless (k \-> u \+ h) -> last_key h < k -> k = fresh h.