End Gapless.
Arguments gp_fresh [A][h] u.
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(* Natmaps with binary range *)
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Section AtVal.
Variable A : Type.
Implicit Type h : natmap (A * A).
Definition atval v t h := oapp snd v (find t h).

Lemma umpleq_atval v t h1 h2 :
gapless h1 -> valid h2 -> [pcm h1 <= h2] -> t <= last_key h1 ->
atval v t h2 = atval v t h1.