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.