Definition ad_monotonic (pf:ad -> ad) :=
forall a a':ad, Nless a a' = true -> Nless (pf a) (pf a') = true.

Lemma alist_of_Map_sorts_1 :
forall (m:Map A) (pf:ad -> ad),
ad_monotonic pf ->
alist_sorted_2
(MapFold1 A (alist A) (anil A) (aapp A)
(fun (a:ad) (y:A) => acons A (a, y) (anil A)) pf m).