Definition alist_sorted_2 (l:alist A) :=
forall m n:nat,
m < n ->
S n <= length l -> Nless (alist_nth_ad m l) (alist_nth_ad n l) = true.

Lemma alist_of_Map_nth_ad :
forall (m:Map A) (pf:ad -> ad) (l:alist A),
l =
MapFold1 A (alist A) (anil A) (aapp A)
(fun (a0:ad) (y:A) => acons A (a0, y) (anil A)) pf m ->
forall n:nat, S n <= length l -> {a' : ad | alist_nth_ad n l = pf a'}.