Fixpoint Map_of_alist (l:alist) : Map A :=
match l with
| nil => M0 A
| (a, y) :: l' => MapPut A (Map_of_alist l') a y
end.

Lemma MapFold_as_fold_1 :
forall (M:Type) (neutral:M) (op:M -> M -> M),
(forall a b c:M, op (op a b) c = op a (op b c)) ->
(forall a:M, op neutral a = a) ->
(forall a:M, op a neutral = a) ->
forall (f:ad -> A -> M) (m:Map A) (pf:ad -> ad),
MapFold1 M neutral op f pf m =
fold_right (fun (r:ad * A) (m:M) => let (a, y) := r in op (f a y) m)
neutral
(MapFold1 alist anil aapp (fun (a:ad) (y:A) => acons (a, y) anil) pf
m).