Fixpoint map_app_list1 (pf : ad -> ad) (l : list ad)
(m : Map A) {struct m} : list ad :=
match m with
| M0 => l
| M1 a y => pf a :: l
| M2 m1 m2 =>
map_app_list1 (fun a0 : ad => pf (Ndouble_plus_one a0))
(map_app_list1 (fun a0 : ad => pf (Ndouble a0)) l m1) m2
end.

Lemma map_app_list1_lemma_4 :
forall (m : Map A) (pf : ad -> ad) (l : list ad) (a : ad),
in_dom _ a m = true -> In (pf a) (map_app_list1 pf l m).