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_2 :
forall (m : Map A) (pf fp : ad -> ad) (l : list ad),
(forall a0 : ad, fp (pf a0) = a0) ->
forall a : ad,
In a (map_app_list1 pf l m) ->
In a l \/ in_dom _ (fp a) m = true /\ pf (fp a) = a.