End FORALL2.
Definition map_fmap (A B:Type) (f:A->B) : map A -> map B :=
fix map_fmap (m:map A) : map B :=
match m with
| Leaf => Leaf _
| Node m1 o m2 =>
let x := match o with
| None => None
| Some x => Some (f x)
end in
Node _ (map_fmap m1) x (map_fmap m2)
end.

Lemma fmap_none : forall A B (m:map A) (f:A->B) x,
m#x = None ->
(map_fmap A B f m)#x = None.