intros A B m f; induction m; simpl; intros;
destruct x; simpl in *; try discriminate; auto.
destruct o; simpl; auto.
discriminate.
Qed.

Lemma fmap_none2 : forall A B (m:map A) (f:A->B) x,
(map_fmap A B f m)#x = None ->
m#x = None.