Ltac map_insert :=
first [
rewrite map_insert; reflexivity
| rewrite <- map_insert; reflexivity
].
Hint Extern 1 (map _ (insert _ _ _) = insert _ _ (map _ _)) =>
map_insert
: map_insert.
Hint Extern 1 (insert _ _ (map _ _) = map _ (insert _ _ _)) =>
map_insert
: map_insert.

(* Specifications: *)
Lemma map_raw_insert:
forall A B (f : A -> B) x e,
map f (raw_insert x None e) = raw_insert x None (map f e).