induction a...
simpl.
unfold flip at 1.
intros.
rewrite <- IHa.
apply product_app.
Qed. (* todo: this shouldn't need induction anymore *)

Lemma concatMap_insert_everywhere_comm x y l: Permutation
(concat (map (insert_everywhere x) (insert_everywhere y l)))
(concat (map (insert_everywhere y) (insert_everywhere x l))).