induction l...
simpl.
rewrite map_map.
simpl.
rewrite IHl. clear IHl.
unfold triple1.
simpl.
rewrite map_map.
simpl...
Qed.

(* Specifications: *)
Lemma insert_everywhere_rsplits (T: Set) (x: T) (l: list T):
insert_everywhere x l =
map (fun x0 => triple0 x0 ++ x :: triple1 x0 :: triple2 x0) (rsplits l) ++ [l ++ [x]].