Proof with auto.
induction l...
simpl.
rewrite IHl. clear IHl.
unfold triple0, triple1, triple2.
repeat rewrite map_map.
rewrite map_app.
rewrite map_map...
Qed.

(* Specifications: *)
Lemma elemsR_map':
forall (A B: Type) (Ra: relation A) (Rb: relation B) (f : A -> B)
(fR: forall x y, Ra x y -> Rb (f x) (f y)) (l l': list A),
elemsR Ra l l' -> elemsR Rb (map f l) (map f l').
Lemma concatMap_insert_everywhere T (x: T) (l: list (list T)):
Permutation
(concat (map (insert_everywhere x) l))
(map (cons x) l ++ concat (map (tail ∘ insert_everywhere x) l)).