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.

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').