intros.
induction H...
apply permuted_trans with l'...
Qed.
(* Lemma Permuted_map_old (A B: Type) (R: relation B) (f: A -> B) x y: Permuted (on f R) x y -> Permuted R (map f x) (map f y). Proof with auto. intros. induction H. simpl... apply permuted_skip... simpl. apply permuted_swap... apply permuted_trans with (map f l')... Qed. *)

Lemma Permuted_map A B (R: relation B) (f: A -> B): Proper (Permuted (on f R) ==> Permuted R) (map f).