induction ls; intuition; simpl in *.
econstructor.
econstructor; intuition.
Qed.

(* Specifications: *)
Lemma list_pred_map_r :
forall (A B : Set)(lsa : list A)(lsb : list B)(P : A -> B -> Prop),
list_pred P lsa lsb ->
forall (C : Set)(f : B -> C),
list_pred (fun a c => exists b, P a b /\ c = f b) lsa (map f lsb).