induction lsa; inversion 1; intuition; subst; simpl in *;
econstructor.
inversion H; clear H; subst.
trivial.
inversion H; clear H; subst.
eauto.
Qed.

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