induction 1; intuition; simpl in *.
econstructor.
econstructor.
econstructor; eauto.
eauto.
Qed.

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