induction lsa1; intuition; simpl in *.
econstructor.
inversion H; clear H; subst.
econstructor; intuition.
eapply IHlsa1; eauto.
inversion H; clear H; subst.
eapply IHlsa1; eauto.
Qed.

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