induction lsa; intuition; simpl in *.
intuition.
destruct lsb.
inversion H.
simpl in H.
intuition.
pairInv.
intuition.
right.
eapply IHlsa.
eauto.
destruct lsb.
inversion H.
simpl in *.
intuition.
pairInv.
intuition.
right.
eapply IHlsa.
eauto.
Qed.

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