induction ls; intuition; simpl in *.
intuition.
intuition.
pairInv.
intuition.
right.
eapply IHls.
eauto.
intuition.
pairInv.
trivial.
eapply IHls; eauto.
Qed.

Lemma list_pred_impl :
forall (A B : Set)(lsa : list A)(lsb : list B) (P1 : A -> B -> Prop),
list_pred P1 lsa lsb ->
forall (P2 : A -> B -> Prop),
(forall a b, P1 a b -> P2 a b) ->
list_pred P2 lsa lsb.