induction ls; intuition; simpl in *.
eapply cons_ne in H0.
destruct H0.
econstructor.
intuition.
intuition.
destruct H2.
intuition.
exists x.
intuition.
trivial.
Qed.

Theorem list_pred_I_in :
forall (A B : Set)(lsa : list A)(lsb : list B),
length lsa = length lsb ->
list_pred (fun a b => In a lsa /\ In b lsb) lsa lsb.