induction ls; intuition; simpl in *.
econstructor.
econstructor.
intuition.
eapply list_pred_impl.
eapply IHls.
intuition.
intuition.
intuition.
intuition.
Qed.

Theorem zip_combine_eq :
forall (A B : Set)(lsa : list A)(lsb : list B),
zip lsa lsb = combine lsa lsb.