intuition.
eapply list_pred_impl.
eapply list_pred_zip_l.
eauto.
eapply list_pred_eq.
eapply list_pred_I.
symmetry.
eapply list_pred_length_eq; eauto.
intuition.
Qed.

Lemma list_pred_map_r_eq :
forall (A B : Set)(ls : list A)(f : A -> B),
list_pred (fun a b => f a = b) ls (map f ls).