induction 1; intuition.
inversion H.
subst.
econstructor.
inversion H1; clear H1; subst.
inversion H2; clear H2; subst.
simpl.
econstructor; intuition.
Qed.

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