induction lsa; destruct lsb; intuition; simpl in *.
econstructor.
discriminate.
discriminate.
econstructor.
intuition.
eapply list_pred_impl.
eapply IHlsa.
omega.
intuition.
intuition.
intuition.
Qed.

Theorem list_pred_fst_split_eq_l :
forall (A B : Set)(a : list (A * B))(b : list A),
list_pred (fun a0 b0 => fst a0 = b0) a b ->
b = fst (split a).