induction ls; intuition; simpl in *.
rewrite IHls.
omega.
Qed.

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