induction 1; intuition; simpl in *.
subst.
destruct a1.
remember (split ls1) as x.
destruct x.
simpl in *.
f_equal.
Qed.

Theorem snd_split_eq_list_pred :
forall (A B : Set)(ls1 : list (B * A))(ls2 : list A),
list_pred (fun a b => snd a = b) ls1 ls2 ->
snd (split ls1) = ls2.