induction lsa; intuition; simpl in *.
destruct lsb; intuition.
f_equal.
eauto.
Qed.

(* Specifications: *)
Theorem list_pred_fst_split_eq :
forall (A B C : Set)(ls1 : list (A * B))(ls2 : list (A * C)),
list_pred (fun a b => fst a = fst b) ls1 ls2 ->
fst (split ls1)  = fst (split ls2).