induction lsa; intuition; simpl in *.
destruct lsb; simpl in *.
omega.
discriminate.
Qed.

Theorem fst_split_app_eq :
forall (A B : Type)(ls1 ls2 : list (A * B)),
fst (split (ls1 ++ ls2)) =
fst (split ls1) ++ fst (split ls2).