induction ls; intuition; simpl in *.
rewrite map_app.
f_equal; eauto.
Qed.

Lemma zip_app :
forall (A B : Set)(lsa1 lsa2 : list A)(lsb1 lsb2 : list B),
length lsa1 = length lsb1 ->
length lsa2 = length lsb2 ->
(zip lsa1 lsb1) ++ (zip lsa2 lsb2) =
(zip (lsa1 ++ lsa2) (lsb1 ++ lsb2)).