induction n; intuition; simpl in *.
destruct ls; simpl in *.
trivial.
omega.
destruct ls.
trivial.
f_equal.
eapply IHn.
simpl in *.
omega.
Qed.

Lemma firstn_app : forall n (A : Set) (ls1 ls2 : list A),
(n <= length ls1)%nat ->
firstn n (ls1 ++ ls2) = firstn n ls1.