induction ls; destruct n1; destruct n2; intuition; simpl in *.
omega.
f_equal.
eapply IHls.
omega.
Qed.

(* Specifications: *)
Theorem nth_firstn :
forall (A : Set)(def : A)(ls : list A)(n1 n2 : nat),
(n1 < n2)%nat ->
nth n1 (firstn n2 ls) def = nth n1 ls def.