induction ls; intuition; simpl in *.
omega.
destruct n; simpl in *; intuition.
discriminate.
eapply IHls; eauto.
omega.
Qed.

Theorem hd_skip_eq_nth :
forall (A : Type)(def : A)(ls : list A)(n : nat),
hd def (skipn n ls) = nth n ls def.