induction ls; intuition; simpl in *.
intuition.
intuition; subst.
exists O.
trivial.
edestruct IHls.
eauto.
exists (S x).
eauto.
Qed.

Theorem nth_skipn_eq :
forall (A : Set)(y x: nat)(ls : list A)(def : A),
nth x (skipn y ls) def = nth (x + y) ls def.