induction ls; intuition; simpl in *.
destruct i.
discriminate.
eapply le_n_S.
eapply IHls.
trivial.
Qed.

(* Specifications: *)
Theorem skipn_S_eq :
forall (A : Set)(ls : list A) n a,
nth_option ls n = Some a ->
skipn n ls = a :: (skipn (S n) ls).