induction ls; intuition; simpl in *.
discriminate.
destruct i.
omega.
eapply lt_n_S.
eauto.
Qed.

(* Specifications: *)
Theorem nth_option_app_None :
forall (A : Set)(ls1 ls2 : list A) i,
nth_option ls1 i = None ->
nth_option (ls1 ++ ls2) i = nth_option ls2 (i - length ls1).