Fixpoint my_delete_nth {A} (n:nat) (xs:list A) : list A :=
 match n with
  | 0%nat => match xs with
             | [] => []
             | _ :: ys => ys
             end
  | S n' => match xs with
            | [] => []
            | y :: ys => y :: my_delete_nth n' ys
            end
  end.

(* Specifications: *)
Lemma my_delete_nth_delete_nth {A}: forall n (l:list A), my_delete_nth n l = delete_nth n l.