(** Test if, starting at position [n], [s1] occurs in [s2]; if
    so it returns the position *)

Fixpoint index (n : nat) (s1 s2 : string) : option nat :=
  match s2, n with
  | EmptyString, O =>
      match s1 with
      | EmptyString => Some O
      | String a s1' => None
      end
  | EmptyString, S n' => None
  | String b s2', O =>
      if prefix s1 s2 then Some O
      else
        match index O s1 s2' with
        | Some n => Some (S n)
        | None => None
        end
   | String b s2', S n' =>
      match index n' s1 s2' with
      | Some n => Some (S n)
      | None => None
      end
  end.

Theorem index_correct3 : forall (n m : nat) (s1 s2 : string), index n s1 s2 = None -> s1 <> EmptyString -> n <= m -> substring m (length s1) s2 <> s1.