Fixpoint positive_serialize p :=
  match p with
  | xI (xI (xI p)) => IOStreamWriter.append (fun _ => serialize x0e)
                                            (fun _ => positive_serialize p)
  | xI (xI (xO p)) => IOStreamWriter.append (fun _ => serialize x0d)
                                            (fun _ => positive_serialize p)
  | xI (xO (xI p)) => IOStreamWriter.append (fun _ => serialize x0c)
                                            (fun _ => positive_serialize p)
  | xI (xO (xO p)) => IOStreamWriter.append (fun _ => serialize x0b)
                                            (fun _ => positive_serialize p)
  | xO (xI (xI p)) => IOStreamWriter.append (fun _ => serialize x0a)
                                            (fun _ => positive_serialize p)
  | xO (xI (xO p)) => IOStreamWriter.append (fun _ => serialize x09)
                                            (fun _ => positive_serialize p)
  | xO (xO (xI p)) => IOStreamWriter.append (fun _ => serialize x08)
                                            (fun _ => positive_serialize p)
  | xO (xO (xO p)) => IOStreamWriter.append (fun _ => serialize x07)
                                            (fun _ => positive_serialize p)
  | xI (xI p) => IOStreamWriter.append (fun _ => serialize x06)
                                       (fun _ => positive_serialize p)
  | xI (xO p) => IOStreamWriter.append (fun _ => serialize x05)
                                       (fun _ => positive_serialize p)
  | xO (xI p) => IOStreamWriter.append (fun _ => serialize x04)
                                       (fun _ => positive_serialize p)
  | xO (xO p) => IOStreamWriter.append (fun _ => serialize x03)
                                       (fun _ => positive_serialize p)
  | xI p => IOStreamWriter.append (fun _ => serialize x02)
                                  (fun _ => positive_serialize p)
  | xO p => IOStreamWriter.append (fun _ => serialize x01)
                                  (fun _ => positive_serialize p)
  | XH => serialize x00
  end.Definition positive_deserialize_step
           (b : byte)
           (s : positive -> positive) :=
  match b with
  | x0e => More (fun p => s (xI (xI (xI p))))
  | x0d => More (fun p => s (xI (xI (xO p))))
  | x0c => More (fun p => s (xI (xO (xI p))))
  | x0b => More (fun p => s (xI (xO (xO p))))
  | x0a => More (fun p => s (xO (xI (xI p))))
  | x09 => More (fun p => s (xO (xI (xO p))))
  | x08 => More (fun p =>  s (xO (xO (xI p))))
  | x07 => More (fun p => s (xO (xO (xO p))))
  | x06 => More (fun p => s (xI (xI p)))
  | x05 => More (fun p => s (xI (xO p)))
  | x04 => More (fun p => s (xO (xI p)))
  | x03 => More (fun p => s (xO (xO p)))
  | x02 => More (fun p => s (xI p))
  | x01 => More (fun p => s (xO p))
  | x00 => Done (s xH)
  | _ => Error
  end.Definition positive_step_aux p :=
  forall (k : positive -> positive) (bytes : list byte),
    ByteListReader.unwrap (ByteListReader.fold positive_deserialize_step k)
                          (IOStreamWriter.unwrap (positive_serialize p) ++ bytes)
    = Some(k p, bytes).

Lemma positive_step : forall (p : positive), positive_step_aux p.