Definition init_ptd :=
  match full_pt in parse_tree head full_word full_sem return
    pt_zipper head full_word full_sem ->
    match head return Type with | T _ => unit | NT _ => pt_dot end
  with
    | Terminal_pt _ _ => fun _ => ()
    | Non_terminal_pt ptl =>
      fun top => build_pt_dot ptl (Non_terminal_pt_ptlz top)
  end Top_ptz.

Lemma init_ptd_cost: S (ptd_cost init_ptd) = pt_size full_pt.