Program Fixpoint pop_ptlz {hole_symbs hole_word hole_sems}
  (ptl:parse_tree_list hole_symbs hole_word hole_sems)
  (ptlz:ptl_zipper hole_symbs hole_word hole_sems):
    { word:_ & { sem:_ &
      (pt_zipper (NT (prod_lhs (ptlz_prod ptlz))) word sem *
       parse_tree (NT (prod_lhs (ptlz_prod ptlz))) word sem)%type } } :=
  match ptlz in ptl_zipper hole_symbs hole_word hole_sems
    return parse_tree_list hole_symbs hole_word hole_sems ->
    { word:_ & { sem:_ &
      (pt_zipper (NT (prod_lhs (ptlz_prod ptlz))) word sem *
       parse_tree (NT (prod_lhs (ptlz_prod ptlz))) word sem)%type } }
  with
    | @Non_terminal_pt_ptlz prod word sem ptz => fun ptl =>
      let sem := uncurry (prod_action prod) sem in
      existT _ word (existT _ sem (ptz, Non_terminal_pt ptl))
    | Cons_ptl_ptlz pt ptlz' => fun ptl =>
      pop_ptlz (Cons_ptl pt ptl) ptlz'
  end ptl.

(* Specifications: *)
Lemma pop_ptlz_cost: forall hole_symbs hole_word hole_sems (ptl:parse_tree_list hole_symbs hole_word hole_sems) (ptlz:ptl_zipper hole_symbs hole_word hole_sems), let 'existT _ word (existT _ sem (ptz, pt)) := pop_ptlz ptl ptlz in ptlz_cost ptlz = ptz_cost ptz.
Lemma pop_ptlz_buffer: forall hole_symbs hole_word hole_sems (ptl:parse_tree_list hole_symbs hole_word hole_sems) (ptlz:ptl_zipper hole_symbs hole_word hole_sems), let 'existT _ word (existT _ sem (ptz, pt)) := pop_ptlz ptl ptlz in ptlz_buffer ptlz = ptz_buffer ptz.
Lemma pop_ptlz_pop_stack_compat_converter: forall A hole_symbs hole_word hole_sems (ptlz:ptl_zipper hole_symbs hole_word hole_sems), arrows_left (map symbol_semantic_type (rev (prod_rhs_rev (ptlz_prod ptlz)))) A = arrows_left (map symbol_semantic_type hole_symbs) (arrows_right A (map symbol_semantic_type (ptlz_past ptlz))).
Lemma pop_ptlz_pop_stack_compat: forall hole_symbs hole_word hole_sems (ptl:parse_tree_list hole_symbs hole_word hole_sems) (ptlz:ptl_zipper hole_symbs hole_word hole_sems) (stack:stack), ptlz_stack_compat ptlz stack -> let action' := eq_rect _ (fun x=>x) (prod_action (ptlz_prod ptlz)) _ (pop_ptlz_pop_stack_compat_converter _ _ _ _ _) in let 'existT _ word (existT _ sem (ptz, pt)) := pop_ptlz ptl ptlz in match pop (ptlz_past ptlz) stack (uncurry action' hole_sems) with | OK (stack', sem') => ptz_stack_compat ptz stack' /\ sem = sem' | Err => True end.