(*  Title:      HOL/Tools/ATP/atp_problem.ML

    Author:     Jia Meng, Cambridge University Computer Laboratory and NICTA

    Author:     Jasmin Blanchette, TU Muenchen

Abstract representation of ATP problems and TPTP syntax.

*)

signature ATP_PROBLEM =

sig

  datatype ('a, 'b) ho_term =

    ATerm of 'a * ('a, 'b) ho_term list |

    AAbs of ('a * 'b) * ('a, 'b) ho_term

  datatype quantifier = AForall | AExists

  datatype connective = ANot | AAnd | AOr | AImplies | AIff

  datatype ('a, 'b, 'c) formula =

    AQuant of quantifier * ('a * 'b option) list * ('a, 'b, 'c) formula |

    AConn of connective * ('a, 'b, 'c) formula list |

    AAtom of 'c

  datatype 'a ho_type =

    AType of 'a * 'a ho_type list |

    AFun of 'a ho_type * 'a ho_type

  datatype tff_polymorphism = TFF_Monomorphic | TFF_Polymorphic

  datatype tff_explicitness = TFF_Implicit | TFF_Explicit

  datatype thf_flavor = THF_Without_Choice | THF_With_Choice

  datatype format =

    CNF |

    CNF_UEQ |

    FOF |

    TFF of tff_polymorphism * tff_explicitness |

    THF0 of thf_flavor

  datatype formula_kind = Axiom | Definition | Lemma | Hypothesis | Conjecture

  datatype 'a problem_line =

    Decl of string * 'a * 'a ho_type |

    Formula of string * formula_kind

               * ('a, 'a ho_type, ('a, 'a ho_type) ho_term) formula

               * (string, string ho_type) ho_term option

               * (string, string ho_type) ho_term option

  type 'a problem = (string * 'a problem_line list) list

  val tptp_cnf : string

  val tptp_fof : string

  val tptp_tff : string

  val tptp_thf : string

  val tptp_has_type : string

  val tptp_type_of_types : string

  val tptp_bool_type : string

  val tptp_individual_type : string

  val tptp_fun_type : string

  val tptp_product_type : string

  val tptp_forall : string

  val tptp_ho_forall : string

  val tptp_exists : string

  val tptp_ho_exists : string

  val tptp_choice : string

  val tptp_not : string

  val tptp_and : string

  val tptp_or : string

  val tptp_implies : string

  val tptp_if : string

  val tptp_iff : string

  val tptp_not_iff : string

  val tptp_app : string

  val tptp_not_infix : string

  val tptp_equal : string

  val tptp_old_equal : string

  val tptp_false : string

  val tptp_true : string

  val tptp_empty_list : string

  val is_tptp_equal : string -> bool

  val is_built_in_tptp_symbol : string -> bool

  val is_tptp_variable : string -> bool

  val is_tptp_user_symbol : string -> bool

  val mk_anot : ('a, 'b, 'c) formula -> ('a, 'b, 'c) formula

  val mk_aconn :

    connective -> ('a, 'b, 'c) formula -> ('a, 'b, 'c) formula

    -> ('a, 'b, 'c) formula

  val aconn_fold :

    bool option -> (bool option -> 'a -> 'b -> 'b) -> connective * 'a list

    -> 'b -> 'b

  val aconn_map :

    bool option -> (bool option -> 'a -> ('b, 'c, 'd) formula)

    -> connective * 'a list -> ('b, 'c, 'd) formula

  val formula_fold :

    bool option -> (bool option -> 'c -> 'd -> 'd) -> ('a, 'b, 'c) formula

    -> 'd -> 'd

  val formula_map : ('c -> 'd) -> ('a, 'b, 'c) formula -> ('a, 'b, 'd) formula

  val is_format_thf : format -> bool

  val is_format_typed : format -> bool

  val tptp_lines_for_atp_problem : format -> string problem -> string list

  val ensure_cnf_problem :

    (string * string) problem -> (string * string) problem

  val filter_cnf_ueq_problem :

    (string * string) problem -> (string * string) problem

  val declare_undeclared_syms_in_atp_problem :

    string -> string -> (string * string) problem -> (string * string) problem

  val nice_atp_problem :

    bool -> ('a * (string * string) problem_line list) list

    -> ('a * string problem_line list) list

       * (string Symtab.table * string Symtab.table) option

end;

structure ATP_Problem : ATP_PROBLEM =

struct

open ATP_Util

(** ATP problem **)

datatype ('a, 'b) ho_term =

  ATerm of 'a * ('a, 'b) ho_term list |

  AAbs of ('a * 'b) * ('a, 'b) ho_term

datatype quantifier = AForall | AExists

datatype connective = ANot | AAnd | AOr | AImplies | AIff

datatype ('a, 'b, 'c) formula =

  AQuant of quantifier * ('a * 'b option) list * ('a, 'b, 'c) formula |

  AConn of connective * ('a, 'b, 'c) formula list |

  AAtom of 'c

datatype 'a ho_type =

  AType of 'a * 'a ho_type list |

  AFun of 'a ho_type * 'a ho_type

datatype tff_polymorphism = TFF_Monomorphic | TFF_Polymorphic

datatype tff_explicitness = TFF_Implicit | TFF_Explicit

datatype thf_flavor = THF_Without_Choice | THF_With_Choice

datatype format =

  CNF |

  CNF_UEQ |

  FOF |

  TFF of tff_polymorphism * tff_explicitness |

  THF0 of thf_flavor

datatype formula_kind = Axiom | Definition | Lemma | Hypothesis | Conjecture

datatype 'a problem_line =

  Decl of string * 'a * 'a ho_type |

  Formula of string * formula_kind * ('a, 'a ho_type, ('a, 'a ho_type) ho_term) formula

             * (string, string ho_type) ho_term option * (string, string ho_type) ho_term option

type 'a problem = (string * 'a problem_line list) list

(* official TPTP syntax *)

val tptp_cnf = "cnf"

val tptp_fof = "fof"

val tptp_tff = "tff"

val tptp_thf = "thf"

val tptp_has_type = ":"

val tptp_type_of_types = "$tType"

val tptp_bool_type = "$o"

val tptp_individual_type = "$i"

val tptp_fun_type = ">"

val tptp_product_type = "*"

val tptp_forall = "!"

val tptp_ho_forall = "!!"

val tptp_exists = "?"

val tptp_ho_exists = "??"

val tptp_choice = "@+"

val tptp_not = "~"

val tptp_and = "&"

val tptp_or = "|"

val tptp_implies = "=>"

val tptp_if = "<="

val tptp_iff = "<=>"

val tptp_not_iff = "<~>"

val tptp_app = "@"

val tptp_not_infix = "!"

val tptp_equal = "="

val tptp_old_equal = "equal"

val tptp_false = "$false"

val tptp_true = "$true"

val tptp_empty_list = "[]"

fun is_tptp_equal s = (s = tptp_equal orelse s = tptp_old_equal)

fun is_built_in_tptp_symbol s =

  s = tptp_old_equal orelse not (Char.isAlpha (String.sub (s, 0)))

fun is_tptp_variable s = Char.isUpper (String.sub (s, 0))

val is_tptp_user_symbol = not o (is_tptp_variable orf is_built_in_tptp_symbol)

fun raw_polarities_of_conn ANot = (SOME false, NONE)

  | raw_polarities_of_conn AAnd = (SOME true, SOME true)

  | raw_polarities_of_conn AOr = (SOME true, SOME true)

  | raw_polarities_of_conn AImplies = (SOME false, SOME true)

  | raw_polarities_of_conn AIff = (NONE, NONE)

fun polarities_of_conn NONE = K (NONE, NONE)

  | polarities_of_conn (SOME pos) =

    raw_polarities_of_conn #> not pos ? pairself (Option.map not)

fun mk_anot (AConn (ANot, [phi])) = phi

  | mk_anot phi = AConn (ANot, [phi])

fun mk_aconn c phi1 phi2 = AConn (c, [phi1, phi2])

fun aconn_fold pos f (ANot, [phi]) = f (Option.map not pos) phi

  | aconn_fold pos f (AImplies, [phi1, phi2]) =

    f (Option.map not pos) phi1 #> f pos phi2

  | aconn_fold pos f (AAnd, phis) = fold (f pos) phis

  | aconn_fold pos f (AOr, phis) = fold (f pos) phis

  | aconn_fold _ f (_, phis) = fold (f NONE) phis

fun aconn_map pos f (ANot, [phi]) = AConn (ANot, [f (Option.map not pos) phi])

  | aconn_map pos f (AImplies, [phi1, phi2]) =

    AConn (AImplies, [f (Option.map not pos) phi1, f pos phi2])

  | aconn_map pos f (AAnd, phis) = AConn (AAnd, map (f pos) phis)

  | aconn_map pos f (AOr, phis) = AConn (AOr, map (f pos) phis)

  | aconn_map _ f (c, phis) = AConn (c, map (f NONE) phis)

fun formula_fold pos f =

  let

    fun fld pos (AQuant (_, _, phi)) = fld pos phi

      | fld pos (AConn conn) = aconn_fold pos fld conn

      | fld pos (AAtom tm) = f pos tm

  in fld pos end

fun formula_map f (AQuant (q, xs, phi)) = AQuant (q, xs, formula_map f phi)

  | formula_map f (AConn (c, phis)) = AConn (c, map (formula_map f) phis)

  | formula_map f (AAtom tm) = AAtom (f tm)

fun is_format_thf (THF0 _) = true

  | is_format_thf _ = false

fun is_format_typed (TFF _) = true

  | is_format_typed (THF0 _) = true

  | is_format_typed _ = false

fun string_for_kind Axiom = "axiom"

  | string_for_kind Definition = "definition"

  | string_for_kind Lemma = "lemma"

  | string_for_kind Hypothesis = "hypothesis"

  | string_for_kind Conjecture = "conjecture"

fun strip_tff_type (AFun (AFun _, _)) =

    raise Fail "unexpected higher-order type in first-order format"

  | strip_tff_type (AFun (ty1, ty2)) = strip_tff_type ty2 |>> cons ty1

  | strip_tff_type ty = ([], ty)

fun general_string_for_type ty =

  let

    fun str _ (AType (s, [])) = s

      | str _ (AType (s, tys)) = s ^ "(" ^ commas (map (str false) tys) ^ ")"

      | str rhs (AFun (ty1, ty2)) =

        str false ty1 ^ " " ^ tptp_fun_type ^ " " ^ str true ty2

        |> not rhs ? enclose "(" ")"

  in str true ty end

fun string_for_type (THF0 _) ty = general_string_for_type ty

  | string_for_type (TFF _) ty =

    (case strip_tff_type ty of

       ([], ty) => general_string_for_type ty

     | ([ty1], ty2) => general_string_for_type (AFun (ty1, ty2))

     | (tys, ty) =>

       "(" ^ space_implode (" " ^ tptp_product_type ^ " ")

       (map general_string_for_type tys) ^ ") " ^ tptp_fun_type ^ " " ^

       general_string_for_type ty)

  | string_for_type _ _ = raise Fail "unexpected type in untyped format"

fun string_for_quantifier AForall = tptp_forall

  | string_for_quantifier AExists = tptp_exists

fun string_for_connective ANot = tptp_not

  | string_for_connective AAnd = tptp_and

  | string_for_connective AOr = tptp_or

  | string_for_connective AImplies = tptp_implies

  | string_for_connective AIff = tptp_iff

fun string_for_bound_var format (s, ty) =

  s ^

  (if is_format_typed format then

     " " ^ tptp_has_type ^ " " ^

     (ty |> the_default (AType (tptp_individual_type, []))

         |> string_for_type format)

   else

     "")

fun string_for_term _ (ATerm (s, [])) = s

  | string_for_term format (ATerm (s, ts)) =

    if s = tptp_empty_list then

      (* used for lists in the optional "source" field of a derivation *)

      "[" ^ commas (map (string_for_term format) ts) ^ "]"

    else if is_tptp_equal s then

      space_implode (" " ^ tptp_equal ^ " ") (map (string_for_term format) ts)

      |> is_format_thf format ? enclose "(" ")"

    else

      (case (s = tptp_ho_forall orelse s = tptp_ho_exists,

             s = tptp_choice andalso format = THF0 THF_With_Choice, ts) of

         (true, _, [AAbs ((s', ty), tm)]) =>

         (* Use syntactic sugar "!" and "?" instead of "!!" and "??" whenever

            possible, to work around LEO-II 1.2.8 parser limitation. *)

         string_for_formula format

             (AQuant (if s = tptp_ho_forall then AForall else AExists,

                      [(s', SOME ty)], AAtom tm))

       | (_, true, [AAbs ((s', ty), tm)]) =>

         (*There is code in ATP_Translate to ensure that Eps is always applied

           to an abstraction*)

         tptp_choice ^ "[" ^ s' ^ " : " ^ string_for_type format ty ^ "] : " ^

           string_for_term format tm ^ ""

         |> enclose "(" ")"

       | _ =>

         let val ss = map (string_for_term format) ts in

           if is_format_thf format then

             "(" ^ space_implode (" " ^ tptp_app ^ " ") (s :: ss) ^ ")"

           else

             s ^ "(" ^ commas ss ^ ")"

         end)

  | string_for_term (format as THF0 _) (AAbs ((s, ty), tm)) =

    "(^[" ^ s ^ " : " ^ string_for_type format ty ^ "] : " ^

    string_for_term format tm ^ ")"

  | string_for_term _ _ = raise Fail "unexpected term in first-order format"

and string_for_formula format (AQuant (q, xs, phi)) =

    string_for_quantifier q ^

    "[" ^ commas (map (string_for_bound_var format) xs) ^ "] : " ^

    string_for_formula format phi

    |> enclose "(" ")"

  | string_for_formula format

        (AConn (ANot, [AAtom (ATerm ("=" (* tptp_equal *), ts))])) =

    space_implode (" " ^ tptp_not_infix ^ tptp_equal ^ " ")

                  (map (string_for_term format) ts)

    |> is_format_thf format ? enclose "(" ")"

  | string_for_formula format (AConn (c, [phi])) =

    string_for_connective c ^ " " ^

    (string_for_formula format phi |> is_format_thf format ? enclose "(" ")")

    |> enclose "(" ")"

  | string_for_formula format (AConn (c, phis)) =

    space_implode (" " ^ string_for_connective c ^ " ")

                  (map (string_for_formula format) phis)

    |> enclose "(" ")"

  | string_for_formula format (AAtom tm) = string_for_term format tm

fun the_source (SOME source) = source

  | the_source NONE =

    ATerm ("inference",

           ATerm ("isabelle", []) :: replicate 2 (ATerm ("[]", [])))

fun string_for_format CNF = tptp_cnf

  | string_for_format CNF_UEQ = tptp_cnf

  | string_for_format FOF = tptp_fof

  | string_for_format (TFF _) = tptp_tff

  | string_for_format (THF0 _) = tptp_thf

fun string_for_problem_line format (Decl (ident, sym, ty)) =

    string_for_format format ^ "(" ^ ident ^ ", type,\n    " ^ sym ^ " : " ^

    string_for_type format ty ^ ").\n"

  | string_for_problem_line format (Formula (ident, kind, phi, source, info)) =

    string_for_format format ^ "(" ^ ident ^ ", " ^ string_for_kind kind ^

    ",\n    (" ^ string_for_formula format phi ^ ")" ^

    (case (source, info) of

       (NONE, NONE) => ""

     | (SOME tm, NONE) => ", " ^ string_for_term format tm

     | (_, SOME tm) =>

       ", " ^ string_for_term format (the_source source) ^

       ", " ^ string_for_term format tm) ^ ").\n"

fun tptp_lines_for_atp_problem format problem =

  "% This file was generated by Isabelle (most likely Sledgehammer)\n\

  \% " ^ timestamp () ^ "\n" ::

  maps (fn (_, []) => []

         | (heading, lines) =>

           "\n% " ^ heading ^ " (" ^ string_of_int (length lines) ^ ")\n" ::

           map (string_for_problem_line format) lines)

       problem

(** CNF (Metis) and CNF UEQ (Waldmeister) **)

fun is_problem_line_negated (Formula (_, _, AConn (ANot, _), _, _)) = true

  | is_problem_line_negated _ = false

fun is_problem_line_cnf_ueq (Formula (_, _, AAtom (ATerm ((s, _), _)), _, _)) =

    is_tptp_equal s

  | is_problem_line_cnf_ueq _ = false

fun open_conjecture_term (ATerm ((s, s'), tms)) =

    ATerm (if is_tptp_variable s then (s |> Name.desymbolize false, s')

           else (s, s'), tms |> map open_conjecture_term)

  | open_conjecture_term _ = raise Fail "unexpected higher-order term"

fun open_formula conj =

  let

    (* We are conveniently assuming that all bound variable names are

       distinct, which should be the case for the formulas we generate. *)

    fun opn (pos as SOME true) (AQuant (AForall, _, phi)) = opn pos phi

      | opn (pos as SOME false) (AQuant (AExists, _, phi)) = opn pos phi

      | opn pos (AConn (ANot, [phi])) = mk_anot (opn (Option.map not pos) phi)

      | opn pos (AConn (c, [phi1, phi2])) =

        let val (pos1, pos2) = polarities_of_conn pos c in

          AConn (c, [opn pos1 phi1, opn pos2 phi2])

        end

      | opn _ (AAtom t) = AAtom (t |> conj ? open_conjecture_term)

      | opn _ phi = phi

  in opn (SOME (not conj)) end

fun open_formula_line (Formula (ident, kind, phi, source, info)) =

    Formula (ident, kind, open_formula (kind = Conjecture) phi, source, info)

  | open_formula_line line = line

fun negate_conjecture_line (Formula (ident, Conjecture, phi, source, info)) =

    Formula (ident, Hypothesis, mk_anot phi, source, info)

  | negate_conjecture_line line = line

exception CLAUSIFY of unit

(* This "clausification" only expands syntactic sugar, such as "phi => psi" to

   "~ phi | psi" and "phi <=> psi" to "~ phi | psi" and "~ psi | phi". We don't

   attempt to distribute conjunctions over disjunctions. *)

fun clausify_formula pos (phi as AAtom _) = [phi |> not pos ? mk_anot]

  | clausify_formula pos (AConn (ANot, [phi])) = clausify_formula (not pos) phi

  | clausify_formula true (AConn (AOr, [phi1, phi2])) =

    (phi1, phi2) |> pairself (clausify_formula true)

                 |> uncurry (map_product (mk_aconn AOr))

  | clausify_formula false (AConn (AAnd, [phi1, phi2])) =

    (phi1, phi2) |> pairself (clausify_formula false)

                 |> uncurry (map_product (mk_aconn AOr))

  | clausify_formula true (AConn (AImplies, [phi1, phi2])) =

    clausify_formula true (AConn (AOr, [mk_anot phi1, phi2]))

  | clausify_formula true (AConn (AIff, phis)) =

    clausify_formula true (AConn (AImplies, phis)) @

    clausify_formula true (AConn (AImplies, rev phis))

  | clausify_formula _ _ = raise CLAUSIFY ()

fun clausify_formula_line (Formula (ident, kind, phi, source, info)) =

    let

      val (n, phis) = phi |> try (clausify_formula true) |> these |> `length

    in

      map2 (fn phi => fn j =>

               Formula (ident ^ replicate_string (j - 1) "x", kind, phi, source,

                        info))

           phis (1 upto n)

    end

  | clausify_formula_line _ = []

fun ensure_cnf_problem_line line =

  line |> open_formula_line |> negate_conjecture_line |> clausify_formula_line

fun ensure_cnf_problem problem =

  problem |> map (apsnd (maps ensure_cnf_problem_line))

fun filter_cnf_ueq_problem problem =

  problem

  |> map (apsnd (map open_formula_line

                 #> filter is_problem_line_cnf_ueq

                 #> map negate_conjecture_line))

  |> (fn problem =>

         let

           val lines = problem |> maps snd

           val conjs = lines |> filter is_problem_line_negated

         in if length conjs = 1 andalso conjs <> lines then problem else [] end)

(** Symbol declarations **)

(* TFF allows implicit declarations of types, function symbols, and predicate

   symbols (with "$i" as the type of individuals), but some provers (e.g.,

   SNARK) require explicit declarations. The situation is similar for THF. *)

val atype_of_types = AType (`I tptp_type_of_types, [])

val bool_atype = AType (`I tptp_bool_type, [])

val individual_atype = AType (`I tptp_individual_type, [])

fun default_type pred_sym =

  let

    fun typ 0 = if pred_sym then bool_atype else individual_atype

      | typ ary = AFun (individual_atype, typ (ary - 1))

  in typ end

fun add_declared_syms_in_problem_line (Decl (_, sym, _)) = insert (op =) sym

  | add_declared_syms_in_problem_line _ = I

fun declared_syms_in_problem problem =

  fold (fold add_declared_syms_in_problem_line o snd) problem []

fun undeclared_syms_in_problem declared problem =

  let

    fun do_sym name ty =

      if member (op =) declared name then I else AList.default (op =) (name, ty)

    fun do_type (AType (name, tys)) =

        do_sym name (K atype_of_types) #> fold do_type tys

      | do_type (AFun (ty1, ty2)) = do_type ty1 #> do_type ty2

    fun do_term pred_sym (ATerm (name as (s, _), tms)) =

        is_tptp_user_symbol s

        ? do_sym name (fn _ => default_type pred_sym (length tms))

        #> fold (do_term false) tms

      | do_term _ (AAbs ((_, ty), tm)) = do_type ty #> do_term false tm

    fun do_formula (AQuant (_, xs, phi)) =

        fold do_type (map_filter snd xs) #> do_formula phi

      | do_formula (AConn (_, phis)) = fold do_formula phis

      | do_formula (AAtom tm) = do_term true tm

    fun do_problem_line (Decl (_, _, ty)) = do_type ty

      | do_problem_line (Formula (_, _, phi, _, _)) = do_formula phi

  in

    fold (fold do_problem_line o snd) problem []

    |> filter_out (is_built_in_tptp_symbol o fst o fst)

  end

fun declare_undeclared_syms_in_atp_problem prefix heading problem =

  let

    fun decl_line (x as (s, _), ty) = Decl (prefix ^ s, x, ty ())

    val declared = problem |> declared_syms_in_problem

    val decls =

      problem |> undeclared_syms_in_problem declared

              |> sort_wrt (fst o fst)

              |> map decl_line

  in (heading, decls) :: problem end

(** Nice names **)

fun empty_name_pool readable_names =

  if readable_names then SOME (Symtab.empty, Symtab.empty) else NONE

fun pool_fold f xs z = pair z #> fold_rev (fn x => uncurry (f x)) xs

fun pool_map f xs =

  pool_fold (fn x => fn ys => fn pool => f x pool |>> (fn y => y :: ys)) xs []

val no_qualifiers =

  let

    fun skip [] = []

      | skip (#"." :: cs) = skip cs

      | skip (c :: cs) = if Char.isAlphaNum c then skip cs else c :: keep cs

    and keep [] = []

      | keep (#"." :: cs) = skip cs

      | keep (c :: cs) = c :: keep cs

  in String.explode #> rev #> keep #> rev #> String.implode end

(* Long names can slow down the ATPs. *)

val max_readable_name_size = 20

(* "equal" is reserved by some ATPs. "op" is also reserved, to avoid the

   unreadable "op_1", "op_2", etc., in the problem files. "eq" is reserved to

   ensure that "HOL.eq" is correctly mapped to equality (not clear whether this

   is still necessary). *)

val reserved_nice_names = [tptp_old_equal, "op", "eq"]

fun readable_name full_name s =

  if s = full_name then

    s

  else

    s |> no_qualifiers

      |> perhaps (try (unprefix "'"))

      |> Name.desymbolize (Char.isUpper (String.sub (full_name, 0)))

      |> (fn s =>

             if size s > max_readable_name_size then

               String.substring (s, 0, max_readable_name_size div 2 - 4) ^

               string_of_int (hash_string full_name) ^

               String.extract (s, size s - max_readable_name_size div 2 + 4,

                               NONE)

             else

               s)

      |> (fn s => if member (op =) reserved_nice_names s then full_name else s)

fun nice_name (full_name, _) NONE = (full_name, NONE)

  | nice_name (full_name, desired_name) (SOME the_pool) =

    if is_built_in_tptp_symbol full_name then

      (full_name, SOME the_pool)

    else case Symtab.lookup (fst the_pool) full_name of

      SOME nice_name => (nice_name, SOME the_pool)

    | NONE =>

      let

        val nice_prefix = readable_name full_name desired_name

        fun add j =

          let

            val nice_name =

              nice_prefix ^ (if j = 0 then "" else string_of_int j)

          in

            case Symtab.lookup (snd the_pool) nice_name of

              SOME full_name' =>

              if full_name = full_name' then (nice_name, the_pool)

              else add (j + 1)

            | NONE =>

              (nice_name,

               (Symtab.update_new (full_name, nice_name) (fst the_pool),

                Symtab.update_new (nice_name, full_name) (snd the_pool)))

          end

      in add 0 |> apsnd SOME end

fun nice_type (AType (name, tys)) =

    nice_name name ##>> pool_map nice_type tys #>> AType

  | nice_type (AFun (ty1, ty2)) = nice_type ty1 ##>> nice_type ty2 #>> AFun

fun nice_term (ATerm (name, ts)) =

    nice_name name ##>> pool_map nice_term ts #>> ATerm

  | nice_term (AAbs ((name, ty), tm)) =

    nice_name name ##>> nice_type ty ##>> nice_term tm #>> AAbs

fun nice_formula (AQuant (q, xs, phi)) =

    pool_map nice_name (map fst xs)

    ##>> pool_map (fn NONE => pair NONE

                    | SOME ty => nice_type ty #>> SOME) (map snd xs)

    ##>> nice_formula phi

    #>> (fn ((ss, ts), phi) => AQuant (q, ss ~~ ts, phi))

  | nice_formula (AConn (c, phis)) =

    pool_map nice_formula phis #>> curry AConn c

  | nice_formula (AAtom tm) = nice_term tm #>> AAtom

fun nice_problem_line (Decl (ident, sym, ty)) =

    nice_name sym ##>> nice_type ty #>> (fn (sym, ty) => Decl (ident, sym, ty))

  | nice_problem_line (Formula (ident, kind, phi, source, info)) =

    nice_formula phi #>> (fn phi => Formula (ident, kind, phi, source, info))

fun nice_problem problem =

  pool_map (fn (heading, lines) =>

               pool_map nice_problem_line lines #>> pair heading) problem

fun nice_atp_problem readable_names problem =

  nice_problem problem (empty_name_pool readable_names)

end;