(*  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 | AFun of 'a ho_type * 'a ho_type

   datatype thf_flavor = Without_Choice | With_Choice

   datatype format =

     CNF |

     CNF_UEQ |

     FOF |

     TFF |

     THF 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 | AFun of 'a ho_type * 'a ho_type

 datatype thf_flavor = Without_Choice | With_Choice

 datatype format =

   CNF |

   CNF_UEQ |

   FOF |

   TFF |

   THF 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 aux pos (AQuant (_, _, phi)) = aux pos phi

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

       | aux pos (AAtom tm) = f pos tm

   in aux 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 (THF _) = true

   | is_format_thf _ = false

 fun is_format_typed TFF = true

   | is_format_typed (THF _) = 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 (AType s, ty)) = strip_tff_type ty |>> cons s

   | strip_tff_type (AFun (AFun _, _)) =

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

   | strip_tff_type (AType s) = ([], s)

 fun string_for_type (THF _) ty =

     let

       fun aux _ (AType s) = s

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

           aux false ty1 ^ " " ^ tptp_fun_type ^ " " ^ aux true ty2

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

     in aux true ty end

   | string_for_type TFF ty =

     (case strip_tff_type ty of

        ([], s) => s

      | ([s'], s) => s' ^ " " ^ tptp_fun_type ^ " " ^ s

      | (ss, s) =>

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

        tptp_fun_type ^ " " ^ s)

   | 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 ^ " " ^

          string_for_type format (ty |> the_default (AType tptp_individual_type))

        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 = 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 THF _) (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 (THF _) = 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 (AFun (ty1, ty2)) = fold do_type [ty1, ty2]

       | do_type (AType name) = do_sym name (K atype_of_types)

     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) = nice_name name #>> 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;