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

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

  datatype quantifier = AForall | AExists

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

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

    ATyQuant of quantifier * ('b * 'd list) list * ('a, 'b, 'c, 'd) formula |

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

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

    AAtom of 'c

  datatype 'a ho_type =

    AType of 'a * 'a ho_type list |

    AFun of 'a ho_type * 'a ho_type |

    APi of 'a list * 'a ho_type

  type term_order =

    {is_lpo : bool,

     gen_weights : bool,

     gen_prec : bool,

     gen_simp : bool}

  datatype polymorphism = Monomorphic | Polymorphic

  datatype tptp_explicitness = TPTP_Implicit | TPTP_Explicit

  datatype thf_choice = THF_Without_Choice | THF_With_Choice

  datatype thf_defs = THF_Without_Defs | THF_With_Defs

  datatype atp_format =

    CNF |

    CNF_UEQ |

    FOF |

    TFF of polymorphism * tptp_explicitness |

    THF of polymorphism * tptp_explicitness * thf_choice * thf_defs |

    DFG of polymorphism

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

  datatype 'a problem_line =

    Class_Decl of string * 'a * 'a list |

    Type_Decl of string * 'a * int |

    Sym_Decl of string * 'a * 'a ho_type |

    Class_Memb of string * ('a * 'a list) list * 'a ho_type * 'a |

    Formula of string * formula_role

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

               * (string, string ho_type) ho_term option

               * (string, string ho_type) ho_term list

  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_pi_binder : 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 isabelle_info_prefix : string

  val isabelle_info : string -> int -> (string, 'a) ho_term list

  val extract_isabelle_status : (string, 'a) ho_term list -> string option

  val extract_isabelle_rank : (string, 'a) ho_term list -> int

  val inductionN : string

  val introN : string

  val inductiveN : string

  val elimN : string

  val simpN : string

  val non_rec_defN : string

  val rec_defN : string

  val rankN : string

  val minimum_rank : int

  val default_rank : int

  val default_term_order_weight : int

  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 bool_atype : (string * string) ho_type

  val individual_atype : (string * string) ho_type

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

  val mk_aconn :

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

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

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

  val formula_fold :

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

    -> 'e -> 'e

  val formula_map :

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

  val is_format_higher_order : atp_format -> bool

  val lines_for_atp_problem :

    atp_format -> term_order -> (unit -> (string * int) list) -> 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 declared_in_atp_problem : 'a problem -> ('a list * 'a list) * 'a list

  val nice_atp_problem :

    bool -> atp_format -> ('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 * 'b list) * ('a, 'b) ho_term list |

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

datatype quantifier = AForall | AExists

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

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

  ATyQuant of quantifier * ('b * 'd list) list * ('a, 'b, 'c, 'd) formula |

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

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

  AAtom of 'c

datatype 'a ho_type =

  AType of 'a * 'a ho_type list |

  AFun of 'a ho_type * 'a ho_type |

  APi of 'a list * 'a ho_type

type term_order =

  {is_lpo : bool,

   gen_weights : bool,

   gen_prec : bool,

   gen_simp : bool}

datatype polymorphism = Monomorphic | Polymorphic

datatype tptp_explicitness = TPTP_Implicit | TPTP_Explicit

datatype thf_choice = THF_Without_Choice | THF_With_Choice

datatype thf_defs = THF_Without_Defs | THF_With_Defs

datatype atp_format =

  CNF |

  CNF_UEQ |

  FOF |

  TFF of polymorphism * tptp_explicitness |

  THF of polymorphism * tptp_explicitness * thf_choice * thf_defs |

  DFG of polymorphism

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

datatype 'a problem_line =

  Class_Decl of string * 'a * 'a list |

  Type_Decl of string * 'a * int |

  Sym_Decl of string * 'a * 'a ho_type |

  Class_Memb of string * ('a * 'a list) list * 'a ho_type * 'a |

  Formula of string * formula_role

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

             * (string, string ho_type) ho_term option

             * (string, string ho_type) ho_term list

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_pi_binder = "!>"

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 = "[]"

val isabelle_info_prefix = "isabelle_"

val inductionN = "induction"

val introN = "intro"

val inductiveN = "inductive"

val elimN = "elim"

val simpN = "simp"

val non_rec_defN = "non_rec_def"

val rec_defN = "rec_def"

val rankN = "rank"

val minimum_rank = 0

val default_rank = 1000

val default_term_order_weight = 1

(* Currently, only SPASS 3.8ds can process Isabelle metainformation. *)

fun isabelle_info status rank =

  [] |> rank <> default_rank

        ? cons (ATerm ((isabelle_info_prefix ^ rankN, []),

                       [ATerm ((string_of_int rank, []), [])]))

     |> status <> "" ? cons (ATerm ((isabelle_info_prefix ^ status, []), []))

fun extract_isabelle_status (ATerm ((s, []), []) :: _) =

    try (unprefix isabelle_info_prefix) s

  | extract_isabelle_status _ = NONE

fun extract_isabelle_rank (tms as _ :: _) =

    (case List.last tms of

       ATerm ((_, []), [ATerm ((rank, []), [])]) => the (Int.fromString rank)

     | _ => default_rank)

  | extract_isabelle_rank _ = default_rank

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)

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

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

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 (ATyQuant (_, _, 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 (ATyQuant (q, xs, phi)) = ATyQuant (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 strip_atype (APi (tys, ty)) = strip_atype ty |>> apfst (append tys)

  | strip_atype (AFun (ty1, ty2)) = strip_atype ty2 |>> apsnd (cons ty1)

  | strip_atype ty = (([], []), ty)

fun is_function_atype ty = snd (strip_atype ty) <> AType (tptp_bool_type, [])

fun is_predicate_atype ty = not (is_function_atype ty)

fun is_nontrivial_predicate_atype (AType _) = false

  | is_nontrivial_predicate_atype ty = is_predicate_atype ty

fun is_format_higher_order (THF _) = true

  | is_format_higher_order _ = false

fun is_format_typed (TFF _) = true

  | is_format_typed (THF _) = true

  | is_format_typed (DFG _) = true

  | is_format_typed _ = false

fun tptp_string_for_role Axiom = "axiom"

  | tptp_string_for_role Definition = "definition"

  | tptp_string_for_role Lemma = "lemma"

  | tptp_string_for_role Hypothesis = "hypothesis"

  | tptp_string_for_role Conjecture = "conjecture"

fun tptp_string_for_app _ func [] = func

  | tptp_string_for_app format func args =

    if is_format_higher_order format then

      "(" ^ space_implode (" " ^ tptp_app ^ " ") (func :: args) ^ ")"

    else

      func ^ "(" ^ commas args ^ ")"

fun flatten_type (APi (tys, ty)) = APi (tys, flatten_type ty)

  | flatten_type (ty as AFun (ty1 as AType _, ty2)) =

    (case flatten_type ty2 of

       AFun (ty' as AType (s, tys), ty) =>

       AFun (AType (tptp_product_type,

                    ty1 :: (if s = tptp_product_type then tys else [ty'])), ty)

     | _ => ty)

  | flatten_type (ty as AType _) = ty

  | flatten_type _ =

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

val dfg_individual_type = "iii" (* cannot clash *)

val suffix_type_of_types =

  suffix (" " ^ tptp_has_type ^ " " ^ tptp_type_of_types)

fun str_for_type format ty =

  let

    val dfg = case format of DFG _ => true | _ => false

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

        if dfg andalso s = tptp_individual_type then dfg_individual_type else s

      | str _ (AType (s, tys)) =

        let val ss = tys |> map (str false) in

          if s = tptp_product_type then

            ss |> space_implode

                      (if dfg then ", " else " " ^ tptp_product_type ^ " ")

               |> (not dfg andalso length ss > 1) ? enclose "(" ")"

          else

            tptp_string_for_app format s ss

        end

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

        (str false ty1 |> dfg ? enclose "(" ")") ^ " " ^

        (if dfg then "" else tptp_fun_type ^ " ") ^ str true ty2

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

      | str _ (APi (ss, ty)) =

        if dfg then

          "[" ^ commas ss ^ "], " ^ str true ty

        else

          tptp_pi_binder ^ "[" ^ commas (map suffix_type_of_types ss) ^ "]: " ^

          str false ty

  in str true ty end

fun string_for_type (format as THF _) ty = str_for_type format ty

  | string_for_type format ty = str_for_type format (flatten_type ty)

fun tptp_string_for_quantifier AForall = tptp_forall

  | tptp_string_for_quantifier AExists = tptp_exists

fun tptp_string_for_connective ANot = tptp_not

  | tptp_string_for_connective AAnd = tptp_and

  | tptp_string_for_connective AOr = tptp_or

  | tptp_string_for_connective AImplies = tptp_implies

  | tptp_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 tptp_string_for_term _ (ATerm ((s, []), [])) = s

  | tptp_string_for_term format (ATerm ((s, tys), ts)) =

    (if s = tptp_empty_list then

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

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

     else if is_tptp_equal s then

       space_implode (" " ^ tptp_equal ^ " ")

                     (map (tptp_string_for_term format) ts)

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

     else case (s = tptp_ho_forall orelse s = tptp_ho_exists, s = tptp_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. *)

       tptp_string_for_formula format

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

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

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

       (* There is code in "ATP_Problem_Generate" to ensure that "Eps" is always

          applied to an abstraction. *)

       tptp_string_for_app format

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

            tptp_string_for_term format tm ^ ""

            |> enclose "(" ")")

           (map (tptp_string_for_term format) args)

     | _ =>

       tptp_string_for_app format s

           (map (string_for_type format) tys

            @ map (tptp_string_for_term format) ts))

  | tptp_string_for_term (format as THF _) (AAbs (((s, ty), tm), args)) =

    tptp_string_for_app format

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

         tptp_string_for_term format tm ^ ")")

        (map (tptp_string_for_term format) args)

  | tptp_string_for_term _ _ =

    raise Fail "unexpected term in first-order format"

and tptp_string_for_formula format (ATyQuant (q, xs, phi)) =

    tptp_string_for_quantifier q ^

    "[" ^

    commas (map (suffix_type_of_types o string_for_type format o fst) xs) ^

    "]: " ^ tptp_string_for_formula format phi

    |> enclose "(" ")"

  | tptp_string_for_formula format (AQuant (q, xs, phi)) =

    tptp_string_for_quantifier q ^

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

    tptp_string_for_formula format phi

    |> enclose "(" ")"

  | tptp_string_for_formula format

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

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

                  (map (tptp_string_for_term format) ts)

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

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

    tptp_string_for_connective c ^ " " ^

    (tptp_string_for_formula format phi

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

    |> enclose "(" ")"

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

    space_implode (" " ^ tptp_string_for_connective c ^ " ")

                  (map (tptp_string_for_formula format) phis)

    |> enclose "(" ")"

  | tptp_string_for_formula format (AAtom tm) = tptp_string_for_term format tm

fun tptp_string_for_format CNF = tptp_cnf

  | tptp_string_for_format CNF_UEQ = tptp_cnf

  | tptp_string_for_format FOF = tptp_fof

  | tptp_string_for_format (TFF _) = tptp_tff

  | tptp_string_for_format (THF _) = tptp_thf

  | tptp_string_for_format (DFG _) = raise Fail "non-TPTP format"

val atype_of_types = AType (tptp_type_of_types, [])

fun nary_type_decl_type n = funpow n (curry AFun atype_of_types) atype_of_types

fun tptp_string_for_line format (Type_Decl (ident, ty, ary)) =

    tptp_string_for_line format (Sym_Decl (ident, ty, nary_type_decl_type ary))

  | tptp_string_for_line format (Sym_Decl (ident, sym, ty)) =

    tptp_string_for_format format ^ "(" ^ ident ^ ", type,\n    " ^ sym ^

    " : " ^ string_for_type format ty ^ ").\n"

  | tptp_string_for_line format (Formula (ident, kind, phi, source, _)) =

    tptp_string_for_format format ^ "(" ^ ident ^ ", " ^

    tptp_string_for_role kind ^ ",\n    (" ^

    tptp_string_for_formula format phi ^ ")" ^

    (case source of

       SOME tm => ", " ^ tptp_string_for_term format tm

     | NONE => "") ^ ").\n"

fun tptp_lines format =

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

         | (heading, lines) =>

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

           map (tptp_string_for_line format) lines)

fun arity_of_type (APi (tys, ty)) =

    arity_of_type ty |>> Integer.add (length tys)

  | arity_of_type (AFun (_, ty)) = arity_of_type ty ||> Integer.add 1

  | arity_of_type _ = (0, 0)

fun string_of_arity (0, n) = string_of_int n

  | string_of_arity (m, n) = string_of_int m ^ "+" ^ string_of_int n

val dfg_class_inter = space_implode " & "

fun dfg_string_for_formula poly gen_simp info =

  let

    val str_for_typ = string_for_type (DFG poly)

    fun str_for_bound_typ (ty, []) = str_for_typ ty

      | str_for_bound_typ (ty, cls) =

        str_for_typ ty ^ " : " ^ dfg_class_inter cls

    fun suffix_tag top_level s =

      if top_level then

        case extract_isabelle_status info of

          SOME s' =>

          if s' = non_rec_defN then

            s ^ ":lt"

          else if (s' = simpN orelse s' = rec_defN) andalso gen_simp then

            s ^ ":lr"

          else

            s

        | NONE => s

      else

        s

    fun str_for_term top_level (ATerm ((s, tys), tms)) =

        (if is_tptp_equal s then "equal" |> suffix_tag top_level

         else if s = tptp_true then "true"

         else if s = tptp_false then "false"

         else s) ^

        (if null tys then ""

         else "<" ^ commas (map (string_for_type (DFG poly)) tys) ^ ">") ^

        (if null tms then ""

         else "(" ^ commas (map (str_for_term false) tms) ^ ")")

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

    fun str_for_quant AForall = "forall"

      | str_for_quant AExists = "exists"

    fun str_for_conn _ ANot = "not"

      | str_for_conn _ AAnd = "and"

      | str_for_conn _ AOr = "or"

      | str_for_conn _ AImplies = "implies"

      | str_for_conn top_level AIff = "equiv" |> suffix_tag top_level

    fun str_for_formula top_level (ATyQuant (q, xs, phi)) =

        str_for_quant q ^ "_sorts([" ^ commas (map str_for_bound_typ xs) ^

        "], " ^ str_for_formula top_level phi ^ ")"

      | str_for_formula top_level (AQuant (q, xs, phi)) =

        str_for_quant q ^ "([" ^

        commas (map (string_for_bound_var (DFG poly)) xs) ^ "], " ^

        str_for_formula top_level phi ^ ")"

      | str_for_formula top_level (AConn (c, phis)) =

        str_for_conn top_level c ^ "(" ^

        commas (map (str_for_formula false) phis) ^ ")"

      | str_for_formula top_level (AAtom tm) = str_for_term top_level tm

  in str_for_formula true end

fun maybe_enclose bef aft "" = "% " ^ bef ^ aft

  | maybe_enclose bef aft s = bef ^ s ^ aft

fun dfg_lines poly {is_lpo, gen_weights, gen_prec, gen_simp} ord_info problem =

  let

    val str_for_typ = string_for_type (DFG poly)

    fun spair (s, s') = "(" ^ s ^ ", " ^ s' ^ ")"

    fun tm_ary sym ty = spair (sym, string_of_arity (arity_of_type ty))

    fun ty_ary 0 ty = ty

      | ty_ary n ty = "(" ^ ty ^ ", " ^ string_of_int n ^ ")"

    fun fun_typ sym ty = "function(" ^ sym ^ ", " ^ str_for_typ ty ^ ")."

    fun pred_typ sym ty =

      let

        val (ty_vars, tys) =

          strip_atype ty |> fst

          |>> (fn [] => [] | ty_vars => ["[" ^ commas ty_vars ^ "]"])

      in "predicate(" ^ commas (sym :: ty_vars @ map str_for_typ tys) ^ ")." end

    fun str_for_bound_tvar (ty, []) = ty

      | str_for_bound_tvar (ty, cls) = ty ^ " : " ^ dfg_class_inter cls

    fun sort_decl xs ty cl =

      "sort(" ^

      (if null xs then ""

       else "[" ^ commas (map str_for_bound_tvar xs) ^ "], ") ^

      str_for_typ ty ^ ", " ^ cl ^ ")."

    fun subclass_of sub super = "subclass(" ^ sub ^ ", " ^ super ^ ")."

    fun formula pred (Formula (ident, kind, phi, _, info)) =

        if pred kind then

          let val rank = extract_isabelle_rank info in

            "formula(" ^ dfg_string_for_formula poly gen_simp info phi ^

            ", " ^ ident ^

            (if rank = default_rank then "" else ", " ^ string_of_int rank) ^

            ")." |> SOME

          end

        else

          NONE

      | formula _ _ = NONE

    fun filt f = problem |> map (map_filter f o snd) |> filter_out null

    val func_aries =

      filt (fn Sym_Decl (_, sym, ty) =>

               if is_function_atype ty then SOME (tm_ary sym ty) else NONE

             | _ => NONE)

      |> flat |> commas |> maybe_enclose "functions [" "]."

    val pred_aries =

      filt (fn Sym_Decl (_, sym, ty) =>

               if is_predicate_atype ty then SOME (tm_ary sym ty) else NONE

             | _ => NONE)

      |> flat |> commas |> maybe_enclose "predicates [" "]."

    val sorts =

      filt (fn Type_Decl (_, ty, ary) => SOME (ty_ary ary ty) | _ => NONE) @

      [[ty_ary 0 dfg_individual_type]]

      |> flat |> commas |> maybe_enclose "sorts [" "]."

    val classes =

      filt (fn Class_Decl (_, cl, _) => SOME cl | _ => NONE)

      |> flat |> commas |> maybe_enclose "classes [" "]."

    val ord_info = if gen_weights orelse gen_prec then ord_info () else []

    val do_term_order_weights =

      (if gen_weights then ord_info else [])

      |> map (spair o apsnd string_of_int) |> commas

      |> maybe_enclose "weights [" "]."

    val syms = [func_aries, pred_aries, do_term_order_weights, sorts, classes]

    val func_decls =

      filt (fn Sym_Decl (_, sym, ty) =>

               if is_function_atype ty then SOME (fun_typ sym ty) else NONE

             | _ => NONE) |> flat

    val pred_decls =

      filt (fn Sym_Decl (_, sym, ty) =>

               if is_nontrivial_predicate_atype ty then SOME (pred_typ sym ty)

               else NONE

             | _ => NONE) |> flat

    val sort_decls =

      filt (fn Class_Memb (_, xs, ty, cl) => SOME (sort_decl xs ty cl)

             | _ => NONE) |> flat

    val subclass_decls =

      filt (fn Class_Decl (_, sub, supers) =>

               SOME (map (subclass_of sub) supers)

             | _ => NONE) |> flat |> flat

    val decls = func_decls @ pred_decls @ sort_decls @ subclass_decls

    val axioms =

      filt (formula (curry (op <>) Conjecture)) |> separate [""] |> flat

    val conjs =

      filt (formula (curry (op =) Conjecture)) |> separate [""] |> flat

    val settings =

      (if is_lpo then ["set_flag(Ordering, 1)."] else []) @

      (if gen_prec then

         [ord_info |> map fst |> rev |> commas

                   |> maybe_enclose "set_precedence(" ")."]

       else

         [])

    fun list_of _ [] = []

      | list_of heading ss =

        "list_of_" ^ heading ^ ".\n" :: map (suffix "\n") ss @

        ["end_of_list.\n\n"]

  in

    "\nbegin_problem(isabelle).\n\n" ::

    list_of "descriptions"

            ["name({**}).", "author({**}).", "status(unknown).",

             "description({**})."] @

    list_of "symbols" syms @

    list_of "declarations" decls @

    list_of "formulae(axioms)" axioms @

    list_of "formulae(conjectures)" conjs @

    list_of "settings(SPASS)" settings @

    ["end_problem.\n"]

  end

fun lines_for_atp_problem format ord ord_info problem =

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

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

  (case format of

     DFG poly => dfg_lines poly ord ord_info

   | _ => tptp_lines format) problem

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

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

  | is_line_negated _ = false

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

    is_tptp_equal s

  | is_line_cnf_ueq _ = false

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

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

            else (s, s'), tys), 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_line line =

  line |> open_formula_line |> negate_conjecture_line |> clausify_formula_line

fun ensure_cnf_problem problem = problem |> map (apsnd (maps ensure_cnf_line))

fun filter_cnf_ueq_problem problem =

  problem

  |> map (apsnd (map open_formula_line #> filter is_line_cnf_ueq

                 #> map negate_conjecture_line))

  |> (fn problem =>

         let

           val lines = problem |> maps snd

           val conjs = lines |> filter is_line_negated

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

(** Symbol declarations **)

fun add_declared_in_line (Class_Decl (_, cl, _)) = apfst (apfst (cons cl))

  | add_declared_in_line (Type_Decl (_, ty, _)) = apfst (apsnd (cons ty))

  | add_declared_in_line (Sym_Decl (_, sym, _)) = apsnd (cons sym)

  | add_declared_in_line _ = I

fun declared_in_atp_problem problem =

  fold (fold add_declared_in_line o snd) problem (([], []), [])

(** Nice names **)

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"]

(* hack to get the same hashing across Mirabelle runs (see "mirabelle.pl") *)

fun cleanup_mirabelle_name s =

  let

    val mirabelle_infix = "_Mirabelle_"

    val random_suffix_len = 10

    val (s1, s2) = Substring.position mirabelle_infix (Substring.full s)

  in

    if Substring.isEmpty s2 then

      s

    else

      Substring.string s1 ^

      Substring.string (Substring.triml (size mirabelle_infix + random_suffix_len) s2)

  end

fun readable_name protect 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 (cleanup_mirabelle_name 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))

  |> protect

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

  | nice_name protect (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 protect full_name desired_name

        fun add j =

          let

            val nice_name =

              nice_prefix ^ (if j = 1 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 1 |> apsnd SOME end

fun avoid_clash_with_alt_ergo_type_vars s =

  if is_tptp_variable s then s else s ^ "_"

fun avoid_clash_with_dfg_keywords s =

  let val n = String.size s in

    if n < 2 orelse (n = 2 andalso String.sub (s, 0) = String.sub (s, 1)) orelse

       String.isSubstring "_" s then

      s

    else if is_tptp_variable s then

      (* "DL" appears to be a SPASS 3.7 keyword *)

      if s = "DL" then s ^ "_" else s

    else

      String.substring (s, 0, n - 1) ^

      String.str (Char.toUpper (String.sub (s, n - 1)))

  end

fun nice_atp_problem readable_names format problem =

  let

    val empty_pool =

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

    val avoid_clash =

      case format of

        TFF (Polymorphic, _) => avoid_clash_with_alt_ergo_type_vars

      | DFG _ => avoid_clash_with_dfg_keywords

      | _ => I

    val nice_name = nice_name avoid_clash

    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

      | nice_type (APi (names, ty)) =

        pool_map nice_name names ##>> nice_type ty #>> APi

    fun nice_term (ATerm ((name, tys), ts)) =

        nice_name name ##>> pool_map nice_type tys ##>> pool_map nice_term ts

        #>> ATerm

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

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

        ##>> pool_map nice_term args #>> AAbs

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

        pool_map nice_type (map fst xs)

        ##>> pool_map (pool_map nice_name) (map snd xs)

        ##>> nice_formula phi

        #>> (fn ((tys, cls), phi) => ATyQuant (q, tys ~~ cls, phi))

      | 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_line (Class_Decl (ident, cl, cls)) =

        nice_name cl ##>> pool_map nice_name cls

        #>> (fn (cl, cls) => Class_Decl (ident, cl, cls))

      | nice_line (Type_Decl (ident, ty, ary)) =

        nice_name ty #>> (fn ty => Type_Decl (ident, ty, ary))

      | nice_line (Sym_Decl (ident, sym, ty)) =

        nice_name sym ##>> nice_type ty

        #>> (fn (sym, ty) => Sym_Decl (ident, sym, ty))

      | nice_line (Class_Memb (ident, xs, ty, cl)) =

        pool_map nice_name (map fst xs)

        ##>> pool_map (pool_map nice_name) (map snd xs)

        ##>> nice_type ty ##>> nice_name cl

        #>> (fn (((tys, cls), ty), cl) =>

                Class_Memb (ident, tys ~~ cls, ty, cl))

      | nice_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_line lines #>> pair heading) problem

  in nice_problem problem empty_pool end

end;