src/HOL/Tools/ATP/atp_problem.ML

avoid clash of Advanced/simp.thy vs. Misc/simp.thy;

(*  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;